Title: Molecular orbital treatments of the aquo and ammine complexes of iron, cobalt and nickel
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Title: Molecular orbital treatments of the aquo and ammine complexes of iron, cobalt and nickel
Physical Description: vi, 90 l. : illus. ; 28 cm.
Language: English
Creator: Feiler, William Adkins, 1940-
Publication Date: 1965
Copyright Date: 1965
 Subjects
Subject: Molecular orbitals   ( lcsh )
Transition metals   ( lcsh )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
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Thesis: Thesis - University of Florida.
Bibliography: Bibliography: l. 87-89.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
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Bibliographic ID: UF00097853
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000423949
oclc - 11038351
notis - ACH2354

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MOLECULAR ORBITAL TREATMENTS OF THE

AQUO AND AMMINE COMPLEXES

OF IRON, COBALT AND NICKEL


















By
WILLIAM ADKINS FEILER, JR.









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










UNIVERSITY OF FLORIDA
August, 1965











ACKNOWLEDGMENTS


The author wishes to express his appreciation to

Dr. R. C. Stoufer, Chairman of the author's Supervisory

Committee, and to the other members of his Supervisory

Committee.

The author particularly wishes to thank Dr. D. W.

Smith, the personnel at the University of Florida Computer

Center, his wife and Mrs. Edwin Johnston for their

assistance in the completion of this dissertation.

The author gratefully acknowledges the support of

this research by the National Science Foundation under

Grant Number GP 1809, and the Graduate School Fellowship

from the University of Florida.














TABLE OF CONTENTS


Page


ACKNOWLEDGMENTS . .

LIST OF TABLES. . .

LIST OF FIGURES . .

INTRODUCTION. . .

PROCEDURE . . . .

RESULTS AND DISCUSSION.

SUMMARY . .....

APPENDICIES . . .

BIBLIOGRAPHY. . .

BIOGRAPHICAL SKETCH .


. . . . . . . . iv

. . . . . . . . iv
. . . . . . . . Vi


. . . . . & 1


. . . . . . . 26



. . . . . . . . 46

S . . . . . . . 87

S . . . . . . . 90


iii











LIST OF TABLES


Table


1. Group Overlap Integrals . . . . . .
2. Weighting Coefficients for the Smith
Population Analysis . . . . . ..

3. Summary of Calculated lODq Values (CM-1) for
the Indicated Complexes at Self-Consistency
4. Radii of Maximum Probability (r ) for Iron,
Cobalt and Nickel (A) . . . . . .

5. Predicted lODq Values Using the Doubly Charged
Aquo Complexes as Standards . . .
6. lODq' Values for the Aquo Complexes of
Cobalt(II) and (III) (CM-1) . .
7. Valence State Ionization Potential Data for
Iron, Cobalt and Nickel (in 1000 CM-1). .
8. Valence State Ionization Potential for
Nitrogen and Oxygen (in 1000 CM-1). . . .
9. Self-Consistent Parameters for [e(H 2+
9. Self-Consistent Parameters for [Fe(H20)6+ .

11. Self-Consistent Parameters for [Co(H20)6]2+ .
12. Self-Consistent Parameters for [Co(H20)6 3+ .
12. Self-Consistent Parameters for [Co(H20)63 .2

2+
15. Self-Consistent Parameters for [Ni(H20)6] + .
14. Self-Consistent Parameters for [Co(NH )6 2+ an
[co(NH 3)6]3. . . . . . . .

15. Self-Consistent Parameters for [Ni(NH3)632 .


d


S81
. 83


Page

17

23

27


36

38


43


50


51
66

69
72

75
78












INTRODUCTION


Background

The majority of octahedral cobalt(II) complexes are

high-spin with magnetic moments between 4.3 and 5.2 Bohr
(14)
magnetons.1 ) A few octahedral cobalt(II) complexes are

low-spin.30) Recently, there have been reported some octa-

hedral cobalt(II) complexes which have magnetic moments of

approximately three Bohr magnetons8'33) It has been pro-

posed that these unusual magnetic moments are due to a

Boltzmann distribution over the thermally accessible spin
4i 2
states ( T1 and E) near the cross-over point from high-

spin to low-spin.833) This proposal provides a satisfactory

semi-quantitative explanation of these low magnetic moments;

however, it is desirable to have a more complete elucida-

tion of the phenomena involved in these unusual complexes.

This will, hopefully, lead to a better understanding of the

cross-over point and, consequently, a better understanding

of transition metal complexes.

For several decades, the molecular orbital (MO) theory,

especially the linear combination of atomic orbital (LCAO)

form of this theory, has been used as an aid in interpreting

and assigning electronic structures and transitions of many







2

compounds, particularly organic compounds. The principles

are equally applicable to inorganic compounds in general and

to the unusual cobalt(II) complexes in particular. It is

logical then to attempt to use this theory to obtain a

better understanding of the phenomena involved in these

unusual complexes.

Unfortunately, it is not possible (at this time) to

subject the unusual complexes to a molecular orbital treat-

ment for two reasons. They are not sufficiently character-

ized (that is, there is still some doubt as to their con-

figurations and certainly the internuclear distances are not

known), and the molecular orbital methods have been applied

to only a few transition metal complexes (see below).

In this investigation, some approximate molecular

orbital methods have been tested on several transition metal

complexes which are well understood (in terms of electronic

transitions), which have much higher symmetry than the

unusual cobalt(II) complexes and in which the internuclear

distances are known or for which very reasonable estimates

may be made.

Each of the cobalt(II) complexes which has an un-

usual magnetic moment contains immine nitrogen donors8',3)

and, thus, it is desirable to choose compounds for treatment

that contain nitrogen as a donor atom. Furthermore, if the

proposal of a Boltzmann distribution is valid, then the









[Co(H20)6]3+ ion should be investigated, since this ion has
been characterized as low-spin but near the cross-over
point.'2 This compound also affords the opportunity to
investigate the importance of pi bonding which should have
considerable influence on the properties of the unusual
cobalt(II) complexes.

Definition of terms

When the LCAO approximation is substituted into the
Schrodinger equation and the variational theorem is applied,
the Schrodinger equation reduces to a secular determinant
of the form
Hij -Gij E= 0 (1)
where

Hij = f0iHOjd Gj f ij= aSij Sij

/7< i d- I

E is the eigenvalue corresponding with the molecular orbital

=7i 0iCij i bij H is a sum of one elec-
j Cj
tron Hamiltonian operators, %i is a molecular orbital, 0i
is a group orbital, Xi is an atomic orbital, and a is a
numerical constant. Hij is called the coulomb integral if
i=j, and the resonance integral if i \ j. Gij is the group
overlap integral and Sij is the (two center) overlap
integral.









The secular determinant can frequently be partially

diagonalized, by group theoretical methods,(1) into a form
in which the most complicated determinant is small, thus

reducing the work by a considerable amount.

Review of literature

In the past several years there have been reported

several attempts to calculate one-electron molecular orbital

energy levels for transition metal complexes. The procedure

used is a semi-empirical one originally proposed by

Mulliken(22) and first applied to a transition metal com-

pound, the permanganate ion, by Wolfsberg and Helmholz.(37

Ballhausen and Gray(2) modified the procedure of
Wolfsberg and Helmholz for estimating the coulomb and reso-

nance integrals and treated the vanadyl ion. Since then,

the permanganate ion (treated by two different groups of

investigators),(10'36) the Cr+ ion in crystals,(17) VC14,

CuC14 NiC142 and CuF6 ,(15) TiF63- (treated by two
(5,9) 2- 3- (4)
different groups of investigators),(59 VF6 and VF6 ,

pi bonding in sulfones(16) and the hexaamines of Cr3+, Co2+

Co and Ni2+,(7) have been treated using the methods of

Ballhausen and Gray for estimating the resonance integrals.

However, each of these investigators utilizes a different
procedure for estimating the coulomb integrals.










The methods of estimating the coulomb integrals fall

into two classes. One of these methods is to make an edu-

cated guess based on previous estimates and chemical

intuition as to what the proper values should be. In the

other method, the coulomb integral is equated to the nega-

tive of the valence state ionization potential (defined

below) which is calculated by approximating the valence

state as a linear combination of configurations, and using

the observed spectroscopic energies of these configurations

to determine the valence state ionization potential.

Ohno, Tanabe, and Sasaki(23) have used a slightly

different modification of the Wolfsberg-Helmholz method of

estimating the resonance integrals in their treatment of

the iron-porphyrin system.

In order to test some of the approximate methods

mentioned above as well as to build a foundation upon which

further work on the unusual cobalt(II) complexes may be

accomplished, the following complexes have been investiga-

ted: the hexaaquo complexes of iron(II), iron(III),

cobalt(II), cobalt(III), and nickel(II) and the hexaamine

complexes o' cobalt(II), cobalt(III), and nickel(II).

Metal wave functions corresponding to atomic wave functions

of several different integral charges q have been used for

each complex. In all cases the resonance integral has been

estimated by two different methods and the eigenvectors










from each of these methods have been analyzed by two

different methods of population analysis. For the aquo

complexes pi bonding has in turn been included and omitted.

The [Co(H20)6]3+ ion has been treated by the above pro-

cedures and in addition calculations using a slightly dif-

ferent method of estimating the coulomb integral for the

donor atom and calculations at a different internuclear

distance have been made.

The procedure followed is similar to that of Ball-

hausen and Gray(2) but, for the sake of clarity, the entire

procedure is outlined below.










PROCEDURE


Symmetry and wave functions

The symmetry of the ammine and of the aquo complexes
in which pi bonding is neglected is Oh (neglecting hydro-
gens). When pi bonding is included in the aquo complexes,
an assumption must be made about the orientation of the
hydrogen atoms. The assumption will be made that the
symmetry is Th, that is, that water molecules have their

C2v axes collinear with the metal-oxygen bond axis, and the
hydrogen atoms of waters in trans positions lie in a
molecular reflection plane. The donor pi orbitals are
perpendicular to this plane. For simplicity the symmetry
orbitals are classified according to the group Oh throughout.
The internuclear distances used for the compounds
2+_O o
studied are as follows: Fe -(H20) 2.17 A as observed in
Fe (P04)2 4H20,(15) Fe-O(H20) 2.08 A as observed in

(NH4)2[FeC15 20],(34) Co2+-O(H2O) 2.08 A as observed in
[Co(C2H302)2(H20)],() Co -O(H20) 1.98 A (estimated),
Ni2-O(H20) 2.08 A as observed in [Ni(C2H302)2(H20)41]
Co -N(NH ) 2.10 A (estimated), Co (-N ) 2.00 A as ob-
served in K[Co(NH )2(N02)4].5(
The observed distances are from crystal x-ray data.
For the other two values it was estimated that the covalent
7









O
radius of Co(II) is 0.10 A greater than that of Co(III) in

agreement with Pauling's table of octahedral radii.(25)
Further, the Cr-0 internuclear distance in [Cr(H20)6]3+ is

1.97 A~12) and since Co should have a slightly larger
effective radius than Cr the estimate of 1.98 A for the

Co+-0(H20) internuclear distance is reasonable.

A minimal set of atomic orbitals has been used

throughout, viz. 3d, 4s, and 4p for cobalt, 2s and 2p for

the donor.

The metal atomic orbitals are the double zeta functions

of Richardson et al.(28,29) For the donor atoms, the double

zeta functions of Clementi(6) have been used. Richardson

does not give a 4p wave function for Co+ or Fe3 These

have been estimated by graphical extrapolation of the 4p

wave functions for the neutral, singly ionized, and doubly
(2q)
ionized cobalt and iron wave functions of Richardson et al 29

and then normalized to unity.*

Hybridized orbitals have been used for the donor

atoms. The amount of hybridization has been estimated to be


The form of these wave functions is:

R p(Fe) 0.10308X(10.60) 0.36541X(4.17)

+ 1.05510o(1.80)

R4p(Co) 0.10098X(11.05) 0.35555X(4.585)

+ 1.05194X(1.86) .









the same as that in the lone pair orbitals of the free

water wave functions of McWeeny and Ohno(18) and the free

ammonia wave function of Tsuchida and Ohno.(35) These lone

pair orbitals are linear combinations of Slater type wave

functions and include hydrogen Is components as well as

nitrogen or oxygen Is components. It will be assumed that

the ratio of coefficients of the 2s and 2p Slater type

orbitals from these lone pair orbitals can be carried over

to the present calculations and can be used with the double

zeta wave functions of Clementi to form the donor hybrids.

In the oxygen case it was necessary to first form linear

combinations of sigma and of pi symmetry. The resulting

hydrids are

X(N(r) = 0.667435;(2 s) + 0.744667X(2pZ)

X(o ,) 0.836458X(2 s) + 0.548032X(2pz)

where theX's are double zeta atomic wave functions. The

pi symmetry orbital is an atomic 2p orbital:

X(om) = X(2px)
When calculations were made that neglected pi bonding, this

pi symmetry orbital was considered to be localized on the

oxygen atom, that is, non-bonding. In none of the calcula-

tions were any of the hydrogen orbitals included.









Coulomb integrals

The coulomb integrals were estimated by equating them

to the negative of the valence state ionization potentials

(VSIP). The valence state is defined by Moffitt(19) and

extended by Ballhausen and Gray(2) as that hypothetical

state of each atom in a molecular aggregate which has been

separated to an infinite distance, while leaving the orbi-

tals with the same hybridization and electronic population

as was found in the molecular aggregate before the atoms

were separated. The separated atoms may be assigned

fractional charges.

The valence state can be written as a linear combi-

nation of all the electronic states (including the continuum)

possible for an atom or ion. The present calculations

utilize the minimal set of atomic orbitals; therefore we

approximate the valence state by a few observed configura-

tions which involve the minimal set of orbitals.

The valence state energy is calculated from the aver-

age energy of a configuration,(1) which is the weighted

mean of the energies of the terms arising from the configu-

ration, relative to the ground state of the atom or ion in

question. The weighting factor has been taken as equal to

the total degeneracy (spin and orbital) of the term, provided

the J components (J is the orbital angular momentum quantum

number) are not too widely separated. If the J components









are widely separated, the average is taken over the energies

of the J components, weighted by their degeneracies.

For the donor atoms (either nitrogen or oxygen) the

minimal set of observed configurations is comprised of

2s22pn and 2s2pn+l configurations where n = 5 for nitrogen

and n = 4 for oxygen. Another configuration, pn+2, is

possible but has been observed in only one case(20) and is

of very high energy and is therefore not included. A

typical plot of these data is shown in Figure 1.

The linear combination of these configurations is

made so that the electron distribution on the donor atom is

reproduced. The electron distribution on the donor atom is

determined from a population analysis of the ligand wave

functions by the Mulliken(21) method. It would have been

possible to analyze them by the Smith method (see below) for

comparison, but the difference in the calculated VSIPs would

be small.

If pi bonding is neglected, the linear combination

for the donor atom (D) is:

2 2
s(2-Q)C + R p(-Q)Cp + R = a(s2pn-Q)

+ b(spn+l-Q) (2)

where a and b are the coefficients of the linear combination,

Q is the charge on each donor atom, Cs and C are determined

by 2(D, ) = Cs (2s) + Cp~(2p), kR and R are the s and p
























/ A




A 2s22p4-Q
B 2s2p5-Q


1
Initial Charge


Fig. 1.-VSIP of 2s electron on oxygen.









populations respectively, in the orbitals on the donor atom

which are bonded to hydrogens.

The left side of this equation is the valence state

configuration of the donor atom. All positive charge (Q)

on the donor is assumed to arise from the removal of elec-

tron density from the donating sigma orbital only. In the

free donor molecule there are two electrons in this orbital

and (2-Q) electrons when the atom is in its valence state.
2 2
Cs and C2 are, respectively, the probabilities that an

electron in this orbital is an s or a p electron. Rs and R

are determined from the population analysis (see above) in

the same manner.

The right side of eq. (2) is the linear combination

of configurations which approximates the valence state.

The electronic configurations in the parenthesis reflect

the manner in which the configurations vary along the curves

in Figure 1.

Other lines could have been drawn to connect the

points in Figure 1 and in that case, other configurations

would be used to form the linear combination. However,

these are the best lines to use since other lines would

lead to the hypothetical ionization of a fraction of an

electron. This is the case in the investigations of Bedon

et al.,('5) where the lines were extrapolated into regions

of the graphs such that there was only a fraction of an s

electron remaining in the configuration to be ionized.









The coefficients (a and b) are then determined by

equating the s and p populations on both sides of eq. (2).

The VSIP energies for the removal of an s electron from each

configuration are then substituted into the equation

V(s) = aVa + bVb (3)

where V(s) is the valence state ionization potential of an

s electron,'Va is the VSIP of an s electron in the configu-

ration s pn-Q, and Vb is the VSIP of an s electron in the

configuration spn+l-Q. A similar relation holds for V(p).

The VSIP of an electron in the orbital D.- is the

sum of the probabilities of that electron being an s or p

electron times the respective ionization potential of an s

or a p electron in the valence state. Thus

H = -VSIP = -C V(s) C V(p)
D s p
An alternate procedure used to test the sensitivity

of the self-consistent parameters to the estimate of the

coulomb integral for the donor was to remove the same

amount of s and p electron density from each configuration

as was removed from the donating orbital. Thus eq. (2)

becomes
s(2-Q)C2 + R p(2-Q)C2 + R 2-QC2 n-QC 2

+ b(sl-QCs2 n+l-QC 2) (5)

The coefficients (a and b) are then determined as above.

These coefficients were then used in equation (3) to de-

termine the valence state ionization potential of an s









electron and in the similar equation for V(p). It is this

step which makes these estimates inaccurate. However, this

procedure consistently raises the coulomb integral of the

donor atom by approximately 2000 cm-1 (0.25 eV) so that

some information may be obtained from the results of this

method.

Analogous procedures were used to calculate the VSIP

of an electron from the pi symmetry orbital.

For the metal VSIP's the above method is not suf-

ficient. There are at least six configurations(20) arising

from the minimal set of basis orbitals which contribute to

the valence state for each ion of integral charge, and

there are only four conditions, the d, s, and p populations

and the normalization condition. When an attempt is made to

interpolate for fractional charge as was done for the donor,

the number of unknowns doubles without any increase in the

number of conditions.

As a substitute for this approximation, the procedure

of Ballhausen and Gray(2) has been used for estimating the

VSIP's. In this procedure the various configurations are

arranged in groups of three for each type of electron, that

is, d, s, or p. The groupings are made so that the three

configurations which are most important to the energy of

the state from which a d, s, or p electron is removed, are

included. Curves similar to Figure 1 have been drawn so










that the charge adjustment of the configurations is made by

removing electron density from the d population only. Thus,

for example, the final form for the linear combination which

was used for calculating the valence state ionization po-

tential for a d electron of cobalt is

a(d9-z) + b(d8-zs) + c(d8-Zp) = dsp( (6)

where a, b, and c are the coefficients of the linear combi-

nations, Z is the charge on the cobalt, a, 3, andy are the

occupation numbers of the d, s, and p orbitals, respectively.

In Appendix I are the generalized solutions for the

coefficients of all the linear combinations as well as the

equations that were fitted to the VSIP data of Basch and

Viste.(

Overlap integrals

The group overlap integrals were calculated from two

center overlap integrals by standard group theoretical pro-

cedures.(1) The two center atomic overlap integrals, as

well as the atomic dipole moment integrals (see below) were

evaluated on the IBM 709 at the University of Florida

Computer Center using a program written by Corbato and

Switendick.(27 In Table 1 are tabulated all of the group

-vcrlap integrals. Overlap of donor orbitals on different

sites is assumed to be zero.









TABLE 1

GROUP OVERLAP INTEGRALS


Overlap of
Metal Ligand Dis- Symmetry


Atom Atom tance


Fe1+

Fe2+

Fe3+

Fe1+

Fe2+

Fe3+

Co0+

Col+

Co2+

Col+

Co22+

Co3+
Co3+



Nil+

Ni2+

Ni3+

Col+

Co2+

Co2+

Co 3+

Nil+

Ni-+


2.17

2.17

2.17

2.08

2.08

2.08

2.08

2.08

2.08

1.98

1.98

1.98

1.93

2.08

2.08

2.08

2.10

2.10

2.00

2.00

2.06


alg

0.68363

0.68363

0.68363

0.72561

0.72561

0.72561

0.72471

0.72471

0.72471

0.77367

0.77367

0.77367

0.79692

0.48062

0.48062

0.48062

0.83995

0.83995

0.88513

0.88513

0.86331


0.20188

0.15142

0.11250

0.23009

0.17558

0.13250

0.26142

0.20030

0.15176

0.23185

0.17897

0.13739

0.15055

0.23243

0.17466

0.13221

0.22588

0.17483

0.20056

0.15752

0.20968'


0.58137

0.55751

0.49819

0.59928

0.59437

0.54382

0.40862

0.60560

0.58522

0.62496

0.62643

0.57811

0.60360

0.43098

0.60825

0.57218

0.63824

0.64674

0.67824

0.64472

0.65132


N 2.06 0.86331 0.16308 0.65315


eg tlu tu lu()


0.19405

0.16358

0.13531

0.21305

0.18529

0.15661

0.18628

0.21001

0.17892

0.23339

0.20565

0.17499

0.18948

0.19221

0.20693

0.17117


_


t2g

0.06801

0.04911

0.03540

0.08153

0.05974

0.04364

0.09489

0.06952

0.05079

0.08533

0.06333

0.04726

0.05305

0.08142

0.05950

0.04360









Resonance integral

The approximation of Wolfsberg and Helmholz() and

the modification of this approximation due to Ballhausen

and Gray (2 have been used to calculate the resonance

integrals. The Wolfsberg and Helmholz approximation is

Hij = FGijii + Hjj)/2 (7)

and the Ballhausen and Gray modification is

Hij = -FG i(H.ii 1/2 (8)

where F is an empirical proportionality constant.

Ballhausen and Gray state(2) that the geometric mean

is preferable since the resonance energy is expected to

decrease rapidly as the difference in the coulomb energies

becomes greater. However, in at least one case reported in

the literature(17) in which this approximation was used, it

was necessary to artificially assume a very small charge

dependence for the coulomb integral in order to obtain self-

consistency of the assumed and calculated charge densities.

A charge dependence equal to the ground state ionization

potential would have led to divergence.

Ohno, Tanabe and Sasaki(23) have also modified the

approximation of Wolfsberg and Helmholz to the form

H. + H.. H + H-
Hij = Gj 2 i + i 2 (9)

This modification consists of separating the electrostatic









part, iH. and H from H.. and Hj. This is done in order
jj Jii j j
to fulfill the requirement that the resulting orbital energy

differences should not change, if the zero point of energy

is changed by adding a constant potential to the Hamiltonian.

For these calculations the electrostatic part is zero.

Wolfsberg and Helmholz( originally used an F of

1.67 for sigma orbitals and 2.00 for pi orbitals. Ball-

hausen and Gray(2) have used F = 2.00 for both sigma and pi

orbitals. Since there is no fundamental reason for expecting

that there should be different values for sigma and pi

orbitals except that values near these gave satisfactory

results in energy level calculations of homonuclear diatomic

molecules of first and second row elements, the value of

2.00 seems reasonable and has been used throughout these

calculations. It is obvious that as F is decreased, lODq

will be decreased. Cotton and Haas(7) have varied F in

their calculations and they concluded that it was necessary

to use different F's for different complexes, in order to

obtain agreement with the experimental lODq values. It

seems more reasonable to adopt one value of F for all

complexes, as has been done here.

PoDulation analysis

The methods of population analysis of Mulliken(21)

and of Smith(32) have been used to estimate the electron

distribution in the self-consistent complexes.









In the Mulliken method the electron density of the
molecule is divided into atomic contributions and the co-
efficient of the square of an atomic orbital is identified
with the population of that orbital. Thus, the contribution
of i = kXkbki to the total electron density arising from

ij = L0.0. is

/i = N(j)( i0iCij)2 (10)

where N(j) is the number of electrons in j .
The Mulliken approximation is that the cross product
term should be equally divided between the two centers.
Thus
(ij sijii2 +j2) (11)


The Smith method of population analysis is a modifi-
cation of this procedure in which the cross product term is
distributed so that the total charge and dipole moment of
that term are preserved and, further, so that the atomic
dipoles on the donor atom are also included. Thus

iXj == Xi i2 + jX 2 (12)

Integrating (12) over all space gives

i + j = Sij (13)

Multiplying (12) by the dipole moment operator Z (measured
from the donor to the metal atom) gives









Xi2ab 2/X2s ZX2d + R = af/XZXZ dr

+ b/j ZX2pd (14)

where Xj is the metal wave function and X. = aX2s + b 2p

and R is the internuclear distance. To account for the
atomic dipoles on the donor, it is assumed that

?(2sX2Kp = 2 + xmxj2 (15)
This is integrated over all space and then multiplied by
the dipole moment operator Z and integrated to give

m = (2ab/R) X2sZ X2pd and X = 1 Xm (16)

The 2p orbital was used for the donor wave function since
the product Xp2s 2p has the appearance of a p wave function.
The donor portion of the cross product term could have been
distributed between the 2s and 2p orbitals; however, no
attempt was made to do this because the electron density
was removed from the 2s and 2p orbitals and the ratio s/p
was kept constant. The same procedure was followed for
analyzing the pi symmetry terms. It must be noted that this
derivation involves only two center orbitals whereas the
orbitals in this investigation are group orbitals. This,
however, is easily handled as the various integrals may
readily be transformed (by group theoretical procedures)(1)
into group integrals either before or after the various
X's have been determined. After having done this, approxi-
mations (12) and (14) are substituted into eq. (10), and
the same procedure as for the Mulliken method is followed.









The procedure for transforming the dipole moment

integrals into two center overlap integrals is found in

Appendix II. In Table 2 the numerical values of the vari-

ous weighting factors (%'s) are tabulated. (The figures

in the column headed hybrid are the k's of eq. (16).)

Self-consistency procedure

Since the coulomb and hence the resonance integrals

are dependent upon the electron population and hence the

charges, it is necessary to make an iterative calculation.

Populations of the d, s, and p orbitals on the metal are

assumed, and from these the changes are calculated. The

populations and charges determine the coulomb integrals and,

consequently, the resonance integrals. This information

together with the group overlap integrals is then substi-

tuted into the secular determinant (eq. 1) and the eigen-

values and eigenvectors calculated. Subsequently, the

eigenvectors are analyzed and the populations of the d, s,

and p orbitals calculated. If the assumed and calculated

populations differ by more than a predetermined amount

(maximum was 0.001), new populations are assumed, and the

entire procedure is repeated.

A computer program was written by the author to carry

out this procedure. Eight forms of this program were used

during this investigation. A copy of one of these programs

is in Appendix III. The modifications that are necessary









TABLE 2

WEIGHTING COEFFICIENTS FOR THE SMITH POPULATION ANALYSIS


Metal Atom Dist.
Donor Atom (AO) alg eg tlu tiu(I) t2o Hybrid

Fel+ 0.04765 0.02716 0.03857 0.03085 0.01853 0.14952
0 0.23144 0.07376 0.37252 0.10636 0.02957 0.85048

Fe2+ 0.04765 0.02329 0.05522 0.03405 0.01479 0.14952
2.17
0 0.23144 0.05242 0.33900 0.08162 0.01993 0.85048

Fe317 0.04765 0.01948 0.06225 0.03304 0.01166 0.14952
0 0.23144 0.03677 0.29003 0.06264 0.01338 0.85048

Fel+ 2.08 0.04923 0.02897 0.03534 0.03365 0.02175 0.15599
0 0.24699 0.08607 0.38841 0.11700 0.03590 0.84401

Fe2+ 2.08 0.04923 0.02181 0.05389 0.03826 0.01762 0.15599
0 0.24699 0.06598 0.36640 0.09276 0.02462 0.84401

Fe3+ 0.04923 0.02165 0.06289 0.03786 0.01407 0.15599
2.08
0 0.24699 0.04460 0.32165 0.07288 0.01679 0.84401

CoO 2.08 0.05229 0.03036 0.00945 0.01600 0.02390 0.15599
0 0.24357 0.10035 0.27949 0.11572 0.04320 0.84401

Co 208 0.05229 0.02696 0.03931 0.03502 0.01948 0.15599
2.08
0 0.24357 0.07318 0.38891 0.11348 0.02968 0.84401

Co2+ 2.08 0.05229 0.02325 0.05697 0.03851 0.01564 0.15599
0 0.24357 0.05263 0.35684 0.08800 0.02028 0.84401

Col+ 0.05413 0.02869 0.03460 0.03862 0.02330 0.16387
0 1.98 0.26172 0.08723 0.40731 0.12642 0.03704 0.83613

Co2' 1 0.05413 0.02537 0.05430 0.04385 0.01900 0.16387
1.98
0 0.26172 0.06412 0.38866 0.10157 0.02578 0.83613

Co3+ 0.05413 0.02203 0.06479 0.04358 0.01541 0.16387
0 0.2617. 0.04693 0.34400 0.08015 0.01801 0.83613

Co3+ 0.05483 0.02308 0.06374 0.04690 0.01706 0.16811
0 1.93 0.27051 0.05220 0.36306 0.08708 0.02045 0.83189

Nil+ 0.07322 0.02837 0.01103 0.01762 0.02162 0.15599
0 2.08 0.12299 0.08785 0.29371 0.11829 0.03596 0.84401









TABLE 2 Continued


Metal Atom Dist.
Donor Atom (AO) alg egt t tu lu (t) t2g Hybrid


0.07322 0.02501 0.04245 0.03596 0.01745 0.15599
0.12299 0.06232 0.38765 0.11036 0.02462 0.84401

0.07322 0.02317 0.06007 0.03857 0.01396 0.15599
0.12299 0.04473 0.34453 0.08246 0.01686 0.84401


Ni2+
0

Ni3+
0

Col+
N

Co2+
N

Co2+
N

Co3+
N

Nil+
N

Ni2+
N


--- 0.19458
--- 0.80452

--- 0.19458
--- 0.80452

--- 0.20431
--- 0.79569

--- 0.20431
--- 0.79569

--- 0.19836
-- 0.80164

--- 0.19836
--- 0.80164


0.08241 0.02966 0.05964 ---
0.26050 0.08328 0.39167 ---

0.08241 0.02592 0.07386 ---
2.10 0.26050 0.06149 0.38346 ---

0.08669 0.02693 0.06976 ---
2.00 0.27465 0.07335 0.40982

0.08669 0.02374 0.07858
S 0.27465 0.05502 0.37730 ---

S 0.08762 0.02804 0.06067 ---
0.26482 0.07680 0.39988 ---

0.08762 0.02452 0.07577 ---
0.26482 0.05702 0.38608 ---









for different metal or donor atoms are indicated. The

program incorporates a subroutine (JACFUL) written by F.

Prosser(26) to diagonalize the secular determinant. It is

to be noted that after one cycle, the new assumed metal

orbital populations (if necessary) are the average of the

previously assumed (input) and the calculated (output) metal

populations. For successive iterations the input and output

populations are compared. If the difference is less than

0.001 then nothing is done. If the difference between the

input and output is larger than the difference of the input

and output from the previous iteration then the new input

is the average of the input from the previous iteration and

the present iteration. If the new difference is less than

the old difference but greater than 0.001, then the new

input is the average of the present input and output metal

orbital populations. This procedure gave rapid convergence

in most instances and was more satisfactory than a straight

cyclic procedure in which the output populations of one

cycle are used as input for the next cycle. The straight

cyclic procedure was slower to converge and occasionally

gave large oscillations which made it difficult or impossible

to reach self-consistency. The few cases where the procedure

described above failed to give self-consistency are indi-

cated (by a's) in Table 3 and discussed below.












RESULTS AND DISCUSSION


The calculated values for 1ODq, which is the

difference in energies of the t2g (or t2g when pi bonding

is omitted) and the e* molecular orbital levels, are sum-
g
marized for the eight different methods indicated by the

column headings in Table 3. The complete self-consistent

results are listed in Appendix IV in Table 9 to Table 17.

An inspection of Table 3 reveals that certain

methods did not give satisfactory results (indicated by

a's and b's). These are the Ballhausen-Gray-Smith method

which leads to divergence and, when pi bonding is included,

both methods which use the Mulliken population analysis.

These four methods must be rejected for reasons given below.

One of the reasons that the Ballhausen-Gray-Smith

method sometimes diverges lies in the method of estimating

the resonance integral. Fenske(9) has pointed out that in

using the Ballhausen-Gray approximation if the absolute

value of one of the coulomb integrals is much greater than

the other, the energy of the antibonding orbital is drasti-

cally affected while the energy of the bonding orbital is

of essentially the same energy as the energy of the lowest

coulomb integral. This causes the eigenvector of the anti-

bonding orbital to have a large percentage metal character,
26











000
0 CD 0






000
cooo

cN -4
No n y


0


0
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C4


000
<3- <
Co r^
Co v cr


000

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\D C 0

000
000
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00


















000
r-.
cro

















O O
AOC4







000
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-4-4





000















0
c0 c'J
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rC) C)
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on
000



















oo

10
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000
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00
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co -4
Or".


000

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mee r
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000
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co o
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COc .-I
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000
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000
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000
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0
0 It n
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u3)
Lr)


000
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000
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'T- r-co

C4 ni-4


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00
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coco
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00


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00






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-4 -4


- C CM4 CMJ


+J

c'4

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Z


0
*
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-4 4-i ca

+ + + +
CMn4 m n mn

0 0 0 0




0 0 0 0
u 0 0 0
0 ^ ^


0
o


-4



O 0



o 0
1-1






0


co ,



CO
O 0

00
a) co
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0
0 o
u o


o C)
00




0 o
4N 0

r0 4


a) u *



0
C




'4- 4-J
- a

aa
o C




0
3 -4


--
CO a


.0C ) '*0
cf Z 0.. 4J

oC Q I

0 c *. 4
a- u
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-4 *H 0 'o 4.
CO314 Q
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>d 0)
.. .. r

~ C'~ CO .


E4-



CO





x
C)
-4



0-


0
U









and as a consequence, the metal has a high electron density.

The Smith population analysis heightens this effect (by

giving more electron density to the metal than does the

Iulliken analysis) so that when these antibonding orbitals

are occupied (as in iron(II), cobalt(II), and nickel(II)),

there is such a high electron density on the metal that its

coulomb integral becomes positive before self-consistency is

achieved, and the calculations, therefore, diverge.

When pi bonding is included and the Mulliken method

of population analysis is used the results often do not

correspond to physical reality. This is because a scramb-

ling of the molecular orbital energy levels has occurred

and the highest and next highest energy levels which are

occupied are respectively a t2* and a t.* molecular orbi-

tals rather than an e* and a t This means that the
g 2g
(predicted) lowest energy transition is permitted whereas

the observed transition is of such low intensity that it

is taken to be a forbidden transition.

The reason for this behavior may be understood by

inspecting the Mulliken population analysis. In this

method the majority of the electron density in the bond is
2 2
given to the donor atom since CD> (c.f. eq. (10)).

If we start with a neutral metal atom and the charge of the

complex distributed over the donor atoms, then, after a few

iterations, the use of the Mulliken analysis gives an









appreciable positive charge to the metal causing the coulomb

integral to be low, and at the same time the decreased posi-

tive charge on the donor atom causes its coulomb integral to

be high compared with the initial values. Often in the

present calculations, the two coulomb integrals became

essentially equal and the scrambling described above occurred.

There are four methods remaining from which a best

method should be chosen, preferably one which is applicable

to all types of complexes. Three of these methods exclude

pi bonding while only one method includes pi bonding. From

an inspection of the total energies (sum of one-electron

energies) of the complexes as calculated by the various

methods (Appendix IV), it is apparent that the inclusion of

pi bonding lowers the total energy of the aquo complexes by

approximately 2000 kK (250 eV) for the iron complexes, 1200

IK (150 eV) for the cobalt complexes, and 400 kK(50 eV) for

the nickel complex. This means that there is an appreciable

mixing of the pi symmetry orbital from the ligand with the

metal orbitals, and hence, pi bonding should be included.

Thus a best method, of those which were tried, has been

found -- the Wolfsberg and Helmholz approximation for the

resonance integral coupled with the Smith method of popula-

tion analysis. It is necessary to add at this point that

although the total energy is a minimum when self-consistency

is achieved, a particular iteration may result in a lower







31

total energy than that obtained from one or more succeeding

iterations. It may also be noted that the Wolfsberg-Helm-
holz-Smith method does not give the lowest total energy of

those methods which include pi bonding. However, these
other methods have been eliminated as a result of previous

considerations.

It is now necessary to compare the calculated values

for lODq with those which have been experimentally determined.

The experimental values are(11) [Fe(H20)6]2+ 10,400 cm-1

[Fe(H20)6]3+ 13,700 cm-1, [Co(H20)6 2+ 9,700 cm-1

[Co(H20)6]3+ 19,100 cm-1, [Ni(H20)6 2+ 8,500 cm-1

[Co(NH3)6]2+ 10,500 cm-1, [Co(NH3)6]3+ 23,500 cm-1, and

[Ni(NH)6]2+ 10,800 cm-1

From an inspection of Table 3 it is apparent that

almost any value of 1ODq may be obtained depending on the

choice of the charge (q) of the free ion metal wave function.

The next question is, "Is it possible to chose q empirically

so that starting from this one piece of information, for

each metal, it is possible to determine all of its experi-

mental 1ODq values?" In order to do this it is necessary
to refer to Figures 2 through 5 where log lODq is plotted

as a function of the log of ro, the radius of maximum
probability of the 3d orbitals. These radii of maximum
probability are tabulated in Table 4. They were obtained































0











1.183 1.520 1.475

log r

Fig. 2.-Log lODq vs log r0 for the iron-aquo complex-
e2
es. Odd numbers are Fe2+, even numbers are
Fe3+1; + 2 W S Pi included, 3 + 4 W S No Pi,
5 + 6 W M No Pi, 7 + 8 B M No Pi.


















50-




r4
's 3









log ro
2numers are Co0-

0

oNo Pi
0

2

10-


8-

1.127 1.257 1.400 1.565
log r0

Fig. 3.-Log 10Dq vs log ro for the cobalt-aquo
complexes. Odd numbers are Co 2+, even
numbers are Co3+ ;l + 2 W S Pi, 3 + 4
W S No Pi, 5 + 6 W M No Pi, 7 + 8 B M
No Pi.



















40-








0
rl




0

t0
o
10-




7-


1'.200 1.535 1:485

Fig. 4.-Log 1ODq vs log ro for [Ni(H20)62).
1 W S Pi, 2 W S No Pi, 3 W M No Pi, 4 B M
No Pi.


























H6








O






1.127 1.200 1.257 1.335 1.400
log ro
02








Fig. 5.-Log 1ODq vs log ro for the cobalt and nickel
ammine complexes. 1,2,3 Co2+; 4,5,6 Co3+ ;
7,8,9 Ni2+; 1,4,7 W S; 2,5,8 W M; 3,6,9 B M.









by plotting r2[R(r)]2 as a function of r (where r is the

radius of the wave function and R (r) is the radial portion
of the 3d wave function) and visually picking the maximum of

r2 [R(r)2.

TABLE 4

RADII OF MAXIMUM PROBABILITY (r ) FOR IRON, COBALT
AND NICKEL (A)

Metal Atom q = 0 1 2 3

Fe 1.475 1.320 1.183

Co 1.565 1.400 1.257 1.127

Ni 1.485 1.335 1.200 -

It was empirically determined that a plot of log lODq

vs log r gives a good straight line for all of the results,

and it is of interest to note that all of these lines have

virtually the same slope.
Since it has been determined that the Wolfsberg and

Helmholz resonance integral approximation coupled with the

Smith population analysis (pi bonding included) gives the

best results in terms of its ability to handle pi bonding,
this method will be used in an attempt to extrapolate from
a single starting point for each metal atom, all of the

observed lODq values. Let the double charged aquo complexes
be used as this starting point. It is observed (Appendix










IV) that although there is a gradual increase in self-

consistent charges as q increases (that is, as the size of

the metal wave function decreases), there is a large dif-

ference between the self-consistent charges Z on the metal

atoms of the doubly charged and triply charged complexes

regardless of which q is used. If it is assumed that the

self-consistent charge Z is independent of q and an average

is taken over Z for the three values of q, then it is

possible to determine how much the wave functions of the

complexes have shrunk in loosing one electron, that is,

the increase in q on going from a doubly to a triply charged

complex. Further, it is assumed that the radius of maximum

probability which gives the experimental 1ODq values for the

doubly charged aquo complexes is equal to the radius r0 of

the metal atom in this complex and more important corres-

ponds to Z, the average of the calculated self-consistent

charges. These charges Z are all approximately zero while

the radius of the free metal ion 3d orbital which gives the

experimental lODq corresponds to a charge q of approximately

two for the free atom wave function. Thus the free atom

wave functions shrink when the complex is formed.

The value of q for the self-consistent complexes is

assumed to be larger or smaller than qo, the value of q

for the doubly charged aquo complexes, by the difference

(AZ) in the average of the calculated self-consistent charges










of the two complexes (that is, q qg + AZ). When this

charge adjustment is made, the lODq values listed in Table

5 are predicted.

TABLE 5

PREDICTED 10Dq VALUES USING THE DOUBLY CHARGED
AQUO COMPLEXES AS STANDARDS

Complex Pi No Pi
(Experimental 10Dq) WS WS WM B M

[Fe (H20) 3]2 Z 0.06 0.09 0.75 1.02
standard 10Dq 10,400 10,400 10,400 10,400

[Fe(H2)6]3+ Z 0.60 0.74 1.09 1.27
(13,700 cm-1) 10Dq 14,200 16,700 20,000 21,000


[Co(H20)6]2+ Z -0.04 -0.01 0.70 0.96
standard 10Dq 9,700 9,700 9,700 9,700

[Co(H20)6]2+ Z 0.59 0.63 0.93 1.14
(19,100 cm-1) 10Dq 13,600 16,400 18,700 18,000

[Co(NH3)6] 2+ Z a -0.45 0.38 0.61
(10,500 cm-1) 10Dq 11,500 11,000 13,900 13,400

[Co(NH3)6]3+ Z a 0.20 0.58 0.76
(23,500 cm-1) 10Dq 20,500 19,000 23,400 25,000


[Ni(H20)6]2+ Z -0.04 -0.02 0.76 1.00
standard 10Dq 8,500 8,500 8,500 8,500

[Ni(NH3)6]2+ Z a -0.49 0.29 0.41
(10,800 cm-1) 10Dq 11,800 10,800 13,900 13,500


aUsing the self-consistent charge of W S no pi.










Several observations may be made on the results

presented in this table. The most apparent of these is

that the so-called best method selected above does not

extrapolate with good agreement to the experimental lODq

values of cobalt(III). However, reasonable agreement is

obtained for the other complexes (within 10%). It is also

apparent that it is possible to extrapolate with almost as

good over-all agreement using the methods which do not in-

clude pi bonding but are acceptable except for the total

energy arguments. An observation on the self-consistent

charges is that in going from the aquo to the ammine com-

plexes the metal wave functions shrink and, in the extrapo-

lating procedure, produce a larger lODq value for the

ammine complexes than for the aquo complexes in agreement

with experiment. This is also in qualitative agreement

with arguments based upon the cloud expanding power of the

nephelauxetic series.

However, several other factors become apparent upon

closer scrutiny. With the Wolfsberg-Helmholz-Smith method

it is possible to extrapolate to acceptable estimates of the

experimental lODq values so long as the spin multiplicity

does not decrease. From iron(II) to iron(III) the extrapo-

lation is from quintet to sextet spin multiplicity but the

difficulty arises in going from high-spin cobalt(II) to

low-spin cobalt(III). Unfortunately, the extrapolated








40

estimate is lower than the experimental value and any attempt

to correct for this change in spin multiplicity will decrease

the predicted 10Dq values. The correction which could be

applied is to subtract the pairing energy of cobalt(III)

(21,000 cm1 for the free ion) from the predicted 10Dq.

Another observation that can be made is that the

Wolfsberg-Helmholz-Mulliken method with pi bonding omitted

does extrapolate with excellent agreement from high-spin to

low-spin cobalt complex but does not give very good agreement

when the spin multiplicity remains constant. However, no

significance was attached to this observation since it is

desired to have a single method for the treatment of all

complexes. A correction factor could conceivably be

applied to bring all of these predicted 10Dq values into

agreement with the experimental values but a different

correction factor could, with as much justification, be

applied to the Wolfsberg-Helmholz-Smith method and the total

energy arguments above strongly suggest that pi bonding

should be included.

It must be added at this point that if the doubly

charged ammines of cobalt or nickel had been chosen as the

reference compounds, the extrapolation would not have been

as good as the above results. It may be that the cause of

this is that it is necessary to start with the weaker ligand

field and extrapolate to the stronger field strength. Future

work with other ligands should clarify this point.










In the procedure section it was stated that an

alternant procedure for estimating the coulomb integral of

the donor atom was used in order to determine the sensitivity

of the experimental parameters to the magnitude of this

estimate. This alternant procedure raises the coulomb

integral of the donor by approximately 2000 cm1 (0.25 eV)

but, as is apparent from Table 3, has little effect on lODq.

(The change is less than 200 cm-1.) Further, the self-

consistent charges, the total energies, and the metal

orbital populations are essentially the same for both pro-

cedures (c.f. Table 12 and Table 17 of Appendix IV).

However, under certain conditions the estimate for

the coulomb integral does become important. This is evi-

denced by a comparison of the present results with those of

Cotton and Haas,(7) who used the Ballhausen-Gray-Mulliken

scheme, but with coulomb integrals much smaller than those

of this investigation. For example, for [Co(N3)6 2+ Cotton

and Haas find lODq 10,910 cm-1 using a d coulomb integral

of-63,250 cm1. Our estimate for the coulomb integral is

-154,810 cm-1, which gives 10Dq = 32,670 cm-1, but we cannot
reproduce their coulomb integrals.

As a test of the sensitivity of the experimental
parameters to the internuclear distance, the [Co(H20)6J3+

ion was treated (with the alternant donor atom coulomb
integral estimate) using an internuclear distance f .93
integral estimate) using an internuclear distance of 1.95 A.









O
This is 0.05 A less than the distance arrived at in the

first section of the procedure. However, this difference

should be much greater than the error in the first estimate.

Table 3 shows that, although the lODq values are raised by

approximately 2000 cm-1, the shortened internuclear dis-

tance does not affect any of the conclusions concerning the

various methods. With reference to the extrapolation of

Table 5 it is apparent that the increase in lODq caused by

the decreased distance is in the right direction and would

improve the extrapolation for the Wolfsberg-Helmholz-Smith

method but is not a sufficient shift to cause the extrapo-

lation to give agreement within 10 per cent.

To this point, the 10Dq results have all been ob-

tained by using Koopmans' theorem, that is, the observed

transitions are said to be equal to the difference in the

energies of the one electron molecular orbitals. If instead

of this procedure, the total energies of the self-consistent

ground and first excited states are calculated, then the

difference in these two energies should be equal to the

first observed transition (lODq').

In Table 6 are listed lODq' values that were calcu-

lated in this manner. All calculations use the free atom

doubly charged wave functions for cobalt. The first observed

transition for [Co(H20)62] is the Tlg 4 T1g(F) transition









TABLE 6

lODq' VALUES FOR THE AQUO COMPLEXES OF COBALT(II)
AND (III) (C-1)


No Pi Pi
Complex W S W M W S W M

[Co(H20)612+ 12,650 11,200 11,500 20,930
[Co(H20)6] 3+ 14,960 9,150 30,510 82,450


and is observed at 8,100 cm-1. For [Co(H20)6]3+ this

transition is the 1Ag g Tlg transition which is observed
1ig
at 16,600 cm-1
If it is assumed that lODq' depends upon the size of
the wave function in the same manner as does 1ODq then it
should be possible to use the first observed transition of

one of the complexes as a standard and extrapolate to the
observed lODq' value of the other complex. If this is
done using the observed lODq' value for [Co(H20)6 2+ as a
standard, the predicted 1ODq' value for [Co(H20)6] is
16,000 cm-1 in good agreement with the experimental value.
This result shows that it is possible to extrapolate an
experimental parameter from high-spin cobalt(II) to low-
spin cobalt(III). It remains for future investigation to
prove, or disprove, the general applicability of this dif-
ferent method.










A modification that could be applied to the Wolfsberg-

Helmholz-Smith procedure in the case of cobalt(III) would be

to subtract the pairing energy of cobalt(III) from the esti-

mate of the metal d orbital coulomb integral. This procedure

would correct for the change in spin multiplicity encountered

in extrapolating from cobalt(II) to cobalt(III). Qualita-

tively this modification will improve the extrapolation

since lODq is increased if the metal orbital coulomb

integral is decreased. However, no calculations have been

made using this modification.












SUMMARY


The hexaamine complexes of cobalt and nickel have

been subjected to four different methods of approximate

molecular orbital calculations and the hexaaquo complexes

of iron, cobalt, and nickel to eight methods. For the

aquo complexes pi symmetry orbitals on the donor atoms have

been treated as bonding and as non-bonding orbitals.

Several different free atom wave functions have been used

for each complex. The effect of changing the estimate for

the donor atom coulomb integral and the effect of changing

the internuclear separation have also been investigated.

The results indicate that the method which uses the

Wolfsberg and Helmholz approximation for the resonance

integral and the Smith method of population analysis is the

preferred method, and further the results indicate that pi

bonding should be included when pi symmetry orbitals are

available. Using this method, it is possible to choose

appropriate metal wave functions semi-empirically and

extrapolate to 10Dq values of other compounds with good

agreement unless the spin multiplicity is decreased. In

such a case a modified procedure is necessary and several

methods have been proposed.




























APPENDICES









APPENDIX I

The generalized linear combinations used for calcu-
lating the valence state ionization potentials (VSIP's) for

the metal atoms and their generalized solutions for the

coefficients are as follows:

For the VSIP of a d electron

a(dn-Z) + b(cn-l-Zs) + c(dn-l-Zp) dasp

a = [a-P(n-l-Z) 2 (n-l-Z)]/(n-Z)

b
c =6

For the VSIP of an s electron

a(dn-l-Zs) + b(dn-2-Zs2) + c(d-2-Zsp) = ds Pp

a = [2a + P(Z+2-n) + (Z+2-n)]/(n-Z)

b = [-c + P(n-l-Z) ]/(n-Z)

c =

For the VSIP of a p electron

a(dn--p) + b(dn-2-Z2) + c(dn-2-Zsp) = dps p
a = [2a + P(Z+2-n) + 6(Z+2-n)]/(n-Z)

b = [-a p + (n-l-Z)]/(n-Z)

c = P

where a, b and c are the coefficients of the linear combi-

nation, a, P and X are the assumed d, s and p populations,
Z = n-a-P- the charge on the metal and n = 8 for iron,

9 for cobalt and 10 for nickel.










The linear combination for the donor atoms and the

generalized solutions are as follows:

v w2 n-Q n+1-
s = a(2s22p-Q) + b(2s2pn+lQ)

a = [(n+l-Q)v w]/(n+2-Q)

b = v 2a


where v (2-Q)C2 + Rs w = (2-Q)C 2 + Q is the

charge on each donor atom, n = 3 for nitrogen and n = 4

for oxygen. Cs, C Rs and R are the same as in the text.

The VSIP data of Basch and Viste(3) are listed in

Table 7 for the metal atoms and Table 8 for the donor atoms.

The equations below were used in the computer program for

calculating the VSIP's. The configuration is the initial

configuration; the other letter is the electron lost. The

data in Table 7 were fitted to a straight line while the

data in Table 8 were fitted to a parabolic curve except in

one case.











dn d
dn-1 sd



dn-1 s s

dn-2 s2s

dn-2 sps

dn-1 p P

an-2 p2p
dn-2 sp
d1- spp


Fe(n=8)
-100.OZ-41.9

-115.3Z-70.5

-115.7Z-81.2
- 71.2Z-57.3

- 80.4Z-68.3

- 74.7Z-81.4

- 58.1Z-29.9
- 65.1Z-39.7

- 65.1Z-40.3


2s22pn s


2s2pn+ls


2s22pn p


2s22pn+lp


N(n=3)

- 9.75(Q+5.65385)2
105.4683

- 8.95(Q+5.87430)2
82.8415

-12.6(Q+4.48016)2

146.5049

-23.4(Q+1.58333)2

+ 70.7375


0(n=4)

- 9.7(Q+6.49485)2
+ 148.3752

+ 9.6(Q-9.20833)2

-1064.1166

-12.95(Q+4.91699)2

+ 185.6892

-147.0 Q 126.4


Co(n=9)
-105.1Z-44.8
-120.1Z-75.6

-119.4Z-88.4

- 73.9Z-59.1
- 82.5Z-70.5

- 78.6Z-84.0

- 59.5Z-30.7
- 68.2Z-40.7

- 68.2Z-40.8


Ni(n=10)
-109.7Z-47.6
-124.9Z-80.9

-122.4Z-95.9

- 74.6Z-60.8
- 84.4Z-72.3

- 83.OZ-86.0
- 60.8Z-31.4

- 71.7Z-41.6

- 71.7Z-40.9












TABLE 7


VALENCE STATE IONIZATION POTENTIAL DATA FOR IRON,
(IN 1000 CM-1)


COBALT AND NICKEL


Chg. 0 --> +1 +1 --> +2
n 3dn 3dn-14s 3dn-14p 3dn-1 3dn-24s 3dn-24p

Ionization of a 3d electron

Fe 8 '41.9 70.0 81.2 141.9 185.3 196.9
Co 9 44.8 75.6 88.4 149.8 195.7 207.8
Ni 10 47.6 80.9 95.9 157.3 205.8 218.3




Chg. 0 +1 +1 ---> +2
n dn-ls dn-2s2 dn-2sp dn-2s dn-3s2 dn-3sp

Ionization of a 4s electron

Fe 8 57.3 68.3 81.4 128.5 148.7 156.1
Co 9 59.1 70.5 89.0 133.0 153.0 162.6
Ni 10 60.8 72.3 86.0 137.2 156.7 169.0



Chg. 0 -- +1 +1 --> +2
n dn-lp dn-2p2 dn-2sp dn-2p dn-3p2 dn-3sp

Ionization of a 4p electron

Fe 8 29.9 39.7 40.3 88.0 105.4
Co 9 30.7 40.7 40.8 90.2 109.0
Ni 10 31.4 41.6 40.9 92.2 112.6

















TABLE 8

VALENCE STATE IONIZATION POTENTIAL FOR NITROGEN AND OXYGEN
(IN 1000 Crl1)

0-> +1 +1---> +2 +2 ---> +3
N 2s2spn-1 2s2pn 2s2spn-2 2s2pn-1 2s22pn-3 2s2pn-2

Ionization of a 2s electron

N 4 206.2 226.0 326.2 340.1 465.7 472.1

0 5 260.8 473.3 396.5 417.3 551.6 565.3





0 -> +1 +1 -> +2 +2 ----> +3
N 2s22pn-1 2s2pn 2s22pn-2 2s2pn-1 2s22pn-3 2s2pn-2

Ionization of a 2p electron

N 4 106.4 129.4 231.9 226.9 382.6 371.2

0 5 127.4 126.4 267.7 273.4 433.9 422.9








APPENDIX II

The dipole moment integrals necessary for the Smith
method of population analysis may be transformed into two
center overlap integrals by multiplication of a second
orbital by a numerical constant. The numerical constant
and the form of the second orbital, which is used for the
overlap integral, are determined by the form of the product
of the dipole moment operator (Z) and one of the orbitals
in the dipole moment integrals. The transformations that
were used in this investigation are listed below:

1 X(f1 100 -Vh X (1200
100 2 200 2 200 20
z (2oo) = 210oo) o q1o)


z X (1210)


Z X(f211)


V2- K(210 X1 (210
+ C(532o) + 22 3oo0

6- X (21211
21 321
211


where X(jnIm) is a double zeta wave functions of Clement:

and n' l'n/ ) are the wave functions of Clementi with an
angular portion correspc-ding to the quantum numbers n ,
I' and m' and a radial portion with the orbital exponents
corresponding to the original wave function.


i














APPENDIX III

Computer Program

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P.- co C, 0 '0r- C, 0 0
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SUBROUTINE U[AG(H,S,EC)
DIMENSION H(6,6) ,S(6,6), (6) ,C(6,6),X(6) ,V(6,6) ,D(6,6)
IVDL(6,6),G(6,6),GG(6,6),B(6,6),VD(6,6)
COMMON M
00 1 J=I,M
U I 1 l=1,M
H(J,I)=H(I,J)
I S(J,I)=S(I,J)
(ALL VALVEC (S,X,V)
DO 12 I=L,M
12 D( I)=1.0/SQRTF(X(1))
100 2 I=1,M
00 2 J=I,M
VI)( I,J)=0.0
DU 3 K=i,M
3 VD(I ,J)=VD( ,J)+(V( ,K)*1)(K,J))
2 CONTINUE
0O0 4 I=l,M
1)0 4 J=L,M
VOL( ,J)=0.0
DO 5 K=l,M
b VDL(I,J)=VOL( I ,J)+(VO( 1,K)*V J K))
4 CONTINUE
00 6 I=1,M
DO 6 J=I,M
G( ,J)=0.0
UO 7 K=1,M
7 G(I,J)=G(I,J)+(V L( I ,K)*IH(K,J))
6 CONTINUE
00 8 I=i,M
UU 8 J=1,M
GG(I,J)=O.0
00 9 K=I,M
9 GG(1,J)=GG(1,J)+(G( I,K)VDL(K,J))
8 CONIINUC
CALL VALVCC (GG,E,B)








61



DO 10 I=1,M
Do 0t J=1,M
C(I,J)=0.O
DO 11 K=1,M
11 C( I,J)=C(I,J)+(VDL( ,K)*B(K,J) )
10 CONIINUL
RETURN
NDI)












SUiBOUTINE VALVEC(A,EVAL,VECTOR)
DIMENSION A(6,6),EVAL(6),VECTUR(6,6),ARRAY(400),VCI (400)
CUMMONM
K=0
DO 10 I=1,M
00 10 J=I,M
K=K+l
10 ARRAY(K)=A(1,J)
CALL JACFUL(M,ARRAY,VEC, I I'ER,TRACE
L=l
DO 20 L=l=1,M
LVAL(L- )ARRAY(L)
20 L=L+M+1
K1=0
OU 30 11=l,M
00 30 JL=l,M
KI=1-KL
30 VECIOR(JI, I)=VECT(KI)
RE URN
CND













SUBROUTINE ADJAL (UELLI)EL2,N)
DIMENSION Dl)L(99),DELO(99)
COMMON M,KAL,KBE,KGA
KAL=KAL+-
N=O
DEL(KAL)=DELI
ODLD(KAL)=ABSF( DELl-DEL2)
IF(UELO(KAL)-.O01) 10,10,1
I IF(KAL-I) 2,2,3
2 DILL=(DELL+UCL2)/2.
GO TU 9
3 IF(DELD(KAL)-DELD(KAL-1)) 2,4,4
4 OELL=(DEL(KAL-i)+(OEL(KAL))/2.
9 N=1
10 RETURN
END
SUB OUr[NE AUJBE (DELI,DEL2,N)
DIMENSION ODL(99),DELD(99)
COMMON M,KAL,KBE,KGA
KBE=KBE+i
N=0
OEL(KBE)=DEL
DELD(KBC)=ADSH DELI-DEL2)
IF(DELD(KBE)-.01) 10,10,1
1 I[(KBE-1) 2,2,3
2 DLLI=(DLLI(+)EL2)/2.
GO TO 9
3 IF(UELU(KDE)-DELO(KBE-1)) 2,4,4
4 UELl=(D[L(KBh-1)+DEL(KBE)J/2.
9 N=1
10 RErURN
END
SUBROUTLNE ADJGA ()ELI ,UEL2,N)
DIMENSION DEL(99),DOCL (99]
COMMON M,KAL,KBE,KGA
KGA=KGA+1








64



N=O
DEL(KGA)=DEL1
DELD(KGA)=ABSF(DELi-DCL2)
IF(DELU(KGA)-.001) 10,10,1
1 IF(KGA-1) 2,2,3
2 DELI=(DEL1+WDL2)/2.
GU [U 9
3 IF(DELD(KGA)-DELU(KGA-1)) 2,4,4
4 DELI=(DEL(KGA-1)+DEL(KGA))/2.
9 N=1
0 RETURN
END









APPENDIX IV

The following is th. format that has been used for

the tables of this appendix.


Name of complex Wavefunction of metal atom Experimental
lODq (cm-1)

No pi : Pi bonding has been omitted

or pi : Pi bonding has been included

Method of estimating the resonance integral

Smith-population analysis Mulliken-population analysis

Self-consistent popula-
tions of metal d, s and p
orbitals. Same as for Smith.

Self-consistent charge on
the metal atom.

Calculated lODq (cm-1)

Calculated total energy
(kK) (kK = kiloKaiser =
103 cml)

a: This method does not lead to convergence.

b: The energy of the pi symmetry orbital on the
donor atom is above the energy of the d orbital on the
metal atom and the energy levels have been scrambled.

c: Same as b but the energy levels have not been
scrambled.
















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BIBLIOGRAPHY


1. C. J. Ballhausen, Introduction to Ligand Field Theory
(McGraw-Hill Book Co., Inc., New York, 1962). p. 174.

2. C. J. Ballhausen and H. B. Gray, Inorg. Chem., 1, 111
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3. H. Basch and A. Viste, Private Communication (published
in C. J; Ballhausen and H. B. Gray, Molecular Orbital
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9. R. F. Fenske, Inorg. Chem. 4, 33 (1965).
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P. 75.
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89

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BIOGRAPHICAL SKETCH


William Adkins Feiler, Jr. was born October 9, 1940,

in Paducah, Kentucky. After attending elementary and junior

high public schools in Paducah, Kentucky, he was graduated

from Paducah Tilghman High School in June, 1958. In June,

1962, he received the degree of Bachelor of Science in

Chemical Engineering from the University of Kentucky,

Lexington, Kentucky.

In September, 1962, he entered the Graduate School

of the University of Florida. During his tenure, he has

held a graduate teaching assistantship in the Department of

Chemistry, a graduate school fellowship, and a research

assistantship on a project supported by the National Science

Foundation.

Mr. Feiler is married to the former Patricia Carolyn

Botner. He is a member of Pi Kappa Alpha, Alpha Chi Sigma,

Gamma Sigma Epsilon, and the American Chemical Society.










This dissertation was prepared under the direction

of the chairman of the candidate's supervisory committee

and has been approved by all members of that committee. It

was submitted to the Dean of the College of Arts and

Sciences and to the Graduate Council, and was approved as

partial fulfillment of the requirements for the degree of

Doctor of Philosophy.


August 14, 1965


Dean, College of Arts and Sciences



Dean, Graduate School


Supervisory Committee:


Chairman /




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