Citation
Localized orbitals in chemistry

Material Information

Title:
Localized orbitals in chemistry
Creator:
Culberson, John Christopher
Publisher:
[s.n.]
Publication Date:
Language:
English
Physical Description:
viii, 79 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Atoms ( jstor )
Centroids ( jstor )
Eggshells ( jstor )
Electrons ( jstor )
Geometry ( jstor )
Ionization potentials ( jstor )
Molecular orbitals ( jstor )
Molecules ( jstor )
Orbitals ( jstor )
Space probes ( jstor )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
Molecular orbitals ( lcsh )
Rare earth metals ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

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

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
001025258 ( ALEPH )
18007580 ( OCLC )
AFA7194 ( NOTIS )
AA00004838_00001 ( sobekcm )

Downloads

This item has the following downloads:


Full Text











LOCALIZED ORBITALS
IN CHEMISTRY










BY

JOHN CHRISTOPHER CULBERSON


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

UNIVERSITY OF FLORIDA


1987




LOCALIZED ORBITALS
IN CHEMISTRY
BY
JOHN CHRISTOPHER CULBERSON
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA TN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1987


ACKNOWLEDGEMENT
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.
11


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
to.
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
else.
iii


TABLE OF CONTENTS
Page
ACKNOWLEDGMENT ii
LIST OF TABLES v
LIST OF FIGURES vi
ABSTRACT vii
INTRODUCTION... 1
CHAPTER ONE LOCALIZED ORBITALS 3
Background 3
Double Projector Localization 6
Fermi Localization 9
Boys Localization 34
CHAPTER TWO LANTHANIDE CHEMISTRY 41
Background 41
Model 43
Procedures 61
Results 62
CONCLUSION 71
BIBLIOGRAPHY 73
BIOGRAPHICAL SKETCH 79
IV


LIST OF TABLES
Page
1-1 Probe electron points for furanone 18
1-2 Boys and Fermi hole centroids for C^H^02 19
1-3 Probe electron points for methlyactetylene 22
1-4 Boys and Fermi hole centroids for CHCH^ 23
1-5 Orbital centroids for BF^ 26
1-6 Eigenvalues and derivatives for BF^ using the
Boys method 27
1-7 Probe electron points for BF^ 28
1-8 Orbital centroids for BF^ 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 SmCl9,
EuC12 and YbCl2 .... 67
2-8 Geometry of CeiNO^)^ ion 68
_2
2-9 Population analysis of CeiNO^)^ ion 70
v


LIST OF FIGURES
Page
1-1 Fermi mobility function for t^CO 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)^ bonding orbital 38
1-6 Ni(CO)^ 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^)^ 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
LOCALIZED ORBITALS
IN CHEMISTRY
by
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 /-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)^.
Vll


The Intermediate Neglect of Differential Overlap (INDO) method is
extended to the /-orbitals, and the intrinsic localization of the /-
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
_2
CeiNO^)^ ion are presented. The model is also used to calculate the
ionization potentials for the above compounds. Due to the localized
nature of /-orbitals the crystal field splittings 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 (Cl) calculation must be done to insure that the converged
state is indeed the lowest energy state. The localized nature of the /-
orbitals in conjunction with the double projector localization method
may be used to isolate the /-orbitals in order to calculate only a Cl
restricted within the /-manifold.
viii


INTRODUCTION
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 /-orbitals should also fall into this
category.
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
1


2
such methods on several systems of chemical interest. Chapter two deals
with expanding the INDO method so that the prediction of /-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 /-orbitals, causes the /-/ 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 /-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 /-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 /-orbitals.


CHAPTER ONE
LOCALIZED ORBITALS
Background
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.^ 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
3


4
localized orbitals is their ability to simplify configuration
interaction (Cl) 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
n
G = Z , (1-1)
i = l ii
is maximized or minimized, where the definition of depends
on the localization criterion. One choice for the value of
2-3
is the two-electron repulsion integrals; 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
46
orbital self-extension operator,
2
gii = r12
(1-2)


5
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
5 3
to N Since the SCF procedure itself proceeds as N (semi-empirical)
4
or N (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 second category of
localization.
We claim that the underlying physical basis of localization is the
Fermi hole. The Fermi hole provides a diroot(non-itera five) method


6
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
12
a search of the Fermi hole mobility function. 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 /-orbital degeneracies are a
problem. The DP method allows one to separate the /-orbitals from the
other metal orbitals and use a small Cl 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
k
like to study n-rc 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 Cl may then be used to study
k k
only the n-Jt transitions and thereby elucidate the n-rt spectra.


7
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 Cl is
then required to restore the localized nature of the d type molecular
orbitals. Such a large Cl can be avoided by first doing a DP
localization.
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 it orbitals to obtain t orbitals,
this is not desirable since the a and it spectra will now be mixed and
more difficult to interpret. The it orbitals may be removed using the DP
"k
method, the remaining orbitals localized, and the n-it spectra
calculated using a small singles Cl. 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 anc* a set r
localized "pattern" orbitals where r is less than or equal to m.
These "pattern" orbitals are projected out of the set by
m
| r> = E I $.><$. 11 > (1-3)
' a .. 1 l l' a
1 = 1
J*
for a = 1 to r. These {|TO}^, are then symmetrically orthogonalized


8
T'+Y' = A (1-4)
-1/2
0 = Y' A (1-5)
and are projected out of the original set
#' = $ ( 1 I |9 ><0 | ) (1-6)
a 1 a a1
The matrix A' is formed and diagonalized
U = U+A U = X (1-7)
The X matrix will have r near zero eigenvalues corresponding to the
{0^)i that have been projected out. These eigenvalues and the
corresponding columns of U are removed. The new set of orthonormalized
orbitals fY J1!1 r is formed from
a 1
-1/7
Y = *'UX (1-8)
This set is an orthogonal complement to the set |0 >, but has no
a
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,
V+Y+FYV = er (1-9)
Y' = YV (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.


9
Fermi Localization
Background
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 hole^ ^ and the Fermi orbital.^^ Unlike localization
methods based on iterative optimization of some criterion of
localization,^ 6,15,16 tjlg 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
17-19
into localized orbitals, 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
20-25
localized orbitals such as the PCLIO method 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(r1;r2) = p(rx) 2 p2( ^, r2>/p( r2), (1-11)
where p(r^) is the diagonal portion of the first order reduced density
matrix and p(r^;r2) is the corresponding part of the second order
26
reduced density matrix. For special case of a closed shell SCF


10
wavefunction, the natural representation of the Fermi hole is the
13 14
absolute square of the Fermi orbital
A(r1;r2) = |f(r:;r2)|2. (1-12)
The Fermi orbital is given by
f(ri;r2) = [2/p(r2)]1/2 E gi(r1)g.(r2),
i
d-13)
where the orbitals g^(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(r^;r2) is interpreted as a function
of r^ which is parametrically dependent upon the position of a probe
electron located at r2-
12 13 27 28
Previous work 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
12 27 28
electron is measured by the Fermi hole mobility function,
F(r) = Fx(r) + F (r) + Fz(r) (1-14)
where
Fv(r) = ^2
p
I h
i>j
3vJ
&i
3v
d-15)
for v = x, y or z. This may be compared to


11
F0(p) = (3n/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)-Fg(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
29-33
Daudel. Regions where F(r) = F(p) resemble boundaries between
loges.
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.^ 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
, 4-6,15
localization.
Localized Orbitals Based on the Fermi hole
Equation 1-13 provides a direct relationship between a set of
canonical SCF orbitals g^(r) and a localized orbital fj(r) = f(r,r^)
where r^ is a point in a region where F(rj) < Fg(p(r^)). 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 n j = 1 to N, each of
which is located in a region where F(r_.) < Fg(p(r^)). Ideally, each of
these points should correspond to a minimum of F(r) or F(r)-Fg(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.


12
Figure 1-1: The fermi hole mobility function F(r) for the HCO based on
the geometry and double zeta basis set of ref. 4l. 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.


13
Figure 1-2: 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.


14
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.


15
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.


16
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
33
symmetric orthogonalization. The resulting unitary transformation is
given by
U = (TT+)~1/2T, (1-17)
where
T.. = gj(ri)/(p(r.)/2)1/2. (1-18)
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
34
problems. 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.


17
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, C^H^C^, and its derivatives are useful
35-37
reagents in 2+2 photochemical cycloadditions. The canonical SCF
molecular orbital for the furanone molecule were calculated with an STO-
38
3G basis set and the geometry specified in Table 1-1. The molecular
geometry was restricted to Csymmetry, 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=0 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


18
Table 1-1 : Molecular geometry and probe electron points for the
furanone (C^H^C^)- 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
0.0
0.0
0.0
Atom C£
0.0
2.589
0.0
Atom
2.671
3.355
0.0
Atom
4.363
0.940
0.0
Atom 0^
2.534
-1.071
0.0
Atom C>2
3.427
5.548
0.0
Atom
-1.802
-1.040
0.0
Atom H2
-1.802
3.629
0.0
Atom
5.560
0.858
1.698
Atom H.
4
5.560
0.858
-1.698
Cf-Cz bond 1
0.0
1.295
2.000
^1~^2 kncl 2
0.0
1.295
-2.000
k011^
1.336
2.972
0.0
C0-C. bond
3.517
2.148
0.0
C^-0^ bond
3.449
-0.066
0.0
Cj-0^ bond
1.267
-0.536
0.0
C2-O2 bond 1
3.049
4.452
2.000
C3-O2 bond 2
3.049
4.452
-2.000
0^ lone pair 1
2.654
-1.821
0.660
lone pair 2
2.654
-1.821
-0.660
C>2 lone pair 1
2.707
6.298
0.0
C>2 lone pair 2
4.267
5.698
0.0


19
Table 1-2 : Orbital centroids for localized orbitals determined by the
Fermi hole method and by the orbital centroid method for the
furanone molecule (C^H^O^. All coordinates are given in
bohr
Orbital
Fermi hole method
Centroid criterion
X
Y
Z
X
Y
Z
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.O
2.671
3.355
0.0
C4 K shell
4.362
0.940
0.0
4.326
0.939
0.0
0X 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
C^-H^ 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
C,-H0 bond
4 3
5.149
0.895
1.132
5.155
0.888
1.153
C.-H. bond
4 4
5.149
0.895
-1.132
5.155
0.888
-1.153
crc2 bond 1
0.008
1.401
0.635
0.030
1.410
0.599
CrC2 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
C^-C^ bond
3.542
2.190
0.0
3.565
2.149
0.0
C--0, bond
3.268
-0.243
0.0
3.243
-0.225
0.0
C.-0. 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-O2 bond 2
3.093
4.584
-0.548
3.114
4.612
-0.511
lone pair 1
2.554
-1.283
0.440
2.594
-1.311
0.472
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
3.970
5.589
0.0
3.970
5.600
0.0


20
determined by the orbital centroid criterion.^ ^>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 T^g of less than 10-^, where TRM<, is the RMS
value of the off-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


21
orbital centroid criterion, over 200 iterations were required to
determine a set of orbitals which satisfied a very weak criterion of
34
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
38 39
determined by an ST0-5G basis set and an experimental geometry.
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 single bond. The remaining three points were located two
bohr from the 1^ rotation axis at a point midway between the and
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


22
Table 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
bohr.
Position
X
Y
Z
Atom
0.0
-1.140
0.0
Atom C2
0.0
1.140
0.0
Atom C^
0.0
3.897
0.0
Atom
0.0
-3.145
0.0
Atom
-1.961
4.438
0.0
Atom
0.980
4.438
1.698
Atom
0.980
4.438
-1.698
bond 1
-2.0
0.0
0.0
^1^2 bn^ ^
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


23
Table 1-4 : Orbital centroids for localized orbitals determined by the
Fermi hole method and by the orbital centroid method for
the methylacetylene molecule. All coordinates are given
in bohr.
Orbital
Fermi hole method
Centroid criterion
X
Y
Z
X
Y
Z
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
C^-H^ ^on<^
0.0
-2.486
0.0
0.0
-2.502
0.0
C3-H2 bond
-1.302
4.270
0.0
-1.316
4.272
0.0
C^-H^ bond
0.651
4.270
1.127
0.658
4.272
1.139
C--H, bond
3 4
0.651
4.270
-1.127
0.658
4.272
-1.139
^l-^2 bnd
-0.713
-0.003
0.0
-0.692
-0.U13
0.0
1-C2 bond 2
0.356
-0.003
0.617
0.346
-0.013
0.599
f'l-^2 *30nc^ ^
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


24
orbital centroid method required 26 minutes starting from the canonical
SCF molecular orbitals or 19 minutes using the Fermi localized orbitals
8
as an initial guess to reach T^M<, of 10 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 axis. The
point group of the orbital centroids for this conformation is C~,.
oh
The "three-up" conformation is generated by rotating the set of lone
pair orbitals on each fluorine atom in the pinwheel conformation by 90


25
degrees about each F-B axis. The point group for the orbital centroids
of this conformation is C^v- The "up-up-dovn" 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 BF^ were calculated based on the
double-zeta basis set and geometry tabulated by Snyder and Basch.^
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
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


26
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
bohr.
Orbital
up-up-up (C3v)
up-up-down (Cg)
X
Y
Z
X
Y
Z
B-F^ 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-F^ bond
-0.862
-1.494
-0.072
-0.862
-1.494
0.069
F^ 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
?2 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
F^ 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


27
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 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.
Configurations
up-up-up
up-up down
pinwheel
Fermi hole
Sum
69.445360
69.444985
69.438724
69.341094
X1
-0.019521
-0.018174
+0.017234
+0.031444
X2
-0.019850
-0.018914
+0.016238
+0.029932
*3
-0.019850
-0.019095
+0.016236
+0.029928
X4
-0.155480
-0.153827
-0.193521
-0.195502
X5
-0.155484
-0.155181
-0.ln4366
-0.195524


28
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 C^)- All
coordinates are given in bohr.
Position
X
Y
Z
B-F^ bond
1.223
0.0
0.0
B-F9 bond
-0.611
1.059
0.0
B-F^ bond
-0.611
-1.059
0.0
F^ 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
F£ 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
F^ 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


29
Table 1-8 : Orbital centroids for localized orbitals determined by the
Fermi hole method and by the orbital centroid method for
the boron Lrifluoiide molecule. The four innershell
orbitals have been excluded from these calculations. All
coordinates are given in bohr.
Orbital
Fermi hole
Centroid criterion
X
Y
Z
X
Y
Z
B-F^ bond
1.713
0.0
0.0
1.720
0.028
0.0
B-F^ bond
-0.857
1.484
0.0
-0.884
1.475
0.0
B-F^ bond
-0.857
1.484
0.0
-0.835
1.504
0.0
F^ lone 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
F^ 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


30
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
criterion.
We do not intend to attribute any special physical significance to
any of the lone pair configurations for BF^. 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.
Conclusions
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.^^


31
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
i
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
localized.
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


32
Al A2
such as molecular mechanics 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, spectroscopic constants, and other
46 47
properties. 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
A3 A7
their utility in subsequent calculations. In order to apply the
12 28
current method to such molecules, the Fermi hole mobility function
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)-Fp(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 Fq(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 Fq(p).


33
Consequently, it is not possible to represent methylacetylene with a
structure like H-C=C=CH^ 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
problems.
Boys Localization
The most widely used form of localized orbitals are those orbitals
5 6
based on the method of Boys. 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 computational step, but the Boys method may be


34
formulated in terms of products of molecular dipole integrals. The
dipole integrals are one-electron integrals therefore the transformation
. 3
is on the order of N 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 unitary transformation relating a delocalized set of
orbitals to a localized set, but the form of this transformation is in
general unknown. The common method foi 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 f1 one can increase the localization by
doing a unitary transformation IT*",
fI + 1 = U1^ (1-19)
where f^+^ is the resulting set of more localized orbitals. The
original matrix formulation is based on a sequence of pairwise
2
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-l)/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
rotations.
Leonard and Luken have developed a second order method that does all
of the N(N-l)/2 rotations at once rather than one at a time.''"* The use
of a second ofder method may have the additional benefit of improving
the convergence difficulties encountered in iterative methods. Their
method is outlined below.


35
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+)~1/2. (1-21)
The matrix R will be defined as
R = NT, (1-22)
where
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 {f^, . f^) one produces a
set of more localized orbitals {f^, . f^). The new value of the
localization, G', is given by
N
G' = Z (i'i'.i'i') (1-25)
i'=l
The new f' orbitals can be thought of in terms of a pertubative
expansion,
f' = f + E t.f + £ tTtTf (1-26)
I I,J
to second order. The t^ matrix does 2x2 rotations that mix in portions
of all the occupied orbitals into orbital f'. The third term t^-tj is
the product of a pair of 2x2 rotations. When this form of the f'


36
orbitals is substituted into the G' equation, you obtain
G' = Gq + I tjGjd) + Z tjt^d.J) (1-27)
Leonard and Luken^ include the G^ second order term to accelerate
convergence when one is in the quadratic region; only the first order
term 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.
Results
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 fot the Ni(CO)^ ion. In a recent experimental paper by Reutt
et al., the photoelectron spectroscopy of Ni(C0)^is 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 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 t (banana) bonds, (3) nickel carbon bonds, and


37
(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 Cl
calculation to see if one could predict the photoelectron spectra for
Ni(C0)4.
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.


38
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.


39
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.


40
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.


CHAPTER TWO
LANTHANIDE CHEMISTRY
Background
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
/-element chemistry, but much modern work is concerned with organo-/-
49
element reactions, and the use of lanthanides and actinides as very
specific catalysts.Unlike the corresponding chemistry involving
the d metals, very little explanation is offered for much of this
chemistry.
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
/-orbitals as valence orbitals is known, although experience is now
being gained on the use of / orbitals as polarization orbitals.
Finally, the /-orbital elements are sufficiently heavy that relativistic
effects become important. Very few ab initio molecular orbital studies
52 "
have been reported on /-orbital systems. Extended Huckel calculation,
53
however, have been successful in explaining some of this chemistry.
Scattered wave and DVM Xa studies of /-orbital systems have also proven
effective, especially in examining the photoelectron spectroscopy of
reasonably complex systems.
We examine an Intermediate Neglect of Differential Overlaps (INDO)
technique for use in calculating properties of /-orbital complexes. At
the Self-Consistent Field (SCF) level this technique executes as rapidly
41


42
!
on a computer as does the Extended Huckel method, and considerably more
rapid than the scattered wave Xa method. Since the electrostatics of
the INDO method are realistically represented, molecular geometries can
be obtained using gradient methods.^ 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 thr^e systems. With
configuration interaction (Cl) this model should also be useful in
examining the UV-visible spectra of /-orbital complexes. Preliminary
studies of /-orbital chemistry using an INDO model have been disclosed
58
by Clack and Warren and, more recently, by Li-Min, Jing-Quing, Guang-
. 59
Xian and Xiu Zhen. 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 efiects 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 /-orbitals aie generally tight, and ligand field splittings thus
small, a great many states differing only in their /-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?


43
Model
The INDO model Hamiltonian that we use was first disclosed by Pople
and collaborators,*^ and then adjusted for spectroscopy*^ and extended
6 2 6 A ,
to the transition metal series. The details of this model are
6 2 6
published elsewhere. 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
k. k
"core" integrals and the Slater-Condon F and G integrals that are used
for the formation of one-center two-electron integrals. The evaluation
of these integrals using experimental information has traditionally made
' 65 66
this model highly successful in predicting optical properties.
We employ 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(/). These parameters will be chosen to give satisfactory
geometries of model systems. Another choice is one that gives good
prediction:' of UV-visible spectroscopy.^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 approximation*^ F(n) =
IP(n) EA(n), (IP = Ionization Potential, EA = Electron Affinity) and
the two-electron two-center integral from one of the more successful
functions established for this purpose.


44
At the SCF level, we seek solutions to the pseudo-eigenvalue problem
F C = C e
(2-1)
with F, the Fock or energy matrix, C, the matrix compound of Molecular
Orbital (MO) coefficients, and e, a diagonal matrix of MO eigenvalues.
The above equation is for the closed shell case (all electrons paired)
The unrestJicted Hartree Fock case is discussed in detail elsewhere,*^
71-73
as is the open shell restricted case. Although nearly all /-
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:
,AA
= ir
AA
'MU
S P
[ c A
crA
(UU | crX) j (ya|yX)
(2-2a)
FAA
MV
+ £ PCTCT (UU| a c B
BM
E Nx
a, X
(U v|aX) j (ya|vX)
1
2
ba(u) + Bb(v)
S ir P (yy I vv)
y\i 2 yv v 1 '
y *v
A*B
(2-2b)
(2-2c)
where
(yv|uX) = Jdx(l)dx(2) yi) Xv(l) 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


45
P
yv
MO
£ c C n
ya \>a a
a
(2-4)
AB
with n the occupation of MO $ n = 0,1,2. In Eq. (2-2), F refers
3 3 3 MV
to a matrix element with AO centered on atom "A". The core
integral
C (x£l \ ^ T + ',Alxi) (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 U will
remove the necessity for explicit consideration of this term. The bar
over an orbital in an integral,
_A_A
such as (y y | 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;^ i.e. (yAvA|, X^X^* The last term in Eq. (2-2a)
represents the attraction between an electron in distribution X^* X^ and
all nuclei but A. The rationale for replacing integral
A A
(yA|RB1|uA) (¡j y |SBSB) (2-6)
is given elsewhere, and compensates for neglected two center inner
27 28
shell-valence shell repulsion and neglected valence orbital
28 29
(symmetrical) orthogonalization. Zg is the core charge of atom B


46
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.
13-15
S of Eq. (2-2c) is related to the overlap matrix D, and is
given by
Syv 2 ^U(l)v(l) gy(l)\j(l) 0*(l)|v(l)) (2-7)
1=0
where is the Eulerian transformation factor required to rotate
from the local diatomic system to the molecular system, (y(l) |\>(1)) are
the sigma (1=0), pi(l=l), delta(l=2) or phi (1=3) components to the
overlap in the local system, and 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,
61-63 68
and set all / = 1 except between p symmetry orbitals viz.
S pp = 1 267papcj (pcTlpCT) + -585gpitpJt(prtlpTt) (2'8)
+ n-585Vpn
Basis Set
In general ZDO methods choose a basis set of Slater Type Orbitals
(STO)
1/2
rn-1e~^r y!J (9, ) (2-9a)
where Y^(0,4>) are the real, normalized spherical harmonics. Atomic
orbitals Xy are expressed as fixed contractions of these {Rn],m}
nlm
LILI
2n+l
2n!


47
(2-9b)
In general a single R function describes the s and p orbitals for
most atoms. The d orbitals of the transition metals, hovever, require
at least a double-C type function (tvo terms in 2-9b) for an accurate
description of both their inner and outer regions. For the lanthanides
59
wo have examined basis sets suggested by Li Le-Min et al., by Bender
78
79
and Davidson,
and by Clementi and Roetti.
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 1MD0. 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
81
program of Desclaux. From these wavefunctions jcdial expectation
2 3
values , and are calculated for 6s, 6p, 5d and 4/
functions. The 6s, 5d and 4f wavefunctions were obtained by Dirac-Fock
calculations on the promoted, 4/m_^5d^6s^ configuration; the 6p from
calculations in which a 5d electron was promoted, 4/n_^6s^6p'*.
Wavefunctions for the mono-positive ions are obtained from 4fm_^3d'*'6s'*
and 4/m_^6.1 6p^ respectively. A generalized Newton procedure was then
used to determin exponents (C) and coefficients for a given set of
2 3
, and with functions of the form of Eq. (29b). Again, as
in the transition metal atoms, we found that a single C function fits
the ns and np atomic functions well in the regions where bonding is


48
important, but the (n-l)d, and now the (n-2)/ 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 4/ 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 4/ and
5d, now with increased shielding, results. After some experimentation
we use the Dirac-Fock values obtained from the mono-positive ions. The
basis set adopted is given in Table 2-1.
The 4/ and 5d functions are quite compact. At typical bonding
distance (4/4/|yy) and (5d5d |yy) are essentially R~^. Because of this
we calculate all two-cenrer two-electron integrals with the values in
Table 2-1. This value is chosen to match the accurate F Slater-Condon
Factors obtained from the numerical atomic calculations by a single
exponent, via
F (4/4/) = 0.200905 C (4/) (2-10a)
F(5d5d) = 0.164761 C (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 IX, and
this procedure is much simpler.
Core Integrals
The average energy of a configuration of an atom or ion is given.


49
Single vs. Double Zeta 4-ST0 Orbital Amplitude
0.20
0.17
0.13
So.io
K
0.07
0.03
0.00
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
r in a.u.
Figure 2-1: Radial vavefunction for the 4f orbital of Ce with single-C
and double-C Slater type orbitals (STOs).


50
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).


Table 2-1 :Slater type orbital (STO) basis functions for the lanthanide atoms.
The single C functions are listed for the if, 5d, 6s and 6p orbitals
along with the double C functions for the 4/ and 5d orbitals.
Atom
Single C
Double C
Exponents
Exponents
Coe fficient s
if
5d
6 s
6p
if
5d
4/
5d
Ce
4.439
2.061
1.548
1.361
6.118
2.522
3.077
1.581
.7159
.4575
. 5003
. 6334
Pr
4.657
2.102
1.571
1.380
6.393
2.646
3.126
1.598
.7175
. 4 542
.5104
.6257
Nd
4.681
2.138
1.593
1.398
6.648
2.75"
3.169
1.612
. 7196
.4 510
. 5202
.6182
Pm
5.053
2.171
1.614
1.415
6.119
2.858
3.208
1.623
. 7220
. 4479
. 5295
.6112
S m
5.236
2.201
1.635
1.432
7.120
2.951
3.244
1.633
. 7247
.4450
.5238
.6046
Eu
5.414
2.229
1.656
1.448
7.342
3.038
3.278
1.640
. 7275
. 4421
. 5466
. 5986
Gd
5.565
2.243
1.678
1.463
7.523
3.091
3.288
1.634
.7332
. 4375
. 5562
. 5916
Tb
5.717
2.254
1.700
1.478
7.707
3.148
3.298
1.628
.7379
. 4337
. 5648
. 5857
Dy
5.869
2.262
1.723
1.492
7.892
3.206
3.307
1.621
. 7421
.4304
. 5724
.5808
II o
6.019
2.269
1.745
1 507
8.076
3.264
3.316
1.613
. 7461
.4273
. 5791
. 5768
El-
6.168
2.274
1.767
1.521
8.258
3.320
3.324
1.605
. 7498
.4243
. 5852
. 5735
Tm
6.316
2.277
1.789
1.535
8.438
3.375
3.332
1.596
. 7534
.4 214
. 5905
. 5709
Yb
6.462
2.279
1.812
1.549
8.616
3.426
3.338
1.587
. 7570
.4186
. 5952
. 5690
Lu
6.607
2.278
1.834
1.562
8.791
3.480
3.343
1.577
. 7605
.4159
. 5994
.5676


52
k irijii
s p
dn/q
= k U + mU + nU, + qU.. + k<^k ^ W +
00 pp dd 1 // 2 ss
ss
(2-11)
ni-1) v + n^~1) v + u +
rt vt -r n *r,jj'r n wx.c-r kmW + knW +
2 pp 2 dd 2 // sp sd
kqVs/ + mnWpd + mqVpf + nqWd/
with j, the average two electron energy of a pair of electrons in
orbitals and Xj given by
Vss = F(ss)
Wpp = F(pp) 2/25F2(pp)
(2-12)
W
dd
= F(dd) 2/63F2(dd) 2/63 F4(dd)
Vff = F(//) 4/195F2(//) 2/143 F^(ff) 100/5577 F6(//)
Vsp = F(sp) l/6G1(sp)
Vsd = F(sd) 1/10 G2(sd)
Ws/ = F(sf) 1/14 G3(s/)
W
pd
= F(pd) 1/15 GX(pd) 3/70 G3(pd)
Vpf = F(pf) 3/70 G2(p/) 2/63 G4(p/)
Udf = F(df) 3/70 G1(d/) 2/105 G3(d;f) 5/231 G5(d/)
The core integrals H_, Eq. (2-5), are then evaluated by removing an
electron from orbital X^ > 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).22^24
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 /n_3d
2 -ji 2 2
s while the remaining lanthanide atoms have the structure j s .
Two processes are then possible for 6s electron ionization:
I /n-3d1s2 - /n_3d1s1 + (s)
II s2 - /1"2s1 + (s)


53
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 Us<, 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
85 86
Brewer and then smoothed by a quadratic fit throughout the series.
For completeness, we also give the values of process II.
ji3 1 2
The lowest configuration containing a 5d electron is j asz
throughout the series, and 5d ionizations are obtained from
III Z1-3 dV -> f"3s2 + (d)
The ionization potentials for the 6p can be obtained from two
processes:
IV fH~Z s1 p3 -> Z^'s1 + (p)
V Z1"3 s2p -* /1_3s2 + (p)
Ionization from process IV is nearly constant at 3.9 eV, from V at
2 2
4.6 eV. The j ^s configuration is lower for all the lanthanides
_ 3 2
except Ce(fdsp), and Gd and Tb(Z* s p). Using the 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 Z1-3 ds2 -* Z^ds2 + (/)
VII Z1-2 s2 - Z1_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)
Uff (VI) = IP(VI) (m-4) Vff 2Vff Wd/ (2-13a)
Ujj (VII) = IP(VII) (m-3) W-. 2\Jsf (2-13b)


Table 2-2 ¡Smoothed Ionization potentials for processes I-VII and configuration
mixing coefficients derived from Brewer's tables. The entries in the
column labeled 1 are the mixing coefficients for the configuration
fn ^ds^, in column 2 the /n ^s^ configuration mixing coefficients are
listed.
Atom
Smoothed Ionization Potentials
Mixing
Coefficients
Process
s
P
d
f
1
2
I
II
IV
V
III
VI
VII
Ce
5.93
5.36
3.69
4.60
6.74
12.17
7.24
.7558
. 244 2
Pi-
5.89
5.42
3.76
4.62
6.77
12.79
7.27
.2764
.7236
Nd
5.93
5.48
3.82
4.67
6.77
13.35
7.29
. 1772
. 8228
Pm
5.98
5.55
3.87
4.65
6.74
13.85
7.31
.1465
. 8535
Sm
6.01
5.63
3.91
4 70
6.73
14.27
7.35
.0557
. 9423
Eu
6.10
5.69
3.95
4 74
6.68
14.62
7 39
. 02 36
. 9764
Gd
6.17
5.77
3.99
4.76
6.61
14.90
7.4a
. 90 37
. 0936
Tb
6.25
5.85
4.03
4 80
6.54
15.13
7.49
. ad21
. 5179
Dy
6.34
5.93
4.05
4.84
6.42
15.27
7.55
.1564
.8436
Ho
6.42
6.01
4.08
4 89
6 .30
15.35
7.61
.1533
. 8467
Eu
6.52
6.09
4.10
4 92
6 .16
15.36
7.69
.1661
. 8339
Tm
6.62
4.17
4.12
4.97
6.10
15.30
7.76
.0731
. 9269
Yb
6.73
6.26
4.13
5.02
5.96
15.18
7.84
.0273
.9727
Lu
6.77



5.31
16.11
1.000


55
Unlike the analogous situation for the 6s and 6p orbitals, use of
Eq. (2-13a) to find U^, and use of this value in Eq. (2-13b) to predict
IP(VII) is not successful, and would require the scaling of the large
F(ff) integral often performed in methods parameterized on molecular
61,63,67
spectroscopy.
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 U^. We create a two-by-
two interaction matrix
E/1 3ds2) X V
,7 rw.cn-2_2>
\
(r \
L1
CNj
= 0
(2-14a)
2 2
where V is an empirical mixing parameter, and and determines the
relative amounts of each of the two configurations that are important.
2
The exact value of V would depend on a given molecular situation.
is then given by
9 X 2
(2-14b)
l.x2
V c /c d sZ) ECf"-V)
X - 2V
(2-14c)
E(/ 3d s2) E(,f 2s2)
n2
2V
+ 1
The values of appear in Table 2-2, where we have used the values of
_j_3 ^ ^ 2
E(j ds ) and E(j s ) obtained for the promotion energies of
85
Brewer and a fixed value of V = 0.02 au. Then could be obtained
from


56
Uff = Cx2 IJ^(VI) + C22 tl^(VTI) (2-15)
n_2 2
In the case of the 3d orbitals this valence bond mixing between 3dn-is
and 3dn ^s was important in obtaining reasonable geometric
63
predictions, an observation now confirmed in careful ab initio
87
studies. For the lanthanide complexes of this study the 4/ 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 /-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 4/ and 5d atomic orbitals. However,
the values of the promotion energies we obtain from Brewer are so
different than those that we obtain from oul own numerical calculations
on the average energy of a configuration, Table 2-3, that for the moment
we choose a 76% : 24% mix of Ei/111 ^ds2) : EC/171 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(/), 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


57
Table 2-3 : Average configuration energy from Dirac-Fock
on the /n_^ds^ and the configurations
lanthanide atoms.
calcula tions3
for all the
Atom
Average
Configuration
Energy
/-v
Ce
-8853.71494569
-8853.64980000
Pr
-9230.41690970
-9.130.37981848
Nd
-9616.94751056
-9616.93446923
Pm
-10013.4526061
-10013.4606378
Sm
-10420.0710475
-10420.0970615
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.


58
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


59
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.
It has been argued that the 4/ orbitals are not used in the chemical
55 56
bonding of those complexes except in the more covalent cases. 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 MF^, MCl^, MBr^ and MI^ and for the comparative values obtained
for CeF^ and CeF^. This is indicated in Table 2-4 by the large values
of |B(f)|. The latter values are a consequence of the fact that the f-
orbitals are tighter than one usually expects for orbitals important in
chemica] bonding. Use of 5d orbitals alone will predict the trends in
these two series, but underestimates the range of values experimentally
observed.
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
(yy |v\>) or (yv|vy)
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
88
this type might be rotationally averaged, but from a study of spectra
63
it appears that all one-center integrals should be evaluated. For
example, in the metallocenes the integral (d 2 2 d Id d ) is
x -y xy1 xy yz
required to separate the two transitions that arise from the e^(d) ->
e (d) transitions that lead to the '*'E1 and excited states. In
4g lg 2g


60
addition, it appears that the inclusion of all one-center integrals
improves the predictions of angles about atoms with s,p,d basis
89 90
sets 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 format only the additional integrals
need be included. These integrals are generated in explicit form via a
computer program that we have used in the past^ and they have also been
89
recently published by Schulz et al. To our knowledge all these
integrals do not appear in the literature for s,p,d and / basis,
although we have checked those of (yy|\nj) and (y\>|\iyl against the
83
formulas of Fanning and Fitzpatrick.
Integrals of the form (yu|\>v) and (yv|vy) can be obtained thiuugh
1c
atomic spectroscopy, and their components, F and G evaluated via
least square fits
(yy | vv) = E ak pk
k
(yv|vy) = E bk clC
k
lc k.
These F and G 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 |d d )). Integrals of the "R" type,
x y yz xz xy
however, cannot be evaluated in this manner; viz.(sd|dd), (sp|pd),
(sd|pp), (sd|//), (s/|d/), (pp|p/), (dd|p/), (pd|d/), (sd|p/), (pd|s/),
(sp|d/), and (p/|//). 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 F value obtained from the Fock-Dirac numerical
k k. k
calculations. All F G and R' integral for k > 0 are then scaled by


61
2/3. This value of the scaling is obtained from a comparison of the
calculated and empirically obtained^^2-94 F^(//) ancj
J/
values that implies 0.66 + 0.04. Empirically obtained values of G (/d)
and F (/d) are far more uncertain and are much smaller, and are thus not
used to obtain this scaling value between calculated and experimental
values.
At this point it seems appropriate to point out the differences of the
59
present INDO model to that suggested by Li Le-Min et_ al. In the
latter formalism only the conventional one-center two-electron integrals
are included leading to rotational variance. In addition, the
Volfsberg-Helmholz approach is used for the resonance integral B,
B = (IP(i)+IP( j ) )S^/2. No geometry optmization has been reported
59
within their model. Further differences are the restriction to
single-C STOs and the smoothing of the valence orbital ionization
59
potentials for the lanthanides via Anno-type expressions.
Procedures
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,^ using either the
restricted or unrestricted Hartree Fock formalism. All UHF calculations
62
are followed by simple annihilation.
Self-consistent field convergence is a problem with many of these
systems. For this reason electrons are assigned to molecular orbitals
that are principally / in nature according to the number of /-electrons
in the system, and the symmetry of the system. Orbitals with large


62
lanthanide 5d character are sought and assigned no electrons. A
procedure is then adopted that extrapolates a new density for a given
93
Fock matrix based on a Mulliken population analysis of each SCF cycle.
Often this procedure is not successful. In such cases all /
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, Cl is performed to check the stability of the SCF, and the
appropriateness of the forced electron assignment to obtain the desired
96
state.
Results
The geometries of CeCl^ and LuCl^ 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 CeF^ is also listed
in Table 2-5. One can see the agreement with experiment is good in all
cases.
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 ^ a.u./bohr, the angles are converged
only to +3. We note, however, that all are predicted non-planar, in
54b 97 98
agreement with experiments.


63
Table 2-5 : Geometry and ionization potentials for Cerium and
Lutetium trihalides. Cerium tetrafluoride is also
included in this table. The bond distances are given
in angstroms^ angles in degrees and IPs in eV.
Experimental^ results are also shown where available.
Molecule
Bond
Distance
Bond
Angle
Ionization
Potential
INDO
Exp.
INDO
Exp.
INDO
Exp.
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
Cel3
2.844
2.927
119.8

9.9

CeF.
4
2.099
2.040
109.5
109.5


LuF3
2.045
2.020
107.4

b
19.0
LuC13
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)
Lul3
2.726
2.771
115.6
114.5
17.7
(16.2 18.1)
a) References 54b, 97 and 98. Estimated values for CeF-,,CeI~, and
LuF3 from Ref 103. ^ 4
b) The SCF calculation on the ion of LuF~ would not converge
therefore no IP is reported.


64
The experimental range of the bond lengths from LnF^ to Lnl^ is
greater than we calculate. Our predicted values for the trif. I uorides
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 in going from CeF^ to CeF^ is also smaller than the
0.14 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 using the ASCF method, and only the first IP is
54b 99
calculated. Experimentally 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 calculations'3^*3
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,
CeX^, X = F, Cl, Br with a maximum value for the chloride, a feature
5 A b
noticeably missing in the DVM Xa results.
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
geometries'3^*3^^ 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.


65
Table 2-6 : Geometries and Ionization Potentials (IPs) for the
lanthanide trichlorides. Bond distances are reported in
angstroms, bond angles in degrees and IPs in eV.
Experimental111 results are also given where available.
Atom
Bond
Distance
Bond
Angle
Ionization
Potential
INDO
Exp.
INDO
Exp.
INDO
Exp.
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.


66
To test the applicability of our model to lanthanide atoms not
formally charged +3, we calculated the geometries and IPs for SmC^,
EUCI2 and YbC^ 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.
_2
Ve chose Ce(NO^)^ 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.
Table 2-9 shows a population study of this complex. Although there is
some /-orbital participation, it appears that this unusual twelve
coordinate T^ structure results from electrostatic forces between the
ligands and the relatively large size of the Ce(IV) ion.


67
Table 2-7 : Geometry and ionization potential for SmC^, EuC^, and
YbC^- Bond distances are given in angstroms, bond angles
in degrees, and ionization potentials in eV. Experimental
results are listed where available.
Molecule
Bond
Distance
Bond
Angle
Ionization
Potential
INDO
Exp.
INDO
Exp.
INDO
Exp.
S111CI2
2.584

143.3
13015
5.3

EuCI,,
2.576

143.2
135+15
6.6

YbCl2
2.400

120.2
12605
3.2

a) Reference 100.


68
_2
Table 2-8 : Average bond distances and bond angles for CeiNO^)^ ion
INDO optimized geometry and the X-ray crystal structure3.
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.
Geometric
Parameter
INDO
Exp.
r(Ce-0c)
2.554
2.508
r(N-0c)
1.256
1.282
r(N-0n)
1.237
1.235
9(0 -N-0 )
c c'
121.5
114.5
9(0 -Ce-0 )
c c
50.9
50.9
a) Reference 101.


CE(N03)6.
_2
Figure 2-3: Plot of the twelve coordinate Ce(NO^)6 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.


70
_2
Table 2-9 : Population analysis of CeiNO^)^ The oxygen atoms that are
coordinated to the cerium are indicated by 0c>The Wyberg
bond index is also given. A Wyberg index of 1.00 is
characteristic of a single bond.
Atom
Orbital
Atomic
Population
Spin
Density
Total
Valence
Ce
s
0.20
0.00

P
0.30
0.00

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
c
Net
-0.40
0.00
1.58
0
Net
-0.48
0.00
1.80
Bond
Wyberg
Bond Index
Ce 0
c
0.40
N 0
c
1.37
N 0
1.22


CONCLUSIONS
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-electron terms, and
including all of the two-elecfron 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, Fiu 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 4/ orbitals, and
one need not invoke London forces to explain this observation. Again
the potential for bending is very flat.
Within this model, /-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 LnF^, LnCl^, LnBr^,
71


72
and Lul^, and CeF^ and CeF^ are reproduced without /-orbital
participation, the range of values calculated is considerably improved
when /-orbitals are allowed to parrici pate. For the twelve coordinate
-2 102
Ce(NO^)^ complex reported here, /-orbital participation appears
minor. A stable complex of near symmetry is obtained regardless of
the /-orbital interaction.


BIBLIOGRAPHY
1. V. Fock, Z. Physik 61, 126 (1930).
2. C. Edmiston and K. Ruedenberg, Rev. Mod. Phys. 35, 457 (1963).
3. C. Edmiston and K. Ruedenberg, J. Chem. Phys. 43, S97 (1965).
4. S.F. Boys, in Quantum Theory of Atoms, Molecules and the Solid
State, p.253, Lowdin P., Ed., New York: Academic 1966.
5. S.F. Boys, Rev. Mod. Phys. 32, 306 (1960).
6. J.M. Foster and S.F. Boys, Rev. Mod. Phys. !32, 300 (1960).
7. E.P. Vigner, F. Seitz, Phys. Rev. 43, 804 (1933); 46, 509 (1934).
8. J.C. Slater, Phys. Rev. 81, 385 (1951).
9. G. Sperber, Int. J. Quantum Chem. 5, 177, 189 (1971); 6, 881
(1972).
10. R.J. Boyd and C.A. Coulson, J. Phys. B7, 1805 (1974).
11. I.L. Cooper and N.M. Pounder, Int. J. Quantum Chem. 17, 759 (1980).
12. W.L. Luken and J.C. Culberson, Int. J. Quant. Chem. Symp. 16, 265
(1982). ~
13. W.L. Luken and D.N. Beratan, Theoret. Chim. Acta (Berl.) 61, 265
(1982). ~
14. W.L. Luken, Int. J. Quantum Chem. 22, 889 (1982).
15. J.M. Leonard and W.L. Luken, Theoret. Chim. Acta (Berl.) 62, 107
(1982).
16. J.M. Leonard and W.L. Luken, Int. J. Quantum Chem. 25, 355 (1984).
17. V. Magnasco and A. Perico, J. Chem. Phys. 47^, 971 (1967).
18. W.S. Vervoerd, Chem. Phys. 44, 151 (1979).
19. T.A. Claxton, Chem. Phys. 52, 23 (1980).
20. S. Diner, J.P. Malrieu, P. Claverie, and F. Jordan, Chem. Phys.
Lett. 2, 319 (1968).
21. S. Diner, J.P. Malrieu, and P. Claverie, Theoret. Chim. Acta
(Berl.) 13, 1 (1969).
22. J.P. Malrieu, P. Claverie and S. Diner, Theoret. Chim. Acta (Berl.)
13, 18(1969).
73


74
23. S. Diner, F.P. Malrieu, F. Jordan and M. Gilbert, Theoret. Chim.
Acta (Berl.) 15, 100 (1969).
24. F. Jordan, M. Gilbert, J.P. Malrieu and U. Pincelli, Theoret. Chim
Acta (Berl.) 15, 211 (1969).
25. J.M. Cullen and M.C. Zerner, Int. J. Quantum Chem. 22, 497 (1982).
26. P.0. Lowdin, Phys. Rev. 97 > 1474 (1955) Adv. Chem. Phys. 2, 207
(1959).
27. W.L. Luken and D.N. Beratan, Electron Correlation and the Chemical
Bond. Durham, NC: Freewater Production, Duke University, 1980.
28. W.L. Luken and J.C. Culberson, in Local Density Approximations in
Quantum Chemistry and Solid State Physics, J.P. Dahl, J. Avery,
Eds., New York: Plenum 1984.
29. R. Daudel, Compt. Rend. Acad. Sci. 237, 601 (1953).
30. R. Daudel, H. Brion and S. Odoit, J. Chem. Phys. 23, 2080 (1955).
31. E.V. Ludena, Int. J. Quantum Chem. 9, 1069 (1975).
32. R.F.W. Bader and M.E. Stephens, J. Amer. Chem. Soc. 97, 7391
(1975).
33. P.0. Lowdin, J. Chem. Phys. 18, 365 (1950).
34. D.A. Kleier, T.A. Halgren, J.H. Hall and V. Lipscomb, J. Chem.
Phys. 61, 3905 (1974).
35. S.U. Baldwin and J.M. Wilkinson, Tetrahedron Lett. 20, 2657 (1979)
36. S.W. Baldwin and J.E. Fredericks, Tetrahedron Lett. 23, 1235
(1982).
37. S.U. Baldwin and H.R. Blomquist, Tetrahedron Lett. 23, 3883 (1982)
38. W.J. Hehre, R.F. Stewart, and J.A. Pople, J. Chem. Phys. 51, 2657
(1969).
39. G. Herzberg, in Molecular Spectra and Molecular Structure III,
Electronic Spectra and Electronic Structure of Polyatomic
Molecules, New York: Van Nostrand Reinhold, 1966.
40. L.C. Snyder and H. Basch, in Molecular Uavefunctions and
Properties, New York: Wiley-Interscience, 1972.
41. E.L. Eliel, N.L. Allinger, S.J. Angyal and G.A.L. Morrison,
Conformational Analysis, New York: Wiley-Interscience, 1965.
42.
J.E. Williams, P.J. Stang and P. Schleyer, Ann. Rev. Phys. Chem.
19, 531 (1968).


75
43. 0. Sinanoglu and B. Skutnik, Chem. Phys. Lett. 1, 699 (1968).
44. W. Kutzelnigg, Israel J. Chem. 19, 193 (1980).
45. M. Schnidler and W. Kutzelnigg, J. Chem. Phys. 76, 1919 (1982).
46. L.A. afie and P.L. Polaravapu, Chem. Phys. 75, 2935 (1981).
47. R. Lavery, C. Etchebest and A. Pullman, Chem. Phys. Lett. 85, 266
(1982).
48. J.E. Reutt, L.S. Wang, Y.T. Lee and D.A. Shirley, Chem. Phys. Lett.
126, 399 (1986).
49. T.J. Marks and I.L. Fragal, Fundamental and Technological Aspects
of Organo-^-Element Chemistry, NATO ASI Series, C155 Reidel
Dordrecht, 1985.
50. T.J. Marks, Acc. Chem. Res. 9, 223 (1976); T.J. Marks, Adv. Chem.
Ser. 150, 232 (1976); T.J. Marks, Prog. Inorg. Chem. 24, 51 (1978).
51. H. Schumann and W. Genthe in Handbook on the Physics and Chemisty
of Rare Earths, North Holland, Amsterdam, 1984, Chpt. 53., H.
Schumann, Angew. Chem. 96, 475 (1984).
52. P.J. Hay, W.R. Wadt, L.R. Kahn, R.C. Raffenetti and D.W. Phillips,
J. Chem. Phys. 70, 1767 (1979); W.R. Wadt, J. Amer. Chem. Soc. 103,
6053 (1981).
53. J.V. Ortiz and R. Hoffmann, Inorg. Chem. 24, 2095 (1985); P. Pyykko
and L.L. Lohr, Jr., Inorg. Chem. 20, 1950 (1981); C.E. Myers, L.J.
Norman II and L.M. Loew, Inorg. Chem. 17, 5443 (1983).
54. a: D.E. Ellis, Actinides in Perspective, ed N.M. Edelstein,
Pergamon (1982).
b: B. Ruscic, G.L. Goodman and J. Berkowitz, J. Chem. Phys. 78,
5443 (1983).
55. N. Rosch and A. Streitvieser, Jr., J. Amer. Chem. Soc. 105, 7237
(1983); N. Rosch, Inorg. Chim. Acta 94, 297 (1984); A.
Streitvieser, Jr., S.A. Kinsley, J.T. Rigbee, I.L. Fragala, E.
Ciliberto and N. Rosch, J. Amer. Chem. Soc. 107, 7786 (1985).
56. D. Hohl and N. Rosch, Inorg. Chem. 25, 2711 (1986); D. Hohl, D.E.
Ellis and N. Rosch, Inorg Chim. Acta, to be published.
57. J.D. Head and M.C. Zerner, Chem. Phys. Letters 122, 264 (1985).
58. D.W. Clack and K.D. Warren, J. Organomet. Chem. 122, c28 (1976).
Li Le-Min, Ren Jing-Qing, Xu Guang-Xian and Wong Xiu-Zhen, Intern.
J. Quantum Chem. 23, 1305 (1983). Ren Jing-Qing and Xu Guang-Xian,
Inter. J. Quantum Chem. 26, 1017 (1986).
59.


76
60. J.A. Pople D.L. Beveridge and P.A. Dobosh J. Chem. Phys. 47, 158
(1967).
61. J.E. Ridley and M.C. Zerner, Theor. Chem. Acta 32, 111 (1973).
62. A.D. Bacon and M.C. Zerner, Theor. Chem. Acta 53, 21 (1979).
63. M.C. Zerner, G.H. Loew, R.F. Kirchner and U.T. Mueller-Uesterhoff,
J. Amer. Chem. Soc. 102, 589 (1980).
64. W. Anderson, W.D. Edwards and M.C. Zerner, Inorg. Chem. 25, 2728
(1986).
65. See, i.e., J.C. Slater, Quantum Theory of Atomic Structure, Vol. 1
and Vol. 2, New York, McGraw Hill, 1960.
66. See, i.e., R.J. Parr, Quantum Theory of Molecular Electronic
Structure, New York, Benjamin, 1963.
67. R. Pariser and R. Parr, J. Chem. Phys. 21, 767 (1953).
68. J. Del Bene and H.H. Jaff, J. Chem. Phys. 48, 1807 (1968).
69. N. Mataga and K. Nishimoto, Z. Phys. Chem. (Frankfurt am Main)13,
140 (1957).
70. K. Ohno, Theoret. Chim. Acta 2, 568 (1964); G. Klopman, J Amer.
Chem. Soc. 87, 3300 (1965).
71. W.D. Edwards and M.C. Zerner, to be published.
72. A. Veillard, Computational Techniques in Quantum Chemistry and
Molecular Physics, ed. G. H. F. Diercksen, B. T. Sutcliffe, A.
Veillard ,Eds. D. Reidel, Dordrecht, The Netherlands (1975).
73. E. Davidson, Chem. Phys. Letters 21, 565 (1973).
74. J.A. Pople and G.A. Segal, J. Chem. Phys. 43, S136 (1965).
75. M.C. Zerner, Mol. Phys. 23, 963 (1972).
76. P. Coffey, Inter. J. Quantum Chem. 8, 263 (1974).
77. J.A. Pople, D.P. Santry and G.A. Segal, J. Chem. Phys. 43, S129
(1965).
78. C.F. Bender and E.R. Davidson, J. Inorg. Nucl. Chem. 42, 721
(1980).
79. E. Clement! and C. Roetti, Atomic Data and Nuc. Data Tables 14, 177
(1974).
80. P. Knappe, Diplom-Chemikers Thesis, Department of Chemistry
Technical University of Munich, Munich, Germany.


77
81. J.P. Desclaux, Comp. Phys. Commun. 9, 31 (1975).
82. G. Karlsson and M.C. Zerner, Intern J. Quantum Chemistry 7, 35
(1973).
83. M.O.Fanning and N.J. Fitzpatrick, Intern. J. Quantum Chem. 28, 1339
(1980).
84. M.C. Zerner in Approximate Methods in Quantum Chemistry and Solid
State Physics, ed. F. Herman, New Tiork, Plenum Press (1972).
85. a. L. Brewer, J. Opt. Soc. Amer. 61, 1101 (1971).
b. L. Brewer, J. Opt. Soc. Amer. 61, 1666 (1971).
86. W.C. Martin, J. Phys. Chem. Ref. Data 3, 771 (1974). J. Sugat, J.
Opt. Soc. Amer. 56, 1189 (1966).
87. C.W. Bauschlicher and P.S. Bagus, J. Chem. Phys. 81, 5889 (1984).
88. R.D. Brown, B.H. James and M.F. O'Dwyer Theor. Chem. Acta 17, 264
(1970): W. Th. A.M. Van der Lugt. Intern. J. Quantum Chem. 6, 859
(1972): J.J. Kaufman and R. Prednen, Intern. J. Quantum ChemT 6,
231 (1977).
89. J. Schulz, R. Iffert and K. Jug, Inter. J. Quantum Chem. 27, 461
(1985).
90. J.C. Culberson and M.C. Zerner, unpublished results.
91. J. Almlof, University of Stockholm, Inst, of Physics (USIF Reports
72-09,74-29).
92. H.D. Arnberger, W. Jahn and N.M. Edelstein, Spectrochem. Acta. 41A.
465: Ibid, in press.
93. H.D. Arnberger, H. Schultze and N.M. Edelstein, Spectrochem. Acta.
41A, 713 (1985).
94. N. Edelstein in Fundamental and Technical Aspects of Organo-f-
Element Chemistry, ed. T.J. Marks and I.L. Fragala, NATO ASO C155,
Reidel, Dordrecht (1985).
95. M.C. Zerner and M. Hehenberger, Chem. Phys. Letters 62, 550 (1979).
96. J.C. Culberson and M.C. Zerner, in preparation.
97. K.S. Krasnov, G.V. Girichev, N.I. Giricheva, V.M. Petrov, E.Z.
Zasorin, N.I. Popenko, Seventh Austin Symp. on Gasphase Molecular
Structure, Austin Texas, p. 88 (1978).
98. N.I. Popenko, E.Z. Zasorin. V.P. Spiridonov and A.A. Ivanov, Iporg.
Chim Acta 31, L371 (1978).
99. E.P.F. Lee, A.W. Potts and J.E. Bloor, Proc. R. Soc. Lond. A 381,
373 (1982).


78
100. C.W. DeKock, R.D. Wesley and D.D. Radtke, High Temp. Sci. 4, 41
(1972); I.R. Beattie, J.S. Ogden and R.S. Wyatt, J. Chem. Soc.
Dalton Trans., 2343 (1983).
101. T.A. Beineke and J. Del Gaudio, Inorganic Chemistry 7, No. 4, 715
(1968).
102. K.S. Krasnov, N.I. Giricheva and G.V. Girichev, Zhurnal Strukturnoi
Khimii 17, 667 (1976).


BIOGRAPHICAL SKETCH
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 /-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.
79


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presenetation and is fully
adeqate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
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 presenetation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
i
N. Yngve Ohrn
Professor t>£ Chemistry and Physics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presenetation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
/ n 1
(.C-AtLhz-zi. '
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 presenetation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presenetation 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


Full Text

PAGE 1

/2&$/,=(' 25%,7$/6 ,1 &+(0,675< %< -2+1 &+5,6723+(5 &8/%(5621 $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 2) 7+( 81,9(56,7< 2) )/25,'$ 71 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$

PAGE 2

$&.12:/('*(0(17 ZRXOG OLNH WR WKDQN P\ SDUHQWV IRU WKHLU VXSSRUW DQG JXLGDQFH WKURXJKRXW P\ OLIH 0DU\ .D\ \RXU LQVSLUDWLRQ LQILQLWH SDWLHQFH DQG ZLOOLQJQHVV WR ZDLW NHSW PH JRLQJ 7KDQNV JR WR %LOO /XNHQ ZKR WDXJKW PH WKH EDVLFV LQ TXDQWXP FKHPLVWU\ DV ZHOO DV LQWURGXFLQJ PH WR WKH FRPSXWHU DV D OHDUQLQJ WRRO ,Q DGGLWLRQ %LOO /XNHQ JDYH PH DQ LQVLJKW LQWR WKH DFDGHPLF ZRUOG )LQDOO\ ZRXOG OLNH WR WKDQN %LOO DQG 0DUJH IRU EHLQJ RXU IULHQGV ZRXOG OLNH WR WKDQN 0LFKDHO & =HUQHU IRU DOORZLQJ PH WR XVH WKH VNLOOV OHDUQHG DW 'XNH DQG WHDFKLQJ PH PRUH TXDQWXP FKHPLVWU\ 7KH IUHHGRP KH JDYH PH WR H[SORUH VRPH RI P\ RZQ LGHDV DV ZHOO DV EHLQJ JXLGHG RFFDVLRQDOO\ ZDV GHHSO\ DSSUHFLDWHG 7KH HQWLUH =HUQHU IDPLO\ PDGH RXU WLPH DW 473 HQMR\DEOH 7KDQNV JR WR P\ *HUPDQ KRVW 'U 1RWNHU 5RVFK IRU DOORZLQJ PH WR FRPH WR *HUPDQ\ 0\ WKDQNV WR 3HWHU DQG 0RQLFD .QDSSH IRU KHOSLQJ 0DU\ .D\ DQG PH GXULQJ RXU HQWLUH VWD\ LQ *HUPDQ\ :H ZRXOG OLNH WR WKDQN )UDX %URZQ IRU PDNLQJ XV IHHO DW KRPH 2QH EHQHILW RI EHLQJ D JUDGXDWH VWXGHQW DW WKH 4XDQWXP 7KHRU\ 3URMHFW LV WKH ZLGH YDULHW\ RI SHRSOH \RX PHHW 2QH RI WKH PRVW HQOLJKWHQLQJ H[SHULHQFHV ZDV WR PHHW DQG WDNH FODVVHV IURP 'U 1 < 2KUQ 7KDQN \RX IRU JLYLQJ PH D QHZ SHUVSHFWLYH RQ TXDQWXP FKHPLVWU\ ZRXOG OLNH WKDQN WR 3XUYLV ,,, IRU DOORZLQJ PH WR KHOS LQ GHVLJQLQJ WKH &' SURJUDP DQG JLYLQJ PH SOHQW\ RI H[SHULHQFH GHEXJJLQJH[SDQGLQJ WKH ,1'2 FRGH RQFH D GD\
PAGE 3

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nW XQGHUVWDQG VRPHWKLQJ XQWLO \RX FDQ H[SODLQ LW WR VRPHRQH HOVH 7KDQNV JR WR $ODQ VKRZLQJ PH WKDW VRPH WKHRU\ FDQ VWLOO EH GRQH RQ D SLHFH RI SDSHU $OO RI WKH PHPEHUV RI WKH FOXEKRXVH KDYH SURYLGHG PH ZLWK DQ DWPRVSKHUH FRQGXFLYH WR WKH IUHH H[FKDQJH RI LGHDV RQ TXDQWXP WKHRU\ DQG HYHU\WKLQJ HOVH LLL

PAGE 4

7$%/( 2) &217(176 3DJH $&.12:/('*0(17 LL /,67 2) 7$%/(6 Y /,67 2) ),*85(6 YL $%675$&7 YLL ,1752'8&7,21 &+$37(5 21( /2&$/,=(' 25%,7$/6 %DFNJURXQG 'RXEOH 3URMHFWRU /RFDOL]DWLRQ )HUPL /RFDOL]DWLRQ %R\V /RFDOL]DWLRQ &+$37(5 7:2 /$17+$1,'( &+(0,675< %DFNJURXQG 0RGHO 3URFHGXUHV 5HVXOWV &21&/86,21 %,%/,2*5$3+< %,2*5$3+,&$/ 6.(7&+ ,9

PAGE 5

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
PAGE 6

/,67 2) ),*85(6 3DJH )HUPL PRELOLW\ IXQFWLRQ IRU WA&2 'LIIHUHQFH EHWZHHQ PRELOLW\ IXQFWLRQ DQG HOHFWURQ JDV FRUUHFWLRQ )HUPL KROH SORW IRU IRUPDOGHK\GH %R\V ORFDOL]HG RUELWDO IRU IRUPDOGHK\GH 1L&2fA ERQGLQJ RUELWDO 1L&2fA QRQERQGLQJ RUELWDO 1L&2fA DQWLERQGLQJ RUELWDO 6LQJOH DQG GRXEOH & EDVLV VHW SORW $YHUDJH YDOXH RI U YHUVXV DWRPLF QXPEHU 3OXWR SORW RI &H12AfA

PAGE 7

$EVWUDFW RI 'LVVHUWDWLRQ 3UHVHQWHG WR WKH *UDGXDWH 6FKRRO RI WKH 8QLYHUVLW\ RI )ORULGD LQ 3DUWLDO )XOILOOPHQW RI WKH 5HTXLUHPHQWV IRU WKH 'HJUHH RI 'RFWRU RI 3KLORVRSK\ /2&$/,=(' 25%,7$/6 ,1 &+(0,675< E\ -RKQ &KULVWRSKHU &XOEHUVRQ 0D\ &KDLUPDQ 0LFKDHO & =HUQHU 0DMRU 'HSDUWPHQW &KHPLVWU\ 7KH ORFDOL]HG RUELWDOV GLVFXVVHG KHUH ZLOO EH GLYLGHG LQWR WZR FODVVHV f LQWULQVLFDOO\ ORFDOL]HG RUELWDOV ZKHUH WKH ORFDOL]DWLRQ LV GXH SULPDULO\ WR V\PPHWU\ RU HQHUJ\ FRQVLGHUDWLRQV IRU H[DPSOH WUDQVLWLRQ PHWDO GRUELWDOV RU ODQWKDQLGH RUELWDOV DQG f RUELWDOV ZKLFK PXVW EH ORFDOL]HG DIWHU D VHOIFRQVLVWHQW ILHOG 6&)f FDOFXODWLRQ ,Q WKH ODWWHU FDVH WZR QHZ PHWKRGV RI ORFDOL]DWLRQ WKH )HUPL DQG WKH GRXEOH SURMHFWRU PHWKRGV DUH SUHVHQWHG KHUH 7KH )HUPL PHWKRG SURYLGHV D PHDQV IRU WKH QRQLWHUDWLYH ORFDOL]DWLRQ RI 6&) RUELWDOV ZKLOH WKH GRXEOH SURMHFWRU DOORZV RQH WR GHVFULEH ZKDW DWRPLF IXQFWLRQV WKH ORFDOL]HG RUELWDOV ZLOO FRQWDLQ 7KH WKLUG ORFDOL]DWLRQ SURFHGXUH GHVFULEHG LV WKH VHFRQG RUGHU %R\V PHWKRG RI /HRQDUG DQG /XNHQ 7KLV PHWKRG LV XVHG WR H[SODLQ WKH SKRWRGLVVRFLDWLRQ SURGXFWV RI 1L&2fA 9OO

PAGE 8

7KH ,QWHUPHGLDWH 1HJOHFW RI 'LIIHUHQWLDO 2YHUODS ,1'2f PHWKRG LV H[WHQGHG WR WKH RUELWDOV DQG WKH LQWULQVLF ORFDOL]DWLRQ RI WKH RUELWDOV LV H[DPLQHG 7KLV H[WHQVLRQ LV FKDUDFWHUL]HG E\ D EDVLV VHW REWDLQHG IURP UHODWLYLVWLF 'LUDF)RFN DWRPLF FDOFXODWLRQV DQG WKH LQFOXVLRQ RI DOO RQHFHQWHU WZRHOHFWURQ LQWHJUDOV $SSOLFDWLRQV RI WKLV PHWKRG WR WKH ODQWKDQLGH KDOLGHV DQG WKH WZHOYH FRRUGLQDWH B &HL12AfA LRQ DUH SUHVHQWHG 7KH PRGHO LV DOVR XVHG WR FDOFXODWH WKH LRQL]DWLRQ SRWHQWLDOV IRU WKH DERYH FRPSRXQGV 'XH WR WKH ORFDOL]HG QDWXUH RI RUELWDOV WKH FU\VWDO ILHOG VSOLWWLQJV LQ WKHVH FRPSRXQGV DUH H[WUHPHO\ VPDOO OHDGLQJ WR 6&) FRQYHUJHQFH SUREOHPV ZKLFK DUH DGGUHVVHG KHUH (YHQ ZKHQ WKH 6&) KDV FRQYHUJHG D VPDOO FRQILJXUDWLRQ LQWHUDFWLRQ &Of FDOFXODWLRQ PXVW EH GRQH WR LQVXUH WKDW WKH FRQYHUJHG VWDWH LV LQGHHG WKH ORZHVW HQHUJ\ VWDWH 7KH ORFDOL]HG QDWXUH RI WKH RUELWDOV LQ FRQMXQFWLRQ ZLWK WKH GRXEOH SURMHFWRU ORFDOL]DWLRQ PHWKRG PD\ EH XVHG WR LVRODWH WKH RUELWDOV LQ RUGHU WR FDOFXODWH RQO\ D &O UHVWULFWHG ZLWKLQ WKH PDQLIROG YLLL

PAGE 9

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

PAGE 10

VXFK PHWKRGV RQ VHYHUDO V\VWHPV RI FKHPLFDO LQWHUHVW &KDSWHU WZR GHDOV ZLWK H[SDQGLQJ WKH ,1'2 PHWKRG VR WKDW WKH SUHGLFWLRQ RI RUELWDOV EHLQJ ORFDOL]HG RUELWDOV PD\ EH YHULILHG DQG VR WKDW WKH XQLTXH ERQGLQJ DQG VSHFWURVFRS\ RI WKHVH FRPSRXQGV PD\ EH H[DPLQHG %\ DGDSWLQJ WKH ,1'2 PHWKRG ZH PD\ QRZ H[SDQG RXU VWXGLHV WR LQFOXGH WKH FKHPLVWU\ RI WKH ODQWKDQLGHV DQG DFWLQLGHV 7KH FKHPLVWU\ RI WKH ODQWKDQLGHV DQG DFWLQLGHV LV GLIIHUHQW IURP WKH FKHPLVWU\ RI WKH FRUUHVSRQGLQJ GRUELWDO FKHPLVWU\ 7KH FRPSDFW ORFDOL]HGf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

PAGE 11

&+$37(5 21( /2&$/,=(' 25%,7$/6 %DFNJURXQG 7KH REVHUYDEOH SURSHUWLHV RI DQ\ ZDYHIXQFWLRQ FRPSRVHG RI D VLQJOH 6ODWHU GHWHUPLQDQW DUH LQYDULDQW WR D XQLWDU\ WUDQVIRUPDWLRQ RI WKH RUELWDOV RFFXSLHG LQ WKH ZDYHIXQFWLRQA %HFDXVH RI WKLV LQYDULDQFH WKH REVHUYDEOH SURSHUWLHV RI D FORVHGVKHOO VHOIFRQVLVWHQW ILHOG 6&)f ZDYHIXQFWLRQ PD\ EH GHVFULEHG XVLQJ FDQRQLFDO RUELWDOV RU DQ\ VHW RI RUELWDOV UHODWHG WR WKH FDQRQLFDO RUELWDOV E\ D XQLWDU\ WUDQVIRUPDWLRQ &DQRQLFDO RUELWDOV DUH TXLWH XVHIXO LQ SRVW+DUWHH)RFN FDOFXODWLRQV IRU VHYHUDO UHDVRQV &DQRQLFDO PROHFXODU RUELWDOV &02Vf DUH REWDLQHG GLUHFWO\ E\ PDWUL[ GLDJRQDOL]DWLRQ IURP WKH 6&) SURFHGXUH LWVHOI 7KH FDQRQLFDO RUELWDOV IRUP LUUHGXFLEOH UHSUHVHQWDWLRQV RI WKH PROHFXODU SRLQW JURXS 6LQFH WKH V\PPHWU\ LV PDLQWDLQHG DOO VXEVHTXHQW FDOFXODWLRQV PD\ EH VLPSOLILHG E\ WKH XVH RI V\PPHWU\ 6SHFWURVFRSLF VHOHFWLRQ UXOHV DUH GHWHUPLQHG XVLQJ WKH FDQRQLFDO RUELWDOV .RRSPDQnV WKHRUHP ZKLFK UHODWHV RUELWDO HQHUJLHV WR PROHFXODU LRQL]DWLRQ SRWHQWLDOV DQG HOHFWURQ DIILQLWLHV LV EDVHG HQWLUHO\ RQ WKH XVH RI FDQRQLFDO RUELWDOV /RFDOL]HG RUELWDOV /02Vf DOORZ IRU WKH ZDYHIXQFWLRQ WR EH LQWHUSUHWHG LQ WHUPV RI ERQG RUELWDOV ORQHSDLU RUELWDOV DQG LQQHU VKHOO RUELWDOV FRQVLVWHQW ZLWK WKH /HZLV VWUXFWXUHV OHDUQHG LQ IUHVKPHQ FKHPLVWU\ 8QOLNH &02V /02V PD\ EH WUDQVIHUUHG LQWR RWKHU ZDYHIXQFWLRQV DV DQ LQLWLDO JXnVV WKHUHE\ UHGXFLQJ WKH HIIRUW QHHGHG WR SURGXFH ZDYHIXQFWLRQV IRU ODUJH PROHFXOHV 7KH PRVW LPSRUWDQW XVH RI

PAGE 12

ORFDOL]HG RUELWDOV LV WKHLU DELOLW\ WR VLPSOLI\ FRQILJXUDWLRQ LQWHUDFWLRQ &Of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f L O LL LV PD[LPL]HG RU PLQLPL]HG ZKHUH WKH GHILQLWLRQ RI LL_JA_LL! GHSHQGV RQ WKH ORFDOL]DWLRQ FULWHULRQ 2QH FKRLFH IRU WKH YDOXH RI LM,JLMAL,NO! LV WKH WZRHOHFWURQ UHSXOVLRQ LQWHJUDOV IRU WKLV FKRLFH WKH VXP LV PD[LPL]HG 7KLV PHWKRG LV UHIHUUHG WR DV WKH (GPLVWRQ 5XHGHQEHUJ (5f PHWKRG 3HUKDSV WKH PRVW SRSXODU FKRLFH RI D ORFDOL]DWLRQ PHWKRG LV WKH %R\V PHWKRG LQ ZKLFK WKH J RSHUDWRU LV WKH f§ RUELWDO VHOIH[WHQVLRQ RSHUDWRU JLL U f

PAGE 13

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f RU 1 DE LQLWLRf DQG WKH V\VWHPV VWXGLHG KHUH DUH ODUJH ZH ZLOO QRW FRQVLGHU WKH (5 PHWKRG RI ORFDOL]DWLRQ DQ\ IXUWKHU 7KH VDPH LQWHJUDO WUDQVIRUPDWLRQ SUREOHP LV HQFRXQWHUHG IRU WKH %R\V PHWKRG EXW VLQFH WKH LQWHJUDOV LQYROYHG DUH GLSROH RQHHOHFWURQf LQWHJUDOV WKH SUREOHP LV PXFK VLPSOHU 6LQFH WKH ORFDOL]DWLRQ FULWHULD DUH VR GLIIHUHQW WKHUH LV QR UHDVRQ WR H[SHFW GLIIHUHQW PHWKRGV WR \LHOG RUELWDOV WKDW DUH VLPLODU EXW LQ JHQHUDO WKH /02V DUH TXLWH VLPLODU IRU WKH %R\V DQG (5 PHWKRGV 7KHVH RUELWDO VLPLODULWLHV OHDG WR WKH VHFRQG FDWHJRU\ RI ORFDOL]DWLRQ :H FODLP WKDW WKH XQGHUO\LQJ SK\VLFDO EDVLV RI ORFDOL]DWLRQ LV WKH )HUPL KROH 7KH )HUPL KROH SURYLGHV D GLURRWQRQLWHUD ILYHf PHWKRG

PAGE 14

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nV QHHGV $V PHQWLRQHG DERYH WKH )HUPL PHWKRG PD\ EH FODVVLILHG LQ WKLV FDWHJRU\ EXW DQRWKHU PHWKRG ZDV GHYHORSHG HVSHFLDOO\ IRU WKLV SXUSRVH RQH WKDW ZH KDYH FDOOHG WKH GRXEOH SURMHFWRU '3f PHWKRG 7KLV PHWKRG KDV EHHQ XVHG LQ FRQMXQFWLRQ ZLWK WKH RWKHU PHWKRGV DERYH WR KHOS SUHGLFW WKH ORZHVW HQHUJ\ VWDWH RI ODQWKDQLGH FRQWDLQLQJ FRPSRXQGV ZKHUH RUELWDO GHJHQHUDFLHV DUH D SUREOHP 7KH '3 PHWKRG DOORZV RQH WR VHSDUDWH WKH RUELWDOV IURP WKH RWKHU PHWDO RUELWDOV DQG XVH D VPDOO &O WR GHWHUPLQH WKH JURXQG VWDWH RI WKH PROHFXOH 'RXEOH 3URMHFWRU 7KH GRXEOH SURMHFWRU '3f PHWKRG RI ORFDOL]DWLRQ LV DQ H[WUHPHO\ XVHIXO PHWKRG IRU ORFDOL]LQJ RUELWDOV ZKHQ WKH IRUP RI WKH ORFDOL]HG RUELWDOV LV NQRZQ RU VXVSHFWHG LQ DGYDQFH )RU H[DPSOH LI RQH ZRXOG fN OLNH WR VWXG\ QUF WUDQVLWLRQV LQ D PROHFXOH D IXOO ORFDOL]DWLRQ QHHG QRW EH GRQH WKH GRXEOH SURMHFWRU PD\ EH XVHG WR LVRODWH ORFDOL]Hf WKH QW\SH RUELWDOV $ VXEVHTXHQW VPDOO VLQJOHV &O PD\ WKHQ EH XVHG WR VWXG\ N N RQO\ WKH Q-W WUDQVLWLRQV DQG WKHUHE\ HOXFLGDWH WKH QUW VSHFWUD

PAGE 15

$QRWKHU H[DPSOH LQYROYHV WKH ORFDOL]DWLRQ RI WKH GRUELWDOV LQ D WUDQVLWLRQ PHWDO FRPSOH[ %HFDXVH RI DFFLGHQWDO GHJHQHUDFLHV EHWZHHQ PHWDO GRUELWDOV DQG OLJDQG PROHFXODU RUELWDOV 02Vf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r D VHW U ORFDOL]HG SDWWHUQ RUELWDOV ZKHUH U LV OHVV WKDQ RU HTXDO WR P 7KHVH SDWWHUQ RUELWDOV DUH SURMHFWHG RXW RI WKH VHW E\ P U! ( ! f n D O On D -r IRU D WR U 7KHVH ^_72`A DUH WKHQ V\PPHWULFDOO\ RUWKRJRQDOL]HG

PAGE 16

7n
PAGE 17

)HUPL /RFDOL]DWLRQ %DFNJURXQG 7KLV VHFWLRQ SUHVHQWV D PHWKRG IRU WUDQVIRUPLQJ D VHW RI FDQRQLFDO 6&) RUELWDOV LQWR D VHW RI ORFDOL]HG RUELWDOV EDVHG RQ WKH SURSHUWLHV RI WKH )HUPL KROHA A DQG WKH )HUPL RUELWDOAA 8QOLNH ORFDOL]DWLRQ PHWKRGV EDVHG RQ LWHUDWLYH RSWLPL]DWLRQ RI VRPH FULWHULRQ RI ORFDOL]DWLRQA WMOJ PHWKRG SUHVHQWHG KHUH SURYLGHV D GLUHFW QRQn LWHUDWLYHf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f SU[f S A U!S Uf f ZKHUH SUAf LV WKH GLDJRQDO SRUWLRQ RI WKH ILUVW RUGHU UHGXFHG GHQVLW\ PDWUL[ DQG SUAUf LV WKH FRUUHVSRQGLQJ SDUW RI WKH VHFRQG RUGHU UHGXFHG GHQVLW\ PDWUL[ )RU VSHFLDO FDVH RI D FORVHG VKHOO 6&)

PAGE 18

ZDYHIXQFWLRQ WKH QDWXUDO UHSUHVHQWDWLRQ RI WKH )HUPL KROH LV WKH DEVROXWH VTXDUH RI WKH )HUPL RUELWDO f $UUf _IUUf_ f 7KH )HUPL RUELWDO LV JLYHQ E\ IULUf >SUf@ ( JLUfJUf L Gf ZKHUH WKH RUELWDOV JAUf DUH HLWKHU WKH FDQRQLFDO 6&) PROHFXODU RUELWDOV RU DQ\ VHW UHODWHG WR WKH FDQRQLFDO 6&) PROHFXODU RUELWDOV E\ D XQLWDU\ WUDQVIRUPDWLRQ 7KH )HUPL RUELWDO IUAUf LV LQWHUSUHWHG DV D IXQFWLRQ RI UA ZKLFK LV SDUDPHWULFDOO\ GHSHQGHQW XSRQ WKH SRVLWLRQ RI D SUREH HOHFWURQ ORFDWHG DW U 3UHYLRXV ZRUN f f f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f f )Uf )[Uf ) Uf )]Uf f ZKHUH )YUf A S K L!M YtL Y Gf IRU Y [ \ RU ] 7KLV PD\ EH FRPSDUHG WR

PAGE 19

)Sf QfSf f ZKLFK SURYLGHV DQ HVWLPDWH RI WKH )HUPL KROH LQ D XQLIRUP GHQVLW\ HOHFWURQ JDV 7KH )HUPL KROH PRELOLW\ IXQFWLRQ )Uf IRU WKH IRUPDOGHK\GH PROHFXOH LV VKRZQ LQ )LJ 7KH GLIIHUHQFH )Uf)JSf LV VKRZQ LQ )LJ 5HJLRQV ZKHUH )Uf )Sf WKDW LV WKH )HUPL KROH LV OHVV VHQVLWLYH WR WKH SRVLWLRQ RI WKH SUREH HOHFWURQ WKDQ LW ZRXOG EH LQ DQ HOHFWURQ JDV RI WKH VDPH GHQVLW\ PD\ EH FRPSDUHG WR WKH ORJHV SURSRVHG E\ 'DXGHO 5HJLRQV ZKHUH )Uf )Sf UHVHPEOH ERXQGDULHV EHWZHHQ ORJHV :KHQ WKH SUREH HOHFWURQ LV ORFDWHG LQ D UHJLRQ ZKHUH )Uf )Sf WKH )HUPL RUELWDO LV IRXQG WR UHVHPEOH D ORFDOL]HG RUELWDO GHWHUPLQHG E\ FRQYHQWLRQDO PHWKRGVA 7KLV VLPLODULW\ LV GHPRQVWUDWHG E\ )LJV DQG ZKLFK FRPSDUH D )HUPL KROH IRU WKH IRUPDOGHK\GH PROHFXOH ZLWK D ORFDOL]HG RUELWDO GHWHUPLQHG E\ WKH RUELWDO FHQWURLG FULWHULRQ RI ORFDOL]DWLRQ /RFDOL]HG 2UELWDOV %DVHG RQ WKH )HUPL KROH (TXDWLRQ SURYLGHV D GLUHFW UHODWLRQVKLS EHWZHHQ D VHW RI FDQRQLFDO 6&) RUELWDOV JAUf DQG D ORFDOL]HG RUELWDO IMUf IUUAf ZKHUH UA LV D SRLQW LQ D UHJLRQ ZKHUH )UMf )JSUAff ,Q RUGHU WR WUDQVIRUP D VHW RI 1 FDQRQLFDO 6&) RUELWDOV LQWR D VHW RI 1 ORFDOL]HG RUELWDOV LW LV QHFHVVDU\ WR VHOHFW 1 SRLQWV Q M WR 1 HDFK RI ZKLFK LV ORFDWHG LQ D UHJLRQ ZKHUH )UBf )JSUAff ,GHDOO\ HDFK RI WKHVH SRLQWV VKRXOG FRUUHVSRQG WR D PLQLPXP RI )Uf RU )Uf)JSf 7KLV FRQGLWLRQ KRZHYHU LV QRW FULWLFDO EHFDXVH WKH )HUPL KROH LV UHODWLYHO\ LQVHQVLWLYH WR WKH SRVLWLRQ RI WKH SUREH HOHFWURQ ZKHQ WKH SUREH HOHFWURQ LV ORFDWHG LQ RQH RI WKHVH UHJLRQV

PAGE 20

)LJXUH 7KH IHUPL KROH PRELOLW\ IXQFWLRQ )Uf IRU WKH +f&2 EDVHG RQ WKH JHRPHWU\ DQG GRXEOH ]HWD EDVLV VHW RI UHI O 7KH ORFDWLRQV RI WKH QXFOHL DUH LQGLFDWHG E\ f VLJQV 7KH FRQWRXUV UHSUHVHQW PRELOLW\ IXQFWLRQ YDOXHV RI DQG DWRPLF XQLWV 7KH FRQWRXUV LQFUHDVH IURP QHDU WKH FRUQHUV WR RYHU LQ UHJLRQV HQFORVLQJ WKH FDUERQ DQG R[\JHQ QXFOHL (DFK QXFOHXV LV ORFDWHG DW D ORFDO PLQLPXP RI WKH PRELOLW\ IXQFWLRQ

PAGE 21

)LJXUH 7KH GLIIHUHQFH EHWZHHQ WKH )HUPL KROH PRELOLW\ IXQFWLRQ )Uf DQG WKH HOHFWURQ JDV DSSUR[LPDWLRQ IRU WKH +A&2 PROHFXOH 7KH FRQWRXUV UHSUHVHQW YDOXHV RI DQG LQ DGGLWLRQ WR WKRVH LQGLFDWHG LQ ILJXUH WKH FRQWRXUV UHSUHVHQWLQJ QHJDWLYH YDOXHV DQG ]HUR DUH LQGLFDWHG E\ EURNHQ OLQHV (DFK QXFOHXV LV ORFDWHG DW D ORFDO PLQLPXP

PAGE 22

)LJXUH 7KH IHUPL KROH IRU WKH IRUPDOGHK\GH PROHFXOH GHWHUPLQHG E\ D SUREH HOHFWURQ ORFDWHG DW RQH RI WKH SURWRQV 7KH FRQWRXUV LQGLFDWH HOHFWURQ GHQVLW\ RI DQG HOHFWURQV SHU FXELF ERKU

PAGE 23

)LJXUH 7KH ORFDOL]HG RUELWDO IRU WKH &+ ERQG RI D IRUPDOGHK\GH PROHFXOH GHWHUPLQHG E\ WKH RUELWDO FHQWULRG FULWHULRQ IRU ORFDOL]DWLRQ 7KH HOHFWURQLF GHQVLW\ FRQWRXUV DUH WKH VDPH DV LQ ILJXUH

PAGE 24

$ VHW RI 1 )HUPL RUELWDOV GHWHUPLQHG E\ (T LV QRW JHQHUDOO\ RUWKRQRUPDO (DFK PHPEHU RI WKLV VHW KRZHYHU LV XVXDOO\ YHU\ VLPLODU WR RQH PHPEHU RI DQ RUWKRQRUPDO VHW RI FRQYHQWLRQDO ORFDOL]HG RUELWDOV &RQVHTXHQWO\ WKH RYHUODS EHWZHHQ D SDLU RI )HUPL RUELWDOV LV XVXDOO\ YHU\ VPDOO DQG D VHW RI 1 )HUPL RUELWDOV PD\ HDVLO\ EH FRQYHUWHG LQWR DQ RUWKRQRUPDO VHW RI ORFDOL]HG RUELWDOV E\ PHDQV RI WKH PHWKRG RI V\PPHWULF RUWKRJRQDOL]DWLRQ 7KH UHVXOWLQJ XQLWDU\ WUDQVIRUPDWLRQ LV JLYHQ E\ 8 77fa7 f ZKHUH 7 JMULfSUff f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

PAGE 25

$GGLWLRQDO SRLQWV IRU WKH SUREH HOHFWURQ PD\ XVXDOO\ EH GHWHUPLQHG EDVHG RQ WKH PROHFXODU JHRPHWU\ 7KH PLGSRLQW EHWZHHQ WZR ERQGHG DWRPV RWKHU WKDQ K\GURJHQf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f ERQGV VLPLODU WR WKRVH GHWHUPLQHG E\ RWKHU PHWKRGV IRU WUDQVIRUPLQJ FDQRQLFDO 6&) RUELWDOV LQWR ORFDOL]HG RUELWDOV $V VKRZQ LQ 7DEOH WKH FHQWURLGV RI WKH RUELWDOV GHWHUPLQHG E\ WKH )HUPL KROH DUH YHU\ FORVH WR WKRVH RI WKH ORFDOL]HG RUELWDOV 3DJH

PAGE 26

7DEOH 0ROHFXODU JHRPHWU\ DQG SUREH HOHFWURQ SRLQWV IRU WKH IXUDQRQH &A+A&Af 7KH ILUVW WHQ SRLQWV LQGLFDWH WKH PROHFXODU JHRPHWU\ XVHG LQ WKHVH FDOFXODWLRQV 7KH WZHOYH DGGLWLRQDO SUREH HOHFWURQ SRVLWLRQV ZHUH GHWHUPLQHG DV GHVFULEHG LQ WKH WH[W $OO FRRUGLQDWHV DUH JLYHQ LQ ERKU 3RVLWLRQ ; < = $WRP $WRP &e $WRP $WRP $WRP A $WRP &! $WRP $WRP + $WRP $WRP + &I&] ERQG AaA NrQFO NA && ERQG &AA ERQG &MA ERQG &2 ERQG &2 ERQG A ORQH SDLU ORQH SDLU &! ORQH SDLU &! ORQH SDLU

PAGE 27

7DEOH 2UELWDO FHQWURLGV IRU ORFDOL]HG RUELWDOV GHWHUPLQHG E\ WKH )HUPL KROH PHWKRG DQG E\ WKH RUELWDO FHQWURLG PHWKRG IRU WKH IXUDQRQH PROHFXOH &A+A2A $OO FRRUGLQDWHV DUH JLYHQ LQ ERKU f 2UELWDO )HUPL KROH PHWKRG &HQWURLG FULWHULRQ ; < = ; < = & VKHOO & VKHOO & VKHOO 82 & VKHOO ; VKHOO VKHOO &A+A ERQG &+ ERQG &+ ERQG &+ ERQG FUF ERQG &U& ERQG && ERQG &A&A ERQG & ERQG & ERQG &a ERQG &2 ERQG ORQH SDLU ORQH SDLU ORQH SDLU ORQH SDLU

PAGE 28

GHWHUPLQHG E\ WKH RUELWDO FHQWURLG FULWHULRQA A! /LNHZLVH WKH ORFDOL]HG RUELWDOV GHWHUPLQHG E\ WKH )HUPL KROH ZHUH IRXQG WR EH YHU\ FORVH WR WKRVH GHWHUPLQHG E\ WKH RUELWDO FHQWURLG FULWHULRQ (DFK RI WKH ORFDOL]HG RUELWDOV GHWHUPLQHG E\ WKH )HUPL KROH PHWKRG ZDV IRXQG WR KDYH DQ RYHUODS RI WR ZLWK RQH RI WKH ORFDOL]HG RUELWDOV GHWHUPLQHG E\ WKH RUELWDO FHQWURLG FULWHULRQ 7KH UHPDLQLQJ RII GLDJRQDOf RYHUODS LQWHJUDOV EHWZHHQ WKHVH WZR VHWV RI ORFDOL]HG RUELWDOV ZHUH IRXQG WR KDYH D URRW PHDQ VTXDUH 506f YDOXH RI 7KH WUDQVIRUPDWLRQ RI D VHW RI FDQRQLFDO 6&) RUELWDOV WR DQ RUWKRQRUPDO VHW RI ORFDOL]HG RUELWDOV GHWHUPLQHG E\ WKH )HUPL KROH UHTXLUHG PLQXWHV RQ D 3'3 FRPSXWHU 7KH RUELWDO FHQWURLG %R\Vf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f ERQGV UHODWLYH WR WKH WKUHH &+ ERQGV RI WKH PHWK\O JURXS ,Q FDOFXODWLRQV EDVHG RQ WKH

PAGE 29

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f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f 3DJH

PAGE 30

7DEOH 0ROHFXODU JHRPHWU\ DQG SUREH HOHFWURQ SRVLWLRQV IRU WKH PHWK\ODFHW\OHQH PROHFXOH 7KH ILUVW VHYHQ SRLQWV LQGLFDWH WKH ORFDWLRQV RI WKH QXFOHL $OO FRRUGLQDWHV DUH JLYHQ LQ ERKU 3RVLWLRQ ; < = $WRP $WRP & $WRP &A $WRP $WRP $WRP $WRP ERQG AA ErQA A && ERQG && ERQG

PAGE 31

7DEOH 2UELWDO FHQWURLGV IRU ORFDOL]HG RUELWDOV GHWHUPLQHG E\ WKH )HUPL KROH PHWKRG DQG E\ WKH RUELWDO FHQWURLG PHWKRG IRU WKH PHWK\ODFHW\OHQH PROHFXOH $OO FRRUGLQDWHV DUH JLYHQ LQ ERKU 2UELWDO )HUPL KROH PHWKRG &HQWURLG FULWHULRQ ; < = ; < = & VKHOO & VKHOO & VKHOO &A+A ARQA &+ ERQG &A+A ERQG &+ ERQG AOA ErQG 8 & ERQG InOA rQFA A && ERQG

PAGE 32

RUELWDO FHQWURLG PHWKRG UHTXLUHG PLQXWHV VWDUWLQJ IURP WKH FDQRQLFDO 6&) PROHFXODU RUELWDOV RU PLQXWHV XVLQJ WKH )HUPL ORFDOL]HG RUELWDOV f§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a RK 7KH WKUHHXS FRQIRUPDWLRQ LV JHQHUDWHG E\ URWDWLQJ WKH VHW RI ORQH SDLU RUELWDOV RQ HDFK IOXRULQH DWRP LQ WKH SLQZKHHO FRQIRUPDWLRQ E\

PAGE 33

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f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

PAGE 34

7DEOH 2UELWDO FHQWURLGV IRU ORFDOL]HG RUELWDOV GHWHUPLQHG E\ WKH RUELWDO FHQWURLG PHWKRG IRU WKH ERURQ WULIOXRULGH PROHFXOH 7KH IRXU LQQHUVKHOO RUELWDOV KDYH EHHQ H[FOXGHG IURP WKHVH FDOFXODWLRQV $OO FRRUGLQDWHV DUH JLYHQ LQ ERKU 2UELWDO XSXSXS &Yf XSXSGRZQ &Jf ; < = ; < = %)A ERQG %) ERQG %)A ERQG )A ORQH SDLU ORQH SDLU ORQH SDLU ORQH SDLU ORQH SDLU ORQH SDLU )A ORQH SDLU ORQH SDLU ORQH SDLU

PAGE 35

7DEOH 9DOXHV RI WKH RUELWDO FHQWURLG FULWHULRQ DQG WKH VHFRQG GHULYDWLYHV RI WKH RUELWDO FHQWURLG FULWHULRQ IRU YDULRXV FRQIRUPDWLRQV RI ORFDOL]HG RUELWDOV IRU WKH ERURQ WULIOXRULGH PROHFXOH 7KH URZ ODEHOOHG VXP LQGLFDWHG WKH VXP RI WKH VTXDUHV RI WKH RUELWDO FHQWULRGV IRU HDFK RI WKH FRQIRUPDWLRQV 7KH IROORZLQJ URZV VKRZ WKH ILYH KLJKHVW PRVW SRVLWLYHf HLJHQYDOXHV RI WKH FRUUHVSRQGLQJ KHVVLDQ PDWUL[ 7KH ILUVW DQG VHFRQG FROXPQV FRUUHVSRQG WR WKH ORFDOL]HG RUELWDOV GHVFULEHG LQ GHVFULEHG LQ 7DEOH 7KH WKLUG DQG IRXUWK FROXPQV FRUUHVSRQG WR ORFDOL]HG RUELWDOV GHVFULEHG LQ 7DEOH 7KH JUDGLHQW YHFWRUV DUH ]HUR IRU WKH ILUVW WKUHH FROXPQV &RQILJXUDWLRQV XSXSXS XSXS GRZQ SLQZKHHO )HUPL KROH 6XP ; ; r ; ; OQ

PAGE 36

7DEOH 3UREH HOHFWURQ SRVLWLRQV IRU WKH ERURQ WULIOXRULGH PROHFXOH 7KH ILUVW WKUHH SRLQWV DUH ORFDWHG DW WKH PLGSRLQW RI WKH %) ERQGV 7KH UHPDLQLQJ SRLQWV KDYH EHHQ FKRVHQ LQ WKH SLQZKHHO FRQIRUPDWLRQ V\PPHWU\ &Af $OO FRRUGLQDWHV DUH JLYHQ LQ ERKU 3RVLWLRQ ; < = %)A ERQG %) ERQG %)A ERQG )A ORQH SDLU ORQH SDLU ORQH SDLU )e ORQH SDLU ORQH SDLU ORQH SDLU )A ORQH SDLU ORQH SDLU ORQH SDLU

PAGE 37

7DEOH 2UELWDO FHQWURLGV IRU ORFDOL]HG RUELWDOV GHWHUPLQHG E\ WKH )HUPL KROH PHWKRG DQG E\ WKH RUELWDO FHQWURLG PHWKRG IRU WKH ERURQ /ULIOXRLLGH PROHFXOH 7KH IRXU LQQHUVKHOO RUELWDOV KDYH EHHQ H[FOXGHG IURP WKHVH FDOFXODWLRQV $OO FRRUGLQDWHV DUH JLYHQ LQ ERKU 2UELWDO )HUPL KROH &HQWURLG FULWHULRQ ; < = ; < = %)A ERQG %)A ERQG %)A ERQG )A ORQH SDLU ORQH SDLU ORQH SDLU n ) ORQH SDLU ORQH SDLU ORQH SDLU )A ORQH SDLU ORQH SDLU ORQH SDLU

PAGE 38

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

PAGE 39

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f GHVFULEHG E\ DQ REYLRXV VHW RI FKHPLFDO ERQGV ORQH SDLU RUELWDOV DQG LQQHUVKHOO DWRPLF RUELWDOV 7KLV LV UHIOHFWHG LQ WKH VXFFHVV RI PHWKRGV

PAGE 40

$O $ VXFK DV PROHFXODU PHFKDQLFV f IRU SUHGLFWLQJ WKH JHRPHWULHV RI FRPSOH[ PROHFXOHV 7KH ORFDOL]HG RUELWDOV RI VXFK PROHFXOHV DUH XQOLNHO\ WR EH WKH REMHFWV RI PXFK LQWHUHVW LQ WKHPVHOYHV EXW WKH\ PD\ EH XVHIXO LQ WKH FDOFXODWLRQ RI RWKHU SURSHUWLHV RI D PROHFXOH VXFK DV WKH FRUUHODWLRQ HQHUJ\ VSHFWURVFRSLF FRQVWDQWV f DQG RWKHU SURSHUWLHV f 7KH VHOHFWLRQ RI D VHW RI SUREH HOHFWURQ SRVLWLRQV IRU RQH RI WKHVH PROHFXOHV LV VLPSOH DQG XQDPELJXRXV DQG WKH PHWKRG SUHVHQWHG KHUH KDV VLJQLILFDQW SUDFWLFDO DGYDQWDJHV FRPSDUHG WR DOWHUQDWLYH PHWKRGV IRU WUDQVIRUPLQJ FDQRQLFDO 6&) PROHFXODU RUELWDOV LQWR ORFDOL]HG PROHFXODU RUELWDOV )RU VRPH PROHFXOHV WKH SDWWHUQ RI ERQGLQJ PD\ QRW EH XQLTXH RU LW PD\ QRW EH HQWLUHO\ REYLRXV HYHQ ZKHQ WKH JHRPHWU\ LV NQRZQ )RU H[DPSOH WZR RU PRUH DOWHUQDWLYH UHVRQDQFHf VWUXFWXUHV PD\ EH LQYROYHG LQ WKH HOHFWURQLF VWUXFWXUH RI VXFK PROHFXOHV 7KH ORFDOL]HG RUELWDOV RI VXFK PROHFXOHV PD\ EH RI LQWHUHVW LQ WKHPVHOYHV LQ RUGHU WR FKDUDFWHUL]H WKH HOHFWURQLF VWUXFWXUH RI VXFK PROHFXOHV LQ DGGLWLRQ WR $ $ WKHLU XWLOLW\ LQ VXEVHTXHQW FDOFXODWLRQV ,Q RUGHU WR DSSO\ WKH FXUUHQW PHWKRG WR VXFK PROHFXOHV WKH )HUPL KROH PRELOLW\ IXQFWLRQ f PXVW EH XVHG WR UHVROYH DQ\ DPELJXLWLHV ZKLFK PD\ DULVH LQ WKH VHOHFWLRQ RI WKH SUREH HOHFWURQ SRVLWLRQV ,I WZR RU PRUH ERQGLQJ VFKHPHV DUH SRVVLEOH WKH SRVLWLRQV RI WKH SUREH HOHFWURQV VKRXOG EH FKRVHQ WR SURYLGH WKH PLQLPXP YDOXHV RI WKH )HUPL KROH PRELOLW\ IXQFWLRQV )Uf RU WKH PRELOLW\ IXQFWLRQ GLIIHUHQFH )Uf)SSf ,Q WKH FDVH RI PHWK\ODFHW\OHQH IRU H[DPSOH WKH && VLQJOH ERQG PD\ EH GHWHUPLQHG E\ D VLQJOH SRLQW PLGZD\ EHWZHHQ WKH FDUERQ DWRPV ZKHUH )Uf LV OHVV WKDQ )TSf $Q\ DWWHPSW WR UHSUHVHQW WKLV SRUWLRQ RI WKH PROHFXOH ZLWK D GRXEOH ERQG ZRXOG UHTXLUH SODFLQJ D SUREH HOHFWURQ DZD\ IURP WKH && D[LV LQ D UHJLRQ ZKHUH )Uf LV JUHDWHU WKDQ )TSf

PAGE 41

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f 7KH LQWHJUDO WUDQVIRUPDWLRQ SURFHGXUH LQ DQ\ ORFDOL]DWLRQ SURFHGXUH FDQ EH D WLPH OLPLWLQJ VWHS ,Q WKH (GPLVWRQ5HXGHQEHUJ PHWKRG WKH WZR HOHFWURQ UHSXOVLRQ LQWHJUDOV PXVW EH WUDQVIRUPHG DQ FRPSXWDWLRQDO VWHS EXW WKH %R\V PHWKRG PD\ EH

PAGE 42

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r I, 8A f ZKHUH IAA LV WKH UHVXOWLQJ VHW RI PRUH ORFDOL]HG RUELWDOV 7KH RULJLQDO PDWUL[ IRUPXODWLRQ LV EDVHG RQ D VHTXHQFH RI SDLUZLVH URWDWLRQV DV SURSRVHG E\ (GPLVWRQ DQG 5HXGHQEHUJ ,Q WKLV SURFHGXUH 1 RUELWDOV DUH ORFDOL]HG E\ URWDWLQJ D SDLU RI RUELWDOV WKHQ D VHFRQG SDLU RI RUELWDOV LV URWDWHG HWF XQWLO DOO 11Of SDLUV RI RUELWDOV KDYH EHHQ URWDWHG 6LQFH WKH URWDWLRQV DUH GRQH LQ D VSHFLILF RUGHU WKH ORFDOL]HG RUELWDOV REWDLQHG ZLOO EH GHSHQGHQW RQ WKH RUGHU RI URWDWLRQV /HRQDUG DQG /XNHQ KDYH GHYHORSHG D VHFRQG RUGHU PHWKRG WKDW GRHV DOO RI WKH 11Of URWDWLRQV DW RQFH UDWKHU WKDQ RQH DW D WLPHnnrf 7KH XVH RI D VHFRQG RIGHU PHWKRG PD\ KDYH WKH DGGLWLRQDO EHQHILW RI LPSURYLQJ WKH FRQYHUJHQFH GLIILFXOWLHV HQFRXQWHUHG LQ LWHUDWLYH PHWKRGV 7KHLU PHWKRG LV RXWOLQHG EHORZ

PAGE 43

7KH 1[1 XQLWDU\ WUDQVIRUPDWLRQ PDWUL[ IRU WKH ORFDOL]DWLRQ FDQ EH ZULWWHQ DV 8 :5 f ZKHUH 9 LV D SRVLWLYH GHILQLWH PDWUL[ GHILQHG E\ 9 55fa f 7KH PDWUL[ 5 ZLOO EH GHILQHG DV 5 17 f ZKHUH 7 W f DQG W LV DQ DQWLV\PPHWULF PDWUL[ W W f 7KH 1 PDWUL[ LV D GLDJRQDO PDWUL[ ZKLFK QRUPDOL]HV WKH FROXPQV RI 5 %\ DSSOLFDWLRQ RI 8 WR D VHW RI RUELWDOV ^IA IAf RQH SURGXFHV D VHW RI PRUH ORFDOL]HG RUELWDOV ^IA IAf 7KH QHZ YDOXH RI WKH ORFDOL]DWLRQ *n LV JLYHQ E\ 1 *n = LnLnLnLnf f Ln O 7KH QHZ In RUELWDOV FDQ EH WKRXJKW RI LQ WHUPV RI D SHUWXEDWLYH H[SDQVLRQ In I ( WI e W7W7I f ,WR VHFRQG RUGHU 7KH WA PDWUL[ GRHV [ URWDWLRQV WKDW PL[ LQ SRUWLRQV RI DOO WKH RFFXSLHG RUELWDOV LQWR RUELWDO In 7KH WKLUG WHUP WAWM LV WKH SURGXFW RI D SDLU RI [ URWDWLRQV :KHQ WKLV IRUP RI WKH In

PAGE 44

RUELWDOV LV VXEVWLWXWHG LQWR WKH *n HTXDWLRQ \RX REWDLQ *n *T WM*MGf = WMWAG-f f /HRQDUG DQG /XNHQA LQFOXGH WKH *A VHFRQG RUGHU WHUP WR DFFHOHUDWH FRQYHUJHQFH ZKHQ RQH LV LQ WKH TXDGUDWLF UHJLRQ RQO\ WKH ILUVW RUGHU WHUP LV FDOFXODWHG LQLWLDOO\ DQG XQWLO WKH TXDGUDWLF UHJLRQ LV HQFRXQWHUHG ,Q SUDFWLFH VWDQGDUG SURFHGXUHV IRU ORFDOL]DWLRQ RIWHQ FDQ WDNH VHYHUDO KXQGUHG LWHUDWLRQV WR FRQYHUJH 7KHVH VHFRQG RUGHU SURFHGXUH GHVFULEHG DERYH VHOGRP WDNHV PRUH WKDQ F\FOHV DQG WKH HQHUJ\ RI RUELWDOV UHODWHG E\ V\PPHWU\ &+ ERQGV LQ EHQ]HQH HWFf LV XVXDOO\ UHSURGXFHG WR VHYHQ VLJQLILFDQW ILJXUHV 5HVXOWV 2QH H[DPSOH RI WKH %R\V PHWKRG RI /HRQDUG DQG /XNHQ ZDV JLYHQ LQ WKH )HUPL KROH PHWKRG VHFWLRQ LQ WKLV VHFWLRQ ZH ZLOO VKRZ WKH ORFDOL]HG RUELWDOV IRW WKH 1L&2fA LRQ ,Q D UHFHQW H[SHULPHQWDO SDSHU E\ 5HXWW HW DO WKH SKRWRHOHFWURQ VSHFWURVFRS\ RI 1L&f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n SURFHGXUH 7KH ORFDOL]DWLRQ EUHDNV WKH RUELWDOV LQWR VHYHUDO FODVVHV f R[\JHQ ORQH SDLUV f FDUERQ R[\JHQ W EDQDQDf ERQGV f QLFNHO FDUERQ ERQGV DQG

PAGE 45

f QLFNHO GRUELWDOV RI WZR W\SHV QDPHO\ ( DQG 7 W\SH RUELWDOV 7KH ORFDOL]DWLRQ PD\ DOVR EH GRQH RQ WKH XQRFFXSLHG RUELWDOV WKLV VHSDUDWHV WKH XQRFFXSLHG RUELWDOV LQWR WZR VHWV f QLFNHO FDUERQ DQWLERQGV DQG f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f 7KH XVH RI DQ\ ORFDOL]HG RUELWDO WHFKQLTXH GRHV QRW DGG RU VXEWUDFW LQIRUPDWLRQ IURP WKH RYHUDOO ZDYHIXQFWLRQ 7KHVH PHWKRGV RQO\ GLYLGH RUELWDOV LQWR PRUH FKHPLFDO SLHFHV DOORZLQJ IRU HDVLHU LQWHUSUHWDWLRQ RI H[SHULPHQWDO UHVXOWV

PAGE 46

)LJXUH $Q LVRYDOXH ORFDOL]HG RUELWDO SORW RI D QLFNHO FDUERQ ERQGLQJ RUELWDO LQ WKH 1L&2fr PROHFXOH 7KH GDVKHG OLQHV LQGLFDWH DQ RUELWDO DPSOLWXGH RI DX SHU FXELF ERKU 7KH VROLG OLQHV LQGLFDWH DQ RUELWDO DPSOLWXGH RI DX SHU ERKU

PAGE 47

)LJXUH $Q LVRYDOXH ORFDOL]HG RUELWDO SORW RI D QLFNHO FDUERQ QRQn ERQGLQJ RUELWDO LQ WKH 1L&2fA PROHFXOH 7KH GDVKHG OLQHV LQGLFDWH DQ RUELWDO DPSOLWXGH RI DX SHU FXELF ERKU 7KH VROLG OLQHV LQGLFDWH DQ RUELWDO DPSOLWXGH RI DX SHU ERKU

PAGE 48

)LJXUH $Q LVRYDOXH ORFDOL]HG RUELWDO SORW RI D QLFNHO FDUERQ DQWLn ERQGLQJ RUELWDO LQ WKH 1L&2fA PROHFXOH 7KH GDVKHG OLQHV LQGLFDWH DQ RUELWDO DPSOLWXGH RI DX SHU FXELF ERKU 7KH VROLG OLQHV LQGLFDWH DQ RUELWDO DPSOLWXGH RI DX SHU ERKU

PAGE 49

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f WHFKQLTXH IRU XVH LQ FDOFXODWLQJ SURSHUWLHV RI RUELWDO FRPSOH[HV $W WKH 6HOI&RQVLVWHQW )LHOG 6&)f OHYHO WKLV WHFKQLTXH H[HFXWHV DV UDSLGO\

PAGE 50

} RQ D FRPSXWHU DV GRHV WKH ([WHQGHG +XFNHO PHWKRG DQG FRQVLGHUDEO\ PRUH UDSLG WKDQ WKH VFDWWHUHG ZDYH ;D PHWKRG 6LQFH WKH HOHFWURVWDWLFV RI WKH ,1'2 PHWKRG DUH UHDOLVWLFDOO\ UHSUHVHQWHG PROHFXODU JHRPHWULHV FDQ EH REWDLQHG XVLQJ JUDGLHQW PHWKRGVA 6LQFH WKH ,1'2 PHWKRG ZH H[DPLQH FRQWDLQV DOO RQHFHQWHU WZRHOHFWURQ WHUPV LW LV DOVR FDSDEOH RI \LHOGLQJ WKH HQHUJLHV RI YDULRXV VSLQ VWDWHV LQ WKUAH V\VWHPV :LWK FRQILJXUDWLRQ LQWHUDFWLRQ &Of WKLV PRGHO VKRXOG DOVR EH XVHIXO LQ H[DPLQLQJ WKH 89YLVLEOH VSHFWUD RI RUELWDO FRPSOH[HV 3UHOLPLQDU\ VWXGLHV RI RUELWDO FKHPLVWU\ XVLQJ DQ ,1'2 PRGHO KDYH EHHQ GLVFORVHG E\ &ODFN DQG :DUUHQ DQG PRUH UHFHQWO\ E\ /L0LQ -LQJ4XLQJ *XDQJ ;LDQ DQG ;LX =KHQ 7KH PHWKRG ZH H[DPLQH ZLOO GLIIHU IURP WKHLU PHWKRGRORJ\ LQ VHYHUDO DUHDV DV GLVFXVVHG EHORZ 6HYHUDO SUREOHPV XQLTXH WR DQ ,1'2 WUHDWPHQW RI WKHVH V\VWHPV PXVW EH FRQVLGHUHG DQG ZH KDYH YHU\ OLWWOH DE LQLWLR ZRUN WR JXLGH XV $V PHQWLRQHG ZKDW UROH GR UHODWLYLVWLF HILHFWV SOD\" $OWKRXJK ZH PLJKW KRSH WR SDUDPHWHUL]H VFDODU FRQWULEXWLRQV WKURXJK WKH FKRLFH RI RUELWDOV DQG SVHXGRSRWHQWLDO SDUDPHWHUV VSLQ RUELW FRXSOLQJ RIWHQ ODUJHU WKDQ FU\VWDO ILHOG HIIHFWV ZLOO QHHG WR EH FRQVLGHUHG DW VRPH ODWHU VWDJH 6LQFH RUELWDOV DLH JHQHUDOO\ WLJKW DQG OLJDQG ILHOG VSOLWWLQJV WKXV VPDOO D JUHDW PDQ\ VWDWHV GLIIHULQJ RQO\ LQ WKHLU RUELWDO SRSXODWLRQV OLH YHU\ FORVH LQ HQHUJ\ 7KHVH QHDU GHJHQHUDFLHV RIWHQ SUHYHQW DXWRPDWLF 6&) FRQYHUJHQFH D SUREOHP ZLWK ZKLFK ZH PXVW GHDO IRU DQ HIIHFWLYH PRGHO 7KH QDWXUH RI WKH YDOHQFH EDVLV VHW LWVHOI LV LQ TXHVWLRQ $UH WKH ILOOHG S DQG WKH YDFDQW S RI WKH ODQWKDQLGHV ERWK UHTXLUHG IRU D SURSHU GHVFULSWLRQ RI WKHLU FRPSRXQGV"

PAGE 51

0RGHO 7KH ,1'2 PRGHO +DPLOWRQLDQ WKDW ZH XVH ZDV ILUVW GLVFORVHG E\ 3RSOH DQG FROODERUDWRUVrA DQG WKHQ DGMXVWHG IRU VSHFWURVFRS\rA DQG H[WHQGHG $ WR WKH WUDQVLWLRQ PHWDO VHULHV 7KH GHWDLOV RI WKLV PRGHO DUH SXEOLVKHG HOVHZKHUH f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n WKLV PRGHO KLJKO\ VXFFHVVIXO LQ SUHGLFWLQJ RSWLFDO SURSHUWLHV f :H HPSOR\ LQ WKLV PRGHO RQH VHW RI SXUH SDUDPHWHUV WKH UHVRQDQFH RU %Nf SDUDPHWHUV IRU HDFK ODQWKDQLGH DWRP ZH GHFLGHG WR XVH %Vf %Sf %Gf DQG %f 7KHVH SDUDPHWHUV ZLOO EH FKRVHQ WR JLYH VDWLVIDFWRU\ JHRPHWULHV RI PRGHO V\VWHPV $QRWKHU FKRLFH LV RQH WKDW JLYHV JRRG SUHGLFWLRQn RI 89YLVLEOH VSHFWURVFRS\A7KHVH YDOXHV VHOGRP GLIIHU PXFK IURP WKRVH FKRVHQ WR UHSURGXFH PROHFXODU JHRPHWU\ ,Q WKLV LQLWLDO ZRUN DOO WZRFHQWHU WZRHOHFWURQ LQWHJUDOV UHTXLUHG IRU WKH ,1'2 PRGHO +DPLOWRQLDQ DUH FDOFXODWHG RYHU WKH FKRVHQ EDVLV VHW DV DUH WKH RQHFHQWHU WZRHOHFWURQ )r LQWHJUDOV $Q DOWHUQDWH FKRLFH ZRXOG EH RQH WKDW IRFXVHV RQ PROHFXODU VSHFWURVFRS\ ,Q VXFK D FDVH DQG RQH WKDW ZH KDYH WR LQYHVWLJDWH VXEVHTXHQWO\ WKH RQH FHQWHU WZR HOHFWURQ )r FRXOG EH FKRVHQ IURP WKH 3DULVHU DSSUR[LPDWLRQrA )rQf ,3Qf ($Qf ,3 ,RQL]DWLRQ 3RWHQWLDO ($ (OHFWURQ $IILQLW\f DQG WKH WZRHOHFWURQ WZRFHQWHU LQWHJUDO IURP RQH RI WKH PRUH VXFFHVVIXO IXQFWLRQV HVWDEOLVKHG IRU WKLV SXUSRVH

PAGE 52

$W WKH 6&) OHYHO ZH VHHN VROXWLRQV WR WKH SVHXGRHLJHQYDOXH SUREOHP ) & & H f ZLWK ) WKH )RFN RU HQHUJ\ PDWUL[ & WKH PDWUL[ FRPSRXQG RI 0ROHFXODU 2UELWDO 02f FRHIILFLHQWV DQG H D GLDJRQDO PDWUL[ RI 02 HLJHQYDOXHV 7KH DERYH HTXDWLRQ LV IRU WKH FORVHG VKHOO FDVH DOO HOHFWURQV SDLUHGf 7KH XQUHVW-LFWHG +DUWUHH )RFN FDVH LV GLVFXVVHG LQ GHWDLO HOVHZKHUHrA DV LV WKH RSHQ VKHOO UHVWULFWHG FDVH $OWKRXJK QHDUO\ DOO RUELWDO V\VWHPV DUH RSHQ VKHOO FRQVLGHUDWLRQ RI WKH FORVHG VKHOO FDVH GHPRQVWUDWHV WKH UHTXLUHG WKHRU\ DQG LV FRQVLGHUDEO\ VLPSOHU :LWKLQ WKH ,1'2 PRGHO HOHPHQWV RI ) DUH JLYHQ E\ $$ LU $$ n08 f 6 3 >M;@ F $ FU$ 88 FU;f M \D_\;f Df )$$ 09 e 3&7&7 88_U[f f ( =% 8_V%V%f D F % %0 ( 1[ D ; 8 Y_D;f M \D_Y;f EDXf %EYf 6 LU 3 \\ YYf \?L \Y Y n \ rY $r% Ef Ff ZKHUH \Y_X;f -G[OfG[f \Lf ;YOf ;rf ;[f f 3 LV WKH ILUVW RUGHU GHQVLW\ PDWUL[ DQG VLQFH RQH DVVXPHV WKDW WKH $WRPLF 2UELWDO $2f EDVLV ^;A` LV RUWKRQRUPDO LW LV LGHQWLFDO WR WKH FKDUJH DQG ERQG RUGHU PDWUL[ JLYHQ E\

PAGE 53

3 \Y 02 e F & Q \D ?!D D D f $% ZLWK Q WKH RFFXSDWLRQ RI 02 Q ,Q (T f ) UHIHUV 09 WR D PDWUL[ HOHPHQW ;A_I_[A! ZLWK $2 FHQWHUHG RQ DWRP $ 7KH FRUH LQWHJUDO & ‘ [eO ? A 7 n$O[Lf f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r 7KH ODVW WHUP LQ (T Df UHSUHVHQWV WKH DWWUDFWLRQ EHWZHHQ DQ HOHFWURQ LQ GLVWULEXWLRQ ;Ar ;A DQG DOO QXFOHL EXW $ 7KH UDWLRQDOH IRU UHSODFLQJ LQWHJUDO $ $ \$_5%_X$f cM \ _6%6%f f LV JLYHQ HOVHZKHUH DQG FRPSHQVDWHV IRU QHJOHFWHG WZR FHQWHU LQQHU VKHOOYDOHQFH VKHOO UHSXOVLRQ f DQG QHJOHFWHG YDOHQFH RUELWDO V\PPHWULFDOf RUWKRJRQDOL]DWLRQ f =J LV WKH FRUH FKDUJH RI DWRP %

PAGE 54

DQG LV HTXDO WR WKH QXPEHU RI HOHFWURQV RI QHXWUDO DWRP % WKDW DUH H[SOLFLWO\ FRQVLGHUHG LH IRU FDUERQ IRU LURQ IRU FHULXP HWF 6 RI (T Ff LV UHODWHG WR WKH RYHUODS PDWUL[ DQG LV JLYHQ E\ 6\Y A8OfYOf J\Of?MOf rOf_YOff f ZKHUH LV WKH (XOHULDQ WUDQVIRUPDWLRQ IDFWRU UHTXLUHG WR URWDWH IURP WKH ORFDO GLDWRPLF V\VWHP WR WKH PROHFXODU V\VWHP \Of _?!ff DUH WKH VLJPD f SLO Of GHOWDO f RU SKL f FRPSRQHQWV WR WKH RYHUODS LQ WKH ORFDO V\VWHP DQG DUH HPSLULFDO ZHLJKWLQJ IDFWRUV FKRVHQ WR EHVW UHSURGXFH WKH PROHFXODU RUELWDO HQHUJ\ VSUHDG IRU PRGHO DE LQLWLR FDOFXODWLRQV :H KDYH PDGH OLWWOH XVH RI WKLV I IDFWRU DQG VHW DOO H[FHSW EHWZHHQ S V\PPHWU\ RUELWDOV YL] f f 6 SS f pSDSFM SF7OS&7f rJSLWS-WSUWOS7Wf nf Q9SQ S OSLO %DVLV 6HW ,Q JHQHUDO ='2 PHWKRGV FKRRVH D EDVLV VHW RI 6ODWHU 7\SH 2UELWDOV 672f UQHaAU \W!f Df ZKHUH
PAGE 55

Ef ,Q JHQHUDO D VLQJOH 5 IXQFWLRQ GHVFULEHV WKH V DQG S RUELWDOV IRU PRVW DWRPV 7KH G RUELWDOV RI WKH WUDQVLWLRQ PHWDOV KRYHYHU UHTXLUH DW OHDVW D GRXEOH& W\SH IXQFWLRQ WYR WHUPV LQ Ef IRU DQ DFFXUDWH GHVFULSWLRQ RI ERWK WKHLU LQQHU DQG RXWHU UHJLRQV )RU WKH ODQWKDQLGHV ZR KDYH H[DPLQHG EDVLV VHWV VXJJHVWHG E\ /L /H0LQ HW DO E\ %HQGHU DQG 'DYLGVRQ DQG E\ &OHPHQWL DQG 5RHWWL ,Q WKH ODWWHU FDVH WKH WZR PDMRU FRQWULEXWRUV RI (T Ef LQ WKH YDOHQFH RUELWDOV RI WKH GRXEOH& DWRPLF FDOFXODWLRQV ZHUH VHOHFWHG DQG WKHVH IXQFWLRQV ZHUH UHQRUPDOL]HG ZLWK IL[HG UDWLR WR \LHOG WKH UHTXLUHG QRGHOHVV GRXEOH& IXQFWLRQV IRU 0' :H ZHUH XQDEOH ZLWK DQ\ RI WKHVH FKRLFHV WR GHYHORS D V\VWHPDWLF PRGHO XVHIXO IRU SUHGLFWLQJ PROHFXODU JHRPHWULHV VHH ODWHU GLVFXVVLRQ RI UHVRQDQFH LQWHJUDOVf :H KDYH DGDSWHG WKH IROORZLQJ SURFHGXUH RQ VHOHFWLQJ DQ HIIHFWLYH EDVLV VHW .QDSSH DQG 5RVFK FDOFXODWHG WKH ODQWKDQLGHV DQG WKHLU PRQRSRVLWLYH LRQV XVLQJ WKH QXPHULFDO 'LUDF)RFN UHODWLYLVWLF DWRPLF SURJUDP RI 'HVFODX[ )URP WKHVH ZDYHIXQFWLRQV MFGLDO H[SHFWDWLRQ YDOXHV U! U DQG U DUH FDOFXODWHG IRU V S G DQG IXQFWLRQV 7KH V G DQG I ZDYHIXQFWLRQV ZHUH REWDLQHG E\ 'LUDF)RFN FDOFXODWLRQV RQ WKH SURPRWHG PBAGAVA FRQILJXUDWLRQ WKH S IURP FDOFXODWLRQV LQ ZKLFK D G HOHFWURQ ZDV SURPRWHG QBAVASnrf :DYHIXQFWLRQV IRU WKH PRQRSRVLWLYH LRQV DUH REWDLQHG IURP IPBAGnrnVnrf DQG PBAf SA UHVSHFWLYHO\ $ JHQHUDOL]HG 1HZWRQ SURFHGXUH ZDV WKHQ XVHG WR GHWHUPLQ H[SRQHQWV &f DQG FRHIILFLHQWV IRU D JLYHQ VHW RI U! U DQG U ZLWK IXQFWLRQV RI WKH IRUP RI (T f§Ef $JDLQ DV LQ WKH WUDQVLWLRQ PHWDO DWRPV ZH IRXQG WKDW D VLQJOH & IXQFWLRQ ILWV WKH QV DQG QS DWRPLF IXQFWLRQV ZHOO LQ WKH UHJLRQV ZKHUH ERQGLQJ LV

PAGE 56

LPSRUWDQW EXW WKH QOfG DQG QRZ WKH Qf UHTXLUH DW OHDVW WZR WHUPV LQ WKH H[SDQVLRQ RI (T Ef 7KLV LV GHPRQVWUDWHG IRU WKH &H LRQ LQ )LJXUH ZKHUH LW LV VKRZQ WKDW D VLQJOH& H[SDQVLRQ LV SRRU IRU WKH RXWHU UHJLRQ RI WKH IXQFWLRQ ,Q )LJXUH WKH YDOXH RI U! LV SORWWHG YHUVXV DWRPLF QXPEHU 7KH FRQWUDFWLRQ RI WKH V DQG S RUELWDOV GXH WR UHODWLYLVWLF HIIHFWV ') YV +)f LV TXLWH DSSDUHQW KHUH DQG LV D FRQVHTXHQFH RI WKH WKH JUHDWHU FRUH SHQHWUDWLRQ RI WKHVH RUELWDOV 6XEVHTXHQW H[SDQVLRQ RI WKH DQG G QRZ ZLWK LQFUHDVHG VKLHOGLQJ UHVXOWV $IWHU VRPH H[SHULPHQWDWLRQ ZH XVH WKH 'LUDF)RFN YDOXHV REWDLQHG IURP WKH PRQRSRVLWLYH LRQV 7KH EDVLV VHW DGRSWHG LV JLYHQ LQ 7DEOH 7KH DQG G IXQFWLRQV DUH TXLWH FRPSDFW $W W\SLFDO ERQGLQJ GLVWDQFH _\\f DQG GG _\\f DUH HVVHQWLDOO\ 5aA %HFDXVH RI WKLV ZH FDOFXODWH DOO WZRFHQUHU WZRHOHFWURQ LQWHJUDOV ZLWK WKH YDOXHV LQ 7DEOH 7KLV YDOXH LV FKRVHQ WR PDWFK WKH DFFXUDWH )r 6ODWHU&RQGRQ )DFWRUV REWDLQHG IURP WKH QXPHULFDO DWRPLF FDOFXODWLRQV E\ D VLQJOH H[SRQHQW YLD )r f & f Df )rGGf & Gf Ef )rVVf & Vf Ff )rSSf & Sf Gf 7KH HUURU LQ FDOFXODWLQJ WZRFHQWHUHG WZRHOHFWURQ LQWHJUDOV DW W\SLFDO ERQGLQJ GLVWDQFHV ZLWK WKLV VLQJOH& DSSUR[LPDWLRQ LV ZHOO XQGHU ,; DQG WKLV SURFHGXUH LV PXFK VLPSOHU &RUH ,QWHJUDOV 7KH DYHUDJH HQHUJ\ RI D FRQILJXUDWLRQ RI DQ DWRP RU LRQ LV JLYHQ

PAGE 57

6LQJOH YV 'RXEOH =HWD 67 2UELWDO $PSOLWXGH 6RLR U LQ DX )LJXUH 5DGLDO YDYHIXQFWLRQ IRU WKH I RUELWDO RI &H ZLWK VLQJOH& DQG GRXEOH& 6ODWHU W\SH RUELWDOV 672Vf

PAGE 58

)LJXUH $YHUDJH YDOXH RI U IRU WKH YDOHQFH RUELWDOV RI WKH ODQWKDQLGHV IURP D UHODWLYLVWLF FDOFXODWLRQ ')f DQG D QRQ UHODWLYLVWLF FDOFXODWLRQ +)f

PAGE 59

7DEOH 6ODWHU W\SH RUELWDO 672f EDVLV IXQFWLRQV IRU WKH ODQWKDQLGH DWRPV 7KH VLQJOH & IXQFWLRQV DUH OLVWHG IRU WKH LI G V DQG S RUELWDOV DORQJ ZLWK WKH GRXEOH & IXQFWLRQV IRU WKH DQG G RUELWDOV $WRP 6LQJOH & 'RXEOH & ([SRQHQWV ([SRQHQWV &RH IILFLHQW V LI G V S LI G G &H 3U 1G f 3P 6 P (X *G 7E '\ ,, R (O 7P
PAGE 60

N LULMLL V S GQT N 8 P8 Q8 T8 NAN A : SS GG VV VV f QLr1f Y QAaf Y X UW YW U Q rUMMnU Q Z[FU NP: NQ: SS GG VS VG NT9V PQ:SG PT9SI QT:G ZLWK M WKH DYHUDJH WZR HOHFWURQ HQHUJ\ RI D SDLU RI HOHFWURQV LQ RUELWDOV DQG ;M JLYHQ E\ 9VV )rVVf :SS )rSSf )SSf f : GG )rGGf )GGf )GGf 9II )rf )f )AIIf )f 9VS )rVSf O*VSf 9VG )rVGf *VGf :V )rVIf *Vf : SG )rSGf *;SGf *SGf 9SI )rSIf *Sf *Sf 8GI )rGIf *Gf *GIf *Gf 7KH FRUH LQWHJUDOV +B (T f DUH WKHQ HYDOXDWHG E\ UHPRYLQJ DQ HOHFWURQ IURP RUELWDO ;A DQG HTXDWLQJ WKH GLIIHUHQFH LQ FRQILJXUDWLRQ HQHUJ\ EHWZHHQ FDWLRQ DQG QHXWUDO WR WKH DSSURSULDWH REVHUYHG ,3Qf :H SUHIHU WKLV SURFHGXUH UDWKHU WKDQ WKDW VXJJHVWHG E\ RWKHUV WKDW DYHUDJH WKH YDOXH REWDLQHG IURP ,3Qf DQG ($QffAfp 7KHUH DUH D JUHDW PDQ\ ORZ O\LQJ FRQILJXUDWLRQV RI WKH ODQWKDQLGH DWRPV DQG WKHLU LRQV 7KH ORZHVW WHUPV RI &H *G DQG /X FRPH IURP QBG ML V ZKLOH WKH UHPDLQLQJ ODQWKDQLGH DWRPV KDYH WKH VWUXFWXUH M V 7ZR SURFHVVHV DUH WKHQ SRVVLEOH IRU V HOHFWURQ LRQL]DWLRQ QGV } QBGV Vf ,, V } V Vf

PAGE 61

7KH LRQL]DWLRQ HQHUJ\ RI D V HOHFWURQ IURP LV V\VWHPDWLFDOO\ H9 ODUJHU WKDQ WKDW REWDLQHG IURP ,, :KHQ FRPELQHG ZLWK (T f WKH HVWLPDWH IRU 8V GLIIHU E\ OHVV WKDQ H9 7KDW LV FKRRVLQJ WKH YDOXHV RI SURFHVV WKH XVH RI (T f SUHGLFWV WKH YDOXHV RI SURFHVV ,, ZLWKLQ H9 :H WKXV FKRRVH WKH YDOXHV RI SURFHVV VKRZQ LQ 7DEOH 7KHVH YDOXHV DUH REWDLQHG IURP WKH SURPRWLRQ HQHUJLHV RI %UHZHU f DQG WKHQ VPRRWKHG E\ D TXDGUDWLF ILW WKURXJKRXW WKH VHULHV )RU FRPSOHWHQHVV ZH DOVR JLYH WKH YDOXHV RI SURFHVV ,, MLf§ 7KH ORZHVW FRQILJXUDWLRQ FRQWDLQLQJ D G HOHFWURQ LV M DV] WKURXJKRXW WKH VHULHV DQG G LRQL]DWLRQV DUH REWDLQHG IURP ,,, = G9 IV Gf 7KH LRQL]DWLRQ SRWHQWLDOV IRU WKH S FDQ EH REWDLQHG IURP WZR SURFHVVHV ,9 I+a= V S =AnV Sf 9 = VS r BV Sf ,RQL]DWLRQ IURP SURFHVV ,9 LV QHDUO\ FRQVWDQW DW H9 IURP 9 DW f H9 7KH M AV FRQILJXUDWLRQ LV ORZHU IRU DOO WKH ODQWKDQLGHV B B H[FHSW &HIGVSf DQG *G DQG 7E=rf V Sf 8VLQJ WKH LRQL]DWLRQ SRWHQWLDOV RI SURFHVV ,9 DQG (T f ZH SUHGLFW WKH YDOXHV RI SURFHVV 9 WR ZLWKLQ H9 :H GR QRW FRQVLGHU WKLV HUURU VLJQLILFDQW DQG WKXV XVH WKH VPRRWKHG YDOXHV IURP ,9 JLYHQ LQ 7DEOH 7KH YDOXHV IURP SURFHVV 9 DUH DOVR JLYHQ LQ WKH WDEOH IRU FRPSDULVRQ )RU D I RUELWDO LRQL]DWLRQ ZH FRQVLGHU WKH WZR SURFHVVHV 9, = GV r =AGV f 9,, = V } =BV If FRPSDUH ZLWK DQG ,,f $V VHHQ LQ 7DEOH WKH YDOXHV IRUP WKH WZR SURFHVVHV DUH YHU\ GLIIHUHQW )URP (T f 8II 9,f ,39,f Pf 9II 9II :G Df 8MM 9,,f ,39,,f Pf : ?-VI Ef

PAGE 62

7DEOH c6PRRWKHG ,RQL]DWLRQ SRWHQWLDOV IRU SURFHVVHV ,9,, DQG FRQILJXUDWLRQ PL[LQJ FRHIILFLHQWV GHULYHG IURP %UHZHUnV WDEOHV 7KH HQWULHV LQ WKH FROXPQ ODEHOHG DUH WKH PL[LQJ FRHIILFLHQWV IRU WKH FRQILJXUDWLRQ IQ AGVA LQ FROXPQ WKH Q AVA FRQILJXUDWLRQ PL[LQJ FRHIILFLHQWV DUH OLVWHG $WRP 6PRRWKHG ,RQL]DWLRQ 3RWHQWLDOV 0L[LQJ &RHIILFLHQWV 3URFHVV V 3 G I ,, ,9 9 ,,, 9, 9,, &H 3L 1G 3P 6P (X *G D 7E DG '\ +R (X 7P
PAGE 63

8QOLNH WKH DQDORJRXV VLWXDWLRQ IRU WKH V DQG S RUELWDOV XVH RI (T Df WR ILQG 8A DQG XVH RI WKLV YDOXH LQ (T Ef WR SUHGLFW ,39,,f LV QRW VXFFHVVIXO DQG ZRXOG UHTXLUH WKH VFDOLQJ RI WKH ODUJH )rIIf LQWHJUDO RIWHQ SHUIRUPHG LQ PHWKRGV SDUDPHWHUL]HG RQ PROHFXODU f VSHFWURVFRS\ f $V ZLWK WKH WUDQVLWLRQ PHWDO QG RUELWDOV ZH PLJKW HQYLVLRQ WKH IROORZLQJ SURFHGXUH :H DVVXPH WKDW WKH ODQWKDQLGH DWRP LQ D PROHFXOH LV D ZHDNO\ SHUWXUEHG DWRP 7KH ORZHVW HQHUJ\ FRQILJXUDWLRQ RI WKH DWRP VKRXOG WKDQ EH PRVW LPSRUWDQW LQ GHWHUPLQLQJ 8A :H FUHDWH D WZRE\ WZR LQWHUDFWLRQ PDWUL[ ( GVf ; 9 UZFQB! ? U ? / &1M Df ZKHUH 9 LV DQ HPSLULFDO PL[LQJ SDUDPHWHU DQG DQG GHWHUPLQHV WKH UHODWLYH DPRXQWV RI HDFK RI WKH WZR FRQILJXUDWLRQV WKDW DUH LPSRUWDQW 7KH H[DFW YDOXH RI 9 ZRXOG GHSHQG RQ D JLYHQ PROHFXODU VLWXDWLRQ LV WKHQ JLYHQ E\ ; Ef fO[ 9 B F F G V=f (&I9f ; 9 Ff (r G Vf (Ir Vf Q 9 7KH YDOXHV RI DSSHDU LQ 7DEOH ZKHUH ZH KDYH XVHG WKH YDOXHV RI BMB A A (M GV f DQG (M V f REWDLQHG IRU WKH SURPRWLRQ HQHUJLHV RI %UHZHU DQG D IL[HG YDOXH RI 9 DX 7KHQ FRXOG EH REWDLQHG IURP

PAGE 64

8II &[ ,-A9,f & WOA97,f f QB ,Q WKH FDVH RI WKH G RUELWDOV WKLV YDOHQFH ERQG PL[LQJ EHWZHHQ GQLV DQG GQ AV ZDV LPSRUWDQW LQ REWDLQLQJ UHDVRQDEOH JHRPHWULF SUHGLFWLRQV DQ REVHUYDWLRQ QRZ FRQILUPHG LQ FDUHIXO DE LQLWLR VWXGLHV )RU WKH ODQWKDQLGH FRPSOH[HV RI WKLV VWXG\ WKH RUELWDOV DUH TXLWH FRPSDFW DQG WKLV YDOHQFH ERQG PL[LQJ GRHV QRW JUHDWO\ DIIHFW JHRPHWULHV +RZHYHU WKH FDOFXODWLRQ RI LRQL]DWLRQ SRWHQWLDOV WKDW UHVXOW LQ VWDWHV ZLWK UHGXFHG RUELWDOV RFFXSDWLRQ LV LQIOXHQFHG 7KHUH DUH PDQ\ UHILQHPHQWV RQH FDQ PDNH LQ WKH IRUPDWLRQ RI D PL[LQJ PDWUL[ VXFK DV (T Df 2QH PLJKW EH WR PDNH 9 GHSHQGHQW RQ WKH FDOFXODWHG SRSXODWLRQ RI WKH DQG G DWRPLF RUELWDOV +RZHYHU WKH YDOXHV RI WKH SURPRWLRQ HQHUJLHV ZH REWDLQ IURP %UHZHU DUH VR GLIIHUHQW WKDQ WKRVH WKDW ZH REWDLQ IURP RXO RZQ QXPHULFDO FDOFXODWLRQV RQ WKH DYHUDJH HQHUJ\ RI D FRQILJXUDWLRQ 7DEOH WKDW IRU WKH PRPHQW ZH FKRRVH D b b PL[ RI (L AGVf (& Vf IRU DOO WKH DWRPV RI WKH VHULHV 7KLV PL[ JLYHV UHDVRQDEOH JHRPHWULHV DQG LRQL]DWLRQ SRWHQWLDOV IRU DOO PROHFXOHV RI WKLV VWXG\ )XUWKHU UHILQHPHQWV ZLOO UHTXLUH PRUH DFFXUDWH DWRPLF SURPRWLRQ HQHUJLHV DQG QXPHULFDO H[SHULHQFH ZLWK WKH PRGHO 5HVRQDQFH 3DUDPHWHUV %Nf (DFK ODQWKDQLGH DWRP KDV WKUHH %ANf YDOXHV %Vf %Sf %Gf DQG %f DQG WKRVH ZH FKRRVH DUH VXPPDUL]HG LQ 7DEOH 7KH\ DUH REWDLQHG E\ ILWWLQJ WKH JHRPHWULHV RI WKH WULKDOLGHV DQG WKH PRUH FRYDOHQW ELVF\FORSHQWDGLHQ\OV WR EH UHSRUWHG HOVHZKHUH %RQG OHQJWKV DUH PRVW VHQVLWLYH WR %Gf DQG ERQG DQJOHV WR %Sf 7KHVH DQJOHV FDQ EH UHSURGXFHG VROHO\ RQ D EDVLV VHW LQFOXGLQJ S RUELWDOV DQG ZH KDYH EHHQ DEOH WR REWDLQ VDWLVIDFWRU\ FRPSDULVRQV ZLWK

PAGE 65

7DEOH $YHUDJH FRQILJXUDWLRQ HQHUJ\ IURP 'LUDF)RFN RQ WKH QBAGVA DQG WKH FRQILJXUDWLRQV ODQWKDQLGH DWRPV FDOFXOD WLRQV IRU DOO WKH $WRP $YHUDJH &RQILJXUDWLRQ (QHUJ\ Y &H 3U 1G 3P 6P (X *G 7E '\ +R (U 7P
PAGE 66

7DEOH 5HVRQDQFH LQWHJUDOV % YDOXHVf IRU WKH /DQWKDQLGH DWRPV LQ H 9 7KH EHWD IRU WKH VRUELWDO LV VHW HTXDO WR WKH EHWD IRU WKH SRUELWDO $WRP %Vf %Sf %Gf %If &H 3U 1G 3P 6P (X *G 7E '\ +R (U 7P
PAGE 67

H[SHULPHQW ZLWKRXW WKH QHFHVVLW\ RI LQFOXGLQJ WKH S RUELWDOV 2Q WKH RWKHU KDQG RUELWDOV RI S V\PPHWU\ GR VHHP WR EH UHTXLUHG IRU DFFXUDWH SUHGLFWLRQV RI JHRPHWU\ f ,W KDV EHHQ DUJXHG WKDW WKH RUELWDOV DUH QRW XVHG LQ WKH FKHPLFDO ERQGLQJ RI WKRVH FRPSOH[HV H[FHSW LQ WKH PRUH FRYDOHQW FDVHV f )URP WKH SUHVHQW VWXG\ ZH DUH OHDG WR WKH FRQFOXVLRQ WKDW VRPH DOEHLW VPDOO FRQWULEXWLRQ LV UHTXLUHG RI WKHVH RUELWDOV WR REWDLQ WKH H[FHOOHQW DJUHHPHQW EHWZHHQ H[SHULPHQWDO DQG FDOFXODWHG ERQG OHQJWKV IRU WKH VHULHV 0)A 0&OA 0%UA DQG 0,A DQG IRU WKH FRPSDUDWLYH YDOXHV REWDLQHG IRU &H)A DQG &H)A 7KLV LV LQGLFDWHG LQ 7DEOH E\ WKH ODUJH YDOXHV RI _%If_ 7KH ODWWHU YDOXHV DUH D FRQVHTXHQFH RI WKH IDFW WKDW WKH I RUELWDOV DUH WLJKWHU WKDQ RQH XVXDOO\ H[SHFWV IRU RUELWDOV LPSRUWDQW LQ FKHPLFD@ ERQGLQJ 8VH RI G RUELWDOV DORQH ZLOO SUHGLFW WKH WUHQGV LQ WKHVH WZR VHULHV EXW XQGHUHVWLPDWHV WKH UDQJH RI YDOXHV H[SHULPHQWDOO\ REVHUYHG 7ZR (OHFWURQ ,QWHJUDOV 6HYHUDO GLIIHUHQW LQWHUSUHWDWLRQV KDYH EHHQ JLYHQ WR WKH ,1'2 VFKHPH 7KH VLPSOHVW RI WKRVH VFKHPHV LV WR LQFOXGH RQO\ RQHFHQWHUHG LQWHJUDOV RI WKH &RXORPE RU H[FKDQJH W\SH \\ _Y?!f RU \Y_Y\f )RU DQ VS EDVLV WKHVH DUH FRPSOHWH )RU DQ VSG RU VSGI EDVLV WKH\ DUH QRW DQG WKH RPLVVLRQ RI WKH UHPDLQLQJ LQWHJUDOV ZLOO OHDG WR URWDWLRQDO YDULDQFH 7R UHVWRUH URWDWLRQDO LQYDULDQFH LQWHJUDOV RI WKLV W\SH PLJKW EH URWDWLRQDOO\ DYHUDJHG EXW IURP D VWXG\ RI VSHFWUD LW DSSHDUV WKDW DOO RQHFHQWHU LQWHJUDOV VKRXOG EH HYDOXDWHG )RU H[DPSOH LQ WKH PHWDOORFHQHV WKH LQWHJUDO G G ,G G f LV [ \ [\ [\ \] UHTXLUHG WR VHSDUDWH WKH WZR WUDQVLWLRQV WKDW DULVH IURP WKH HAGf Hf Gf WUDQVLWLRQV WKDW OHDG WR WKH nrn( DQG H[FLWHG VWDWHV ,Q J OJ J

PAGE 68

DGGLWLRQ LW DSSHDUV WKDW WKH LQFOXVLRQ RI DOO RQHFHQWHU LQWHJUDOV LPSURYHV WKH SUHGLFWLRQV RI DQJOHV DERXW DWRPV ZLWK VSG EDVLV VHWV f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f DQG \?!_?L\O DJDLQVW WKH IRUPXODV RI )DQQLQJ DQG )LW]SDWULFN ,QWHJUDOV RI WKH IRUP \X_?!Yf DQG \Y_Y\f FDQ EH REWDLQHG WKLXXJK f F DWRPLF VSHFWURVFRS\ DQG WKHLU FRPSRQHQWV ) DQG HYDOXDWHG YLD OHDVW VTXDUH ILWV \\ YYf ( DN SN N \Y_Y\f ( EN FO& N OF N 7KHVH ) DQG FDQ WKHQ EH XVHG WR HYDOXDWH DOO LQWHJUDOV RI WKH ) RU W\SH HYHQ WKRVH WKDW GR QRW DSSHDU LQ DWRPLF VSHFWUD EHFDXVH RI KLJK V\PPHWU\ LH G G _G G ff ,QWHJUDOV RI WKH 5 W\SH [ f§\ \] [] [\ KRZHYHU FDQQRW EH HYDOXDWHG LQ WKLV PDQQHU YL]VG_GGf VS_SGf VG_SSf VG_f V_Gf SS_Sf GG_Sf SG_Gf VG_Sf SG_Vf VS_Gf DQG S_f )RU WKLV UHDVRQ ZH HYDOXDWH DOO RQHFHQWHU WZR HOHFWURQ LQWHJUDOn RI WKH ODQWKDQLGHV XVLQJ WKH EDVLV VHW RI 7DEOH ZKLFK \LHOGV WKH H[DFW )r YDOXH REWDLQHG IURP WKH )RFN'LUDF QXPHULFDO N N N FDOFXODWLRQV $OO ) DQG 5n LQWHJUDO IRU N DUH WKHQ VFDOHG E\

PAGE 69

7KLV YDOXH RI WKH VFDOLQJ LV REWDLQHG IURP D FRPSDULVRQ RI WKH FDOFXODWHG DQG HPSLULFDOO\ REWDLQHGApfA )Af DQFM YDOXHV WKDW LPSOLHV (PSLULFDOO\ REWDLQHG YDOXHV RI Gf DQG ) Gf DUH IDU PRUH XQFHUWDLQ DQG DUH PXFK VPDOOHU DQG DUH WKXV QRW XVHG WR REWDLQ WKLV VFDOLQJ YDOXH EHWZHHQ FDOFXODWHG DQG H[SHULPHQWDO YDOXHV $W WKLV SRLQW LW VHHPV DSSURSULDWH WR SRLQW RXW WKH GLIIHUHQFHV RI WKH SUHVHQW ,1'2 PRGHO WR WKDW VXJJHVWHG E\ /L /H0LQ HWB DO ,Q WKH ODWWHU IRUPDOLVP RQO\ WKH FRQYHQWLRQDO RQHFHQWHU WZRHOHFWURQ LQWHJUDOV DUH LQFOXGHG OHDGLQJ WR URWDWLRQDO YDULDQFH ,Q DGGLWLRQ WKH 9ROIVEHUJ+HOPKRO] DSSURDFK LV XVHG IRU WKH UHVRQDQFH LQWHJUDO % %f ,3Lf,3 M f f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

PAGE 70

ODQWKDQLGH G FKDUDFWHU DUH VRXJKW DQG DVVLJQHG QR HOHFWURQV $ SURFHGXUH LV WKHQ DGRSWHG WKDW H[WUDSRODWHV D QHZ GHQVLW\ IRU D JLYHQ )RFN PDWUL[ EDVHG RQ D 0XOOLNHQ SRSXODWLRQ DQDO\VLV RI HDFK 6&) F\FOH 2IWHQ WKLV SURFHGXUH LV QRW VXFFHVVIXO ,Q VXFK FDVHV DOO RUELWDOV DUH FRQVLGHUHG GHJHQHUDWH DQG WKH\ DUH HTXDOO\ RFFXSLHG LQ WKH KLJKHVW VSLQ FRQILJXUDWLRQ XVLQJ WKH 5+) RSHQ VKHOO PHWKRGA 7KHVH YHFWRUV RUELWDOVf DUH WKHQ VWRUHG DQG WKH 6&) UHSHDWHG ZLWK WKH VSHFLILF I RUELWDO DVVLJQPHQWV DV GHVFULEHG DERYH ,Q FDVHV RI VORZ FRQYHUJHQFH D VLQJOHV RU VPDOO VLQJOHV DQG GRXEOHV &O LV SHUIRUPHG WR FKHFN WKH VWDELOLW\ RI WKH 6&) DQG WKH DSSURSULDWHQHVV RI WKH IRUFHG HOHFWURQ DVVLJQPHQW WR REWDLQ WKH GHVLUHG f VWDWH 5HVXOWV 7KH JHRPHWULHV RI &H&OA DQG /X&OA ZHUH XVHG WR GHWHUPLQH DQ RSWLPDO VHW RI UHVRQDQFH LQWHJUDOV DQG FRQILJXUDWLRQDO PL[LQJ FRHIILFLHQWV 1R IXUWKHU ILWWLQJ ZDV SHUIRUPHG DQG WKXV WKH VWUXFWXUHV RI DOO RWKHU FRPSRXQGV DUH SUHGLFWLRQV 7KH UHVRQDQFH SDUDPHWHUV IRU WKH RWKHU ODQWKDQLGHV ZHUH GHWHUPLQHG E\ LQWHUSRODWLRQ IURP WKH YDOXHV IRU &H DQG /X VHH 7DEOH f 7KH ,1'2 RSWLPL]HG JHRPHWULHV DV ZHOO DV WKH UHPDLQLQJ FHULXP DQG OXWHWLXP WULKDOLGHV DUH OLVWHG LQ 7DEOH ,Q DGGLWLRQ WR WKH WULKDOLGHV UHSRUWHG WKH JHRPHWU\ RI &H)A LV DOVR OLVWHG LQ 7DEOH 2QH FDQ VHH WKH DJUHHPHQW ZLWK H[SHULPHQW LV JRRG LQ DOO FDVHV 7KH SRWHQWLDO HQHUJ\ RI WKH WULKDOLGHV DV D IXQFWLRQ RI WKH RXW RI SODQH DQJOH LV YHU\ IODW $OWKRXJK ZH KDYH RSWLPL]HG DOO VWUXFWXUHV XQWLO WKH JUDGLHQWV DUH EHORZ A DXERKU WKH DQJOHV DUH FRQYHUJHG RQO\ WR r :H QRWH KRZHYHU WKDW DOO DUH SUHGLFWHG QRQSODQDU LQ E DJUHHPHQW ZLWK H[SHULPHQWV f f

PAGE 71

7DEOH *HRPHWU\ DQG LRQL]DWLRQ SRWHQWLDOV IRU &HULXP DQG /XWHWLXP WULKDOLGHV &HULXP WHWUDIOXRULGH LV DOVR LQFOXGHG LQ WKLV WDEOH 7KH ERQG GLVWDQFHV DUH JLYHQ LQ DQJVWURPVA DQJOHV LQ GHJUHHV DQG ,3V LQ H9 ([SHULPHQWDOA UHVXOWV DUH DOVR VKRZQ ZKHUH DYDLODEOH 0ROHFXOH %RQG 'LVWDQFH %RQG $QJOH ,RQL]DWLRQ 3RWHQWLDO ,1'2 ([S ,1'2 ([S ,1'2 ([S &H) f§ &H&O &H%U &HO f§ f§ &H) f§ f§ /X) f§ E /X& f /X%U f /XO f Df 5HIHUHQFHV E DQG (VWLPDWHG YDOXHV IRU &H)&H,a DQG /X) IURP 5HI A Ef 7KH 6&) FDOFXODWLRQ RQ WKH LRQ RI /X)a ZRXOG QRW FRQYHUJH WKHUHIRUH QR ,3 LV UHSRUWHG

PAGE 72

7KH H[SHULPHQWDO UDQJH RI WKH ERQG OHQJWKV IURP /Q)A WR /QOA LV JUHDWHU WKDQ ZH FDOFXODWH 2XU SUHGLFWHG YDOXHV IRU WKH WULI XRULGHV DQG WULFKORULGHV DUH LQ JRRG DJUHHPHQW ZKLOH ERQG OHQJWKV IRU WKH WULEURPLGHV DQG WULLRGLGHV DUH WRR VKRUW 6LQFH WKHVH DUH WKH PRUH SRODUL]DEOH DWRPV LW LV SRVVLEOH WKDW FRQILJXUDWLRQ LQWHUDFWLRQ ZLOO KDYH LWV ODUJHVW DIIHFW RQ WKHVH V\VWHPV 7KH FDOFXODWHG FKDQJH LQ ERQG OHQJWK RI ƒ LQ JRLQJ IURP &H)A WR &H)A LV DOVR VPDOOHU WKDQ WKH ƒ REVHUYHG ,RQL]DWLRQ SRWHQWLDOV ,3Vf DUH DOVR UHSRUWHG LQ 7DEOH ,Q DOO FDVHV WKH ,1'2 YDOXHV IDOO ZLWKLQ WKH H[SHULPHQWDO UDQJHV 7KHVH YDOXHV DUH FDOFXODWHG XVLQJ WKH $6&) PHWKRG DQG RQO\ WKH ILUVW ,3 LV E FDOFXODWHG ([SHULPHQWDOO\ f WKHVH YDOXHG DUH VRPHZKDW XQFHUWDLQ EXW WKH\ DUH VSOLW E\ ERWK FU\VWDO ILHOG HIIHFWV DQG E\ WKH ODUJH VSLQ RUELW FRXSOLQJ QRW \HW LQFOXGHG LQ RXU FDOFXODWLRQV +RZHYHU WKH ODWWHU LQWHUDFWLRQ LV WUHDWHG LPSOLFLWO\ LQ WKH '90 ;D FDOFXODWLRQVnAr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nArfAA DUH YHU\ ZHOO UHSURGXFHG E\ WKH ,1'2 FDOFXODWLRQV 7KH ,1'2 ,3V UHSURGXFH WKH FKDUDFWHULVWLF : SDWWHUQ RI WKH ODQWKDQLGH DWRPV DQG IDOO ZLWKLQ WKH H[SHULPHQWDO UDQJHV

PAGE 73

7DEOH *HRPHWULHV DQG ,RQL]DWLRQ 3RWHQWLDOV ,3Vf IRU WKH ODQWKDQLGH WULFKORULGHV %RQG GLVWDQFHV DUH UHSRUWHG LQ DQJVWURPV ERQG DQJOHV LQ GHJUHHV DQG ,3V LQ H9 ([SHULPHQWDO UHVXOWV DUH DOVR JLYHQ ZKHUH DYDLODEOH $WRP %RQG 'LVWDQFH %RQG $QJOH ,RQL]DWLRQ 3RWHQWLDO ,1'2 ([S ,1'2 ([S ,1'2 ([S &H 3U f 1G f§ 3P f§ f§ f§ 6P f§ f§ f (X f§ f§ f§ *G f 7E f '\ f§ f§ f +R f (U f§ f§ f 7P f§ f§ f
PAGE 74

7R WHVW WKH DSSOLFDELOLW\ RI RXU PRGHO WR ODQWKDQLGH DWRPV QRW IRUPDOO\ FKDUJHG ZH FDOFXODWHG WKH JHRPHWULHV DQG ,3V IRU 6P&A (8&, DQG
PAGE 75

7DEOH *HRPHWU\ DQG LRQL]DWLRQ SRWHQWLDO IRU 6P&A (X&A DQG
PAGE 76

B 7DEOH $YHUDJH ERQG GLVWDQFHV DQG ERQG DQJOHV IRU &HL12AfA LRQ ,1'2 RSWLPL]HG JHRPHWU\ DQG WKH ;UD\ FU\VWDO VWUXFWXUH 'LVWDQFHV DUH LQ DQJVWURPV DQG DQJOHV LQ GHJUHHV 7KH F VXEVFULSW RQ WKH R[\JHQ DWRPV GHQRWHV WKH WKDW R[\JHQ LV ERQGHG WR WKH FHULXP DQG WKH Q VXEVFULSW VLJQLILHV D QRQ ERQGHG R[\JHQ *HRPHWULF 3DUDPHWHU ,1'2 ([S U&HFf U1Ff U1Qf 1 f F Fn &H f F F Df 5HIHUHQFH

PAGE 77

&(1f B )LJXUH 3ORW RI WKH WZHOYH FRRUGLQDWH &H12Af LRQ 1LWURJHQV DQG DUH DERYH WKH SODQH RI WKH SDSHU ZKLOH QLWURJHQV DQG OLH EHORZ WKH SODQH RI WKH SDSHU

PAGE 78

B 7DEOH 3RSXODWLRQ DQDO\VLV RI &HL12AfA f 7KH R[\JHQ DWRPV WKDW DUH FRRUGLQDWHG WR WKH FHULXP DUH LQGLFDWHG E\ F!7KH :\EHUJ ERQG LQGH[ LV DOVR JLYHQ $ :\EHUJ LQGH[ RI LV FKDUDFWHULVWLF RI D VLQJOH ERQG $WRP 2UELWDO $WRPLF 3RSXODWLRQ 6SLQ 'HQVLW\ 7RWDO 9DOHQFH &H V f§ 3 f§ G f§ I f§ 1HW 1 1HW F 1HW 1HW %RQG :\EHUJ %RQG ,QGH[ &H F 1 F 1

PAGE 79

&21&/86,216 :H GHYHORS DQ ,QWHUPHGLDWH 1HJOHFW RI 'LIIHUHQWLDO 2YHUODS ,1'2f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
PAGE 80

DQG /XOA DQG &H)A DQG &H)A DUH UHSURGXFHG ZLWKRXW RUELWDO SDUWLFLSDWLRQ WKH UDQJH RI YDOXHV FDOFXODWHG LV FRQVLGHUDEO\ LPSURYHG ZKHQ RUELWDOV DUH DOORZHG WR SDUULFL SDWH )RU WKH WZHOYH FRRUGLQDWH &H12AfA FRPSOH[ UHSRUWHG KHUH RUELWDO SDUWLFLSDWLRQ DSSHDUV PLQRU $ VWDEOH FRPSOH[ RI QHDU V\PPHWU\ LV REWDLQHG UHJDUGOHVV RI WKH RUELWDO LQWHUDFWLRQ

PAGE 81

%,%/,2*5$3+< 9 )RFN = 3K\VLN f & (GPLVWRQ DQG 5XHGHQEHUJ 5HY 0RG 3K\V f & (GPLVWRQ DQG 5XHGHQEHUJ &KHP 3K\V 6 f 6) %R\V LQ 4XDQWXP 7KHRU\ RI $WRPV 0ROHFXOHV DQG WKH 6ROLG 6WDWH S /RZGLQ 3 (G 1HZ
PAGE 82

6 'LQHU )3 0DOULHX ) -RUGDQ DQG 0 *LOEHUW 7KHRUHW &KLP $FWD %HUOf f ) -RUGDQ 0 *LOEHUW -3 0DOULHX DQG 8 3LQFHOOL 7KHRUHW &KLP $FWD %HUOf f -0 &XOOHQ DQG 0& =HUQHU ,QW 4XDQWXP &KHP f 3 /RZGLQ 3K\V 5HY f $GY &KHP 3K\V f :/ /XNHQ DQG '1 %HUDWDQ (OHFWURQ &RUUHODWLRQ DQG WKH &KHPLFDO %RQG 'XUKDP 1& )UHHZDWHU 3URGXFWLRQ 'XNH 8QLYHUVLW\ :/ /XNHQ DQG -& &XOEHUVRQ LQ /RFDO 'HQVLW\ $SSUR[LPDWLRQV LQ 4XDQWXP &KHPLVWU\ DQG 6ROLG 6WDWH 3K\VLFV -3 'DKO $YHU\ (GV 1HZ
PAGE 83

6LQDQRJOX DQG % 6NXWQLN &KHP 3K\V /HWW f : .XW]HOQLJJ ,VUDHO &KHP f 0 6FKQLGOHU DQG : .XW]HOQLJJ &KHP 3K\V f /$ “DILH DQG 3/ 3RODUDYDSX &KHP 3K\V f 5 /DYHU\ & (WFKHEHVW DQG $ 3XOOPDQ &KHP 3K\V /HWW f -( 5HXWW /6 :DQJ <7 /HH DQG '$ 6KLUOH\ &KHP 3K\V /HWW f 70DUNV DQG ,/ )UDJDO£ )XQGDPHQWDO DQG 7HFKQRORJLFDO $VSHFWV RI 2UJDQRA(OHPHQW &KHPLVWU\ 1$72 $6, 6HULHV & 5HLGHO 'RUGUHFKW 70DUNV $FF &KHP 5HV f 70DUNV $GY &KHP 6HU f 70DUNV 3URJ ,QRUJ &KHP f + 6FKXPDQQ DQG : *HQWKH LQ +DQGERRN RQ WKH 3K\VLFV DQG &KHPLVW\ RI 5DUH (DUWKV 1RUWK +ROODQG $PVWHUGDP &KSW + 6FKXPDQQ $QJHZ &KHP f 3+D\ :5 :DGW /5 .DKQ 5& 5DIIHQHWWL DQG ': 3KLOOLSV &KHP 3K\V f :5 :DGW $PHU &KHP 6RF f -9 2UWL] DQG 5 +RIIPDQQ ,QRUJ &KHP f 3 3\\NNR DQG // /RKU -U ,QRUJ &KHP f &( 0\HUV /1RUPDQ ,, DQG /0 /RHZ ,QRUJ &KHP f D '( (OOLV $FWLQLGHV LQ 3HUVSHFWLYH HG 10 (GHOVWHLQ 3HUJDPRQ f E % 5XVFLF */ *RRGPDQ DQG %HUNRZLW] &KHP 3K\V f 1 5RVFK DQG $ 6WUHLWYLHVHU -U $PHU &KHP 6RF f 1 5RVFK ,QRUJ &KLP $FWD f $ 6WUHLWYLHVHU -U 6$ .LQVOH\ -7 5LJEHH ,/ )UDJDOD ( &LOLEHUWR DQG 1 5RVFK $PHU &KHP 6RF f +RKO DQG 1 5RVFK ,QRUJ &KHP f +RKO '( (OOLV DQG 1 5RVFK ,QRUJ &KLP $FWD WR EH SXEOLVKHG -' +HDG DQG 0& =HUQHU &KHP 3K\V /HWWHUV f ': &ODFN DQG .' :DUUHQ 2UJDQRPHW &KHP F f /L /H0LQ 5HQ -LQJ4LQJ ;X *XDQJ;LDQ DQG :RQJ ;LX=KHQ ,QWHUQ 4XDQWXP &KHP f 5HQ -LQJ4LQJ DQG ;X *XDQJ;LDQ ,QWHU 4XDQWXP &KHP f

PAGE 84

-$ 3RSOH '/ %HYHULGJH DQG 3$ 'RERVK &KHP 3K\V f -( 5LGOH\ DQG 0& =HUQHU 7KHRU &KHP $FWD f $' %DFRQ DQG 0& =HUQHU 7KHRU &KHP $FWD f 0& =HUQHU *+ /RHZ 5) .LUFKQHU DQG 87 0XHOOHU8HVWHUKRII $PHU &KHP 6RF f : $QGHUVRQ :' (GZDUGV DQG 0& =HUQHU ,QRUJ &KHP f 6HH LH -& 6ODWHU 4XDQWXP 7KHRU\ RI $WRPLF 6WUXFWXUH 9RO DQG 9RO 1HZ
PAGE 85

-3 'HVFODX[ &RPS 3K\V &RPPXQ f .DUOVVRQ DQG 0& =HUQHU ,QWHUQ 4XDQWXP &KHPLVWU\ f 02)DQQLQJ DQG 1)LW]SDWULFN ,QWHUQ 4XDQWXP &KHP f 0& =HUQHU LQ $SSUR[LPDWH 0HWKRGV LQ 4XDQWXP &KHPLVWU\ DQG 6ROLG 6WDWH 3K\VLFV HG ) +HUPDQ 1HZ 7LRUN 3OHQXP 3UHVV f D / %UHZHU 2SW 6RF $PHU f E / %UHZHU 2SW 6RF $PHU f :& 0DUWLQ 3K\V &KHP 5HI 'DWD f 6XJDW 2SW 6RF $PHU f &: %DXVFKOLFKHU DQG 36 %DJXV &KHP 3K\V f 5' %URZQ %+ -DPHV DQG 0) 2n'Z\HU 7KHRU &KHP $FWD f : 7K $0 9DQ GHU /XJW ,QWHUQ 4XDQWXP &KHP f -.DXIPDQ DQG 5 3UHGQHQ ,QWHUQ 4XDQWXP &KHP7 f 6FKXO] 5 ,IIHUW DQG -XJ ,QWHU 4XDQWXP &KHP f -& &XOEHUVRQ DQG 0& =HUQHU XQSXEOLVKHG UHVXOWV $OPORI 8QLYHUVLW\ RI 6WRFNKROP ,QVW RI 3K\VLFV 86,) 5HSRUWV f +' $UQEHUJHU : -DKQ DQG 10 (GHOVWHLQ 6SHFWURFKHP $FWD $ ,ELG LQ SUHVV +' $UQEHUJHU + 6FKXOW]H DQG 10 (GHOVWHLQ 6SHFWURFKHP $FWD $ f 1 (GHOVWHLQ LQ )XQGDPHQWDO DQG 7HFKQLFDO $VSHFWV RI 2UJDQRI (OHPHQW &KHPLVWU\ HG 70DUNV DQG ,/ )UDJDOD 1$72 $62 & 5HLGHO 'RUGUHFKW f 0& =HUQHU DQG 0 +HKHQEHUJHU &KHP 3K\V /HWWHUV f -& &XOEHUVRQ DQG 0& =HUQHU LQ SUHSDUDWLRQ .6 .UDVQRY *9 *LULFKHY 1, *LULFKHYD 90 3HWURY (= =DVRULQ 1, 3RSHQNR 6HYHQWK $XVWLQ 6\PS RQ *DVSKDVH 0ROHFXODU 6WUXFWXUH $XVWLQ 7H[DV S f 1, 3RSHQNR (= =DVRULQ 93 6SLULGRQRY DQG $$ ,YDQRY ,SRUJ &KLP $FWD / f (3) /HH $: 3RWWV DQG -( %ORRU 3URF 5 6RF /RQG $ f

PAGE 86

&: 'H.RFN 5' :HVOH\ DQG '' 5DGWNH +LJK 7HPS 6FL f ,5 %HDWWLH -6 2JGHQ DQG 56 :\DWW &KHP 6RF 'DOWRQ 7UDQV f 7$ %HLQHNH DQG 'HO *DXGLR ,QRUJDQLF &KHPLVWU\ 1R f .6 .UDVQRY 1, *LULFKHYD DQG *9 *LULFKHY =KXUQDO 6WUXNWXUQRL .KLPLL f

PAGE 87

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nV JXLGDQFH ,Q DGGLWLRQ WR WKH RUELWDO FKHPLVWU\ GHWDLOHG LQ WKLV WKHVLV D PDMRU SRUWLRQ RI KLV WLPH DW WKH 8QLYHUVLW\ RI )ORULGD ZDV VSHQW H[SORULQJ WKH XVH RI HOHFWURVWDWLF SRWHQWLDOV (3Vf DQG H[DPLQLQJ ELRFKHPLFDO SUREOHPV XVLQJ (3V :KLOH DW WKH 8QLYHUVLW\ RI )ORULGD KH ZDV JLYHQ WKH FKDQFH WR JR WR *HUPDQ\

PAGE 88

, FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQHWDWLRQ DQG LV IXOO\ DGHTDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 0LFKDHO & =HUQHU &KDLUPDQ 3URIHVVRU RI &KHPLVWU\ FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQHWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ L 1
PAGE 89

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

PAGE 90

81,9(56,7< 2) )/25,'$


LOCALIZED ORBITALS
IN CHEMISTRY
BY
JOHN CHRISTOPHER CULBERSON
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA TN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1987

ACKNOWLEDGEMENT
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.
11

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
to.
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
else.
iii

TABLE OF CONTENTS
Page
ACKNOWLEDGMENT ii
LIST OF TABLES v
LIST OF FIGURES vi
ABSTRACT vii
INTRODUCTION... 1
CHAPTER ONE LOCALIZED ORBITALS 3
Background 3
Double Projector Localization 6
Fermi Localization 9
Boys Localization 34
CHAPTER TWO LANTHANIDE CHEMISTRY 41
Background 41
Model 43
Procedures 61
Results 62
CONCLUSION 71
BIBLIOGRAPHY 73
BIOGRAPHICAL SKETCH 79
IV

LIST OF TABLES
Page
1-1 Probe electron points for furanone 18
1-2 Boys and Fermi hole centroids for C^H^02 19
1-3 Probe electron points for methlyactetylene 22
1-4 Boys and Fermi hole centroids for CHCH^ 23
1-5 Orbital centroids for BF^ 26
1-6 Eigenvalues and derivatives for BF-. using the
Boys method 27
1-7 Probe electron points for BF^ 28
1-8 Orbital centroids for BF^ 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 SmCl9,
EuC12 and YbCl2 .... 67
2-8 Geometry of CeiNO^)^ ion 68
_2
2-9 Population analysis of CeiNO^)^ ion 70
v

LIST OF FIGURES
Page
1-1 Fermi mobility function for t^CO 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)^ bonding orbital 38
1-6 Ni(CO)^ non-bonding orbital 39
1-7 Ni(CO)^ anti-bonding orbital 40
2-1 Single and double Í, basis set plot 49
2-2 Average value of r versus atomic number 50
2-3 Pluto plot of Ce(NO^)^ 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
LOCALIZED ORBITALS
IN CHEMISTRY
by
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 /-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)^.
Vll

The Intermediate Neglect of Differential Overlap (INDO) method is
extended to the /-orbitals, and the intrinsic localization of the /-
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
_2
CeiNO^)^ ion are presented. The model is also used to calculate the
ionization potentials for the above compounds. Due to the localized
nature of /-orbitals the crystal field splittings 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 (Cl) calculation must be done to insure that the converged
state is indeed the lowest energy state. The localized nature of the /-
orbitals in conjunction with the double projector localization method
may be used to isolate the /-orbitals in order to calculate only a Cl
restricted within the /-manifold.
viii

INTRODUCTION
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 /-orbitals should also fall into this
category.
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
1

2
such methods on several systems of chemical interest. Chapter two deals
with expanding the INDO method so that the prediction of /-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 /-orbitals, causes the /-/ 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 /-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 /-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 /-orbitals.

CHAPTER ONE
LOCALIZED ORBITALS
Background
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.^ 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
3

4
localized orbitals is their ability to simplify configuration
interaction (Cl) 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
n
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
2-3
is the two-electron repulsion integrals; 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
4—6
orbital self-extension operator,
2
gii = r12
(1-2)

5
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
5 , 3
to N . Since the SCF procedure itself proceeds as N (semi-empirical)
4
or N (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 second category of
localization.
We claim that the underlying physical basis of localization is the
Fermi hole. The Fermi hole provides a dire?t(non-iterative) method

6
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
12
a search of the Fermi hole mobility function. 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 /-orbital degeneracies are a
problem. The DP method allows one to separate the /-orbitals from the
other metal orbitals and use a small Cl 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
•k
like to study n-rc 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 Cl may then be used to study
k k
only the n-Jt transitions and thereby elucidate the n-rt spectra.

7
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 Cl is
then required to restore the localized nature of the d type molecular
orbitals. Such a large Cl can be avoided by first doing a DP
localization.
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 it orbitals to obtain t orbitals,
this is not desirable since the a and it spectra will now be mixed and
more difficult to interpret. The it orbitals may be removed using the DP
"k
method, the remaining orbitals localized, and the n-it spectra
calculated using a small singles Cl. 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 anc* a set r
localized "pattern" orbitals {T^}^, where r is less than or equal to m.
These "pattern" orbitals are projected out of the set by
m
| TO = Z I $.><$. 1T > (1-3)
1 a . i i i1 a
1 = 1
J*
for a = 1 to r. These {|TO}^, are then symmetrically orthogonalized

8
T'+Y' = A , (1-4)
-1/2
0 = Y' A ¿ . (1-5)
and are projected out of the original set
*' = * ( 1 - I |9 ><0 |) . (1-6)
a 1 a a1
The matrix A' is formed and diagonalized
U = U+A U = X . (1-7)
The X matrix will have r near zero eigenvalues corresponding to the
{0^)i that have been projected out. These eigenvalues and the
corresponding columns of U are removed. The new set of orthonormalized
orbitals fY J1!1 r is formed from
a 1
-1/7
Y = *'UX . (1-8)
This set is an orthogonal complement to the set |0 >, but has no
a
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,
V+Y+FYV = er , (1-9)
Y' = YV . (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.

9
Fermi Localization
Background
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 hole^ ^ and the Fermi orbital.^^ Unlike localization
methods based on iterative optimization of some criterion of
localization,^ 6,15,16 tjlg 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
17-19
into localized orbitals, 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
20-25
localized orbitals such as the PCLIO method 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(r1;r2) = p(rx) - 2 p2( ^, r2>/p( r2), (1-11)
where p(r^) is the diagonal portion of the first order reduced density
matrix and p(r^;r2) is the corresponding part of the second order
26
reduced density matrix. For special case of a closed shell SCF

10
wavefunction, the natural representation of the Fermi hole is the
13 14
absolute square of the Fermi orbital ’
A(r1;r2) = |f(r:;r2)|2. (1-12)
The Fermi orbital is given by
f(ri;r2) = [2/p(r2)]1/2 Z g.(r1)g.(r2),
i
d-13)
where the orbitals g^(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(r^;r2) is interpreted as a function
of r^ which is parametrically dependent upon the position of a probe
electron located at r2-
12 13 27 28
Previous work ’ ’ ’ 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
12 27 28
electron is measured by the Fermi hole mobility function, ’ ’
F(r) = Fx(r) + F (r) + Fz(r) (1-14)
where
Fv(r) = ^2
p
I u
3vJ
&i
3v
i>j
d-15)
for v = x, y or z. This may be compared to

11
F0(p) = (3n/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)-Fg(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
29-33
Daudel. Regions where F(r) = F(p) resemble boundaries between
loges.
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.^^ 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
, 4-6,15
localization.
Localized Orbitals Based on the Fermi hole
Equation 1-13 provides a direct relationship between a set of
canonical SCF orbitals g^(r) and a localized orbital fj(r) = f(r,r^)
where r^ is a point in a region where F(rj) < Fg(p(r^)). 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 n j = 1 to N, each of
which is located in a region where F(r_.) < Fg(p(r^)). Ideally, each of
these points should correspond to a minimum of F(r) or F(r)-Fg(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.

12
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. 4l. 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.

13
Figure 1-2: 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.

14
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.

15
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.

16
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
33
symmetric orthogonalization. The resulting unitary transformation is
given by
U = (TT+)~1/2T, (1-17)
where
T.. = gj(ri)/(p(r.)/2)1/2. (1-18)
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
34
problems. 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.

17
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, C^H^C^, and its derivatives are useful
35-37
reagents in 2+2 photochemical cycloadditions. The canonical SCF
molecular orbital for the furanone molecule were calculated with an STO-
38
3G basis set and the geometry specified in Table 1-1. The molecular
geometry was restricted to Csymmetry, 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=0 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

18
Table 1-1 : Molecular geometry and probe electron points for the
furanone (C^H^C^)- 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
0.0
0.0
0.0
Atom C£
0.0
2.589
0.0
Atom
2.671
3.355
0.0
Atom
4.363
0.940
0.0
Atom 0^
2.534
-1.071
0.0
Atom C>2
3.427
5.548
0.0
Atom
-1.802
-1.040
0.0
Atom H2
-1.802
3.629
0.0
Atom
5.560
0.858
1.698
Atom H.
4
5.560
0.858
-1.698
Cf-Cz bond 1
0.0
1.295
2.000
^1~^2 k°ncl 2
0.0
1.295
-2.000
^onc'
1.336
2.972
0.0
C0-C. bond
3 4
3.517
2.148
0.0
C^-0^ bond
3.449
-0.066
0.0
Cj-0^ bond
1.267
-0.536
0.0
C^-02 bond 1
3.049
4.452
2.000
C3-O2 bond 2
3.049
4.452
-2.000
0^ lone pair 1
2.654
-1.821
0.660
lone pair 2
2.654
-1.821
-0.660
C>2 lone pair 1
2.707
6.298
0.0
C>2 lone pair 2
4.267
5.698
0.0

19
Table 1-2 : Orbital centroids for localized orbitals determined by the
Fermi hole method and by the orbital centroid method for the
furanone molecule (C^H^O^. All coordinates are given in
bohr.
Orbital
Fermi hole method
Centroid criterion
X
Y
Z
X
Y
Z
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.O
2.671
3.355
0.0
C4 K shell
4.362
0.940
0.0
4.326
0.939
0.0
0X 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
C^-H^ 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
C,-H0 bond
4 3
5.149
0.895
1.132
5.155
0.888
1.153
C.-H. bond
4 4
5.149
0.895
-1.132
5.155
0.888
-1.153
crc2 bond 1
0.008
1.401
0.635
0.030
1.410
0.599
CrC2 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
C^-C^ bond
3.542
2.190
0.0
3.565
2.149
0.0
C--0, bond
3.268
-0.243
0.0
3.243
-0.225
0.0
C.-0. 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-O2 bond 2
3.093
4.584
-0.548
3.114
4.612
-0.511
lone pair 1
2.554
-1.283
0.440
2.594
-1.311
0.472
0^ 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
3.970
5.589
0.0
3.970
5.600
0.0

20
determined by the orbital centroid criterion.^ ^>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 T^g of less than 10-^, where TRM<, is the RMS
value of the off-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

21
orbital centroid criterion, over 200 iterations were required to
determine a set of orbitals which satisfied a very weak criterion of
34
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
38 39
determined by an ST0-5G basis set and an experimental geometry.
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 single bond. The remaining three points were located two
bohr from the 1’^ rotation axis at a point midway between the and
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

22
Table 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
bohr.
Position
X
Y
Z
Atom
0.0
-1.140
0.0
Atom C2
0.0
1.140
0.0
Atom C^
0.0
3.897
0.0
Atom
0.0
-3.145
0.0
Atom
-1.961
4.438
0.0
Atom
0.980
4.438
1.698
Atom
0.980
4.438
-1.698
C^-CL bond 1
-2.0
0.0
0.0
^1^2 bon^ ^
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

23
Table 1-4 : Orbital centroids for localized orbitals determined by the
Fermi hole method and by the orbital centroid method for
the methylacetylene molecule. All coordinates are given
in bohr.
Orbital
Fermi hole method
Centroid criterion
X
Y
Z
X
Y
Z
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
C^-H^ bond
0.0
-2.486
0.0
0.0
-2.502
0.0
C3-H2 bond
-1.302
4.270
0.0
-1.316
4.272
0.0
C^-H^ bond
0.651
4.270
1.127
0.658
4.272
1.139
C--H, bond
3 4
0.651
4.270
-1.127
0.658
4.272
-1.139
^l-^2 b°nd
-0.713
-0.003
0.0
-0.692
-0.U13
0.0
¿1-C2 bond 2
0.356
-0.003
0.617
0.346
-0.013
0.599
f'l-^2 *30nc^ ^
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

24
orbital centroid method required 26 minutes starting from the canonical
SCF molecular orbitals or 19 minutes using the Fermi localized orbitals
—8
as an initial guess to reach T^M<, of 10 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 axis. The
point group of the orbital centroids for this conformation is C~,.
oh
The "three-up" conformation is generated by rotating the set of lone
pair orbitals on each fluorine atom in the pinwheel conformation by 90

25
degrees about each F-B axis. The point group for the orbital centroids
of this conformation is C^v- The "up-up-dovn" 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 BF^ were calculated based on the
double-zeta basis set and geometry tabulated by Snyder and Basch.^
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
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

26
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
bohr.
Orbital
up-up-up (C3v)
up-up-down (Cg)
X
Y
Z
X
Y
Z
B-F^ 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-F^ bond
-0.862
-1.494
-0.072
-0.862
-1.494
0.069
F^ 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
F^ 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

27
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 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.
Configurations
up-up-up
up-up down
pinwheel
Fermi hole
Sum
69.445360
69.444985
69.438724
69.341094
X1
-0.019521
-0.018174
+0.017234
+0.031444
X2
-0.019850
-0.018914
+0.016238
+0.029932
*3
-0.019850
-0.019095
+0.016236
+0.029928
X4
-0.155480
-0.153827
-0.193521
-0.195502
X5
-0.155484
-0.155181
-0.ln4366
-0.195524

28
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 C^)- All
coordinates are given in bohr.
Position
X
Y
Z
B-F^ bond
1.223
0.0
0.0
B-F9 bond
-0.611
1.059
0.0
B-F^ bond
-0.611
-1.059
0.0
F^ 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
F£ 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
F^ 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

29
Table 1-8 : Orbital centroids for localized orbitals determined by the
Fermi hole method and by the orbital centroid method for
the boron Lrifluoiide molecule. The four innershell
orbitals have been excluded from these calculations. All
coordinates are given in bohr.
Orbital
Fermi hole
Centroid criterion
X
Y
Z
X
Y
Z
B-F^ bond
1.713
0.0
0.0
1.720
0.028
0.0
B-F^ bond
-0.857
1.484
0.0
-0.884
1.475
0.0
B-F^ bond
-0.857
1.484
0.0
-0.835
1.504
0.0
F^ lone 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
y2 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
F^ 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

30
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
criterion.
We do not intend to attribute any special physical significance to
any of the lone pair configurations for BF^. 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.
Conclusions
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.^^

31
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
i
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
localized.
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

32
Al A2
such as molecular mechanics ’ 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, spectroscopic constants, ’ and other
46 47
properties. ’ 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
A3 A7
their utility in subsequent calculations. In order to apply the
12 28
current method to such molecules, the Fermi hole mobility function ’
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)-Fp(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 Fq(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 Fq(p).

33
Consequently, it is not possible to represent methylacetylene with a
structure like H-C=C=CH^ 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
problems.
Boys Localization
The most widely used form of localized orbitals are those orbitals
5 6
based on the method of Boys. ’ 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 computational step, but the Boys method may be

34
formulated in terms of products of molecular dipole integrals. The
dipole integrals are one-electron integrals therefore the transformation
. 3
is on the order of N , 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 unitary transformation relating a delocalized set of
orbitals to a localized set, but the form of this transformation is in
general unknown. The common method foi 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 f1 one can increase the localization by
doing a unitary transformation IT*",
fI + 1 = U1^ (1-19)
where f^+^ is the resulting set of more localized orbitals. The
original matrix formulation is based on a sequence of pairwise
2
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-l)/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
rotations.
Leonard and Luken have developed a second order method that does all
of the N(N-l)/2 rotations at once rather than one at a time.''"*’ The use
of a second ofder method may have the additional benefit of improving
the convergence difficulties encountered in iterative methods. Their
method is outlined below.

35
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+)~1/2. (1-21)
The matrix R will be defined as
R = NT, (1-22)
where
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 {f^, . . . , f^) one produces a
set of more localized orbitals {f^, . . . , f^). The new value of the
localization, G', is given by
N
G' = Z (i'i'.i'i') (1-25)
i'=l
The new f' orbitals can be thought of in terms of a pertubative
expansion,
f' = f + E t.f + £ tTt.f (1-26)
I I,J
to second order. The t^ matrix does 2x2 rotations that mix in portions
of all the occupied orbitals into orbital f'. The third term t^-tj is
the product of a pair of 2x2 rotations. When this form of the f'

36
orbitals is substituted into the G' equation, you obtain
G' = Gq + I tjGjd) + Z tItJG2(I,J) . (1-27)
Leonard and Luken^ include the G^ second order term to accelerate
convergence when one is in the quadratic region; only the first order
term 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.
Results
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 fot the Ni(CO)^ ion. In a recent experimental paper by Reutt
et al., the photoelectron spectroscopy of Ni(C0)^is 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 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 t (banana) bonds, (3) nickel carbon bonds, and

37
(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 Cl
calculation to see if one could predict the photoelectron spectra for
Ni(C0)4.
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.

38
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.

39
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.

40
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.

CHAPTER TWO
LANTHANIDE CHEMISTRY
Background
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
/-element chemistry, but much modern work is concerned with organo-/-
49
element reactions, and the use of lanthanides and actinides as very
specific catalysts.Unlike the corresponding chemistry involving
the d metals, very little explanation is offered for much of this
chemistry.
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
/-orbitals as valence orbitals is known, although experience is now
being gained on the use of / orbitals as polarization orbitals.
Finally, the /-orbital elements are sufficiently heavy that relativistic
effects become important. Very few ab initio molecular orbital studies
52 "
have been reported on /-orbital systems. Extended Huckel calculation,
53
however, have been successful in explaining some of this chemistry.
Scattered wave and DVM Xa studies of /-orbital systems have also proven
effective, especially in examining the photoelectron spectroscopy of
i „ 54-56
reasonably complex systems.
We examine an Intermediate Neglect of Differential Overlaps (INDO)
technique for use in calculating properties of /-orbital complexes. At
the Self-Consistent Field (SCF) level this technique executes as rapidly
41

42
»!
on a computer as does the Extended Huckel method, and considerably more
rapid than the scattered wave Xa method. Since the electrostatics of
the INDO method are realistically represented, molecular geometries can
be obtained using gradient methods.^ 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 thr^e systems. With
configuration interaction (Cl) this model should also be useful in
examining the UV-visible spectra of /-orbital complexes. Preliminary
studies of /-orbital chemistry using an INDO model have been disclosed
58
by Clack and Warren and, more recently, by Li-Min, Jing-Quing, Guang-
. 59
Xian and Xiu Zhen. 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 efiects 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 /-orbitals aie generally tight, and ligand field splittings thus
small, a great many states differing only in their /-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?

43
Model
The INDO model Hamiltonian that we use was first disclosed by Pople
and collaborators,*^ and then adjusted for spectroscopy*^ and extended
6 2 6 A ,
to the transition metal series. - The details of this model are
6 2 6
published elsewhere. ” 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
k. k
"core" integrals and the Slater-Condon F and G integrals that are used
for the formation of one-center two-electron integrals. The evaluation
of these integrals using experimental information has traditionally made
' 65 66
this model highly successful in predicting optical properties. ’
We employ 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(/). These parameters will be chosen to give satisfactory
geometries of model systems. Another choice is one that gives good
prediction:' of UV-visible spectroscopy.^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 approximation*^ F°(n) =
IP(n) - EA(n), (IP = Ionization Potential, EA = Electron Affinity) and
the two-electron two-center integral from one of the more successful
functions established for this purpose.

44
At the SCF level, we seek solutions to the pseudo-eigenvalue problem
F C = C e
(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 unrestJicted Hartree Fock case is discussed in detail elsewhere,*^
71-73
as is the open shell restricted case. Although nearly all /-
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:
,AA
= ir
AA
'MU
• E P
[ c A
crX
(UU | crX) - j (ya|yX)
(2-2a)
FAA
MV
+ £ PCTCT (uu|ffff) - E ZB (UÜ|sBsB)
a c B
BM
E Nx
a, X
(U v|aX) - j (ya|vX)
1
2
BA(u) + bb(v)
S - T P (MU Ivo)
UM 2 UV v 1 '
y*v
A¿B
(2-2b)
(2-2c)
where
(yv|uX) = Jdx(l)dx(2) yi) Xv(l) r‘J 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

45
P
yv
MO
£ c C n
ya \>a a
a
(2-4)
AB
with n the occupation of MO $ , n = 0,1,2. In Eq. (2-2), F refers
3. 3. 3. MV
to a matrix element with AO centered on atom "A". The core
integral
C ■ (x£l - \ ^ - T + ',Alxi) (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 U will
remove the necessity for explicit consideration of this term. The bar
over an orbital in an integral,
_A_A
such as (y y | 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;^ i.e. (yAvA|, X^X^* The last term in Eq. (2-2a)
represents the attraction between an electron in distribution X^* X^ and
all nuclei but A. The rationale for replacing integral
A A
(yA|RB1|uA) (¡] y |SBSB) (2-6)
is given elsewhere, and compensates for neglected two center inner
27 28
shell-valence shell repulsion ’ and neglected valence orbital
28 29
(symmetrical) orthogonalization. ’ Zg is the core charge of atom B

46
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.
13-15
S of Eq. (2-2c) is related to the overlap matrix D, and is
given by
Syv 2 ^U(l)v(l) gy(l)\j(l) 0*(l)|v(l)) (2-7)
1=0
where £y (]_) ) is the Eulerian transformation factor required to rotate
from the local diatomic system to the molecular system, (y(l) |\>(1)) are
the sigma (1=0), pi(l=l), delta(l=2) or phi (1=3) components to the
overlap in the local system, and 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,
61-63 68
and set all / = 1 except between p symmetry orbitals viz. ” ’
S pp = 1 ‘ 267®papcr (pcTlpCT) + °-585gpitpJt(prtlpTt) (2'8)
+ n-585gpitpn (p" lpíl >
Basis Set
In general ZDO methods choose a basis set of Slater Type Orbitals
(STO)
1/2
rn_1e"Cr y!J (9, ) (2-9a)
where Y^(0,4>) are the real, normalized spherical harmonics. Atomic
orbitals are expressed as fixed contractions of these
nlm
Lm
2n+l
2n!

47
*U * S 3nlm Rnlm <2-9t»
In general a single R , function describes the s and p orbitals for
n J m
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
wo have examined basis sets suggested by Li Le-Min et^ al. , by Bender
and Davidson,^ and by Clementi and Roetti.^ 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 INDO. 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
81
program of Desclaux. From these wavefunctions jndial expectation
2 3
values , and are calculated for 6s, 6p, 5d and 4/
functions. The 6s, 5d and 4/ wavefunctions were obtained by Dirac-Fock
calculations on the promoted, 4/11 ^5d^6s^ configuration; the 6p from
3 2 1
calculations in which a 5d electron was promoted, 4jI-'>6s op .
Wavefunctions for the mono-positive ions are obtained from 4,fm~^3d'''6s'*'
and 4/m ^6.‘^6p^ respectively. A generalized Newton procedure was then
used to determin exponents (C) and coefficients a ^ for a given set of
2 3
, and with functions of the form of Eq. (2-9b). Again, as
in the transition metal atoms, we found that a single C function fits
the ns and np atomic functions well in the regions where bonding is

48
important, but the (n-l)d, and now the (n-2)/ 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 4/ and
5d, now with increased shielding, results. After some experimentation
we use the Dirac-Fock values obtained from the mono-positive ions. The
basis set adopted is given in Table 2-1.
The 4/ and 5d functions are quite compact. At typical bonding
distance (4/4/|yy) and (5d5d |yy) are essentially R~^. Because of this
we calculate all two-cenrer two-electron integrals with the values in
Table 2-1. This value is chosen to match the accurate F° Slater-Condon
Factors obtained from the numerical atomic calculations by a single
exponent, via
F°(4/4/) = 0.200905 C (4/) (2-10a)
F°(5d5d) = 0.164761 C (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 IX, and
this procedure is much simpler.
Core Integrals
The average energy of a configuration of an atom or ion is given.

49
Single vs. Double Zeta 4Í-ST0 Orbital Amplitude
0.20
0.17
0.13
So.io
K
0.07
0.03
0.00
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
r in a.u.
Figure 2-1: Radial vavefunction for the 4f orbital of Ce with single-C
and double-C Slater type orbitals (STOs).

50
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).

Table 2-1 :Slater type orbital (STO) basis functions for the lanthanide atoms.
The single C functions are listed for the 4/, 5d, 6s and 6p orbitals
along with the double C functions for the 4f and 5d orbitals.
At oro
Single C
Double C
Exponents
Exponents
Coe fficient s
if
5d
6 s
6p
if
5d
if
5d
Ce
4.439
2.061
1.548
1.361
6.118
2.522
3.077
1.581
.7159
.4575
. 5003
. 6334
Pr
4.657
2.102
1.571
1.380
6.393
2.646
3.126
1.598
.7175
. 4 542
.5104
.6257
Nd
4.681
2.138
1.593
1.398
6.648
2.75"’
3.169
1.612
. 7196
.4510
. 5202
.6182
Pm
5.053
2.171
1.614
1.415
6.119
2.858
3.208
1.623
. 7220
. 4479
. 5295
.6112
S m
5.236
2.201
1.635
1.432
7.120
2.951
3.244
1.633
. 7247
.4450
.5238
.6046
Eu
5.414
2.229
1.656
1.448
7.342
3.038
3.278
1.640
. 7275
. 4421
. 5466
. 5986
Gd
5.565
2.243
1.678
1.463
7.523
3.091
3.288
1.634
.7332
. 4375
. 5562
. 5916
Tb
5.717
2.254
1.700
1.478
7.707
3.148
3.298
1.628
.7379
. 4337
. 5648
. 5857
Dy
5.869
2.262
1.723
1.492
7.892
3.206
3.307
1.621
. 7421
.4304
. 5724
.5808
Ho
6.019
2.269
1.745
1 . 507
8.076
3.264
3.316
1.613
. 7461
.4273
. 5791
. 5768
El-
6.168
2.274
1.767
1.521
8.258
3.320
3.324
1.605
. 7498
.4243
. 5852
. 5735
Tro
6.316
2.277
1.789
1.535
8.438
3.375
3.332
1.596
. 7534
.4214
. 5905
. 5709
Yb
6.462
2.279
1.812
1.549
8.616
3.426
3.338
1.587
. 7570
.4186
. 5952
. 5690
Lu
6.607
2.278
1.834
1.562
8.791
3.480
3.343
1.577
. 7605
.4159
. 5994
.5676

52
skpmdn/> = k U + mU + nU, + qU.. + k^k ^ W +
r 00 pp dd 1 ff 2 ss
ss
(2-11)
niâ„¢-1) v + n^~1) v + q^q-1) u +
rt vt -r n n wx-c_r kmW + knV +
2 pp 2 dd 2 // sp sd
kqVs/ + mnWpd + mqVpf + nqWd/
with j, the average two electron energy of a pair of electrons in
orbitals and Xj given by
Vss = F°(ss)
Wpp = F°(pp) - 2/25F2(pp)
(2-12)
W
dd
= F°(dd) - 2/63F2(dd) - 2/63 F4(dd)
Vff = F°(//) - 4/195F2(//) - 2/143 F4(//) - 100/5577 F6(//)
Wsp = F°(sp) - l/6G1(sp)
Wsd = F°(sd) - 1/10 G2(sd)
Wsf = F°(sf) - 1/14 G3(s/)
V
pd
= F°(pd) - 1/15 GX(pd) - 3/70 G3(pd)
Vpf = F°(pf) - 3/70 G2(p/) - 2/63 G4(p/)
Vdf = F°(d^.) - 3/70 G3(d/) - 2/105 G3(d;f) - 5/231 G5(d/)
The core integrals H_, Eq. (2-5), are then evaluated by removing an
electron from orbital X^, 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).22’^2’®4
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 /n_3d
2 -ji 2 2
s , while the remaining lanthanide atoms have the structure j s .
Two processes are then possible for 6s electron ionization:
I /n-3d1s2 -» /n_3d1s1 + (s)
II s2 -» /n_2s1 + (s)

53
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 Us<, 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
85 86
Brewer ’ and then smoothed by a quadratic fit throughout the series.
For completeness, we also give the values of process II.
J1-3 1 2
The lowest configuration containing a 5d electron is j JdJs
throughout the series, and 5d ionizations are obtained from
III Z1-3 dV -» fh2 + (d)
The ionization potentials for the 6p can be obtained from two
processes:
IV f~2 s1 p^ -> /""‘s1 + (p)
V Z1"3 s2p -* /1_3s2 + (p)
Ionization from process IV is nearly constant at 3.9 eV, from V at
-2 2
4.6 eV. The f s configuration is lower for all the lanthanides
3 2
except Ce(fdsp), and Gd and Tb(j ^- s p). Using the 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 Z1-3 ds2 Z^W + (/)
VII Z1-2 s2 Z1_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)
Uff (VI) = IP(VI) - (m-4) Vff - 2Wff - (2- 13a)
Ufj: (VII) = IP(VII) - (m-3) - 2Us/
(2-13b)

Table 2-2 ¡Smoothed Ionization potentials for processes I-VII and configuration
mixing coefficients derived from Brewer's tables. The entries in the
column labeled 1 are the mixing coefficients for the configuration
fn ^ds^, in column 2 the /n ^s^ configuration mixing coefficients are
listed.
Atom
Smoothed Ionization Potentials
Mixing
Coefficients
Process
s
P
d
f
1
2
I
II
IV
V
III
VI
VII
Ce
5.93
5.36
3.69
4.60
6.74
12.17
7.24
.7558
. 244 2
Pi-
5.89
5.42
3.76
4.62
6.77
12.79
7.27
.2764
.7236
Nd
5.93
5.48
3.82
4.67
6.77
13.35
7.29
. 1772
. 8228
Pm
5.98
5.55
3.87
4.65
6.74
13.85
7.31
.1465
. 8535
Sm
6.01
5.63
3.91
4 . 70
6.73
14.27
7.35
.0557
. 9423
Eu
6.10
5.69
3.95
4 . 74
6.68
14.62
7 . 39
. 02 36
. 9764
Gd
6.17
5.77
3.99
4.76
6.61
14.90
7.4a
.9037
. 0936
Tb
6.25
5.85
4.03
4 . 80
6.54
15.13
7.49
. ad21
. 5179
Dy
6.34
5.93
4.05
4.84
6.42
15.27
7.55
.1564
.8436
HO
6.42
6.01
4.08
4 . 89
6 .30
15.35
7.61
.1533
. 8467
El*
6.52
6.09
4.10
4 . 92
6 .16
15.36
7.69
.1661
. 8339
Tm
6.62
4.17
4.12
4.97
4.10
15.30
7.76
.0731
. 9269
Yb
6.73
6.26
4.13
5.02
5.96
15.18
7.84
.0273
.9727
Lu
6.77
—
—
—
5.31
16.11
1.000

55
Unlike the analogous situation for the 6s and 6p orbitals, use of
Eq. (2-13a) to find U^, and use of this value in Eq. (2-13b) to predict
IP(VII) is not successful, and would require the scaling of the large
F°(ff) integral often performed in methods parameterized on molecular
„ 61,63,67
spectroscopy. ’
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 U^. We create a two-by-
two interaction matrix
E(3ds2) - A V
„ rw J!n-2_2.
\
(r \
L1
S
CNJ
= 0
(2-14a)
2 2
where V is an empirical mixing parameter, and and C2 determines the
relative amounts of each of the two configurations that are important.
2
The exact value of V would depend on a given molecular situation.
is then given by
9 x 2
(2-14b)
‘1.x2
V _ c /c - E a - 2V
1-2 2,
(2-14c)
E(/° 3d s2) - E(f° 2s2)
n2
2V
+ 1
The values of appear in Table 2-2, where we have used the values of
_j_3 ^ ^ 2
E(j ds ) and E(j s ) obtained for the promotion energies of
85
Brewer and a fixed value of V = 0.02 au. Then U^ could be obtained
from

56
Vff = Cx2 IJ^(VI) + C22 U^(VTI) (2-15)
n_2 2
In the case of the 3d orbitals this valence bond mixing between 3dn-is
and 3dn ^s was important in obtaining reasonable geometric
63
predictions, an observation now confirmed in careful ab initio
87
studies. For the lanthanide complexes of this study the 4/ 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 /-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 4/ and 5d atomic orbitals. However,
the values of the promotion energies we obtain from Brewer are so
different than those that we obtain from oul own numerical calculations
on the average energy of a configuration, Table 2-3, that for the moment
we choose a 76Z : 24% mix of Ei/111 ^ds2) : EC/"1 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(/), 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

57
Table 2-3 : Average configuration energy from Dirac-Fock
on the /^^ds^ and the f' configurations
lanthanide atoms.
calcula tions3
for all the
Atom
Average
Configuration
Energy
f-w
/-v
Ce
-8853.71494569
-8853.64980000
Pr
-9230.41690970
-9.130.37981848
Nd
-9616.94751056
-9616.93446923
Pm
-10013.4526061
-10013.4606378
Sm
-10420.0710475
-10420.0970615
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.

58
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

59
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. ’
It has been argued that the 4/ orbitals are not used in the chemical
55 56
bonding of those complexes except in the more covalent cases. ’ 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 MF^, MCl^, MBr^ and MI^ and for the comparative values obtained
for CeF^ and CeF^. This is indicated in Table 2-4 by the large values
of |B(f)|. The latter values are a consequence of the fact that the f-
orbitals are tighter than one usually expects for orbitals important in
chemica] bonding. Use of 5d orbitals alone will predict the trends in
these two series, but underestimates the range of values experimentally
observed.
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
(yy |v\>) or (yv|vy)
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
88
this type might be rotationally averaged, but from a study of spectra
63
it appears that all one-center integrals should be evaluated. For
example, in the metallocenes the integral (d 2 2 d Id d ) is
x -y xy1 xy yz
required to separate the two transitions that arise from the e^(d) ->
e„ (d) transitions that lead to the '*'E1 and excited states. In
4g lg 2g

60
addition, it appears that the inclusion of all one-center integrals
improves the predictions of angles about atoms with s,p,d basis
89 90
sets ’ 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 format only the additional integrals
need be included. These integrals are generated in explicit form via a
computer program that we have used in the past^ and they have also been
recently published by Schulz et al. To our knowledge all these
integrals do not appear in the literature for s,p,d and / basis,
although we have checked those of (uu|na>) and (y\>|\iyl against the
83
formulas of Fanning and Fitzpatrick.
Integrals of the form (yu|\>v) and (yv|vy) can be obtained thi uugh
• 1c
atomic spectroscopy, and their components, F and G , evaluated via
least square fits
(yy | vv) = E ak pk
k
(yv|vy) = E bk ck
k
lc k.
These F and G 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 |d d )). Integrals of the "R" type,
x — y yz xz xy
however, cannot be evaluated in this manner; viz.(sd|dd), (sp|pd),
(sd|pp), (sd|//), (s/|d/), (pp|p/), (dd|p/), (pd|d/), (sd|p/), (pd|s/),
(sp|d/), and (p/|//). 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 F° value obtained from the Fock-Dirac numerical
k k. k
calculations. All F , G and R integral for k > 0 are then scaled by

61
2/3. This value of the scaling is obtained from a comparison of the
calculated and empirically obtained^®’^2-94 F^(//) ancj
J/
values that implies 0.66 + 0.04. Empirically obtained values of G (/d)
and F (/d) are far more uncertain and are much smaller, and are thus not
used to obtain this scaling value between calculated and experimental
values.
At this point it seems appropriate to point out the differences of the
59
present INDO model to that suggested by Li Le-Min et_ al. In the
latter formalism only the conventional one-center two-electron integrals
are included leading to rotational variance. In addition, the
Volfsberg-Helmholz approach is used for the resonance integral B,
B^ = (IP(i)+IP(j ))S^j/2. No geometry optmization has been reported
59
within their model. Further differences are the restriction to
single-C STOs and the smoothing of the valence orbital ionization
59
potentials for the lanthanides via Anno-type expressions.
Procedures
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,"^ using either the
restricted or unrestricted Hartree Fock formalism. All UHF calculations
62
are followed by simple annihilation.
Self-consistent field convergence is a problem with many of these
systems. For this reason electrons are assigned to molecular orbitals
that are principally / in nature according to the number of /-electrons
in the system, and the symmetry of the system. Orbitals with large

62
lanthanide 5d character are sought and assigned no electrons. A
procedure is then adopted that extrapolates a new density for a given
93
Fock matrix based on a Mulliken population analysis of each SCF cycle.
Often this procedure is not successful. In such cases all /
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 / orbital assignments as described above.
In cases of slow convergence, a singles or small singles and
doubles, Cl is performed to check the stability of the SCF, and the
appropriateness of the forced electron assignment to obtain the desired
„ . 96
state.
Results
The geometries of CeCl^ and LuCl^ 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 CeF^ is also listed
in Table 2-5. One can see the agreement with experiment is good in all
cases.
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 ^ a.u./bohr, the angles are converged
only to ±3°. We note, however, that all are predicted non-planar, in
54b 97 98
agreement with experiments. ’ ’

63
Table 2-5 : Geometry and ionization potentials for Cerium and
Lutetium trihalides. Cerium tetrafluoride is also
included in this table. The bond distances are given
in angstroms^ angles in degrees and IPs in eV.
Experimental^ results are also shown where available.
Molecule
Bond
Distance
Bond
Angle
Ionization
Potential
INDO
Exp.
INDO
Exp.
INDO
Exp.
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
Cel3
2.844
2.927
119.8
—
9.9
—
CeF.
4
2.099
2.040
109.5
109.5
—
—
LuF3
2.045
2.020
107.4
—
b
19.0
LuC13
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)
Lul3
2.726
2.771
115.6
114.5
17.7
(16.2 - 18.1)
a) References 54b, 97 and 98. Estimated values for CeF~,CeI~, and
LuF3 from Ref 103. ^ 4
b) The SCF calculation on the ion of LuF~ would not converge
therefore no IP is reported.

64
The experimental range of the bond lengths from LnF^ to Lnl^ 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 Á in going from CeF^ to CeF^ is also smaller than the
0.14 Á 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 using the ASCF method, and only the first IP is
54b 99
calculated. Experimentally ’ 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 calculations"3^*3
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,
CeX^, X = F, Cl, Br with a maximum value for the chloride, a feature
5 A b
noticeably missing in the DVM Xa results.
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
geometries'3^*3’^^ 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.

65
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.
Experimental111 results are also given where available.
Atom
Bond
Distance
Bond
Angle
Ionization
Potential
INDO
Exp.
INDO
Exp.
INDO
Exp.
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.

66
To test the applicability of our model to lanthanide atoms not
formally charged +3, we calculated the geometries and IPs for SmC^,
EUCI2 and YbC^ 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.
_2
Ve chose Ce(NO^)^ 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.
Table 2-9 shows a population study of this complex. Although there is
some /-orbital participation, it appears that this unusual twelve
coordinate T^ structure results from electrostatic forces between the
ligands and the relatively large size of the Ce(IV) ion.

67
Table 2-7 : Geometry and ionization potential for SmC^, EuC^, and
YbC^- Bond distances are given in angstroms, bond angles
in degrees, and ionization potentials in eV. Experimental
results are listed where available.
Molecule
Bond
Distance
Bond
Angle
Ionization
Potential
INDO
Exp.
INDO
Exp.
INDO
Exp.
S111CI2
2.584
—
143.3
130±15
5.3
—
EuCI,,
2.576
—
143.2
135+15
6.6
—
YbCl2
2.400
—
120.2
126±05
3.2
—
a) Reference 100.

68
_2
Table 2-8 : Average bond distances and bond angles for CeiNO^)^ ion
INDO optimized geometry and the X-ray crystal structure3.
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.
Geometric
Parameter
INDO
Exp.
r(Ce-0c)
2.554
2.508
r(N-0c)
1.256
1.282
r(N-0n)
1.237
1.235
9(0 -N-0 )
c c'
121.5
114.5
9(0 -Ce-0 )
c c
50.9
50.9
a) Reference 101.

CE(N03)6.
_2
Figure 2-3: Plot of the twelve coordinate Ce(NO^)6 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.

70
_2
Table 2-9 : Population analysis of CeiNO^)^ • The oxygen atoms that are
coordinated to the cerium are indicated by 0c>The Wyberg
bond index is also given. A Wyberg index of 1.00 is
characteristic of a single bond.
Atom
Orbital
Atomic
Population
Spin
Density
Total
Valence
Ce
s
0.20
0.00
—
P
0.30
0.00
—
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
c
Net
-0.40
0.00
1.58
0
Net
-0.48
0.00
1.80
Bond
Wyberg
Bond Index
Ce - 0
c
0.40
N - 0
c
1.37
N - 0
1.22

CONCLUSIONS
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-electron terms, and
including all of the two-elecfron 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, FSu 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 4/ orbitals, and
one need not invoke London forces to explain this observation. Again
the potential for bending is very flat.
Within this model, /-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 LnF^, LnCl^, LnBr^,
71

72
and Lul^, and CeF^ and CeF^ are reproduced without /-orbital
participation, the range of values calculated is considerably improved
when /-orbitals are allowed to parrici pate. For the twelve coordinate
-2 102
Ce(NO^)^ complex reported here, /-orbital participation appears
minor. A stable complex of near symmetry is obtained regardless of
the /-orbital interaction.

BIBLIOGRAPHY
1. V. Fock, Z. Physik 61, 126 (1930).
2. C. Edmiston and K. Ruedenberg, Rev. Mod. Phys. 35, 457 (1963).
3. C. Edmiston and K. Ruedenberg, J. Chem. Phys. 43, S97 (1965).
4. S.F. Boys, in Quantum Theory of Atoms, Molecules and the Solid
State, p.253, Lowdin P., Ed., New York: Academic 1966.
5. S.F. Boys, Rev. Mod. Phys. 32, 306 (1960).
6. J.M. Foster and S.F. Boys, Rev. Mod. Phys. !32, 300 (1960).
7. E.P. Vigner, F. Seitz, Phys. Rev. 43, 804 (1933); 46, 509 (1934).
8. J.C. Slater, Phys. Rev. 81, 385 (1951).
9. G. Sperber, Int. J. Quantum Chem. 5, 177, 189 (1971); 6, 881
(1972).
10. R.J. Boyd and C.A. Coulson, J. Phys. B7, 1805 (1974).
11. I.L. Cooper and N.M. Pounder, Int. J. Quantum Chem. 17, 759 (1980).
12. W.L. Luken and J.C. Culberson, Int. J. Quant. Chem. Symp. 16, 265
(1982). ~
13. W.L. Luken and D.N. Beratan, Theoret. Chim. Acta (Berl.) 61, 265
(1982). ~
14. W.L. Luken, Int. J. Quantum Chem. 22, 889 (1982).
15. J.M. Leonard and W.L. Luken, Theoret. Chim. Acta (Berl.) 62, 107
(1982). —
16. J.M. Leonard and W.L. Luken, Int. J. Quantum Chem. 25, 355 (1984).
17. V. Magnasco and A. Perico, J. Chem. Phys. 47^, 971 (1967).
18. W.S. Vervoerd, Chem. Phys. 44, 151 (1979).
19. T.A. Claxton, Chem. Phys. 52, 23 (1980).
20. S. Diner, J.P. Malrieu, P. Claverie, and F. Jordan, Chem. Phys.
Lett. 2, 319 (1968).
21. S. Diner, J.P. Malrieu, and P. Claverie, Theoret. Chim. Acta
(Berl.) 13, 1 (1969).
22. J.P. Malrieu, P. Claverie and S. Diner, Theoret. Chim. Acta (Berl.)
13, 18(1969).
73

74
23. S. Diner, F.P. Malrieu, F. Jordan and M. Gilbert, Theoret. Chim.
Acta (Berl.) 15, 100 (1969).
24. F. Jordan, M. Gilbert, J.P. Malrieu and U. Pincelli, Theoret. Chim
Acta (Berl.) 15, 211 (1969).
25. J.M. Cullen and M.C. Zerner, Int. J. Quantum Chem. 22, 497 (1982).
26. P.0. Lowdin, Phys. Rev. 97 > 1474 (1955) Adv. Chem. Phys. 2, 207
(1959).
27. W.L. Luken and D.N. Beratan, Electron Correlation and the Chemical
Bond. Durham, NC: Freewater Production, Duke University, 1980.
28. W.L. Luken and J.C. Culberson, in Local Density Approximations in
Quantum Chemistry and Solid State Physics, J.P. Dahl, J. Avery,
Eds., New York: Plenum 1984.
29. R. Daudel, Compt. Rend. Acad. Sci. 237, 601 (1953).
30. R. Daudel, H. Brion and S. Odoit, J. Chem. Phys. 23, 2080 (1955).
31. E.V. Ludena, Int. J. Quantum Chem. 9, 1069 (1975).
32. R.F.W. Bader and M.E. Stephens, J. Amer. Chem. Soc. 97, 7391
(1975).
33. P.0. Lowdin, J. Chem. Phys. 18, 365 (1950).
34. D.A. Kleier, T.A. Halgren, J.H. Hall and V. Lipscomb, J. Chem.
Phys. 61, 3905 (1974).
35. S.W. Baldwin and J.M. Wilkinson, Tetrahedron Lett. 20, 2657 (1979)
36. S.W. Baldwin and J.E. Fredericks, Tetrahedron Lett. 23, 1235
(1982).
37. S.W. Baldwin and H.R. Blomquist, Tetrahedron Lett. 23, 3883 (1982)
38. W.J. Hehre, R.F. Stewart, and J.A. Pople, J. Chem. Phys. 51, 2657
(1969).
39. G. Herzberg, in Molecular Spectra and Molecular Structure III,
Electronic Spectra and Electronic Structure of Polyatomic
Molecules, New York: Van Nostrand Reinhold, 1966.
40. L.C. Snyder and H. Basch, in Molecular Wavefunctions and
Properties, New York: Wiley-Interscience, 1972.
41. E.L. Eliel, N.L. Allinger, S.J. Angyal and G.A.L. Morrison,
Conformational Analysis, New York: Wiley-Interscience, 1965.
42.
J.E. Williams, P.J. Stang and P. Schleyer, Ann. Rev. Phys. Chem.
19, 531 (1968).

75
43. 0. Sinanoglu and B. Skutnik, Chem. Phys. Lett. 1, 699 (1968).
44. W. Kutzelnigg, Israel J. Chem. 19, 193 (1980).
45. M. Schnidler and W. Kutzelnigg, J. Chem. Phys. 76, 1919 (1982).
46. L.A. Ñafie and P.L. Polaravapu, Chem. Phys. 75, 2935 (1981).
47. R. Lavery, C. Etchebest and A. Pullman, Chem. Phys. Lett. 85, 266
(1982).
48. J.E. Reutt, L.S. Wang, Y.T. Lee and D.A. Shirley, Chem. Phys. Lett.
126, 399 (1986).
49. T.J. Marks and I.L. Fragalá, Fundamental and Technological Aspects
of Organo-^-Element Chemistry, NATO ASI Series, C155 Reidel
Dordrecht, 1985.
50. T.J. Marks, Acc. Chem. Res. 9, 223 (1976); T.J. Marks, Adv. Chem.
Ser. 150, 232 (1976); T.J. Marks, Prog. Inorg. Chem. 24, 51 (1978).
51. H. Schumann and W. Genthe in Handbook on the Physics and Chemisty
of Rare Earths, North Holland, Amsterdam, 1984, Chpt. 53., H.
Schumann, Angew. Chem. 96, 475 (1984).
52. P.J. Hay, W.R. Wadt, L.R. Kahn, R.C. Raffenetti and D.W. Phillips,
J. Chem. Phys. 70, 1767 (1979); W.R. Wadt, J. Amer. Chem. Soc. 103,
6053 (1981).
53. J.V. Ortiz and R. Hoffmann, Inorg. Chem. 24, 2095 (1985); P. Pyykko
and L.L. Lohr, Jr., Inorg. Chem. 20, 1950 (1981); C.E. Myers, L.J.
Norman II and L.M. Loew, Inorg. Chem. 17, 5443 (1983).
54. a: D.E. Ellis, Actinides in Perspective, ed N.M. Edelstein,
Pergamon (1982).
b: B. Ruscic, G.L. Goodman and J. Berkowitz, J. Chem. Phys. 78,
5443 (1983).
55. N. Rosch and A. Streitvieser, Jr., J. Amer. Chem. Soc. 105, 7237
(1983); N. Rosch, Inorg. Chim. Acta 94, 297 (1984); A.
Streitvieser, Jr., S.A. Kinsley, J.T. Rigbee, I.L. Fragala, E.
Ciliberto and N. Rosch, J. Amer. Chem. Soc. 107, 7786 (1985).
56. D. Hohl and N. Rosch, Inorg. Chem. 25, 2711 (1986); D. Hohl, D.E.
Ellis and N. Rosch, Inorg Chim. Acta, to be published.
57. J.D. Head and M.C. Zerner, Chem. Phys. Letters 122, 264 (1985).
58. D.W. Clack and K.D. Warren, J. Organomet. Chem. 122, c28 (1976).
Li Le-Min, Ren Jing-Qing, Xu Guang-Xian and Wong Xiu-Zhen, Intern.
J. Quantum Chem. 23, 1305 (1983). Ren Jing-Qing and Xu Guang-Xian,
Inter. J. Quantum Chem. 26, 1017 (1986).
59.

76
60. J.A. Pople D.L. Beveridge and P.A. Dobosh J. Chem. Phys. 47, 158
(1967).
61. J.E. Ridley and M.C. Zerner, Theor. Chem. Acta 32, 111 (1973).
62. A.D. Bacon and M.C. Zerner, Theor. Chem. Acta 53, 21 (1979).
63. M.C. Zerner, G.H. Loew, R.F. Kirchner and U.T. Mueller-Uesterhoff,
J. Amer. Chem. Soc. 102, 589 (1980).
64. W. Anderson, W.D. Edwards and M.C. Zerner, Inorg. Chem. 25, 2728
(1986).
65. See, i.e., J.C. Slater, Quantum Theory of Atomic Structure, Vol. 1
and Vol. 2, New York, McGraw Hill, 1960.
66. See, i.e., R.J. Parr, Quantum Theory of Molecular Electronic
Structure, New York, Benjamin, 1963.
67. R. Pariser and R. Parr, J. Chem. Phys. 21, 767 (1953).
68. J. Del Bene and H.H. Jaffé, J. Chem. Phys. 48, 1807 (1968).
69. N. Mataga and K. Nishimoto, Z. Phys. Chem. (Frankfurt am Main)13,
140 (1957). —
70. K. Ohno, Theoret. Chim. Acta 2, 568 (1964); G. Klopman, J Amer.
Chem. Soc. 87, 3300 (1965).
71. W.D. Edwards and M.C. Zerner, to be published.
72. A. Veillard, Computational Techniques in Quantum Chemistry and
Molecular Physics, ed. G. H. F. Diercksen, B. T. Sutcliffe, A.
Veillard ,Eds. D. Reidel, Dordrecht, The Netherlands (1975).
73. E. Davidson, Chem. Phys. Letters 21, 565 (1973).
74. J.A. Pople and G.A. Segal, J. Chem. Phys. 43, S136 (1965).
75. M.C. Zerner, Mol. Phys. 23, 963 (1972).
76. P. Coffey, Inter. J. Quantum Chem. 8, 263 (1974).
77. J.A. Pople, D.P. Santry and G.A. Segal, J. Chem. Phys. 43, S129
(1965).
78. C.F. Bender and E.R. Davidson, J. Inorg. Nucl. Chem. 42, 721
(1980).
79. E. Clement! and C. Roetti, Atomic Data and Nuc. Data Tables 14, 177
(1974). —
80. P. Knappe, Diplom-Chemikers Thesis, Department of Chemistry
Technical University of Munich, Munich, Germany.

77
81. J.P. Desclaux, Comp. Phys. Commun. 9, 31 (1975).
82. G. Karlsson and M.C. Zerner, Intern J. Quantum Chemistry 7, 35
(1973).
83. M.O.Fanning and N.J. Fitzpatrick, Intern. J. Quantum Chem. 28, 1339
(1980).
84. M.C. Zerner in Approximate Methods in Quantum Chemistry and Solid
State Physics, ed. F. Herman, New Tiork, Plenum Press (1972).
85. a. L. Brewer, J. Opt. Soc. Amer. 61, 1101 (1971).
b. L. Brewer, J. Opt. Soc. Amer. 61, 1666 (1971).
86. W.C. Martin, J. Phys. Chem. Ref. Data 3, 771 (1974). J. Sugat, J.
Opt. Soc. Amer. 56, 1189 (1966).
87. C.W. Bauschlicher and P.S. Bagus, J. Chem. Phys. 81, 5889 (1984).
88. R.D. Brown, B.H. James and M.F. O'Dwyer Theor. Chem. Acta YJ_, 264
(1970): U. Th. A.M. Van der Lugt. Intern. J. Quantum Chem. 6, 859
(1972): J.J. Kaufman and R. Prednen, Intern. J. Quantum ChemT 6,
231 (1977).
89. J. Schulz, R. Iffert and K. Jug, Inter. J. Quantum Chem. 27, 461
(1985).
90. J.C. Culberson and M.C. Zerner, unpublished results.
91. J. Almlof, University of Stockholm, Inst, of Physics (USIF Reports
72-09,74-29).
92. H.D. Arnberger, W. Jahn and N.M. Edelstein, Spectrochem. Acta. 41A.
465: Ibid, in press.
93. H.D. Arnberger, H. Schultze and N.M. Edelstein, Spectrochem. Acta.
41A, 713 (1985).
94. N. Edelstein in Fundamental and Technical Aspects of Organo-/-
Element Chemistry, ed. T.J. Marks and I.L. Fragala, NATO ASO C155,
Reidel, Dordrecht (1985).
95. M.C. Zerner and M. Hehenberger, Chem. Phys. Letters 62, 550 (1979).
96. J.C. Culberson and M.C. Zerner, in preparation.
97. K.S. Krasnov, G.V. Girichev, N.I. Giricheva, V.M. Petrov, E.Z.
Zasorin, N.I. Popenko, Seventh Austin Symp. on Gasphase Molecular
Structure, Austin Texas, p. 88 (1978).
98. N.I. Popenko, E.Z. Zasorin. V.P. Spiridonov and A.A. Ivanov, Iporg.
Chim Acta 31, L371 (1978).
99. E.P.F. Lee, A.W. Potts and J.E. Bloor, Proc. R. Soc. Lond. A 381,
373 (1982).

78
100. C.W. DeKock, R.D. Wesley and D.D. Radtke, High Temp. Sci. 4, 41
(1972); I.R. Beattie, J.S. Ogden and R.S. Wyatt, J. Chem. Soc.
Dalton Trans., 2343 (1983).
101. T.A. Beineke and J. Del Gaudio, Inorganic Chemistry 7, No. 4, 715
(1968).
102. K.S. Krasnov, N.I. Giricheva and G.V. Girichev, Zhurnal Strukturnoi
Khimii 17, 667 (1976).

BIOGRAPHICAL SKETCH
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 /-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.
79

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presenetation and is fully
adeqate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
/ ‘ *•- ~~ \
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 presenetation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
i
N. Yngve Ohrn
Professor t>£ Chemistry and Physics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presenetation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
/ • • ' n 1
¿(.iC-AtLhzil. !?_•_ ]—
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 presenetation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presenetation 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

UNIVERSITY OF FLORIDA
3 1262 08554 1588



UNIVERSITY OF FLORIDA
3 1262 08554 1588