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 (CM1) for
the Indicated Complexes at SelfConsistency
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) (CM1) . .
7. Valence State Ionization Potential Data for
Iron, Cobalt and Nickel (in 1000 CM1). .
8. Valence State Ionization Potential for
Nitrogen and Oxygen (in 1000 CM1). . . .
9. SelfConsistent Parameters for [e(H 2+
9. SelfConsistent Parameters for [Fe(H20)6+ .
11. SelfConsistent Parameters for [Co(H20)6]2+ .
12. SelfConsistent Parameters for [Co(H20)6 3+ .
12. SelfConsistent Parameters for [Co(H20)63 .2
2+
15. SelfConsistent Parameters for [Ni(H20)6] + .
14. SelfConsistent Parameters for [Co(NH )6 2+ an
[co(NH 3)6]3. . . . . . . .
15. SelfConsistent 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
highspin with magnetic moments between 4.3 and 5.2 Bohr
(14)
magnetons.1 ) A few octahedral cobalt(II) complexes are
lowspin.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 crossover point from high
spin to lowspin.833) This proposal provides a satisfactory
semiquantitative 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
crossover 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 lowspin but near the crossover
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 oneelectron molecular orbital
energy levels for transition metal complexes. The procedure
used is a semiempirical 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 WolfsbergHelmholz method of
estimating the resonance integrals in their treatment of
the ironporphyrin 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 metaloxygen 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) FeO(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),
Ni2O(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 xray 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 Cr0 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, nonbonding. 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(2Q)C + R p(Q)Cp + R = a(s2pnQ)
+ b(spn+lQ) (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 2s22p4Q
B 2s2p5Q
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 (2Q) 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 pnQ, and Vb is the VSIP of an s electron in the
configuration spn+lQ. 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 selfconsistent 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(2Q)C2 + R p(2Q)C2 + R 2QC2 nQC 2
+ b(slQCs2 n+lQC 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 cm1 (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(d9z) + b(d8zs) + c(d8Zp) = 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 selfconsistent 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).)
Selfconsistency 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 selfconsistency. The few cases where the procedure
described above failed to give selfconsistency 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 selfconsistent
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 BallhausenGraySmith 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 BallhausenGraySmith
method sometimes diverges lies in the method of estimating
the resonance integral. Fenske(9) has pointed out that in
using the BallhausenGray 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
co
C4
000
<3 <
Co r^
Co v cr
000
0 C4 4
\D C 0
000
000
40
00
000
r.
cro
O O
AOC4
000
'0 C4
44
000
0
c0 c'J
o C) r)
rC) C)
1n C4
on
000
oo
10
OMr
000
%O C0 14
00
see c
0 0
co 4
Or".
000
:r Co4 r
NNW
mee r
oONl
000
CM 4\ 0
co o
co o a
000
00 C
OO
COc .I
<3C enCM
000
co 0 co 0
in CM en
enCM "4
000
coO\l
000
(N t L
0
0 It n
O T
Cto
on
u3)
Lr)
000
< cco
C CM
000
smo
>rl
'T rco
C4 ni4
000
C) CM C)
1 ,1 
N ) \
'IT C4 1
00
co C
coco
CN .4
00
n o
1
co
o o
S0
e o
00
00
oo
O'3 o
0 0
coo
4 4
(dm( (0( me3
QlN~
rlr
+
\0
C..)
O
cM
a
v
O
u
 coo I I I I
r44 4
0 0 C
in co
000
4
O
r
c at a
cl I I 
000
AAA UN
000
c~cO
cn 4c
00 1
0C0
000
uucON
000 c
mcow
00
o o
nq en
eo o
N co
"H M
rCM e
C~M 
000
'T co %
M It
c4r0
4n OCN CC (M Oe4 n o rMN C4 n
0
i
4
,4
0
SII 0
oa
mo 1 o
II o o
I I *
00 0
4 0
cOii .i
00
14 
c, '
c CO I I
0 .0 .0
44
0 0 0
S0 0
 4 4
O O
0 0
0 0
0 0
T4
0 0 0
on o o
1 r4 4
0 0
0 0 0
0 0
0 0
l4 4
0 0
o oo
0 0
4 4
 C CM4 CMJ
+J
c'4
'
Z
0
*
4
4 4i 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
11
0
co ,
CO
O 0
00
a) co
4IJ
0
0 o
u o
o C)
00
0 o
4N 0
r0 4
a) u *
0
C
'4 4J
 a
aa
o C
0
3 4

CO a
.0C ) '*0
cf Z 0.. 4J
oC Q I
0 c *. 4
a u
4 4r.
4 *H 0 'o 4.
CO314 Q
~ C~ ) Ia
>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 selfconsistency 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 oneelectron
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 selfconsistency
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 WolfsbergHelm
holzSmith 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 cm1
[Fe(H20)6]3+ 13,700 cm1, [Co(H20)6 2+ 9,700 cm1
[Co(H20)6]3+ 19,100 cm1, [Ni(H20)6 2+ 8,500 cm1
[Co(NH3)6]2+ 10,500 cm1, [Co(NH3)6]3+ 23,500 cm1, and
[Ni(NH)6]2+ 10,800 cm1
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 ironaquo 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 cobaltaquo
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 selfconsistent 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
selfconsistent 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 selfconsistent
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 selfconsistent 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 selfconsistent 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 cm1) 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 cm1) 10Dq 13,600 16,400 18,700 18,000
[Co(NH3)6] 2+ Z a 0.45 0.38 0.61
(10,500 cm1) 10Dq 11,500 11,000 13,900 13,400
[Co(NH3)6]3+ Z a 0.20 0.58 0.76
(23,500 cm1) 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 cm1) 10Dq 11,800 10,800 13,900 13,500
aUsing the selfconsistent 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 socalled 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 overall agreement using the methods which do not in
clude pi bonding but are acceptable except for the total
energy arguments. An observation on the selfconsistent
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 WolfsbergHelmholzSmith 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 highspin cobalt(II) to
lowspin 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
WolfsbergHelmholzMulliken method with pi bonding omitted
does extrapolate with excellent agreement from highspin to
lowspin 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 WolfsbergHelmholzSmith 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 cm1.) 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 BallhausenGrayMulliken
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 cm1 using a d coulomb integral
of63,250 cm1. Our estimate for the coulomb integral is
154,810 cm1, which gives 10Dq = 32,670 cm1, 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 cm1, 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 WolfsbergHelmholzSmith
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 selfconsistent
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) (C1)
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 cm1. For [Co(H20)6]3+ this
transition is the 1Ag g Tlg transition which is observed
1ig
at 16,600 cm1
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 cm1 in good agreement with the experimental value.
This result shows that it is possible to extrapolate an
experimental parameter from highspin 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
HelmholzSmith 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 nonbonding 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 semiempirically 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(dnZ) + b(cnlZs) + c(dnlZp) dasp
a = [aP(nlZ) 2 (nlZ)]/(nZ)
b
c =6
For the VSIP of an s electron
a(dnlZs) + b(dn2Zs2) + c(d2Zsp) = ds Pp
a = [2a + P(Z+2n) + (Z+2n)]/(nZ)
b = [c + P(nlZ) ]/(nZ)
c =
For the VSIP of a p electron
a(dnp) + b(dn2Z2) + c(dn2Zsp) = dps p
a = [2a + P(Z+2n) + 6(Z+2n)]/(nZ)
b = [a p + (nlZ)]/(nZ)
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 = naP 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 nQ n+1
s = a(2s22pQ) + b(2s2pn+lQ)
a = [(n+lQ)v w]/(n+2Q)
b = v 2a
where v (2Q)C2 + Rs w = (2Q)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
dn1 sd
dn1 s s
dn2 s2s
dn2 sps
dn1 p P
an2 p2p
dn2 sp
d1 spp
Fe(n=8)
100.OZ41.9
115.3Z70.5
115.7Z81.2
 71.2Z57.3
 80.4Z68.3
 74.7Z81.4
 58.1Z29.9
 65.1Z39.7
 65.1Z40.3
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(Q9.20833)2
1064.1166
12.95(Q+4.91699)2
+ 185.6892
147.0 Q 126.4
Co(n=9)
105.1Z44.8
120.1Z75.6
119.4Z88.4
 73.9Z59.1
 82.5Z70.5
 78.6Z84.0
 59.5Z30.7
 68.2Z40.7
 68.2Z40.8
Ni(n=10)
109.7Z47.6
124.9Z80.9
122.4Z95.9
 74.6Z60.8
 84.4Z72.3
 83.OZ86.0
 60.8Z31.4
 71.7Z41.6
 71.7Z40.9
TABLE 7
VALENCE STATE IONIZATION POTENTIAL DATA FOR IRON,
(IN 1000 CM1)
COBALT AND NICKEL
Chg. 0 > +1 +1 > +2
n 3dn 3dn14s 3dn14p 3dn1 3dn24s 3dn24p
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 dnls dn2s2 dn2sp dn2s dn3s2 dn3sp
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 dnlp dn2p2 dn2sp dn2p dn3p2 dn3sp
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 2s2spn1 2s2pn 2s2spn2 2s2pn1 2s22pn3 2s2pn2
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 2s22pn1 2s2pn 2s22pn2 2s2pn1 2s22pn3 2s2pn2
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 correspcding 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
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=1KL
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( DELlDEL2)
IF(UELO(KAL).O01) 10,10,1
I IF(KALI) 2,2,3
2 DILL=(DELL+UCL2)/2.
GO TU 9
3 IF(DELD(KAL)DELD(KAL1)) 2,4,4
4 OELL=(DEL(KALi)+(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 DELIDEL2)
IF(DELD(KBE).01) 10,10,1
1 I[(KBE1) 2,2,3
2 DLLI=(DLLI(+)EL2)/2.
GO TO 9
3 IF(UELU(KDE)DELO(KBE1)) 2,4,4
4 UELl=(D[L(KBh1)+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(DELiDCL2)
IF(DELU(KGA).001) 10,10,1
1 IF(KGA1) 2,2,3
2 DELI=(DEL1+WDL2)/2.
GU [U 9
3 IF(DELD(KGA)DELU(KGA1)) 2,4,4
4 DELI=(DEL(KGA1)+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 (cm1)
No pi : Pi bonding has been omitted
or pi : Pi bonding has been included
Method of estimating the resonance integral
Smithpopulation analysis Mullikenpopulation analysis
Selfconsistent popula
tions of metal d, s and p
orbitals. Same as for Smith.
Selfconsistent charge on
the metal atom.
Calculated lODq (cm1)
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.
I
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BIBLIOGRAPHY
1. C. J. Ballhausen, Introduction to Ligand Field Theory
(McGrawHill Book Co., Inc., New York, 1962). p. 174.
2. C. J. Ballhausen and H. B. Gray, Inorg. Chem., 1, 111
(1962).
3. H. Basch and A. Viste, Private Communication (published
in C. J; Ballhausen and H. B. Gray, Molecular Orbital
Theory (W. A. Benjamin, Inc., New York, 1964). p. 120.
4. H. D. Bedon, W. E. Hatfield, S. M. Homer and S. Y.
Tyree, Inorg. Chem., 4, 743 (1965).
5. H. D. Bedon, S. M. Horner and S. Y. Tyree, Inorg. Chem.
2, 647 (1964).
6. E. Clementi, J. Chem. Phys., 40, 1944 (1964).
7. F. A. Cotton and T. E. Haas, Inorg. Chem., 3, 1004
(1964).
8. E. A. Clevenger, "Thesis1961." The University of
Florida.
9. R. F. Fenske, Inorg. Chem. 4, 33 (1965).
10. R. F. Fenske and C. C. Sweeney, Inorg. Chem., 2, 1105
(1964).
11. K. Jorgensen, Report to the X Solvay Council, Brussels,
May, 1956, Absorption Spectra of Complexes with Unfilled
dShells.
12. W. H. Kleiner, J. Chem. Phys., 20, 1784 (1952).
13. T. Ito and H. Mori, Acta Cryst., 4, 412 (1951).
14. J. Lewis and R. G. Wilkins, Eds., Modern Coordination
Chemistry (Interscience Publishers, Inc., New York,
1960). p. 406.
15. L. L. Lohr and W. N. Lipscomb, Inorg. Chem., 2, 911
(1963).
16. L. L. Lohr and W. N. Lipscomb, J. Am. Chem. Soc., 85,
846 (1963).
17. L. L. Lohr and W. N. Lipscomb, J. Chem. Phys., 38,
1607 (1963).
18. R. McWeeny and K. Ohno, Proc. Roy. Soc. (London), A255,
367 (1960).
19. W. Moffitt, Rept. Progr. Phys., 17, 173 (1954).
20. C. E. Moore, Atomic Energy Levels, U. S. Nat'l. Bur.
Standards, Circular 467, 1949 and 1952.
21. R. S. Mulliken, J. Chem. Phys., 23, 1833 (1955).
22. R. S. Mulliken, J. chim.. phys., 46, 497 (1949).
23. K. Ohno, Y. Tanabe and F. Sasaki, Theoret. Chim. Acta
(Berl.), 1, 378 (1963).
24. L. E. Orgel, An Introduction to Transition Metal
Chemistry (John Wiley and Sons, Inc., New York, 1960).
P. 75.
25. L. Pauling, The Nature of the Chemical Bond, 3rd Ed.
(Cornell University Press, Ithaca, New York, 1960).
26. Quantum Chemistry Program Exchange, number 4. (1963)
27. Quantum Chemistry Program Exchange, number 29. (1964)
28. J. W. Richardson, W. C. Nieuwpoort, R. R. Powell and
W. F. Edgell, J. Chem. Phys., 36, 1057 (1962).
29. J. W. Richardson, R. R. Powell, W. C. Nieuwpoort, J.
Chem. Phys., 38, 796 (1963).
30. B. D. Sarma and J. C. Bailar, Jr., J. Am. Chem. Soc.,
7Z, 5476 (1955).
31. J. C. Slater, Quantum Theory of Atomic Structure, Vol.
II (McGrawHill Book Co., Inc., New York, 1960).
Appendix 21.
32. D. W. Smith, Prog. Rept. 6, Wave Mechanics Group,
Univ. of Oxford (1960). p. 20. Unpublished.
89
33. R. C. Stoufer, W. B. Hadley and D. H. Busch, J. Am.
Chem. Soc., 8_, 3732 (1961).
34. Tables of Interatomic Distances and Configurations
in Molecules and Ions, Special Publication No. 11, The
Chemical Society, London, 1958.
35. A. Tsuchida and K. Ohno, J. Chem. Phys., 22, 600 (1963).
36. A. Viste and H. B. Gray, Inorg. Chem., 1, 1113 (1964).
37. M. Wolfsberg and L. Helmholz, J. Chem. Phys., 20, 837
(1952).
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 /
