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Transition state search and geometry optimization in chemical reactions

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Transition state search and geometry optimization in chemical reactions
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Cardenas-Lailhacar, Cristian E., 1957-
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ix, 170 leaves : ill. ; 29 cm.

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Algorithms ( jstor )
Ammonia ( jstor )
Coordinate systems ( jstor )
Energy ( jstor )
Geometric angles ( jstor )
Geometry ( jstor )
Harmonic functions ( jstor )
Potential energy ( jstor )
Reactants ( jstor )
Saddle points ( jstor )
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Thesis:
Thesis (Ph.D.)--University of Florida, 1998.
Bibliography:
Includes bibliographical references (leaves 162-168).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Cristian E. Cardenas-Lailhacar.

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TRANSITION STATE SEARCH AND GEOMETRY
OPTIMIZATION IN CHEMICAL REACTIONS










By

CRISTIAN E. CARDENAS-LAILHACAR


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1998



















To my beloved girls, my wife Alejandra and our
daughters Francisca Javiera and Catalina Sofia















ACKNOWLEDGMENTS


I would like to thank my advisor Prof. Michael C. Zerner for his support, criticism

and teaching. I have been always amazed by his never-ending creativity and enthusiasm:

you can do it Cristian!; it was always a must. With his very busy schedule, no wonder

we couldn't interact more, but somehow he managed to find the time to discuss my

progress. He allowed me the time to go ahead with my own ideas, sometimes just to

prove to me that I was going in the wrong direction.

In the past few years I have had the opportunity to interact with many people in

the Quantum Theory Project, specially in the Zerner research group. Among them I

would like to thank Dr. Krassimir Stavrev, Dr. Toomas Tamm, Dr. Marshall Cory,

Dr. Guillermina Estiu, Dr. Igor Zilberberg and Dr. Wagner B. De Almeida. Out of

the Zerner group, my gratitude for many hours of great science and friendship is given

to Dr. Agustin Diz, Dr. Keith Runge, Dr. Ajith Perera, Dr. Steven Gwaltney, and

many others that I am probably missing.

I would also like to thank Sandy Weakland, Leann Golemo and Judy Parker from

the QTP staff for whom I will hold warmest remembrances.

Friends have always been important for me and my family: Marta and Pradeep

Raval, Judy and Marshall Odham, Deborah and Ricardo Cavallino, Sue and Dale

Kirmsee, Marcela and Augie Diz, Guillermina Estiu and Luis Bruno-Blanch are friends

we will never forget.

My family has always been important to me: my sister Marie-Helene and my








brother Bernard have been good siblings and friends. I always felt lucky to have the

parents God gave me, Eduardo and Helene, who gave me so much and ask nothing in

return. Few are the occasions that as a friend and a husband I have to express, in a

public way, the deep and eternal gratitude that I have for the patience, encouragement,

support and love of my wife Alejandra, in whose eyes I saw my future and to whom I

gave so little but owe so much. I dedicate this work, my love and my life to her and

to our daughters, as they are everything to me.














TABLE OF CONTENTS

ACKNOWLEDGMENTS ..................................... iii

ABSTRACT ............................................. viii

CHAPTERS

1. INTRODUCTION ......................................... 1

2. REVIEW OF METHODS FOR GEOMETRY OPTIMIZATION AND
TRANSITION STATE SEARCH .......................... 14
Introduction . . . . 14


Geometry Optimization Methods .....
Newton and Quasi Newton Methods
The Line Search Technique .....
The Simplex (Amoeba) Technique .
Restricted Step Method ........
Rational Functions (RFO) ......
Reaction Path Following Method .
The Hellmann-Feynman Theorem..
Transition State Search Methods .....
Tntrnri 1 Cti Af


. . . 16
. . . 16
. . . 18
. . .... 20
. . . 22
. . . 2 3
. . . 24
. . . 24
. . . 2 5
95


11 L /)J ), ../ o. o .. ). .. .
Simple Monte Carlo and Simulated Annealing Algorithms .
Synchronous-Transit Methods (LST & QST) ...........
Minimax / Minimi Method ........................
The Chain and Saddle Methods .....................
Cerjan-Miller .................................
Schlegel's Algorithm ...........................
The Normalization Technique or E Minimization .........
Augmented Hessian .............................
Norm of the Gradient Square Method (NGSM) ..........
Gradient Extremal ..............................
Gradient Extremal Paths (GEP) .....................
Constrained Internal Coordinates ...................
The Image Potential Intrinsic Reaction Coordinate (IPIRC) .
The Constrained Optimization Technique ..............
Gradient-Only Algorithms ........................


. .... 26
. .... 29
. .. 32
. 33
. 35
. 38
. 39
..... 40
..... 42
. 43
. 48
..... 49
..... 49
. .. 50
. 50








3. HISTORY OF GEOMETRY OPTIMIZATION AND TRANSITION STATE
SEARCH IN SEMI-EMPIRICAL AND AB-INITIO PROGRAM
PA CKA GES .. .. ... .. ...... .. .. .. ... .... .... ....


Brief Historical Overview


NDDO and MNDO ...........................
M O PA C . . . . .
AMI .... ...... ........................
P M 3 . . . . .
Z IN D O . . . . .
AM PAC (Version 2.1) .........................
GAUSSIAN 94 .............................
H O N D O . . . . .
ACES II (Version 1.0) .........................

4. THE LINE-THEN-PLANE MODEL ..................
Introduction . . . .
The Line-Then-Plane (LTP) Search Technique ...........
The A lgorithm .. .. ..... .... ... .. .. ... .. .. ..
Minimizing in Perpendicular Directions: Search for Minima ...
Projector Properties ...........................
LTP Convergency ............................
T he Step . . . . .
D efault Step .. . .. .
U pdated Step .. .... .... ... .. .. ... ... .. ..
Newton-Raphson-Like Step ...................
Hammond's-Postulate-Adapted LTP Methods ............
Introduction . . . .
Hammond-Adapted LTP Procedure (HALTP) .........
Restricted Hammond Adapted LTP (RHALTP) .........

5. GEOMETRY OPTIMIZATION .....................
Introduction . . . .
ARROBA: A Line-Then-Plane Geometry Optimization Technique
The A lgorithm ..............................
Minima in Perpendicular Directions .................
Convergency . . . .
A Proposed Global Minima Search Algorithm ...........


. 54








6. APPLICATIONS .............................
Introduction . . . .
Model Potential Functions for Transition State ..........
The Halgren-Lipscomb Potential Function ...........
The Cerjan-Miller Potential Function ...............
The Hoffman-Nord-Ruedenberg Potential Function .....
The Culot-Dive-Nguyen-Ghuysen Potential Function .
A Midpoint Transition State Potential Function ........
A Potential Function with a Minimum ..............
Summary of Results .........................
The Step .................................
Step Size Dependence ........................
Hammond and Restricted Hammond Adapted LTP Models
Summary of Results .........................
Molecular Cases for Transition State ..................
Introduction ...............................
Inversion of Water ..........................
Symmetric Inversion of Ammonia (NH3) ...........
Asymmetric Inversion of Ammonia (NH3) ..........
Rotated Symmetric Inversion of Ammonia (NH3) ......


Hydrogen Cyanide: HCN -- CNH
Formic Acid ...............
Methyl Imine ............
Thermal Retro [2+2] Cycloaddition
Hammond Adapted LTP Results .
Summary of Results ........
Model Potential Function for ARROBA
Introduction .............
Model Potential Function ......
Step Size Dependence .......
Summary of Results ........
Molecular Case for ARROBA: Water

7. CONCLUSIONS AND FUTURE WORK

BIBLIOGRAPHY ...............

BIOGRAPHICAL SKETCH .........


Reac


..... ......

. . .
ztion of Oxetane .

. ., .
. ., .
. . ., ,
.... .......
. . ., o
. ., .


. 89
. 89
. 90
. 90
. 90
. 91
....... 94
. 94
. 96
. 96
. 103
. .. 105
. 105
. 110
. ......112
. 112
. 112
. 116
. .. 122
. ..... 124
. 125
. .. 132
. 136
. 141
. 147
. ..... 147
. 150
. 150
. 150
. 15 1
. 153
. 153


..................... 162


. 169











Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy.


TRANSITION STATE SEARCH AND GEOMETRY
OPTIMIZATION IN CHEMICAL REACTIONS

By

Cristian Cardenas-Lailhacar

August, 1998


Chairman: Michael C. Zerner
Major Department: Chemistry


The research presented in this thesis involves the development of procedures for
finding transition states in chemical reactions as well as techniques to optimize the
geometries that are involved in their calculations. A procedure for finding transition
states (TS) that does not require the evaluation of second derivatives (Hessian) during
the search is proposed. The procedure is based on connecting a series of points that
represent products Pi and reactants Ri. From these points, conservative steps along
the difference vector from Pi toward Ri, and from Ri toward Pi, are taken, until
the two points coalesce. Although the initial points of the set, Po and Ro, represent
specifically the product and the reactant, other Pi and Ri are determined by minimization
in hyperplanes that are perpendicular to Pi-I and Ri-i, simultaneously. In order to test the
accuracy of the methodology proposed here, the technique has been applied to seven
well-known potential functions, and the results compared with those obtained from
other well-known procedures. Most methods that search for transition states require
an accurate evaluation of the Hessian as they proceed uphill from either product to
reactant, or from reactant to product. These procedures are both costly in computer
time and in memory storage.








The line-then-plane (LTP) methods described here do not need the accurate cal-
culation of the Hessian except for the last step in which its signature usually has to
be checked. This particular one point could also be probed numerically. This feature

potentially allows the study of much larger chemical systems.

When this LTP technique is applied to molecular reactions, the results compare
closely with those derived from the application of other models. The proposed LTP
geometry optimization procedure, after being tested in a model potential function, has

been used together with the LTP technique to define a general procedure to find the
global minimum.

It is shown that, because of the Newton-Raphson nature of the step taken, the LTP
procedures will converge unequivocally to the TS on any continuous surface. The same
applies for the minima searching in the geometry optimization procedures.















CHAPTER 1
INTRODUCTION



Since the beginning of human reasoning symbols have played a key role in a

colossal attempt to try to describe our universe, the cosmos. The history of chemistry

takes us back to the symbols of fire, water, air and earth that were extensively used by

alchemists around the 13th century and attributed to Plato's polyhedral symbols. Also

involved in this historical perspective are Empedocles of Agrigent (-440 BC), Thales

of Miletus (-600 BC), Anaximenes (546 BC) and Heraclitus (-500 BC) who claimed

fire to be a basic element. Few of these symbols are still with us. The symbol for fire

(heat), A, is the only one used in chemistry [1].

But alchemy has evolved into chemistry as the scientific method has replaced the

old beliefs such as the transmutation of matter (an idea that still today is among some

chemists' inquiries). We are still wondering about the inner secrets of matter. Chemists

are confronted with many questions, one of the most important being the description

of how atoms are held together in molecules, and how they interact with each other to

produce new compounds, that is, how chemical reactions proceed. Quantum chemistry

has become a powerful tool to assess such a goal.

As quantum chemists we are interested, in general, in accounting for the properties

of excited as well as ground states. In the case of chemical reactions, the aim is to

understand and describe the laws of nature that control them. To this end, algorithms

are constructed, to reproduce features of a chemical reaction on the computer, and






2

tested in their goodness by calculating observable quantities that are finally compared

with the experiment.

The potential energy surface (PES) is the cornerstone of all theoretical studies of

reaction mechanisms in relation to the chemical reactivity. Topographic features of the

PESs are strongly associated with experimental observations of the chemical reaction.

A lowest energy path connecting reactants (R) and products (P) (selected ones) on

the surface is a concept that can be associated with the mechanism through which the

reaction, theoretically, occurs. The association of these pathways with valleys among

mountains is as unavoidable as practical and allows us to understand the feasibility of

a reaction.

The maxima along the path related to the reaction mechanism are essential for

understanding of the energetics of the processes under study. These particular points,

which have been called transition states, tell us about the type of reaction with which

we are confronted. An insurmountable mountain along another pathway tells us that

the associated reaction connected is not feasible. On the other hand, the presence of

two transition states is the theoretical equivalent to competing reactions in a test tube,

whereas a shallow minimum may confirm the existence of a postulated intermediate.

The variety of reaction mechanisms is enormous, and, consequently it is essential to

have a good understanding of the properties that are general and common to all potential

energy surfaces.

Chemical reactivity is the main subject of chemistry. The goal is to predict the

products that are most likely to be obtained according to the interactions among the

participating species. In 1889 Svante Arrhenius initiated the study of transition states

by expressing the sensitivity of the reaction rate to temperature through his famous

relation [2]. Later, in 1931, with the development of molecular reactivity theories and






3

particularly with the work of Michael Polanyi and Henry Eyring [3], the goal was to

formulate relations for the kinetics of reactions. These theories introduced concepts

such as activation barrier and transition state. With the coming of quantum chemistry,

rational techniques for the prediction of molecular structures (geometry optimization)

and mechanisms of reactions (transition states) became available. Results obtained

today reveal the spectacular degree of refinement that quantum chemical theory has

achieved, even on occasions competing with experimental measurements for accuracy.

If we plot all the positions and energies of reactants as they evolve to products, we

will obtain a potential energy surface (PES). The TS may be like a volcano between

two valleys (for a single transition state) or a rugged mountain range (for more than

one TS). This multidimensional surface contains many paths with different mountain

passages (energy barriers) through which reactants must move to become products. In

the particular path that the reaction follows, the transition state is the point of highest

energy between reactants and products. This classical view of transition states has

evolved today into a broader definition: the full range of configurations the reactants

can take as they evolve to products [4]. This difference is mainly due to how we look

at the TS, that is, as one point or a realm of reaction rates in a potential energy surface

of 3N-6 dimensions (with N being the number of atoms).

Some reactions can go from reactants to products without passing through a

transition state via a minimum energy pathway, making the location of the TS a very

unpleasant task particularly for experimentalists. This area of the PES is called a seam,

that is, a region of the PES that is penetrated by another one. This behaviour happens

when the energies of the ground and excited states are so close that the system can

bypass the transition state. A representative example of this phenomena is the internal

rotation of stilbene [5].






4

Chemical reactions are classified according to the difference in energy between the

Rs and Ps, that is, endothermic and exothermic reactions, according to:


AE E, ER (1)


where AE' < 0 and AE' > 0 respectively. The reactions are studied along

a given reaction coordinate (RC) which tells us how to step, the walking direction

and the evolution of the reaction from reactants to products (initial and final situation)

and vice-versa. Somewhere between Rs and Ps, there is a maximum in energy that is

unavoidable, the transition state, a very unstable conformation that will transform itself

into reactants or products according to the initial conditions of the trajectory [6].

What is the appearance of the transition state? What bonds are broken and formed?

What structural changes are occurring in the system and why? Unfortunately, after

almost three decades, quantum chemistry still does not have effective algorithms to

solve this problem, and even today the most common recipe is to locate and describe

transition states from chemical intuition, that is to say, experience.

The usual starting point is to optimize both reactants and products geometries:

minima in the PES. The TS is then a maximum situated between them. However, there

are exceptions to this scheme [5, 6]. In Figure 1.1 we show the internal rotation of

hydrogen persulfide, for which the reactants and the products (trans and cis isomers,

respectively) have been fully optimized at a fixed dihedral angle of Ca = 0 and 180',

respectively, as chemical intuition will indicate. In turn, the TS is expected to be located,

at a higher energy, around midway between reactants and products. Here the expected

TS (located at a = 930) turns out to be a minimum and the initial reactant and product

maxima along the reaction coordinate. Which is then a minimum and which a TS?























2




0




-2 i




-4




-6




-8
0 50 100 150 200

Figure 1. 1. Hydrogen persulfide internal rotation. The continuous line represents Ab-Initio (STO-3G) calculations, whereas

the dashed line is for the potential energy surface as obtained through a symmetry adapted technique that we developed [5].








Energy


TS


Global
Minimum


P0

Reaction Coordinate

Figure 1.2. The Multiple Minima problem. Energy versus a given reaction
coordinate showing local minima, reactants (R), products (P), intermediates of reaction
(I), transition state (TS) and the global minimum.









On the other hand, when optimizing geometries the problem as to which is a

local and which a global minimum, as shown in Figure 1.2, is still a hazard for big

molecular systems. If a given geometry is optimized, chances are that the structure soon






7
will become trapped in the energy minimum of the potential energy surface closest to

the starting conformation. How then is one to find the desired global minimum? One

way would be to use brute force, namely to change systematically the value of a

givenvariable across the surface. An alternative is to perform a systematic search by

covering conformation space with a fine mesh, but this requires too many calculations.

Another interesting way to address the problem is to think of minimization algorithms

as cooling molecular structures to 0' Kelvin, then by a warming-up process the system

is taken to a higher energy position in the PES, and the search can continue in another

region of the N-dimensional conformational energy surface.

This overview simply tells us that we still do not have algorithms that are efficient

enough to solve this optimization problem, not to mention the expense in terms of

number of iterations necessary to obtain this minimum (when found), that is, computer

time. Transition states have only an ephemeral existence (vide infra) that lies in the

femtosecond scale as shown in the cosmic time scale in Figure 1.3 (if we were able

to live 32 million years, the transition state would last only for a few seconds of our

lives). Worthy of mention is the time-resolved experimental work of Zewail [7] and

collaborators who, by using femtosecond (ultrafast) laser techniques, observed reaction

dynamics of small molecular systems. Nevertheless, TS are attainable by quantum

mechanics, whereas experimentally they can only be inferred indirectly. This distinction

has motivated theoretical chemists to develop new and powerful models to search for

transition states [8]. New methods appear frequently in the literature [5, 9-16] and,

as we will see in the next chapters, the mathematical tools as well as the models

sometimes seem to be directly proportional to the number of scientists devoted to

tackling the problem.

But none of these procedures is as yet utterly convincing or generally successful.










Big Bang


Origin of Life


Dinosaurs


Jesus Christ


Columbus
Discovers
America


Eye Response


Molecular
Rotations


Transition State


1018



1015

1012




109




106


103


1


10-3


10-6


10-9



10-12


10-15


Milky Way


Age of Earth


Pyramids


Australopithecus


Newton: Principia
Mathematica


Year


Day Hour




Second


1 m sec


1 p sec


1 v sec



1 7r sec


1 0 sec


Flash Photolysis
(1949)


(1950)
(1960) Lasser
(1966)



(1970)


Femtosecond
(1985)


Figure 1.3. Cosmic time scale for transition state (in seconds).






9

Consequently it becomes very important to examine new algorithms capable of ac-

curately finding minima and transition states. The algorithms usually found in the

literature can be divided in two general kinds; those (the cheaper, and usually less

accurate, ones) that use only gradients and can give a quick, but rough, idea of the

transition state location, and those (more sophisticated ones) that use gradients and

Hessians (more expensive but also more accurate, when successful). The most efficient

algorithms to find TSs use second derivative matrices which require great computational

effort. This fact alone is a powerful incentive to try to develop new procedures that

do not require the Hessian.


The determination of TS structures is more difficult than the structure of equilibrium

geometries, partly because minima are intrinsically easier to locate and also because

often no apriori knowledge is available about TS structures.


For a given structure Xe to be a TS of a reaction it must fulfill the following

conditions, according to McIver and Komornicki [8]:




X, must be a stationary point, which means that all gradients (g) of the energy

evaluated at this point must be zero: g(Xe) = 0.


The force constant matrix (H) at the transition state must have one and only

one negative eigenvalue H(Xe).


The transition state must be the highest energy point on a continuous curve

connecting reactants and products.


The point identified as the transition state (Xe) must be the lowest energy point

which satisfies the above three conditions.






10

The computation of the energy and its derivatives of the system under study is

essential for our purposes. For this, the Born-Oppenheimer Approximation is used

which is based on the assumption that, given a molecular system, the nuclei are much

heavier than the electrons remaining clamped. As a consequence the kinetic energy of

the nuclei is neglected and the repulsion between nuclei is considered to be a constant.

This approximation gives rise to the electronic Hamiltonian, which in atomic units for

N electrons and M nuclei is

1H -- NV? A7-ZN M (2)
lr V2 5, + E E(2

= 2i= ==1 i=1 j>i

where V? is the Laplacian operator (derivatives respect to the coordinates of the i th

electron), Z0 is the atomic number of nucleus c, ri, is the distance between the ith

electron and nucleus o, whereas rij is the distance between electrons i and j In this

Hamiltonian we identify the first term as to be the kinetic energy of the electrons, the

second term represents the Coulomb attraction between electrons i and nuclei a and

the third term addresses for the repulsion between electrons.

The energy (E) of the system comes from the solution of Schrodinger's equation

that we wish to solve using our Hamiltonian operator: 7R = E'I where IQ is the

wave function we use to represent the system under study.

The algorithms that we used for a transition state (TS) search and geometry

optimization, as well, are generally based on a truncated Taylor series expansion of

the energy


E E + qtg + IqtHq+.. (3)


and of the gradient


g=go+qH






11

with q the coordinates, g the gradient (first derivative of the energy with respect to

coordinates q) and H the Hessian (second derivative matrix of the energy with respect

to coordinates q). First derivatives for any wave-function generally can be acquired

analytically in about the same time as the energy. Analytical second derivatives, on

the other hand, involve at least coupled perturbed Hartree-Fock (CPHF) algorithms.

These have, in general, a fifth-order dependence on the size of the basis set, that cannot

be avoided if the Hessian is required to find minima, and are imperative to insure the

location of a TS.

Today modern procedures try to avoid the evaluation of the Hessian as this is a real

bottleneck in the calculation in terms not only of computer time as well as memory

storage. Consequently, algorithms that update the Hessian (or its inverse), that is,

procedures that use a guess of the Hessian and information of the actual and previous

structure give a "good enough" estimate of the real Hessian after a few iterations.

When the initial Hessian is chosen to be the identity (or other approximate) matrix,

the procedure is said to be a quasi-Newton one. On the other hand, "true" Newton

procedures are those that use a calculated Hessian.

Update procedures in turn are known to be of two types (see for example [17-19]):

Rank 1: G, = Gn-1+ + W,

(5)

Rank 2: G,, Gn-1 + W + V,

where W, and V, are corrections to the initial Hessian or its inverse G,,-1 at cycle n.

To rank 1 correspond update procedures such as the one by Murtagh and Sargent (MS)

[20], while a popular rank 2 method was constructed by Broyden-Fletcher-Goldfarb and

Shanno (BFGS) [21], Davidon-Fletcher-Powell (DFP) [22] and Greenstadt [23].






12

It is germane to note that rank 2 update procedures can be regarded as being a

rank 1 update of an already rank-l-updated Hessian (or its inverse). A great deal of

work has been carried out lately in this field: the more recent papers combine rank 2

update procedures [24].

The determination of the minimum energy conformations of reacting species is

handled more or less routinely except for very large systems with multiple minima.

Transition states are not as easy to find as minima. Moreover, most algorithms that

we will describe in the next chapter do not always succeed in the search for transition

states because of the following general reasons:



It is difficult to insure movement along a surface that exactly meets the

conditions of a simple saddle point.

In general, little a priori knowledge of the transition state structure is

available.

Wave functions for a TS may be considerably more complex than those

describing minima.



Some procedures make a guess of the TS and perform a Newton-Raphson mini-

mization of the energy. Unfortunately this technique is not reliable because it can lead

back to R, P or to a TS. Many different algorithms for these tasks are available in the

literature, with good reviews found in references [9, 19, 25].

In this work we show methods to find transition states based on a continuous

walking of fixed step along a line connecting R and P, assuming that their structures






13

are known, and utilizing methods to optimize geometries (GOPT) based on the initial

and the newly generated structure. No previous knowledge of the TS is necessary.

In chapter 2 we review the existing literature on geometry optimization and transi-

tion state search algorithms in terms of advantages and disadvantages, starting with a

description of the line-search technique that accounts for parameters used in the major-

ity of the models. We emphasize the disadvantages as they account for costly failures

which these procedures suffer.

Chapter 3 starts with a brief historical overview of semi-empirical molecular orbital

theories. Next, some semi-empirical and ab-initio program packages are examined in

terms of their TS and GOPT capabilities.

Chapter 4 introduces the line-then-plane (LTP) procedure. The algorithm is de-

scribed discussing its convergence to the TS and how the step should be taken. Alter-

native algorithms, in the basis of Hammond's postulate, are discussed. We concentrate

on some properties of LTP, for example its dependence on the size of the steps in terms

of the number of energy evaluations required to find a maxima or minima.

In chapter 5, ARROBA, a new LTP geometry optimization procedure, is presented.

The main features of this technique are studied through a model potential function and

a molecular example. Finally, an algorithm is proposed to solve the multiple minima

problem.

In chapter 6 the LTP technique is tested with some potential energy functions and

molecular systems for both transition states and geometry optimization problems.

Finally, in chapter 7 we summarize results, draw conclusions, and set the stage for

future systematic work in this area.















CHAPTER 2
REVIEW OF METHODS FOR GEOMETRY
OPTIMIZATION AND TRANSITION STATE SEARCH


Introduction


In general, optimization techniques for finding stationary points on PESs can be

classified (avoiding details for simplicity), as [17-19]



Without Gradient,

With Gradient: Numerical or Analytical, or

With Numerical or Analytical Gradient and Numerical Hessian.



With the exception of the first method, these algorithms are all based on a truncation

of a Taylor series expansion of the energy and of the gradient as was given in Eqns.

(3) and (4), respectively, in the previous chapter.

The general scheme is complete when the characteristics of the stationary point are

included, that is, zero gradient (g = 0). In practice the gradient should be smaller than

a preestablished threshold at the critical points which, from Eqn. (4), yields


g -qH. (6)


From here the new coordinates are


q =-gH- (7)

14






15

and the step (s) is taken as a fraction of q



s = aq VaE /0


The bottle neck for all procedures that search for minima or maxima is the evaluation

of the Hessian matrix (H), as this is time consuming and requires storage. As mentioned

in the previous chapter, the way around this problem is to use techniques that update the

second derivatives matrix as MS, BFGS, DFP, and so on. At this point, the techniques

used to find the critical points of interest are classified as (exact) Newton-Raphson if no

approximations are used for the evaluation of H, that is, numerical or analytical second

derivatives are used in the search. Procedures that use update techniques for the Hessian

are said to be of the Newton-Raphson-like type. Whereas, if the identity matrix (I) is

used, then we are in the steepest descent regime. This should not be confused with

those techniques that use the identity matrix as a starting point to update the Hessian

as these, when converged, show a second derivative which not only is not the identity

matrix but, moreover, is an Hermitian matrix that is very close to the analytical one.

First derivatives for any wave function generally can be acquired analytically in

about the same time as the energy; methods that use numerical gradients typically are

not competitive. On the other hand, the analytical evaluation of second derivatives

involves, at the very least, a Coupled Perturbed Hartree-Fock (CPHF) procedure and

is, in general, an N5 procedure where N is the size of the basis set. Therefore, second

derivative methods are costly even at the single determinant level, and even more so at

the CI level. However, a good feeling of the topology of the potential energy surface

is needed to locate maxima and minima.






16

Geometry Optimization Methods



Newton and Quasi Newton Methods


From the Taylor series expansion of the energy (equation (3)) we write for the

gradient


g(x + ) = g(x) + H(x)b (9)


with 6 the infinitesimal step in the search direction. A minimum implies that

g(x + 8) = 0. Thus a relation for the search direction s can be written as


as = -(H-1)g(x) = &". (10)


For "a true" Newton method, H is the exact Hessian matrix, while for the quasi-

Newton procedure the Hessian can be a matrix that approximates the second derivatives

[26, 27], commonly the identity. The line search parameter a can be determined by a

variety of algorithms that reduce the energy along the search direction. This type of

procedure will be discussed in detail in the next section.

When the Hessian is positive and exact, the Newton method shows a quadratic

convergence to a local minimum if expanded in the quadratic region. The problem

is that this procedure requires an explicit calculation of the second derivative matrix,

demands the extra computational effort previously discussed and, moreover, is accurate

only in the quadratic region.

The quasi-Newton methods depend on information obtained about the Hessian

during the search. The gradients calculated at different geometries are used to build

an approximate inverse Hessian G = H-1 using the quasi-Newton conditions (Eqn.








(6)). This is
7 = g(x + 6) g(x)

S= G(11)



as a constraint to obtain a relation to get a matrix product for the search direction s.

This is


G(x + 6) = G(x) + U (12)


and

s = -G(x + 6)g (13)

where U is a correction to the inverse Hessian G. Thus the approximate Hessian

is updated every geometry cycle. In this way some time is spent evaluating the

approximate Hessian or its inverse, but this task requires a small fraction of the time

that the evaluation of the Hartree-Fock gradient takes. One of the most successful

relations for updating the Hessian is the one developed by Broyden, Fletcher, Goldfarb

and Shanno (BFGS) [21]. Many of the procedures available for finding minima differ

on how they evaluate or choose s. For computational details on these methods see

Kuester and Mize [28].

New procedures to optimize geometry are the subject of study by many research

groups and one can see specialized journals frequently publishing works devoted to this

problem. Rather than follow a textbook classification, here we summarize only those

procedures that have proved to be more stable and are most often used. We will start

by summarizing the line search technique, as it is widely used by several minimization

techniques. Time-dependent, statistical mechanics and variational procedures will not

be addressed nor discussed here (except Monte Carlo techniques).






18

Reviews of geometry optimization and transition state search methods can be found

in the literature [13, 18, 23-25].

The Line Search Technique

The fundamental idea of this strategy is to look for the appropriate displacement

along a search direction. Suppose that at a given point Xk we found a search direction

determined by

Sk = -Gk9k (14)

where k indexes the cycle and the inverse of the Hessian Gk = H1 is already

updated (for example using BFGS). Then the next point is given by Xk.+ Xk + 0Zksk

Here the parameter 0Zk comes from the line search and has a value such that the

decrease in energy is reasonable (Figure 2.1). This is

E(Xk+l) < E(xk) or E(Xk+l) E(Xk) < (15)


where c is a given energy threshold. But an exact line search will be able to find the exact

value of ak for which E(Xk+1) along the line defined by Sk is a minimum. Among the

many procedures for performing these calculations, one of the most efficient approaches

is to perform an "efficient partial line search," in which a reasonable decrease in the

function is obtained when an appropriate value of alpha is selected [29].

The energy along Sk is written as

E(xk + ask) = E(Xk) + agt(xk) Sk + 1 2StHSk (16)

and will be lowered provided that the descent condition is satisfied: g9(xk) sk < 0.

From here the sign for the search direction is selected and the minimum is found by

direct differentiation

E(Xk + (= E 9t(Xk+ a4).sk=0 (17)


















S k+1


S


Figure 2.1. Representation of the line search technique in a potential energy surface.
gk and 9k+1 (continuous arrows) are the directions of greatest slope at points Xk and
Xk+1, respectively, and are orthogonal to the tangents to the surface (dashed lines) at
these points. The point on Sk represents the result of a partial line search, whereas on
Sk+1 the exact line search and the partial line search results coincide.


where the null value characterizes the exact line search. Now the energy at Xk+1 is

required to approach the Xk-kSk value. Then the gradient g(Xk+l) = g(xk + 0ksk)

is evaluated and a left extreme test is performed as


0

(18)


If a=O an exact line search is performed, and if uT=l any reduction of the scalar product


(g(Xk-11sk) is acceptable.


I (9(Xk+l)lSk) <- -(7(9(Xk)1,9k)






20

If the left test fails, ak is too low, then a new value of alpha is estimated as

-new (ak ak)E'(ak) (19)
k E'(al)- E'(ak)

where a' is the initial value of the interval (a', a) used to examine the kth line search

cycle.

Finally the energy E(xTk) -* E(Xk + a',Sk) is evaluated and the test repeated

with ok O k and ak -4 ak"' until the condition is fulfilled. The partial line search

stops and a new search direction is sought.1

The Simplex (Amoeba) Technique

Designed by Nelder and Mead [30], this procedure has as one of its main charac-

teristics its geometrical behaviour and that it only requires the evaluation of the energy,

having as major drawback the large amount of energy evaluations necessary to find

the minimum. The different behaviours, according to possible different topological

situations, that the algorithm might have are shown in Figure 2.2.

For molecular systems, a simplex is a 3N dimensional geometrical structure with

N vertices (in 3 dimensions it is a tetrahedron). In the multidimensional complex

topography of a molecular system, simplex requires only 1 (3N-dimensional) point q.

from which the procedure will walk downhill reaching a minimum (probably a local

one).

The algorithm starts with q. defining the initial simplex (the other N points qi) as


qi = q. + f0pi (20)


where pi are the 3N unit vectors and f3 is a constant that is a guess for the length of

the problem that can, in turn, have different values for its x,y,z components.

1. Further details can be found in the literature [15, 29].
























(c)










/ %
I -\


(a


Figure 2.2. The Simplex method. The different steps that can be taken by the
algorithm according to the topology of the potential energy surface. In all cases the
original simplex is represented by solid lines. The generated simplex, represented (for
all cases) by dashed lines, can be (a) A reflection in the opposite side of the triangle
that contains the lowest energy point (L) to which the highest energy point (H) does
not belong, (b) A reflection plus an expansion, (c) A contraction along the dimension
represented by the highest energy point and the point where the triangle oppossite to it
is crossed (X), or (d) A contraction along all dimensions.






22

The first steps are spent moving a projection of the highest energy point through

the opposite face (to which it does not belong) that contains the point of lowest energy.

This step is known as to be a reflection because it is taken in such a way that the volume

of the initial simplex is held constant. Then the algorithm extends the new simplex in

a given direction in order to take larger steps. When in the surroundings of a minima,

the algorithm contracts itself in a transverse direction trying to softly spread down to

the valley. The procedure can also, in these situations, contract itself in all directions

in order to find a perhaps final tortuous minimum.

The search stops when the magnitude of the last displacement vector is fractionally

smaller than a given threshold. In addition, it is customary to examine the decrease in

energy such that simplex stops if this difference is smaller than a given threshold.

Restricted Step Method

Included here for historical and review reasons, Greenstadt has studied the Relative

Efficiency of Gradient Methods up to the ones available until 1970 [23, 31].

This procedure rewrites the Taylor series expansion of the energy of Eqn. (3) as

E-E, = AE = q + IqtHq + ... (21)
2

Now the following Lagrangian (L) is introduced

= [qfq h] (22)

where q is the difference in coordinates (actual and previous cycle), A is Lagrange's

multiplier and h is the trust radius (or confidence region) for the stepping. The square

bracket factor ensures that the search remains in the quadratic region.

The first derivative of the Lagrangian gives

= 0 = g + Hq Aq (23)
aq








from here

q -[H AI1g (24)

where I is the identity matrix. This is a Newton-Raphson-like procedure which is fully

recovered when A = 0 By construction this technique has shown to be useful for

transition state search as well.

Rational Functions (RFO)

Created by Banerjee and his coworkers [32], this is a procedure in which the

energy is written in a normalized form as
AE = E- E, qtg 2 qtHq (25)

1 + qtSq

where S is a step matrix. Next step is to augment all the components of this rational

function such that, and by using Eqn. (25) the difference in energy now becomes

1~qt 1 H g 1)
-2g 0q
AE q (26)
q*1S g 1
g 0 q

As usual, the next move is to ask for the first derivative of the change in energy with

respect to the coordinates to be zero ( &AE/aq 0 ) to obtain the following

eigenvalue equation

H g 1) A 1) (27)
gt 0 q A q
from which two sets of equations are obtained
Hq + g = Aq
(28)
9tq = A
The first of these two relations gives a Newton-Raphson-like step

q (H- Al) 'g (29)


which is, as before, recovered for A = 0.








Reaction Path Following Method

This method requires one to know the TS, which will connect in a steepest descent

fashion, the TS with reactants and products in order to draw a reaction path. First

developed by Gonzalez and Schlegel [10], they have included modifications from the

very beginning in order to consider the effects due to atomic masses. In subsequent

articles they proposed a "Modified Implicit Trapezoid Method", which is a contribution

on the way of obtaining the final coordinates used by this method, namely: "The

Constrained Optimization" by Gear [33]. The method will be described in the TS

section when we discuss Schlegel's procedure as it is related to the search of maxima

and mimima.

The Hellmann-Feynman Theorem

The idea [34] is to obtain gradients to be used to optimize molecular geometries.

To accomplish this, the starting point is Schroedinger's equation


HI') = E[oI') (30)


or,

(IHIT) = E0('1)= E, (31)

where the wave function is assumed to be normalized to unity (IF IT) = 1. The

demonstration of the theorem starts by taking the first derivative of the energy with

respect to a given (set of) coordinate(s) "q" as follows

OFO 111 l)+K l 1H P) & =O (32)
aqo O aq H (9q


If the two first terms vanish,


H I T) + T H 0(33)








then the gradient will be simply


O /(34)


This is a very appealing relation since OH/Dq is easily obtained. Experience tells

us that this scheme works only for very, very good wave functions. It is obvious why

since, for an exact eigenfunction, equation (33) can be written as


Eo (9 1 = ] 0 (35)


This step concludes the proof because the bracket involves the first derivative of a

constant, due to the normalization condition, and consequently the term vanishes.

Transition State Search Methods

Introduction

It is clear that procedures to locate TS and geometry optimization (GOPT) algo-

rithms are intimately related. Techniques to find minima have been much more suc-

cessful than those developed for locating TS. Their success is based on the relative ease

in following downhill searches, such as with the steepest descent type of algorithms.

Because of this success the problem of locating TS repeatedly has been approached as

one of dealing with the location of minima. In general, such methods choose a higher

energy point and from there walk downhill, stopping at a local minimum where the

signature of the Hessian is checked (one and only one negative vibrational mode). But

when studying a reaction mechanism, knowledge of the lowest energy reaction path is

of great use but expensive. To accomplish this the recipe is to, by sitting at the found

TS, follow a downhill path to reactants and products.

Other methods use a mirror-image technique. Consider the picture of reactants and

products (usually only one of them) as minima and TS as a lowest energy maxima






26

between them. Placing a mirror at the TS the image obtained is the one of two maxima

(reactants and products) and a minimum (original TS). Consequently the problem now

is to find that minimum. Again downhill methods are used with the pitfalls described

above.

A collection of the procedures reviewed here, showing their general features,

advantages and disadvantages can be found in Table 2.1.


Simple Monte Carlo and Simulated Annealing Algorithms


The Metropolis Monte Carlo algorithm [35] has proven very successful in evaluating

equilibrium properties of systems. The bulk properties are simulated from a small

physically meaningful number of particles (N) such that the fluctuations in the calculated

value of a property, usually a thermodynamic observable, are minimized.

The interactions of N particles are described by a potential energy function, say

U(r, Q) where r is the distance separating the particles while Q represents any

other coordinates (Eulerian angles for example) on which the potential may depend.

Therefore, the potential interaction between particles AE is written as


ij ,-.Va % ( 3 6 )



Monte Carlo method

N particles are placed in a system of volume V such that the macroscopic density is

kept constant. The initial configuration of the particles is arbitrary, a flexibility which

is a tremendous advantage of this method.

The position of any particle i with a position ri = (xi, yi, zi) is chosen randomly

and moved according to









Xi= xi + bnil Yi = yi + bn2


(37)
Zi = zi + bnL3 ...(4 = Qi + f n,



where b and f are chosen step sizes for r and Q and {ni } is a set of random numbers

(for all i c 7/ ) in the interval [-1, 1]. The particle always stays in the cube such that

surface effects are reduced and the density of particles (p) is always constant. The new

conformation energy AE' is calculated according to Eqn. (36).

If AE' < AE then the new arrangement of the particle is accepted, and the

calculation is continued from the newer configuration. On the other hand if AE' > AE

then a probability (P) is calculated as
P = exp[-(AE' AE)/KBT]
(38)

where KB and T refer to Boltzman's constant and the temperature, respectively. Thus

one of the following conditions is met:

If P > 6 e [0, 1] the move is accepted

If P < 8 E [0, 1] the move is rejected

In the first case the algorithm continues as described above. In the second case,

where the new configuration is rejected, the particle is returned to its initial position

and a new particle is chosen randomly and moved according to Eqn. (37) (6 is a

random number).

The method is repeated until no further configurations are accepted. The system is

said to have reached its equilibrium configuration when this criterion is satisfied.

The efficiency in reaching the minimum by Metropoli's algorithm depends on the

number of moves allowed for the displacement of particles. A more accurate equilibrium






28

configuration of the particles can be determined if the number of random moves allowed

is large. These methods are not competitive with gradient methods in obtaining local

minima but, as discussed below, Monte Carlo allows us to leave a region of local

minimum for the global one.

Simulated Annealing method

Simulated Annealing is similar to the Monte Carlo algorithm. The difference is

that the probability P is evaluated as

P = exp[-(AE' AE)/T*]
(39)


where T* is a parameter with energy units. The potential energy surface is scanned in

a finite number of moves using the random process described above for a given value

of T*. Then T* is varied using an annealing factor a as follows


T*+I = QT* < a <1 (40)


where i is the number of steps allowed. The search process is then repeated with the

new value of T*.

As T* decreases, areas of the surface closer to the minimum are scanned and if

any minima had been missed by the search using the previous value of T*, the method

can now lock itself onto a lower minimum. As the number of cycles allowed for the

annealing step increases, the search for the global minimum of lower energy becomes

more efficient. This flexibility in being able to "anneal" the PES is one of the assets

of the simulated annealing method.

Differing from gradient techniques, in which the displacements are generally within

a small region of the PES, the random displacements in annealing enable the search

to tunnel out of local minima in which the algorithm could have been trapped. The






29

evaluation of the global minimum can be assured provided that the algorithm has been

allowed a large enough number of random moves for each value of T* [35]. A

discussion of these algorithms, with applications, can be found in the Allison and

Tildesley's text [36].


Synchronous-Transit Methods (LST & QST)


Initially proposed by Halgren and Lipscomb [37], the Linear Synchronous Transit

(LST) and the Quadratic Synchronous Transit (QST) methods treat "Forward" and

"Reverse" processes equivalently, generating a continuous path between specified R

and P. The main features of this technique are schematically shown in Figure 2.3.

The LST pathway is constructed by considering a linearly interpolated internuclear

distance connecting reactants and products and estimating a TS that is improved by

minimizing the energy with respect to all perpendicular coordinates. Finally the reaction

path is approximated using a parabolic path between R and P, that is, the QST path

giving a good estimation of the TS location.

The path coordinate (PC) (steps) is defined as: PC=DR/(DR+DP), where DR and

DP are a measure of the distance to the path-limiting-structure, obtained as

De : (Xf ,)2 + (ym- y)l + (zZ )
[N i=1 (41)



where Q and q = R,P (reactants or products), N is the number of atoms and m stands

for, in accordance to the principle of least motion (PLM) [38], optimized structures

of reactants and products that are re-oriented relative to each other in terms of rigid

translations and rotations such that the sums over squares, for all corresponding atom

coordinate differences (equation (41)), reaches a minima.






30

Intramolecular distances R.0 must vary simultaneously between the path-limiting

structures RB and RpO. To avoid limitations in the method, provision must be taken

to meet one of the following two conditions:

a) Linear Condition:
-(1 -f)R R,+ fRP 0O rvo --3 --

b) Parabolic Condition:
(i) ABfCf 0 R(.)A A + B.- f + C -f' z< 1N(3


where
=R-R R _
a3

(44)
C[R" R I'Mr M. ./P P 1)]



This ensures that the following conditions are satisfied
f = 0 _,, R(,1) ao
f ~ O -- / d

f-Ro RP (45)
=PM R(') = RM

where f is the interpolation parameter, i refers to interpolated quantities, PM denotes

the value for the path coordinate and IR" a < f = 1, N} are the atomic distances

of some intermediate structure on the path. Geometries of the synchronous transit

path may be evaluated in terms of interatomic distances from equations (42) and (43).

In practice, linearly/parabolic interpolated Cartesian coordinates between path-limiting

structures at maximum coincidence are subsequently refined so as to minimize
N-1 N
[R(') 3 R(1'- X E T ~i- W(,7)]
a ~ ~el [Wa~ ,, w=jyz ) (46)



























L$T

QST,1T.


TS2%,














Figure 2.3. Potential energy surface representating Halgren-Lipscomb's TS search
technique. The continuous line connecting reactants (R) and products (P), represents
the LST with a maxima (TS) in TSI. The dashed line represents the QST that passes
through a transition state TS2. The QST path has all the features required to represent
the minimum energy reaction path. TS, and TS2 are connected through a parabola.
This issue has been discussed by Jensen through the Minimax/Minimi procedure [39].






32

where (i) stands for interpolated and (e) for evaluated (calculated) quantities referring

to the evaluated (updated) Cartesian coordinates. The weighting factor (1/R )4

ensures a close reproduction of bond distances, whereas the 10-6 factor is proposed to

suppress rigid translations and rotations, between the interpolated and calculated points

(W.0 and W, respectively).

This procedure can be used for molecules with N>3 since the number of interatomic

distances exceed the number of 3N-6 internal degrees of freedom for non linear

molecules.

Cartesian coordinates are then submitted to the PLM to associate a unique path

coordinate and the total energy is computed. A variation off will produce a continuous

energy path called, depending on the path, LST or QST. For example, in a uni-molecular

reaction, the path will usually connect both limiting structures via some maximum path,

whose structure can be determined using Eqns. (42) and (43).

Alternative algorithms have been developed maximizing along a path of known

form and minimization perpendicular to the path [8, 25, 39]. In particular, Jensen

has lately introduced a variation of this procedure namely the MINIMAX / MINIMI

procedure [39], that is briefly discussed below.



Minimax / Minimi Method


Based on the Synchronous Transit method, the minimax/minimi method is a pro-

cedure for the location of transition states and stable intermediates [38]. It is based on

the idea that a simple parabolic transit path cannot provide a correct description of the

true minimum energy path, as suggested by Halgren and Lipscomb [37], if this path

shows frequently changing sign of curvature.






33

An essential supposition states that to find a new quadratic synchronous transit

maximum with higher energy after exhaustive orthogonal minimization is too expensive.

Consequently, it is assumed that any practical method must explicitly take into account

the influence of each geometric modification on the new transit path maximum.

It is then suggested that a straightforward way to proceed is the following: A change

in a structure corresponding to the transit maximum (a minimum) under investigation

will be accepted only if the resulting new path maximum (minimum) is of lower energy.

Successive geometry optimizations (GOPT) of all internal coordinates will consequently

lead to the lowest QST maximum, the transition state (TS), or to an intermediate (a

local minimum) and is, because of this, called the MINIMAX / MINIMI optimization

procedure [39].

A major drawback of this procedure is that an extra parabolic line minimization

along the QST path, at each level of the parameter optimization, is needed. However,

the procedure has the advantage that unexpected intermediates (MINIMAX) will be

uncovered and that extreme shifts of the path coordinate may be obtained.

The Chain and Saddle Methods

The aim of this algorithm is to ensure stability towards a transition state. Stepping

along the vector gradient field of an arbitrary continuous path between reactants and

products, leads to a limiting path where the highest energetic point is considered the

saddle point or, in the case of a multi-step mechanism, the highest energy transition

state will be located. Figure 2.4 shows the behaviour of this technique.

The algorithm consists in replacing a chain of points C(n) (,..p, ..., _P)

running from R to P by a new chain C(n+l) at iteration n+l. In order to maintain the

connectivity of the path, each distance between two successive points is restricted to a
















































Figure 2.4. Potential energy surface representating the Saddle TS search technique.
The dashed line connecting reactants (R) and products (P), represents the displacement
vector from which identical fractions are taken as steps. At each step the energy is
minimized in perpendicular directions (doted lines) to obtain new sets of projected
coordinates that represents the minimum energy reaction path. The process is repeated
until a maximum (TS) is found.






35

given length (in AMPAC [40] this length is 0.3 A). The iteration consists of skipping

the current highest point of the chain along either a descending or ascending path. In

the first case the energetic relaxation of the whole path is insured, while on the second

an interpolation of a point along the path is performed. New points are inserted as soon

as a link length becomes too long. The successive evaluations of the gradient are used

to update a quadratic local estimate of the potential, providing quadratic termination

properties. This would make this procedure very computer time consuming.

Although this seems to be a Hessian free algorithm, it is not because a differentiation

of a first order expansion of the energy is used [41]. Its recent appearance and lack of

verification will exclude this procedure from Table 2.1. For extended references also

see [24, 40-42].

Cerjan-Miller

This is essentially an uphill procedure [43] that is able to generate the reaction path

coordinate by connecting a transition state with a minimum on the potential surface,

schematically shown in Figure 2.5. It considers the Lagrangian function:


(sA) =Eo+stg+IstHs+ A(A2-sts) (47)


where A is the Lagrange multiplier, s is a fixed step size (the radius of the hypersurface),

g is the gradient and H is the Hessian matrix. The extrema are determined by the

conditions



as a 0 0 (48)





















*










-- -- -- - -

---- --N ---
-+



% I II -.. .. .'













Figure 2.5. Potential energy surface representating Cerjan-Miller's technique. The
arrows represent the step size A~ of the the trusted radius (dashed lines). The search
starts from the minimum (qj, reactants for example) and climbs up-hill towards the
transition state (TS), from where a minimization is carried out to connect q, and TS
with the new minima q2 (products).








which gives the two following relations
A2 sts = 0 A gt(H -A 2g
= --f (H A l(4 9


where A is evaluated for a given value of s. From Eqn. (49) the step size is obtained.

E(s) is given then by

E(s) = E(A) = E + gt(H I)-(IH AI) (H AI) -g (50)

Now the unitary matrix U that diagonalizes H, is introduced: UtHU = At

this point a new parameter "d" is defined: d = Ut .g then we write Eqn. (49) as
SF '?(A 1, )
A2= (A) = T _" E(A)-E=Z (A 2

where the kappas (K) are positive values for minima. Now the assumption that one

is seated at a local minimum is made by saying that A = Ao, then it follows that

E(Ao) E, > 0 that is, the step s generated is indeed uphill in this direction. For A

= 0 the increment s (on Eqn. (49)) is the Newton Raphson step s = -H-19.

Finally the step for walking uphill to a transition state from the minimum of the

potential surface is given by equation (51) (where A =Ao, if A, > 0). It must be noted

that Ao is the local minimum of the function A2(A), in other words, it is the root of

dA2(A) F 0 (52)
dA (A-K)3

In a general case, the function A2(A) will have F-I local minima. Cerjan and Miller

suggest picking up the smallest value of A, that is, the smallest root of equation (52),

corresponding to the softest mode.

Draw backs of this procedure are the use of second derivatives, the use of too many

steps when approaching the transition state and the coupling between the step and the

curvature radii of the surface in the actual point. This last issue is important as the

next step might not encounter a minimum.








Schlegel's Algorithm

This is essentially a gradient algorithm [44], proposed in 1982 by Bernhard

Schlegel2, in which the "right inertia of the approximate Hessian matrix" is obtained

by adjusting the sign of inadequate eigenvalues." The sign of the smallest positive

eigenvalue is changed if in the search of the TS no negative eigenvalue is present. On

the other hand, if sundry negative eigenvalues happen, all of them are replaced by their

absolute value (except the smallest one). Given the stationary condition Vqk(s) = 0

the quasi-Newton step, at cycle k in the step direction (s), is thus modified according to

9k ,v Ijb= Ibklb < 0 < <..-< b' (53)
i=1 1

where the b. terms are eigenvalues of the Hessian Hk V is the eigenvector basis, 9k

is the gradient and k is the cycles index. This is then related to Greenstadt's proposal3,

that is used in a minimization process. His idea is to reverse the ascendant/descendant

character of the search direction. Nevertheless, in areas of large curvature, the resulting

direction is not necessarily the opposite of the initial one, if the investigated region is

far from an extremum and thus may be incorrect. A scaling factor is used to modulate

this effect.

If the quasi-Newton search direction of Eqn. (53) exceeds the maximum allowed

step Rmax, its length is set to this maximum value. This change requires the addition

of a shift parameter A obtained by the search of an extremum of the quadratic function
qk(g). In practice, the shift parameter A is obtained by minimizing the function


(11 Sk(A) I- Rmax)2 (54)

2. We will keep here the super k indices used by Schlegel, according to Powell's notation; see:
M.J.D. Powell, Math. Prog. 1:26 (1971).
3. For details see J. Greenstadt, Math. Comp. 21, 360 (1967) ; Y. Bard Nonlinear Parameter
Estimation. Academic Press, New York, 1974, p. 91-94.






39
The radius Rmax is updated using a trust region method. The step direction is
71 k
(b' ~( A)V '(55)

When implementing trust region methods, the minimization of Eqn. (54) is

performed by determining the zero of its first derivative using a Newton-Raphson

procedure. However, the convergence threshold of such an algorithm is guided by

a zero value of Eqn. (54). Given that the minimum of this function is not necessarily

associated with a zero function value, the procedure may fail.

Besides, this Newton-Raphson search of a zero value of the first derivative implies

that the parameter A lies in the open interval ]bl,b2[. Thus, concerning Eqn. (55), the

step sk is uphill along the first eigenvector Vk and down-hill along all the others.

The Normalization Technique or E Minimization

Developed by Dewar and co-workers [41, 42], the Normalization Technique is a

root search technique rather than a saddle point location. Only convergence to a zero of

the gradient is ensured, not necessarily the TS. Moreover, the procedure has been shown

to require a good initial guess. In fact, if the PES is tortuous, stability problems appears

and the procedure requires a large number of energy evaluations to be successful.

Originally implemented in the closed-shell version of MNDO, the geometry of

reactants (R) and products (P) is defined (in 3N-6 coordinates) as R = ai and

P Z bi. A reaction coordinate (D) is defined as:

R- P = D b(-b)2 (56)

where D is reduced subject to the condition that the structure with lower energy is

moved to approach the TS. The following procedure is used:






40

1) Obtain the optimized geometry of R and P.

2) Evaluate the energy of R and P then, defining the origin on the higher energy

structure, the geometry of the other species is expressed in terms of its new origin as:


Za'= (ai-bi) -(a) (57)
iii

3) Modify geometry of lower energy structures to select a new distance4 D' to

reduce the difference between R and P as: ai = T a'D'/D

4) Optimize R's geometry such that D is held constant at D'.

5) If D is small enough, then stop; otherwise go to step 3.

Caution must be taken in ensure that one geometry (for example products) can

be obtained from the other (for example reactants) by a continuous deformation [16,

17, 42, 44-46]. The first work of Komornicki and McIver [46] is also known as the

Normalization Energy Minimization or as the Gradient Norm Minimization technique.

Pertinent previous work of Komornicki and McIver is cited in their last 2 articles in

the literature [8, 16, 47].

Augmented Hessian

The Augmented Hessian procedure was originally proposed by Lengsfield [19, 48]

for MCSCF calculations, and further developed by Nguyen and Case [49] and later on

by many other groups [33, 50]. Augmented Hessian is essentially an uphill walking

algorithm, implemented in the ZINDO package by Zerner and co-workers [19, 26]. The

search direction s is found by diagonalizing the "Augmented Hessian"

( z )( )=A(") (58)


4. Typically D' = 0.950D






41

For a down hill search, A is the lowest eigenvalue, c a parameter that can be varied

to give the required step lengths, and the lowest normalized eigenvector: v2 + 2 1.

From Eqn. (58) we get two equations
Hv + ~g Av(H A)1/ = -cvog
H+ = -(H A)-lcyog (59)
agtv AOt

solving for the step size we get


s = -(H AI)-'g /cvjJ (60)

where H is the exact Hessian matrix or, as suggested by Zerner et al. [19], an

approximate matrix of the Hessian if H is not available. The step direction is obtained by

writing s in terms of the gradient (g) the eigenvalues and eigenvectors of the Hessian

(Ai and jvi) respectively):

Ivi) (v 1g) (61)


To ensure an uphill search direction, a specific eigenvector of H that overlaps strongly

with the uphill search direction, is chosen such that A, is scaled using scaling factors

and the search direction s is obtained as

1nv) (nvKg) Vi) (VIg) (62)
A'X A A
where the scaling factor n is chosen as: n VA/A and 4 A2/4; A is chosen


to lie between A1 < A < A2. Thus the step is scaled accordingly to the curvature of the

quadratic region. Finally, the stationary point Xe is found switching to the Norm of the

Gradient Square Method (NGSM, see below) when H develops a negative eigenvalue

[19].

Although this model has the advantage of being precise, it is expensive to compute

since the exact Hessian is required. It has to be pointed out that Jensen and Jorgensen






42

[51] developed this method for MCSCF optimization of excited states. Further devel-

opments were carried out by Zerner and his co-workers [27].

Norm of the Gradient Square Method (NGSM)

The sum of the squares of the gradient (g), written as


= g? 1(gg) (63)


is minimized [18] as was initially suggested by McIver and Komornicki [16]. The

Taylor series expansion will be:


o-, + = o-S + oKS + -Sk-UsK + ... (64)


and
SI II
(TI+1 = U/ + U-KS, (65)


where k indexes the cycles and s, is the step, defined as-


s, = X,4+1 Xh- (66)


From Eqn. (65), an extreme point for the function a is one in which u'+1 = 0 then

(II-)-1 ,
s -K) o (67)


where
9 (o- gi
Ujk -- j- 2 gi 5X,

(68)
,, 0o a2 gi Ua&y)
o-jk O~cX -2 E ioix ~ ~
0+ O"XJ
or in matrix form


,T =2Hg =2[C+(69)


(69)








It must be noticed that here
a2gi
Cjk = rax.jaxk (70)

contains the third derivative a3E/aXiOXjaXk and becomes less important as gi -* 0,

that is, as an extreme point is approached.

From Eqn. (67) we have

s [2(C+HH)] 2Hg [ HH] Hg. (71)

When C -4 0

s -H-19 (72)

which is the Newton-Raphson equation for an extremum point provided that C is

sufficiently small (locally, near an extreme point it must always be correct).

The "object" function being reduced from equation (64) is o (not E) and the line

search condition is: o,+1 < an.

This method can be applied to find any stationary point and will not necessarily

find local minima with respect to the energy: Rather, one usually increases the energy

of the nearest stationary point and then minimize it with this technique.

Gradient Extremal

This model, first proposed by Ruedenberg [52] and further developed by others

[53], uses gradient extremals which are defined as lines on the mass scaled potential

energy surface E(x) having the property that, at each point xo, its molecular gradient

g(xo) is a minimum with respect to variations within the contour subspace, for example,

along a contour of E(x) constant. Figure 2.6 shows the behaviour of this procedure as it

steps uphill, whereas Figure 2.7 shows how minima and maxima get connected through

the gradient extremal.






44

The procedure starts by introducing the Lagrangian multiplier A


a [gtg 2A(E K.)] /x = 0. (73)


By differentiating Eqn. (73) the following eigenvalue equation is obtained


H(x)g(x) = A(x)g(x) (74)


This is perhaps the most important contribution of this technique, as it states that the

gradient is an eigenvector of the Hessian. A simple interpretation of this expression is

that 2Hg is proportional to the gradient g at the point x. Moreover, g is orthogonal to the

contour subspace at gradient extremals, since g is orthogonal to the contour subspace.

It is assumed that the potential energy, its gradient and Hessian are calculated

explicitly at each iteration. Setting the geometry of the k'th iteration, say xk a step

Sk is determined, where it is possible to write: xk+1 = Xk. + sk. The second order

total energy at this point is approximated as

E(2)(Xk+l) = E(Xk) + gT(Xk)Sk + I s kH(Xk)Sk (75)


and the actual energy at this point (with no approximations) is

E(Xk+l) = E(2)(xk+l) + R (76)


where R contains higher order terms in Sk. Steps are taken with confidence if:

E(2) (Xk+l) -4 E(Xk+1). A quantitative measure of this approach to agreement may

be obtained from the ratio r as

=12
r-= [E(Xk+l) E(94)] = 1 + R(77)
[E(2)(Xk+1) E(Xk)] [E(2) (l) -E(Xk)] (7

If r-+ 1, the third-order terms are negligible and the second-order expansion is considered

to be exact. The chosen step size should then depend on how close r is to unity.






45

A trust region with radius h is introduced, within which the second-order expansion

approximates the exact potential surface, and the trust radius is updated according to

the size of r.

The step direction (Sk) is obtained by using the extremal of the second-order surface.

The steps in the walk are determined assuming that in the trust region the gradient

extremal of the second-order surface will describe accurately the gradient extremal of

the exact surface. In the quadratic region we have

H(x) =H A (x)=- -= g(X) =g+HX (78)

where H and A are constant. It is assumed that the origin is the center of expansion.

Substitution of Eqn. (78) in (74) gives

(H AI)Hx = -(H AI) (79)

which reproduces the Newton-Raphson step equation if (H AI)-1 exists. Let v be the

eigenvector of H belonging to A (the eigenvector along the reaction path): (H-AI)v=0.

If A is non-degenerate then (H-Al) is non-singular on the orthogonal complement of

V. Thus, the following projector is introduced: P I vvt and Eqn. (79) is

now written as

PHx = -Pg Px -PH-19 (80)

The solution for this relation, assuming that H is non-singular, is

x(a) = -PH-lg + av. (81)

The gradient extremal x(a) (alpha is an arbitrary real parameter) for the second-

order surface defines a straight line which is parallel to the eigenvector 'V passing

through the solution of the projected Newton equation PH-lg where oav is the step in

our Newton-Raphson scheme. If now the Hessian has the desired number of negative

eigenvalues




















q,


g


-PGg








*TS














Figure 2.6. Potential energy surface representing the gradient extremal uphill walk
(bold arrow) that will connect stationary points, that is, all minima and transition states.
























TS1


Figure 2.7. Potential energy surface representing gradient extremal, unique lines
(bold) connecting stationary points, that is, all minima (ql, q2 and q3) and transition
states (TS1 and TS2).






48

(only one for a true TS), then the stationary point of the surface is used as the next

iteration point Xk+l. On the other hand, if the stationary point is outside the trust

region or if the Hessian has not the desired index, then the gradient-extremal point on

the boundary of what becomes the next iteration will point downhill. The gradient-

extremal point on the boundary is determined by varying oz in Eqn. (81) to obtain a

step length equal to the current trust radio h.

Although results are promising for this procedure, H has the specific drawback

that the step must be inside or on the boundary of the trust region and those steps are

conservative. The gradient extremal has been found to bifurcate also during such a

walk. It is important to note that the usefulness of the gradient extremal is related to

the fact that there are unique lines connecting stationary points, as shown in Figure 2.7.

This, together with the fact that these lines are locally characterized, makes gradient

extremals potentially very useful for exploring potential energy surfaces and for some

uses in molecular dynamics. Unfortunately, applications of this technique have not

been reported yet.


Gradient Extremal Paths (GEP)


The original idea of Gradient Extremal Paths is due to J. Pancir [54] with subsequent

testing by Muller [55]. A formal mathematical definition was given by Basilevsky [56].

Hoffman et al. [52] discussed the nature of GEPs with emphasis on their usefulness

in molecular dynamics. They showed that third-order derivatives are very important to

characterize GEPs. Jorgensen et al. [31 d] were among the first in developing algorithms

to find TSs in chemical reactions using second order GEP. Recent developments and

applications have appeared for GEP [10, 12, 13, 62a]. Use of GEP to obtain molecular

vibrations, as well as a good review of this model have been discussed by Almlof [57].








Constrained Internal Coordinates

Internal Coordinates [19, 58] are often preferred over Cartesians because they allow

valence bond parameters (bond lengths, bond angles) to be constrained in a physically

meaningful way as the remaining structure parameters are optimized.

Such procedures can be summarized in accordance to the following three steps:

Series of minimizations constraining some coordinates.

TS is the Emax with respect to the unconstrained coordinate(s).

Energy is minimized with respect to all other coordinates.

An advantage of this procedure is that the Hessian is not required to reach the saddle

point. A major drawback is that an important reaction coordinate must be identified

in advance.

The Image Potential Intrinsic Reaction Coordinate (IPIRC)

Designed by Sun and Ruedenberg [12d], IPIRC is a transformation of Fukui's

Intrinsic Reaction Coordinates [59, 60] transition state search procedure converted into

an algorithm that searches for minima. IRC was originally proposed by Fukui [59]

and later developed by others [60]. Andres et al. [61] applied the IRC to the addition

reaction of CO2 to CH3NHCONH2 using different semiempirical methods and Ab-

Initio basis sets.

The strategy of this technique is as follows:

1) Diagonalize the inverse of the Hessian matrix: CtH-'C = A.

2) Organize the eigenvalues of the diagonal matrix A in decreasing order:

Al1 > A2 > ..... A,.

3) Change the sign of the smallest eigenvalue A,

4) Undiagonalize A and procede to minimize using a steepest descent procedure.






50

As a consequence, the transition state structure now becomes a minima (to be

sought) and the original minima (starting conformation) becomes a higher energy

structure from which the down-hill walk (minimization) will start.

The Constrained Optimization Technique

Constructed by Muller and Brown [62], the constrained optimization technique opti-

mizes consecutively the geometry through a given pre-established coordinate. Abashkin

and his collaborators as well as others [41, 63, 64], have proposed a mixture of tech-

niques and implement this idea into DFT calculations. The main contribution of their

algorithm is that they solve the problem of the constrained optimization by explicitly

eliminating one of the variables using the constraint condition.

Gradient-Only Algorithms

A gradient-only algorithm recently was explored by Quapp [13]. It has as a major

drawback its apparent necessity of a large number of steps to find the saddle point. Its

success relies on the small size of the step it takes but, as a consequence, convergence

is very slow. Figure 2.7 shows the main features of this procedure.

The algorithm starts by stating a new definition of the valley pathway: A point q

belongs to a -/-minimum energy path (-yMEP) if the gradient condition g(q) = g(q-)

holds, and is used to compare differences of gradient vectors. The new coordinates are

given by: q = q + yg(q). We immediately recognize the steepest descent like

relation to obtain the new coordinates in this uphill walk (where the Hessian has been

replaced by the identity matrix). Here -y is a step length parameter (not coming from

a line search).

An asymptotic steepest descent path is defined as the geometrical space in which

many steepest descent lines, from the left and the right side, converge into the stream






51

bed of the valley ground whose shape will be followed by the 'yMEP. The points close

to these path are shown in Figure 2.2.

The situations to be encountered are as follows: If the point q.o, is at the left of the

7MEP, then the negative gardient of q11 will point it back to the right. Conversely,

if the point q,. is displaced to the right then the negative value of the gradient at qlr

will point to the left. The idea is that this gradients can be used to correct the steps

as they go apart from -/MEP.

The algorithm, which needs a step length (s) and a tolerance (t) to start, is as follows:

1) Optimize starting geometry q. that it is not necessarily a minimum: Ig (qo) I

0.

2) Choose a step length and a tolerance (t) such that: Set counter i = 0 and t <<

s with t < 1.

3) Predict a step in a steepest descent fashion: qi+l = qi + -yg(qi).

4) If lg(qi+l)l < T then STOP, meaning that a saddle point has been located

(T is a given threshold).

5) Get a scaling factor () (in braket notation): f = (9(qi+l)jg(qi)).

Here a backwards checking is performed:

If f > 1 t then: seti=i+ 1 Go To step 3). Else:

6) Correction to the step: qij+ = q1,+l yg(qi+l), set i = i + 1 and Go

To step 3).

The technique seems to work well if t < 10-2. This procedure is not competitive,

for example, with the Approximate LTP technique of Cardenas-Lailhacar and Zerner

[14], which requires one-fourth as many energy evaluations to get the same results.
















































Figure 2.7. Quapp's only gradient procedure. Three different points on and in
the neighborhood of a minimum energy path (MEP) are shown. The gradients (uphill
arrows) are shown for points q.1, q o and q.,. For points q1i, qj, and qji the negative
gradients (down hill arrows) are drawn. The uphill steps are corrected using the gradient
vectors -9(qll) and -g(qiL,)


AL


\qot


/r
qor\,








Table 2.1. Features, Advantages and Disadvantages of some of the most used
Transition State search techniques available today in program packages.


Model


Simple
Monte Carlo


Simulated
Annealing



Synchronous
Transit Path
(LST and QST)


Cerjan-Miller




Schlegel




Minimax / Minimi


Energy Min or
Normalization
Augmented
Hessian

Gradient Extremal



Constrained Int.
Coordinates
Squared Norm of
the Gradient


Partics. in volume.
Arbitrary initial
Config.: p = cte
Reduced
Temperature to
evaluate
probability. p = cte
LST: Line
connects R and P.
QST: Max LST
fitted to a Parabola
Evaluate Hessian
to define uphill
path. Lagrange
multiplier is used
Right inertia of
App. H obtained
by fixing
eigenvalues sign
Successive Opt of
Int. Coords. of a
given Symmetry
Distance between
R and P is used
Search Dir founded
diagonalizing the
Approx. Aug. H
Stationary points
in PES connected
by stream beds
Selection of RC
(bond length)
Newton step-like
search direction


Advantages
Initial Config of the
system is arbitrary


Flexibility to anneal
the P.E.S.



Simple assumptions
about reaction path
simplifies the search


Walks up-hill from
minimum to TS
essentially in an
automatic way
Reverse up/down
search direction,
refined by a factor

Hints unexpected
Intermediates or
extreme shifts
Very simple and
cheap procedure
Precise, few cycles
needed to Minimize
the gradient
Unique lines (g)
that connect
stationary points
g is not needed to
reach TS
Nearest stationary
point uncovered


Disadvantages
Random walk
needs large number
of moves
Need large number
of random moves



If path is curved
QST might not
converge


Frequent H matrix
calculation makes
it expensive


Fails if number of
iterations needed is
large. Downhill
step 1 dimension.
Extra parabolic line
minimization along
QST
Needs good initial
guess for TS
Evaluation of the H
matrix is expensive


Complications
happen if Gradient
Extremal bifurcates
Identify suitable
Reac. Coord.
Costly evaluation
of the Hessian


matrix















CHAPTER 3
HISTORY OF GEOMETRY OPTIMIZATION AND TRANSITION STATE
SEARCH IN SEMI-EMPIRICAL AND AB-INITIO PROGRAM PACKAGES


The relations describing the approximations involved in each model will not be

examined here because this is not a comprehensive review. We refer the reader to the

original works [65].

Brief Historical Overview

Semi-empirical molecular orbital theories are mainly based on approximations to the

Hartree-Fock equations. The first of the Zero Differential Overlap (ZDO) methods that

has historical importance is the 7r---electron method developed in 1931 by Htickel [66].

It is still used today to demonstrate important qualitative features of delocalized systems.

In the early 1950's, Pariser, Parr, and Pople developed the PPP theory [67], which

while only of historical importance, had great influence on future procedures. This

technique was the first to describe molecular electronic spectroscopy with any degree

of accuracy and generality. Related to these procedures, in 1952 Dewar developed

the Perturbational Molecular Orbital (PMO) theory [68] (also a ir electron method),

calibrated directly on the energies of model organic compounds. The accuracy of this

method was remarkable [69].

Pople and his co-workers, in 1965, extended the ZDO method to all valence

electrons [70]. The impact that such an approximation had in the formation of the Fock

matrix gave rise to new methods such as the Complete Neglect of Differential Overlap

(CNDO), Intermediate Neglect of Differential Overlap (INDO), Neglect of Differential






55

Diatomic Overlap (NDDO) schemes. New modifications and approximations introduced

in these procedures, have produced revised methods such as CNDO/1, CNDO/2,

CNDO/S, 1NDO/1, INDO/2, and INDO/S.SThe first gradient method was introduced by

Komornicki and McIver [2] in CNDO and this important work had enormous impact.

The MINDO/3 model [71] (a Modified INDO model) was developed by Dewar and

his collaborators in 1995. This technique was designed to reproduce experimental

properties such as molecular geometries, heats of formation, dipole moments and

ionization potentials. This method has prove to be remarkably useful. Introduced

much later, MINDO/3 has an automatic geometry optimization procedure which was

a contribution of tremendous impact at that time the Davidon, Fletcher and Powell

(DFP) [22] algorithm.

The SINDOl model by Jug et al. has proven to be very accurate in reproducing

geometries as well as other properties as binding energies, ionization potentials, and

dipole moments [72]. The geometry optimization part of SINDO1 was implemented at

the Quantum Theory Project (University of Florida) by Hans Peter Schluff, and uses

the BFGS procedure [21, 27] as developed by Head and Zerner [19].


NDDO and MNDO

Proposed by M. Dewar and W. Thiel in 1977 [73], the Modified Neglect of

Differential Overlap (MNDO) model was introduced as the first NDDO method. Today

it uses the gradient norm minimization procedure for finding the transition state, whereas

for optimizing geometries it employs a variation of the DFP algorithm [22].

MNDO, as well as the majority of other ab-initio and semi-empirical programs, is

subject to improvements which generate a proliferation of related programs and methods

5. S stands for Spectroscopy; parameter sets are modified to reproduce electronic spectra.






56

such as MNDOC [74], initially parametrized only for H, C, N and 0 [75]. More recently,

it has been suggested by M. Kolb and W. Thiel [76] that an improvement to the MNDO

model can be achieved by the explicit inclusion of valence shell orthogonalization

corrections, penetration integrals, and effective core potentials (ECP's) in the one-

center part of the core Hamiltonian matrix. Their results shows good improvement in

the location of TS over such methods as MNDO, AMI and PM3. MNDO originally

was parametrized on experimental molecular geometries, heats of formation, dipole

moments, and ionization potentials.


MOPAC

In 1983 Stewart [77] wrote a semi-empirical molecular orbital program (MOPAC)

containing both MINDO/3 and MNDO models, allowing geometry optimization and TS

location using a Reaction Coordinate gradient minimization procedure, introduced by

Komornicki and McIver [8] and vibrational frequency calculations.

AM1

The Austin Model 1 (AM1) developed in 1985 by Dewar and his group [41], was

created as a consequence of shortcomings in the MNDO model (spurious interatomic

repulsions, inability to reproduce hydrogen bonding accurately). Minima and TS

location are the same as for MNDO.

PM3

The Parametric Method Number 3 (PM3) introduced by Stewart [78], is the third

parametrization of the original MNDO model. As in AM1, PM3 is also a NDDO

method using a modified core-core repulsion term that we will not describe here. PM3

and AMI differ from each other in that PM3 treats the one-center, two-electron integrals

as pure parameters. This choice implies that in PM3 all quantities that enter the Fock






57
matrix and the total energy expression have been treated as pure parameters. This, in

turn, is proving to be a disadvantage as many anomalies are beginning to appear as a

consequence of parameters that are not physically reasonable. Geometry is searched

using the Saddle technique [40, 41].


ZINDO

This package of programs implemented by Zerner and co-workers [26], contains the

INDO/1 and INDO/S models. ZINDO is constructed to perform a series of calculations

based on different models, namely PPP, EHT, IEHT, CNDO/1, CNDO/2, INDO/1,

INDO/2 and MNDO. It includes techniques to examine geometry [17, 19, 79] using the

Line Search (see chapter 2), Newton-Raphson, Augmented Hessian, Minimize Norm

Square of Gradient, and other techniques. This allows the user to select from a variety

of search types as well as updating procedures (e.g. BFGS, Murtagh-Sargent (MS),

DFP, and Greenstadt). For TS structures search, the "Augmented Hessian" procedure

developed by Nguyen and Case [49] has been implemented by Zerner et al. [19].

Although this is a very effective method, it requires the use of second derivatives,

again making it very time consuming. The Gradient Extremal method of Ruedenberg

et. al. [52] is very effective, but requires the exact evaluation of the Hessian. The LTP

algorithm [14] has recently been implemented; it makes use of up-date techniques like

BFGS [21] to treat the second derivative matrix, although it uses reactants and products

for the search. Widely used, the BFGS update procedure for geometry optimization and

TS search was born in this versatile package; and is now used in Gaussian, HONDO,

Gamess as well as AMPAC.






58

AMPAC (Version 2.1)


This molecular orbital package, a product of Dewar's research group, is a new im-

proved version of the original AMPAC, containing the semiempirical Hamiltonians for

MINDO/3, MNDO and AM. It uses the BFGS algorithm for geometry optimization,

while for the search of the TS uses the Chain method [9, 40].

The Chain method needs, in order to maintain the connectivity of the path, to restrict

each distance between two successive points to a given length which in AMPAC is 0.3

A.

GAUSSIAN 94

A further development of its previous versions (Gaussian 76, 80, 82, 86, 88-92)

[80], Gaussian 946 is a connected system of programs for performing semi-empirical

and ab-initio molecular orbital (MO) calculations. Gaussian 88-92 includes:

Semi-empirical calculations using the CNDO, INDO, MINDO/3, MNDO and

AM1 model Hamiltonians.

Automated geometry optimization to either minima or saddle points [15, 20, 22,

44, 74], numerical differentiation to produce force constants and reaction path following

[15], and so on.

The option in Gaussian for "Optimizing for a Transition State" is sensitive to the

curvature of the surface. In the best case, in which the optimization begins in a region

known to have the correct curvature (there is a specific option for this in the menu) and

steps into a region of undesirable curvature, the full optimization option (available as

a control option for the calculations) can be used. This is quite expensive in computer

6. A version of this program-package, with parts rewritten by Czismadia and coworkers, is
also called Monster-Gauss because of the tremendous amount of calculations that can perform
as well as being monstrous in length because of its ab-initio block.






59

time but the full Newton-Raphson procedure, already implemented in the program, with

good second derivatives at every point will reach a stationary point of correct curvature

very reliably if started in the desired region (line searches can be conducted with second

derivatives at every point). If a stationary region is not carefully selected, it will simply

find the nearest extreme point.

An eigenvalue-following, mode-walking optimization method [74, 81] can be

requested by another option (OPT=EF) [43, 82] that is available for both minima and TS,

with second, first, or no analytic derivatives as indicated by internal options (CalcAll,

CalcFC, default or EnOnly). This choice is often superior to the Berny7 method, but

has a dimensioning limit of variables (50 active variables). By default, the lowest mode

is followed. This default is correct when already in a region of correct curvature and

when the softest mode is to be followed uphill. Other options of interest, in connection

with GOPT and TS calculations, are:

Freezing Variables During Optimization: Frozen variables are only retained for

Berny optimizations.

Curvature Testing: By default the curvature (number of negative eigenvalues) is

checked for the transition state optimization. If the number is not correct (1 for a TS),

the job is aborted. Here the search for a minimum will succeed because the steepest

descent part of the algorithm will keep the optimization moving downward. On the other

hand, a TS optimization has little hope if the curvature at the current point is wrong.

Murtagh-Sargent Optimization: This method almost always converges slower than

the Berny algorithm. It is reliable for minima only.

Berny or Intrinsic Reaction Coordinate (IRC) method: This is an algorithm

designed for finding minima mentioned here because it is often used in the Gaussian

7. Berny stands, with tenderness from the Gaussian people, for Bernard Schlegel.






60

package. With IRC the reaction path leading down from a TS is examined using the

method of Gonzalez and (Berny) Schlegel [15]. In this procedure, the geometry is

optimized at each point along the reaction path. All other options which control the

details of geometry optimizations can be used with IRC.

Although Gaussian 88-92 has many optimization options that can be used in

combination with one another, it will be enough for our purposes to note that this

package of programs uses Newton-Raphson, Murtagh-Sargent, Fletcher-Powell and

Berny's (by default) methods for optimization, whereas for TS search it uses the Cerjan-

Miller algorithm [43] and the Linear Synchronous Transit method (LST) [37].


HONDO


This package written by Michel Dupuis and his co-workers [58] uses algorithms

that take advantage of analytic energy derivatives. The Cerjan-Milller algorithm [43] is

implemented with an updating of the Hessian matrix. This algorithm has proven efficient

provided that a "good" second derivative matrix is used [59] (it has been found that a

force constant calculated with a small basis set at the starting geometry is adequate).

As an option, the HONDO program allows the user to use the "Distinguished

Reaction Coordinate" approach. This approach consists of a series of optimizations

with an appropriately chosen coordinate being frozen at adequately chosen values

(for further details see the "Constrained Internal Coordinates" [19, 59] outlined the

preceeding chapter).

After this, and an inspection of the potential energy curve, it is possible to guess

the TS structure. At this point, a geometry optimization of the guessed TS is suggested

by using the BFGS algorithm implemented in the program. The method can be used in

conjunction with all SCF wavefunctions, as implemented for the geometry optimization.






61

All other options on HONDO assume that the TS structure is known as well as the

vibrational mode corresponding to the imaginary frequency. Then the gradient norm is

minimized [18] to find the nearest extrema.


ACES II (Version 1.0)

This package of programs for performing ab-initio calculations, developed in the

early 1990's by Bartlett and co-workers [83], contains geometry optimization algorithms

that are all based on the Newton-Raphson method, in which the step direction and size

are related to the first and second derivatives of the molecular potential energy. In

almost all calculations the exact Hessian is not evaluated but approximated. By default

ACES II geometry optimization starts with a very crude estimate of the Hessian in

which all force constants for bonded interactions are set to 1 hartree/bohr2, all bending

force constants are set to 0.25 hartree/bohr2, and all torsional force constants are set to

0.10 hartree/bohr2. An alternative Hessian is used for some small systems, allowing

the use of an In the search for a minimum, the method implemented in this package

can be used when the initial structure is in a region where the second derivative matrix

index is nonzero. Moreover, a very efficient minimization scheme, particularly if the

Hessian is available, is included in this package of programs, namely, a Morse-adjusted

Newton-Raphson search for a minimum.

For the TS search ACES II uses the Cerjan-Miller algorithm [43]. This involves

following an eigenvector of the Hessian matrix (that corresponds to a negative eigen-

value) to locate the stationary point, ensuring that it will stay within the region of

the TS. Finally, to ensure that a TS has been obtained, the vibrational frequencies are

evaluated by taking finite differences.















CHAPTER 4
THE LINE-THEN-PLANE MODEL



Introduction


With the exception of the Synchronous Transit [37] and the Normalization technique

[41] models both of which consider the distance between reactants (R) and products

(P) while searching in a linear fashion for the TS to finally minimize the maxima found

all other procedures discussed in chapter 2 use up-hill methods. Those procedures start

from reactants through a second order expansion of the energy in terms of a Taylor

series.

For comparison we have collected in Table 2.1 the procedures reviewed in chapter

2, emphasizing their principal features, advantages and, especially, their disadvantages.

If we focus on the disadvantages, we notice a general trend in the problems that appear

when searching for a TS (which are similar to the geometry optimization ones):

Costly evaluation of the Hessian matrix.

Difficulties in identifying an appropriate reaction coordinate.

Requirement of a good initial guess of the TS.

Convergence achievement.

Large number of moves needed (i.e. random procedures).

Because of these problems, a better method to find the TS should consider both

reactants and products because both contain information on the appropriate saddle point.






63

Here we present a procedure that is based on a continuous walking from R to P

(and vice-versa) with a fixed step length, along a line connecting them. By minimizing

the energy at those new points a new line is drawn and the procedure is repeated until

a pre-established criterion to find the maxima (or minima) is fulfilled.

The procedure that we present is very simple and has been designed to overcome

the problems enumerated above. Thus, we have built up a strategy to find the TS, which

makes use of the line search technique, that has the advantage of using a reduced number

of calculations, has a simple and convenient expression of projected coordinates, does

not require evaluation of the Hessian matrix, considers an intermediate of reaction, and

involves the idea of finding the TS(s) starting simultaneously from R and P.

The algorithm does not require the evaluation of the Hessian. As a result, it is much

faster in its execution than most of the methods presently in use, and it is applicable

for searching the potential energy surfaces of rather large systems. The procedure is

not completely unlike the Saddle procedure of Dewar, Healy and Stewart [42] or the

line procedure of Halgren and Lipscomb [37], but does differ in a rather substantive

way. In this new technique, the line direction is allowed to change during the walk,

initially from a line connecting product and reactant to points that represents them.

These representative points are determined through minimization of all the coordinates

that are perpendicular to the connecting line. The efficiency of this procedure rests upon

the observation that it is faster and easier to minimize repeatedly in the N-1 directions

than it is to evaluate N(N + 1)/2 second derivatives, where N represents the number

of variables (coordinates) to be searched.

When the interest is to focus on the shape of the reaction path (mechanism) we

suggest, as an alternative simpler strategy, to find the TS(s) so that, when there, the

reaction path is constructed by using a down-hill procedure to reactants and products.






64

However we do not recommend this sequence. We will introduce, unstead, the LTP

algorithm and discuss its properties, the step and the speed-up of the procedure by

using Hammond's postulate. Finally, Hammond's postulate adapted LTP techniques

are discussed.


The Line-Then-Plane (LTP) Search Technique


This procedure, originally conceived to find TS in chemical reactions, requires

knowledge of reactants (R) and products (P); no previous knowledge of the TS is

necessary. It makes partial use of both the line search technique and the search for

minima in perpendicular directions that have been discussed already. Figure 4.1 shows

the behavior of this technique.

As in the Saddle method of Dewar [40-42] we begin by calculating the structures

of the reactant R and product P. A difference vector di = Pi Ri is defined


d (XP + (Yp + (Zp -ZR) ]1 (82)


with i = 0, 1, 2, 3, ... and we walk a fraction of the way from Ri to Pi along -di, and

from Pi to Ri along di The structures at these two points are minimized in the plane

(i.e. all directions) perpendicular to di, defining the new points Ri+j and Pi+I. A new

difference vector di+ = Pj+ Ri+: is then defined and the procedure repeated.

The steps along di are conservative initially, but increased as a percentage of the norm

of di, that is dd, as the TS is approached. The BFGS technique [18, 21, 26, 27]

is used to minimize the energy in the hyperplane perpendicularly to di.

At this point we distinguish the Exact and the Approximate LTP procedures:

Exact LTP: The BFGS technique is used to minimize the energy in the hyperplane

perpendicular to the direction di.





























# (i>n)

P2 P TS

di



(a) (b)

Figure 4.1. (a) General scheme for the Line-Then-Plane (LTP) procedure where the scanning is performed between the
last two minimal points found, from both R and P. (b) A sequential transverse view of the planes containing the projected
and the in-line (di) points.






66
Approximate LTP: No up-date of H is performed. For the minimization of the

energy, in perpendicular directions to di, we use the identity matrix as the Hessian (PH

= P1 = P) only once (steepest descent), so that the projected coordinates depends only

on the gradient gi and on the projector Pdi (i.e., i = 0), see below.

The Algorithm

We adopt the following algorithm, shown graphically in Figures 4.1 a-b:

Step 1) Calculate Reactant and Product geometries RPo and PP (the super index p stands

for projected coordinates). Set counters i = j = 0 (see below).

Step 2) Define the difference vector di VP R If the norm d~d, < T (a

given threshold), stop. Otherwise,

Step 3) Examine ot(R)di and -o-t(Pi_)di, where uT(x) = (E/di)x

Set R = Rp .P' = P (83)

(the super index 1 stands for variables in the line that connects the corresponding R's

and P's) unless,

i) If oUt(RP) d, < 0 set P = R' and R' = R- (84)


or,

ii) If ct(PP) di < 0 set R = PP and Pl = P1-1 (85)



Step 4) Walk along di from R' to P$, and vice-versa, a fixed step of length: si di/Ni

Set Ri1 = Ri + si and Pi+1 = P Si (86)

unless,
If T < dtdi < sjsJ set Ni= Ni/2 and j=

If N < 2 set N,= 2
(87)






67

Step 5) As in Step 3), examine or4(R +)si and O-f(P.,)si


If J7t(R'+,) si < 0 set P+ R and Ri+ = i
S7+1 R1Z+


If -o-t(P+,) si < 0 set R+= P+1


and P41+ = Pi


(88)


(89)


Step 6) Minimize in the hyperplanes perpendicular to di containing R and P.+1, to

obtain projected points Ri+, and PP+?, respectively. Set i = i+1 and go to Step 2).

Although one can demonstrate that this procedure must lead to the Transition State

on a continuous potential energy surface E(x) if the steps are conservative enough, the

norm of the gradient (g) and the Hessian (H) are examined at convergence in order to

insure that the converged point on E(x) has the right inertia, (i.e. g = 0) and H has

one and only one negative eigenvalue.

To accelerate convergence to the TS, we might add to Step 3) a further test that

becomes useful as the TS is approached,

Step 3) iii)

If Alts(k) = 0t(P',) Sk (t(RP,) Si < T' and,

AE(k)=E(Pk)- E(Rk)
Then, Set: qi = (P, Ri)/2, evaluate E(qi), g(qi) and H(qi)

Else, If gt(qi)g(qi) < gt(Pi)g(Pi) and gt(qi)di < 0
(91)
Then qi replaces Pi or,

If gt(qi)g(qi) < gt(Ri)g(Ri) and -gt(qi)di < 0
(92)
Then qi replaces Ri.
Then, go to Step 2), Else, go to Step 4).

Here qi refers to the coordinates that represents a conformation that is very close

to the TS structure.






68

Based on our experience, the choice N, 10 generates a conservative initial step

and suitable thresholds are T = T' = T" T.. = 10-4 arbitrary units. We have

studied the variation of the step size with the number of energy evaluations needed to

converge to the saddle point, as is described in a subsequent section.

The strategy delineated above is also successful even for systems which have

intermediate structures between R and P. The tests indicated in equations (84-85)

under Step 3) and (88) and (89) under Step 5) disclose potential turning points, caused

either by a too large a step from R toward P or P toward R.

The reaction path can be approximated by connecting all points Ri and Pi. An

approximate and faster procedure would be to quit in Steps 3) or 5) thereby avoiding

the reset of coordinates between consecutive steps.

Then the displacement di can now be divided in smaller parts (say 4) and the

procedure continued as before. The last half is now submitted to a perpendicular

minima line search founding a last point Xe, the TS.

In general, the TS is said to be found if the gradient norm is zero and if the Hessian

has one and only one negative eigenvalue, respectively. As for LTP, the transition will

be considered to be found when the norm of the displacement vector di is smaller

than a pre-established convergency threshold T, (usually T, < 10-3). Nevertheless, the

general conditions are checked at the estimated saddle point (i.e. (7(Xe) = 0).


Minimizing in Perpendicular Directions: Search for Minima


The coordinates perpendicular to the direction di are obtained by projection, and

the energy in the hyperplane minimized using the BFGS algorithm as developed by

Head and Zerner [27]. It has to be pointed out that translations and rotations must be

eliminated from G = H-1 as they represent zero eigenvalues of H, in order to construct






69
a projector free of them. This requirement has been included in the ZINDO program

package [26] as part of the implementation of the LTP techniques.

This procedure is restricted to the projected coordinates

(I)
Pd-(Xii = =Pd Gi+,Pdigi+, (93)

where


N (94)
is the step (coordinates) along the line connecting projected R's and P's.

In a more compact way, we can write Eqn. (93) as

(xi+ xiL)Pdi a GPd, Pd (95)

or
x(pi) (1) p
ia gil G"ip1 (96)


where the projector perpendicular to di is defined as [17]:
P& = I di d!
d= dd' (97)
i ,i
In these equations, a is the line search parameter which determines how far along

the direction si+] of equation (94) one should proceed. For the simple test cases

studied in the next chapter, there is but one perpendicular direction, and we set a = 0.3

for all i which is a more conservative value than that recommended by Zerner and his

collaborators (ao = 0.4, and all other ai = 1.0) [17, 19, 27].

It can be demonstrated easily that, computationally, it is much more convenient to

project out only the forces rather than project the forces and the second derivative matrix

at the same time. Consequently the new projected coordinates are now obtained as

= i+) a Gi+ gi~ (98)


where the inverse Hessian G is updated using any appropriate technique.








Projector Properties

The projector Pdi must be well behaved (i.e. it has to fulfil the conditions of being

idempotent and hermitian).



Idempotency: P = p2

We start from the definition of the projector:


P=I-


ddt
didd


(99)


consequently


p2 = I ]t I ddt ]
dtdj I dtdJ


ddt



dtd



2ddt
dtd


ddt
dtd


ddt ddtddt
dtdd + dtddtd


ddtddt
dtddtd


ddt
d-d


=P


Hermiticity: P = Pt


pt I dddJ]

[ddl I
-d dJ=


(102)
dtd
q. e. d.


(100)


q.e.d.


P I ddt
dtd


(101)






71

LTP Convergency


Consider the coordinates difference


q = Xi+1 X*


(< c)


(103)


where X* represents a maximum, the TS (X* = XTS).

Defining, for the neighborhood of the TS X*:


lir q = 0 = E (q) = a iff 3 c E R a cq.
-4. O


(104)


The gradient around a given point X,:


g(X. + qK) = g(X-) + Hq, + 0(Iq ]2)


(105)


but q = -q, then:


g(X X, + X*) = g(X,) + Hq, + (1%,12)


(106)


g(X*) gK HnqH + e(lqh.2)


If X, is too close to X* (with Hk with only one negative eigenvalue), the considered

region of the space exists by continuity of the Hessian H. Consequently, the / th

iteration exists.

Projecting from the left by HK 3-


Hrg9(X*) = 0 = H-1g9 H-'Hcq, + 8(1q,


(107)


= s, which is a Newton-Raphson like step.


but: g,.s








Finally:

0 q, + e(1q, 2) q'+1 + e (Iq, 12) (108)

but according to our original definition:

3c E R/lq+ll- clqJ2 (109)

If Xh, is very close to X* for which:



qJ < a/c 0 < a < 1 (110)



by induction, and because X, -- X*, the iteration is defined and it exists for all

K and q,- --+ 0. Consequently, by construction, LTP always will go uphill in the

search for a maximum.

To ensure that the new projected points R and P are perpendicular to the reaction

coordinate di, we must show now that the energy is a minimum in these directions.

Consider the second-order expansion for the Energy

E(x) = E(xo) + qtg + 1qfHq (111)


From the gradient expansion

g = g0 + Hq and q = -H-19 or g = -Hq. (112)

Introducing g in the equation for the energy, we get

E(x) = E(xo) 1 qHq (113)
2

which demonstrates that the energy in perpendicular directions to the step is minimized

by a steepest-descent-like term in which H is positive definite.

From these, we conclude that a second (or higher) order LTP iteration converges.






73

The Step


A good step will provide a good starting point for the next step, such that

the maximization will converge without problems in a reduced number of iterations

along the chosen direction. In general, almost all algorithms take their steps without

considering previous information about the PES.

In developing LTP, three ways of stepping were studied. The first stepping method

is a superimposed step given by a fraction (1/N) of the displacement vector between

projected products and reactants coordinates. The second stepping method is based on

a proportionality relationship between the actual and previous step. It is shown that

this choice will locate the TS (not its final position) at most at half of the size of N,

that is, around N/2 LTP cycles, because the final displacements are very small. The

stepping method is based on the knowledge of information about the PES given by the

current projected point (reactants or products) where the value of N is then estimated

by relating the LTP step to the Newton-Raphson one (since LTP is a Newton-Rapson

like algorithm).


Default Step


In LTP, the step (si) is a fraction of the current displacement vector (di)


si = di/N (114)


where N is a number greater than one. For the first iteration N = 10 (an arbitrary choice

suggested after many test calculations) and thereafter the distance between current

reactants and products is checked to be not less than a given threshold (say 10-3),

otherwise N is reset to 2.5.








Updated Step

A convenient decision on how the step should be taken comes from an algorithm

that will decide automatically what the value of N should be for the new LTP cycle

once the displacement vector is known. To accomplish this the next step is redefined

as to be directly proportional to the previous one:


si+l = di+l/Ni+, c di/Ni = si (115)

Now the problem at hand is an estimation of the value of Ni+1 and consequently the

next move. For this, we consider the following relation between the next (i+]) and

the previous (i) steps


di+-Ni+, = Adi/Ni (116)

where 0 < A < 1. Projecting now from the left by df we obtain,


Ni+1 = Ni ddi (117)

when A = 1.

Alternately, it might be better to consider a relation with a penalty function on it.

This can be written easily as

_di+l_ "di [ ('dfli2 ( 4s)2(118)
N+ N ( +

where Xi, and Xip are the difference vectors between the new projected reactant and

product, and their corresponding coordinates in the line (step from where the searches

start).

Newton-Raphson-Like Step

Consider now the usual LTP step. We want to take a non-arbitrary step based

on previous knowledge of the curvature of the region in which we are walking.






75

Furthermore, we want, at any cycle, the LTP step (sLTp) to be as well behaved as

the Newton-Raphson (SNR) one


1d = s p = SNR = g (119)

where a is a term that comes from the line search technique and the gradient (g) and the

displacement (d) are column vectors. Note that only the absolute value of N should be

considered. This is because the direction of the walk as defined by the LTP algorithm, is

positive when going from reactants to products and is negative in the opposite direction.

Projecting from the left by the gradient complex conjugate (gt)

INgtd agtHlg (120)


we derive,

T () td (121)
a9gH-19

It has been suggested, and shown, by Zerner and his co-workers [17, 19, 27], that

for the initial Newton-Raphson step a good choice is to set a = 0.4 and the inverse

of the Hessian as the identity matrix. Consequently we can have an approximation to

the estimation of N as

N= (5)9td (122)

This stepping might not be convenient when searching in the vicinity of the saddle

point because the denominator will be too small and N will be too large.

Hammond's-Postulate-Adapted LTP Methods

Introduction

It might be argued that LTP, because of its twofold search (reactants and products

at the same time), requires too many steps or that it needs twice the amount of effort






76

(steps) required by other algorithms such as augmented Hessian [19, 48, 49]. Hence,

the Augmented Hessian method will be extensively used for comparison. This concern,

and the desire to have an algorithm that will move faster and efficiently towards the

TS, brought us to the approximate LTP procedure ennunciated in the previous section.

However, and by construction, this lack of specific information about the curvature of

the potential energy surface provided by the Hessian can be a drawback.

With these problems and goals in mind, we recall Hammond's postulate (HP) [6],

which states that the TS will resemble more the initial reactants (Ro) or products (Po)

according to whether the initial or the final state, is higher in energy. However, we

have already mentioned some not uncommon examples for which HP fails.

In this section we study the inclusion of HP in order to save some computational

efforts by reducing the number of steps. We will do this by adapting LTP to Hammond's

postulate and consequently generate two more LTP like procedures, the Hammond-

Adapted-Line-Then-Plane procedures (HALTP) and the Restricted HALTP (RHALTP)

procedures. For these, the energy of both initial reactants (R.) and products (Po) (E&o

and Ep., respectively) will be considered.

Hammond-Adapted LTP Procedure (HALTP)


Two situations need to be considered:

a) If ER. >_ Epo This is the original (exact and approximate) LTP as described

above.

b) If ER. < Epo Reset to a new set of coordinates (prime): R' = P, and

Po = Ro.

The situation described in b) is shown in Figure 4.2, after which LTP will continue

as before. This particular situation can also be seen as if the search starts from the






77

original products. The advantage of this adaptation lies precisely in a reduction on the

amount of energy evaluations (LTP cycles) as now LTP will start searching from the

geometry of highest energy.

Restricted Hammond Adapted LTP (RHALTP)

In this case the same two situations depicted before are analyzed where the concept

of Hammond's postulate is now strictly enforced. The first subcase still leave us with

the classical LTP (ER, > Epo), but the second subcase (with ER, < Ep as

condition) is now modified as follows:

RHALTP I. If (ERo < Epo) then, do not move the initial products.

This means that the coordinates of the starting products, characterized by Po, are

held constant.

This choice will allow the reactants to move uphill faster towards the TS by being

lifted by the products, as shown in Figure 4.3. This possibility is of particular interest

when one is concerned with following the path of the reaction under study. The idea

is tested in the next chapter for the inversion of ammonia reaction.

RHALTP II. If (ER0 < Ep) then, do not move the initial reactants.

This time we consider that the reactants, characterized by Ro, remain as the initial

ones lifting the products towards the TS, as shown in Figure 4.4. The idea is tested,

again in chapter 6, for the non-symmetric inversion of ammonia reaction.

It has to be pointed out that RHALTP I and RHALTP II are not the same procedure

with different label for reactants and products (and of technique), because the energetics

of the changed coordinates are completely different.

















































Figure 4.2. Hammond-Adapted-Line-Then-Plane (HALTP) technique in which the
search starts from the set of coordinates of higher energy according to: ER. < Epo.



















































Figure 4.3. Products Restricted-Hammond-Adapted-Line-Then-Plane technique
(RHALTP I). The coordinates of products characterized by P0 are held constant, lifting
the reactants towards the TS.



















































Figure 4.4. Reactants Restricted-Hammond-Adapted-Line-Then-Plane technique
(RHALTP II). The coordinates of reactants characterized by P0 are held constant, lifting
the products towards the TS.















CHAPTER 5
GEOMETRY OPTIMIZATION



Introduction


Almost all the procedures discussed in chapter 2 use Steepest Descent methods

to search for minima through a second-order expansion of the energy in terms of a

Taylor series.

From the Geometry Optimization procedures reviewed in chapter 2, the general

behavior of problems in the search of minima becomes clear:



Costly evaluation of the Hessian matrix.

Large number of moves needed (i.e. random procedures).

Convergence problems.



Although several procedures are available, there are still other problems, such as

the loss of information about the curvature when the Hessian is not considered. In this

way, and as is the case for TS search, the development of new techniques will rely on

experimentation, namely that the model must show acceptable behavior on a variety of

test functions, chosen to represent the different features of a typical problem.

Because of these problems, it seems that a better method to find the minima (hope-

fully the global minimum) must consider the initial geometry plus a generated second






82
one (only at the initial step). Therefore, in addition to position and/or displacement vec-

tors, the displacement vector between the two initial points should also be considered.

A procedure that is based on the Line-Then-Plane technique (LTP), that is, a

continuous walking from the lowest energy point through a line connecting the two

lowest energy points, is proposed. By minimizing the energy at the new point a new line

is drawn between the new point and the one from which the projection was performed.

The procedure is repeated until a preestablished criterion to find the minimum is fulfilled.

Figure 5.1 illustrates the behavior of this procedure.

The same features already described for TS search with LTP are valid here, that

is, this is a procedure that does not require the evaluation of the Hessian. As a result,

the proposed method is much swifter in its execution than most of the methods used

today, and is applicable for searching for minima in potential energy surfaces of rather

large systems. In this technique, the line direction is allowed to change during the

down-hill walk, initially from a line connecting the starting geometries that represent

them. These points are determined through minimization of all the coordinates that are

perpendicular to the connecting line. The efficiency of this procedure rests upon the

observation (as for LTP), that it is quicker and easier to minimize repeatedly in the N-1

directions than it is to evaluate N(N + 1)/2 second derivatives, where N represents the

number of variables (coordinates) to be searched.



ARROBA: A Line-Then-Plane Geometry Optimization Technique


This procedure requires a single input geometry from which a second set of

coordinates will be generated only in the first step. The down-hill walk starts by

determining the lowest energy point, making partial use of the line search technique






83

and the search for minima in perpendicular directions. As introduced previously, a new

projected minima is then found. Figure 5.1 illustrates the behavior of this idea.

As in the "Amoeba," or "Simplex" method of Nelder and Mead [30], we begin by

calculating the structures of the initial point and a second one generated as:


q2 = q1 + /3 (123)


where /3 is a 3N dimensional unitary vector (where N = number of atoms) scaled by

three different factors ((k, X and ip, for the x, y and z components, respectively), one

can make /3 a constant (but /3 $ 0). Any of these choices will be the initial guess for

the problem and will depend on the size of the system.

Once a second initial point is generated, energies (E) and gradients (g) are evaluated

for both initila points (ql, El, g, and q2, E2, 92). The strategy then is as follows:

A difference vector di q, q, is defined




L-] [(xi~l ), "-(il + (Zi+l Zi)2]1/ (124)


with n = 0, 1, 2, 3, ...

The structure of lowest energy of these two points will be minimized in the plane

perpendicular to di, defining a new point qi+2 and i is reset to i = i + 1. A new difference

vector di = qi+l qi is then defined and the procedure is repeated. The norm

of di (i.e. didj ), is checked for convergence as the minimum is approached. The

BFGS technique [18, 21, 26, 27] is used to minimize the energy in the hyperplane

perpendicular to di.

As for the search for maxima, we differentiate between the Exact and the Approx-

imate ARROBA procedures:






84

Exact ARROBA: The BFGS technique is used to minimize the energy in the

hyperplane perpendicular to the direction di.

Approximate ARROBA: No up-date of H is performed. For the minimization of

the energy, in directions perpendicular to di, we use the identity matrix as the Hessian

(PH = P1 = P) only once (constrained steepest descent), so that the projected coordinates

depend only on the gradient gi and on the projector Pd, (i.e., i = 0), as described below.


Figure 5.1. Schematic representation of ARROBA, an adapted Line-Then-Plane
technique for geometry optimization. The input coordinates (ql), the initially generated
one (q2), the general zigzag behavior of the procedure and the found minima (qmi)
are shown.






85

The Algorithm


We adopt the following algorithm, shown graphically in Figure 5.1:

Step 1) Calculate initial and new generated geometries qjL and q2. Set counter i = 1.

Step 2) Define the difference vector di for which its norm didi is greater than T (a

given threshold), else the program will stop:



i) If Ei+, < Ei then di -qi+ qi (125)


ii) If Ei,+, > Ei then d qi qi+, (126)

Else: Stop, and check for convergency: ddi > T.

Step 3) Minimize in the hyperplanes perpendicular to di containing E,+ to obtain

projected points Ei+. The point from where the perpendicular minimization starts is

that one with the lowest energy. We set i = i + 1, accept the new point if and only

if Ei+2 < Ei+,, Else: go to Step 2).

Step 4) The new projected point coordinates are given by


If Ei+1 < Ei then =- a gi+1 ?,(17


where the upper script (p) stands for projected variables using the projector as showed

in the preceding chapter.

It can be shown that this procedure must lead to a minimum that, according to its

location in the PES, might be a local or a global minimum, provided the surface is

continuous. The norm of the gradient and the Hessian are examined at convergence in

order to ensure that the converged point on E(q) has the right inertia. The minimum

is said to be found if the gradient norm is zero and if all the Hessian eigenvalues are

positive.






86

Minima in Perpendicular Directions


As in the LTP method, ARROBA uses coordinates perpendicular to the direction di,

obtained by projection. The energy in the perpendicular hyperplane is then minimized

using the BFGS algorithm as developed by Head and Zemer [27].

Again, it is computationally much more convenient to project only the forces rather

than project them and the Hessian matrix at the same time. Consequently the new

projected coordinates are obtained as

x(P=) = (/) aPg Gi+1 (128)
i+1 Xi+I1 -- O i+1


where the inverse Hessian is now updated using any appropriate technique.


Convergency


To ensure that the new projected points R and P are perpendicular to the reaction

coordinate d, we must show that the energy is a minimum in these directions.

Again consider the second order expansion for the energy


E(x) = E(xo) + qtg + 1 q*Hq (129)
2

and the gradient expansion


g = g0 + Hq (130)


and

q = -H-lg or g -Hq (131)

which is the quasi-Newton condition. Introducing g in the equation for the energy we get


E(x) = E(xo) qfHq (132)
2






87

which demonstrates that the energy in perpendicular directions to the step is minimized

by a steepest-descent-like term in which H is positive definite.

From these considerations, we conclude that a second (or higher) order for the

ARROBA iterations converges. This minimization procedure has the advantage of using

a reduced number of calculations, particularly in the case of the Approximate technique,

and does not use the Hessian. It is guaranteed to step down-hill. The minimum found

will be a local minima. The search for the global minimum is discussed below.

A Proposed Global Minima Search Algorithm


As discussed in chapter 1, when looking for minima it is very desirable for a

procedure to be able to find the global minimum, especially for large molecular systems

(proteins, enzymes) for which the most widely used current procedure is the Monte Carlo

model often requiring thousands of energy evaluations.

Here we propose a procedure that will have a behavior like Monte Carlo, but does

not depend on the temperature and that does not need as many calculations. It uses

a jump-out technique, as the warm-up part of the Monte Carlo techniques to take the

system out of the local well in which it is trapped.

The algorithm requires a control option from the input file that allows the user to

perform several ARROBA calculations. The strategy is as follows:

1) Make an ARROBA minima search.

2) Set counter i = 1 and label the new minimum as: qi (n is the internal

ARROBA counter).

3) Construct a displacement vector (ri) between the minimum found and the input

geometry Xo:


ri = qi Xo .


(133)






88

4) Get a new displacement vector ri orthogonal to ri, that is, in braket notation:


(iIri) = 0. (134)


5) Obtain a new initial set of coordinates Xo:


X = ri + Xo (135)


6) Check for maximum allowed number of searches M:


If i < M Go To Step 1 Else Stop (136)


where M is a pre-established maximum number of iterations.















CHAPTER 6
APPLICATIONS


Introduction


The ideas discussed in the previous chapters have been tested by two different

approaches. One approach involves the use of two-dimensional model potential func-

tions to test the behavior of the LTP procedures and compare the results with reports

on other methods in the literature [14]. Six model potential functions are examined.

The Hammond-Adpated LTP technique has also been tested on three of these functions

and, the Restricted-Hammond-Adapted procedures were investigated on a 7th potential

function. Finally, using the potential functions, the LTP accuracy and convergence

dependence on the step size have been studied.

The LTP method was also tested on several molecular systems: the inversion

reaction of water, the symmetric and the non-symmetric isomerization reactions of

ammonia, a rotated inversion reaction of ammonia, the hydrogen cyanide isomerization

reaction, the formic acid 1,3 sigmatropic shift reaction, the methyl imine isomerization

and the thermal retro [2+2] cycloaddition reaction of Oxetane. The accuracy has also

been examined in terms of the step and number of energy evaluations required to find

the TS in the molecular examples.

For the study of those systems, the Intermediate Neglect of Differential Overlap

(INDO) technique [84] has been used, at the Restricted-Hartree-Fock (RHF) level [85]

within the ZINDO program package [26]. The minimization procedures are, of course,






90

limited to no particular energy function, provided it is continuous. The results were

compared with those of the Augmented Hessian (AH) technique that uses the same

INDO Hamiltonian but evaluates the Hessian at each iteration. All the above-mentioned

procedures were implemented in the ZINDO [26] suite of programs.

Model Potential Functions for Transition State

LTP procedures have been tested on six model potential functions which are

traditionally used to examine TS searching procedures. The first two, the Halgren-

Lipscomb and Cerjan-Miller potential functions, have their TS located closer to the

reactant than to the product. The next two potential surfaces, the Hoffman-Nord-

Ruedenberg and Culot-Dive-Ghuysen, have the TS located closer to the products than

the reactants. The fifth potential function has a TS located midway between reactants

and products, and the sixth PES has a steep minimum located in the products region.

The results of these tests are collected in Tables 6.1, 6.2 and 6.3 and are discussed below.

The Halgren-Lipscomb Potential Function

The Halgren-Lipscomb potential function [37, 39]:

EHL(xy) y(x- (5/3)232 + 4(xy 4)2 + x y (137)

has two minima (we have chosen points (1.328, 3.012) and (3.0, 1.333) for reactants

and products respectively) and one first order saddle point (2.0, 2.0). Figure 6.1 shows

the shape of this surface as well as the points obtained with the LTP procedure. Notice

that both LTP procedures, Exact and Approximate, walk uphill using the same points.

The Cerjan-Miller Potential Function

Cerjan and Miller's function [43]:

ECM(Xy) = (a by2)X2 exp(- 2) + y(13)
2(18






91

has two symmetric TSs located at points ( 1, 0) and a minimum at point (0, 0). As R

and P coordinates we have selected points (0,0) and (2.7, 0.05) respectively.

For this procedure, an accurate Hessian is required. Others that have used this

function include Simons et al. [31], with a Fletcher-based surface algorithm [22],

Banerjee et al. [32], with a rational function optimization algorithm, and Abashkin and

Russo [86], with a constrained optimization procedure. All of these previous studies

have used a = b = c = 1 with the exception of Simons and his coworkers, who used

a = c = 1, b = 1.2. Figure 6.2 shows the behavior of our procedure when applied to

this potential energy surface. Here both procedures walk towards the TS and are very

close to each other (notice the scale on the Y axis).

The Hoffman-Nord-Ruedenberg Potential Function

Hoffman et al. [52] have used the model surface function:


EHNR(xy) = (xy2 yx2 + X2 + 2y 3)/2 (139)


to test their gradient extremal procedure. This function has also been tested by Schlegel

[10]. As other algorithms, previously mentioned, these methods require an accurate

evaluation of the second derivatives.

The function has two saddle points TS1 (- 0.8720, 0.7105) and TS2 (3.1352, 1.2487).

In order to test the LTP procedures the points (1, 1) and (5.4980, 1.2874) have been

chosen as the R and P coordinates respectively. From these points, a walk towards the

TS has been performed. Figure 6.3 shows the behavior of our suggested procedure in

this potential function surface.

We note the somewhat chaotic behaviour of the Approximate procedure in the

products region, due to its inherent lack of information of the quadrature of the surface.




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TRANSITION STATE SEARCH AND GEOMETRY
OPTIMIZATION IN CHEMICAL REACTIONS
By
CRISTIAN E. CARDEN AS -L AILH AC AR
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA

To my beloved girls, my wife Alejandra and our
daughters Francisca Javiera and Catalina Sofia

ACKNOWLEDGMENTS
I would like to thank my advisor Prof. Michael C. Zerner for his support, criticism
and teaching. I have been always amazed by his never-ending creativity and enthusiasm:
you can do it Cristian/; it was always a must. With his very busy schedule, no wonder
we couldn’t interact more, but somehow he managed to find the time to discuss my
progress. He allowed me the time to go ahead with my own ideas, sometimes just to
prove to me that I was going in the wrong direction.
In the past few years I have had the opportunity to interact with many people in
the Quantum Theory Project, specially in the Zerner research group. Among them I
would like to thank Dr. Krassimir Stavrev, Dr. Toomas Tamm, Dr. Marshall Cory,
Dr. Guillermina Estiu, Dr. Igor Zilberberg and Dr. Wagner B. De Almeida. Out of
the Zerner group, my gratitude for many hours of great science and friendship is given
to Dr. Agustin Diz, Dr. Keith Runge, Dr. Ajith Perera, Dr. Steven Gwaltney, and
many others that I am probably missing.
I would also like to thank Sandy Weakland, Leann Golemo and Judy Parker from
the QTP staff for whom I will hold warmest remembrances.
Friends have always been important for me and my family: Marta and Pradeep
Raval, Judy and Marshall Odham, Deborah and Ricardo Cavallino, Sue and Dale
Kirmsee, Marcela and Augie Diz, Guillermina Estiu and Luis Bruno-Blanch are friends
we will never forget.
My family has always been important to me: my sister Marie-Helene and my
iii

brother Bernard have been good siblings and friends. I always felt lucky to have the
parents God gave me, Eduardo and Helene, who gave me so much and ask nothing in
return. Few are the occasions that as a friend and a husband I have to express, in a
public way, the deep and eternal gratitude that I have for the patience, encouragement,
support and love of my wife Alejandra, in whose eyes I saw my future and to whom I
gave so little but owe so much. I dedicate this work, my love and my life to her and
to our daughters, as they are everything to me.
IV

TABLE OF CONTENTS
ACKNOWLEDGMENTS iii
ABSTRACT viii
CHAPTERS
1. INTRODUCTION 1
2. REVIEW OF METHODS FOR GEOMETRY OPTIMIZATION AND
TRANSITION STATE SEARCH 14
Introduction 14
Geometry Optimization Methods 16
Newton and Quasi Newton Methods 16
The Line Search Technique 18
The Simplex (Amoeba) Technique 20
Restricted Step Method 22
Rational Functions (RFO) 23
Reaction Path Following Method 24
The Hellmann-Feynman Theorem 24
Transition State Search Methods 25
Introduction 25
Simple Monte Carlo and Simulated Annealing Algorithms 26
Synchronous-Transit Methods (LST & QST) 29
Minimax / Minimi Method 32
The Chain and Saddle Methods 33
Cerjan-Miller 35
Schlegel’s Algorithm 38
The Normalization Technique or E Minimization 39
Augmented Hessian 40
Norm of the Gradient Square Method (NGSM) 42
Gradient Extremal 43
Gradient Extremal Paths (GEP) 48
Constrained Internal Coordinates 49
The Image Potential Intrinsic Reaction Coordinate (IPIRC) 49
The Constrained Optimization Technique 50
Gradient-Only Algorithms 50

3. HISTORY OF GEOMETRY OPTIMIZATION AND TRANSITION STATE
SEARCH IN SEMI-EMPIRICAL AND AB-INITIO PROGRAM
PACKAGES 54
Brief Historical Overview 54
NDDO and MNDO 55
MOPAC 56
AMI 56
PM 3 56
ZINDO 57
AMPAC (Version 2.1) 58
GAUSSIAN 94 58
HONDO 60
ACES II (Version 1.0) 61
4. THE LINE-THEN-PLANE MODEL 62
Introduction 62
The Line-Then-Plane (LTP) Search Technique .64
The Algorithm 66
Minimizing in Perpendicular Directions: Search for Minima 68
Projector Properties 70
LTP Convergency 71
The Step 73
Default Step 73
Updated Step 74
Newton-Raphson-Like Step 74
Hammond’s-Postulate-Adapted LTP Methods 75
Introduction 75
Hammond-Adapted LTP Procedure (HALTP) 76
Restricted Hammond Adapted LTP (RHALTP) 77
5. GEOMETRY OPTIMIZATION 81
Introduction 81
ARROBA: A Line-Then-Plane Geometry Optimization Technique 82
The Algorithm 85
Minima in Perpendicular Directions 86
Convergency 86
A Proposed Global Minima Search Algorithm 87
vi

6. APPLICATIONS 89
Introduction 89
Model Potential Functions for Transition State 90
The Halgren-Lipscomb Potential Function 90
The Cerjan-Miller Potential Function 90
The Hoffman-Nord-Ruedenberg Potential Function 91
The Culot—Dive—Nguyen—Ghuysen Potential Function 94
A Midpoint Transition State Potential Function 94
A Potential Function with a Minimum 96
Summary of Results 96
The Step 103
Step Size Dependence 105
Hammond and Restricted Hammond Adapted LTP Models 105
Summary of Results 110
Molecular Cases for Transition State 112
Introduction 112
Inversion of Water 112
Symmetric Inversion of Ammonia (NH3) 116
Asymmetric Inversion of Ammonia (NH3) 122
Rotated Symmetric Inversion of Ammonia (NH3) 124
Hydrogen Cyanide: HCN -> CNH 125
Formic Acid 132
Methyl Imine 136
Thermal Retro [2+2] Cycloaddition Reaction of Oxetane 141
Hammond Adapted LTP Results 147
Summary of Results 147
Model Potential Function for ARROBA 150
Introduction 150
Model Potential Function 150
Step Size Dependence 151
Summary of Results 153
Molecular Case for ARROBA: Water 153
7. CONCLUSIONS AND FUTURE WORK 158
BIBLIOGRAPHY 162
BIOGRAPHICAL SKETCH 169
vii

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy.
TRANSITION STATE SEARCH AND GEOMETRY
OPTIMIZATION IN CHEMICAL REACTIONS
By
Cristian Cardenas-Lailhacar
August, 1998
Chairman: Michael C. Zerner
Major Department: Chemistry
The research presented in this thesis involves the development of procedures for
finding transition states in chemical reactions as well as techniques to optimize the
geometries that are involved in their calculations. A procedure for finding transition
states (TS) that does not require the evaluation of second derivatives (Hessian) during
the search is proposed. The procedure is based on connecting a series of points that
represent products Pi and reactants Ri. From these points, conservative steps along
the difference vector from Pi toward Ri, and from Ri toward Pi, are taken, until
the two points coalesce. Although the initial points of the set, Po and Ro, represent
specifically the product and the reactant, other Pi and Ri are determined by minimization
in hyperplanes that are perpendicular to Pi 1 and Ri 1, simultaneously. In order to test the
accuracy of the methodology proposed here, the technique has been applied to seven
well-known potential functions, and the results compared with those obtained from
other well-known procedures. Most methods that search for transition states require
an accurate evaluation of the Hessian as they proceed uphill from either product to
reactant, or from reactant to product. These procedures are both costly in computer
time and in memory storage.
viii

The line-then-plane (LTP) methods described here do not need the accurate cal¬
culation of the Hessian except for the last step in which its signature usually has to
be checked. This particular one point could also be probed numerically. This feature
potentially allows the study of much larger chemical systems.
When this LTP technique is applied to molecular reactions, the results compare
closely with those derived from the application of other models. The proposed LTP
geometry optimization procedure, after being tested in a model potential function, has
been used together with the LTP technique to define a general procedure to find the
global minimum.
It is shown that, because of the Newton-Raphson nature of the step taken, the LTP
procedures will converge unequivocally to the TS on any continuous surface. The same
applies for the minima searching in the geometry optimization procedures.
IX

CHAPTER 1
INTRODUCTION
Since the beginning of human reasoning symbols have played a key role in a
colossal attempt to try to describe our universe, the cosmos. The history of chemistry
takes us back to the symbols of fire, water, air and earth that were extensively used by
alchemists around the 13th century and attributed to Plato’s polyhedral symbols. Also
involved in this historical perspective are Empedocles of Agrigent (“440 BC), Thales
of Miletus (“600 BC), Anaximenes (546 BC) and Heraclitus (“500 BC) who claimed
fire to be a basic element. Few of these symbols are still with us. The symbol for fire
(heat), A, is the only one used in chemistry [1],
But alchemy has evolved into chemistry as the scientific method has replaced the
old beliefs such as the transmutation of matter (an idea that still today is among some
chemists’ inquiries). We are still wondering about the inner secrets of matter. Chemists
are confronted with many questions, one of the most important being the description
of how atoms are held together in molecules, and how they interact with each other to
produce new compounds, that is, how chemical reactions proceed. Quantum chemistry
has become a powerful tool to assess such a goal.
As quantum chemists we are interested, in general, in accounting for the properties
of excited as well as ground states. In the case of chemical reactions, the aim is to
understand and describe the laws of nature that control them. To this end, algorithms
are constructed, to reproduce features of a chemical reaction on the computer, and
1

2
tested in their goodness by calculating observable quantities that are finally compared
with the experiment.
The potential energy surface (PES) is the cornerstone of all theoretical studies of
reaction mechanisms in relation to the chemical reactivity. Topographic features of the
PESs are strongly associated with experimental observations of the chemical reaction.
A lowest energy path connecting reactants (R) and products (P) (selected ones) on
the surface is a concept that can be associated with the mechanism through which the
reaction, theoretically, occurs. The association of these pathways with valleys among
mountains is as unavoidable as practical and allows us to understand the feasibility of
a reaction.
The maxima along the path related to the reaction mechanism are essential for
understanding of the energetics of the processes under study. These particular points,
which have been called transition states, tell us about the type of reaction with which
we are confronted. An insurmountable mountain along another pathway tells us that
the associated reaction connected is not feasible. On the other hand, the presence of
two transition states is the theoretical equivalent to competing reactions in a test tube,
whereas a shallow minimum may confirm the existence of a postulated intermediate.
The variety of reaction mechanisms is enormous, and, consequently it is essential to
have a good understanding of the properties that are genera] and common to all potential
energy surfaces.
Chemical reactivity is the main subject of chemistry. The goal is to predict the
products that are most likely to be obtained according to the interactions among the
participating species. In 1889 Svante Arrhenius initiated the study of transition states
by expressing the sensitivity of the reaction rate to temperature through his famous
relation [2], Later, in 1931, with the development of molecular reactivity theories and

3
particularly with the work of Michael Polanyi and Henry Eyring [3], the goal was to
formulate relations for the kinetics of reactions. These theories introduced concepts
such as activation barrier and transition state. With the coming of quantum chemistry,
rational techniques for the prediction of molecular structures (geometry optimization)
and mechanisms of reactions (transition states) became available. Results obtained
today reveal the spectacular degree of refinement that quantum chemical theory has
achieved, even on occasions competing with experimental measurements for accuracy.
If we plot all the positions and energies of reactants as they evolve to products, we
will obtain a potential energy surface (PES). The TS may be like a volcano between
two valleys (for a single transition state) or a rugged mountain range (for more than
one TS). This multidimensional surface contains many paths with different mountain
passages (energy barriers) through which reactants must move to become products. In
the particular path that the reaction follows, the transition state is the point of highest
energy between reactants and products. This classical view of transition states has
evolved today into a broader definition: the full range of configurations the reactants
can take as they evolve to products [4], This difference is mainly due to how we look
at the TS, that is, as one point or a realm of reaction rates in a potential energy surface
of 3N-6 dimensions (with N being the number of atoms).
Some reactions can go from reactants to products without passing through a
transition state via a minimum energy pathway, making the location of the TS a very
unpleasant task particularly for experimentalists. This area of the PES is called a seam,
that is, a region of the PES that is penetrated by another one. This behaviour happens
when the energies of the ground and excited states are so close that the system can
bypass the transition state. A representative example of this phenomena is the internal
rotation of stilbene [5],

4
Chemical reactions are classified according to the difference in energy between the
Rs and Ps, that is, endothermic and exothermic reactions, according to:
AE° = EP - E« (1)
where AEa < 0 and AE° > 0 , respectively. The reactions are studied along
a given reaction coordinate (RC) which tells us how to step, the walking direction
and the evolution of the reaction from reactants to products (initial and final situation)
and vice-versa. Somewhere between Rs and Ps, there is a maximum in energy that is
unavoidable, the transition state, a very unstable conformation that will transform itself
into reactants or products according to the initial conditions of the trajectory [6],
What is the appearance of the transition state? What bonds are broken and formed?
What structural changes are occurring in the system and why? Unfortunately, after
almost three decades, quantum chemistry still does not have effective algorithms to
solve this problem, and even today the most common recipe is to locate and describe
transition states from chemical intuition, that is to say, experience.
The usual starting point is to optimize both reactants and products geometries:
minima in the PES. The TS is then a maximum situated between them. However, there
are exceptions to this scheme [5, 6]. In Figure 1.1 we show the internal rotation of
hydrogen persulfide, for which the reactants and the products (trans and cis isomers,
respectively) have been fully optimized at a fixed dihedral angle of a = 0 and 180°,
respectively, as chemical intuition will indicate. In turn, the TS is expected to be located,
at a higher energy, around midway between reactants and products. Here the expected
TS (located at a = 93°) turns out to be a minimum and the initial reactant and product
maxima along the reaction coordinate. Which is then a minimum and which a TS?

4
Figure 1.1. Hydrogen persulfide internal rotation. The continuous line represents Ab-Initio (STO-3G) calculations, whereas
the dashed line is for the potential energy surface as obtained through a symmetry adapted technique that we developed [5],

6
Energy
Figure 1.2. The Multiple Minima problem. Energy versus a given reaction
coordinate showing local minima, reactants (R), products (P), intermediates of reaction
(I), transition state (TS) and the global minimum.
On the other hand, when optimizing geometries the problem as to which is a
local and which a global minimum, as shown in Figure 1.2, is still a hazard for big
molecular systems. If a given geometry is optimized, chances are that the structure soon

7
will become trapped in the energy minimum of the potential energy surface closest to
the starting conformation. How then is one to find the desired global minimum? One
way would be to use brute force, namely to change systematically the value of a
givenvariable across the surface. An alternative is to perform a systematic search by
covering conformation space with a fine mesh, but this requires too many calculations.
Another interesting way to address the problem is to think of minimization algorithms
as cooling molecular structures to 0° Kelvin, then by a warming-up process the system
is taken to a higher energy position in the PES, and the search can continue in another
region of the N-dimensional conformational energy surface.
This overview simply tells us that we still do not have algorithms that are efficient
enough to solve this optimization problem, not to mention the expense in terms of
number of iterations necessary to obtain this minimum (when found), that is, computer
time. Transition states have only an ephemeral existence (vide infra) that lies in the
femtosecond scale as shown in the cosmic time scale in Figure 1.3 (if we were able
to live 32 million years, the transition state would last only for a few seconds of our
lives). Worthy of mention is the time-resolved experimental work of Zewail [7] and
collaborators who, by using femtosecond (ultrafast) laser techniques, observed reaction
dynamics of small molecular systems. Nevertheless, TS are attainable by quantum
mechanics, whereas experimentally they can only be inferred indirectly. This distinction
has motivated theoretical chemists to develop new and powerful models to search for
transition states [8]. New methods appear frequently in the literature [5, 9-16] and,
as we will see in the next chapters, the mathematical tools as well as the models
sometimes seem to be directly proportional to the number of scientists devoted to
tackling the problem.
But none of these procedures is as yet utterly convincing or generally successful.

8
Big Bang
Dinosaurs
Origin of Life
Jesus Christ
Columbus
Discovers
America
Eye Response
Molecular
Rotations
Transition State
1018
1015
1012
10 9
10 6
10 3
1
10- 3
10- 6
10- 9
10-12
10-15
Age of Earth
Pyramids
Year
Day - Hour
Second
1 m sec
1 ¡j, sec
1 v sec
1 7r sec
1 (f) sec
Milky Way
Australopithecus
Newton: Principia
Mathematica
Flash Photolysis
(1949)
(1950)
(1960) Lasser
(1966)
(1970)
Femtosecond
(1985)
Figure 1.3. Cosmic time scale for transition state (in seconds).

9
Consequently it becomes very important to examine new algorithms capable of ac¬
curately finding minima and transition states. The algorithms usually found in the
literature can be divided in two general kinds; those (the cheaper, and usually less
accurate, ones) that use only gradients and can give a quick, but rough, idea of the
transition state location, and those (more sophisticated ones) that use gradients and
Hessians (more expensive but also more accurate, when successful). The most efficient
algorithms to find TSs use second derivative matrices which require great computational
effort. This fact alone is a powerful incentive to try to develop new procedures that
do not require the Hessian.
The determination of TS structures is more difficult than the structure of equilibrium
geometries, partly because minima are intrinsically easier to locate and also because
often no apriori knowledge is available about TS structures.
For a given structure %e to be a TS of a reaction it must fulfill the following
conditions, according to Mclver and Komornicki [8]:
• Xc must be a stationary point, which means that all gradients (g) of the energy
evaluated at this point must be zero: g(Xe) = 0.
• The force constant matrix (H) at the transition state must have one and only
one negative eigenvalue H(Xe).
• The transition state must be the highest energy point on a continuous curve
connecting reactants and products.
• The point identified as the transition state (Xe) must be the lowest energy point
which satisfies the above three conditions.

10
The computation of the energy and its derivatives of the system under study is
essential for our purposes. For this, the Born—Oppenheimer Approximation is used
which is based on the assumption that, given a molecular system, the nuclei are much
heavier than the electrons remaining clamped. As a consequence the kinetic energy of
the nuclei is neglected and the repulsion between nuclei is considered to be a constant.
This approximation gives rise to the electronic Hamiltonian, which in atomic units for
N electrons and M nuclei is
1 N N M „ N M
« = ~£v?-££A + ££:r:
(2)
i=l ¿=1 n— 1 u* ?'=1 j>i ^
where V? is the Laplacian operator (derivatives respect to the coordinates of the i th
electron), Za is the atomic number of nucleus a, r,;a is the distance between the ith
electron and nucleus a, whereas r,;?- is the distance between electrons i and j . In this
Hamiltonian we identify the first term as to be the kinetic energy of the electrons, the
second term represents the Coulomb attraction between electrons i and nuclei a and
the third term addresses for the repulsion between electrons.
The energy (E) of the system comes from the solution of Schrodinger’s equation
that we wish to solve using our Hamiltonian operator: 7i\i = E\k , where i' is the
wave function we use to represent the system under study.
The algorithms that we used for a transition state (TS) search and geometry
optimization, as well, are generally based on a truncated Taylor series expansion of
the energy
E = E0 + q^g + -qTHq + ...
1
fl
(3)
and of the gradient
g = ga + qH
(4)

11
with q the coordinates, g the gradient (first derivative of the energy with respect to
coordinates q) and H the Hessian (second derivative matrix of the energy with respect
to coordinates q). First derivatives for any wave-function generally can be acquired
analytically in about the same time as the energy. Analytical second derivatives, on
the other hand, involve at least coupled perturbed Hartree-Fock (CPHF) algorithms.
These have, in general, a fifth-order dependence on the size of the basis set, that cannot
be avoided if the Hessian is required to find minima, and are imperative to insure the
location of a TS.
Today modern procedures try to avoid the evaluation of the Hessian as this is a real
bottleneck in the calculation in terms not only of computer time as well as memory
storage. Consequently, algorithms that update the Hessian (or its inverse), that is,
procedures that use a guess of the Hessian and information of the actual and previous
structure give a “good enough” estimate of the real Hessian after a few iterations.
When the initial Hessian is chosen to be the identity (or other approximate) matrix,
the procedure is said to be a quasi-Newton one. On the other hand, “true” Newton
procedures are those that use a calculated Hessian.
Update procedures in turn are known to be of two types (see for example [17-19]):
Rank 1 : Gn — Gn-\ + Wn
(5)
Rank 2 : Gn — Gn—\ + Wn + Vn
where W„ and V„ are corrections to the initial Hessian or its inverse Gn-1 at cycle n.
To rank 1 correspond update procedures such as the one by Murtagh and Sargent (MS)
[20], while a popular rank 2 method was constructed by Broyden-Fletcher-Goldfarb and
Shanno (BFGS) [21], Davidon-Fletcher-Powell (DFP) [22] and Greenstadt [23],

12
It is germane to note that rank 2 update procedures can be regarded as being a
rank 1 update of an already rank-1-updated Hessian (or its inverse). A great deal of
work has been carried out lately in this field: the more recent papers combine rank 2
update procedures [24].
The determination of the minimum energy conformations of reacting species is
handled more or less routinely except for very large systems with multiple minima.
Transition states are not as easy to find as minima. Moreover, most algorithms that
we will describe in the next chapter do not always succeed in the search for transition
states because of the following general reasons:
• It is difficult to insure movement along a surface that exactly meets the
conditions of a simple saddle point.
• In genera], little a priori knowledge of the transition state structure is
available.
• Wave functions for a TS may be considerably more complex than those
describing minima.
Some procedures make a guess of the TS and perform a Newton-Raphson mini¬
mization of the energy. Unfortunately this technique is not reliable because it can lead
back to R, P or to a TS. Many different algorithms for these tasks are available in the
literature, with good reviews found in references [9, 19, 25],
In this work we show methods to find transition states based on a continuous
walking of fixed step along a line connecting R and P, assuming that their structures

13
are known, and utilizing methods to optimize geometries (GOPT) based on the initial
and the newly generated structure. No previous knowledge of the TS is necessary.
In chapter 2 we review the existing literature on geometry optimization and transi¬
tion state search algorithms in terms of advantages and disadvantages, starting with a
description of the line-search technique that accounts for parameters used in the major¬
ity of the models. We emphasize the disadvantages as they account for costly failures
which these procedures suffer.
Chapter 3 starts with a brief historical overview of semi-empirical molecular orbital
theories. Next, some semi-empirical and ab-initio program packages are examined in
terms of their TS and GOPT capabilities.
Chapter 4 introduces the line-then-plane (LTP) procedure. The algorithm is de¬
scribed discussing its convergence to the TS and how the step should be taken. Alter¬
native algorithms, in the basis of Hammond’s postulate, are discussed. We concentrate
on some properties of LTP, for example its dependence on the size of the steps in terms
of the number of energy evaluations required to find a maxima or minima.
In chapter 5, ARROBA, a new LTP geometry optimization procedure, is presented.
The main features of this technique are studied through a model potential function and
a molecular example. Finally, an algorithm is proposed to solve the multiple minima
problem.
In chapter 6 the LTP technique is tested with some potential energy functions and
molecular systems for both transition states and geometry optimization problems.
Finally, in chapter 7 we summarize results, draw conclusions, and set the stage for
future systematic work in this area.

CHAPTER 2
REVIEW OF METHODS FOR GEOMETRY
OPTIMIZATION AND TRANSITION STATE SEARCH
Introduction
In general, optimization techniques for finding stationary points on PESs can be
classified (avoiding details for simplicity), as [17-19]
• Without Gradient,
• With Gradient: Numerical or Analytical, or
• With Numerical or Analytical Gradient and Numerical Hessian.
With the exception of the first method, these algorithms are all based on a truncation
of a Taylor series expansion of the energy and of the gradient as was given in Eqns.
(3) and (4), respectively, in the previous chapter.
The general scheme is complete when the characteristics of the stationary point are
included, that is, zero gradient (g = 0). In practice the gradient should be smaller than
a preestablished threshold at the critical points which, from Eqn. (4), yields
9 = -qH (6)
From here the new coordinates are
q - -i/H-1 (7)
14

15
and the step (s) is taken as a fraction of q
s = «q V« 6 5ft / 0 < a < 1 . (8)
The bottle neck for all procedures that search for minima or maxima is the evaluation
of the Hessian matrix (H), as this is time consuming and requires storage. As mentioned
in the previous chapter, the way around this problem is to use techniques that update the
second derivatives matrix as MS, BFGS, DFP, and so on. At this point, the techniques
used to find the critical points of interest are classified as (exact) Newton-Raphson if no
approximations are used for the evaluation of H, that is, numerical or analytical second
derivatives are used in the search. Procedures that use update techniques for the Hessian
are said to be of the Newton-Raphson-like type. Whereas, if the identity matrix (I) is
used, then we are in the steepest descent regime. This should not be confused with
those techniques that use the identity matrix as a starting point to update the Hessian
as these, when converged, show a second derivative which not only is not the identity
matrix but, moreover, is an Hermitian matrix that is very close to the analytical one.
First derivatives for any wave function generally can be acquired analytically in
about the same time as the energy; methods that use numerical gradients typically are
not competitive. On the other hand, the analytical evaluation of second derivatives
involves, at the very least, a Coupled Perturbed Hartree-Fock (CPHF) procedure and
is, in general, an N5 procedure where N is the size of the basis set. Therefore, second
derivative methods are costly even at the single determinant level, and even more so at
the Cl level. However, a good feeling of the topology of the potential energy surface
is needed to locate maxima and minima.

16
Geometry Optimization Methods
Newton and Quasi Newton Methods
From the Taylor series expansion of the energy (equation (3)) we write for the
gradient
(9)
with ó the infinitesimal step in the search direction. A minimum implies that
g(x. + S) = 0. Thus a relation for the search direction s can be written as
as = -(H_1)^(x) = 6 . (10)
For “a true” Newton method, H is the exact Flessian matrix, while for the quasi-
Newton procedure the Hessian can be a matrix that approximates the second derivatives
[26, 27], commonly the identity. The line search parameter a can be determined by a
variety of algorithms that reduce the energy along the search direction. This type of
procedure will be discussed in detail in the next section.
When the Hessian is positive and exact, the Newton method shows a quadratic
convergence to a local minimum if expanded in the quadratic region. The problem
is that this procedure requires an explicit calculation of the second derivative matrix,
demands the extra computational effort previously discussed and, moreover, is accurate
only in the quadratic region.
The quasi-Newton methods depend on information obtained about the Hessian
during the search. The gradients calculated at different geometries are used to build
an approximate inverse Hessian G — H-1 using the quasi-Newton conditions (Eqn.

17
(6)). This is
7 = g(x + S) - g(x)
6 = Gj (11)
as a constraint to obtain a relation to get a matrix product for the search direction s.
This is
G(x + 5) = G(x) + U (12)
and
s = —G(x + 8)g (13)
where U is a correction to the inverse Hessian G. Thus the approximate Hessian
is updated every geometry cycle. In this way some time is spent evaluating the
approximate Hessian or its inverse, but this task requires a small fraction of the time
that the evaluation of the Hartree-Fock gradient takes. One of the most successful
relations for updating the Hessian is the one developed by Broyden, Fletcher, Goldfarb
and Shanno (BFGS) [21]. Many of the procedures available for finding minima differ
on how they evaluate or choose s. For computational details on these methods see
Kuester and Mize [28].
New procedures to optimize geometry are the subject of study by many research
groups and one can see specialized journals frequently publishing works devoted to this
problem. Rather than follow a textbook classification, here we summarize only those
procedures that have proved to be more stable and are most often used. We will start
by summarizing the line search technique, as it is widely used by several minimization
techniques. Time-dependent, statistical mechanics and variational procedures will not
be addressed nor discussed here (except Monte Carlo techniques).

18
Reviews of geometry optimization and transition state search methods can be found
in the literature [13, 18, 23-25],
The Line Search Technique
The fundamental idea of this strategy is to look for the appropriate displacement
along a search direction. Suppose that at a given point Xk we found a search direction
determined by
Sk = -Gkgk (14)
where k indexes the cycle and the inverse of the Hessian Gk = H^1 is already
updated (for example using BFGS). Then the next point is given by xk+\ — xk + aksk
. Here the parameter ak comes from the line search and has a value such that the
decrease in energy is reasonable (Figure 2.1). This is
E(sjfc+i) < E(xjfe) or E(arfc+1) - E(xk) < e (15)
where e is a given energy threshold. But an exact line search will be able to find the exact
value of Q!k for which E(xk+i) along the line defined by Sk is a minimum. Among the
many procedures for performing these calculations, one of the most efficient approaches
is to perform an “efficient partial line search,” in which a reasonable decrease in the
function is obtained when an appropriate value of alpha is selected [29].
The energy along Sk is written as
E(xk + ask) = E(xk) + ag\xk) • sk + (16)
and will be lowered provided that the descent condition is satisfied: g\xk) â–  Sk < 0 .
From here the sign for the search direction is selected and the minimum is found by
direct differentiation
E(xk + amsk)
(17)

19
Figure 2.1. Representation of the line search technique in a potential energy surface.
gk and gk+i (continuous arrows) are the directions of greatest slope at points Xk and
Xk+i> respectively, and are orthogonal to the tangents to the surface (dashed lines) at
these points. The point on Sk represents the result of a partial line search, whereas on
Sk+i the exact line search and the partial line search results coincide.
where the null value characterizes the exact line search. Now the energy at Xk+i is
required to approach the x^—o^Slc value. Then the gradient g(xk+\) = g(xk + aksk)
is evaluated and a left extreme test is performed as
I (g{xk+i)\sk) \< -a(g(xk)\sk) , 0 < a < 1 (18)
If cr=0 an exact line search is performed, and if cr=l any reduction of the scalar product
(g(xk+l)\sk) is acceptable.

20
If the left test fails, ak is too low, then a new value of alpha is estimated as
* new _ i
ak — ak +
(ak - a\)E (ak)
E'(aJ)-E'(a*)
(19)
where a1 is the initial value of the interval (o^, o|) used to examine the kth line search
cycle.
Finally the energy E(xk) —> E(xk + akewsk) is evaluated and the test repeated
with ak —> a.k and ak —> a™11’ until the condition is fulfilled. The partial line search
stops and a new search direction is sought.1
The Simplex (Amoeba) Technique
Designed by Nelder and Mead [30], this procedure has as one of its main charac¬
teristics its geometrical behaviour and that it only requires the evaluation of the energy,
having as major drawback the large amount of energy evaluations necessary to find
the minimum. The different behaviours, according to possible different topological
situations, that the algorithm might have are shown in Figure 2.2.
For molecular systems, a simplex is a 3N dimensional geometrical structure with
N vertices (in 3 dimensions it is a tetrahedron). In the multidimensional complex
topography of a molecular system, simplex requires only 1 (3N-dimensional) point q0
from which the procedure will walk downhill reaching a minimum (probably a local
one).
The algorithm starts with qG defining the initial simplex (the other N points q,;) as
qi = Qo + PQi
(20)
where g¿ are the 3N unit vectors and ¡3 is a constant that is a guess for the length of
the problem that can, in turn, have different values for its x,y,z components.
1. Further details can be found in the literature [15, 29].

21
Figure 2.2. The Simplex method. The different steps that can be taken by the
algorithm according to the topology of the potential energy surface. In all cases the
original simplex is represented by solid lines. The generated simplex, represented (for
all cases) by dashed lines, can be (a) A reflection in the opposite side of the triangle
that contains the lowest energy point (L) to which the highest energy point (H) does
not belong, (b) A reflection plus an expansion, (c) A contraction along the dimension
represented by the highest energy point and the point where the triangle oppossite to it
is crossed (X), or (d) A contraction along all dimensions.

22
The first steps are spent moving a projection of the highest energy point through
the opposite face (to which it does not belong) that contains the point of lowest energy.
This step is known as to be a reflection because it is taken in such a way that the volume
of the initial simplex is held constant. Then the algorithm extends the new simplex in
a given direction in order to take larger steps. When in the surroundings of a minima,
the algorithm contracts itself in a transverse direction trying to softly spread down to
the valley. The procedure can also, in these situations, contract itself in all directions
in order to find a perhaps final tortuous minimum.
The search stops when the magnitude of the last displacement vector is fractionally
smaller than a given threshold. In addition, it is customary to examine the decrease in
energy such that simplex stops if this difference is smaller than a given threshold.
Restricted Step Method
Included here for historical and review reasons, Greenstadt has studied the Relative
Efficiency of Gradient Methods up to the ones available until 1970 [23, 31].
This procedure rewrites the Taylor series expansion of the energy of Eqn. (3) as
E - E0 = AE - q1# + ^q^Hq + • • •
(21)
Now the following Lagrangian (£) is introduced
(22)
where q is the difference in coordinates (actual and previous cycle), A is Lagrange’s
multiplier and h is the trust radius (or confidence region) for the stepping. The square
bracket factor ensures that the search remains in the quadratic region.
The first derivative of the Lagrangian gives
(23)

23
from here
q = - [H - AI] 1g
(24)
where I is the identity matrix. This is a Newton-Raphson-like procedure which is fully
recovered when A = 0 . By construction this technique has shown to be useful for
transition state search as well.
Rational Functions (RFO)
Created by Banerjee and his coworkers [32], this is a procedure in which the
energy is written in a normalized form as
qTtf + |qfHq
A E = E - En
(25)
1 + q^Sq
where S is a step matrix. Next step is to augment all the components of this rational
function such that, and by using Eqn. (25) the difference in energy now becomes
A E =
H g
g o
* o
s g
g o
q O
(26)
As usual, the next move is to ask for the first derivative of the change in energy with
respect to the coordinates to be zero ( dAE/dq = 0 ) to obtain the following
eigenvalue equation
H g
gi 0
q 1) = A | q l)
(27)
from which two sets of equations are obtained
Hq Eg — Aq
g'q - A
The first of these two relations gives a Newton-Raphson-like step
(28)
q — - (h - AI)
-1
(29)
which is, as before, recovered for A = 0.

24
Reaction Path Following Method
This method requires one to know the TS, which will connect in a steepest descent
fashion, the TS with reactants and products in order to draw a reaction path. First
developed by Gonzalez and Schlegel [10], they have included modifications from the
very beginning in order to consider the effects due to atomic masses. In subsequent
articles they proposed a "Modified Implicit Trapezoid Method", which is a contribution
on the way of obtaining the final coordinates used by this method, namely: "The
Constrained Optimization" by Gear [33]. The method will be described in the TS
section when we discuss Schlegel’s procedure as it is related to the search of maxima
and mimima.
The Hellmann-Feynman Theorem
The idea [34] is to obtain gradients to be used to optimize molecular geometries.
To accomplish this, the starting point is Schroedinger’s equation
|H|$) = E0\V)
(30)
or,
= E0{9\V) = E0
(31)
where the wave function is assumed to be normalized to unity ('I'I'I') = 1. The
demonstration of the theorem starts by taking the first derivative of the energy with
respect to a given (set of) coordinate(s) “9” as follows
0Eo _ /¿FI/
dq \ dq
H
* > + (tf
H
dV\
~dq)
+ (tf
<9H
dq
= 0
(32)
If the two first terms vanish,
H
H
£>-»
5T
dq
(33)

25
then the gradient will be simply
OE
dq
<9H
dq
(34)
This is a very appealing relation since <9H¡dq is easily obtained. Experience tells
us that this scheme works only for very, very good wave functions. It is obvious why
since, for an exact eigenfunction, equation (33) can be written as
(35)
This step concludes the proof because the bracket involves the first derivative of a
constant, due to the normalization condition, and consequently the term vanishes.
Transition State Search Methods
Introduction
It is clear that procedures to locate TS and geometry optimization (GOPT) algo¬
rithms are intimately related. Techniques to find minima have been much more suc¬
cessful than those developed for locating TS. Their success is based on the relative ease
in following downhill searches, such as with the steepest descent type of algorithms.
Because of this success the problem of locating TS repeatedly has been approached as
one of dealing with the location of minima. In general, such methods choose a higher
energy point and from there walk downhill, stopping at a local minimum where the
signature of the Hessian is checked (one and only one negative vibrational mode). But
when studying a reaction mechanism, knowledge of the lowest energy reaction path is
of great use but expensive. To accomplish this the recipe is to, by sitting at the found
TS, follow a downhill path to reactants and products.
Other methods use a mirror-image technique. Consider the picture of reactants and
products (usually only one of them) as minima and TS as a lowest energy maxima

26
between them. Placing a mirror at the TS the image obtained is the one of two maxima
(reactants and products) and a minimum (original TS). Consequently the problem now
is to find that minimum. Again downhill methods are used with the pitfalls described
above.
A collection of the procedures reviewed here, showing their general features,
advantages and disadvantages can be found in Table 2.1.
Simple Monte Carlo and Simulated Annealing Algorithms
The Metropolis Monte Carlo algorithm [35] has proven very successful in evaluating
equilibrium properties of systems. The bulk properties are simulated from a small
physically meaningful number of particles (N) such that the fluctuations in the calculated
value of a property, usually a thermodynamic observable, are minimized.
The interactions of N particles are described by a potential energy function, say
U(r,fl) , where r is the distance separating the particles while Cl represents any
other coordinates (Eulerian angles for example) on which the potential may depend.
Therefore, the potential interaction between particles AE is written as
AE = £••£■•£ U(ri,rj,Cti, Qj, ...N)
n,rj..N Sli,ílj..N (36)
Monte Carlo method
N particles are placed in a system of volume V such that the macroscopic density is
kept constant. The initial configuration of the particles is arbitrary, a flexibility which
is a tremendous advantage of this method.
The position of any particle i with a position r¡ — (Xi,yi,Zi) is chosen randomly
and moved according to

27
X¡ - xí + ¿711 Y¡ = Vi + bv.2
/
Zi = zí + 6n3 .... = Í2¿ + fnK
(37)
where b and f are chosen step sizes for r and O and {n¿} is a set of random numbers
(for all i e Z ) in the interval [-1, 1], The particle always stays in the cube such that
surface effects are reduced and the density of particles (p) is always constant. The new
conformation energy AE is calculated according to Eqn. (36).
If AE < AE then the new arrangement of the particle is accepted, and the
calculation is continued from the newer configuration. On the other hand if AE* > AE
then a probability (P) is calculated as
P = exp[-{AE' - AE)/KíjT]
(38)
where Kp and T refer to Boltzman’s constant and the temperature, respectively. Thus
one of the following conditions is met:
If P > 6 , <5 6 [0,1] the move is accepted
If P < 6 , 6 e [0,1] the move is rejected
In the first case the algorithm continues as described above. In the second case,
where the new configuration is rejected, the particle is returned to its initial position
and a new particle is chosen randomly and moved according to Eqn. (37) (6 is a
random number).
The method is repeated until no further configurations are accepted. The system is
said to have reached its equilibrium configuration when this criterion is satisfied.
The efficiency in reaching the minimum by Metrópoli’s algorithm depends on the
number of moves allowed for the displacement of particles. A more accurate equilibrium

28
configuration of the particles can be determined if the number of random moves allowed
is large. These methods are not competitive with gradient methods in obtaining local
minima but, as discussed below, Monte Carlo allows us to leave a region of local
minimum for the global one.
Simulated Annealing method
Simulated Annealing is similar to the Monte Carlo algorithm. The difference is
that the probability P is evaluated as
F = exp[-(AE - AE)/T*]
(39)
where T* is a parameter with energy units. The potential energy surface is scanned in
a finite number of moves using the random process described above for a given value
of T*. Then T* is varied using an annealing factor a as follows
T*+1 = aT* , 0 < a < 1 (40)
where i is the number of steps allowed. The search process is then repeated with the
new value of T*.
As T* decreases, areas of the surface closer to the minimum are scanned and if
any minima had been missed by the search using the previous value of T*, the method
can now lock itself onto a lower minimum. As the number of cycles allowed for the
annealing step increases, the search for the global minimum of lower energy becomes
more efficient. This flexibility in being able to “anneal” the PES is one of the assets
of the simulated annealing method.
Differing from gradient techniques, in which the displacements are generally within
a small region of the PES, the random displacements in annealing enable the search
to tunnel out of local minima in which the algorithm could have been trapped. The

29
evaluation of the global minimum can be assured provided that the algorithm has been
allowed a large enough number of random moves for each value of T* [35]. A
discussion of these algorithms, with applications, can be found in the Allison and
Tildesley’s text [36].
Synchronous-Transit Methods (LST & QST)
Initially proposed by Halgren and Lipscomb [37], the Linear Synchronous Transit
(LST) and the Quadratic Synchronous Transit (QST) methods treat “Forward” and
“Reverse” processes equivalently, generating a continuous path between specified R
and P. The main features of this technique are schematically shown in Figure 2.3.
The LST pathway is constructed by considering a linearly interpolated internuclear
distance connecting reactants and products and estimating a TS that is improved by
minimizing the energy with respect to all perpendicular coordinates. Finally the reaction
path is approximated using a parabolic path between R and P, that is, the QST path
giving a good estimation of the TS location.
The path coordinate (PC) (steps) is defined as: PC=DR/(DR+DP), where DR and
DP are a measure of the distance to the path-limiting-structure, obtained as
DQ =
1 N
- ^2(xr - Xf)2 + (Yâ„¢ - Yqf + (Zf - Zq)2
n i/2
i—l
(41)
where Q and q = R,P (reactants or products), N is the number of atoms and m stands
for, in accordance to the principle of least motion (PLM) [38], optimized structures
of reactants and products that are re-oriented relative to each other in terms of rigid
translations and rotations such that the sums over squares, for all corresponding atom
coordinate differences (equation (41)), reaches a minima.

30
Intramolecular distances R0;^ must vary simultaneously between the path-limiting
structures R^g and R^g. To avoid limitations in the method, provision must be taken
to meet one of the following two conditions:
a) Linear Condition:
R% = (l-f)R% + fRra3 0 < / < 1 ; a b) Parabolic Condition:
R®A = A + B-f + C-f2 0 < / < 1 ; a < 0 = l, N (43)
where
a = r5„
c =
B = R„0 - R% - C
R% - (1 - PMX„ - PM ■ (44)
This ensures that the following conditions are satisfied
/ = 0
-
i?6)
na.3
— na3
/ = 1
-
R{*1
OLfJ
= R*3
(45)
/ = PM
Ral
— no8
where / is the interpolation parameter, i refers to interpolated quantities, PM denotes
the value for the path coordinate and {R^a ; a < (3 = 1, A7'} are the atomic distances
of some intermediate structure on the path. Geometries of the synchronous transit
path may be evaluated in terms of interatomic distances from equations (42) and (43).
In practice, linearly/parabolic interpolated Cartesian coordinates between path-limiting
structures at maximum coincidence are subsequently refined so as to minimize
N —1 N / , \ 4 _ 2 N
a= 1 B=a+1 '■‘t/vf
d(1) c>(e)
lxn(i 11 a 3
+ KG'3 X
E EK-
wP
-i 2
ui=x,y,z n=l
(46)

31
Figure 2.3. Potential energy surface representating Halgren-Lipscomb’s TS search
technique. The continuous line connecting reactants (R) and products (P), represents
the LST with a maxima (TS) in TSj. The dashed line represents the QST that passes
through a transition state TS2. The QST path has all the features required to represent
the minimum energy reaction path. TS] and TS2 are connected through a parabola.
This issue has been discussed by Jensen through the Minimax/Minimi procedure [39].

32
where (i) stands for interpolated and (e) for evaluated (calculated) quantities referring
to the evaluated (updated) Cartesian coordinates. The weighting factor (1 /F&l)4
ensures a close reproduction of bond distances, whereas the 10—6 factor is proposed to
suppress rigid translations and rotations, between the interpolated and calculated points
(Wa^ and Wa \ respectively).
This procedure can be used for molecules with N>3 since the number of interatomic
distances exceed the number of 3N-6 internal degrees of freedom for non linear
molecules.
Cartesian coordinates are then submitted to the PLM to associate a unique path
coordinate and the total energy is computed. A variation of/will produce a continuous
energy path called, depending on the path, LST or QST. For example, in a uni-molecular
reaction, the path will usually connect both limiting structures via some maximum path,
whose structure can be determined using Eqns. (42) and (43).
Alternative algorithms have been developed maximizing along a path of known
form and minimization perpendicular to the path [8, 25, 39]. In particular, Jensen
has lately introduced a variation of this procedure namely the MINIMAX / MINIMI
procedure [39], that is briefly discussed below.
Minimax / Minimi Method
Based on the Synchronous Transit method, the minimax/minimi method is a pro¬
cedure for the location of transition states and stable intermediates [38]. It is based on
the idea that a simple parabolic transit path cannot provide a correct description of the
true minimum energy path, as suggested by Halgren and Lipscomb [37], if this path
shows frequently changing sign of curvature.

33
An essential supposition states that to find a new quadratic synchronous transit
maximum with higher energy after exhaustive orthogonal minimization is too expensive.
Consequently, it is assumed that any practical method must explicitly take into account
the influence of each geometric modification on the new transit path maximum.
It is then suggested that a straightforward way to proceed is the following: A change
in a structure corresponding to the transit maximum (a minimum) under investigation
will be accepted only if the resulting new path maximum (minimum) is of lower energy.
Successive geometry optimizations (GOPT) of all internal coordinates will consequently
lead to the lowest QST maximum, the transition state (TS), or to an intermediate (a
local minimum) and is, because of this, called the MINIMAX / MINIMI optimization
procedure [39].
A major drawback of this procedure is that an extra parabolic line minimization
along the QST path, at each level of the parameter optimization, is needed. However,
the procedure has the advantage that unexpected intermediates (MINIMAX) will be
uncovered and that extreme shifts of the path coordinate may be obtained.
The Chain and Saddle Methods
The aim of this algorithm is to ensure stability towards a transition state. Stepping
along the vector gradient field of an arbitrary continuous path between reactants and
products, leads to a limiting path where the highest energetic point is considered the
saddle point or, in the case of a multi-step mechanism, the highest energy transition
state will be located. Figure 2.4 shows the behaviour of this technique.
The algorithm consists in replacing a chain of points C(n) = (R, P)
running from R to P by a new chain C(n+1) at iteration n+1. In order to maintain the
connectivity of the path, each distance between two successive points is restricted to a

34
Figure 2.4. Potential energy surface representating the Saddle TS search technique.
The dashed line connecting reactants (R) and products (P), represents the displacement
vector from which identical fractions are taken as steps. At each step the energy is
minimized in perpendicular directions (doted lines) to obtain new sets of projected
coordinates that represents the minimum energy reaction path. The process is repeated
until a maximum (TS) is found.

35
given length (in AMPAC [40] this length is 0.3 Á). The iteration consists of skipping
the current highest point of the chain along either a descending or ascending path. In
the first case the energetic relaxation of the whole path is insured, while on the second
an interpolation of a point along the path is performed. New points are inserted as soon
as a link length becomes too long. The successive evaluations of the gradient are used
to update a quadratic local estimate of the potential, providing quadratic termination
properties. This would make this procedure very computer time consuming.
Although this seems to be a Hessian free algorithm, it is not because a differentiation
of a first order expansion of the energy is used [41]. Its recent appearance and lack of
verification will exclude this procedure from Table 2.1. For extended references also
see [24, 40-42],
Cerjan-Miller
This is essentially an uphill procedure [43] that is able to generate the reaction path
coordinate by connecting a transition state with a minimum on the potential surface,
schematically shown in Figure 2.5. It considers the Lagrangian function:
£(s. A) = E0 + s[g 4- -s^Hs + — — s's) (47)
where A is the Lagrange multiplier, s is a fixed step size (the radius of the hypersurface),
g is the gradient and H is the Hessian matrix. The extrema are determined by the
conditions
(48)

36
Figure 2.5. Potential energy surface representating Cerjan-Miller’s technique. The
arrows represent the step size A of the the trusted radius (dashed lines). The search
starts from the minimum (qj, reactants for example) and climbs up-hill towards the
transition state (TS), from where a minimization is carried out to connect q¡ and TS
with the new minima q2 (products).

37
which gives the two following relations
where A is evaluated for a given value of s. From Eqn. (49) the step size is obtained.
E(s) is given then by
E(s) = E(\) = E0 + g^H-)l) 1 (^H - Al) (H - Al) \ . (50)
Now the unitary matrix U that diagonalizes H, is introduced: UtHU = k . At
this point a new parameter “d” is defined: d = U* • g , then we write Eqn. (49) as
(51)
where the kappas (k) are positive values for minima. Now the assumption that one
is seated at a local minimum is made by saying that A = A0, then it follows that
E(Á0) — E0 > 0 , that is, the step s generated is indeed uphill in this direction. For A
= 0 the increment s (on Eqn. (49)) is the Newton Raphson step s = — H-1# .
Finally the step for walking uphill to a transition state from the minimum of the
potential surface is given by equation (51) (where A =A0, if A0 > 0). It must be noted
that A0 is the local minimum of the function A2(A), in other words, it is the root of
(52)
In a general case, the function A2(A) will have F-l local minima. Cerjan and Miller
suggest picking up the smallest value of A, that is, the smallest root of equation (52),
corresponding to the softest mode.
Draw backs of this procedure are the use of second derivatives, the use of too many
steps when approaching the transition state and the coupling between the step and the
curvature radii of the surface in the actual point. This last issue is important as the
next step might not encounter a minimum.

38
Schlegel’s Algorithm
This is essentially a gradient algorithm [44], proposed in 1982 by Bernhard
Schlegel2, in which the “right inertia of the approximate Hessian matrix” is obtained
by “ adjusting the sign of inadequate eigenvalues.” The sign of the smallest positive
eigenvalue is changed if in the search of the TS no negative eigenvalue is present. On
the other hand, if sundry negative eigenvalues happen, all of them are replaced by their
absolute value (except the smallest one). Given the stationary condition W/"(s) = 0
the quasi-Newton step, at cycle k in the step direction (s), is thus modified according to
71
**--E?h* I 1^1 = < 0 < 62 < ■ • - < (53)
2 — 1 1
where the b• terms are eigenvalues of the Hessian Hk , V is the eigenvector basis, is the gradient and k is the cycles index. This is then related to Greenstadt’s proposal3,
that is used in a minimization process. His idea is to reverse the ascendant/descendant
character of the search direction. Nevertheless, in areas of large curvature, the resulting
direction is not necessarily the opposite of the initial one, if the investigated region is
far from an extremum and thus may be incorrect. A scaling factor is used to modulate
this effect.
If the quasi-Newton search direction of Eqn. (53) exceeds the maximum allowed
step Rmax, its length is set to this maximum value. This change requires the addition
of a shift parameter A obtained by the search of an extremum of the quadratic function
qk(g). In practice, the shift parameter A is obtained by minimizing the function
(II /(A) || -Rmax)2 • (54)
2. We will keep here the super k indices used by Schlegel, according to Powell’s notation; see:
M.J.D. Powell, Math. Prog. 1:26 (1971).
3. For details see J. Greenstadt, Math. Comp. 21, 360 (1967) ; Y. Bard Nonlinear Parameter
Estimation. Academic Press, New York, 1974, p. 91-94.

39
The radius Rmax is updated using a trust region method. The step direction is
n
h
(55)
When implementing trust region methods, the minimization of Eqn. (54) is
performed by determining the zero of its first derivative using a Newton-Raphson
procedure. However, the convergence threshold of such an algorithm is guided by
a zero value of Eqn. (54). Given that the minimum of this function is not necessarily
associated with a zero function value, the procedure may fail.
Besides, this Newton-Raphson search of a zero value of the first derivative implies
that the parameter A lies in the open interval ]bj,b2[. Thus, concerning Eqn. (55), the
step sk is uphill along the first eigenvector Vj* and down-hill along all the others.
The Normalization Technique or E Minimization
Developed by Dewar and co-workers [41, 42], the Normalization Technique is a
root search technique rather than a saddle point location. Only convergence to a zero of
the gradient is ensured, not necessarily the TS. Moreover, the procedure has been shown
to require a good initial guess. In fact, if the PES is tortuous, stability problems appears
and the procedure requires a large number of energy evaluations to be successful.
Originally implemented in the closed-shell version of MNDO, the geometry of
reactants (R) and products (P) is defined (in 3N-6 coordinates) as R = a> and
i
P = J2 h- A reaction coordinate (D) is defined as:
(56)
where D is reduced subject to the condition that the structure with lower energy is
moved to approach the TS. The following procedure is used:

40
1) Obtain the optimized geometry of R and P.
2) Evaluate the energy of R and P then, defining the origin on the higher energy
structure, the geometry of the other species is expressed in terms of its new origin as:
a'i = - b¡) -* D = (a'>)2 (57)
1 i i
3) Modify geometry of lower energy structures to select a new distance4 D’ to
reduce the difference between R and P as: a¿ = /D .
4) Optimize R’s geometry such that D is held constant at D\
5) If D is small enough, then stop; otherwise go to step 3.
Caution must be taken in ensure that one geometry (for example products) can
be obtained from the other (for example reactants) by a continuous deformation [16,
17, 42, 44-46]. The first work of Komomicki and Mclver [46] is also known as the
Normalization Energy Minimization or as the Gradient Norm Minimization technique.
Pertinent previous work of Komornicki and Mclver is cited in their last 2 articles in
the literature [8, 16, 47].
Augmented Hessian
The Augmented Hessian procedure was originally proposed by Lengsfield [19, 48]
for MCSCF calculations, and further developed by Nguyen and Case [49] and later on
by many other groups [33, 50]. Augmented Hessian is essentially an uphill walking
algorithm, implemented in the ZINDO package by Zerner and co-workers [19, 26]. The
search direction s is found by diagonalizing the “Augmented Hessian”
4. Typically D’ = 0.95»D

41
For a down hill search, A is the lowest eigenvalue, a a parameter that can be varied
to give the required step lengths, and the lowest normalized eigenvector: v2 + /32 = 1.
From Eqn. (58) we get two equations
FI v + a(3g = \v
ag^v — X/3
solving for the step size we get
f (H — X)v - —afig
\u=-(R-\r1apg
s = -(H - AI)-1# = u/a/3
(59)
(60)
where H is the exact Hessian matrix or, as suggested by Zerner et al. [19], an
approximate matrix of the Hessian if H is not available. The step direction is obtained by
writing s in terms of the gradient (g) the eigenvalues and eigenvectors of the Hessian
(A,; and |u¿) respectively):
(61)
To ensure an uphill search direction, a specific eigenvector of H that overlaps strongly
with the uphill search direction, is chosen such that \x is scaled using scaling factors
and the search direction s is obtained as
\nvx)(nvx\g) v- \vi)(vi\g)
s~ a;.-A 2s A/ -A { }
7
where the scaling factor n is chosen as: n — y/\x/X'x and \'x — A-2/4; A is chosen
to lie between Ai < A < A2. Thus the step is scaled accordingly to the curvature of the
quadratic region. Finally, the stationary point Xe is found switching to the Norm of the
Gradient Square Method (NGSM, see below) when H develops a negative eigenvalue
[19].
Although this model has the advantage of being precise, it is expensive to compute
since the exact Hessian is required. It has to be pointed out that Jensen and Jorgensen

42
[51] developed this method for MCSCF optimization of excited states. Further devel¬
opments were carried out by Zerner and his co-workers [27],
Norm of the Gradient Square Method (NGSM)
The sum of the squares of the gradient (g), written as
a = = (g\g) (63)
i
is minimized [18] as was initially suggested by Mclver and Komornicki [16]. The
Taylor series expansion will be:
°Vt-l — &K + G +
-sJ erKsh-, + ...
(64)
and
°K+1 — aK + °KSK
(65)
where k indexes the cycles and sK is the step, defined as-
5 k = X/s-i-l XK
(66)
From Eqn. (65), an extreme point for the function a is one in which erK+1 = 0 then
. = -K)
» 1
(67)
where
' _ da _ dgi
ajk ~ dXi ~ 2^ 9idXi
(68)
" d2a
ajk =
dXjdXk
o d29i , da dgi
2^{gi -~~~ +
dXjdXk dXjdXk
or in matrix form
a — 2H g
2 [C + HH]
(69)

43
It must be noticed that here
(70)
contains the third derivative d¿E/dXjdXjdXk and becomes less important as g¡ —► 0,
that is, as an extreme point is approached.
From Eqn. (67) we have
-l
-l
s =
2(C + HH) 2H (71)
When C -> 0 :
s=
(72)
which is the Newton-Raphson equation for an extremum point provided that C is
sufficiently small (locally, near an extreme point it must always be correct).
The “object” function being reduced from equation (64) is a (not E) and the line
search condition is: aK+1 < aK.
This method can be applied to find any stationary point and will not necessarily
find local minima with respect to the energy: Rather, one usually increases the energy
of the nearest stationary point and then minimize it with this technique.
Gradient Extremal
This model, first proposed by Ruedenberg [52] and further developed by others
[53], uses gradient extremals which are defined as lines on the mass scaled potential
energy surface E(jc) having the property that, at each point jc0, its molecular gradient
g(jc0) is a minimum with respect to variations within the contour subspace, for example,
along a contour of E(jc) constant. Figure 2.6 shows the behaviour of this procedure as it
steps uphill, whereas Figure 2.7 shows how minima and maxima get connected through
the gradient extremal.

44
The procedure starts by introducing the Lagrangian multiplier A
d[g^g-2\{E-K)\/dx = 0 . (73)
By differentiating Eqn. (73) the following eigenvalue equation is obtained
H(s)ff(&) = X(x)g(x) . (74)
This is perhaps the most important contribution of this technique, as it states that the
gradient is an eigenvector of the Hessian. A simple interpretation of this expression is
that 2Hg is proportional to the gradient g at the point x. Moreover, g is orthogonal to the
contour subspace at gradient extremals, since g is orthogonal to the contour subspace.
It is assumed that the potential energy, its gradient and Hessian are calculated
explicitly at each iteration. Setting the geometry of the k’th iteration, say x^ , a step
Sk is determined, where it is possible to write: xk+\ = xk + sk- The second order
total energy at this point is approximated as
E(2)(zifc+i) = E(xk) + gT(xk)sk + (75)
and the actual energy at this point (with no approximations) is
E(z*+1) = E^2\xk+i) + R (76)
where R contains higher order terms in s^. Steps are taken with confidence if:
—>• E(sfc+i). A quantitative measure of this approach to agreement may
be obtained from the ratio r as
r = [E(^+i)-E(.t,.)] = R
[EMfe+iJ-E^)] [E»)(I(+i) - E(arjt)]
If r—>1, the third-order terms are negligible and the second-order expansion is considered
to be exact. The chosen step size should then depend on how close r is to unity.

45
A trust region with radius h is introduced, within which the second-order expansion
approximates the exact potential surface, and the trust radius is updated according to
the size of r.
The step direction (Sk) is obtained by using the extremal of the second-order surface.
The steps in the walk are determined assuming that in the trust region the gradient
extremal of the second-order surface will describe accurately the gradient extremal of
the exact surface. In the quadratic region we have
H(.t) = H ; \{x) — A -> g(x) = g + H.t (78)
where H and A are constant. It is assumed that the origin is the center of expansion.
Substitution of Eqn. (78) in (74) gives
(H - A I) Ha; = — (H - XI) g (79)
which reproduces the Newton-Raphson step equation if (H —AI)-1 exists. Let v be the
eigenvector of H belonging to A (the eigenvector along the reaction path): (H-AI)t>=0.
If A is non-degenerate then (H-AI) is non-singular on the orthogonal complement of
v. Thus, the following projector is introduced: P = I — vv* and Eqn. (79) is
now written as
PH.x = —P g —» Pa; = — PH-1# (80)
The solution for this relation, assuming that H is non-singular, is
x(a) — — PH-1g + av . (81)
The gradient extremal x(a) (alpha is an arbitrary real parameter) for the second-
order surface defines a straight line which is parallel to the eigenvector v passing
through the solution of the projected Newton equation PH—1g where av is the step in
our Newton-Raphson scheme. If now the Hessian has the desired number of negative
eigenvalues

46
Figure 2.6. Potential energy surface representing the gradient extremal uphill walk
(bold arrow) that will connect stationary points, that is, all minima and transition states.

47
Figure 2.7. Potential energy surface representing gradient extremal, unique lines
(bold) connecting stationary points, that is, all minima (qi, q2 and q3) and transition
states (TSi and TS2).

48
(only one for a true TS), then the stationary point of the surface is used as the next
iteration point x^+j. On the other hand, if the stationary point is outside the trust
region or if the Hessian has not the desired index, then the gradient-extremal point on
the boundary of what becomes the next iteration will point downhill. The gradient-
extremal point on the boundary is determined by varying a in Eqn. (81) to obtain a
step length equal to the current trust radio h.
Although results are promising for this procedure, H has the specific drawback
that the step must be inside or on the boundary of the trust region and those steps are
conservative. The gradient extremal has been found to bifurcate also during such a
walk. It is important to note that the usefulness of the gradient extremal is related to
the fact that there are unique lines connecting stationary points, as shown in Figure 2.7.
This, together with the fact that these lines are locally characterized, makes gradient
extremals potentially very useful for exploring potential energy surfaces and for some
uses in molecular dynamics. Unfortunately, applications of this technique have not
been reported yet.
Gradient Extremal Paths (GEP)
The original idea of Gradient Extremal Paths is due to J. Pancir [54] with subsequent
testing by Muller [55]. A formal mathematical definition was given by Basilevsky [56].
Hoffman et al. [52] discussed the nature of GEPs with emphasis on their usefulness
in molecular dynamics. They showed that third-order derivatives are very important to
characterize GEPs. Jorgensen et al. [3 Id] were among the first in developing algorithms
to find TSs in chemical reactions using second order GEP. Recent developments and
applications have appeared for GEP [10, 12, 13, 62a]. Use of GEP to obtain molecular
vibrations, as well as a good review of this model have been discussed by Almlof [57].

49
Constrained Internal Coordinates
Internal Coordinates [19, 58] are often preferred over Cartesians because they allow
valence bond parameters (bond lengths, bond angles) to be constrained in a physically
meaningful way as the remaining structure parameters are optimized.
Such procedures can be summarized in accordance to the following three steps:
• Series of minimizations constraining some coordinates.
• TS is the Emax with respect to the unconstrained coordinate(s).
• Energy is minimized with respect to all other coordinates.
An advantage of this procedure is that the Hessian is not required to reach the saddle
point. A major drawback is that an important reaction coordinate must be identified
in advance.
The Image Potential Intrinsic Reaction Coordinate (IPIRC)
Designed by Sun and Ruedenberg [ 12d], IPIRC is a transformation of Fukui’s
Intrinsic Reaction Coordinates [59, 60] transition state search procedure converted into
an algorithm that searches for minima. IRC was originally proposed by Fukui [59]
and later developed by others [60]. Andres et al. [61] applied the IRC to the addition
reaction of CO2 to CH3NHCONH2 using different semiempirical methods and Ab-
Initio basis sets.
The strategy of this technique is as follows:
1) Diagonalize the inverse of the Hessian matrix: C^H_1C = A.
2) Organize the eigenvalues of the diagonal matrix A in decreasing order:
Ai > A2 > An.
3) Change the sign of the smallest eigenvalue A„ .
4) Undiagonalize A and procede to minimize using a steepest descent procedure.

50
As a consequence, the transition state structure now becomes a minima (to be
sought) and the original minima (starting conformation) becomes a higher energy
structure from which the down-hill walk (minimization) will start.
The Constrained Optimization Technique
Constructed by Muller and Brown [62], the constrained optimization technique opti¬
mizes consecutively the geometry through a given pre-established coordinate. Abashkin
and his collaborators as well as others [41, 63, 64], have proposed a mixture of tech¬
niques and implement this idea into DFT calculations. The main contribution of their
algorithm is that they solve the problem of the constrained optimization by explicitly
eliminating one of the variables using the constraint condition.
Gradient-Only Algorithms
A gradient-only algorithm recently was explored by Quapp [13]. It has as a major
drawback its apparent necessity of a large number of steps to find the saddle point. Its
success relies on the small size of the step it takes but, as a consequence, convergence
is very slow. Figure 2.7 shows the main features of this procedure.
The algorithm starts by stating a new definition of the valley pathway: A point q
belongs to a 7—minimum energy path (7MEP) if the gradient condition g(q) = g(q7)
holds, and is used to compare differences of gradient vectors. The new coordinates are
given by: q7 = q + jg(q). We immediately recognize the steepest descent like
relation to obtain the new coordinates in this uphill walk (where the Hessian has been
replaced by the identity matrix). Here 7 is a step length parameter (not coming from
a line search).
An asymptotic steepest descent path is defined as the geometrical space in which
many steepest descent lines, from the left and the right side, converge into the stream

51
bed of the valley ground whose shape will be followed by the 7MEP. The points close
to these path are shown in Figure 2.2.
The situations to be encountered are as follows: If the point qG/ is at the left of the
7MEP, then the negative gardient of qx/ will point it back to the right. Conversely,
if the point qor is displaced to the right then the negative value of the gradient at qir
will point to the left. The idea is that this gradients can be used to correct the steps
as they go apart from 7MEP.
The algorithm, which needs a step length (s) and a tolerance (t) to start, is as follows:
1) Optimize starting geometry qG that it is not necessarily a minimum: |g(q0)| 7^
0.
2) Choose a step length and a tolerance (t) such that: Set counter i = 0 and t «
s with t < 1.
3) Predict a step in a steepest descent fashion: q,;+i = q, + 7(?(q¿).
4) If |#(q¿+i)| < T then STOP, meaning that a saddle point has been located
(T is a given threshold).
5) Get a scaling factor (/) (in braket notation): / = (p(q¿+i)|p(q¿)).
Here a backwards checking is performed:
If / > 1 — f then: set i = i + 1 , Go To step 3). Else:
6) Correction to the step: q¿+i = q7+i — ryg(q¿+i), set i - i + 1 and Go
To step 3).
The technique seems to work well if t < 10—2. This procedure is not competitive,
for example, with the Approximate LTP technique of Cardenas-Lailhacar and Zerner
[14], which requires one-fourth as many energy evaluations to get the same results.

52
Figure 2.7. Quapp’s only gradient procedure. Three different points on and in
the neighborhood of a minimum energy path (MEP) are shown. The gradients (uphill
arrows) are shown for points qD;, qG and qor- For points qu, qi, and qi,. the negative
gradients (down hill arrows) are drawn. The uphill steps are corrected using the gradient
vectors -g{qu) and -tf(qir) .

53
Table 2.1. Features, Advantages and Disadvantages of some of the most used
Transition State search techniques available today in program packages.
Model Advantages Disadvantages
Simple
Monte Carlo
Simulated
Annealing
Synchronous
Transit Path
(LST and QST)
Cerjan-Miller
Schlegel
Minimax / Minimi
Energy Min or
Normalization
Augmented
Hessian
Gradient Extremal
Constrained Int.
Coordinates
Squared Norm of
the Gradient
Parties, in volume.
Arbitrary initial
Config.: p = cte
Reduced
Temperature to
evaluate
probability, p = cte
LST: Line
connects R and P.
QST: Max LST
fitted to a Parabola
Evaluate Hessian
to define uphill
path. Lagrange
multiplier is used
Right inertia of
App. H obtained
by fixing
eigenvalues sign
Successive Opt of
Int. Coords, of a
given Symmetry
Distance between
R and P is used
Search Dir founded
diagonalizing the
Approx. Aug. H
Stationary points
in PES connected
by stream beds
Selection of RC
(bond length)
Newton step-like
search direction
Initial Config of the
system is arbitrary
Flexibility to anneal
the P.E.S.
Simple assumptions
about reaction path
simplifies the search
Walks up-hill from
minimum to TS
essentially in an
automatic way
Reverse up/down
search direction,
refined by a factor
Hints unexpected
Intermediates or
extreme shifts
Very simple and
cheap procedure
Precise, few cycles
needed to Minimize
the gradient
Unique lines (g)
that connect
stationary points
g is not needed to
reach TS
Nearest stationary
point uncovered
Random walk
needs large number
of moves
Need large number
of random moves
If path is curved
QST might not
converge
Frequent H matrix
calculation makes
it expensive
Fails if number of
iterations needed is
large. Downhill
step - 1 dimension.
Extra parabolic line
minimization along
QST
Needs good initial
guess for TS
Evaluation of the H
matrix is expensive
Complications
happen if Gradient
Extremal bifurcates
Identify suitable
Reac. Coord.
Costly evaluation
of the Hessian
matrix

CHAPTER 3
HISTORY OF GEOMETRY OPTIMIZATION AND TRANSITION STATE
SEARCH IN SEMI-EMPIRICAL AND AB-INITIO PROGRAM PACKAGES
The relations describing the approximations involved in each model will not be
examined here because this is not a comprehensive review. We refer the reader to the
original works [65].
Brief Historical Overview
Semi-empirical molecular orbital theories are mainly based on approximations to the
Hartree-Fock equations. The first of the Zero Differential Overlap (ZDO) methods that
has historical importance is the 7r—electron method developed in 1931 by Hückel [66],
It is still used today to demonstrate important qualitative features of delocalized systems.
In the early 1950’s, Pariser, Parr, and Pople developed the PPP theory [67], which
while only of historical importance, had great influence on future procedures. This
technique was the first to describe molecular electronic spectroscopy with any degree
of accuracy and generality. Related to these procedures, in 1952 Dewar developed
the Perturbational Molecular Orbital (PMO) theory [68] (also a tt electron method),
calibrated directly on the energies of model organic compounds. The accuracy of this
method was remarkable [69].
Pople and his co-workers, in 1965, extended the ZDO method to all valence
electrons [70]. The impact that such an approximation had in the formation of the Fock
matrix gave rise to new methods such as the Complete Neglect of Differential Overlap
(CNDO), Intermediate Neglect of Differential Overlap (INDO), Neglect of Differential
54

55
Diatomic Overlap (NDDO) schemes. New modifications and approximations introduced
in these procedures, have produced revised methods such as CNDO/1, CNDO/2,
CNDO/S, INDO/1, INDO/2, and INDO/S.5The first gradient method was introduced by
Komornicki and Mclver [2] in CNDO and this important work had enormous impact.
The MINBO/3 model [71] (a Modified INDO model) was developed by Dewar and
his collaborators in 1995. This technique was designed to reproduce experimental
properties such as molecular geometries, heats of formation, dipole moments and
ionization potentials. This method has prove to be remarkably useful. Introduced
much later, MINDO/3 has an automatic geometry optimization procedure which was
a contribution of tremendous impact at that time — the Davidon, Fletcher and Powell
(DFP) [22] algorithm.
The SINDOl model by Jug et al. has proven to be very accurate in reproducing
geometries as well as other properties as binding energies, ionization potentials, and
dipole moments [72], The geometry optimization part of SINDOl was implemented at
the Quantum Theory Project (University of Florida) by Hans Peter Schluff, and uses
the BFGS procedure [21, 27] as developed by Head and Zerner [19].
NDDO and MNDO
Proposed by M. Dewar and W. Thiel in 1977 [73], the Modified Neglect of
Differential Overlap (MNDO) model was introduced as the first NDDO method. Today
it uses the gradient norm minimization procedure for finding the transition state, whereas
for optimizing geometries it employs a variation of the DFP algorithm [22].
MNDO, as well as the majority of other ab-initio and semi-empirical programs, is
subject to improvements which generate a proliferation of related programs and methods
5. S stands for Spectroscopy; parameter sets are modified to reproduce electronic spectra.

56
such as MNDOC [74], initially parametrized only for H, C, N and O [75]. More recently,
it has been suggested by M. Kolb and W. Thiel [76] that an improvement to the MNDO
model can be achieved by the explicit inclusion of valence shell orthogonalization
corrections, penetration integrals, and effective core potentials (ECP’s) in the one-
center part of the core Hamiltonian matrix. Their results shows good improvement in
the location of TS over such methods as MNDO, AMI and PM3. MNDO originally
was parametrized on experimental molecular geometries, heats of formation, dipole
moments, and ionization potentials.
MOPAC
In 1983 Stewart [77] wrote a semi-empirical molecular orbital program (MOPAC)
containing both MINDO/3 and MNDO models, allowing geometry optimization and TS
location using a Reaction Coordinate gradient minimization procedure, introduced by
Komornicki and Mclver [8] and vibrational frequency calculations.
AMI
The Austin Model 1 (AMI) developed in 1985 by Dewar and his group [41], was
created as a consequence of shortcomings in the MNDO model (spurious interatomic
repulsions, inability to reproduce hydrogen bonding accurately). Minima and TS
location are the same as for MNDO.
PM 3
The Parametric Method Number 3 (PM3) introduced by Stewart [78], is the third
parametrization of the original MNDO model. As in AMI, PM3 is also a NDDO
method using a modified core-core repulsion term that we will not describe here. PM3
and AMI differ from each other in that PM3 treats the one-center, two-electron integrals
as pure parameters. This choice implies that in PM3 all quantities that enter the Fock

57
matrix and the total energy expression have been treated as pure parameters. This, in
turn, is proving to be a disadvantage as many anomalies are beginning to appear as a
consequence of parameters that are not physically reasonable. Geometry is searched
using the Saddle technique [40, 41].
ZINDO
This package of programs implemented by Zerner and co-workers [26], contains the
INDO/1 and INDO/S models. ZINDO is constructed to perform a series of calculations
based on different models, namely PPP, EHT, IEHT, CNDO/1, CNDO/2, INDO/1,
INDO/2 and MNDO. It includes techniques to examine geometry [17, 19, 79] using the
Line Search (see chapter 2), Newton-Raphson, Augmented Hessian, Minimize Norm
Square of Gradient, and other techniques. This allows the user to select from a variety
of search types as well as updating procedures (e.g. BFGS, Murtagh-Sargent (MS),
DFP, and Greenstadt). For TS structures search, the “Augmented Hessian” procedure
developed by Nguyen and Case [49] has been implemented by Zerner et al. [19].
Although this is a very effective method, it requires the use of second derivatives,
again making it very time consuming. The Gradient Extremal method of Ruedenberg
et. al. [52] is very effective, but requires the exact evaluation of the Hessian. The LTP
algorithm [14] has recently been implemented; it makes use of up-date techniques like
BFGS [21] to treat the second derivative matrix, although it uses reactants and products
for the search. Widely used, the BFGS update procedure for geometry optimization and
TS search was born in this versatile package; and is now used in Gaussian, HONDO,
Gamess as well as AMPAC.

58
AMPAC (Version 2.1)
This molecular orbital package, a product of Dewar’s research group, is a new im¬
proved version of the original AMPAC, containing the semiempirical Hamiltonians for
MINDO/3, MNDO and AMI. It uses the BFGS algorithm for geometry optimization,
while for the search of the TS uses the Chain method [9, 40],
The Chain method needs, in order to maintain the connectivity of the path, to restrict
each distance between two successive points to a given length which in AMPAC is 0.3
o
A.
GAUSSIAN 94
A further development of its previous versions (Gaussian 76, 80, 82, 86, 88-92)
[80], Gaussian 946 is a connected system of programs for performing semi-empirical
and ab-initio molecular orbital (MO) calculations. Gaussian 88-92 includes:
• Semi-empirical calculations using the CNDO, INDO, MINDO/3, MNDO and
AMI model Hamiltonians.
• Automated geometry optimization to either minima or saddle points [15, 20, 22,
44, 74], numerical differentiation to produce force constants and reaction path following
[15], and so on.
The option in Gaussian for “Optimizing for a Transition State” is sensitive to the
curvature of the surface. In the best case, in which the optimization begins in a region
known to have the correct curvature (there is a specific option for this in the menu) and
steps into a region of undesirable curvature, the full optimization option (available as
a control option for the calculations) can be used. This is quite expensive in computer
6. A version of this program-package, with parts rewritten by Czismadia and coworkers, is
also called Monster-Gauss because of the tremendous amount of calculations that can perform
as well as being monstrous in length because of its ab-initio block.

59
time but the full Newton-Raphson procedure, already implemented in the program, with
good second derivatives at every point will reach a stationary point of correct curvature
very reliably if started in the desired region (line searches can be conducted with second
derivatives at every point). If a stationary region is not carefully selected, it will simply
find the nearest extreme point.
An eigenvalue-following, mode-walking optimization method [74, 81] can be
requested by another option (OPT=EF) [43, 82] that is available for both minima and TS,
with second, first, or no analytic derivatives as indicated by internal options (CalcAll,
CalcFC, default or EnOnly). This choice is often superior to the Berny7 method, but
has a dimensioning limit of variables (50 active variables). By default, the lowest mode
is followed. This default is correct when already in a region of correct curvature and
when the softest mode is to be followed uphill. Other options of interest, in connection
with GOPT and TS calculations, are:
• Freezing Variables During Optimization: Frozen variables are only retained for
Berny optimizations.
• Curvature Testing: By default the curvature (number of negative eigenvalues) is
checked for the transition state optimization. If the number is not correct (1 for a TS),
the job is aborted. Here the search for a minimum will succeed because the steepest
descent part of the algorithm will keep the optimization moving downward. On the other
hand, a TS optimization has little hope if the curvature at the current point is wrong.
• Murtagh-Sargent Optimization: This method almost always converges slower than
the Berny algorithm. It is reliable for minima only.
• Berny or Intrinsic Reaction Coordinate (IRC) method: This is an algorithm
designed for finding minima mentioned here because it is often used in the Gaussian
7. Berny stands, with tenderness from the Gaussian people, for Bernard Schlegel.

60
package. With IRC the reaction path leading down from a TS is examined using the
method of Gonzalez and (Bemy) Schlegel [15]. In this procedure, the geometry is
optimized at each point along the reaction path. All other options which control the
details of geometry optimizations can be used with IRC.
Although Gaussian 88-92 has many optimization options that can be used in
combination with one another, it will be enough for our purposes to note that this
package of programs uses Newton-Raphson, Murtagh-Sargent, Fletcher-Powell and
Berny’s (by default) methods for optimization, whereas for TS search it uses the Cerjan-
Miller algorithm [43] and the Linear Synchronous Transit method (LST) [37].
HONDO
This package written by Michel Dupuis and his co-workers [58] uses algorithms
that take advantage of analytic energy derivatives. The Cerjan-Milller algorithm [43] is
implemented with an updating of the Hessian matrix. This algorithm has proven efficient
provided that a “good” second derivative matrix is used [59] (it has been found that a
force constant calculated with a small basis set at the starting geometry is adequate).
As an option, the HONDO program allows the user to use the “Distinguished
Reaction Coordinate” approach. This approach consists of a series of optimizations
with an appropriately chosen coordinate being frozen at adequately chosen values
(for further details see the “Constrained Internal Coordinates” [19, 59] outlined the
preceeding chapter).
After this, and an inspection of the potential energy curve, it is possible to guess
the TS structure. At this point, a geometry optimization of the guessed TS is suggested
by using the BFGS algorithm implemented in the program. The method can be used in
conjunction with all SCF wavefunctions, as implemented for the geometry optimization.

61
All other options on HONDO assume that the TS structure is known as well as the
vibrational mode corresponding to the imaginary frequency. Then the gradient norm is
minimized [18] to find the nearest extrema.
ACES II (Version 1.0)
This package of programs for performing ab-initio calculations, developed in the
early 1990’s by Bartlett and co-workers [83], contains geometry optimization algorithms
that are all based on the Newton-Raphson method, in which the step direction and size
are related to the first and second derivatives of the molecular potential energy. In
almost all calculations the exact Hessian is not evaluated but approximated. By default
ACES II geometry optimization starts with a very crude estimate of the Hessian in
which all force constants for bonded interactions are set to 1 hartree/bohr2, all bending
force constants are set to 0.25 hartree/bohr2, and all torsional force constants are set to
0.10 hartree/bohr2. An alternative Hessian is used for some small systems, allowing
the use of an . In the search for a minimum, the method implemented in this package
can be used when the initial structure is in a region where the second derivative matrix
index is nonzero. Moreover, a very efficient minimization scheme, particularly if the
Hessian is available, is included in this package of programs, namely, a Morse-adjusted
Newton-Raphson search for a minimum.
For the TS search ACES II uses the Cerjan-Miller algorithm [43], This involves
following an eigenvector of the Hessian matrix (that corresponds to a negative eigen¬
value) to locate the stationary point, ensuring that it will stay within the region of
the TS. Finally, to ensure that a TS has been obtained, the vibrational frequencies are
evaluated by taking finite differences.

CHAPTER 4
THE LINE-THEN-PLANE MODEL
Introduction
With the exception of the Synchronous Transit [37] and the Normalization technique
[41] models both of which consider the distance between reactants (R) and products
(P) while searching in a linear fashion for the TS to finally minimize the maxima found
all other procedures discussed in chapter 2 use up-hill methods. Those procedures start
from reactants through a second order expansion of the energy in terms of a Taylor
series.
For comparison we have collected in Table 2.1 the procedures reviewed in chapter
2, emphasizing their principal features, advantages and, especially, their disadvantages.
If we focus on the disadvantages, we notice a general trend in the problems that appear
when searching for a TS (which are similar to the geometry optimization ones):
• Costly evaluation of the Hessian matrix.
• Difficulties in identifying an appropriate reaction coordinate.
• Requirement of a good initial guess of the TS.
• Convergence achievement.
• Large number of moves needed (i.e. random procedures).
Because of these problems, a better method to find the TS should consider both
reactants and products because both contain information on the appropriate saddle point.
62

63
Here we present a procedure that is based on a continuous walking from R to P
(and vice-versa) with a fixed step length, along a line connecting them. By minimizing
the energy at those new points a new line is drawn and the procedure is repeated until
a pre-established criterion to find the maxima (or minima) is fulfilled.
The procedure that we present is very simple and has been designed to overcome
the problems enumerated above. Thus, we have built up a strategy to find the TS, which
makes use of the line search technique, that has the advantage of using a reduced number
of calculations, has a simple and convenient expression of projected coordinates, does
not require evaluation of the Hessian matrix, considers an intermediate of reaction, and
involves the idea of finding the TS(s) starting simultaneously from R and P.
The algorithm does not require the evaluation of the Hessian. As a result, it is much
faster in its execution than most of the methods presently in use, and it is applicable
for searching the potential energy surfaces of rather large systems. The procedure is
not completely unlike the Saddle procedure of Dewar, Healy and Stewart [42] or the
line procedure of Halgren and Lipscomb [37], but does differ in a rather substantive
way. In this new technique, the line direction is allowed to change during the walk,
initially from a line connecting product and reactant to points that represents them.
These representative points are determined through minimization of all the coordinates
that are perpendicular to the connecting line. The efficiency of this procedure rests upon
the observation that it is faster and easier to minimize repeatedly in the N-l directions
than it is to evaluate N(N + l)/2 second derivatives, where N represents the number
of variables (coordinates) to be searched.
When the interest is to focus on the shape of the reaction path (mechanism) we
suggest, as an alternative simpler strategy, to find the TS(s) so that, when there, the
reaction path is constructed by using a down-hill procedure to reactants and products.

64
However we do not recommend this sequence. We will introduce, unstead, the LTP
algorithm and discuss its properties, the step and the speed-up of the procedure by
using Hammond’s postulate. Finally, Hammond’s postulate adapted LTP techniques
are discussed.
The Line-Then-Plane (LTP) Search Technique
This procedure, originally conceived to find TS in chemical reactions, requires
knowledge of reactants (R) and products (P); no previous knowledge of the TS is
necessary. It makes partial use of both the line search technique and the search for
minima in perpendicular directions that have been discussed already. Figure 4.1 shows
the behavior of this technique.
As in the Saddle method of Dewar [40—42] we begin by calculating the structures
of the reactant R and product P. A difference vector d¡ — — Ri is defined
di =
(Xp - X*)? + (Yp - YR)J + (Zp - ZR)1
1/2
(82)
with i = 0, 1,2, 3, ... and we walk a fraction of the way from R¿ to P, along —d¿, and
from Pi to R, along d, . The structures at these two points are minimized in the plane
(i.e. all directions) perpendicular to d¿, defining the new points R/+i and P,+i. A new
difference vector d¿+1 — P¿+1 — R,+1 is then defined and the procedure repeated.
The steps along d¿ are conservative initially, but increased as a percentage of the norm
of d¿, that is Jdjd,, as the TS is approached. The BFGS technique [18, 21, 26, 27]
is used to minimize the energy in the hyperplane perpendicularly to d¿.
At this point we distinguish the Exact and the Approximate LTP procedures:
• Exact LTP: The BFGS technique is used to minimize the energy in the hyperplane
perpendicular to the direction d,.

(a)
Figure 4.1. (a) General scheme for the Line-Then-Plane (LTP) procedure where the scanning is performed between the
last two minimal points found, from both R and P. (b) A sequential transverse view of the planes containing the projected
and the in-line (d¡) points.
ON

66
• Approximate LTP: No up-date of H is performed. For the minimization of the
energy, in perpendicular directions to d¡, we use the identity matrix as the Hessian (PH
= PI = P) only once (steepest descent), so that the projected coordinates depends only
on the gradient g¡ and on the projector Pat (i.e., i - 0), see below.
The Algorithm
We adopt the following algorithm, shown graphically in Figures 4.1 a-b:
Step 1) Calculate Reactant and Product geometries R„ and Po (the super index p stands
for projected coordinates). Set counters i = j = 0 (see below).
Step 2) Define the difference vector d,; = Pf — R? . If the norm d|d; < T (a
given threshold), stop. Otherwise,
Step 3) Examine a^R^Jd,; and — Set R| - Rf ; P[ = Pf (83)
(the super index l stands for variables in the line that connects the corresponding R’s
and P’s) unless,
i) If ct^r?) d¿ < 0 set P- — Rf and R\ — R¿_1 (84)
or,
ii) If - Step 4) Walk along d¿ from R- to P •, and vice-versa, a fixed step of length: s,: = d¿¡Ni
Set R[+1 = R[ + s¡ and P¡+i = P¡ — s¡ (86)
unless,
If T < djd¿ < s?-s| set N¡: = Ni ¡2 and j = i
If Ni < 2
set Ni = 2
(87)

67
Step 5) As in Step 3), examine ct^(rí+1)sí and — If ^(h'+J si < 0 set, P/+i - R-+1 and R-+1 = R- (88)
If - (tt(p'+1) s¿ < 0 set R|:+1 = P-+1 and P-+1 = P- (89)
Step 6) Minimize in the hyperplanes perpendicular to d, containing R-+1 and Pj:+1, to
obtain projected points R?+1 and P?+1 respectively. Set i = i+1 and go to Step 2).
Although one can demonstrate that this procedure must lead to the Transition State
on a continuous potential energy surface E(x) if the steps are conservative enough, the
norm of the gradient (g) and the Hessian (H) are examined at convergence in order to
insure that the converged point on E(x) has the right inertia, (i.e. g = 0) and H has
one and only one negative eigenvalue.
To accelerate convergence to the TS, we might add to Step 3) a further test that
becomes useful as the TS is approached,
Step 3) iii)
If AaU(k) = a\p‘k+l) sk - AE(k) - E(Pfe) - E(Rfc) < T", k = i, i - i and i - 2 (90)
Then, Set: q¿ = (P¿ — R,;)/2, evaluate E(q,;), g(cy) and. H(q¿)
Else, If í7t(q¿)'?(qí;) < g\Pi)g(Pi) and - (7T(q,;)d,: < 0
(91)
Then q¡ replanes P¿ or,
If 5T(q/)p(q¿) < 7t(Rl)á'(R¿) and -^(q^d,- < 0
(92)
Then q¡ replaces R¿.
Then, go to Step 2), Else, go to Step 4).
Here q, refers to the coordinates that represents a conformation that is very close
to the TS structure.

68
Based on our experience, the choice A0 = 10 generates a conservative initial step
and suitable thresholds are T — T = T = T =10 arbitrary units. We have
studied the variation of the step size with the number of energy evaluations needed to
converge to the saddle point, as is described in a subsequent section.
The strategy delineated above is also successful even for systems which have
intermediate structures between R and P. The tests indicated in equations (84-85)
under Step 3) and (88) and (89) under Step 5) disclose potential turning points, caused
either by a too large a step from R toward P or P toward R.
The reaction path can be approximated by connecting all points R¿ and P¿. An
approximate and faster procedure would be to quit in Steps 3) or 5) thereby avoiding
the reset of coordinates between consecutive steps.
Then the displacement d, can now be divided in smaller parts (say 4) and the
procedure continued as before. The last half is now submitted to a perpendicular
minima line search founding a last point Xe, the TS.
In general, the TS is said to be found if the gradient norm is zero and if the Hessian
has one and only one negative eigenvalue, respectively. As for LTP, the transition will
be considered to be found when the norm of the displacement vector di is smaller
than a pre-established convergency threshold Tc (usually Tc < 10~3). Nevertheless, the
general conditions are checked at the estimated saddle point (i.e. a(Xe) = 0).
Minimizing in Perpendicular Directions: Search for Minima
The coordinates perpendicular to the direction d, are obtained by projection, and
the energy in the hyperplane minimized using the BFGS algorithm as developed by
Head and Zerner [27]. It has to be pointed out that translations and rotations must be
eliminated from G = H—1 as they represent zero eigenvalues of H, in order to construct

69
a projector free of them. This requirement has been included in the ZINDO program
package [26] as part of the implementation of the LTP techniques.
This procedure is restricted to the projected coordinates
Pd,(x/+i s*+i) — Pd¡(xi+i X¡+i) — «P d, Gj+jP dt9i+!
where
Si
?+i
:d, = X-
N~‘ ,:+1
is the step (coordinates) along the line connecting projected R’s and P’s.
In a more compact way, we can write Eqn. (93) as
,(0 \Pd, _ TV
(93)
(94)
or
(*+i - = -«G¿+;
(.Pi) _ „(0 _ J?i r>Pi
x¿+í = x*+i
« 9iU Gf
i+i
(95)
(96)
where the projector perpendicular to d, is defined as [17]:
d/d-
= I
djd,
(97)
In these equations, a is the line search parameter which determines how far along
the direction sj+1 of equation (94) one should proceed. For the simple test cases
studied in the next chapter, there is but one perpendicular direction, and we set a = 0.3
for all i which is a more conservative value than that recommended by Zerner and his
collaborators (o:0 = 0.4, and all other a.\ = 1.0) [17, 19, 27].
It can be demonstrated easily that, computationally, it is much more convenient to
project out only the forces rather than project the forces and the second derivative matrix
at the same time. Consequently the new projected coordinates are now obtained as
M) ~ Jl) a Gi+1 gf+1 (98)
x,-+i = x,+i
where the inverse Hessian G is updated using any appropriate technique.

70
Projector Properties
The projector P^, must be well behaved (i.e. it has to fulfil the conditions of being
idempotent and hermitian).
Idempotency: P = P2
We start from the definition of the projector:
dd)
consequently
P = I -
d)d
p2 -
dd)'
d.d)'
1 ~~ dü_
(99)
dd) dd) dd)dd)
- did, ~ did, + dJddJd,
2 dd) dd)dd)
~dJd + d)dd)d
2dd) ddt
I -
d)d d U
(100)
_i-í£ = p
dU
Hermiticity: P = P^
Pt =
I -
= I
P = I
dcü_y
d)d._
dXd,
dd)
~d)d
- I
dd)
d)d,
- P
q.e.d.
q.e.d.
(101)
(102)

71
LTP Convergency
Consider the coordinates difference
q = Xi+1 - X* (< c ) (103)
where X* represents a maximum, the TS (X* = Xts)-
Defining, for the neighborhood of the TS X*:
lim q =0 =>■ ©(q) - a iff 3c G 3?/la I < cq . (104)
■S—>o V /ii
The gradient around a given point XK:
but qK = —qK , then:
g(XK - XK + X*) - g(XK) + HKqK + 0(|qK|2)
(106)
g(X*) = 9* - H„qK + 0(|qK|2) .
If XK is too close to X* (with Hk with only one negative eigenvalue), the considered
region of the space exists by continuity of the Hessian H. Consequently, the k th
iteration exists.
Projecting from the left by H”1 :
H^VX*) = 0 = H^y - H-'H^q, + 0(|q„|2) (107)
but: g,.H, 1 =
— sK, which is a Newton-Raphson like step.

72
Finally:
O = sK - qK + 0(|qK.|2) = - q'K+1 + 0(|qK|2) (108)
but according to our original definition:
3 c € ft/lqjc+il < c|q'K|2 . (109)
If XK is very close to X* , for which:
|q| < a/c , 0 < a < 1 (110)
by induction, and because XK —> X*, the iteration is defined and it exists for all
k and |qh-| —> 0. Consequently, by construction, LTP always will go uphill in the
search for a maximum.
To ensure that the new projected points R and P are perpendicular to the reaction
coordinate di, we must show now that the energy is a minimum in these directions.
Consider the second-order expansion for the Energy
E(x) = E(x0) + qTp + ^qjHq . (Ill)
From the gradient expansion
9 = 9o + Hq and q = -H-1p or g = -Hq . (112)
Introducing g in the equation for the energy, we get
E(x) = E(x0) - ^ q'Hq (113)
which demonstrates that the energy in perpendicular directions to the step is minimized
by a steepest-descent-like term in which H is positive definite.
From these, we conclude that a second (or higher) order LTP iteration converges.

73
The Step
A good step will provide a good starting point for the next step, such that
the maximization will converge without problems in a reduced number of iterations
along the chosen direction. In general, almost all algorithms take their steps without
considering previous information about the PES.
In developing LTP, three ways of stepping were studied. The first stepping method
is a superimposed step given by a fraction (1/N) of the displacement vector between
projected products and reactants coordinates. The second stepping method is based on
a proportionality relationship between the actual and previous step. It is shown that
this choice will locate the TS (not its final position) at most at half of the size of N,
that is, around N/2 LTP cycles, because the final displacements are very small. The
stepping method is based on the knowledge of information about the PES given by the
current projected point (reactants or products) where the value of N is then estimated
by relating the LTP step to the Newton-Raphson one (since LTP is a Newton-Rapson
like algorithm).
Default Step
In LTP, the step (s¿) is a fraction of the current displacement vector (d,)
s i = d i/N (114)
where N is a number greater than one. For the first iteration N = 10 (an arbitrary choice
suggested after many test calculations) and thereafter the distance between current
reactants and products is checked to be not less than a given threshold (say 10-3),
otherwise N is reset to 2.5.

74
Updated Step
A convenient decision on how the step should be taken comes from an algorithm
that will decide automatically what the value of N should be for the new LTP cycle
once the displacement vector is known. To accomplish this the next step is redefined
as to be directly proportional to the previous one:
Si+i = di+i/Ni+1 ex di/Ni = Si . (115)
Now the problem at hand is an estimation of the value of Ni+1 and consequently the
next move. For this, we consider the following relation between the next (i+1) and
the previous (/) steps
d¿+1/jV¿+1 = Ad i/Ni (116)
where 0 < A < 1. Projecting now from the left by dj we obtain,
Ni+1 - Ni-^
d¡di+1
djdi
(117)
when A = 1.
Alternately, it might be better to consider a relation with a penalty function on it.
This can be written easily as
WrlA r í\<¡t.«A2
(118)
I A|d* |
(sU)2
Ni
.(xlRXip)'2 + (4S02.
N,
where xír and Xip are the difference vectors between the new projected reactant and
product, and their corresponding coordinates in the line (step from where the searches
start).
Newton-Raphson-Like Step
Consider now the usual LTP step. We want to take a non-arbitrary step based
on previous knowledge of the curvature of the region in which we are walking.

75
Furthermore, we want, at any cycle, the LTP step (sltp) to be as well behaved as
the Newton-Raphson (snr) one
= «H ~lg
(119)
where a is a term that comes from the line search technique and the gradient (g) and the
displacement (d) are column vectors. Note that only the absolute value of N should be
considered. This is because the direction of the walk as defined by the LTP algorithm, is
positive when going from reactants to products and is negative in the opposite direction.
Projecting from the left by the gradient complex conjugate (g^)
^7fd = ag'H lg
(120)
we derive
(121)
It has been suggested, and shown, by Zerner and his co-workers [17, 19, 27], that
for the initial Newton-Raphson step a good choice is to set a = 0.4 and the inverse
of the Hessian as the identity matrix. Consequently we can have an approximation to
the estimation of N as
(122)
This stepping might not be convenient when searching in the vicinity of the saddle
point because the denominator will be too small and N will be too large.
Hammond’ s-Postulate-Adapted LTP Methods
Introduction
It might be argued that LTP, because of its twofold search (reactants and products
at the same time), requires too many steps or that it needs twice the amount of effort

76
(steps) required by other algorithms such as augmented Hessian [19, 48, 49]. Hence,
the Augmented Hessian method will be extensively used for comparison. This concern,
and the desire to have an algorithm that will move faster and efficiently towards the
TS, brought us to the approximate LTP procedure ennunciated in the previous section.
However, and by construction, this lack of specific information about the curvature of
the potential energy surface provided by the Hessian can be a drawback.
With these problems and goals in mind, we recall Hammond’s postulate (HP) [6],
which states that the TS will resemble more the initial reactants (R0) or products (P0)
according to whether the initial or the final state, is higher in energy. However, we
have already mentioned some not uncommon examples for which HP fails.
In this section we study the inclusion of HP in order to save some computational
efforts by reducing the number of steps. We will do this by adapting LTP to Hammond’s
postulate and consequently generate two more LTP like procedures, the Hammond-
Adapted-Line-Then-Plane procedures (HALTP) and the Restricted HALTP (RHALTP)
procedures. For these, the energy of both initial reactants (R0) and products (P0) {Ero
and Epn, respectively) will be considered.
Hammond-Adapted LTP Procedure (HALTP)
Two situations need to be considered:
a) If ERo > Epo : This is the original (exact and approximate) LTP as described
above.
b) If ERo < Epo : Reset to a new set of coordinates (prime): li0 — P0 and
P0 = R0.
The situation described in b) is shown in Figure 4.2, after which LTP will continue
as before. This particular situation can also be seen as if the search starts from the

77
original products. The advantage of this adaptation lies precisely in a reduction on the
amount of energy evaluations (LTP cycles) as now LTP will start searching from the
geometry of highest energy.
Restricted Hammond Adapted LTP (RHALTP)
In this case the same two situations depicted before are analyzed where the concept
of Hammond’s postulate is now strictly enforced. The first subcase still leave us with
the classical LTP (Epo > Epo), but the second subcase (with Epo < Epo as
condition) is now modified as follows:
RHALTP I. If (Epo < Epo) then, do not move the initial products.
This means that the coordinates of the starting products, characterized by P0, are
held constant.
This choice will allow the reactants to move uphill faster towards the TS by being
lifted by the products, as shown in Figure 4.3. This possibility is of particular interest
when one is concerned with following the path of the reaction under study. The idea
is tested in the next chapter for the inversion of ammonia reaction.
RHALTP II. If (Ero < Epo) then, do not move the initial reactants.
This time we consider that the reactants, characterized by R0, remain as the initial
ones lifting the products towards the TS, as shown in Figure 4.4. The idea is tested,
again in chapter 6, for the non-symmetric inversion of ammonia reaction.
It has to be pointed out that RHALTP I and RHALTP II are not the same procedure
with different label for reactants and products (and of technique), because the energetics
of the changed coordinates are completely different.

18

19

80
Figure 4.4. Reactants Restricted-Hammond-Adapted-Line-Then-Plane technique
(RHALTP II). The coordinates of reactants characterized by P0 are held constant, lifting
the products towards the TS.

CHAPTER 5
GEOMETRY OPTIMIZATION
Introduction
Almost all the procedures discussed in chapter 2 use Steepest Descent methods
to search for minima through a second-order expansion of the energy in terms of a
Taylor series.
From the Geometry Optimization procedures reviewed in chapter 2, the general
behavior of problems in the search of minima becomes clear:
• Costly evaluation of the Hessian matrix.
• Large number of moves needed (i.e. random procedures).
• Convergence problems.
Although several procedures are available, there are still other problems, such as
the loss of information about the curvature when the Hessian is not considered. In this
way, and as is the case for TS search, the development of new techniques will rely on
experimentation, namely that the model must show acceptable behavior on a variety of
test functions, chosen to represent the different features of a typical problem.
Because of these problems, it seems that a better method to find the minima (hope¬
fully the global minimum) must consider the initial geometry plus a generated second
81

82
one (only at the initial step). Therefore, in addition to position and/or displacement vec¬
tors, the displacement vector between the two initial points should also be considered.
A procedure that is based on the Line-Then-Plane technique (LTP), that is, a
continuous walking from the lowest energy point through a line connecting the two
lowest energy points, is proposed. By minimizing the energy at the new point a new line
is drawn between the new point and the one from which the projection was performed.
The procedure is repeated until a preestablished criterion to find the minimum is fulfilled.
Figure 5.1 illustrates the behavior of this procedure.
The same features already described for TS search with LTP are valid here, that
is, this is a procedure that does not require the evaluation of the Hessian. As a result,
the proposed method is much swifter in its execution than most of the methods used
today, and is applicable for searching for minima in potential energy surfaces of rather
large systems. In this technique, the line direction is allowed to change during the
down-hill walk, initially from a line connecting the starting geometries that represent
them. These points are determined through minimization of all the coordinates that are
perpendicular to the connecting line. The efficiency of this procedure rests upon the
observation (as for LTP), that it is quicker and easier to minimize repeatedly in the N-l
directions than it is to evaluate N(N + l)/2 second derivatives, where N represents the
number of variables (coordinates) to be searched.
ARROBA: A Line-Then-Plane Geometry Optimization Technique
This procedure requires a single input geometry from which a second set of
coordinates will be generated only in the first step. The down-hill walk starts by
determining the lowest energy point, making partial use of the line search technique

83
and the search for minima in perpendicular directions. As introduced previously, a new
projected minima is then found. Figure 5.1 illustrates the behavior of this idea.
As in the “Amoeba,” or “Simplex” method of Nelder and Mead [30], we begin by
calculating the structures of the initial point and a second one generated as:
(123)
where /3 is a 3N dimensional unitary vector (where N = number of atoms) scaled by
three different factors {(f), x and if), for the x, y and z components, respectively), one
can make fl a constant (but /3 ^ 0). Any of these choices will be the initial guess for
the problem and will depend on the size of the system.
Once a second initial point is generated, energies (E) and gradients (g) are evaluated
for both initila points (qi, E\, g\ and q2, E2, g^)- The strategy then is as follows:
A difference vector d¿ = q2 — qx is defined
(In —
(Xi+1 - Xi)l + (Yi+1 - Yi)l + (Zi+1 - Z,
i)n
(124)
with n = 0, 1, 2, 3, ...
The structure of lowest energy of these two points will be minimized in the plane
perpendicular to d;, defining a new point q¡+2 and i is reset to i = i + 1. A new difference
vector d, = q¿+1 — q¿ is then defined and the procedure is repeated. The norm
of d¡ (i.e. \Jdjdj ), is checked for convergence as the minimum is approached. The
BFGS technique [18, 21, 26, 27] is used to minimize the energy in the hyperplane
perpendicular to d¿.
As for the search for maxima, we differentiate between the Exact and the Approx¬
imate ARROBA procedures:

84
• Exact ARROBA: The BFGS technique is used to minimize the energy in the
hyperplane perpendicular to the direction d,.
• Approximate ARROBA: No up-date of H is performed. For the minimization of
the energy, in directions perpendicular to d¿, we use the identity matrix as the Hessian
(PH = PI = P) only once (constrained steepest descent), so that the projected coordinates
depend only on the gradient g¡ and on the projector P¿. (i.e., i = 0), as described below.
Figure 5.1. Schematic representation of ARROBA, an adapted Fine-Then-Plane
technique for geometry optimization. The input coordinates (qi), the initially generated
one (q2), the general zigzag behavior of the procedure and the found minima (qmjn)
are shown.

85
The Algorithm
We adopt the following algorithm, shown graphically in Figure 5.1:
Step 1) Calculate initial and new generated geometries qi and q2. Set counter i = 1.
Step 2) Define the difference vector d, for which its norm d-d¿ is greater than T (a
given threshold), else the program will stop:
If
E¡+1
< Ei
then
d¿
= q*+i - q>
(125)
If
■£¿+1
> Ei
then
d,:
= q* - qi+i
(126)
Else: Stop, and check for convergency: d}d,; > T.
Step 3) Minimize in the hyperplanes perpendicular to d,- containing E¡+1 to obtain
projected points E{+2. The point from where the perpendicular minimization starts is
that one with the lowest energy. We set i = i + 1, accept the new point if and only
if E¡-1-2 < Ei+ls Else: go to Step 2).
Step 4) The new projected point coordinates are given by
If Ei+1 < Ei then qp+2 = qf+1 - a g?+1 Gpi+1 (127)
where the upper script (p) stands for projected variables using the projector as showed
in the preceding chapter.
It can be shown that this procedure must lead to a minimum that, according to its
location in the PES, might be a local or a global minimum, provided the surface is
continuous. The norm of the gradient and the Hessian are examined at convergence in
order to ensure that the converged point on E(q) has the right inertia. The minimum
is said to be found if the gradient norm is zero and if all the Hessian eigenvalues are
positive.

86
Minima in Perpendicular Directions
As in the LTP method, ARROBA uses coordinates perpendicular to the direction d;,
obtained by projection. The energy in the perpendicular hyperplane is then minimized
using the BFGS algorithm as developed by Head and Zemer [27],
Again, it is computationally much more convenient to project only the forces rather
than project them and the Hessian matrix at the same time. Consequently the new
projected coordinates are obtained as
,{Pi)
'i+1
a gf+1 G?;+i
(128)
where the inverse Hessian is now updated using any appropriate technique.
Convergency
To ensure that the new projected points R and P are perpendicular to the reaction
coordinate d, we must show that the energy is a minimum in these directions.
Again consider the second order expansion for the energy
E(x) = E(xo) + q^g + i q'Hq (129)
and the gradient expansion
9 = 9o + Hq (130)
and
q = -H"1.,
or
g = -Hq
(131)
which is the quasi-Newton condition. Introducing g in the equation for the energy we get
q'Hq
E(x) = E(x o)
(132)

87
which demonstrates that the energy in perpendicular directions to the step is minimized
by a steepest-descent-like term in which H is positive definite.
From these considerations, we conclude that a second (or higher) order for the
ARROBA iterations converges. This minimization procedure has the advantage of using
a reduced number of calculations, particularly in the case of the Approximate technique,
and does not use the Hessian. It is guaranteed to step down-hill. The minimum found
will be a local minima. The search for the global minimum is discussed below.
A Proposed Global Minima Search Algorithm
As discussed in chapter 1, when looking for minima it is very desirable for a
procedure to be able to find the global minimum, especially for large molecular systems
(proteins, enzymes) for which the most widely used current procedure is the Monte Carlo
model often requiring thousands of energy evaluations.
Here we propose a procedure that will have a behavior like Monte Carlo, but does
not depend on the temperature and that does not need as many calculations. It uses
a jump-out technique, as the warm-up part of the Monte Carlo techniques to take the
system out of the local well in which it is trapped.
The algorithm requires a control option from the input file that allows the user to
perform several ARROBA calculations. The strategy is as follows:
1) Make an ARROBA minima search.
2) Set counter i = 1 and label the new minimum as: q¿ = Xn (n is the internal
ARROBA counter).
3) Construct a displacement vector (r¿) between the minimum found and the input
geometry
r¡ = q¿ - Xo ■
(133)

88
4)Get a new displacement vector r- orthogonal to r¿, that is, in braket notation:
(r¿|r¿) = 0 .
(134)
5)Obtain a new initial set of coordinates xc
Xo = r¿ + Xo
(135)
6)Check for maximum allowed number of searches M:
If i < M Go To Step i Else Stop (136)
where M is a pre-established maximum number of iterations.

CHAPTER 6
APPLICATIONS
Introduction
The ideas discussed in the previous chapters have been tested by two different
approaches. One approach involves the use of two-dimensional model potential func¬
tions to test the behavior of the LTP procedures and compare the results with reports
on other methods in the literature [14]. Six model potential functions are examined.
The Hammond-Adpated LTP technique has also been tested on three of these functions
and, the Restricted-Hammond-Adapted procedures were investigated on a 7lh potential
function. Finally, using the potential functions, the LTP accuracy and convergence
dependence on the step size have been studied.
The LTP method was also tested on several molecular systems: the inversion
reaction of water, the symmetric and the non-symmetric isomerization reactions of
ammonia, a rotated inversion reaction of ammonia, the hydrogen cyanide isomerization
reaction, the formic acid 1,3 sigmatropic shift reaction, the methyl imine isomerization
and the thermal retro [2+2] cycloaddition reaction of Oxetane. The accuracy has also
been examined in terms of the step and number of energy evaluations required to find
the TS in the molecular examples.
For the study of those systems, the Intermediate Neglect of Differential Overlap
(INDO) technique [84] has been used, at the Restricted-Hartree-Fock (RHF) level [85]
within the ZINDO program package [26], The minimization procedures are, of course,
89

90
limited to no particular energy function, provided it is continuous. The results were
compared with those of the Augmented Hessian (AH) technique that uses the same
INDO Hamiltonian but evaluates the Hessian at each iteration. All the above-mentioned
procedures were implemented in the ZINDO [26] suite of programs.
Model Potential Functions for Transition State
LTP procedures have been tested on six model potential functions which are
traditionally used to examine TS searching procedures. The first two, the Halgren-
Lipscomb and Cerjan-Miller potential functions, have their TS located closer to the
reactant than to the product. The next two potential surfaces, the Hoffman-Nord-
Ruedenberg and Culot-Dive-Ghuysen, have the TS located closer to the products than
the reactants. The fifth potential function has a TS located midway between reactants
and products, and the sixth PES has a steep minimum located in the products region.
The results of these tests are collected in Tables 6.1, 6.2 and 6.3 and are discussed below.
The Halgren-Lipscomb Potential Function
The Halgren-Lipscomb potential function [37, 39]:
EHL(x,y) = [(.t - yf - (5/3)2]2 + 4(xy - 4)2 + x - y (137)
has two minima (we have chosen points (1.328, 3.012) and (3.0, 1.333) for reactants
and products respectively) and one first order saddle point (2.0, 2.0). Figure 6.1 shows
the shape of this surface as well as the points obtained with the LTP procedure. Notice
that both LTP procedures, Exact and Approximate, walk uphill using the same points.
The Cerjan-Miller Potential Function
Cerjan and Miller’s function [43]:
ECM(x,y) = (a - by2)x2 exp(—x2) + ^y2
(138)

91
has two symmetric TSs located at points (±1,0) and a minimum at point (0, 0). As R
and P coordinates we have selected points (0,0) and (2.7, 0.05) respectively.
For this procedure, an accurate Hessian is required. Others that have used this
function include Simons et al. [31], with a Fletcher-based surface algorithm [22],
Banerjee et al. [32], with a rational function optimization algorithm, and Abashkin and
Russo [86], with a constrained optimization procedure. All of these previous studies
have used a = b = c = 1 with the exception of Simons and his coworkers, who used
a = c = \, b = 1.2. Figure 6.2 shows the behavior of our procedure when applied to
this potential energy surface. Here both procedures walk towards the TS and are very
close to each other (notice the scale on the Y axis).
The Hoffman-Nord-Ruedenberg Potential Function
Hoffman et al. [52] have used the model surface function:
EHNR(x,y) = (xy2 - yxr + x2 + 2y - 3)/2 (139)
to test their gradient extremal procedure. This function has also been tested by Schlegel
[10], As other algorithms, previously mentioned, these methods require an accurate
evaluation of the second derivatives.
The function has two saddle points TS] (- 0.8720, 0.7105) and TS2 (3.1352, 1.2487).
In order to test the LTP procedures the points (1, 1) and (5.4980, 1.2874) have been
chosen as the R and P coordinates respectively. From these points, a walk towards the
TS has been performed. Figure 6.3 shows the behavior of our suggested procedure in
this potential function surface.
We note the somewhat chaotic behaviour of the Approximate procedure in the
products region, due to its inherent lack of information of the quadrature of the surface.

2
Figure 6.1. The Halgren-Lipscomb Potential Energy Surface EHi(x,y) = [(a; — y)2 — (5/3)2] + 4(xy — 4)2 +
x — y showing the full procedure. The Exact and Approximate LTP procedures are represented by crosses and asterisks
respectively, walking together (same coordinates) towards TS. The TS is represented by the point where R and P meet.

Figure 6.2. Cerjan-Miller ECM(x,y) = (a — by~)x2 exp(—x2) + cy2/2 potential energy function, with a = b = c =
1. Here the transition state (black diamond for Exact LTP and bold cross for the Approximate LTP) in the 3 dimensional surface
is located in the reactants region. The Exact and Approximate LTP methods (represented by crosses and asterisks) satisfy the
threshold positions before reaching the same point (see text and Table 6.2).

94
On the other hand, what seems to be a gap on steps in the uphill walk towards the
transition state, is nothing but the convergence acceleration mechanism at work.
The Culot—Dive—Nguyen—Ghuysen Potential Function
Proposed by Culot et al. [11], this test function has, in the range interval [-5, 5], 1
maximum, 4 minima and 3 first-order Saddle Points (points (1); (2), (3), (4), (5); (6),
(7) and (8) respectively, identified as in reference [11]). LTP was tested using as R, P
and TS points (3), (2) and (7) whose respective coordinates are (3.585, -1.850), (3.0,
2.0) and (3.4, 0.1). The function is:
Ecdng{x,'Ij) = {x2 + y - ll)2 + (a: + y2 - 7)2 (140)
In Figure 6.4 the behavior of our procedure for this potential surface is represented.
As was the case for the Cerjan-Miller surface, both procedures, Exact and Approximate,
walk towards the TS and have in common the same projected points.
A Midpoint Transition State Potential Function
A simple function:
EMP?s(x,y) = [0 - 1)(x - 2)]2 + [(y - 1 )(y - 2)]2 (141)
was used in order to test the ability of our procedure to step up-hill towards the TS in a
flat PES. This function has two minima: (1.0, 1.0) used as reactants (R) and (0, 2.0) as
products (P) . The TS is located in the midpoint between reactants and products (1.5,
1.5) with a potential energy of 0.125 (arbitrary units). The behavior of our algorithm on
this potential surface is shown in Figure 6.5, where it can be seen that both techniques
walk towards the TS using the same projected points.

Figure 6.3. The Hoffman-Ross-Ruedenberg potential energy function EHNR(.x, y) = {xy2 - yx2 + + 2y - 3)/2.
Exact LTP (cross) and Approximate LTP (asterisk) approach the TS (bold square) located in the products region, although the
steps of the approximate method deviate considerably from the reaction coordinate.

96
A Potential Function with a Minimum
This potential energy surface:
EMin(x,y) = x4 + y4 + 2 xy (142)
has the property of having a very steep minimum located at (0.7071 , -0.7071),
very close to the starting products coordinates (1.0, -1.0). The reactant is located at
(0.2, 0.0). Figure 6.6 demonstrates the downhill behavior of our procedure.
The Approximate procedure starts with erratic behaviour on the products side, which
is corrected by the reactants side as the minimum is approached. Here both techniques
walk in their own fashion showing their individual characteristics.
Summary of Results
Table 6.1 summarizes the number of function evaluations required by the LTP
procedures along with the results of other authors. The comparison is based in the
simple “counting” (when possible) of the number of points on the appropriate figures
of the corresponding works, assuming that these are the number of steps required to find
the TS. Table 6.2 shows our results for the exact LTP as well as the values obtained
using the Approximate LTP procedures.
It should be remarked that a better answer is obtained when a fraction of the smallest
slope between reactants and products is used as an initial value of a for both procedures.
Because of the nature of the steps advocated, the TS will never be missed, or
passed, provided that the steps along the vector connecting P¿ and R¿ are small enough.
A clear demonstration of this is given graphically in Figure 4.2b (chapter 4), where it
can be seen that the last point, the saddle point: TS, is a unique point at which R¿ = P,-.
Moreover, it can be demonstrated easily that, because of the Newton-Raphson nature
of the step chosen [14], the LTP procedure will always converge to the TS.

Figure 6.4. Culot’s et. al. EcoNa(x,y) = (x2 + y — ll)2 + (x + y2 — 7)2 potential energy function.
Here the is represented by the highest in energy bold asterisk. Notice how both (Exact (crosses) and Approximate (asterisks))
LTP procedures walk uphill with the same coordinates and direction. The TS is located in the Products region.

2 2
Figure 6.5. Representation of EMPTS(x,y) = [(x — l)(x — 2)] + [(y — l)(y — 2)] potential energy function. The
TS is located exactly in the midpoint between reactants and products. Again, both procedures (crosses for the Exact LTP and
asterisks for the Approximate LTP) walk uphill together using the same coordinates and direction.

E(x,y)
Figure 6.6. Potential energy function with a minima EMin(x,y) — a;4 + y4 + 2xy . Here the minimum Min (bold
bullet) is found in the products region. Again, the continuous line represents the path followed by the Exact LTP procedure
whereas the dashed line stands for the Approximate procedure path. Note: the Approximate LTP (X) deviates considerably from
the intrinsic reaction coordinate, but does find the TS as does the Exact LTP (crosses), see text.

100
Table 6.1. Number of function evaluations (NFE) required to find the TS for
Cerjan-Miller [43] (CM) and Hoffman-Nord-Ruedenberg [12] (HNR) potential function
surfaces (see text). Values in parentheses are for the Approximate LTP procedure.
Model
NFEcm
Lagrange Multipliers [43]
8
Soft-Mode Analytical Hessian [31]
12
Soft-Mode Updated H [31]
23
Stiff-Mode Analytical Hessian [31]
6
Stiff-Mode Updated H [31]
7
C.G. + q-N.M.a [87]
10
RFOb [25a]
14 and 13
RFOb [32]
13c
P-RFOd [32]
13
RFO + H Updated [32]
16
P-RFO + H Updated [32]
15
Constrained Opt. Tech. [9]
12
Gradient Extremal [10] (HNR)
11
This Work :
NFEcm
NFEhnr
20 (22)
20 (26)
a. Conjugated Gradients + quasi-Newton Minimization methods.
b. Rational Function Optimization.
c. a = c = 1, b = 1.2.
d. Partitioned RFO.
In practice, we are maximizing along a line d¿, and it is easy to insure that
d^E/ddJ is negative when R¿ = P¿. The perpendicular searches utilize the BFGS
procedure starting with a positive Hessian. Since this procedure cannot change the
signature of the Hessian it will either minimize in all perpendicular directions (all other
d2 E / dx^2 > 0) or it will fail.

101
Table 6.2. Coordinates and potential energies for the Exact and Approximate LTP
procedures (first and second rows respectively), compared with the Expected results for
the TS for the six potential function surfaces (gradient norm less than 10-4, a = 0.3
with N0 = 10). Also displayed are the number of steps used to find the TS and its
location (region0). The average deviations for these procedures are Exact LTP: Ax =
0.0013, Ay = 0.0037 and AE = 0.0018; Approximate LTP: Ax = 0.0015, Ay = 0.0043
and AE = 0.0018 (arbitrary units).
Expect
X
LTP
Expect
Y
LTP
Expect
E(x,y)
LTP
Steps-
Region0
E3HL
2.0000
1.9955
2.0000
2.0045
7.7161
7.7066
14-R
1.9955
2.0045
7.7066
14
Ecm
1.0000
1.0003
0.0000
0.0024
0.3679
0.3679
10-R
0.9983
0.0069
0.3679
11
Ehnr
3.1352
3.1358
1.2487
1.2489
0.9707
0.9707
10-P
3.1353
1.2487
0.9707
13
E3CDNG
3.3852
3.3834
0.0739
0.0879
13.3119
13.3105
15-P
3.3834
0.0879
13.3105
15
EbMPTS
1.5000
1.5000
1.5000
1.5000
0.1250
0.1250
15-MP
1.5000
1.5000
0.1250
15
1.5000
1.5000
0.1250
4
Emíii
0.7071
0.7078
-0.7071
-0.7061
-0.5000
-0.5000
10-P
0.7079
-0.7070
-0.5000
13
a. Only Ehl shows an appreciable error for the threshold, 0.0095 units. For the function value,
only Ecndg shows an appreciable percentage error in y of only 0.014.
b. For Empts the third row shows results obtained using the simple test according to condition iii
on Step 3), whose principal feature is the reduced number of steps (4) used to find the Transition State.
c. R - Reactants, P = Products, MP = Midpoint between initial reactants and products.
On the other hand, it can be seen from Figures 6.1-6.6 that both the exact
and the approximate LTP procedures, walk uphill in the same direction with similar

102
coordinates. In particular, the coordinates are the same for both approaches in Halgren
and Lipscomb’s, Culot’s et. al. and MPTS potential functions, almost the same for
Cerjan and Miller’s function, close for Ruedenberg’s et. al. functions, and a little
bit erratic for the function that presents a minimum very close to the products region
(Figure 6.6). In this case the jumpy behavior of the approximate LTP procedure is
caused by a Hessian that is far from the projector itself. In spite of this, the results
are remarkably good. We may speculate that the Approximate LTP procedure may not
accurately follow an intrinsic reaction path initially but will become more accurate as
the TS is approached and, because of the size of the step, it will find the TS.
Finally, large separations between consecutive points, as in Cerjan-Miller’s (Figure
6.2), Hoffman, Nord and Ruedenberg (Figure 6.3) and the Function with a Minima
(Figure 6.6) potential energy functions, are due to the accelerating convergence condi¬
tions to the TS, established in Steps 3) and 5) of the algorithm. As for the case of the
Hammond-Adapted-Line-Then-Plane technique, we found this, as expected, to result
in a small reduction of the number of steps that are needed to find the TS, because
it just decides from which initial set of coordinates (reactants or products) the search
has to start.
We have tested the Restricted-Hammond-Adapted-LTP procedures on 3 of the pre¬
ceding potential energy functions, (i.e. Cerjan and Miller, Hoffman-Nord-Ruedenberg
and Culot-Dive-Nguyen-Ghuysen). The results of these tests are shown in Table 6.3.
The reduction in number of steps expected is small, when compared with previous
calculations (Table 6.2), because this is a very simple and modest improvement of the
algorithm. In fact, the changes are only noticed at the beginning of the search and at
each time that an LTP acceleration in the up-hill direction is performed (changes in
slopes for example).

103
Table 6.3. Coordinates and potential energies for both Restricted-Hammond-
Adapted-Line-Then-Plane (RHALTP) procedures (Exact and Approximate, models, first
and second rows for each potential function, respectively), compared with the Expected
results for the TS for three selected potential function surfaces (gradient norm less than
10^ , a = 0.3 with N0 = 10). Also displayed are the number of steps used to find
the TS. The average deviation for the Exact LTP procedure for this threshold is Ax =
0.0014, Ay = 0.0044 and AE = 0.0001 (arbitrary units), whereas for the Approximate
LTP procedure is Ax = 0.0023, Ay = 0.0068 and AE = 0.0000 (arbitrary units).
X
Y
E(x,y)
Steps
Expect
LTP
Expect
LTP
Expect
LTP
EacM
1.0000
0.9994
0.0000
0.0124
0.3679
0.3679
9
0.9994
0.0172
0.3679
9
Ehnr
3.1352
3.1369
1.2487
1.2490
0.9707
0.9707
18
3.1288
1.2456
0.9707
14
Ecdng
3.3852
3.3870
0.0739
0.0734
13.3119
13.3121
16
3.3852
0.0739
13.3119
16
a. For the independent coordinates, for the function value only Ecm shows an appreciable percentage
error, though the error in y is only 0.0124 and 0.0172 for the Exact and Approximate LTP procedures,
respectively.
The Step
As discussed in the preceding chapter, we have studied different ways of stepping
in order to get better insight of the potential energy surface and, in this way be capable
of choosing the best response from our techniques, as for all of them the stepping will
be the same.
We have applied these ideas to the Midpoint Symmetric Potential Function:
Eu?T.(x,y) = [(x - l)(x - 2}]2 + [(„ - l}(y - 2)]2
(143)

104
Table 6.4. Estimation of the size of the step based on information of the previous
one as established on Chapter 6, Case 2 (see text). The results come from applying
this stepping to the midpoint symmetric potential function (EMPTS(x, y)). Notice that
the size of the step () has been kept almost constant no matter the size of the
displacement.
i
Nj




1
10.00
14.04
1.72
197.00
2
8.10
11.39
1.72
129.65
159.6
3
6.04
8.51
1.40
72.45
96.68
4
3.98
5.61
1.72
31.52
47.78
5
1.98
2.84
1.44
8.05
15.68
and summarize the results of the application of both relations derived from Case 2 of
the preceding chapter in Table 6.4. It is interesting to notice that the sizes of the steps
are approximately the same for each iteration. But what is much more interesting is
that, according to the results the TS should be found at the 5th iteration, because the
last iteration established that N{+1 ~ 2. For any size of the initial N, the number
of LTP cycles required to find the TS will always be N/2, provided that N is within
a safe range of convergence to the saddle point, and that the potential energy surface
has a quadratic behavior.
The same test was performed on Cerjan Miller’s potential energy function:
ECm(x,v) = (a - by2)x2 exp(-x2) + C-y2 (144)
for which the step was kept constant. The results are displayed in Table 6.5. We
notice that at the beginning of the 6th LTP cycle (until the 11th cycle) the step size
is set to 2.5, after the updating of it (N5) gets closer to 2. At the 6th cycle, an until
the 11th one, the coordinates are relabeled because of the acceleration to convergency

105
properties of LTP, narrowing the region to be examined until at the 12th cycle reactants
step the TS is found. The behavior of this alternative stepping is compared against
the classical LTP (Exact) by plotting the trajectories followed by both stepping cases,
as shown in Figure 6.7.
From these results it becomes clear that there is no saving in steps and that both
techniques starts walking using the same points until the 3rd cycle, after which they
slightly separate, suffer from the same acceleration techniques, and find the same TS
at which they meet. Consequently, it can be assumed that a step factor of 1/10 is as
good as any other one, and it will used as default from now on.
Step Size Dependence
The dependence of the convergence of LTP to the TS as a function of the step size
has been studied also. For each case the requirements to locate the TS are the same, that
is, size of the displacement less than 10“3, maximum component of the gradient less
than 10“3 and continuous check of the Hessian signature until one negative eigenvalue
is established. Because potential functions have only 2 dimensions, a molecular case
will be studied (inversion reaction of ammonia), and discussed in a separate section.
The dependency has been examined on the midpoint transition state potential energy
function. The results are shown in Figure 6.8 from where we infer the safe region to
be 7 < N < 10.
Hammond and Restricted Hammond Adapted LTP Models
Quapp [13] has recently presented a new TS search algorithm which has its relevant
feature an emphasis on step searching. The procedure corrects its direction, after each
application, by performing a downhill step. However, it uses the Hessian and needs
many steps in order to locate the TS. Using Quapp’s potential function:

Table 6.5. Evolution of the uphill walk of the Approximate LTP technique applied to Cerjan-Miller’s potential function to
test the effect of the size of the step (Case 2, see text) on the search of the TS. Displayed are the LTP cycles (i), the step factor
(N¡), the geometry, energy and gradients. At and after cycle 6 the coordinates are re-labeled. The LTP found TS (midpoint
between the last projected reactants and products (i = 12)), and the expected one are displayed at the end.
REACTANTS
PRODUCTS
i
Nj
X
Y
E
9(x,y)
X
Y
E
g(x.y)
1
10.00
0.0000
0.0000
0.0000
0.0000, 0.0000
2.7000
0.0500
0.0062
-0.0231, 0.0495
2
8.00
0.2700
0.0063
0.0678
0.4654, 0.0054
2.4302
0.0316
0.0166
-0.0649, 0.0306
3
6.17
0.5399
0.0099
0.2178
0.5715, 0.0056
2.2049
0.0206
0.0378
-0.1317, 0.0191
4
4.17
0.8098
0.0097
0.3404
0.2893, 0.0031
1.9350
0.0137
0.0886
-0.2512, 0.0113
5
2.17
1.0798
0.0097
0.3634
-0.1117, 0.0027
1.6650
0.0098
0.1734
-0.3689, 0.0064
6
2.50
0.9178
0.0097
0.3628
0.0420, 0.0027
0.9718
0.0089
0.3673
0.0420, 0.0024
7
2.50
1.0150
0.0084
0.3677
-0.0219, 0.0022
1.0367
0.0085
0.3669
-0.0529, 0.0023
8
2.50
0.9933
0.0079
0.3679
0.0099, 0.0021
0.9977
0.0079
0.3679
0.0034, 0.0021
9
2.50
1.0046
0.0074
0.3679
-0.0068, 0.0020
1.0081
0.0075
0.3678
-0.0119, 0.0020
10
2.50
1.0005
0.0071
0.3679
-0.0007, 0.0019
1.0018
0.0071
0.3679
-0.0026, 0.0019
11
2.50
0.9990
0.0082
0.3679
0.0015, 0.0022
0.9995
0.0059
0.3679
0.0007, 0.0016
12
0.9995
0.0059
0.3679
0.0007, 0.0016
1.0005
0.0071
0.3679
-0.0007, 0.0019
TSltp
1.0000
0.0065
0.3679
0.0000, 0.0017
TSex
1.0000
0.0000
0.3679
0.0000, 0.0000

Figure 6.7. Representation of the effect of the update on the step (Case 2) in the Approximate LTP TS search in Cerjan-
Miller’s ECM(x,y) = (a — by2)x2 exp(—or) + |y2 potential energy function (we used a = b = c = 1). The different
uphill paths followed by the classical model (continuous line), that is, N = 10 for all cycles, and the updated step factor (dashed
line) techniques are plotted here, showing their convergency to the TS (bold cross) through similar, but not identical, pathways
starting from the same reactants and products (R and P, respectively).

20
18
16
14
12
10
8
6
4
lumb
[<*
of Line-Then-Plane cycles needed to find the TS for the Midpoint Potential energy function:
i)(* - 2)]2 + [(» - i)(y - 2)]2 as a function of the Step Size.
o
oo

109
Table 6.6. Coordinates, potential energies and number of cycles for Quapp’s [13]
potential energy function. A comparison with LTP, HALTP and RHALTP techniques
is shown. The first and second row of all 3 LTP procedures corresponds to Exact and
Approximate procedures, respectively. Calculations were performed using an step of Na
- 10 and a = 0.3. Convergency was achieved when the displacement vector norm was
less than 10"4 at which the gradient norm was less than 10"4 (all units are arbitrary).
The expected results for this potential energy function are those obtained by Quapp.
X
Y
E(x,y)
Cycles
Quapp3
0.00
-1.00
-1.00
130
LTPb
0.02
-1.05
-1.00
59
0.01
-1.06
-1.00
15
HALTPb
0.02
-1.05
-1.00
57
0.02
-1.04
-1.00
13
RHALTP IIC
0.01
-1.04
-1.00
45
0.02
-1.04
-1.00
10
a. These results comes from reference [13]. For his method Quapp used second derivatives.
b. Deviations for both, exact and approximate LTP methods are small and very close among each
other. LTP founds the TS to be located in the Reactants region.
c. Deviations for the exact and approximate of the Products Restricted Hammond Adapted LTP
(RHALTP II) method are small and very close to each other.
EQuaPP(x,y) = 2y + y2 + (y + 0.4a:2) x2 (145)
the difference between the 3 main LTP techniques was also studied, namely LTP,
HALTP and the Restricted HALTP (RHALTP) procedures. The results are shown in
Table 6.6.
The function in question has a saddle point at (0.00,-1.00). For our study, we have
used the points (1.77, -2.55) and (-1.00, -1.00) as reactants and products, respectively.

110
In Figure 6.9 we represent the behavior of both LTP techniques in this potential energy
surface.
Note that all three LTP procedures converge to a very reasonable saddle point in
less than half the number of the steps required by Quapp’s method. The Approximate
LTP methods are particularly interesting as they give the right answer with a relatively
small amount of computational effort.
Summary of Results
Most methods that search for TS require an accurate evaluation of the Hessian as
they proceed uphill from product to reactant, or from both points to the saddle point.
These evaluations of the Hessian are costly in computer time and in storage. The LTP
methods described here do not need the accurate calculation of the Hessian, with the
exception of the last step, at which the Hessian should be calculated in order to check
its signature. This distinction potentially allows the study of much larger chemical
systems. The procedure is characterized by consecutive uphill climbs from reactant to
product and vice-versa, with simultaneous minimization in all perpendicular directions
at each step. Because of the nature of the steps, a TS will never be either missed
nor passed provided that the steps are small enough. It can be shown that, because of
the Newton-Raphson nature of the step that we consider [8], the LTP procedure will
converge to the TS on any continuous surface. Connecting all points P¡ and R¿, will
give an approximation to the reaction path.
On the other hand, the results displayed on Table 6.2 show that the Approximate
LTP procedure (No Hessian) is as accurate as the Exact LTP procedure (BFGS technique
used to up-date the Hessian) in order to find the TS, although connecting the points may
not give an accurate representation of the RC when the approximate method is used.

E(x,y)
Figure 6.9. Representation of Quapp’s potential energy function: EQuapp(x,y) = 2y + y2 + (y + 0.4.7;2) x2.
The different uphill paths followed by the Exact LTP (continuous line) and the Exact RHALTP II (dashed line) techniques
are plotted here, showing their convergency to the TS (bold cross). The plot has been deliberately tilted so the behaviour of
both techniques could be better appreciated.

112
Finally, the testing of variations on the stepping factors has not shown to yield any
improvement, or savings, in the number of cycles needed to find the TS. Consequently,
it will be assumed that a step factor of 1/10 is as good as any other one, and it will
be used as default for all the molecular cases studied in the next section. For each
molecular system studied, the Hessian of the found TS was examined to ensure that it
has one, and only one, negative eigenvalue.
Molecular Cases for Transition State
Introduction
The efficiencies (accuracies) of the LTP procedures are examined by means of
application to several chemical reactions in this section. We first focus on the ability
of the algorithms to find the TS in some selected chemical reactions by means of the
application of the LTP techniques presented in the previous chapter. All the calculations
were performed at the Restricted Hartree-Fock (RHF) model, except for the thermal retro
[2+2] cycloaddition reaction of Oxetane, for which we used the Unrestricted Hartree-
Fock (UHF) technique so to account for breaking-formation of bonds. Calculations to
test the performance of the procedure as a function of the step size then are presented.
Inversion of Water
In Figure 6.10 a scheme of the inversion of water is shown. The search direction
(d) and the evolution of Rs and Ps towards the linear TS are shown. Whereas the
hydrogen-oxygen bond length gets shortened only by 0.0207 Angstroms, the reaction is
characterized by an evolution of the initial 106.81° hydrogen-oxygen-hydrogen angle to
a final one of 179.67°, which is not 180° as expected (although very close) because of
the LTP termination conditions, that is, the size of the displacement between the last 2
projected structures. A detailed LTP cycle-by-cycle uphill walk is shown in Table 6.7.

Products
H
â–²
d
(a)
H
H
H •*-
H â—„
H
Reactants
H
^ â–º H
H
H
(b)
Figure 6.10. Symmetric inversion of water, (a) The search direction (d) is shown, b) Pictorial representation of the uphill
walk of reactants and products towards the linear (bold) TS structure (not to scale).

114
Table 6.7. Cycle-by-cycle uphill walk in the search of the TS for the inversion
of H2O using the Exact LTP technique. Energy, geometry, the distance (square of
the norm of the displacement vector), the maximum component of the gradient and
number of update iterations (UI) required are displayed. The units are Hartrees for the
energy, Angstroms for the oxygen-hydrogen bond length (r) and the distance, degrees
for the hydrogen-oxygen-hydrogen angle (9) and Hartrees/Angstroms for the gradient
component (gmax). The TS is the linear structure for which 9 - 180°. The convergence
criteria was for the maximum component of the gradient (gmax) of the step (in the line
coordinates connecting R 5 and P5) to be smaller than 10-3 (Hartrees/Angstroms). The
energy and geometry of the steps in the line connecting projected coordinates are not
included, but the total number of energy evaluations was 41.
Energy
rOH
#HOH
d¡
9maX
UI
Ro
= Po
-18.122165
0.9952
106.74°
1.4975
0.000000
0
Ri
= Pl
-18.115333
0.9844
122.29°
1.1980
0.000501
3
r2
= P2
-18.102663
0.9798
134.35°
0.9584
0.000170
3
r3
= P3
-18.090434
0.9775
143.76°
0.7667
0.000049
3
r4
= P4
-18.060540
0.9745
172.85°
0.1533
0.000208
3
Rs
= Psa
-18.059076
0.9745
178.57°
0.0307
0.000010
2
R6 step
-18.059017
0.9744
179.71°
0.000193
0
a. At this LTP cycle, and as the distance dj (the square of the norm of the displacement vector) is
smaller than 0.1, a threshold prestablished by the LTP algorithm, the step is amplified by a factor of 2.
The uphill evolution of the LTP search technique towards the linear TS structure
of water is shown in a 3-dimensional plot in Figure 6.11
Finally, the symmetry change when going from the initial Czv reactants (R) and
products (P) to the D^ TS, is shown in Figure 6.12. Note that the TS has a higher
symmetry than R’s and P’s. We shall see that this seems to be the case for molecular
systems when R’s and P’s are mirror images of one another.

TS
Figure 6.11. The uphill path towards the TS, according to the Exact LTP technique, starting from reactants or products
(R,P), for the water inversion reaction.

E (Kcal/mol)
116
32.00-
0.00 -
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
TS
Figure 6.12. Energy and symmetry change between the initial reactants (R),
products (P) and the transition state (TS) found for the inversion of water.
Symmetric Inversion of Ammonia (NH3)
Figure 6.13 shows a 3-dimensional scheme for the inversion of ammonia, including
both the initial reactants and products geometries as well as the TS found. A detailed
LTP cycle-by-cycle uphill walk which includes energy, geometry and the number of
iterations used to update the Hessian is summarized in Table 6.8, and graphically shown
in Figure 6.14.

Reactants
r= 1.0508 A
0 = 105.59°
H
H ei
N
H
Transition State
r = 1.0381 A
0 = 120.00°
H
Products
r = 1.0508
9 = 105.59°
Figure 6.13. Symmetric inversion reaction of Ammonia. The initial reactants, products and the TS structure found are
shown, r is the nitrogen-hydrogen bond length and 9 is the hydrogen-nitrogen-hydrogen angle.
o<

118
Table 6.8. LTP geometry, energy and number of iterations required to find the TS
for the inversion of NH3. The results are from the Exact Line-Then-Plane technique.
The energy (Hartrees), the optimized geometry in the plane perpendicular to the
walk direction, namely the nitrogen-hydrogen bond length (r, in Angstroms) and the
hydrogen-nitrogen-hydrogen angle (8, in degrees), the distance (angstroms) between R’s
and P’s, the maximum component of the gradient and the number of update iterations
(UI) of the Hessian at each LTP cycle are displayed. The convergence criteria was
that the maximum component of the gradient and the distance to be smaller than 10"3
(Hartrees/Angstroms). The total number of energy evaluations was 31.
Energy
INH
#HNH
di
9max
UI
Ro
= Po
-12.522855
1.0503
105.63°
1.1825
0.000000
0
Rl
= Fl
-12.520988
1.0452
110.54°
0.9460
0.000059
3
r2
II
ls>
-12.517629
1.0421
113.83°
0.7568a
0.000627
2
r3
= P3
-12.507616
1.0383
119.74°
0.1514
0.000206
3
r4
= P4
-12.507098
1.0381
119.99°
0.0303b
0.000480
2
R5 step
-12.507077
1.0381
120.00°
0.000480
0
a. One of the acceleration flags of LTP is turned on such that the step factor now is amplified by 4.
b. At this LTP cycle, and as the distance d4 (the square of the norm of the displacement vector) is
smaller than 0.1, a threshold prestablished by the LTP algorithm, the step is amplified by a factor of 2.
Our results are collected and compared with those belonging to the Augmented
Hessian (AH) technique, in Table 6.9. They show that the TS found by LTP is as good
as the one found using the AH method. The main difference between both procedures
is that AH uses analytical second derivatives and better represents the PES using 7
SCF iterations (see Table 6.9), whereas LTP updates the Hessian using the BFGS [21]
technique (other techniques as Murtagh and Sargent (MS) [20], Davidon, Fletcher and
Powell (DFP) [22], and Greenstadt [23] are also available), and walks uphill towards the
TS from both sides, reactants and products of the reaction.

Figure 6.14.
inversion reaction.
TS
The uphill path towards the TS starting from reactants or products (R,P), for the symmetric ammonia

120
Table 6.9. LTP geometry, energy and number of iterations required to find the
TS for the Symmetric NH3 isomerization reaction. We show results obtained using
the Augmented Hessian (AH) technique, the Exact and Approximate Line-Then-Plane
(LTP) techniques and a final geometry optimization (GOPT) calculation performed on
the TS found by the Approximate LTP procedure, r represents the nitrogen-hydrogen
bond length (in Angstroms), 6 is the hydrogen-nitrogen-hydrogen angle (in degrees)
and E is the energy (in Hartrees). Linally the number of SCL cycles used by each
procedure are also displayed. The convergence criteria was for both the norm of the
gradient and the displacement vector to be smaller than 10-3.
AH
Exact-LTP
App-LTP
App-LTP GOPT
r
1.0382
1.0382
1.0004
1.0380
9
120°
120°
120°
120°
E
-12.507084
-12.507079
-12.498321
-12.507076
Cycles
7
22
10
3
The importance of the last condition was already described and discussed in the
previous chapter. On the other hand, the Approximate LTP obtains a good answer
without losing much in accuracy, using a number of SCF cycles (10) markedly lower
than the one used by LTP (22). However, the App-LTP has an intrinsic problem, it
does not contain enough information about the topography of the PES. Consequently it
requires a final geometry optimization of the TS. When this last calculation is performed,
only 3 extra SCF cycles are required. A substantial improvement is achieved in this
way, as is shown in Table 6.9. Performing this final optimization is not risky at all
because the TS found by App-LTP has the right signature over the Hessian as it is in
the surroundings of the TS.
Finally, and for a better description of this reaction, the symmetry change when

121
going from the initial reactants (R) and products (P) to the TS is shown in Figure 6.15.
It is interesting to note that the TS has a higher symmetry than R’s and P’s. This was
also the case for the invertí on reaction of H2O. We shall see that this will not be the
case for molecular systems where R’s and P’s are a mirror image of one another.
E (Kcal/mol)
9.90-
0.00 -
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
P
Figure 6.15. Schematic representation of the change in energy and symmetry
between the initial reactants (R) and products (P), and the found transition state (TS)
for the inversion reaction of ammonia.

122
Asymmetric Inversion of Ammonia (NH3)
Figure 6.16 shows the geometry for the initial reactants and products as well as
for the TS found. The results are displayed in Table 6.10. It is noticeable that the
TS found from LTP is as good as the one resulting from the application of the AH
method. The differences between both procedures are the same as described in the
previous example. The same comments about the number of SCF calculations, the final
geometry optimization and the symmetry changes between R’s, P’s and TS are valid.
Table 6.10. LTP geometry, energy and number of iterations required to find the
TS for the Asymmetric NH3 Isomerization reaction. We show results obtained using
the Augmented Hessian (AH) technique, the Exact and Approximate Line-Then-Plane
(LTP) techniques and a final geometry optimization (GOPT) calculation performed on
the TS found by the Approximate LTP procedure, r represents the nitrogen-hydrogen
bond length (in Angstroms), 6 is the hydrogen-nitrogen-hydrogen angle (in degrees), E
is the energy (in Hartrees). Finally the number of SCF cycles used by each procedure
are also displayed. The convergence criteria was for both the norm of the gradient and
the displacement vector to be smaller than 10—3.
AH
Exact-LTP
App-LTP
App-LTP-GOPT
r
1.0382
1.0380
1.0367
1.0381
e
120°
120°
120°
O
O
E
-12.507084
-12.507079
-12.507066
-12.507080
Cycles
7
22
10
2

H
H
/
N
r
H
Reactants
r= 1.0501 A
0= 105.70°
Transition State
r= 1.0380 A
0 = 120°
Products
r = 0.9706 A
0 = 119.16°
Figure 6.16. Asymmetric inversion reaction of Ammonia. The initial reactants and products, and the found transition state
structures are shown. Products are energetically higher than reactants. Notice that products are a flattened mirror image
of reactants.
to
OJ

124
Rotated Symmetric Inversion of Ammonia (NH3)
This test was performed to establish whether the LTP procedure is capable of
reaching the well-known planar TS conformation of NH3 by means of a rotation
through 50 degrees of the products (inverted umbrella structure), relative to the reactants
(umbrella conformation) in the plane that contains the hydrogen atoms (Figure 6.17),
by means of a step-by-step procedure during the up-hill walk.
These results displayed in Table 6.11 show that with only a few more SCF
calculations, LTP (exact) is not only able to rotate the reactants and products towards
the TS, but also to eliminate the problem associated with translations and rotations. On
the other hand, the TS found by the approximate technique (Table 6.11) gives a quite
Table 6.11. LTP geometry, energy and number of iterations required to find the
TS for the Rotated Symmetric NH3 Isomerization reaction. We show results obtained
using the Augmented Hessian technique, the exact and approximate Line-Then-Plane
(LTP) techniques and a final geometry optimization (GOPT) calculation performed on
the TS found by the Approximate LTP procedure, r represents the nitrogen-hydrogen
bond length (in Angstroms), 6 is the hydrogen-nitrogen-hydrogen angle (in degrees), E
is the energy (in Hartrees). Finally the number of SCF Iterations (energy evaluations)
used by each procedure are also displayed. The convergence criteria was for both the
norm of the gradient and the displacement vector to be smaller than 1CT3.
AH
Exact-LTP
App-LTP
App-LTP-GOPT
r
1.0382
1.0381
1.0370
1.0381
9
120°
120°
120°
120°
E
-12.507084
-12.507079
-12.507070
-12.507080
SCF-Its
7
26
12
2

125
good answer, and its optimization (App-LTP-GOPT), which requires only two more
energy evaluations gives a final answer as good as does the AH technique. This outcome
is in agreement with the cases studied before.
Hydrogen Cyanide: HCN —> CNH
Figure 6.18 shows a scheme of the HCN proton transfer reaction. The initial
reactant and product and the TS geometries as well as a detailed LTP cycle-by-cycle
uphill walk which includes energy, geometry, and the number of update iterations to
update the Hessian is summarized in Table 6.12.
The LTP uphill walk towards the transition state (TS) starting from reactants (R)
is represented in Figure 6.19. In spite of the tortuous shape of the hypersurface the
algorithm is capable of overcoming it.
A comparison of our results on the search for the TS of the proton transfer type
reaction with those of other research group’s collected in Table 6.13. Zerner et. al. have
used the augmented Hessian (AH) [27] technique, but no bond lengths were reported.
Bell and Crighton [87] on the other hand, have used an INDO Hamiltonian for their
study. However, their results cannot be compared with ours derived from the application
of ZINDO, because both methodologies do not have the same parametrization. Their
results are included here for geometry comparison purposes only.
In relation to the geometry the angular coordinate obtained by Zerner et.al. [18]
is reproduced, the difference being only 0.09°. There is, however, a difference of
0.038 A for the H-C bond and of 3.86° for the HCN angle between our data and
those of Bell and Crighton, which we associate with the different parametrization of the
Hamiltonians. The energy the difference between AH and this work is 0.3 Kcal/mol.

Reactants
r = 1.0508 A
0 = 105.59°
Transition State
r= 1.0381 A°
0 = 120.00°
Products
r = 1.0508
0 = 105.59°
Figure 6.17. Rotated symmetric inversion reaction of Ammonia. The initial reactants and products, and the found transition
state geometries are displayed. Products are a mirror image of the reactants but rotated by 50 degrees, but are geometrically
and energetically identical.
ro
Os
o<

Reactants
Products
r*
H C = ¡SJ
C = N
*1
H
N
â–º
Transition State
Figure 6.18. Proton transfer reaction of Hydrogen Cyanide. The behavior of the LTP technique is shown as reactants and
products suffers structural modifications (arrows), to coalesce finally in the TS.

128
Table 6.12. LTP geometry, energy and number of update iterations (UI) required
to find the TS for the hydrogen cyanide isomerization reaction (HCN —> CNH). The
results are those belonging to the Exact Line-Then-Plane technique. The optimized
geometry in a plane perpendicular to the walk direction, namely the carbon-nitrogen
and hydrogen-carbon bond lengths (in Angstroms) and the hydrogen-carbon-nitrogen
angle (9, in degrees), the energy (Hartrees) at each iteration and the number of update
iterations of the Hessian for each cycle are displayed. The convergence criterion
requires that the maximum component of the gradient should be smaller than 10~3
(Hartrees/Angstroms), whereas the search was started at an angle of 6 = 130°. The
energy and geometry of the steps in the line are not shown, but the total number of
energy evaluations was 32.
Energy
rHC
rCN
thn
^HCN
#HNC
UI
R0
-17.354816
1.0752
1.1840
130.00°
0
Po
-17.313130
1.1991
1.0376
180.00°
0
Ri
-17.304199
1.0766
1.2161
104.65°
4
Pi
-17.312384
1.2024
1.0521
175.57°
4
r2
-17.283882
1.1226
1.2285
77.78°
4
P2
-17.291977
1.2041
1.0074
144.09°
3
r3
-17.283736
1.0953
1.2195
88.72°
3
r4
-17.282221
1.1093
1.2244
82.10°
2
p4
-17.282171
1.1039
1.2232
84.31°
2
Rs
-17.282200
1.1092
1.2247
82.23°
2
Ps
-17.282156
1.1043
1.2233
84.16°
2
In order to obtain further insight on the origin of these differences (that are, however,
very small), we have performed a geometry optimization of the TS found by LTP, which
required only 2 more SCF iterations. As a result, the geometry has improved, but not
significantly. Whereas the difference in the HC bond length decreased only 0.005
Angstroms, the CN bond length increased in 0.002 Angstroms and the HCN angle by

Energy / Kcal/mol
Figure 6.19. The uphill path towards the TS, starting from reactants (R), for the isomerization reaction of hydrogen cyanide.

130
0.22°. On the other hand, the decrease in energy accounts to only 0.3 Kcal/mol. The
number of iterations was not comparable because all the calculations have different
starting geometries for the HCN angle a0 as shown in Table 6.12. However, it is
interesting to notice that the number of SCF calculations was still in the range of
procedures associated with the use of second derivatives.
Table 6.13. Geometry, energy and number of SCF iterations required for the
convergence to transition state for the HCN isomerization reaction. Displayed are
the results obtained through the Augmented Hessian technique, Bell-Crighton and LTP
techniques. Also displayed in the last column is the TS found when an extra geometry
optimization calculation was performed on the LTP transition state already found, tcn
and rHc are the corresponding carbon-nitrogen and hydrogen-carbon bond lengths
(Angstroms), 6 is the hydrogen-carbon-nitrogen angle (in degrees) and E is the energy
(in Hartrees).
AHa
Bell-Crightonb
LTPC
GOPT on LTP
rCN
1.2254
1.2255
1.2239
HfC
1.1397
1.1019
1.1073
e
83°
79.23°
83.09°
83.22°
E
-17.282563
-17.282074
-17.282125
Its
15
11
13
2
a. For this technique Zemer et.al. [18] did not report the geometry of the TS found. Their starting
hydrogen-carbon-nitrogen angle was of 180°.
b. Bell and Crighton used an INDO Hamiltonian that is different from the one used by Zerner and
his coworkers, as well as by LTP. However we included it here in order to compare the geometries. Their
starting hydrogen-carbon-nitrogen angle was 90°. They did not report the energetics of their calculations.
c. The starting hydrogen-carbon-nitrogen angle was of 9 - 160°.

E (Kcal/mol)
131
45.60-
26.16-
0.00 -
Figure 6.20. Schematic representation of the change in energy and symmetry
between the initial reactants (R) and products (P), and the found TS for the isomerization
of hydrogen cyanide.
From Table 6.12 we notice that the TS is located by the second iteration. It is in
the reactants zone, with an HCN angle of 77.78°. From this point, the LTP algorithm
resets the coordinates and searches in a smaller part of the hypersurface, accelerating
convergence. At the next iteration (3) a test is performed at the reactants zone. The TS
is again located in a narrower region of the space, and once more the coordinates are
reset accordingly. A few final iterations are performed and the TS is found.

132
The examination of the change of symmetry in this reaction shows that, in contrast
to the previous cases, the TS has lower symmetry (Cs) than either the reactants or the
products (Coo,,) as shown in Figure 6.19.
Formic Acid
In Figure 6.21 the [1,3] Sigmatropic reaction of Formic Acid under study is shown.
The reaction is characterized by the migration of the hydrogen with its sigma bond in
a 7r carboxilic, i.e. the migration occurs by a shift in the 7r bonds of this metanoic
acid environment.
The energetical and geometrical evolution of the LTP TS search technique for this
reaction is shown in Table 6.14. The TS is characterized by the migrating proton bonded
to both, the source and migration, oxygens, by a oxygen-hydrogen bond length of r24 =
1.2045 Angstroms. Both oxygens form angles with the carbon and the hydrogen atached
to the carbon of #215 = #315 = 129.82°. As expected, both oxygen carbon bonds have
now the same length, r]2 = rj3 = 1.2960 Angstroms, and the oxygen-carbon-oxygen
angle has narrowed by around 23°.
It is assumed that the mechanism of the reaction is a Symmetry-allowed Suprafacial
sigmatropic shift reaction as reactants, products and TS are all in the same molecular
plane.
The LTP uphill search towards the TS starting from reactants or products (R,P) is
represented in Figure 6.22, where the energy has been plotted against the most relevant
geometrical changes during the simulated reaction.
These geometrical rearrengements convey some changes in the symmetry of the
main structures of this reaction. The TS goes to a lower C2), symmetry in relation to
the Coot, of both reactants and products, as shown in Figure 6.23.

o3
2
O
â–º
c
1
H
5
Reactants
Transition State Products
u>
u>
Figure 6.21. The [1,3] Sigmatropic reaction of formic acid. The initial reactants, the transition state found and initial products
structures are shown. Products are a mirror image of reactants. The geometrical details are given in Table 6.13.

Table 6.14. LTP geometry, energy and number of update iterations (UI) required to find the TS for the formic acid [1,3]
Sigmatropic reaction. The results are those belonging to the exact Line-Then-Plane technique. As reactants (R) and products
(P) had the same geometry and energy we group them as: Q¡ = R ¡ = P¡ , for i = 0, 1, 2, 3, 4 . The optimized geometry (bond
lengths r are in Angstroms and angles 6 are in degrees) in a plane perpendicular to the walk direction at each step, the energy
(Kcal/mol) and the number of update iterations of the Hessian for each cycle are displayed. The convergence criterion requires
that the maximum component of the gradient should be smaller than 10“3. The energy and geometry of the steps in the line are
not shown, but the total number of energy evaluations was 37. The labeling of the atoms is as shown in Figure 6.20.
Energy3
r12
r13
T24
$213
$215
$315
$124
UI
Qo
0.0000
1.3298
1.2451
0.9976
123.15°
112.75°
124.10°
113.17°
0
Qi
15.1640
1.3304
1.2527
1.0238
110.01
123.37°
126.63°
89.93°
4
Q2
32.7422
1.3157
1.2883
1.0764
104.82°
128.68°
126.50°
82.72°
3
q3
38.6872
1.3414
1.2421
1.1181
105.33°
126.19°
128.48°
79.18°
3
q4
47.7016
1.3083
1.2917
1.1656
100.55°
130.20°
129.26°
76.60°
2
Rs
49.2611
1.2960
1.2960
1.2045
100.37°
129.82°
129.82°
74.08°
2
a. The origin of the energy (at Q0) is: E = -41.394497 a.u.

Energy / Kcal/mol
TS
Figure 6.22. The uphill path towards the transition state (TS), starting from reactants (R), for the [1,3] Sigmatropic
reaction of formic acid.

136
E (Kcal/mol)
49.26-
0.00
C
ooV
”rT
\
v
\
\
\
\
\
\
\
\
\
\
\
\
\
\.
C
ooV
Figure 6.23. Schematic representation of the change in energy and symmetry
between the initial reactants (R) and products (P), and the transition state (TS) found
for the Sigmatropic shift reaction of formic acid.
Methyl Imine
This is an interesting isomerization reaction that occurs through 2 main mechanisms
as shown in Figure 6.24.
The first mechanism (a) involves the move, in the molecular plane, of the hydrogen
(H5) attached to the nitrogen passing through a planar C2V- The results for the LTP
search for this mechanism are shown in Table 6.15, indicating that the TS is found at

137
the 7th cycle, in the reactants step with a hydrogen-nitrogen-carbon angle #521 closer to
180° as expected. The other significative change in the geometry is that the hydrogen-
nitrogen bond length is shortened during the reaction as disclosed from Table 6.15.
The second mechanism (b) involves the internal rotation of the imine double
bond. This mechanism is performed through a reaction coordinate chosen to be the
dihedral angle a, as shown in Figure 6.24. This mechanisms occurs when the hydrogen
connected to the nitrogen (H5) comes out of the molecular plane passing through a
maxima at a = 90°. The results for this mechanism have been fitted through an
interpolation procedure [5] and are represented here in terms of the dihedral angle a,
which is used as the reaction coordinate. The results for this mechanism are shown in
Table 6.16, in which only the most relevant geometrical changes have been displayed.
Table 6.16. Internal rotation of methyl imine according to mechanism (b) of Figure
6.23. The TS is founded at a dihedral angle of a = 90°. The labeling of the geometrical
parameters is as shown in Figure 6.23. Notice that this internal reaction is symmetric.
a
Energy3
r25
#125
0° = 180°
0.0000
1.0546
112.72°
0
0
II
0
O
(N
8.8409
1.1026
111.74°
40° = 140°
31.2268
1.2242
109.27°
Os
0
0
ll
to
0
0
56.6831
1.3624
106.46°
0
0
0
II
0
0
OO
73.2985
1.4526
104.62°
so
0
0
75.5774
1.4650
104.37°
a. The origin of the energy (at P0) is: E = —40.490650 a.u.

(a)
N,
r H5
H
TS1
r H
H
H,
H,
H
H
Ci — n2
H.
H
N
(b)
H3 «
-â–º c
H ,
1-; - N2
TS-
H,
H,
U)
oo
Reactants Transition States Products
Figure 6.24. The methyl imine isomerization reaction. The initial reactants, the 2 possible transition states and the initial
products structures are shown for the 2 pathways for this reaction, a) Transition state (TSj) found by the Exact LTP technique,
showing its uphill behaviour, b) A second transition state (TSj) due to the internal rotation of the double bond. Products are
a mirror image of reactants. The geometrical details are given in Table 6.15.

Table 6.15. LTP geometry, energy, and number of update iterations (UI) required to find the TS for the methyl imine
isomerization reaction, according to mechanism (a) of Figure 6.23. The results are those belonging to the exact Line-Then-Plane
technique. As reactants (R) and products (P) had the same energy and almost the same geometry, we group them as: Q¡ =
R ¡ = P¡, for i = 0, 1, 2, 3, 4. The optimized geometry (bond lengths r are in Angstroms and angles 6 are in degrees) in
a plane perpendicular to the walk direction at each step, the energy (Kcal/mol) and the number of update iterations of the
Hessian for each cycle are displayed. The convergence criterion requires that the maximum component of the gradient should
be smaller than 10“3. The energy and geometry of the steps in the line are not shown, but the total number of LTP cycles
was 73. The labeling of the atoms is as shown in Figure 6.23.
Energy3
* 12
r13
r14
•'25
#213
#214
#314
#125
UI
Qo
0.0000
1.2884
1.0941
1.0941
1.0546
122.38°
122.38°
115.24°
112.72°
0
Qi
5.5596
1.2812
1.0870
1.1019
1.0323
121.79°
123.02°
115.20°
131.08°
6
Qz
12.6725
1.2766
1.1025
1.0883
1.0291
123.07°
121.90°
115.04°
142.77°
4
q3
18.1732
1.2737
1.1029
1.0895
1.0289
123.09°
122.00°
114.91°
151.05°
5
q4
22.0346
1.2714
1.1027
1.0910
1.0291
123.08°
122.13°
114.80°
157.22°
5
Qs
29.1952
1.2679
1.0996
1.0969
1.0299
122.83°
122.61°
114.56°
175.56°
6
Qb6
29.4964
1.2679
1.0983
1.0978
1.0298
122.73°
122.68°
114.59°
179.11°
3
RC7
29.5083
1.2679
1.0981
1.0980
1.0297
122.71°
122.70°
114.59°
179.82°
0
a. The origin of the energy (at Q0) is: E = -18.888477 a.u.
b. At this cycle the step factor is reseted to 2.5, according to the LTP algorithm.
c. At this cycle the step factor is amplified by a factor of 2, according to the LTP algorithm.

E (Kcal/mol)
140
75.58 -
29.51 -
0.00 -
R
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
Figure 6.25. Relative energy differences between the two studied mechanisms for
the methyl-imine reactions (not to scale). TS] corresponds to the in the molecular plane
saddle point (mechanism (a)), whereas TS2 corresponds to the out of the molecular plane
saddle point (mechanism (b)), according to the scheme shown in Figure 6.24.
The energetical difference between this two possible mechanisms is shown in Figure
6.25, which indicates that the in the molecular plane TS (TSj through mechanism (a))
is lower in energy than the out of the molecular plane TS (TS2 through mechanism
(b)) by more than 46 Kcal/mol.

141
Thermal Retro [2+2] Cycloaddition Reaction of Oxetane
This mechanism considers two molecules reacting to give products. Particular care
was taken on the preparation of the input file such that the molecular planes of reactants
were not the same but parallel among them, and at a distance of 3.0 Angstroms (rjy
= r2g = 3.0).
In Figure 6.26 the mechanism of the reaction is shown as well as the labeling of
the atoms employed. The results of the LTP uphill search are shown in Table 6.17.
The TS found shows an elongation of the ethane Ci-C2 bond length, from 1.3238
Angstroms at the initial reactants structure to 1.4082 Angstroms at the TS, which is
midway between the double and single bond of ethene and the oxo-cycle, respectively.
The same tendency is observed for the CyOg. This changes are expected as the new
single bonds C\-C-¡ and C2-Os are coming to a distance closer to a single bond, while
the original double bonds migrate to intermediate single-double bonds.
Our results, obtained using the Restricted-Hartree-Fock method, allow us to con¬
clude that the reaction occurs in a concerted fashion, that is, the formation and breaking
of bonds occurs simultaneously, as no intermediate of reaction was found.
Step Size Dependency
The symmetric inversion reaction of ammonia has been used to study the variation
on the number of SCF calculations that are necessary to find the TS. We have also
focused our attention on the dependence of the convergence to the TS geometry with
the step size as the Line-Then-Plane algorithm was stepping up-hill. Regardless the
step sizes (N) taken, the accuracy of the TS found was kept as the energy, the nitrogen-
hydrogen bond length and the hydrogen-nitrogen-hydrogen angle were kept constant,
as reported in Table 6.18.

â–º
i
c
2
H
9
I
C8
/I
/
/
//
ZJ
H9 H10 H H
Reactants
Transition State
Products
Figure 6.26. The thermal retro [2+2] cycloaddition reaction of Oxetane mechanism. The initial reactants, the transition state
found and initial products structures are shown. The geometrical details are given in Table 6.17.
142

143
Table 6.17. LTP energy (Kcal/mol), geometry (bond lengths r are in Angstroms and
angles 6 are in degrees) and the number of update iterations (UI) required to find the
TS for the thermal retro [2+2] cycloaddition reaction of Oxetane. The results belong to
the Exact LTP technique. The convergence criterion was that the maximum component
of the gradient and the displacement vector should be smaller than 10-3. The energy
and geometry of the steps in the line are not shown. The total number of LTP cycles
was 47. The labeling of the atoms is as shown in Figure 6.26.
Energy3
Idl
Fl2
r78
r17
r28
#287
UI
Ro
336.6424
3.1202
1.3238
1.2366
3.0000
3.0000
0
Po
0.0000
1.5007
1.3849
1.5000
1.3842
88.65°
0
Ri
337.6691
2.4962
1.3403
1.25.12
2
Pi
49.9408
1.4819
1.3699
1.6498
1.5459
88.87°
2
r2
340.6801
1.9969
1.3537
1.2629
2
P2
135.2269
1.4671
1.3579
1.7697
1.6753
89.07°
2
r3
344.3734
1.5976
1.3646
1.2723
2
P3
215.0582
1.4553
1.3483
1.7788
89.23°
2
r4
3480924
1.2781
1.3733
1.2799
2
p4
280.1598
1.4459
1.3345
2
Rs
351.5258
1.0224
1.3804
1.2859
3
Ps
329.8571
1.4385
1.3345
3
r6
354.5469
0.8178
1.3861
1.2907
2
P6
364.1506
1.4325
1.3296
2
Rb7
365.5244
0.1636
1.4045
1.3063
2
p7
371.1769
1.4138
1.3140
2
RC8
367.8363
1.4082
1.3094
0
a. The origin of the energy (at P0) is: E = -40.490650 a.u.
b. At this cycle the step factor is reseted to 2.5, according to the LTP algorithm.
c. At this cycle the TS is found in the Reactants step. LTP stops.

144
Table 6.18. Number of LTP iterations, SCF calculations, and variation on the
energetics and structure as function of the step size (N) for the symmetric inversion
reaction of ammonia. For all cases the convergence was the same as detailed above.
Energy is in Hartrees, the nitrogen-hydrogen bond length is in Angstroms and the
hydrogen-nitrogen-hydrogen angle (a) is in degrees.
N
LTP-Its
Iterations
Energy
fNH
a
5
5
35
-12.507077
1.0380
120°
6
5
27
-12.507075
1.0376
120°
7
5
29
-12.507077
1.0381
120°
8
6
31
-12.507077
1.0380
120°
9
5
31
-12.507077
1.0380
120°
10
6
29
-12.507077
1.0381
120°
11
7
39
-12.507076
1.0378
120°
12
6
35
-12.507077
1.0381
120°
Figures 6.27 and 6.28 show the variation on the number of SCF calculations and the
change of the nitrogen-hydrogen bond length as a function of the step size, respectively.
The first one shows a wide safe range: 7 < N < 10 maintain the accuracy in the TS
found. On the other hand, for the nitrogen-hydrogen bond length change, it becomes
clear that there is not a big error for values of N smaller than 7 and bigger than 10.
However, it seems that this range is a safe one for a good answer. The same can be
inferred for the number of SCF calculations required to locate TS.

Step Size (N)
Figure 6.27. Number of SCF calculations as a function of the step size (N) for the Symmetric Ammonia Isomerization reaction.

Figure 6.28. Nitrogen-hydrogen bond length (r(N-H)) change as a function of the step size (N) for the Symmetric Ammonia
Inversion reaction.

147
Hammond Adapted LTP Results
When the Hammond-adapted techniques were applied to Quapp’s potential function
(Table 6.6), it was clear that a considerable reduction in the effort (i.e. number of
function evaluations), to reach the TS was achieved. In particular, the approximate-
restricted-Hammond-adapted LTP technique had a considerable reduction (2/3) in the
number of energy evaluations, while maintaining the same accuracy as the exact (LTP)
procedures.
In this section, the LTP technique is compared against the Hamond-adapted
(HALTP), and the two restricted Hammond-adapted LTP (RHALTP I (clamped ini¬
tial reactants) and RHALTP II (clamped initial products)) procedures through the study
of the asymmetric NH3 isomerization reaction. These algorithms are also compared
with the augmented Hessian model. The results displayed in Table 6.19 indicate that
the geometry and the energy of the TS found is kept almost constant by all methods.
The HALTP procedure reduces the number of SCF calculations aproaching the num¬
ber required by AH. On the other hand, RHALTP I requires the same amount of SCF
calculations (7) as AH does. However, RHALTP II requires less effort than any of the
other techniques studied, as it requires only 6 SCF iterations.
Summary of Results
The results of the symmetric inversion of ammonia derived from the application of
the Approximate-LTP method raise the question of whether it would not be cheaper to
use the Approximate method and optimize the TS found instead of using the exact LTP.
In Table 6.11 we see that for the TS search on the symmetric inversion of ammonia
reaction, LTP needs 22 SCF iterations, while the App-LTP + GOPT requires only 13
SCF iterations. This outcome implies that around half of the effort is required by the

148
Approximate-LTP technique to give roughly the same answer. The same behavior is
observed for the Asymmetric inversion of ammonia reaction, according to the results
displayed on Table 6.11.
It seems then that a full line search optimization in the perpendicular direction is not
worthwhile and that a search using an Approximate method (like App-LTP) will be close
enough to the right TS, with a simultaneous substantial reduction on the computational
effort. This reduction becomes even more significant for bigger systems.
This tendency has been already observed and examined by Zerner [17, 18]. The
present work confirms, hence, their observations. The TS found by LTP (when no
approximations are involved) generally does not require any optimization.
As shown in Tables 6.9 to 6.11, the change in the geometry and the energy
of the TS located by LTP for the different types of NH3 reactions studied, do not
improve significantly after optimization (the difference in energy goes down only by
approximately 10% of an already small deviation). The improvement that results from
optimization requires only a few (2 or 3) extra SCF caculations.
It is interesting to note that, in the approximate methods, the main improvement is
related to the substantial reduction of the number of SCF iterations necessary to find
the TS. This is the purpose of designing these methods. It is also clear that, regardless
of the technique that is used, there are no significant changes in any of the relevant
parameters measured and displayed in Table 6.16, that is, the accuracy is kept constant.
The difference between energies and bond lengths obtained with the Augmented Hessian
and the LTP family of procedures are associated whith the inherent use of analytical
second derivatives by the former.
From the Hammond-Adapted and the Restricted-Hammond-Adapted LTP algo-

Table 6.19. Geometry, energy, convergence criterias and number of iterations required to find the TS for the asymmetric
NH3 isomerization reaction. We show results obtained using LTP, Approximate LTP, Hammond Adapted LTP (HALTP), the two
cases of the Restricted Hammond Adapted LTP (RHALTP I and RHALTP II, exact ones) and the Augmented Hessian technique,
r represents the nitrogen-hydrogen bond length (in Angstroms), 0 is the hydrogen-nitrogen-hydrogen angle (in degrees), E is the
energy (in Hartrees) and |g| is the norm of the gradient achived after convergence to the TS. The convergency criteria require
the norm of the gradient and the displacement vector, to be smaller than 10“3. Finally the number of SCF Iterations (energy
evaluations) required by each procedure is also displayed.
LTP
App-LTP
HALTP
RHALTPIa
RHALTP IIb
AH
r
1.0364
1.0367
1.0364
1.0362
1.0362
1.0382
6
0
0
120°
0
0
to
0
0
120°
120°
E
-12.507062
-12.507079
-12.507062
-12.507070
-12.507075
-12.507084
Igl
3.71 * 10'5
4.51 * 10‘5
3.10*10"5
5.13*10“5
5.13*10'5
1.00* 10-5
Its
10
10
8
7
6
7
a. Reactants coordinates are kept constant (initial ones) along all the search.
b. Products coordinates are kept constant (initial ones) along all the search.

150
rithms whose results are shown in Tables 6.6 and 6.16, it becomes clear that these
algorithms represent a substantial reduction in the number of energy evaluations nec¬
essary to find the TS involved. Moreover, in the molecular case of the asymmetric
inversion reaction of ammonia, the RHALTP I is seen to be as competitive as the AH
model, whereas RHALTP II is less expensive. Note that for all LTP techniques, the
structure and energy of the TS found showed no appreciable deviation among each
other. For all the molecular systems studied here, our calculations shows LTP to be
a reliable TS search algorithm.
The same INDO Hamiltonian, basis set and optimization techniques have been used
for all the procedures tested here. Furthermore, they are comparable to those procedures
that employ it, as the Augmented Hessian method, in terms of speed (number of SCF
Iterations), accuracy of the calculated energy and geometry of the found TS.
Model Potential Function for ARROBA
Introduction
With the purpose of using the LTP technique features for search in perpendicular
directions, a procedure to optimize geometries was developed (ARROBA) that was
described in the preceding chapter. The algorithm has been tested on a potential energy
function and on the H2O molecule. The dependence of its accuracy and convergence
on the line search technique parameter (a). The results presented here are, although
preliminary, are promising and encouraging.
Model Potential Function
The behavior of our procedure was tested by means of its application to the
same model potential function that contains a minimum, proposed in the preced¬
ing section and shown in Figure 6.6. We have choosen the initial geometry to

151
be at the point (0,0), from which a new point was generated and the search for
a minimum started. Figure 6.29 shows this potential energy surface and the be¬
havior of the procedure. The minima are located at (+/- 0.7071,-/4-0.7071) with
an energy of -0.5000 . We have used, for the perpendicular search, a = 0.2 for
the first cycle and 1.00 for the rest of the cycles. Both the Exact and Approxi¬
mate ARROBA techniques were checked. It is interesting to notice that both pro¬
cedures, represented by a cross in the contour plot, walk downhill together using the
same coordinates (positions) and direction, finding the minimum in the 10th iteration.
The exact method uses a total of 12 energy evaluations whereas the approximate one
uses only 10. The difference is due to the BFGS update technique used by the Exact
method.
The comparison of Exact and Approximate methodologies seems to indicate, as
was previously observed by Zerner [17, 18], that the effort of updating the Hessian
(or the use of more sophisticated techniques) is not always imperative. For several
cases, approximate algorithms are good enough, providing accuracy is used in the final
stages of the search.
Step Size Dependence
This geometry optimization procedure has among its outstanding features the
automatic generation of steps and direction. There is no constraint for the step as
each new projected point is generated using the line search technique (LST). For this
generation, a Newton-Raphson-like step is taken and the second derivative matrix is
replaced by the projector itself.
We have studied the dependence, and accuracy, of the number of energy evaluations
as a function of the LST parameter (a), which is not related to the LTP step parameter

E(x,y)
Figure 6.29. Potential energy surface with a minima EMin(x,y) = x4 + y4 + 2xy. The starting point is qo(0, 0)
and the minimum founded at qMin(-0-7071, 0.7071), with an energy of E(q0) = -0.5000. All units are arbitrary. Notice that
both procedures (exact and approximate ARROBA) represented by a cross in the contour plot walk downhill together using the
same coordinates and direction, finding the minimum at the 10th iteration.

153
(N), using only the Approximate ARROBA method. Our results are displayed in Table
6.20 and graphically shown in Figure 6.29. The accepted values for the minima of this
function are E = -0.5000 and (x,y) = (+/-0.7071, -/+0.7071) arbitrary units.
Summary of Results
The ARROBA procedure for geometry optimization is accurate in finding the
minima for a wide range of values of alpha. The exception is defined by values of
a in the vicinity of a = 0.3, as can be inferred from Table 6.16. In this range of a
values, the deviation for the energy and coordinates are AE = 0.0036, AX = 0.0344
and AY = 0.0015. The study of the dependence of the step size on a was aimed at
establishing a range of values of alpha for which not only a good response from the
algorithm, in terms of both the energy and the coordinates, but also a reduced number
of energy evaluations that speeds up the calculations was achieved. According to our
research, a convenient, and safe, range of alpha’s to consider will be: 0.18 < ex < 0.25.
Values of a = 0.20 and 0.21 are recomended, as they not only give the best energy
and coordinates to find the minima, but also a small number of energy evaluations, as
is shown in Figure 6.30.
Molecular Case for ARROBA: Water
This simple molecular system has been used as a starting point to test the algorithm.
The results are shown in Table 6.21.
We started with the laziest geometry of 1.0000 Angstrom for the hydrogen-oxygen
bond lengths and an HOH angle of 90°, which corresponds to the structure labeled as
q2 (originally qi) in Table 6.21. As the algorithm starts the relabeling of the initial
geometries is performed according to their relative energies. One of the main features

154
of ARROBA is shown when we note that both hydrogen-oxygen bond lengths are not
identical, but that they become of the same length as the minimum is found.
Next, the goodness of the procedure is examined by comparing it with the steepest
descent technique in Ab-initio and Semi-empirical program packages, in terms of the
quality of the minimum found (geometry) and number of cycles required to converge to
a minimal energey geometry. Table 6.22 shows results for the geometry optimization of
water (all with the same input geometry) ACES II [83] (through a variety of Ab-initio
basis sets), ZINDO and ARROBA. Also displayed is the experimental geometry.
Table 6.20. Variation on the number of energy evaluations (Cycles) with respect to
the line search technique parameter a for the Approximate-ARROBA technique. Also
displayed are the energy and coordinates of the minima found at each value of alpha.
In the last, framed, line of the table we display the values of energy and coordinates
that are expected for the minimum. All units are arbitrary.
Q
Cycles
E(x,y)
X
Y
0.10
36
-0.5000
-0.7070
0.7069
0.15
22
-0.5000
-0.7071
0.7071
0.17
18
-0.5000
-0.7071
0.7071
0.18
16
-0.5000
-0.7071
0.7071
0.19
14
-0.5000
-0.7071
0.7071
0.20
12
-0.5000
-0.7071
0.7071
0.21
11
-0.5000
-0.7074
0.7071
0.22
15
-0.5000
-0.7072
0.7071
0.23
16
-0.5000
-0.7071
0.7071
0.25
14
-0.5000
-0.7069
0.7068
0.30
25
-0.4964
-0.7415
0.7086
Expected3
-0.5000
-0.7071
0.7071
a. This are the energy and coordinates accepted values for the minimum found. The other,
symmetric, minimum for potential energy function is at (x,y) = (0.7071,-0.7071) with the same energy.

155
Table 6.22 indicates that despite the better overall geometry obtained by the
Ab-initio calculations, particularly from the Double Zeta Polarization-Dunning basis
set (DZP-DN), the semi-empirical geometry obtained by ZINDO is very good when
compared with the experimental one. On the other hand, the optimized geometry
coming from ARROBA is certainly encouraging.
We conclude that, despite the apparent extra cost of ARROBA because of the
consecutive elimination of a given direction by perpendicular projection which implies
more SCF calculations, its simplicity is appealing. The results from ARROBA are
indeed encouraging, as it is well known that steepest descent works well when going
downhill for steep slopes, but that along the valley of the minima shows a numerical
disadvantage known as the zigzagging across the valley ground line [13, 88]. This
seems no to be the problem with ARROBA because of its construction.
Table 6.21. Cycle-by-cycle downhill walk in the search for minimum of H2O using
the Approximate ARROBA technique. Energy (Hartrees), geometry (oxygen-hydrogen
bond lengths (r, in Angstroms) and the hydrogen-oxygen-hydrogen angle (0, in degrees))
and the maximum component of the gradient (gmax) at each cycle are displayed. The
convergence criteria was that the displacement vector norm to be smaller than 10—3.
Energy
rOH2
rOH3
#HOH
9max
qi
-18.101384
1.0000
1.0800
90.00°
0.14890
Q2
-18.113190
1.0000
1.0000
90.00°
0.03385
Q3
-18.115546
0.9910
0.9902
92.84°
0.02926
Q4
-18.112815
1.0231
1.0279
92.25°
0.06088
Q5
-18.121658
0.9951
0.9956
102.77°
0.00806
Q6
-18.121647
0.9981
0.9966
102.71°
0.01186
Q7
-18.121975
0.9919
0.9929
104.71°
0.00917

40
35
30
25
20
15
10
5
imt
suit
V)
1 1 i i i
0.1 0.15 0.2 0.25 0.3
Alpha
r of energy evaluations required to find the minimum, as a function of the line search technique
corresponds to the approximate ARROBA technique applied to the potential energy surface with a
= x4 + y4 + 2 xy.

157
Table 6.22. Equilibrium geometry for H2O. We compare SCF results at different
basis sets using the ACES IIa (first six calculations) and ZINDO programs and compare
them with experiment. Here the ARROBA results corresponds to the Approximate
technique, r is the hydrogen-oxygen bond length (in Angstroms) and 6 is the hydrogen-
oxygen-hydrogen angle (in degrees)
r
e
Cycles
STO-3G
1.0135
97.28°
4
4-31G
0.9750
108.93°
4
6-31G*
0.9685
104.00°
4
6-31G**
0.9608
103.87°
4
DZP-DN
0.9624
104.44°
4
DZP-DIF
0.9639
105.02°
4
ZINDO
0.9952
106.74°
5
ARROBA
0.9924
104.71°
7
Experimentb
0.9573
104.50°
a. ACES II is an Ab-initio program package whose main features were described in chapter 3.
b. Results taken from Tables 6.5 and 6.6 of reference [85].

CHAPTER 7
CONCLUSIONS AND FUTURE WORK
We have reviewed some of the most commonly used algorithms for Transition
State (TS) and Geometry Optimization (GOPT) searches and their implementation in
various available program packages (chapters 2 and 3, respectively), with discussion
of their main features. We have pointed out their disadvantages, mainly related to
cost (computer time) associated with the number of energy and/or Hessian evaluations
required or to the fact that one needs to identify a suitable reaction coordinate. Other
procedures need a large number of moves, or a good initial guess of the TS, whereas
several others fail because of the large number of iterations required. In the case of
geometry optimization, the problems, although fewer than for TS search, are similarly
related to the size, initial guess, Hessian evaluations, and number of iterations required.
We suggest that for a model to be successful in the search for the TS, both R and
P should be considered. The LTP procedures suggested here, including the Hammond
Adapted ones, are based on this idea. For GOPT the LTP search features also are
very convenient.
The main difference between the ideas suggested here and the procedures reviewed
in this work, are based and defined by: the simultaneous consideration of products and
reactants, a constrained step size, the simultaneous walking along a line connecting R
and P towards the TS, the energy difference (AE) between the starting structures and a
continuous downhill walk for GOPT, according to a zigzag type of search.
158

159
The concept of considering R and P approaching the TS simultaneously from both
sides of the reaction is mainly based on the idea that each structure contains information
and history about the other (having in common the same TS). The trajectory drawn in
this way can be thought of as the lowest passage from one side of a mountain (R) to
the other (P). The advantage of this idea is that we look to the mountain from both
sides, and not from one. The GOPT algorithm was developed among the same lines,
for which a second set of coordinates is automatically generated allowing for a fast
and insured downhill walk.
The procedures investigated in this work are all characterized by the fact that no
guess of a TS is needed and no evaluation of the second derivative matrix (Hessian) is
required. This is the most important contribution of this work in comparison with the
procedures reviewed in Chapter 2. Moreover, the Hammond-Adapted LTP (HALTP)
procedures require a smaller number of calculations, still finding the TS with the
same accuracy. Because of their simplicity they are very versatile. Avoiding the time
consuming evaluation of the Hessian has been a general goal of this study.
Table 7.1. Summary of general properties and advantages of the Line-Then-Plane
Technique for finding Transition States and to Optimize Geometry
Transition State search Geometry Optimization
• No guess of TS is needed
• R and P are needed
• Reduced number of calculations
• Simple up-hill walk
• Considers intermediate of reaction
• Path is easily approximated
• Updated Hessian procedure
Only 1 initial geometry is needed
Input can be a bad one
Simple down-hill walk
Reduced number of calculations
Updated Hessian procedure

160
Because of the advantages shown in Table 7.1, the procedures presented here will,
at least in part, avoid some of the problems usually found in a TS search.
The procedures proposed in this work may fail only in cases characterized by a very
steep reaction path. In those cases, it appears to be necessary to reconstruct more of the
reaction path to ensure that the TS has been found and to reproduce the reaction path
more accurately. This can be pursued by connecting the found first-order saddle point
with reactants and products by means of a down-hill procedure, such as the steepest-
descent technique (despite the inherent extra cost), especially if one is interested in
kinetic aspects of the reaction. However, as has been advised at the beginning of this
work, lower energy saddle points may exist (that can be found by means of several
different procedures), as well as the many procedures to find them.
Although our procedures are simple and were constructed by inspection of a general
potential energy surface, we have built up a strategy to find the TS that has the advantage
of using a reduced number of calculations, has a simple and convenient writing of
projected coordinates, updates the Hessian matrix, considers an intermediate of reaction,
and involves the idea of finding the TS(s) starting simultaneously from R and P.
We believe that the new LTP TS and geometry optimization search procedures are
well behaved and simple. It appears clear that more insight has to be obtained from
the use of this procedures for the evaluation of bigger and more complicated molecular
systems. We also foresee the inclusion of solvent effects in the TS searching, in order
to analyze how the solvent influences the reaction mechanism.
On the other hand, work on an updating procedure of the projector is needed,
mainly because as the initial guess in the perpendicular directions to the displacement
vector is for now the projector itself. An update of it will speed up the Hessian update
procedure in use.

161
For the geometry optimization procedure, we believe our results to be very promis¬
ing. Some analysis is yet to be done on the line search parameter in order to construct
a step for the perpendicular search adapted to the surface. On the other hand, it can
be argued that ARROBA cannot compete with steepest descent techniques, however
we remind that these techniques work very well for steep slopes but that they show a
zigzagging behavior across an ample flat valley, which is a numerical disadvantage. We
beleive that our only gradient technique, because of its projected features, overcomes
this problem.
To accelerate convergence to maxima or minima, it can be argued that after a couple
of movements, or when a change in the slope is detected, one should use the calculated
projected points and fit them to a parabola to find the saddle points (the vertex of the
parabola). This procedure has two inconveniences: first when we actually tried this
in the model potential functions, the fitting gives the wrong answer either for maxima
or minima mainly because the potential energy surface is somehow tortuous and not
absolutely symmetric. Secondly, this fitting will eliminate the inherent beauty of the
acceleration mechanisms of the LTP procedures.
Finally, it has become evident along the years, and according to the large amount
of work devoted to tackle the search for maxima and minima, that chemical intuition
in this kind of problems is always required. Unfortunately, if it fails one is lost in the
hypersurface of algorithms with no saving recipe available. Consequently, one should
handle it with care.

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BIOGRAPHICAL SKETCH
Cristian is the second son of Eduardo Cardenas-Diaz and Helene Lailhacar-Rochett.
He was born in October the 25th, 1957, in Santiago, Chile.
In his earliest memories he recalls the life in the countryside farm of his grandfather
Luis Lailhacar-Lapeyre (who died too soon for Cristian, who continues to miss him),
where he began with his sister and brother, to love nature, the wind, wide open spaces,
life. At a very early age, he became interested in small creatures; insects soon became
a passion. At 13 years old, he started to study entomology in the National Museum
of Natural History in Santiago. Biology was then very appealing but not physics, and
of course mathematics was out of the question. Although chemistry was not an early
passion, he always felt a particular interest in atoms and their structure. Consequently,
from the early studies of Leonardo Da Vinci, his hero by default, the history of
chemistry, and the atomic bomb soon became an irresistible attraction. Scientists such
as Albert Einstein, Max Planck, Madame Curie, Heisenberg, Ehrenfest, Desiderio Papp
(a Chilean scientist), and others started filling his mind with fascinating ideas and their
stories.
All this turned young Cristian to chemistry, but the implicit good background
in physics and mathematics required was too much and, moreover, his high school
grades in these subjects were not just bad, but terrible. With almost a colossal effort
he overcame this natural barrier, studied chemistry and soon became an assistant
professor of calculus. He decided to write his master’s thesis in quantum chemistry
169

170
at the University of Chile. This opened unimaginable doors for Cristian, who married
Alejandra, his girlfriend of more than 10 years. Then in 1991 he came to the University
of Florida to work on his Ph.D. in chemistry, under the scientific supervision of Dr.
Michael C. Zerner, a good friend.
Uife has been good to Cristian, who has found his marriage blessed with two
beautiful daughters, Francisca and Catalina, and his loving wife Alejandra. Friends
and family have always been a vital part of his life and they have always been there
for him and his family. Being a Catholic, he has always felt a deep devotion to the
church, but for national (Chilean) historical reasons, it has always been the Virgin of El
Carmen, patroness of Chile, to whom he asked for help in hard times and his prayers
for everything in life were devoted.

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Michael C. Zern^ry,Chair
Professor of Chemistry
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
William Dolbier
Professor of Chemistry
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Professor of Chemistry
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
N. Yngve (phrn
Professor of Chemistry

I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
Samuel B. Trickey
Professor of Physics
This dissertation was submitted to the Graduate Faculty of the Department of
Chemistry in the College of Liberal Arts and Sciences and to the Graduate School
and was accepted as partial fulfillment of the requirements for the degree of Doctor
of Philosophy.
August 1998
Dean, Graduate School