Vorontsovo Pole 10, 103064, Moscow, Russia;

at the M.V. Keldysh

Institute for Applied Mathematics RAS

first transition row ions

Calculations on magnetic and optical properties of transition metal
complexes (TMC) is one of important problems of theoretical chemistry.
Semi-empirical [,,,,] as well as
*ab initio* methods
Newton1,Newton2,Pierloot,Bausch,Morokuma,Johansen,Goddard,Janssen were
applied for that, both substantially using the Hartree-Fock-Roothaan (or the
self-consistent field MO-LCAO) approximation. The *ab initio*
calculations are highly time consuming when applied to TMC's. The reason is
a huge number of electronic states to be included into configuration
interaction (CI) procedure, due to poor convergence of the CI series when
the canonical Hartree-Fock molecular orbitals (MO) are used for constructing
the configurations. Thus *ab initio* calculations can be performed only
for relatively small systems, for which nevertheless a considerable
agreement of calculated and experimental transition energies can be obtained
[,]. Semi-empirical methods based on the SCF
approximation are less demanding for computational resources, though results
obtained strongly depend on parametrization of numerous molecular integrals.

The application of semi-empirical and *ab initio* methods based on
Hartree-Fock approximation to the TMC electronic structure calculations is,
however, complicated by violation of the SCF approximation itself. This is
exhibited in some of its important consequences []. Namely,
the *Aufbau* principle and Koopmans' theorem are not fulfilled for the
MO's with considerable weight of the atomic *d*-orbitals; the relaxation
energy for the latter (the difference between the ionization potential
calculated by the Koopmans' theorem and its experimental value) can achieve
10 or even 20 eV for ionization from these MO's, and the MO's being higher
in energy can turn out to be occupied while MO's of lower energy remain
vacant or singly occupied. Also the SCF iteration process often converges
very slowly or oscillates.

All these observations indicate that behavior of *d*-electrons in the TMC
goes beyond the SCF approximation's frames which can be characterized as a
motion of independent electrons in the self consistent field induced by
nuclear cores and by other electrons. By contrast, *d*-electrons in TMC's
are strongly correlated (as compared to those in the ligand orbitals) and
form a well localized separate group. As ground state spin and low-energy
excitations of TMC's mainly depend on *d*-electrons' state [],
account for correlations of the latter is of principal importance for
description of magnetic and optical properties of the TMC's.

For interpreting experimental data and explaning properties of TMC's the
phenomenological crystal field theory (CFT) [,] is
widely used. The latter describes the TMC's in terms of excitations of their
*d*-shells only. According to the CFT the one-electron states in the *d*-shells are split by electrostatic field induced by effective charges
residing in the ligands. The main defect of the CFT is lack of details of
the ligand electronic structure that entails the limitation of the
interaction between the *d*-shell of the central atom and the ligands to
purely electrostatic one. For that reason the *d*-level splitting parameter
10*Dq* is essentially underestimated and the one-electronic splitting
parameters can not be calculated within the CFT itself, and remain
independent parameters of the theory. The ligand field theory (LFT)
Bersuker,Jorgensen partially takes into account the covalent character of
interactions between the ligands and the central ion. However, the LFT
calculations are in fact equivalent to the Hartree-Fock approximation and
reduces to taking into account the MO's symmetry when making linear
combinations with the *d*-orbitals. For this reason splitting
characteristics calculated within the model are different from ones fitting
in experimental data analysis.

For taking into account in the semi-empirical realm qualitative features induced by both electron correlation and covalency effects and to circumvene the problems of the original naive CFT and LFT the effective crystal field (ECF) method was proposed [] which is based on combination of the effective Hamiltonian method and the group function technique Lowdin,McWeeny. Basic features and formulae of this method are given below.

The trial wave function F of electrons in TMC has the form of the
antisymmetrized product of the full CI wave function for *d*-electrons Y_{M}, and the single determinant wave function Y_{L} for remaining
electrons of the TMC:

| (0) |

In order to arrive to the above form of the trial wave function we notice
that the general form of the exact wave function can be presented as a
linear combination of the functions having all possible electron
distributions between the two subsystems singled out. However, in Ref.
SouTchMis it was shown that the wave function eq. (1) with a
fixed number *n*_{d} of electrons in the *d*-shell, which is equal to their
number in the ground state of the corresponding free metal ion must be
obtained from the mentioned linear combination by projecting onto subspace
spanned by the functions of the form eq. (1).

Within the ECF theory the total Hamiltonian is rewritten in the form:

| (0) |

For the charge transfer states the Löwdin partition technique []
as a version of effective Hamiltonian method was used in Ref.[]. After projecting the exact Hamiltonian eq.(2) is replaced by the
effective one acting in the subspace spanned by the functions with the fixed
number *n*_{d} of *d*-electrons. Eigenvalues of the effective Hamiltonian
coincide with exact Hamiltonian eigenvalues by construction. By simple
algebra the explicit form of the effective Hamiltonian is obtained
SouTchMis:

| (0) |

where *P* is the operator projecting to the subspace of the functions with
the fixed number *n*_{d} of the *d*-shell electrons and *n*_{l} = *N*-*n*_{d} is that of
the ligand subsystem electrons; and *Q* = 1-*P*.

The TMC eigenstate energies must be obtained from the relation:

| (0) |

Since the effective Hamiltonian depends on energy, the last equation must be
solved iteratively until convergence in energy is achieved. However, since
the charge transfer states (determining the poles of the above resolvent
term) lay significantly higher in energy than the *d*-shell excitations this
dependence turns out to be weak and can be neglected, so one can set:

| (0) |

| (0) |

In the above system the effective Hamiltonian *H*_{d}^{eff} for the *d*-electron subsystem depends on the wave function of the ligand subsystem F^{l}, and in its turn the effective Hamiltonian *H*_{l}^{eff} for the
ligand susbsytem depends on the *d*-electrons' wave functions F_{n}^{d}.
These equations must be solved self-consistently as well. In the ECF method
Ref.[] the Slater determinant F^{l}, is constructed of
MO's of the ligand subsystem, obtained from the Hartree-Fock equations in
the CNDO/2 approximation for the valence electrons of the ligands. In this
case the transition from the bare Hamiltonian *H*_{l}^{eff} for the ligand
subsystem to the corresponding effective (dressed) Hamiltonian reduces to
renormalization of one-electron parameters related to the transition metal
ion:

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

| (0) |

The effective Hamiltonian for the *d*-shell eq.(8) formally
coincides with the CFT Hamiltonian. The substantial difference is the
covalence contribution eq.(11) to the *d*-shell one-electron
parameters eq.(9), taking into account the effect of virtual charge
transfers between the metal *d*-shell and the ligands. Thereby, the ECF
contains not only electrostatic but also covalence terms coming from the
resonance interactions between the *d*-shell and the ligands.

According to calculations performed in Refs.[,,,,] for the TMC's of divalent cations the
covalence contribution to the splitting parameter 10*Dq* dominates and gives
up to 90 % of the total. This stresses the importance of the procedures
described above aimed to include the covalence interaction in an economic
and transparent fashion into the effective Hamiltonian parameters. The
parameters of the ECF method for the complexes of divalent cations of the
first transition range metals have been found, tested and employed in Refs.[,,,,]. They allowed to describe
correctly the symmetry of the ground states and the optical *d*-*d*-transition
energies with precision up to 1000 cm^{-1} for about a hundred of the
TMC's of the first transition row divalent cations ranging from
hexafluoroanions to porphyrine complexes. Therefore the use of the ECF
allows to improve the semiempirical description of TMC electronic structure
significantly.

The purpose of the present work was to explore the possibility to extend the
parametrization of Refs. [,,,,]
to the complexes of trivalent ions of the first transition row. Let us
remind that when talking about di- and trivalent cations we imply the Werner
complexes for which the *d*-electron subsystem contains the same number of
electrons as the 3*d*-shell of the corresponding isolated cation, *e.g.*
the *d*-electron subsystem of the Fe^{3+} complexes has 3*d*^{5}
configuration with five *d*-electrons.

The whole set of the parameters of the ECF method consists of those of the
CNDO/2 method for the ligand atoms, specific parameters *U*_{dd}, z_{d},
*B*, and *C* (see below) for the *d*-shell, Oleari parameters describing
intraatomic part of the Coulomb interaction between subsystems. In the
present paper only the core attraction of *d*-electrons *U*_{dd} for each
metal ion and the b^{M-L} resonance parameters for each pair metal
(M)-donor atom (L) have been fitted. The b^{M-L} parameters have
been adjusted to fit the calculated excitation energies to the experimental
ones for the complexes with organic amines, pyridine and its derivatives,
and other nitrogen-, and oxygen-containing, ligands, halogen anions *etc*.

In the present paper the ECF calculations on octahedral complexes of
trivalent cations of the first transition row have been performed. Only
those complexes we are considered whose both the geometry and the transition
energies and the corresponding symmetries of the excited states we are
known, namely: V^{3+}, Cr^{3+}, Mn^{3+}, Fe^{3+}, and Co^{3+}.
(The data on Ni^{3+} and Cu^{3+} complexes are not readily available
since the latter are largely unstable.) The geometry structure parameters -
the atomic coordinates, bond lengths, and valence angles - were taken from
the literature (the corresponding references are given in the tables).

The *U*_{dd} and b^{M-L} parameters determining the value of
one-electron splitting parameter 10*Dq* in the *d*-shell and the spectrum of
the *d*-*d*-transitions, were fitted so that the best agreement of calculated
and experimental values of 10*Dq* and of the excitation energies is
achieved. The values of *U*_{dd} for each metal of the considered V, Cr, Fe,
and Co series were modified so that the positive values of D*E*_{dj}
and D*E*_{jd} Eq.(13) are guaranteed in order to ensure the
stability of the system with respect to electron transfer between the
subsystems. After that the b^{M-L} parameters have been fitted to
reach a required agreement between the calculated and experimental values of
the splitting parameters and calculated and experimental excitation energies.

The calculated 10*Dq* values, the ionic and covalent contributions to them
and the respective experimental values are given in Table 1. The fitted
values of the parameters *U*_{dd} and b^{M-L} for the trivalent ions,
the respective parameters for the divalent ions, and also bond length
differences between the di- and trivalent cations complexes (in those cases
when the corresponding data for both complexes are available) are given.
Also the Slater exponents for the valence 4*s*-,4*p*-, and 3*d*-orbitals of
the trivalent ions are given. These quantities, chosen in accordance with
the Burns rules [] accepted in the original ECF method
SouTchMis, are less diffuse than the respective orbitals of the divalent
ions. It should be noticed that the *U*_{dd} values for the trivalent ions
are smaller (by the absolute value) by only a few electronvolts (or a few
percents) than the values of the same parameters employed previously to
describe their divalent analogs. The parameters b^{M-L} for the
trivalent cations are as a rule smaller than those for the divalent ones
with exceptions of hexahydrate complexes of V^{3+} and Co^{3+}. One also
may notice that the values b^{M-L} for all the donor atoms L are
systematically smaller for the trivalent ions, but increase with the atomic
number of the metal like it is for the divalent ions. Since we considered
only one complex of Mn, we had to take the *U*_{dd} value by a few percent
smaller than the absolute value for the divalent manganese ion by analogy
with Cr, Fe, and Co.

We notice also that the fitted values of the parameters are rather close to
the respective values for the divalent ions, which is an additional proof of
consistency of the ECF method permitting a comparatively simple extension to
a new class of objects - complexes of trivalent cations. Covalent
interactions in the considered complexes of the trivalent cations contribute
about 80% to the *d*-states splitting like it was in the case of the
divalent cation complexes.

In Tables 3-7 the results of calculations on octahedral complexes of the
transition metal trivalent cations are given. Such complexes as [VF_{6}]^{3+}, [V(H_{2}O)_{6}]^{3+}, and [CoF_{6}]^{3+} are the Jahn-Teller
complexes. Our assumption that these complexes posess an octahedral symmetry
adds an additional uncertainty to the calculated transition energies, so one
may expect somewhat larger difference between the calculated and
experimental transition energies for these complexes.

Values of the Racah parameters *B* and *C* describing the electron
interaction in the *d*-shell cited in the literature were used in our
calculations. But if such data are absent in the original works, we used the
Racah parameters given in tables in Ref.[].

Inspite all the sources of uncertainty mentioned above, we succeeded in
fitting the b^{M-L} parameters so that the correct ground state term
symmetry and the values of optical electron transitions were obtained with
high accuracy (in most cases up to a few hundreds cm^{-1}). The calculated
values of 10*Dq* are in a good agreement with the experimental data with the
precision of 200-400 cm^{-1}. An exception is the Jahn-Teller vanadium
aquacomplex, but even in this case the difference does not exceed 1000 *cm*^{-1}, which is accurate enough.

Comparison of the transition energies in the *d*-*d*-spectra, calculated by
the ECF method with the parametrization cited in Table 2, with the
experimental absorption band energies demonstrates that calculation with the
fitted parameters leads to results that are in a good agreement with
experiment. The divergences are in the range of 1000 cm^{-1} except for
cyanocomplexes. The calculated transition energies in the latter case
differs from the experimental energies by up to 2000 cm^{-1}. However, two
experimental works concerning iron(III) hexacyanide [,]
give for the same states the energy values differing by *ca.* 1000 *cm*^{-1}. Thus one may conclude that the accuracy of the ECF method with the
given parametrization is comparable with the accuracy of the spectral
methods themselves.

Figures given in Table 2 concerning the differences of the metal - donor
atom separations between the complexes of di- and trivalent cations allows
to notice some regularities. In all cases with the only exception of the
pair of iron hexacyanocomplexes where the Fe-C distance increases by 0.037
Å , the metal - donor atom distances decrease by 0.1-0.3 Å . However,
in the pair of hexaaquacomplexes of cobalt an essentially smaller difference
(0.011 Å ) is observed. We also note that for the majority of cations the
complexes of the trivalent ones posess the total spin values which are
smaller than those for their divalent couterparts. If a complex of a
divalent cation has a low spin the corresponding complex of a trivalent
cations also has a low spin, and *vice versa*. The only exception is
provided by the aquacomplexes of Co. This anomaly in the experimental data
can be reflected in the ECF calculations only if the regularity in the b^{M-L} parameter variation along the row is broken. Right in the
case of the cobalt hexaaquacomplexes pair the difference of b^{M-L}
values is abnormally large (0.7, that is essentially higher than the
difference typical for all other complexes: 0.2-0.4). This fact demands to
perfect our approach, *e.g.* by adjusting a larger number of parameters.
Nevertheless, in the case of the iron hexacyanocomplexes the difference of b^{M-L} appears to be regular in spite of the anomalous difference of
the metal-donor atom separation.

In the present work the ECF method [] is employed for
calculations of the complexes of trivalent ions of the first row transition
metals. It is demonstrated that the description of the *d*-*d*-excitation
spectra of the trivalent ion complexes is possible with the same high
accuracy as in the case of the complexes of respective divalent ions
provided the parameters of the resonance interactions between the *d*-shell
and the ligands are respectively adjusted.

This work is partially supported by the RFBR grant No 99-03-33176, by the Federal Task Programm of Russia 'Integracia' by the grant No A0078, and by the special RAS grant program No 6 for younger scientists by the grant No 120. The work has been performed using the Net Laboratory system netlab of access to quantum chemical software sponsored by the RFBR through the grant No 01-07-90383.

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