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 Permanent Link:
 http://ufdc.ufl.edu/AA00004838/00001
Material Information
 Title:
 Localized orbitals in chemistry
 Creator:
 Culberson, John Christopher
 Publisher:
 [s.n.]
 Publication Date:
 1987
 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 )
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 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1987.
 Bibliography:
 Bibliography: leaves 7378.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by John Christopher Culberson.
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 University of Florida
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 University of Florida
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 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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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 postdoc 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
11 Probe electron points for furanone 18
12 Boys and Fermi hole centroids for C^H^02 19
13 Probe electron points for methlyactetylene 22
14 Boys and Fermi hole centroids for CHCH^ 23
15 Orbital centroids for BF^ 26
16 Eigenvalues and derivatives for BF^ using the
Boys method 27
17 Probe electron points for BF^ 28
18 Orbital centroids for BF^ 29
21 Basis functions for Lanthanide atoms 51
22 Ionization potentials for Lanthanide atoms 54
23 Average configuration energy for Lanthanides... 57
24 Resonance integrals for Lanthanide atoms 58
25 Geometry and ionization potentials for Cerium
and Lutetium trihalides 63
26 Geometry and ionization potentials for
Lanthanide trichlorides 65
27 Geometry and ionization potentials for SmCl9,
EuC12 and YbCl2 .... 67
28 Geometry of CeiNO^)^ ion 68
_2
29 Population analysis of CeiNO^)^ ion 70
v
LIST OF FIGURES
Page
11 Fermi mobility function for t^CO 12
12 Difference between mobility function and
electron gas correction 13
13 Fermi hole plot for formaldehyde 14
14 Boys localized orbital for formaldehyde 15
15 Ni(CO)^ bonding orbital 38
16 Ni(CO)^ nonbonding orbital 39
17 Ni(CO)^ antibonding orbital 40
21 Single and double C basis set plot 49
22 Average value of r versus atomic number 50
23 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 dorbitals or lanthanide /orbitals; and (2) orbitals
which must be localized after a selfconsistent 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 noniterative 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 DiracFock atomic calculations, and the
inclusion of all onecenter twoelectron 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, antibond, 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 dorbitals 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 dd, ligandd 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 dorbital 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 closedshell selfconsistent 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 postHarteeFock 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, lonepair 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 intraorbital electronic
interactions and minimize interorbital 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 , (11)
i = l ii
is maximized or minimized, where the definition of depends
on the localization criterion. One choice for the value of
23
is the twoelectron 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 selfextension operator,
2
gii = r12
(12)
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 lesslocalized set of orbitals to a morelocalized 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 twoelectron 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 (semiempirical)
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 (oneelectron) 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(nonitera 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 nrc transitions in a molecule, a full localization need
not be done, the double projector may be used to isolate (localize) the
ntype orbitals. A subsequent small singles Cl may then be used to study
k k
only the nJt transitions and thereby elucidate the nrt spectra.
7
Another example involves the localization of the dorbitals in a
transition metal complex. Because of accidental degeneracies between
metal dorbitals 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 nit 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 > (13)
' a .. 1 l l' a
1 = 1
J*
for a = 1 to r. These {TO}^, are then symmetrically orthogonalized
8
T'+Y' = A (14)
1/2
0 = Y' A (15)
and are projected out of the original set
#' = $ ( 1 I 9 ><0  ) (16)
a 1 a a1
The matrix A' is formed and diagonalized
U = U+A U = X (17)
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 (18)
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 (19)
Y' = YV (110)
Y' are linear combinations of Y that we can energy order according to
Â£
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
1719
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
2025
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), (111)
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. (112)
The Fermi orbital is given by
f(ri;r2) = [2/p(r2)]1/2 E gi(r1)g.(r2),
i
d13)
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) (114)
where
Fv(r) = ^2
p
I h
i>j
3vJ
&i
3v
d15)
for v = x, y or z. This may be compared to
11
F0(p) = (3n/4)(p/2)2/3 (116)
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. 11. The difference F(r)Fg(p) is shown in Fig. 12.
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
2933
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.
13 and 14 which compare a Fermi hole for the formaldehyde molecule
with a localized orbital determined by the orbital centroid criterion of
, 46,15
localization.
Localized Orbitals Based on the Fermi hole
Equation 113 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 11: 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 12: 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 11. the contours representing negative values and
zero are indicated by broken lines. Each nucleus is located
at a local minimum.
14
Figure 13: 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 14: The localized orbital for the CH bond of a formaldehyde
molecule determined by the orbital centriod criterion for
localization. The electronic density contours are the same
as in figure 13.
16
A set of N Fermi orbitals determined by Eq. 13 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, (117)
where
T.. = gj(ri)/(p(r.)/2)1/2. (118)
In the following three sections, the transformation of canonical SCF
orbitals based on Eqs. 117 and 118 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 RH 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
3537
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 11. 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 11. 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 11 are shown in Table 12. 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 12, the centroids of the orbitals determined by
the Fermi hole are very close to those of the localized orbitals Page
18
Table 11 : 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 CÂ£
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
CfCz 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
C0C. bond
3.517
2.148
0.0
C^0^ bond
3.449
0.066
0.0
Cj0^ bond
1.267
0.536
0.0
C2O2 bond 1
3.049
4.452
2.000
C3O2 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 12 : 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
C2H2 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
C2C3 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
C0, 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
C3O2 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 PDP11/44 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 offdiagonal 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 CC (banana) bonds relative to the
three CH 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 ST05G 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 13. 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 14. 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 14. These are very close to those determined by the Fermi hole
method. The RMS value of the offdiagonal 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 13 : 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
C1C2 bond 3
1.0
0.0
1.732
C2C3 bond
0.0
2.510
0.0
23
Table 14 : 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
C3H2 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
CH, 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
1C2 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
C2C3 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 innershell
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 BF 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 FB 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 "threeup" 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 FB axis. The point group for the orbital centroids
of this conformation is C^v The "upupdovn" conformation is generated
by rotating the set of lone pair orbitals on one of the fluorine atoms
in the "threeup" conformation by 180 degrees about the FB axis. This
conformation has the symmetry of the Cs point group.
The canonical SCF orbitals for BF^ were calculated based on the
doublezeta basis set and geometry tabulated by Snyder and Basch.^
Localized orbitals determined by the orbital centroid criterion were
obtained for the threeup conformation and the upupdown conformation.
The centroids for these orbitals are shown in Table 15. The first five
(most positive) eigenvalues of the hessian matrix for each of these
conformations are shown in Table 16. 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 threeup
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 17. The centroids of
the resulting set of localized orbitals are shown in Table 18. 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 18. As shown in Table 16, three of the eigenvalues of the
hessian matrix were positive at this point, demonstrating that the Page
26
Table 15 : 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
upupup (C3v)
upupdown (Cg)
X
Y
Z
X
Y
Z
BF^ bond
1.725
0.0
0.072
1.725
0.0
0.070
BF2 bond
0.862
1.494
0.072
0.862
1.494
0.070
BF^ 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 16 : 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 15. The third and fourth columns
correspond to localized orbitals described in Table 18.
The gradient vectors are zero for the first three columns.
Configurations
upupup
upup 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 17 : Probe electron positions for the boron trifluoride
molecule. The first three points are located at the
midpoint of the BF bonds. The remaining points have been
chosen in the pinwheel conformation (symmetry C^) All
coordinates are given in bohr.
Position
X
Y
Z
BF^ bond
1.223
0.0
0.0
BF9 bond
0.611
1.059
0.0
BF^ 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
FÂ£ 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 18 : 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
BF^ bond
1.713
0.0
0.0
1.720
0.028
0.0
BF^ bond
0.857
1.484
0.0
0.884
1.475
0.0
BF^ 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 threeup
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 12, 14 and 18, 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 CC 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 CC 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 HC=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
EdmistonReudenberg 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 oneelectron 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^ (119)
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(Nl)/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(Nl)/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, (120)
where V is a positive definite matrix defined by
V = (RR+)~1/2. (121)
The matrix R will be defined as
R = NT, (122)
where
T = 1 + t (123)
and t is an antisymmetric matrix
t+ = t (124)
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') (125)
i'=l
The new f' orbitals can be thought of in terms of a pertubative
expansion,
f' = f + E t.f + Â£ tTtTf (126)
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) (127)
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 (CH 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 dorbitals 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. 15, an isovalue plot of one of the nickel carbon bonds is
shown. The orbital shows a large amplitude near the carbon atom,
indicative of large porbital contributions on the carbon and a
relatively small contribution from the nickel dorbitals. As one can see
from Fig. 16, a sizable contribution to bonding comes from one of the
partially occupied nickel dorbitals. The nickel carbon antibond shown
in Fig. 17 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. 15 and 16. 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 15: An isovalue 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 16: An isovalue 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 17: An isovalue 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 SelfConsistent 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 onecenter twoelectron 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 UVvisible 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 LiMin, JingQuing, 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 pseudopotential 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 twocenter twoelectron integrals.
Subsequent atomic parameters that enter the model are the valence
state ionization potentials used for calculating onecenter oneelectron
k. k
"core" integrals and the SlaterCondon F and G integrals that are used
for the formation of onecenter twoelectron 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 UVvisible spectroscopy.^These values seldom differ
much from those chosen to reproduce molecular geometry.
In this initial work all twocenter twoelectron integrals required
for the INDO model Hamiltonian are calculated over the chosen basis set,
as are the onecenter twoelectron 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 twoelectron twocenter integral from one of the more successful
functions established for this purpose.
44
At the SCF level, we seek solutions to the pseudoeigenvalue problem
F C = C e
(21)
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,*^
7173
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 (yayX)
(22a)
FAA
MV
+ Â£ PCTCT (UU
a c B
BM
E Nx
a, X
(U vaX) j (yavX)
1
2
ba(u) + Bb(v)
S ir P (yy I vv)
y\i 2 yv v 1 '
y *v
A*B
(22b)
(22c)
where
(yvuX) = Jdx(l)dx(2) yi) Xv(l) X*(2) Xx(2) (23)
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
Â£ c C n
ya \>a a
a
(24)
AB
with n the occupation of MO $ n = 0,1,2. In Eq. (22), F refers
3 3 3 MV
to a matrix element with AO centered on atom "A". The core
integral
C (xÂ£l \ ^ T + ',Alxi) (25)
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 noninteracting from the neglected
innershell 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. (22a)
represents the attraction between an electron in distribution X^* X^ and
all nuclei but A. The rationale for replacing integral
A A
(yARB1uA) (Â¡j y SBSB) (26)
is given elsewhere, and compensates for neglected two center inner
27 28
shellvalence 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.
1315
S of Eq. (22c) 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)) (27)
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,
6163 68
and set all / = 1 except between p symmetry orbitals viz.
S pp = 1 267papcj (pcTlpCT) + 585gpitpJt(prtlpTt) (2'8)
+ n585Vpn
Basis Set
In general ZDO methods choose a basis set of Slater Type Orbitals
(STO)
1/2
rn1e~^r y!J (9, ) (29a)
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
(29b)
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 doubleC type function (tvo terms in 29b) for an accurate
description of both their inner and outer regions. For the lanthanides
59
wo have examined basis sets suggested by Li LeMin et al., by Bender
78
79
and Davidson,
and by Clementi and Roetti.
In the latter case, the
two major contributors of Eq. (29b) in the valence orbitals of the
doubleC atomic calculations were selected, and these functions were
renormalized with fixed ratio to yield the required nodeless doubleC
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
monopositive ions using the numerical DiracFock 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 DiracFock
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 monopositive 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 (nl)d, and now the (n2)/ require at least two terms
in the expansion of Eq. (29b). This is demonstrated for the Ce+ ion in
Figure 21, where it is shown that a singleC expansion is poor for the
outer region of the 4/ function.
In Figure 22 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 DiracFock values obtained from the monopositive ions. The
basis set adopted is given in Table 21.
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 twocenrer twoelectron integrals with the values in
Table 21. This value is chosen to match the accurate F SlaterCondon
Factors obtained from the numerical atomic calculations by a single
exponent, via
F (4/4/) = 0.200905 C (4/) (210a)
F(5d5d) = 0.164761 C (5d) (210b)
F(6s6s) = 0.139803 C (6s) (210c)
F(6p6p) = 0.139803 C (6p) (210d)
The error in calculating twocentered twoelectron integrals at typical
bonding distances with this singleC 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 4ST0 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 21: Radial vavefunction for the 4f orbital of Ce with singleC
and doubleC Slater type orbitals (STOs).
50
Figure 22: Average value of r for the valence orbitals of the
lanthanides from a relativistic calculation (DF) and a non
relativistic calculation (HF).
Table 21 :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
(211)
ni1) v + n^~1) v + u +
rt vt r n *r,jj'r n wx.cr 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)
(212)
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. (25), 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 /n3d1s2  /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 22. 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 Z13 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. (211), 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 22. The values
from process V are also given in the table for comparison.
For a f orbital ionization, we consider the two processes
VI Z13 ds2 * Z^ds2 + (/)
VII Z12 s2  Z1_3s2 + (f)
(compare with I and II). As seen .in Table 22 the values form the two
processes are very different. From Eq. (211)
Uff (VI) = IP(VI) (m4) Vff 2Vff Wd/ (213a)
Ujj (VII) = IP(VII) (m3) W. 2\Jsf (213b)
Table 22 Â¡Smoothed Ionization potentials for processes IVII 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. (213a) to find U^, and use of this value in Eq. (213b) 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 twoby
two interaction matrix
E/1 3ds2) X V
,7 rw.cn2_2>
\
(r \
L1
CNj
= 0
(214a)
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
(214b)
l.x2
V c /c d sZ) ECf"V)
X  2V
(214c)
E(/ 3d s2) E(,f 2s2)
n2
2V
+ 1
The values of appear in Table 22, 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) (215)
n_2 2
In the case of the 3d orbitals this valence bond mixing between 3dnis
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. (214a). 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 23, 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 24. They are
obtained by fitting the geometries of the trihalides, and the more
covalent biscyclopentadienyls 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 23 : Average configuration energy from DiracFock
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 24 :Resonance integrals (B values) for the Lanthanide atoms in e
V. The beta for the sorbital is set equal to the beta for
the porbital.
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 24 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 onecentered
integrals of the Coulomb or exchange type
(yy v\>) or (yvvy)
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 onecenter 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 onecenter 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 onecenter 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 (yvvy) can be obtained thiuugh
1c
atomic spectroscopy, and their components, F and G evaluated via
least square fits
(yy  vv) = E ak pk
k
(yvvy) = 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.(sddd), (sppd),
(sdpp), (sd//), (s/d/), (ppp/), (ddp/), (pdd/), (sdp/), (pds/),
(spd/), and (p///). For this reason we evaluate all onecenter two
electron integral' of the lanthanides using the basis set of Table 21,
which yields the exact F value obtained from the FockDirac 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^^294 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 LeMin et_ al. In the
latter formalism only the conventional onecenter twoelectron integrals
are included leading to rotational variance. In addition, the
VolfsbergHelmholz 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
singleC STOs and the smoothing of the valence orbital ionization
59
potentials for the lanthanides via Annotype expressions.
Procedures
The input to the INDO program consists of molecular coordinates and
atomic numbers. Molecular geometries are obtained automatically via a
gradient driven quasiNewton update procedure,^ using either the
restricted or unrestricted Hartree Fock formalism. All UHF calculations
62
are followed by simple annihilation.
Selfconsistent 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 24). The INDO optimized geometries as well as the
remaining cerium and lutetium trihalides are listed in Table 25. In
addition to the trihalides reported, the geometry of CeF^ is also listed
in Table 25. 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 nonplanar, in
54b 97 98
agreement with experiments.
63
Table 25 : 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 25. 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 26. 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 26 : 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
(10.911.2)
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
(13.717.0)
Eu
2.532
113.2
16.4
Gd
2.514
2.489
110.0
113.0
17.7
(15.516.5)
Tb
2.496
2.478
109.8
109.9
13.0
(13.020.5)
Dy
2.479
110.1
14.3
(14.020.0)
Ho
2.464
2.459
112.0
111.2
15.0
(15.520.0)
Er
2.448
110.9
15.6
(11.516.0)
Tm
2.430
108.5
15.9
(15.321.0)
Yb
2.421
109.6
15.9
(15.521.0)
Lu
2.415
2.417
108.2
111.5
18.6
(17.418.7)
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 27. 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 porbital 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 28 and a plot of the optimized geometry is shown as
figure 23. As one can see from Table 28 INDO predicts a geometry that
is in excellent agreement with the experimental crystal structure.
Table 29 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 27 : 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 28 : Average bond distances and bond angles for CeiNO^)^ ion
INDO optimized geometry and the Xray 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(Ce0c)
2.554
2.508
r(N0c)
1.256
1.282
r(N0n)
1.237
1.235
9(0 N0 )
c c'
121.5
114.5
9(0 Ce0 )
c c
50.9
50.9
a) Reference 101.
CE(N03)6.
_2
Figure 23: 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 29 : 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 DiracFock numerical functions on the lanthanide
monocations, and is characterized by the use of atomic ionization
information for obtaining the onecenter oneelectron terms, and
including all of the twoelecfron 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.
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(1985).
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78
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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>Â£ 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
(.CAtLhzzi. '
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

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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 postdoc 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
11 Probe electron points for furanone 18
12 Boys and Fermi hole centroids for C^H^02 19
13 Probe electron points for methlyactetylene 22
14 Boys and Fermi hole centroids for CHCH^ 23
15 Orbital centroids for BF^ 26
16 Eigenvalues and derivatives for BF. using the
Boys method 27
17 Probe electron points for BF^ 28
18 Orbital centroids for BF^ 29
21 Basis functions for Lanthanide atoms 51
22 Ionization potentials for Lanthanide atoms 54
23 Average configuration energy for Lanthanides... 57
24 Resonance integrals for Lanthanide atoms 58
25 Geometry and ionization potentials for Cerium
and Lutetium trihalides 63
26 Geometry and ionization potentials for
Lanthanide trichlorides 65
27 Geometry and ionization potentials for SmCl9,
EuC12 and YbCl2 .... 67
28 Geometry of CeiNO^)^ ion 68
_2
29 Population analysis of CeiNO^)^ ion 70
v
LIST OF FIGURES
Page
11 Fermi mobility function for t^CO 12
12 Difference between mobility function and
electron gas correction 13
13 Fermi hole plot for formaldehyde 14
14 Boys localized orbital for formaldehyde 15
15 Ni(CO)^ bonding orbital 38
16 Ni(CO)^ nonbonding orbital 39
17 Ni(CO)^ antibonding orbital 40
21 Single and double Ã, basis set plot 49
22 Average value of r versus atomic number 50
23 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 dorbitals or lanthanide /orbitals; and (2) orbitals
which must be localized after a selfconsistent 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 noniterative 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 DiracFock atomic calculations, and the
inclusion of all onecenter twoelectron 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, antibond, 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 dorbitals 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 dd, ligandd 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 dorbital 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 closedshell selfconsistent 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 postHarteeFock 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, lonepair 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 intraorbital electronic
interactions and minimize interorbital 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 , (11)
i = l ii
is maximized or minimized, where the definition of depends
on the localization criterion. One choice for the value of
23
is the twoelectron 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
4â€”6
orbital selfextension operator,
2
gii = r12
(12)
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 lesslocalized set of orbitals to a morelocalized 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 twoelectron 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 (semiempirical)
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 (oneelectron) 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(noniterative) 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
â€¢k
like to study nrc transitions in a molecule, a full localization need
not be done, the double projector may be used to isolate (localize) the
ntype orbitals. A subsequent small singles Cl may then be used to study
k k
only the nJt transitions and thereby elucidate the nrt spectra.
7
Another example involves the localization of the dorbitals in a
transition metal complex. Because of accidental degeneracies between
metal dorbitals 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 nit 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 > (13)
1 a . i i i1 a
1 = 1
J*
for a = 1 to r. These {TO}^, are then symmetrically orthogonalized
8
T'+Y' = A , (14)
1/2
0 = Y' A Â¿ . (15)
and are projected out of the original set
*' = * ( 1  I 9 ><0 ) . (16)
a 1 a a1
The matrix A' is formed and diagonalized
U = U+A U = X . (17)
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 . (18)
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 , (19)
Y' = YV . (110)
Y' are linear combinations of Y that we can energy order according to
Â£
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 (nonÂ¬
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
1719
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
2025
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), (111)
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 â€™
A(r1;r2) = f(r:;r2)2. (112)
The Fermi orbital is given by
f(ri;r2) = [2/p(r2)]1/2 Z g.(r1)g.(r2),
i
d13)
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 â€™ â€™ â€™ 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, â€™ â€™
F(r) = Fx(r) + F (r) + Fz(r) (114)
where
Fv(r) = ^2
p
I u
3vJ
&i
3v
i>j
d15)
for v = x, y or z. This may be compared to
11
F0(p) = (3n/4)(p/2)2/3 (116)
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. 11. The difference F(r)Fg(p) is shown in Fig. 12.
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
2933
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.
13 and 14 which compare a Fermi hole for the formaldehyde molecule
with a localized orbital determined by the orbital centroid criterion of
, 46,15
localization.
Localized Orbitals Based on the Fermi hole
Equation 113 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 11: The fermi hole mobility function F(r) for the Hâ€žCO 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 12: 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 11. the contours representing negative values and
zero are indicated by broken lines. Each nucleus is located
at a local minimum.
14
Figure 13: 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 14: The localized orbital for the CH bond of a formaldehyde
molecule determined by the orbital centriod criterion for
localization. The electronic density contours are the same
as in figure 13.
16
A set of N Fermi orbitals determined by Eq. 13 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, (117)
where
T.. = gj(ri)/(p(r.)/2)1/2. (118)
In the following three sections, the transformation of canonical SCF
orbitals based on Eqs. 117 and 118 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 RH 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
3537
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 11. 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 11. 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 11 are shown in Table 12. 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 12, the centroids of the orbitals determined by
the Fermi hole are very close to those of the localized orbitals Page
18
Table 11 : 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 CÂ£
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
CfCz bond 1
0.0
1.295
2.000
^1~^2 kÂ°ncl 2
0.0
1.295
2.000
^onc'
1.336
2.972
0.0
C0C. bond
3 4
3.517
2.148
0.0
C^0^ bond
3.449
0.066
0.0
Cj0^ bond
1.267
0.536
0.0
C^02 bond 1
3.049
4.452
2.000
C3O2 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 12 : 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
C2H2 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
C2C3 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
C0, 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
C3O2 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 PDP11/44 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 offdiagonal 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 CC (banana) bonds relative to the
three CH 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.''""â€™ 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 ST05G 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 13. 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 1â€™^ 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 14. 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 14. These are very close to those determined by the Fermi hole
method. The RMS value of the offdiagonal 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 13 : 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
C1C2 bond 3
1.0
0.0
1.732
C2C3 bond
0.0
2.510
0.0
23
Table 14 : 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
C3H2 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
CH, bond
3 4
0.651
4.270
1.127
0.658
4.272
1.139
^l^2 bÂ°nd
0.713
0.003
0.0
0.692
0.U13
0.0
Â¿1C2 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
C2C3 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
â€”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 innershell
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 BF 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 FB 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 "threeup" 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 FB axis. The point group for the orbital centroids
of this conformation is C^v The "upupdovn" conformation is generated
by rotating the set of lone pair orbitals on one of the fluorine atoms
in the "threeup" conformation by 180 degrees about the FB axis. This
conformation has the symmetry of the Cs point group.
The canonical SCF orbitals for BF^ were calculated based on the
doublezeta basis set and geometry tabulated by Snyder and Basch.^
Localized orbitals determined by the orbital centroid criterion were
obtained for the threeup conformation and the upupdown conformation.
The centroids for these orbitals are shown in Table 15. The first five
(most positive) eigenvalues of the hessian matrix for each of these
conformations are shown in Table 16. 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 threeup
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 17. The centroids of
the resulting set of localized orbitals are shown in Table 18. 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 18. As shown in Table 16, three of the eigenvalues of the
hessian matrix were positive at this point, demonstrating that the Page
26
Table 15 : 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
upupup (C3v)
upupdown (Cg)
X
Y
Z
X
Y
Z
BF^ bond
1.725
0.0
0.072
1.725
0.0
0.070
BF2 bond
0.862
1.494
0.072
0.862
1.494
0.070
BF^ 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 16 : 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 15. The third and fourth columns
correspond to localized orbitals described in Table 18.
The gradient vectors are zero for the first three columns.
Configurations
upupup
upup 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 17 : Probe electron positions for the boron trifluoride
molecule. The first three points are located at the
midpoint of the BF bonds. The remaining points have been
chosen in the pinwheel conformation (symmetry C^) All
coordinates are given in bohr.
Position
X
Y
Z
BF^ bond
1.223
0.0
0.0
BF9 bond
0.611
1.059
0.0
BF^ 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
FÂ£ 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 18 : 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
BF^ bond
1.713
0.0
0.0
1.720
0.028
0.0
BF^ bond
0.857
1.484
0.0
0.884
1.475
0.0
BF^ 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 threeup
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 12, 14 and 18, 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 â€¢
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 â€™ 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, â€™ and other
46 47
properties. â€™ 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 â€™
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 CC 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 CC 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 HC=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. â€™ The integral transformation procedure
in any localization procedure can be a time limiting step. In the
EdmistonReudenberg 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 oneelectron 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^ (119)
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(Nl)/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(Nl)/2 rotations at once rather than one at a time.''"*â€™ 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, (120)
where V is a positive definite matrix defined by
V = (RR+)~1/2. (121)
The matrix R will be defined as
R = NT, (122)
where
T = 1 + t , (123)
and t is an antisymmetric matrix
t+ = t . (124)
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') (125)
i'=l
The new f' orbitals can be thought of in terms of a pertubative
expansion,
f' = f + E t.f + Â£ tTt.f (126)
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) . (127)
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 (CH 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 dorbitals 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. 15, an isovalue plot of one of the nickel carbon bonds is
shown. The orbital shows a large amplitude near the carbon atom,
indicative of large porbital contributions on the carbon and a
relatively small contribution from the nickel dorbitals. As one can see
from Fig. 16, a sizable contribution to bonding comes from one of the
partially occupied nickel dorbitals. The nickel carbon antibond shown
in Fig. 17 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. 15 and 16. 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 15: An isovalue 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 16: An isovalue localized orbital plot of a nickel carbon nonÂ¬
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 17: An isovalue localized orbital plot of a nickel carbon antiÂ¬
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 â€ž 5456
reasonably complex systems.
We examine an Intermediate Neglect of Differential Overlaps (INDO)
technique for use in calculating properties of /orbital complexes. At
the SelfConsistent Field (SCF) level this technique executes as rapidly
41
42
Â»!
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 onecenter twoelectron 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 UVvisible 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 LiMin, JingQuing, 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 pseudopotential 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. â€ 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 twocenter twoelectron integrals.
Subsequent atomic parameters that enter the model are the valence
state ionization potentials used for calculating onecenter oneelectron
k. k
"core" integrals and the SlaterCondon F and G integrals that are used
for the formation of onecenter twoelectron integrals. The evaluation
of these integrals using experimental information has traditionally made
' 65 66
this model highly successful in predicting optical properties. â€™
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 UVvisible spectroscopy.^These values seldom differ
much from those chosen to reproduce molecular geometry.
In this initial work all twocenter twoelectron integrals required
for the INDO model Hamiltonian are calculated over the chosen basis set,
as are the onecenter twoelectron FÂ° 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 FÂ° could be chosen from the Pariser approximation*^ FÂ°(n) =
IP(n)  EA(n), (IP = Ionization Potential, EA = Electron Affinity) and
the twoelectron twocenter integral from one of the more successful
functions established for this purpose.
44
At the SCF level, we seek solutions to the pseudoeigenvalue problem
F C = C e
(21)
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,*^
7173
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
â€¢ E P
[
c A
crX
(UU  crX)  j (yayX)
(22a)
FAA
MV
+ Â£ PCTCT (uuffff)  E ZB (UÃœsBsB)
a c B
BM
E Nx
a, X
(U vaX)  j (yavX)
1
2
BA(u) + bb(v)
S  T P (MU Ivo)
UM 2 UV v 1 '
y*v
AÂ¿B
(22b)
(22c)
where
(yvuX) = Jdx(l)dx(2) yi) Xv(l) râ€˜J X*(2) Xx(2) (23)
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
Â£ c C n
ya \>a a
a
(24)
AB
with n the occupation of MO $ , n = 0,1,2. In Eq. (22), F refers
3. 3. 3. MV
to a matrix element with AO centered on atom "A". The core
integral
C â– (xÂ£l  \ ^  T + ',Alxi) (25)
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 noninteracting from the neglected
innershell 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. (22a)
represents the attraction between an electron in distribution X^* X^ and
all nuclei but A. The rationale for replacing integral
A A
(yARB1uA) (Â¡] y SBSB) (26)
is given elsewhere, and compensates for neglected two center inner
27 28
shellvalence shell repulsion â€™ and neglected valence orbital
28 29
(symmetrical) orthogonalization. â€™ 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.
1315
S of Eq. (22c) 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)) (27)
1=0
where Â£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,
6163 68
and set all / = 1 except between p symmetry orbitals viz. â€ â€™
S pp = 1 â€˜ 267Â®papcr (pcTlpCT) + Â°585gpitpJt(prtlpTt) (2'8)
+ n585gpitpn (p" lpÃl >
Basis Set
In general ZDO methods choose a basis set of Slater Type Orbitals
(STO)
1/2
rn_1e"Cr y!J (9, ) (29a)
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 <29tÂ»
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 doubleC type function (two terms in 29b) for an accurate
description of both their inner and outer regions. For the lanthanides
wo have examined basis sets suggested by Li LeMin et^ al. , by Bender
and Davidson,^ and by Clementi and Roetti.^ In the latter case, the
two major contributors of Eq. (2â€”9b) in the valence orbitals of the
doubleC atomic calculations were selected, and these functions were
renormalized with fixed ratio to yield the required nodeless doubleC
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
monopositive ions using the numerical DiracFock 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 DiracFock
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 monopositive ions are obtained from 4,fm~^3d'''6s'*'
and 4/m ^6.â€˜^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. (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 (nl)d, and now the (n2)/ require at least two terms
in the expansion of Eq. (29b). This is demonstrated for the Ce+ ion in
Figure 21, where it is shown that a singleC expansion is poor for the
outer region of the 4f function.
In Figure 22 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 DiracFock values obtained from the monopositive ions. The
basis set adopted is given in Table 21.
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 twocenrer twoelectron integrals with the values in
Table 21. This value is chosen to match the accurate FÂ° SlaterCondon
Factors obtained from the numerical atomic calculations by a single
exponent, via
FÂ°(4/4/) = 0.200905 C (4/) (210a)
FÂ°(5d5d) = 0.164761 C (5d) (210b)
FÂ°(6s6s) = 0.139803 C (6s) (210c)
FÂ°(6p6p) = 0.139803 C (6p) (210d)
The error in calculating twocentered twoelectron integrals at typical
bonding distances with this singleC 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 4ÃST0 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 21: Radial vavefunction for the 4f orbital of Ce with singleC
and doubleC Slater type orbitals (STOs).
50
Figure 22: Average value of r for the valence orbitals of the
lanthanides from a relativistic calculation (DF) and a non
relativistic calculation (HF).
Table 21 :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"â€™
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
(211)
niâ„¢1) v + n^~1) v + q^q1) u +
rt vt r n n wxc_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 = FÂ°(ss)
Wpp = FÂ°(pp)  2/25F2(pp)
(212)
W
dd
= FÂ°(dd)  2/63F2(dd)  2/63 F4(dd)
Vff = FÂ°(//)  4/195F2(//)  2/143 F4(//)  100/5577 F6(//)
Wsp = FÂ°(sp)  l/6G1(sp)
Wsd = FÂ°(sd)  1/10 G2(sd)
Wsf = FÂ°(sf)  1/14 G3(s/)
V
pd
= FÂ°(pd)  1/15 GX(pd)  3/70 G3(pd)
Vpf = FÂ°(pf)  3/70 G2(p/)  2/63 G4(p/)
Vdf = FÂ°(d^.)  3/70 G3(d/)  2/105 G3(d;f)  5/231 G5(d/)
The core integrals H_, Eq. (25), 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).22â€™^2â€™Â®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 /n3d1s2 Â» /n_3d1s1 + (s)
II s2 Â» /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 22. These values are obtained from the promotion energies of
85 86
Brewer â€™ and then smoothed by a quadratic fit throughout the series.
For completeness, we also give the values of process II.
J13 1 2
The lowest configuration containing a 5d electron is j JdJs
throughout the series, and 5d ionizations are obtained from
III Z13 dV Â» fh2 + (d)
The ionization potentials for the 6p can be obtained from two
processes:
IV f~2 s1 p^ > /""â€˜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. (211), 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 22. The values
from process V are also given in the table for comparison.
For a f orbital ionization, we consider the two processes
VI Z13 ds2 Z^W + (/)
VII Z12 s2 Z1_3s2 + (f)
(compare with I and II). As seen .in Table 22 the values form the two
processes are very different. From Eq. (211)
Uff (VI) = IP(VI)  (m4) Vff  2Wff  (2 13a)
Ufj: (VII) = IP(VII)  (m3)  2Us/
(213b)
Table 22 Â¡Smoothed Ionization potentials for processes IVII 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
â€”
â€”
â€”
5.31
16.11
1.000
55
Unlike the analogous situation for the 6s and 6p orbitals, use of
Eq. (213a) to find U^, and use of this value in Eq. (213b) to predict
IP(VII) is not successful, and would require the scaling of the large
FÂ°(ff) integral often performed in methods parameterized on molecular
â€ž 61,63,67
spectroscopy. â€™
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 twoby
two interaction matrix
E(3ds2)  A V
â€ž rw J!n2_2.
\
(r \
L1
S
CNJ
= 0
(214a)
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
(214b)
â€˜1.x2
V _ c /c  E
a  2V
12 2,
(214c)
E(/Â° 3d s2)  E(fÂ° 2s2)
n2
2V
+ 1
The values of appear in Table 22, 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) (215)
n_2 2
In the case of the 3d orbitals this valence bond mixing between 3dnis
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. (214a). 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 23, 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 24. They are
obtained by fitting the geometries of the trihalides, and the more
covalent biscyclopentadienyls 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 23 : Average configuration energy from DiracFock
on the /^^ds^ and the f' configurations
lanthanide atoms.
calcula tions3
for all the
Atom
Average
Configuration
Energy
fw
/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 24 :Resonance integrals (B values) for the Lanthanide atoms in e
V. The beta for the sorbital is set equal to the beta for
the porbital.
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. â€™
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. â€™ 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 24 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 onecentered
integrals of the Coulomb or exchange type
(yy v\>) or (yvvy)
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 onecenter 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) >
eâ€ž (d) transitions that lead to the '*'E1 and excited states. In
4g lg 2g
60
addition, it appears that the inclusion of all onecenter integrals
improves the predictions of angles about atoms with s,p,d basis
89 90
sets â€™ and considerably improves the predictions of angles about the
lanthanides. For these reasons we include all the onecenter 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 (uuna>) and (y\>\iyl against the
83
formulas of Fanning and Fitzpatrick.
Integrals of the form (yu\>v) and (yvvy) can be obtained thi uugh
â€¢ 1c
atomic spectroscopy, and their components, F and G , evaluated via
least square fits
(yy  vv) = E ak pk
k
(yvvy) = 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 â€” y yz xz xy
however, cannot be evaluated in this manner; viz.(sddd), (sppd),
(sdpp), (sd//), (s/d/), (ppp/), (ddp/), (pdd/), (sdp/), (pds/),
(spd/), and (p///). For this reason we evaluate all onecenter two
electron integral' of the lanthanides using the basis set of Table 21,
which yields the exact FÂ° value obtained from the FockDirac 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^Â®â€™^294 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 LeMin et_ al. In the
latter formalism only the conventional onecenter twoelectron integrals
are included leading to rotational variance. In addition, the
VolfsbergHelmholz 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
singleC STOs and the smoothing of the valence orbital ionization
59
potentials for the lanthanides via Annotype expressions.
Procedures
The input to the INDO program consists of molecular coordinates and
atomic numbers. Molecular geometries are obtained automatically via a
gradient driven quasiNewton update procedure,"^ using either the
restricted or unrestricted Hartree Fock formalism. All UHF calculations
62
are followed by simple annihilation.
Selfconsistent 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
â€ž . 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 24). The INDO optimized geometries as well as the
remaining cerium and lutetium trihalides are listed in Table 25. In
addition to the trihalides reported, the geometry of CeF^ is also listed
in Table 25. 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 Â±3Â°. We note, however, that all are predicted nonplanar, in
54b 97 98
agreement with experiments. â€™ â€™
63
Table 25 : 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
â€”
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
â€”
9.9
â€”
CeF.
4
2.099
2.040
109.5
109.5
â€”
â€”
LuF3
2.045
2.020
107.4
â€”
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 Ã in going from CeF^ to CeF^ is also smaller than the
0.14 Ã observed.
Ionization potentials (IPs) are also reported in Table 25. 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 â€™ 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 26. The experimental
geometries'3^*3â€™^^ 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 26 : 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
(10.911.2)
Nd
2.563
2.545
112.7
â€”
13.3
12.0
Pm
2.556
â€”
112.7
â€”
14.4
â€”
Sm
2.544
â€”
113.0
â€”
15.3
(13.717.0)
Eu
2.532
â€”
113.2
â€”
16.4
â€”
Gd
2.514
2.489
110.0
113.0
17.7
(15.516.5)
Tb
2.496
2.478
109.8
109.9
13.0
(13.020.5)
Dy
2.479
â€”
110.1
â€”
14.3
(14.020.0)
Ho
2.464
2.459
112.0
111.2
15.0
(15.520.0)
Er
2.448
â€”
110.9
â€”
15.6
(11.516.0)
Tm
2.430
â€”
108.5
â€”
15.9
(15.321.0)
Yb
2.421
â€”
109.6
â€”
15.9
(15.521.0)
Lu
2.415
2.417
108.2
111.5
18.6
(17.418.7)
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 27. 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 porbital 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 28 and a plot of the optimized geometry is shown as
figure 23. As one can see from Table 28 INDO predicts a geometry that
is in excellent agreement with the experimental crystal structure.
Table 29 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 27 : 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
â€”
143.3
130Â±15
5.3
â€”
EuCI,,
2.576
â€”
143.2
135+15
6.6
â€”
YbCl2
2.400
â€”
120.2
126Â±05
3.2
â€”
a) Reference 100.
68
_2
Table 28 : Average bond distances and bond angles for CeiNO^)^ ion
INDO optimized geometry and the Xray 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(Ce0c)
2.554
2.508
r(N0c)
1.256
1.282
r(N0n)
1.237
1.235
9(0 N0 )
c c'
121.5
114.5
9(0 Ce0 )
c c
50.9
50.9
a) Reference 101.
CE(N03)6.
_2
Figure 23: 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 29 : Population analysis of CeiNO^)^ â€¢ 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
â€”
P
0.30
0.00
â€”
d
1.32
0.00
â€”
f
1.10
1.00
â€”
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 DiracFock numerical functions on the lanthanide
monocations, and is characterized by the use of atomic ionization
information for obtaining the onecenter oneelectron terms, and
including all of the twoelecfron 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.
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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.
/ â€˜ *â€¢ ~~ \
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>Â£ 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.
/ â€¢ â€¢ ' n 1
Â¿(.iCAtLhzil. !?_â€¢_ ]â€”
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
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