Localized orbitals in chemistry


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Localized orbitals in chemistry
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viii, 79 leaves : ill. ; 28 cm.
Culberson, John Christopher
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Subjects / Keywords:
Molecular orbitals   ( lcsh )
Rare earth metals   ( lcsh )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
bibliography   ( marcgt )
non-fiction   ( marcgt )


Thesis (Ph. D.)--University of Florida, 1987.
Bibliography: leaves 73-78.
Statement of Responsibility:
by John Christopher Culberson.
General Note:
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University of Florida
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Full Text








I would like to thank my parents for their support and guidance

throughout my life. Mary Kay your inspiration, infinite patience and

willingness to wait kept me going.

Thanks go to Bill Luken who taught me the basics.in quantum

chemistry as well as introducing me to the computer as a learning tool.

In addition, Bill Luken gave me an insight into the academic world.

Finally, I would like to thank Bill and Marge for being our friends.

I would like to thank Michael C. Zerner for allowing me to use the

skills I learned at Duke and teaching me more quantum chemistry. The

freedom he gave me to explore some of my own ideas as well as being

guided occasionally was deeply appreciated. The entire Zerner family

made our time at QTP enjoyable.

Thanks go to my German host Dr. Notker Rosch for allowing me to come

to Germany. My thanks to Peter and Monica Knappe for helping Mary Kay

and me during our entire stay in Germany. We would like to thank Frau

Brown for making us feel at home.

One benefit of being a graduate student at the Quantum Theory

Project is the wide variety of people you meet. One of the most

enlightening experiences was to meet and take classes from Dr. N. Y.

Ohrn. Thank you for giving me a new perspective on quantum chemistry.

I would like thank to G. D. Purvis III for allowing me to help in

designing the C3D program and giving me plenty of experience

debugging/expanding the INDO code once a day. Your persistence in

asking the question "Well why do you want to do that?" help me formulate

problems more completely.

No graduate student can ever learn about life in a large research

program without a great post-doc to help him or her along. Dan Edwards

gave me a handle, provided constant assistance, and is a friend to talk


It has been great to be a member of QTP and share in the wealth of

experiences common only to QTP. The Sanibel symposium provided a chance

to meet some of the most unique people in the word. I would like to

thank all of the members of QTP, especially the secretarial staff, for

making my stay here great.

Last but not least thanks to the boys and girls of the clubhouse.

Thanks go to Bill reminding me that learning something does not have to

be boring. Thanks go to Charlie reminding me that you don't understand

something until you can explain it to someone else. Thanks go to Alan

showing me that some theory can still be done on a piece of paper. All

of the members of the clubhouse have provided me with an atmosphere

conducive to the free exchange of ideas on quantum theory and everything




ACKNOWLEDGMENT........................................... ii

LIST OF TABLES........................................... v

LIST OF FIGURES.......................................... vi

ABSTRACT................................................ vii

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


Background ........................... 3

Double Projector Localization........ 6

Fermi Localization................... 9

Boys Localization.................... 34


Background........................... 41

Model................................ 43

Procedures............................ 61

Results.............................. 62

CONCLUSION............................................... 71

BIBLIOGRAPHY............................................. 73

BIOGRAPHICAL SKETCH...................................... 79


1-1 Probe electron points for furanone............. 18

1-2 Boys and Fermi hole centroids for C4H402-....... 19

1-3 Probe electron points for methlyactetylene..... 22

1-4 Boys and Fermi hole centroids for CHCH3........ 23

1-5 Orbital centroids for BF3...................... 26

1-6 Eigenvalues and derivatives for BF3 using the
Boys method .................................... 27

1-7 Probe electron points for BF3.................. 28

1-8 Orbital centroids for BF3...................... 29

2-1 Basis functions for Lanthanide atoms............ 51

2-2 Ionization potentials for Lanthanide atoms..... 54

2-3 Average configuration energy for Lanthanides... 57

2-4 Resonance integrals for Lanthanide atoms....... 58

2-5 Geometry and ionization potentials for Cerium
and Lutetium trihalides......................... 63

2-6 Geometry and ionization potentials for
Lanthanide trichlorides........................ 65

2-7 Geometry and ionization potentials for SmC12,
EuCl2 and YbC 2......................... .. 67
2-8 Geometry of Ce(NO3) 6 ion...................... 68

2-9 Population analysis of Ce(N03 62 ion........... 70
3 6


1-1 Fermi mobility function for H2CO................ 12

1-2 Difference between mobility function and
electron gas correction......................... 13

1-3 Fermi hole plot for formaldehyde................ 14

1-4 Boys localized orbital for formaldehyde......... 15

1-5 Ni(CO)4 bonding orbital......................... 38

1-6 Ni(CO)4 non-bonding orbital..................... 39

1-7 Ni(CO) anti-bonding orbital..................... 40

2-1 Single and double C basis set plot.............. 49

2-2 Average value of r versus atomic number......... 50

2-3 Pluto plot of Ce(NO 2......................... 69

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



John Christopher Culberson

May 1987

Chairman : Michael C. Zerner
Major Department : Chemistry

The localized orbitals discussed here will be divided into two

classes: (1) intrinsically localized orbitals, where the localization is

due primarily to symmetry or energy considerations, for example

transition metal d-orbitals or lanthanide f-orbitals; and (2) orbitals

which must be localized after a self-consistent field (SCF) calculation.

In the latter case, two new methods of localization, the Fermi and the

double projector methods, are presented here. The Fermi method provides

a means for the non-iterative localization of SCF orbitals, while the

double projector allows one to describe what atomic functions the

localized orbitals will contain. The third localization procedure

described is the second order Boys method of Leonard and Luken. This

method is used to explain the photodissociation products of Ni(CO)4.

The Intermediate Neglect of Differential Overlap (INDO) method is

extended to the f-orbitals, and the intrinsic localization of the f-

orbitals is examined. This extension is characterized by a basis set

obtained from relativistic Dirac-Fock atomic calculations, and the

inclusion of all one-center two-electron integrals. Applications of

this method to the lanthanide halides and the twelve coordinate
Ce(NO3)62 ion are presented. The model is also used to calculate the

ionization potentials for the above compounds. Due to the localized

nature of f-orbitals the crystal field splitting in these compounds are

extremely small, leading to SCF convergence problems which are addressed

here. Even when the SCF has converged, a small configuration

interaction (CI) calculation must be done to insure that the converged

state is indeed the lowest energy state. The localized nature of the f-

orbitals in conjunction with the double projector localization method

may be used to isolate the f-orbitals in order to calculate only a CI

restricted within the f-manifold.



Localized orbitals may be defined as either orbitals which are

spatially compact or as molecular orbitals which are dominated by a

single atomic orbital. The use of the terms bond, anti-bond, or lone

pair to describe a set of orbitals are all based on a localized orbital

framework. The use of ball and stick models and hybrid orbitals in every

general chemistry class illustrates the power of localized orbitals as

an aid in the understanding of molecular structure.

Localized orbitals may be divided into two categories. The first

category encompasses orbitals which must be localized. Although, some

orbitals are localized automatically either by their symmetry or by

their energy relation to other orbitals in the molecule, these orbitals

form the basis for the second category of localized orbitals.

Transition metal d-orbitals fall into this second category, and it has

been predicted that the lanthanide f-orbitals should also fall into this


Our understanding of transition metal chemistry is also based on the

concept of orbitals being localized. The excitations that give rise to

the colors of many metal complexes are classified as d-d, ligand-d or

charge transfer. These classifications are based on the fact the d

orbitals are localized allowing for the easy interpretation of spectra.

The success of crystal field theory reinforces the belief that the d-

orbitals are localized.

In the following two chapters, I will examine both of these types of

localized orbitals. The first chapter will deal with methods developed

to obtain localized orbitals from delocalized orbitals, and the use of

such methods on several systems of chemical interest. Chapter two deals

with expanding the INDO method so that the prediction of f-orbitals

being localized orbitals may be verified and so that the unique bonding

and spectroscopy of these compounds may be examined. By adapting the

INDO method, we may now expand our studies to include the chemistry of

the lanthanides and actinides.

The chemistry of the lanthanides and actinides is different from the

chemistry of the corresponding d-orbital chemistry. The compact

(localized) nature of the f-orbitals, causes the f-f spectral

transitions to be characterized by very sharp transitions and the

positions of the transitions are almost unaffected by the ligands

attached to the metal. The f-orbitals are potentially involved in

expanding the valence of lanthanide containing compounds; some

lanthanide molecules have a coordination number of nine and several

twelve coordinate lanthanide compounds are known. Are f-orbitals

required for greater valency, or is the greater valency merely a

consequence of the larger ionic radius of most lanthanides? The study

by quantum chemical methods has been slowed by the size of the

lanthanide containing molecules, but the INDO method lends itself to the

study of large molecules and therefore the choice was made to expand the

INDO model to include f-orbitals.




The observable properties of any wavefunction composed of a single

Slater determinant are invariant to a unitary transformation of the

orbitals occupied in the wavefunction.1 Because of this invariance, the

observable properties of a closed-shell self-consistent field (SCF)

wavefunction may be described using canonical orbitals, or any set of

orbitals related to the canonical orbitals by a unitary transformation.

Canonical orbitals are quite useful in post-Hartee-Fock calculations

for several reasons. Canonical molecular orbitals (CMOs) are obtained

directly by matrix diagonalization from the SCF procedure itself. The

canonical orbitals form irreducible representations of the molecular

point group. Since the symmetry is maintained, all subsequent

calculations may be simplified by the use of symmetry. Spectroscopic

selection rules are determined using the canonical orbitals. Koopman's

theorem, which relates orbital energies to molecular ionization

potentials and electron affinities, is based entirely on the use of

canonical orbitals.

Localized orbitals (LMOs) allow for the wavefunction to be

interpreted in terms of bond orbitals, lone-pair orbitals and inner-

shell orbitals, consistent with the Lewis structures learned in freshmen

chemistry. Unlike CMOs, LMOs may be transferred into other

wavefunctions as an initial gu ss, thereby reducing the effort needed to

produce wavefunctions for large molecules. The most important use of


localized orbitals is their ability to simplify configuration

interaction (CI) calculations. LMOs maximize intra-orbital electronic

interactions and minimize inter-orbital electronic interactions. This

concentrates correlation energy into several large portions instead of

many small portions as given by the CMOs. A major disadvantage of the

use of localized orbitals is the loss of molecular point group symmetry.

Localized orbitals do not transform as an irreducible representation of

the molecular point group. The total wavefunction, of course, does.

Localization methods may be divided into several categories. The

first category of localization is based on an implicit definition of

what a localized orbital should be. An underlying physical basis for

localized orbitals is exploited in the second category of localization.

Localized orbitals may also be produced in accord with the users own

definition of localization.

The implicit definition on which the localized orbitals are produced

differs from method to method but all of these methods proceed in a

similar fashion. A function of the form

G = E , (1-1)
i=l ii

is maximized or minimized, where the definition of depends

on the localization criterion. One choice for the value of

is the two-electron repulsion integrals;2-3 for this

choice the sum G is maximized. This method is referred to as the

Edmiston Ruedenberg (ER) method. Perhaps the most popular choice of a

localization method is the Boys method, in which the g operator is the

orbital self-extension operator,6

gii = 2 (1-2)

This form of the g operator may be recast in terms of the product of two

molecular orbital dipole operators. One can relate this form of

localization to maximizing the distance between the orbital centroids.

Once a localization criterion has been established, the next step is

to construct a transformation matrix to do the localization. Since the

exact nature of the transformation is unknown, an iterative procedure is

used to construct the localized orbitals. This iterative procedure

moves from a less-localized set of orbitals to a more-localized set.

Once a convergence criterion is met i.e., the orbitals do not change

within a given tolerance, the iterative procedure is stopped.

Although localizations using either of the above two methods are

relatively standard, some problems may be encountered. As with any

iterative procedure, convergence difficulties may be encountered. In

the case of the ER method all two-electron repulsion integrals must be

transformed on each iteration, a very time consuming step proportional

to N5. Since the SCF procedure itself proceeds as N (semi-empirical)

or N4 (ab initio) and the systems studied here are large, we will not

consider the ER method of localization any further. The same integral

transformation problem is encountered for the Boys method, but since the

integrals involved are dipole (one-electron) integrals the problem is

much simpler. Since the localization criteria are so different there is

no reason to expect different methods to yield orbitals that are

similar, but in general the LMOs are quite similar for the Boys and ER

methods. These orbital similarities lead to the :;scond category of


We claim that the underlying physiciJ b'as.i of onrdlization is the

Fermi hole. The Fermi hnolo provide : n a dir(-t (nlo-i.tei rtive) method


for transforming canonical orbitals to localized orbitals. The integral

transformations that limit the usefulness of the Boys and ER methods are

also eliminated when using this method. The disadvantage of this method

is the fact that a series of probe points must be generated for the

molecule. These points may be generated using chemical intuition or by
a search of the Fermi hole mobility function.2 The Fermi hole method

of localization may also fall into the final category since it can be

made to pick out a particular localized orbital set.

The final category of localization method allows one to produce

orbitals in accordance with one's needs. As mentioned above, the Fermi

method may be classified in this category, but another method was

developed especially for this purpose, one that we have called the

double projector (DP) method. This method has been used in conjunction

with the other methods above to help predict the lowest energy state of

lanthanide containing compounds where f-orbital degeneracies are a

problem. The DP method allows one to separate the f-orbitals from the

other metal orbitals and use a small CI to determine the ground state of

the molecule.

Double Projector

The double projector (DP) method of localization is an extremely

useful method for localizing orbitals when the form of the localized

orbitals is known or suspected in advance. For example, if one would

like to study n-n transitions in a molecule, a full localization need

not be done, the double projector may be used to isolate (localize) the

n-type orbitals. A subsequent small singles CI may then be used to study

only the n-n transitions and thereby elucidate the n-n spectra.


Another example involves the localization of the d-orbitals in a

transition metal complex. Because of accidental degeneracies between

metal d-orbitals and ligand molecular orbitals (MOs), the atomic d-

orbitals may be spread out in many canonical orbitals. A large CI is

then required to restore the localized nature of the d type molecular

orbitals. Such a large CI can be avoided by first doing a DP


The DP method can also be used to remove orbitals from the orbital

set so that the remaining orbitals may be localized using a standard

localization technique. For example, a common problem with a Boys

localization is the mixing of a and R orbitals to obtain T orbitals,

this is not desirable since the a and n spectra will now be mixed and

more difficult to interpret. The n orbitals may be removed using the DP

method, the remaining orbitals localized, and the n-n spectra

calculated using a small singles CI. The double projector is a

complementary method of localization and is normally used in conjunction

with other traditional methods of localization; therefore, no examples

of its use will be given here.

An outline of the double projector method is given in this section.

Consider a set of m occupied spin orbitals ([im, and a set of r

localized "pattern" orbitals (['i}, where r is less than or equal to m.

These "pattern" orbitals are projected out of the set ('i}l by

IT,> = E I|.X>< IT > (1-3)
a i=l i

for a = 1 to r. These (I'>}1 are then symmetrically orthogonalized

'+' = A (1-4)

S= 6-1/2 (1-5)

and are projected out of the original set {i},

= ( ~ l >

The matrix A' is formed and diagonalized

U V 'U = U+A U = X (1-7)

The X matrix will have r near zero eigenvalues corresponding to the

(8 )1 that have been projected out. These eigenvalues and the

corresponding columns of U are removed. The new set of orthonormalized

orbitals (Y }m"r is formed from
oc 1

Y = 4'UX-1/2 (1-8)

This set is an orthogonal complement to the set je >, but has no

particular physical significance. To obtain a set of orbitals most like

the canonical set, we form F, the Fock matrix, over the Y subset and

diagonalize F,

+Y +FYV = er (1-9)

Y' = YW (1-10)

Y' are linear combinations of Y that we can energy order according to

e These Y' orbitals are the most like the original canonical orbitals

with the "pattern" orbitals removed.


Fermi Localization


This section presents a method for transforming a set of canonical

SCF orbitals into a set of localized orbitals based on the properties of

the Fermi hole7-11 and the Fermi orbital.13'14 Unlike localization

methods based on iterative optimization of some criterion of

localization,2-6,15,16 the method presented here provides a direct (non-

iterative) calculation of the localized orbital transformation matrix.

Consequently, this method avoids the convergence problems which are

possible with iterative transformations.

Unlike the extrinsic methods for transforming canonical SCF orbitals
into localized orbitals,17-19 the method presented here does not depend

on the introduction of a definition of a set of "atomic orbitals". The

method presented here may also be distinguished from applications of

localized orbitals such as the PCLIO method2025 in that the latter

method does not involve SCF orbitals, and it is not concerned with the

transformation of canonical SCF orbitals into localized orbitals.

Properties of the Fermi Hole

The Fermi hole is defined as

A(rl;r2) = p(rl) 2p2(rl,r2)/p(r2), (1-11)

where p(rl) is the diagonal portion of the first order reduced density

matrix and p(rl;r2) is the corresponding part of the second order

reduced density matrix.26 For special case of a closed shell SCF


wavefunction, the natural representation of the Fermi hole is the

absolute square of the Fermi orbital13,14

A(rl;r2) = If(rl;r2)2. (1-12)

The Fermi orbital is given by

f(rl;r2) = 2/p(r2)]1/2 gi(rl)g(r2), (1-13)

where the orbitals gi(r) are either the canonical SCF molecular orbitals

or any set related to the canonical SCF molecular orbitals by a unitary

transformation. The Fermi orbital f(rl;r2) is interpreted as a function

of r1 which is parametrically dependent upon the position of a probe

electron located at r2.

Previous work 12,13,27,28 has demonstrated that the Fermi hole does

not follow the probe electron in a uniform manner. Instead, molecules

are found to possess regions where the Fermi hole is insensitive to the

position of the probe electron. As the probe electron passes through

one of these regions, the Fermi hole remains nearly stationary with

respect to the nuclei. These regions are separated by regions where the

Fermi hole is very sensitive to the position of the probe electron. As

the probe electron passes through one of these regions, the Fermi hole

changes rapidly from one stable form to another.

The sensitivity of the Fermi hole to the position of the probe

electron is measured by the Fermi hole mobility function,12,27,28

F(r) = Fx(r) + F (r) + Fz(r) (1-14)


F(r) =2 j j 2 (1-15)
S2 avj av

for v = x, y or z. This may be compared to

FO(P) = (3i/4)(p/2)2/3 (1-16)

which provides an estimate of the Fermi hole in a uniform density

electron gas.

The Fermi hole mobility function F(r) for the formaldehyde molecule

is shown in Fig. 1-1. The difference F(r)-F0(p) is shown in Fig. 1-2.

Regions where F(r) > F(p) that is, the Fermi hole is less sensitive to

the position of the probe electron than it would be in an electron gas

of the same density, may be compared to the loges proposed by
Daudel. Regions where F(r) = F(p) resemble boundaries between


When the probe electron is located in a region where F(r) < F(p),

the Fermi orbital is found to resemble a localized orbital determined by

conventional methods.2-6,15,16 This similarity is demonstrated by Figs.

1-3 and 1-4 which compare a Fermi hole for the formaldehyde molecule

with a localized orbital determined by the orbital centroid criterion of


Localized Orbitals Based on the Fermi hole

Equation 1-13 provides a direct relationship between a set of

canonical SCF orbitals gi(r) and a localized orbital f.(r) = f(r,r.)

where r. is a point in a region where F(r.) < F0(p(rj)). In order to

transform a set of N canonical SCF orbitals into a set of N localized

orbitals, it is necessary to select N points r. j = 1 to N, each of

which is located in a region where F(rj) < F (p(r.)). Ideally, each of

these points should correspond to a minimum of F(r) or F(r)-FO(p). This

condition, however, is not critical, because the Fermi hole is

relatively insensitive to the position of the probe electron when the

probe electron is located in one of these regions.

Figure 1-1: The fermi hole mobility function F(r) for the H CO based on
the geometry and double zeta basis set of ref. 41. The
locations of the nuclei are indicated by (+) signs. The
contours represent mobility function values of 0.1, 0.25,
0.5, 1.0, 2.0 and 5.0 atomic units. The contours increase
from 0.1 near the corners, to over 5.0 in regions enclosing
the carbon and oxygen nuclei. Each nucleus is located at a
local minimum of the mobility function.


Figure 1-2:

\ /

S(+ / '

S\ \ \I

The difference between the Fermi hole mobility function F(r)
and the electron gas approximation for the H CO molecule.
The contours represent values of 0.0, -0.1, -0.25, -0.5,
-1.0, -2.0 and -5.0, in addition to those indicated in
figure 1-1. the contours representing negative values and
zero are indicated by broken lines. Each nucleus is located
at a local minimum.

Figure 1-3: The fermi hole for the formaldehyde molecule determined by a
probe electron located at one of the protons. The contours
indicate electron density of 0.005, 0.01, 0.02, 0.04, 0.08,
0.16, 0.32, 0.64, 1.28 and 2.56 electrons per cubic bohr.



Figure 1-4: The localized orbital for the C-H bond of a formaldehyde
molecule determined by the orbital centriod criterion for
localization. The electronic density contours are the same
as in figure 1-3.


A set of N Fermi orbitals determined by Eq. 1-3 is not generally

orthonormal. Each member of this set, however, is usually very similar

to one member of an orthonormal set of conventional localized orbitals.

Consequently, the overlap between a pair of Fermi orbitals is usually

very small, and a set of N Fermi orbitals may easily be converted into

an orthonormal set of localized orbitals by means of the method of

symmetric orthogonalization.33 The resulting unitary transformation is

given by

U = (TT+)-1/2T, (1-17)


T. = g.(r.)/(p(r.)/2) (1-18)
J3 3 1 1

In the following three sections, the transformation of canonical SCF

orbitals based on Eqs. 1-17 and 1-18 is demonstrated for each of three

molecules. The first example, a cyclic conjugated enone, represents a

simple case where conventional methods are not expected to have any

special difficulties. The second example, methyl acetylene, is a

molecule for which conventional methods have serious convergence

problems.34 The third example, boron trifluoride, is a pathological

case for the orbital centroid criterion, with a number of local maxima

and saddle points in the potential surface according to the Boys

criterion of localization.

In each case, the first step in the application of this method is

the selection of the set N points. This set always includes the

locations of all of the nuclei in the molecule. For atoms other than

hydrogen, the resulting Fermi orbitals are similar to innershell

localized orbitals. When the probe electron is located on a hydrogen

atom, the Fermi orbital is similar to an R-H bond orbital.


Additional points for the probe electron may usually be determined

based on the molecular geometry. The midpoint between two bonded atoms

(other than hydrogen) tends to yield a Fermi orbital resembling a single

bond. Multiple bonds may be represented with two or three points

located roughly one to two bohr from a point midway between the multiply

bonded atoms, along lines perpendicular to a line joining the nuclei.

Likewise, lone pair orbitals may be determined by points located roughly

one bohr from the nucleus of an atom which is expected to possess lone

pair orbitals.

Application to the Furanone Molecule

The furanone molecule, C4H402, and its derivatives are useful
reagents in 2+2 photochemical cycloadditions. 7 The canonical SCF

molecular orbital for the furanone molecule were calculated with an STO-

3G38 basis set and the geometry specified in Table 1-1. The molecular

geometry was restricted to Cs symmetry, with a planar five membered

ring. Fermi hole localized orbitals were calculated based on the set of

points indicated in Table 1-1. These points include the positions of

the ten nuclei, as well as twelve additional points determined by the

method outlined above.

The centroids of the localized orbitals determined by the points in

Table 1-1 are shown in Table 1-2. The C=C and C=O double bonds are each

represented by a pair of equivalent bent (banana) bonds similar to those

determined by other methods for transforming canonical SCF orbitals into

localized orbitals.

As shown in Table 1-2, the centroids of the orbitals determined by

the Fermi hole are very close to those of the localized orbitals Page

Table 1-1 :

Molecular geometry and probe electron points for the
furanone (C4H402). The first ten points indicate the
molecular geometry used in these calculations. The twelve
additional probe electron positions were determined as
described in the text. All coordinates are given in bohr.

Position X Y Z

Atom C1 0.0 0.0 0.0

Atom C2 0.0 2.589 0.0

Atom C3 2.671 3.355 0.0

Atom C4 4.363 0.940 0.0

Atom 01 2.534 -1.071 0.0

Atom 02 3.427 5.548 0.0

Atom H1 -1.802 -1.040 0.0

Atom H2 -1.802 3.629 0.0

Atom H3 5.560 0.858 1.698

Atom H4 5.560 0.858 -1.698

C1-C2 bond 1 0.0 1.295 2.000

C1-C2 bond 2 0.0 1.295 -2.000

C2-C3 bond 1.336 2.972 0.0

C3-C bond 3.517 2.148 0.0

C4-01 bond 3.449 -0.066 0.0

C1-01 bond 1.267 -0.536 0.0

C3-02 bond 1 3.049 4.452 2.000

C3-02 bond 2 3.049 4.452 -2.000

01 lone pair 1 2.654 -1.821 0.660

01 lone pair 2 2.654 -1.821 -0.660
02 lone pair 1 2.707 6.298 0.0

02 lone pair 2 4.267 5.698 0.0

Table 1-2 :


Orbital centroids for localized orbitals determined by the
Fermi hole method and by the orbital centroid method for the
furanone molecule (C4H402). All coordinates are given in

Fermi hole method Centroid criterion

C1 K shell 0.0 0.001 0.0 0.0 0.0 0.0

C2 K shell 0.0 2.588 0.0 0.0 2.588 0.0

C3 K shell 2.670 3.355 U.0 2.671 3.355 0.0

C4 K shell 4.362 0.940 0.0 4.326 0.939 0.0

01 K shell 2.534 -1.070 0.0 2.533 -1.070 0.0

02 K shell 3.427 5.547 0.0 3.426 5.547 0.0
C1-H1 bond -1.138 -0.742 0.0 -1.171 -0.761 0.0

C2-H2 bond -1.148 3.318 0.0 -1.181 3.331 0.0

C4-H3 bond 5.149 0.895 1.132 5.155 0.888 1.153

C4-H4 bond 5.149 0.895 -1.132 5.155 0.888 -1.153

C1-C2 bond 1 0.008 1.401 0.635 0.030 1.410 0.599

C1-C2 bond 2 0.008 1.401 -0.635 0.030 1.410 -0.599

C2-C3 bond 1.320 3.054 0.0 1.284 3.046 0.0

C3-C4 bond 3.542 2.190 0.0 3.565 2.149 0.0

C4-01 bond 3.268 -0.243 0.0 3.243 -0.225 0.0

C1-01 bond 1.472 -0.627 0.0 1.495 -0.613 0.0

C3-02 bond 1 3.093 4.584 0.548 3.114 4.612 0.511

C3-02 bond 2 3.093 4.584 -0.548 3.114 4.612 -0.511

01 lone pair 1 2.554 -1.283 0.440 2.594 -1.311 0.472

01 lone pair 2 2.554 -1.283 -0.440 2.594 -1.311 -0.472.

02 lone pair 1 3.015 5.886 0.0 3.011 5.892 0.0


02 lone pair 2






determined by the orbital centroid criterion.4-6,15 Likewise, the

localized orbitals determined by the Fermi hole were found to be very

close to those determined by the orbital centroid criterion. Each of

the localized orbitals determined by the Fermi hole method was found to

have an overlap of 0.994 to 0.999 with one of the localized orbitals

determined by the orbital centroid criterion. The remaining (off-

diagonal) overlap integrals between these two sets of localized orbitals

were found to have a root mean square (RMS) value of 0.011734.

The transformation of a set of canonical SCF orbitals to an

orthonormal set of localized orbitals determined by the Fermi hole

required 10 minutes on a PDP-11/44 computer. The orbital centroid

(Boys) method required 140 minutes starting from the canonical SCF

molecular orbitals or 80 minutes, using the Fermi localized orbitals as
an initial guess, to reach TRMS of less than 10- where TRMS is the RMS

value of the oEf-diagonal part of the transformation matrix which

converts the orbitals obtained on one iteration to those of the next

iteration. The orbital centroid criterion calculations reported here

are based on a partially quadratic procedure which requires less time

than conventional localization procedures based on 2X2 rotation.

Application to Methylacetylene

The localized orbitals of methylacetylene are of interest because of

the convergence difficulties encountered in attempts to calculate these

orbitals using iterative localization methods. These difficulties are

caused by the weak dependence of the criterion of localization on the

orientation of the three equivalent C-C (banana) bonds relative to the

three C-H bonds of the methyl group. In calculations based on the


orbital centroid criterion, over 200 iterations were required to

determine a set of orbitals which satisfied a very weak criterion of
convergence. Most of these difficulties may be overcome using the
quadratically convergent method which has been developed recently. As

shown below, however, the localized orbitals based on the Fermi hole

yield nearly equivalent results and require much less effort than even

the quadratically convergent method.

The canonical SCF molecular orbitals for methylacetylene were

determined by an STO-5G basis set38 and an experimental geometry.39

Transformation of the 11 occupied SCF orbitals into localized orbitals

based on the Fermi hole method required the selection of 11 points.

These points are shown in Table 1-3. The positions of the nuclei

provided seven of these points. One point was located at the midpoint

of the C2-C3 single bond. The remaining three points were located two

bohr from the I' rotation axis at a point midway between the C1 and C2

nuclei. These last three points were eclipsed with respect to the

methyl protons.

The centroids of the localized orbitals determined by this method

are shown in Table 1-4. As expected, the triple bond is represented

with three equivalent banana bonds. The centroids of the corresponding

orbitals determined by the orbital centroid criterion are also shown in

Table 1-4. These are very close to those determined by the Fermi hole

method. The RMS value of the off-diagonal part of the overlap matrix

between the localized orbitals determined by the Fermi hole and those

determined by the orbital centroid criterion is 0.012874.

The Fermi hole method required 1.62 minutes to transform the

canonical SCF molecular orbitals into an orthonormal set of localized

molecular orbitals. By comparison, the (quadratically convergent) Page

Tablp 1-3 :

Molecular geometry and probe electron positions for the
methylacetylene molecule. The first seven points indicate
the locations of the nuclei. All coordinates are given in

Position I



Atom C1 0.0 -1.140 0.0

Atom C2 0.0 1.140 0.0

Atom C3 0.0 3.897 0.0

Atom H1 0.0 -3.145 0.0

Atom H2 -1.961 4.438 0.0

Atom H3 0.980 4.438 1.698

Atom H4 0.980 4.438 -1.698

C1-C2 bond 1 -2.0 0.0 0.0

C1-C2 bond 2 1.0 0.0 1.732

C1-C2 bond 3 1.0 0.0 -1.732

C2-C3 bond 0.0 2.510 0.0

Table 1-4 :


Orbital centroids for localized orbitals determined by the
Fermi hole method and by the orbital centroil method for
the methylacetylene molecule. All coordinates are given
in bohr.

Fermi hole method Centroid criterion

C1 K shell 0.0 -1.141 0.0 0.0 -1.140 0.0

C2 K shell 0.0 1.139 0.0 0.0 1.134 0.0

C3 K shell 0.0 3.896 0.0 0.0 3.896 0.0

C1-H1 bond 0.0 -2.486 0.0 0.0 -2.502 0.0

C3-H2 bond -1.302 4.270 0.0 -3.316 4.272 0.0

C3-H3 bond 0.651 4.270 1.127 0.658 4.272 1.139

C3-H4 bond 0.651 4.270 -1.127 0.658 4.272 -1.139

C1-C2 bond 1 -0.713 -0.003 0.0 -0.692 -0.013 0.0

Ci-C2 bond 2 0.356 -0.003 0.617 0.346 -0.013 0.599

C1-C2 bond 3 0.356 -0.003 -0.617 0.346 -0.013 -0.599

C2-C3 bond 0.0 2.457 0.0 0.0 2.498 0.0


orbital centroid method required 26 minutes starting from the canonical

SCF molecular orbitals or 19 minutes using the Fermi localized orbitals
as an initial guess to reach TRMS of 10-8 or less.

Application to Boron Trifluoride

As a further example of the application of the Fermi hole

localization method, localized orbitals were also calculated for the

boron trifluoride molecule. The molecule provides a demonstration of

how characteristics of little or no physical significance can cause

serious convergence difficulties for iterative localization methods. A

localized representation of boron trifluoride includes four inner-shell

orbitals, three boron fluorine bond orbitals, and nine fluorine lone

pair orbitals. The orbital centroid criterion method shows a small

dependence on rotation of each set of three lone pair orbitals about the

corresponding B-F axis. Consequently, the hessian matrix for he

criterion of localization as a function of a unitary transformation of

the orbitals has three very small eigenvalues.

The optimal orientation of the fluorine lone pairs may correspond to

one of several possible conformations. One of these, the "pinwheel"

conformation, has a single lone pair orbital on one of the fluorine

atoms in the plane of the molecule. The other two lone pair orbitals on

this fluorine atom are related to the first lone pair by 120 degree

rotations about the F-B axis. The lone pair orbitals on the other two

fluorines are obtained by 120 degree rotations about the C3 axis. The

point group of the orbital centroids for this conformation is C3h.

The "three-up" conformation is generated by rotating the set of lone

pair orbitals on each fluorine atom in the pinwheel conformation by 90


degrees about each F-B axis. The point group for the orbital centroids

of this conformation is C3v. The "up-up-down" conformation is generated

by rotating the set of lone pair orbitals on one of the fluorine atoms

in the "three-up" conformation by 180 degrees about the F-B axis. This

conformation has the symmetry of the Cs point group.

The canonical SCF orbitals for BF3 were calculated based on the

double-zeta basis set and geometry tabulated by Snyder and Basch.40

Localized orbitals determined by the orbital centroid criterion were

obtained for the three-up conformation and the up-up-down conformation.

The centroids for these orbitals are shown in Table 1-5. The first five

(most positive) eigenvalues of the hessian matrix for each of these

conformations are shown in Table 1-6. All eigenvalues of the hessian

matrix are negative for both of these conformations, indicating that

both conformations are maxima for the sum of squares of the orbital

centroids. The pinwheel conformation, however, was never found.

Consequently, it was not possible to exclude the possibility that the

pinwheel conformation was the global maximum and the three-up

conformation was only a local maximum.

The pinwheel conformation can easily be constructed using the Fermi

hole localization method by selecting an appropriate set of probe

positions. This set of points is shown in Table 1-7. The centroids of

the resulting set of localized orbitals are shown in Table 1-8. When

this set of localized orbitals is used as the starting point, the

orbital centroid method quickly converges to a stationary point with C3h

symmetry. The centroids of the resulting set of orbitals are shown in

Table 1-8. As shown in Table 1-6, three of the eigenvalues of the

hessian matrix were positive at this point, demonstrating that the Page

Table 1-5 :


Orbital centroids for localized orbitals determined by the
orbital centroid method for the boron trifluoride
molecule. The four innershell orbitals have been excluded
from these calculations. All coordinates are given in

up-up-up (C3v) up-up-down (Cs)

B-F1 bond 1.725 0.0 -0.072 1.725 0.0 -0.070

B-F2 bond -0.862 1.494 -0.072 -0.862 1.494 -0.070

B-F3 bond -0.862 -1.494 -0.072 -0.862 -1.494 0.069

F1 lone pair 1 2.445 0.0 0.487 2.446 0.0 0.487

lone pair 2 2.596 0.433 -0.207 2.596 0.433 -0.208

lone pair 3 2.596 -0.433 -0.207 2.596 -0.433 -0.207

F2 lone pair 1 -1.222 2.117 0.487 -1.222 2.118 0.487

lone pair 2 -0.922 2.465 -0.207 -0.922 2.465 -0.208

lone pair 3 -1.674 2.032 -0.207 -1.674 2.031 0.207

F3 lone pair 1 -1.222 -2.117 0.487 -1.223 -2.119 -0.486

lone pair 2 -0.922 -2.465 -0.207 -0.922 -2.464 0.208

lone pair 3 -1.674 -2.032 -0.207 -1.673 -2.031 0.208

Table 1-6 :

Values of the orbital centroid criterion and the second
derivatives of the orbital centroid criterion for various
conformations of localized orbitals for the boron
trifluoride molecule. The row labelled sum indicated the
sum of the squares of the orbital centriods for each of
the conformations. The following rows show the five
highest (most positive) eigenvalues X. of the
corresponding hessian matrix. The first and second
columns correspond to the localized orbitals described in
described in Table 1-5. The third and fourth columns
correspond to localized orbitals described in Table 1-8.
The gradient vectors are zero for the first three columns.


up-up-up up-up down pinwheel Fermi hole

Sum 69.445360 69.444985 69.438724 69.341094

X1 -0.019521 -0.038174 +0.017234 +0.031444

X2 -0.019850 -0.018914 +0.016238 +0.029932

X -0.019850 -0.019095 +0.016236 +0.029928

X4 -0.155480 -0.153827 -0.193521 -0.195502

X5 -0.155484 -0.155181 -0.1n4366 -0.195524

Table 1-7 :

Probe electron positions for the boron trifluoride
molecule. The first three points are located at the
midpoint of the B-F bonds. The remaining points have been
chosen in the pinwheel conformation (symmetry C3h). All
coordinates are given in bohr.

Position X Y Z

B-F1 bond 1.223 0.0 0.0

B-F2 bond -0.611 1.059 0.0

B-F3 bond -0.611 -1.059 0.0

F1 lone pair 1 2.781 0.943 0.0

lone pair 2 2.718 -0.472 0.817

lone pair 3 2.781 -0.472 -0.817

F2 lone pair 1 -2.207 1.937 0.0

lone pair 2 -0.981 2.644 0.817

lone pair 3 -0.981 2.644 -0.817

F3 lone pair 1 -0.573 -2.880 0.0

lone pair 2 -1.799 -2.172 0.817

lone pair 3 -1.799 -2.172 -0.817

Table 1-8 :


Orbital centroids for localized orbitals determined by the
Fermi hole method and by the orbital c:ntroid method for
the boron LrifluoLide molecule. The ufur innershell
orbitals have been excluded from these calculations. All
coordinates are given in bohr.

Fermi hole Centroid criterion

B-F1 bond 1.713 0.0 0.0 1.720 0.028 0.0

B-F2 bond -0.857 1.484 0.0 -0.884 1.475 0.0

B-F3 bond -0.857 1.484 0.0 -0.835 1.504 0.0

F1 lonp pair 1 2.567 0.488 0.0 2.603 0.484 0.0

lone pair 2 2.538 -0.244 0.414 2.520 -0.256 0.410

lone pair 3 2.538 -0.244 -0.414 2.520 -0.2',6 -0.410

F2 lone pair 1 -1.706 1.976 0.0 -1.721 2.012 0.0

lone pair 2 -1.057 2.320 0.414 -1.037 2.310 0.410

lone pair 3 -1.057 2.320 -0.414 -1.037 2.310 -0.410

F3 lone pair 1 -0.861 -2.467 0.0 -0.882 -2.497 0.0

lone pair 2 -1.480 -2.076 0.414 -1.482 -2.054 0.410

lone pair 3 -1.480 -2.076 -0.414 -1.482 -2.054 -0.410

pinwheel conformation is a saddle point with respect to the orbital

centroid criterion. These calculations also indicate that the three-up

conformation is probably the global maximum for the orbital centroid


We do not intend to attribute any special physical significance to

any of the lone pair configurations for BF3. These calculations

demonstrate some of the problems, such as local maxima and saddle

points, which may occur for conventional iterative localization methods.

These calculations demonstrate how the Fermi hole method may be used by

itself to transform the canonical SCF oLbitals into localized orbitals

without any of these difficulties. In addition, these calculations

demonstrate how the Fermi hole method may be used in conjunction with

the orbital centroid method to establish a characteristic of the orbital

centroid criterion which would have been very difficult to establish

using the orbital centroid method alone.


The numerical results presented here demonstrate how the properties

of the Fermi hole may be used to transform canonical SCF molecular

orbitals into a set of localized SCF molecular orbitals. Except for the

symmetric orthogonalization, this method requires no integrals and no

iterative transformations. The localized orbitals obtained from this

method are very similar to the localized orbitals determined by the

orbital centroid criterion. The orbitals determined by the Fermi hole

may be used directly in subsequent calculations requiring localized

orbitals. Alternatively, the orbitals determined by this method may be

used as a starting point for iterative localization procedures.261516


The necessity of providing the set of probe electron positions may

appear to introduce a subjective element into the localized orbitals

determined by the Fermi hole method. Most of the subjective character

to this choice, however, is eliminated by the fact that Fermi hole is

relatively insensitive to the location of the probe electron whenever

the probe electron is located in a region associated with a strongly

localized orbital. This is reflected by the fact that the centroids of

the localized orbitals determined by the Fermi hole method, as shown in

Tables 1-2, 1-4 and 1-8, are much closer to the centroids of the

corresponding localized orbitals determined by the orbital centroid

criterion than they are to the probe electron positions used to

calculate them.

If the electrons are not strongly localized in certain portions of a

molecule, such as in the lone pairs of a fluorine atom, then the Fermi

hole may be more strongly dependent on the location of the probe

electron than where the electrons are strongly localized. In such

cases, the localized orbitals determined by the Fermi hole method may

reflect the locations of the probe electron points more strongly than

they are reflected in well localized regions. In such regions, .however,

there may be no physically meaningful way to.distinguish between the

localized orbitals determined by this method and those determined by any

other method. In these situations, the Fermi hole method may provide a

practical method for avoiding the convergence problems which may be

expected for iterative methods when the electrons are not well


The electronic structure of most common stable molecules may be

described by an obvious set of chemical bonds, lone pair orbitals, and

innershell atomic orbitals. This is reflected in the success of methods

such as molecular mechanics41'42 for predicting the geometries of

complex molecules. The localized orbitals of such molecules are

unlikely to be the objects of much interest in themselves, but they may

be useful in the calculation of other properties of a molecule, such as
43 44,45
the correlation energy,43 spectroscopic constants, 5 and other

properties.447 The selection of a set of probe electron positions for

one of these molecules is simple and unambiguous, and the method

presented here has significant practical advantages compared to

alternative methods for transforming canonical SCF molecular orbitals

into localized molecular orbitals.

For some molecules, the pattern of bonding may not be unique or it

may not be entirely obvious, even when the geometry is known. For

example, two or more alternative (resonance) structures may be involved

in the electronic structure of such molecules. The localized orbitals

of such molecules may be of interest in themselves, in order to

characterize the electronic structure of such molecules, in addition to

their utility in subsequent calculations.43-47 In order to apply the

current method to such molecules, the Fermi hole mobility function228

must be used to resolve any ambiguities which may arise in the selection

of the probe electron positions. If two or more bonding schemes are

possible, the positions of the probe electrons should be chosen to

provide the minimum values of the Fermi hole mobility functions F(r) or

the mobility function difference F(r)-FO(p).

In the case of methylacetylene, for example, the C-C single bond may

be determined by a single point midway between the carbon atoms, where

F(r) is less than FO(p). Any attempt to represent this portion of the

molecule with a double bond would require placing a probe electron away

from the C-C axis, in a region where F(r) is greater than FO(p).


Consequently, it is not possible to represent methylacetylene with a

structure like H-C=C=CH3 without placing one or more probe electron

points in regions where the Fermi hole is unstable.

In extreme cases, even the Fermi hole mobility function may fail to

provide unambiguous positions for the probe electrons. This is expected

in highly conjugated aromatic molecules, metallic conductors, and other

highly delocalized systems. For these electrons, the method presented

here, as well as all other methods for calculating localized orbitals,

are entirely arbitrary. The electronic structure of such a delocalized

system may be represented by an unlimited number of localized

descriptions, each of which is equally valid.

If there is a need for imposing a localized description on a highly

delocalized system, the current method would be no less arbitrary than

existing alternatives. The arbitrariness of the current method would be

manifested in the choice of the probe electron positions for the

delocalized electrons. However, the current method would continue to

provide practical advantages over alternative transformations. These

advantages include the absence of integrals to evaluate, the absence of

iteratively repeated calculations, and the absence of convergence


Boys Localization

The most widely used form of localized orbitals are those orbitals

based on the method of Boys.5'6 The integral transformation procedure

in any localization procedure can be a time limiting step. In the

Edmiston-Reudenberg method the two electron repulsion integrals must be

transformed an N5 computational step, but the Boys method may be


formulated in terms of products of molecular dipole integrals. The

dipole integrals are one-electron integrals therefore the transformation

is on the order of N3, making the Boys localization the method of

choice. One disadvantage of this method is that it is an iterative

method which may be prone to convergence difficulties.

Methods of Boys Localization

There exists a unitaLy transformation relating a delocalized set of

orbitals to a localized set, but the form of this transformation is in

general unknown. The common method fol solving this problem is to do a

series of unitary transformations that increase the degree of

localization of a set of orbitals.

Given a set of orbitals f one can increase the localization by

doing a unitary transformation U ,

fl+1 = Uf1 (1-19)

where f is the resulting set of more localized orbitals. The

original UI matrix formulation is based on a sequence of pairwise
rotations, as proposed by Edmiston and Reudenberg. In this procedure,

N orbitals are localized by rotating a pair of orbitals, then a second

pair of orbitals is rotated, ., etc. until all N(N-1)/2 pairs of

orbitals have been rotated. Since the rotations are done in a specific

order, the localized orbitals obtained will be dependent on the order of


Leonard and Luken have developed a second order method that does all

of the N(N-1)/2 rotations at once rather than one at a time.5 The use

of a second order method may have the additional benefit of improving

the convergence difficulties encountered in iterative methods. Their

method is outlined below.


The NxN unitary transformation matrix for the localization can be

written as

U = WR, (1-20)

where V is a positive definite matrix defined by

V = (RR)-/2. (1-21)

The matrix R will be defined as

R = NT, (1-22)


T = 1 + t (1-23)

and t is an antisymmetric matrix

t+ = -t (1-24)

The N matrix is a diagonal matrix which normalizes the columns of R. By

application of U to a set of orbitals (fl' fN) one produces a

set of more localized orbitals (fi, fN}. The new value of the

localization, G', is given by

G' = (i'i',i'i') (1-25)

The new f' orbitals can be thought of in terms of a pertubative


f' = f + Z t f + Z tt jf (1-26)

to second order. The tI matrix does 2x2 rotations that mix in portions

of all the occupied orbitals into orbital f'. The third term titJ is

the product of a pair of 2x2 rotations. When this form of the f'


orbitals is substituted into the G' equation, you obtain

G' = GO + Z t G1(I) + E tit G2(I,J) .(1-27)

Leonard and Luken5 include the G2 second order term to accelerate

convergence when one is in the quadratic region; only the first order

term G1 is calculated initially, and until the quadratic region is

encountered. In practice, standard procedures for localization often

can take several hundred iterations to converge. These second order

procedure described above seldom takes more than 20 cycles, and the

energy of orbitals related by symmetry (C-H bonds in benzene, etc.) is

usually reproduced to seven significant figures.


One example of the Boys method of Leonard and Luken was given in the

Fermi hole method section, in this section we will show the localized

orbitals foi the Ni(CO)+ ion. In a recent experimental paper by Reutt

et al., the photoelectron spectroscopy of Ni(CO)4is reported in order to

clarify the nature of the transition metal carbonyl bond. Since the

spectrum is interpreted in terms of localized orbitals on both the metal

and the carbonyl groups, a first step in any quantum chemical treatment

of the problem is to localize the orbitals.

This system is somewhat complicated because it has an unpaired

electron. The three T2 orbitals are occupied by five electrons leading

to a triply degenerate ground state. The MOs are localized using the

Leonard and Luken implementation of the Boys' procedure. The

localization breaks the orbitals into several classes; (1) oxygen lone

pairs, (2) carbon oxygen x (banana) bonds, (3) nickel carbon bonds, and


(4) nickel d-orbitals of two types namely E and T2 type orbitals. The

localization may also be done on the unoccupied orbitals; this separates

the unoccupied orbitals into two sets (1) nickel carbon antibonds and

(2) carbon oxygen antibonds.

In Fig. 1-5, an iso-value plot of one of the nickel carbon bonds is

shown. The orbital shows a large amplitude near the carbon atom,

indicative of large p-orbital contributions on the carbon and a

relatively small contribution from the nickel d-orbitals. As one can see

from Fig. 1-6, a sizable contribution to bonding comes from one of the

partially occupied nickel d-orbitals. The nickel carbon antibond shown

in Fig. 1-7 possesses a large node along the internuclear axis. The

nickel carbon bond is expected to be quite weak, because it is composed

of a sum of the two bonding orbitals shown in Figs. 1-5 and 1-6. The

diffuse nature of the photoelectron spectra indicates the population of

additional vibrational modes, resulting from a distortion from a

tetrahedral geometry. The weak nickel carbon bonds would allow for such

a distortion to take place in the ion. A further application of the

Boys method would be to include the localized orbitals into a limited CI

calculation to see if one could predict the photoelectron spectra for


The use of any localized orbital technique does not add or subtract

information from the overall wavefunction. These methods only divide

orbitals into more chemical pieces allowing for easier interpretation of

experimental results.



\ !


Figure 1-5: An iso-value localized orbital plot of a nickel carbon
bonding orbital in the Ni(CO)+ molecule. The dashed lines
indicate an orbital amplitude of -0.05 a.u. per cubic bohr.
The solid lines indicate an orbital amplitude of 0.05 a.u.
per bohr.





Figure 1-6: An iso-value localized orbital plot of a nickel carbon non-
bonding orbital in the Ni(CO)+ molecule. The dashed lines
indicate an orbital amplitude of -0.05 a.u. per cubic bohr.
The solid lines indicate an orbital amplitude of 0.05 a.u.
per bohr.



C \


Figure 1-7: An iso-value localized orbital plot of a nickel carbon anti-
bonding orbital in the Ni(CO), molecule. The dashed lines
indicate an orbital amplitude of -0.05 a.u. per cubic bohr.
The solid lines indicate an orbital amplitude of 0.05 a.u.
per bohr.




The past decade has seen a dramatic increase in interest and

activity in lanthanide and actinide chemistry. Not only has

considerable knowledge been gained in the traditional area of inorganic

f-element chemistry, but much modern work is concerned with organo-f-
element reactions,49 and the use of lanthanides and actinides as very

specific catalysts.551 Unlike the corresponding chemistry involving

the d metals, very little explanation is offered for much of this


The electronic structure of these systems is difficult to calculate

from quantum chemical means for several reasons. Most of the complexes

of real experimental interest are large. In addition, veiy little about

f-orbitals as valence orbitals is known, although experience is now

being gained on the use of f orbitals as polarization orbitals.

Finally, the f-orbital elements are sufficiently heavy that relativistic

effects become important. Very few ab initio molecular orbital studies

have been reported on f-orbital systems. Extended Huckel calculation,

however, have been successful in explaining some of this chemistry.53

Scattered wave and DVM Xa studies of f-orbital systems have also proven

effective, especially in examining the photoelectron spectroscopy of

reasonably complex systems. 56

We examine an Intermediate Neglect of Differential Overlaps (INDO)

technique for use in calculating properties of f-orbital complexes. At

the Self-Consistent Field (SCF) level this technique executes as rapidly


on a computer as does the Extended Huckel method, and considerably more

rapid than the scattered wave Xac method. Since the electrostatics of

the INDO method are realistically represented, molecular geometries can

be obtained using gradient methods.57 Since the INDO method we examine

contains all one-center two-electron terms it is also capable of

yielding the energies of various spin states in three systems. With

configuration interaction (CI) this model should also be useful in

examining the UV-visible spectra of f-orbital complexes. Preliminary

studies of f-orbital chemistry using an INDO model have been disclosed

by Clack and Warren58 and, more recently, by Li-Min, Jing-Quing, Guang-
Xian and Xiu Zhen.59 The method we examine will differ from their

methodology in several areas, as discussed below.

Several problems unique to an INDO treatment of these systems must

be considered, and we have very little ab initio work to guide us. As

mentioned, what role do relativistic effects play? Although we might

hope to parameterize scalar contributions through the choice of orbitals

and pseudo-potential parameters, spin orbit coupling, often larger than

crystal field effects, will need to be considered at some later stage.

Since f-orbitals are generally tight, and ligand field splitting thus

small, a great many states differing only in their f-orbital populations

lie very close in energy. These near degeneracies often prevent

"automatic" SCF convergence, a problem with which we must deal for an

effective model. The nature of the valence basis set itself is in

question. Are the filled 5p and the vacant 6p of the lanthanides both

required for a proper description of their compounds?


The INDO model Hamiltonian that we use was first disclosed by Pople

and collaborators,60 and then adjusted for spectroscopy61 and extended

to the transition metal series.62-64 The details of this model are

published elsewhere.62-6 To extend this model Hamiltonian to the f-

orbital systems we need first a basis set that characterizes the valence

atomic orbitals, and that is subsequently used for calculating the

overlap and the one- and two-center two-electron integrals.

Subsequent atomic parameters that enter the model are the valence

state ionization potentials used for calculating one-center one-electron

"core" integrals and the Slater-Condon Fk and Gk integral that are used

for thr formation of one-center two-electron integrals. The evaluation

of these integrals using experimental information has traditionally made

this model highly successful in predicting optical properties.6

We ,mploy in this model one set of pure parameters, the resonance or

B(k) parameters; for each lanthanide atom we decided to use B(s) = B(p),

B(d) and B(f). These parameters will be chosen to give satisfactory

geometries of model systems. Another choice is one that gives good

prediction: of UV-visible spectroscopy.61,63 These values seldom differ

much from those chosen to reproduce molecular geometry.

In this initial work all two-center two-electron integrals required

for the INDO model Hamiltonian are calculated over the chosen basis set,

as are the one-center two-electron F integrals. An alternate choice

would be one that focuses on molecular spectroscopy. In such a case,

and one that we have to investigate subsequently, the one-center two-

electron F could be chosen from the Pariser approximation67 FO(n) =

IP(n) EA(n), (IP = Ionization Potential, EA = Electron Affinity) and

the two-electron two-center integral from one of the more successful

function established for this purpose.68-70
functions establishedd for this purpose.

At the SCF level, we seek solutions to the pseudo-eigenvalue problem

F C = C (2-1)

with F, the Fock or energy matrix, C, the matrix compound of Molecular

Orbital (MO) coefficients, and s, a diagonal matrix of MO eigenvalues.

The above equation is for the closed shell case (all electrons paired).

The uncestiicted Hartree Fock case is discussed in detail elsewhere,61
as is the open :shell restricted case.71-73 Although nearly all f-

orbital systems are open shell, consideration of the closed shell case

demonstrates the required theory and is considerably simpler.

Within the INDO model, elements of F are given by:

F AA= U + PX[ lax (a (l) (2-2a)

+ E P ( iir a) E ZB (liulsBB)
acB B#A

FA = PaX [(I l) (avX) u.t (2-2b)

FAB 1[

F = A(p) + BB(v) S 2 P' ('iVv) APB (2-2c)


rd -1 *
(iv u ) = Jdr(1)d.(2) Xr(1) X (l) r12 X(2) XX(2) (2-3)

P is the first order density matrix, and since one assumes that the

Atomic Orbital (AO) basis {X} is orthonormal it is identical to the

charge and bond order matrix, given by

P = E C Ca na (2-4)

with n the occupation of MO a n = 0,1,2. In Eq. (2-2), FB refers

to a matrix element with AO X centered on atom "A". The core


UAA ( 1 2 (2-5)

is essentially an atomic term and will be estimated from spectroscopic

data as described below. V is an effective potential that keeps the

valence orbital X orthogonal and non-interacting from the neglected

inner-shell orbitals. The choice of an empirical procedure for UA will

remove the necessity for explicit consideration of this term. The bar

over an orbital in an integral,

such as (0 ii indicates that the orbital X is to be replaced with an s

symmetry orbital of the same quantum number and exponent. The

appearance of such orbitals in the theory is required for rotational

symmetry and compensates for not including other two center integrals of

the NDDO type;74 i.e. (u A A, X#X. The last term in Eq. (2-2a)

represents the attraction between an electron in distribution X* X and

all nluclei but A. The rationale for replacing integral

(U IR1 uA) --- (p S B) (2-6)

is given elsewhere, and compensates for neglected two center inner
shell-valence shell repulsion228 and neglected valence orbital

(symmetrical) orthogonalization. 29 is the core charge of atom B

and is equal to the number of electrons of neutral atom B that are

explicitly considered; i.e. 4 for carbon, 8 for iron, 4 for cerium, etc.
S of Eq. (2-2c) is related to the overlap matrix D, 5 and is

given by

SU = E f (l)V(1) g () ( ((v(1)) (2-7)

where gu(1)v(1) is the Eulerian transformation factor required to rotate

from the local diatomic system to the molecular system, (u(l)jv(l)) are

the sigma (1=0), pi(l=l), delta(l=2) or phi (1=3) components to the

overlap in the local system, and fu(l)v(l) are empirical weighting

factors chosen to best reproduce the molecular orbital energy spread for

model ab initio calculations. We have made little use of this f factor,

and set all f = 1 except between p symmetry orbitals viz.61

S = 1.267gp (palp) + 0.585p (pnpg) (2-8)
r P
+ 0.585gppE (pn Ip )

Basis Set

In general ZDO methods choose a basis set of Slater Type Orbitals


R = -1 rn-l -Cr Y1 (89,) (2-9a)
nim [2n! e

where Y,(9,O) are the real, normalized spherical harmonics. Atomic

orbitals XP are expressed as fixed contractions of these {Rnlm}

X = E anlm Rnlm (2-9b)

In general a single Rn1m function describes the s and p orbitals for

most atoms. The d orbitals of the transition metals, however, require

at least a double-C type function (two terms in 2-9b) for an accurate

description of both their inner and outer regions. For the lanthanides

we have examined basis sets suggested by Li Le-Min et al.,59 by Bender

and Davidson,78 and by Clementi and Roetti.79 In the latter case, the

two major contributors of Eq. (2-9b) in the valence orbitals of the

double-C atomic calculations were selected, and these functions were

renormalized with fixed ratio to yield the required nodeless double-c

functions for IDO. We were unable with any of these choices to develop

a systematic model useful for predicting molecular geometries (see later

discussion of resonance integrals).

We have adapted the following procedure on selecting an effective
80 "
basis set. Knappe and Rosch calculated the lanthanides and their

mono-positive ions using the numerical Dirac-Fock relativistic atomic

program of Desclaux.81 From these wavefunctions jadial expectation

values , and are calculated for 6s, 6p, 5d and 4f

functions. The 6s, 5d and 4f wavefunctions were obtained by Dirac-Fock

calculations on the promoted, 4fm-35d16s2 configuration; the 6p from

calculations in which a 5d electron was promoted, 4fm-36s26p.

Wavefunctions for the mono-positive ions are obtained from 4,-33d 6s

and 4fm-36':6pl respectively. A generalized Newton procedure was then

used to determine exponents () and coefficients anlm for a given set of

, and with functions of the form of Eq. (2-9b). Again, as

in the transition metal atoms, we found that a single function fits

the ns and np atomic functions well in the regions where bonding is

important, but the (n-l)d, and now the (n-2)f require at least two terms

in the expansion of Eq. (2-9b). This is demonstrated for the Ce+ ion in

Figure 2-1, where it is shown that a single-C expansion is poor for the

outer region of the 4f function.

In Figure 2-2 the value of is plotted versus atomic number. The

contraction of the 6s and 6p orbitals due to relativistic effects (DF

vs. HF) is quite apparent here, and is a consequence of the the greater

core penetration of these orbitals. Subsequent expansion of the 4f and

5d, now with increased shielding, results. After some experimentation

we use the Dirac-Fock values obtained from thf: mono-positive ions. The

basis set adopted is given in Table 2-1.

The 4f and 5d functions are quite compact. At typical bonding

distance (4f4f iv) and (5d5dlpu) are essentially RAB. Because of this

we calculate all two-center two-electron integrals with the C1 values in

Table 2-1. This value is chosen to match the accurate e FO Slater-Condon

Factors obtained from the numerical atomic calculations by a single

exponent, via

F(4f4f) = 0.200905 C (4f) (2-10a)

F(5d5d) = 0.164761 (5d) (2-10b)

F(6s6s) = 0.139803 C (6s) (2-10c)

F(6p6p) = 0.139803 C (6p) (2-10d)

The error in calculating two-centered two-electron integrals at typical

bonding distances with this single-c approximation is well under 17, and

this procedure is much simpler.

Core Integrals

The average energy of a configuration of an atom or ion is given

Single vs. Double Zeta 41-STO Orbital Amplitude










1.5 2.0 2.5 3.0 3.5 4.0 4. 60
r in a.U.

Figure 2-1: Radial wavefunction for the 4f orbital of Ce+ with single-C
and double-C Slater type orbitals (STOs).

0 0.6 1.0


*/ \

- '\
<- \


r .5


DF vs. HF Average Values of r for the 4f,5d.6s and 6p Orbital



60 62 64 66
Atomic number

68 70

A HF 8p

a DF- a

x DP 5d
* HW 6d
+ DI-4
o HI 4


Figure 2-2: Average value of r for the valence orbitals of the
lanthanides from a relativistic calculation (DF) and a non-
relativistic calculation (HF).


-.- .- -



w .
4,0 4

0 4.1 .

u 0 0
C 0

0 -a

.4 .a1

0a 1
o 'o "o

r *

C0 A

-4 4J


U0 0

a 3

0 -4
o4 A
4.1 0


0 C

U4 -.4

4 C




cu M

'O& n O r- N -t r- 4 r4 rf4

n .

NJ 4.

* 0

0' 0

fn ca



_1 I 1 I 1 r f

. .

I I I . .


I I I r t r ) t r


1- a 1 w 6

E sk md-q = k Uss+ mUpp + nUdd + qU + -1 W
(Sss pp dd ff 2 ss


m(m-1) + n(n-) + q(-l +kmW + knWs +
2 pp 2 dd 2 ff sp sd

kqWsf + mnWpd + mqWVp + nqVdf

with W.., the average two electron energy of a pair of electrons in


ls Xi

and Xj given by






= F(ss)

= FO(pp)

= Fo(dd)

= F(ff)

= FO(sp)

= FO(sd)

= FO(sf)

= F(pd)

= FO(pf)

= Fo(df)

The core integrals


2/63F2(dd) 2/63 F (dd)

4/195F2(ff) 2/143 F4(ff) 100/5577 F6(ff)


1/10 G2(sd)

1/14 G3(sf)

1/15 G1(pd) 3/70 G3(pd)

3/70 G2(pf) 2/63 G4(pf)

3/70 Gl(df) 2/105 G3(df) 5/231 G5(df)

U.., Eq. (2-5), are then evaluated by removing an

electron from orbital Xi, and equating the difference in configuration

energy between cation and neutral to the appropriate observed IP(n). We

prefer this procedure rather than that suggested by others that average
the value obtained from IP(n) and EA(n).77

There are a great many low lying configurations of the lanthanide

atoms and their ions. The lowest terms of Ce, Gd and Lu come from fn-3d

s 2, while the remaining lanthanide atoms have the structure fn-2 s2

Two processes are then possible for 6s electron ionization:

I fn-3d1s2 fn-d s + (s)

II fn-2 s2 fn-2sl + (s)

The ionization energy of a 6s electron from I is systematically 0.4 -

0.5 eV larger than that obtained from II. When combined with Eq. (2-

11), the estimate for Uss differ by less than 0.1 eV. That is, choosing

the values of process I, the use of Eq. (2-11) predicts the values of

process II within 0.1 eV. We thus choose the values of process I shown

in Table 2-2. These values are obtained from the promotion energies of

Brewer85, 86 and then smoothed by a quadratic fit throughout the series.

For completeness, we also give the values of process II.

The lowest configuration containing a 5d electron is fn-3d s2

throughout the series, and 5d ionizations are obtained from

III fn- d1s2 fn-3s2 + (d)

The ionization potentials for the 6p can be obtained from two


IV fn-2 slp1 4 n-sl + (p)

V fn-3 s2p -3s2 + (p)

Ionization from process IV is nearly constant at 3.9 eV, from V at

4.6 eV. The fn 2s2 configuration is lower for all the lanthanides

except Ce(fdsp), and Gd and Tb(fn-3s2p). Using il ionization

potentials of process IV, and Eq. (2-11), we predict the values of

process V to within 0.2 eV. We do not consider this error significant,

and thus use the smoothed values from IV given in Table 2-2. The values

from process V are also given in the table for comparison.

For a f orbital ionization, we consider the two processes

VI f,-3 ds2 fn-4ds2 + (f)

VII fn-2 s2 n-3s2 + (f)
(compare with I and II). As seen in Table 2-2 the values form the two

processes are very different. From Eq. (2-11)

U (VI) = IP(VI) (m-4) W 2Wf Wdf (2-13a)

U f (VII) = IP(VII) (m-3) Wf 2Wsf (2-13b)


.4 a
41 0 I)
5r. -

3 a a

0 U U-4 41
0 0

o 4 a 4

0 ~


0 In 14 r.
I 00


to 1u 0 0
In I -I

S3 44 -4

a 0

40 C

0.4 4 *

40-4 m

41 U. *
0 N
o C H

O -4 In

N X 1


S>I o0 in In v w Ch w 0) 0% r-
N m 4 0 InN 'C a Mn 1 n %a rN
N in -W i 0% T wa en m -C
N rI co c a m 0

-.4 .4
x u
.-.I I-
S 04 0 v' CN in r. 'o ir '. v1 m '.4 m o
44 Ln %o rI 'o In Imp r4 N IC r1 o

U .
S -4 I "- 0 r in 0 1 mn o 0% IC ( i

,0 0 M :
H 'N 'I N m n Ln 10 In I 'C I' 0 I

1 In a 1 1N 0 ean i-a in '0 0 0 -

.4 I 'o r- qo 1) 0o N 0o 1. 0 'C i

N a

O 1 0%o N i^. .I In m M in o N n
H 0 > 'C 0 o (A 0a C 0 0 0 i

o 'C N 0o n en 0p% r' In en M .4 0% r 'o I

il -.- -- I--__ _

ITn o on no I 0 0 In 10 V a p0 e -
41 % 0. 0 0. 0 w N

0 ID 1 o4 O I I v o .0 >, 0 1 1 .0 -:
41 0. Z* C. Ill Cyi l. u & 0 Q (. F. >1 l

Unlike the analogous situation for the 6s and 6p orbitals, use of

Eq. (2-13a) to find Uff, and use of this value in Eq. (2-13b) to predict

IP(VII) is not successful, and would require the scaling of the large

Fo(ff) integral often performed in methods parameterized on molecular


As with the transition metal nd orbitals we might envision the

following procedure. We assume that the lanthanide atom in a molecule

is a weakly perturbed atom. The lowest energy configuration of the atom

should than be most important in determining Uff. We create a two-by-

two interaction matrix

( n-32 2ds2 C = 0 (2-14a)
S V E(f-2s2) X C2)
2 2
where V is an empirical mixing parameter, and C1 and C22 determines the

relative amounts of each of the two configurations that are important.

The exact value of V would depend on a given molecular situation. C1

is then given by

C = (2-14b)
1 1+X2

X = C/C= E(f -3d s2) E(f"-2s2) + (2-14c)

E(f-3d s2) E(f-2s2 1

The values of C1 appear in Table 2-2, where we have used the values of

E(fm-3ds2) and E(f-2s2) obtained for the promotion energies of

Brewer and a fixed value of V = 0.02 au. Then Uff could be obtained


U = C12 ff(VI) + C22 U ff(VTI) (2-15)

In the case of the 3d orbitals this valence bond mixing between 3dn2 s

and 3d n-s was important in obtaining reasonable geometric

predictions,63 an observation now confirmed in careful ab initio
studies.8 For the lanthanide complexes of this study the 4f orbitals

are quite compact, and this valence bond mixing does not greatly affect

geometries. However, the calculation of ionization potentials that

result in states with reduced f-orbitals occupation is influenced.

There are many refinements one can make in the formation of a

"mixing" matrix such as Eq. (2-14a). One might be to make V dependent

on the calculated population of the 4f and 5d atomic orbitals. However,

the values of the promotion energies we obtain from Brewer are so

different than those that we obtain from out own numerical calculations

on the average energy of a configuration, Table 2-3, that for the moment

we choose a 76% : 24% mix of E(fm-3ds2) : E(fm-2s2) for all the atoms of

the series. This mix gives reasonable geometries and ionization

potentials for all molecules of this study. Further refinements will

require more accurate atomic promotion energies and numerical experience

with the model.

Resonance Parameters, B(k)

Each lanthanide atom has three B (k) values, B(s) = B(p), B(d) and

B(f), and those we choose are summarized in Table 2-4. They are

obtained by fitting the geometries of the trihalides, and the more

covalent bis-cyclopentadienyls to be reported elsewhere.

Bond lengths are most sensitive to B(d) and bond angles to B(p).

These angles can be reproduced solely on a basis set including 6p

orbitals, and we have been able to obtain satisfactory comparisons with

Table 2-3 : Average configuration energy from Dirac-Fock calculations
on the fn-3ds2 and the f" s2 configurations for all the
lanthanide atoms.

Atom Energy

fn-3d s fn-ms

Ce -8853.71494569 -8853.64980000

Pr -9230.41690970 -9230.3/981848

Nd -9616.94751056 -9616.93446923

Pm -10013.4526061 -10013.4606378

Sm -10420.0710475 -10420.0976615

Eu -10836.9533112 -10836.8834715

Gd -11264.0945266 -11264.0439334

Tb -11701.7877496 -11701.7482691

Dy -12150.1565528 -12150.1286785

Ho -12609.3663468 -12609.3484161

Er -13079.5686394 -13079.5585245

Tm -13560.9236801 -13560.9201649

Yb -14053.5770354 -14053.5786047

Lu -14557.7153258 --------

a) Reference 81.


Table 2-4 :Resonance integrals (B values) for the Lanthanide atoms in e
V. The beta for the s-orbital is set equal to the beta for
the p-orbital.

Atom B(s) B(p) B(d) B(f)

Ce -8.00 -8.00 -17.50 -80.00

Pr -7.61 -7.61 -17.58 -80.00

Nd -7.23 -7.23 -17.65 -80.00

Pm -6.85 -6.58 -17.73 -80.00

Sm -6.46 -6.46 -17.81 -80.00

Eu -6.08 -6.08 -17.88 -80.00

Gd -5.69 -5.69 -17.96 -80.00

Tb -5.31 -5.31 -18.04 -80.00

Dy -4.92 -4.92 -18.11 -80.00

Ho -4.54 -4.54 -18.19 -80.00

Er -4.15 -4.15 -18.27 -80.00

Tm -3.77 -3.77 -18.35 -80.00

Yb -3.38 -3.38 -18.42 -80.00

Lu -3.00 -3.00 -18.50 -80.00

experiment without the necessity of including the 5p orbitals. On the

other hand, orbitals of p symmetry do seem to be required for accurate
predictions of geometry.5859

It has been argued that the 4f orbitals are not used in the chemical

bonding of those complexes except in the more covalent cases.55'56 From

the present study we are lead to the conclusion that some, albeit small,

contribution is required of these orbitals to obtain the excellent

agreement between experimental and calculated bond lengths for the

series MF3, MCI3, MBr3 and MI3 and for the comparative values obtained

for CeF3 and CeF4. This is indicated in Table 2-4 by the large values

of IB(f) The latter values are a consequence of the fact that the f-

orbitals are tighter than one usually expects for orbitals important in

chemical bonding. Use of 5d orbitals alone will predict the trends in

these two series, but underestimates the range of values experimentally


Two Electron Integrals

Several different interpretations have been given to the INDO

scheme. The simplest of those schemes is to include only one-centered

integrals of the Coulomb or exchange type

(uulvv) or (uvlvu)
For an s,p basis these are complete. For an s,p,d or s,p,d,f basis they

are not, and the omission of the remaining integrals will lead to

rotational variance. To restore rotational invariance, integrals of
this type might be rotationally averaged,88 but from a study of spectra

it appears that all one-center integrals should be evaluated.6 For

example, in the metallocenes the integral (d 2 2 d xId d ) is

required to separate the two transitions that arise from the elg(d) 4

e (d) transitions that lead to the Elg and E2g excited states. In

addition, it appears that the inclusion of all one-center integrals

improves the predictions of angles about atoms with s,p,d basis

sets89,90 and considerably improves the predictions of angles about the

lanthanides. For these reasons we include all the one-center two-

electron integrals. Since the INDO programs we use process integrals

and their labels in the MOLECULE Format91 only the additional integrals

need he included. These integrals are generated in explicit form via a

computer program that we have used in the past63 and they have also been

recently published by Schulz et al.89 To our knowledge all these

integrals do not appear in the literature for s,p,d and f basis,

although we have checked those of (uujvv) and (uJv\uv) against the

formulas of Fanning and Fitzpatrick.83

Integrals of the form (uujvv) and (uvlvu) can be obtained though

atomic spectroscopy, and their components, Fk and Gk, evaluated via

least square fits

(i~ivv) = C ak Fk
(PIvlu) = T bk Gk

These Fk and Gk can then be used to evaluate all integrals of the "F" or

"G" type, even those that do not appear in atomic spectra because of

high symmetry (i.e. (d 2 2 d zdxz d )). Integrals of the "R" type,
x -y yz xz xy
however, cannot be evaluated in this manner; viz.(sdldd), (splpd),

(sdlpp), (sdlff), (sfldf), (pplpf), (ddlpf), (pdldf), (sdlpf), (pdlsf),

(spldf), and (pflff). For this reason we evaluate all one-center two-

electron integral- of the lanthanides using the basis set of Table 2-1,

which yields the exact FO value obtained from the Fock-Dirac numerical

calculations. All Fk, Gk and Rk integral for k > 0 are then scaled by

2/3. This value of the scaling is obtained from a comparison of the

calculated and empirically obtained5892-94 F2(ff), F4(ff) and F(f)

values that implies 0.66 + 0.04. Empirically obtained values of Gk(fd)

and F (fd) are far more uncertain and are much smaller, and are thus not

used to obtain this scaling value between calculated and experimental


At this point it seems appropriate to point out the differences of the

present INDO model to that suggested by Li Le-Min et al.59 In the

latter formalism only the conventional one-center two-electron integrals

are included leading to rotational variance. In addition, the

Wolfsberg-Helmholz approach is used for the resonance integral B,

B..=(IP(i)+IP(j))S../2. No geometry optmization has been reported

within their model.59 Further differences are the restriction to

single-c STOs and the smoothing of the valence orbital ionization

potentials for the lanthanides via Anno-type expressions.5


The input to the INDO program consists of molecular coordinates and

atomic numbers. Molecular geometries are obtained automatically via a

gradient driven quasi-Newton update procedure,57 using either the

restricted or unrestricted Hartree Fock formalism. All UHF calculations

are followed by simple annihilation.62

Self-consistent field convergence is a problem with many of these

systems. For this reason electrons are assigned to molecular orbitals

that are principally f in nature according to the number of f-electrons

in the system, and the symmetry of the system. Orbitals with large


lanthanide 5d character are sought and assigned no electrons. A

procedure is then adopted that extrapolates a new density for a given

Fock matrix based on a Mulliken population analysis of each SCF cycle.93

Often this procedure is not successful. In such cases all f

orbitals are considered degenerate, and they are equally occupied in the

highest spin configuration using the RHF open shell method. These

vectors (orbitals) are then stored, and the SCF repeated with the

specific f orbital assignments as described above.

In cases of slow convergence, a singles or small singles and

doubles, CI is performed to check the stability of the SCF, and the

appropriateness of the forced electron assignment to obtain the desired



The geometries of CeC13 and LuC13 were used to determine an optimal

set of resonance integrals and configurational mixing coefficients. No

further fitting was performed, and thus the structures of all other

compounds are "predictions". The resonance parameters for the other

lanthanides were determined by interpolation from the values for Ce and

Lu (see Table 2-4). The INDO optimized geometries as well as the

remaining cerium and lutetium trihalides are listed in Table 2-5. In

addition to the trihalides reported, the geometry of CeF4 is also listed

in Table 2-5. One can see the agreement with experiment is good in all


The potential energy of the trihalides as a function of the out of

plane angle is very flat. Although we have optimized all structures

until the gradients are below 10-4 a.u./bohr, the angles are converged

only to 30. We note, however, that all are predicted non-planar, in

agreement with experiments.54b,97,98

Geometry and ionization potentials for Cerium and
Lutetium trihalides. Cerium tetrafluoride is also
included in this table. The bond distances are given
in angstromsg angles in degrees and IPs in eV.
Experimental results are also shown where available.

a) References 54b, 97 and 98.
LuF3 from Ref 103.

Estimated values for CeF3,CeI3, and

b) The SCF calculation on the ion of LuF3 would not converge
therefore no IP is reported.

Table 2-5 :

Bond Bond Ionization
Distance Angle Potential

CeF3 2.204 2.180 106.8 --- 8.4 8.0

CeCl3 2.570 2.569 115.6 111.6 10.0 9.8

CeBr3 2.668 2.722 115.8 115.0 9.6 9.5

CeI3 2.844 2.927 119.8 --- 9.9 ---

CeF4 2.099 2.040 109.5 109.5 --- ---

LuF3 2.045 2.020 107.4 --- b 19.0

LuCl3 2.415 2.417 108.2 111.5 18.6 (17.4 18.7)

LuBr3 2.528 2.561 108.6 114.0 17.8 (16.8 18.4)

LuI3 2.726 2.771 115.6 114.5 17.7 (16.2 18.1)


The experimental range of the bond lengths from LnF3 to Lnl3 is

greater than we calculate. Our predicted values for the trifluorides

and trichlorides are in good agreement, while bond lengths for the

tribromides and triiodides are too short. Since these are the more

polarizable atoms it is possible that configuration interaction will

have its largest affect on these systems. The calculated change in bond

length of 0.11 A in going from CeF3 to CeF4 is also smaller than the

0.14 A observed.

Ionization potentials (IPs) are also reported in Table 2-5. In all

cases the INDO values fall within the experimental ranges. These values

are calculated usinf the ASCF method, and only the first IP is

calculated. Experimentally54b,99 these valued are somewhat uncertain,

but they are split by both crystal field effects, and by the large spin-

orbit coupling not yet included in our calculations. However, the

latter interaction is treated implicitly in the DVM Xa calculations4

based on the Dirac equation. Therefore, the Xa result for the

ionization potentials show better agreement with the experiment in this

aspect, but it is quite remarkable that the present INDO approach is

able to reproduce the experimental trend in the first IP of the series,

CeX3, X = F, Cl, Br with a maximum value for the chloride, a feature
noticeably missing in the DVM Xa results.5

The initial success of the INDO model as implemented here lead us to

calculate both geometries and IPs for the remaining lanthanide

trichlorides. These results are shown in Table 2-6. The experimental

geometries54b,9798 are very well reproduced by the INDO calculations.

The INDO IPs reproduce the characteristic "W" pattern of the lanthanide

atoms, and fall within the experimental ranges.


Table 2-6 : Geometries and Ionization Potentials (IPs) for the
lanthanide trichlorides. Bond distances are reported in
angstroms, bgnd angles in degrees and IPs in eV.
Experimental results are also given where available.

Bond Bond Ionization
Distance Angle Potential

Ce 2.570 2.569 115.6 111.6 10.0 9.8

Pr 2.566 2.553 108.5 110.8 11.8 (10.9-11.2)

Nd 2.563 2.545 112.7 ---- 13.3 12.0

Pm 2.556 ---- 112.7 ---- 14.4 ---

Sm 2.544 ---- 113.0 ---- 15.3 (13.7-17.0)

Eu 2.532 ---- 113.2 ---- 16.4 ---

Gd 2.514 2.489 110.0 113.0 17.7 (15.5-16.5)

Tb 2.496 2.478 109.8 109.9 13.0 (13.0-20.5)

Dy 2.479 ---- 110.1 ---- 14.3 (14.0-20.0)

Ho 2.464 2.459 112.0 111.2 15.0 (15.5-20.0)

Er 2.448 ---- 110.9 ---- 15.6 (11.5-16.0)

Tm 2.430 ---- 108.5 ---- 15.9 (15.3-21.0)

Yb 2.421 ---- 109.6 ---- 15.9 (15.5-21.0)

Lu 2.415 2.417 108.2 111.5 18.6 (17.4-18.7)

a) References 54b, 97 and 99.


To test the applicability of our model to lanthanide atoms not

formally charged +3, we calculated the geometries and IPs for SmCl2,

EuCl2 and YbCl2 molecules. The results are given in Table 2-7. The

INDO model gives optimized geometries that are bent and in good
agreement with experimental results. We note that this bending is a

result of a small amount of p-orbital hybridization. It is not

necessary to invoke London type forces, and thus correlation, to

explain this effect.
We chose Ce(N03)6 as our last example because it is one of the few

known examples of a twelve coordinate metal. The optimized geometry is

summarized in table 2-8 and a plot of the optimized geometry is shown as

figure 2-3. As one can see from Table 2-8 INDO predicts a geometry that

is in excellent agreement with the experimental crystal structure.1

Table 2-9 shows a population study of this complex. Although there is

some f-orbital participation, it appears that this unusual twelve

coordinate Th structure results from electrostatic forces between the

ligands and the relatively large size of the Ce(IV) ion.

Geometry and ionization potential for SmC12, EuCl2, and
YbCl2. Bond distances are given in angstroms, bond angles
in degrees, and ionization potentials in eV. Experimental
resultsa are listed where available.

a) Reference 100.

Table 2-7 :

Bond Bond Ionization
Molecule Distance Angle Potential


SmCI2 2.584 --- 143.3 130+15 5.3

EuCI,, 2.576 143.2 13515 6.6

YbCl2 2.400 --- 120.2 12605 3.2

Table 2-8 :

Average bond distances and bond angles for Ce(N03) 2 ion
INDO optimized geometry and the X-ray crystal structure
Distances are in angstroms and angles in degrees. The c
subscript on the oxygen atoms denotes the that oxygen is
bonded to the cerium and the n subscript signifies a non-
bonded oxygen.

a) Reference 101.

Parameter INDO Exp.

r(Ce-0 ) 2.554 2.508

r(N-Oc) 1.256 1.282

r(N-On) 1.237 1.235

O(0-N-0 ) 121.5 114.5

0(0 -Ce-0 ) 50.9 50.9
C c


Figure 2-3: Plot of the twelve coordinate Ce(NO3)2 ion. Nitrogens 2, 18
and 22 are above the plane of the paper, while nitrogens 6,
10 and 14 lie below the plane of the paper.

Table 2-9 :

Population analysis of Ce(N03)6 The oxygen atoms that are
coordinated to the cerium are indicated by 0 .The Vyberg
bond index is also given. A Wyberg index of 1.00 is
characteristic of a single bond.

Atomic Spin Total
Atom Orbital Population Density Valence

s 0.20 0.00

p 0.30 0.00

Ce d 1.32 0.00

f 1.10 1.00

Net 1.08 1.00 4.80

N Net 0.59 0.00 3.78

0 Net -0.40 0.00 1.58
0 Net -0.48 0.00 1.80


Bond Index

Ce 0 0.40

N 0 1.37

N 0 1.22


We develop an Intermediate Neglect of Differential Overlap (INDO)

method that includes the lanthanide elements. This method uses a basis

set scaled to reproduce Dirac-Fock numerical functions on the lanthanide

mono-cations, and is characterized by the use of atomic ionization

information for obtaining the one-center one-e]r'ctron terms, and

including all of the two-electron integrals. This latter refinement is

required for accurate geometric predictions, some of which are

represented here, and for accurate spectroscopic predictions, to be

reported latter.

We have applied this method to complexes of the lanthanide elements

with the halogens. The geometries calculated for these complexes are in

good agreement with experiment, when experimental values are available.

The trihalides are calculated to be pyramidal in agreement with

observation. The potential for the umbrella mode, however, is very

flat. The dichlorides of Sm, Eu and Yb are all predicted to be bent

even at the SCF level, again in agreement with experiment. This bending

is caused by a small covalent mixing of ungerade 6p and 4f orbitals, and

one need not invoke London forces to explain this observation. Again

the potential for bending is very flat.

Within this model, f-orbitals participation in the bonding of these

ionic compound through covalent effects is small. Nevertheless f-

orbitals participation does contribute to the pyrimidal geometry of .the

trihalides and the bent structure of the dihalides. In addition,

although the trend of bond lengths within the series LnF3, LnCl3, LnBr3,


and Lul3, and CeF3 and CeF4 are reproduced without f-orbital

participation, the range of values calculated is considerably improved

when f-orbitals are allowed to participate. For the twelve coordinate

Ce(N03)62 complex102 reported here, f-orbital participation appears

minor. A stable complex of near Th symmetry is obtained regardless of

the f-orbital interaction.


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Chris Culberson was born in Saint Petersburg, Florida. He graduated

from St. Petersburg Catholic High School. He obtained a Bachelor of

Science degree with honors in chemistry from Eckerd College. He is

married to Mary Kay Terns. After graduating from Eckerd College, he

went to Duke University to study quantum chemistry under the direction

of W. L. Luken. At Duke, the major portion of his research was devoted

to localized orbital methods. Two years later, he transferred to the

University of Florida to continue his studies under Michael C. Zerner's

guidance. In addition to the f-orbital chemistry detailed in this

thesis, a major portion of his time at the University of Florida was

spent exploring the use of electrostatic potentials (EPs) and examining

biochemical problems using EPs. While at the University of Florida, he

was given the chance to go to Germany.

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.

/ 1 / /

Michael C. Zerner, Chairman
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 Ohrn
Professor 6f Chemistry and Physics

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.

Willis B. Person
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 Weltner Jr.
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.


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.

May 1986
Dean, Graduate School

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