CRYSTAL FIELD SPLITTINGS AND OPTICAL SPECTRA OF

TRANSITION METAL MIXED-LIGAND COMPLEXES BY EFFECTIVE HAMILTONIAN METHOD

A. V. Soudackov^{1}, A. L. Tchougreeff^{2}, and I. A. Misurkin

Karpov Institute of Physical Chemistry, Moscow, Russia.

Abstract. Many of important properties of transition metal complexes depend on the low-energy excitation spectrum formed by d-electrons of the central transition metal atom. The spectra of this type are usually fit to the well-known crystal field theory or to the angular overlap model. The result of the fitting is a set of parameters which are considered as characteristics of the electronic structure of the complex such as strength of ligand field or types and extents of metal-ligand bonding. We present here a short account of the effective Hamiltonian method recently developed to calculate the splitting of the d-levels by the ligands and the resulting d − d spectra of transition metal complexes together with some results of its application to the mixed ligand complexes with general formula ML_{4}Z_{2}, where M = V, Co, Ni; L = H_{2}O, NH_{3}, P y; Z = H_{2}O, NCS^{−}, Cl^{−}. Particular attention is paid to the V (H_{2}O)_{4}Cl_{2} and Co(H_{2}O)_{4}Cl_{2} compounds. The former seems to have tetragonal structure, whereas for the latter our method predicts a spatially degenerate ground state for the tetragonal arrangement of the ligands. That must lead to the Jahn - Teller distortion which is actually observed. I. Introduction

Calculations on electronic structure of transition metal complexes (TMC) present a great interest for chemistry. For the direct quantum-chemical calculation of the energies of the d − d excitations both semiempirical [1-5] and ab initio [6-13] methods based upon the Hartree-Fock-Roothaan approximation have been applied. In the case of ab initio calculations with large CI expansions [6-10] or with a special selection of configurations in the MCSCF wavefunction [11, 12] a reasonable agreement of the calculated transition energies with experiment can be obtained. However, such calculations are very cumbersome and for that reason are restricted to relatively simple systems.

1Present address: Lehrstuhl für Theoretische Chemie, Universität Hannover, Hannover, Germany

2Present address: Max-Planck-Institut für Physik komplexer Systeme, Dresden, 01187, Germany

Recently a semiempirical INDO/S − CI method for the calculation of the electronic transition energies has been elaborated and extended to transition metal atoms [2]. It successfully applies to rather complex systems like ferrocene [3] or porphyrin complexes of transition metals [4,5]. However, despite reasonable numerical agreement between the calculated and observed d − d transition energies obtained by this method some fundamental problems remain unsolved. The main complication which arises when semiempirical quantum chemistry addresses transition metal complexes (TMC) is the doubt in the validity of the fundamental quantum chemical approximation, the self consistent field (SCF) approximation, for this class of the molecules. The main problems the SCF approximation encounters are the following (for more detailed discussion see [14] and references terein):

(i) the Koopmans’ theorem is not valid for the states with large contributions from the atomic d-states; (ii) the Aufbauprinzip is frequently violated for the orbitals with significant contributions from the atomic d-states; (iii) the iteration procedures implied by the SCF approximation converge very slowly or oscillate.

The violation of the Koopmans’ theorem is the most spectacular among the problems listed above. This theorem appears as a consequence of the SCF approximation where each electron is treated as if it moved in a mean field induced by nuclei and other electrons. The ionization potentials are then equal to minus energies of electrons in such a potential (minus orbital energies). This picture is an approximation since the mean field itself changes when an electron is removed from the system or added to it. The difference between the minus orbital energy which must be the ionization potential according to the Koopmans’ theorem and the real ionization potential is called relaxation energy. For organic molecules the relaxation energies are usually small and the whole picture remains consistent. In the case of TMC the relaxation energies can reach the values from 10 to 20eV when the levels with some significant contribution from d-orbitals are involved [15]. This suggests that the real behaviour of d-electrons in TMC does not fit in the picture where independent electrons move in the mean field induced by the nuclei and other electrons but by contrast trace any motion of each other very carefully. In other words d-electrons are strongly correlated. At the same time namely d-electrons are responsible for the ground state total spin and for the low energy excitations of TMC whereas the description of electrons in the ligands seems to be less problematic.

Another problem, which is rarely recognized as a problem, is the violation of the Aufbauprinzip (the principle of the occupation of MOs by electrons by two from the bottom), which frequently happens in the semiempirical calculations on TMC [1-4]. In this case as in the case of the Koopmans’ theorem the orbitals which turn out to be unoccupied or singly occupied but have lower orbital energy than some doubly occupied orbitals are the orbitals formed largely by the d atomic orbitals. One can check [16] that the Slater determinants breaking the Aufbauprinzip do not present a minimum of the Hartre - Fock energy functional rather a saddle point. This in its turn causes the well known problems with the convergency of the SCF iterations. However, more important than the convergency problems is the doubtfulness of very idea to parameterize a semiempirical Fock operator for transition metals using throughout the calculation a trial wave function which does not represent a stable solution of the SCF problem.

The instability of the Hartree - Fock solutions for TMC does not create any physical problem by itself. It simply means that the ground state cannot be presented by a single Slater determinant and some other configurations must be added. The problem which arises in this concern is that the CI series taking the Hartree - Fock orbitals as a one-electron basis set converge rather slowly and thus a large number of configurations must be taken.

From the above discussion we see that there are some fundamental problems in the semiempirical description of the electronic structure of TMC. They are related to the highly correlated nature of d-electrons which are ultimately responsible for both the symmetry and the multiplicity of the ground state of TMC and for their low-energy excitation spectrum. In order to find a way out of all the problems outlined above let us discuss first the basic properties related to the TMC electronic structure as they are known from experiment. It should be mentioned first of all that the experimental data concerning the d − d excitation spectra of TMC can be successfully fit to the models equivalent to the crystal field theory (CFT) [17]. That means that specific properties of TMC are determined by the d-electrons of the transition metal ion. Their low energy excitations are responsible for the absorption bands in the optical spectra and for the magnetic properties. The ground state spin depends on the balance between the electron repulsion of d-electrons and their interaction with the ligands which provide some external field. The excitations of the ligands have much larger energies than those in the d-shell and incidentally the ligands have a closed electronic shell so that they cannot directly affect the spin multiplicity of the complex. The features formulated above apparently correspond to the situation covered by the crystal field theory where all the interesting events take place in the d-shell of the transition metal ions whose ligand environment remains inert [17]. The picture provided by the CFT is correct in a large extent. The majority of magnetic and optical experiments on TMC can be interpreted in the framework of the CFT operating with the multiplets of some fixed number of d-electrons in the external field of appropriate symmetry (see for example [18]). The reason why the CFT is that successful is the correct form of the electronic wave function which is implicitly used in it. It obviously takes the wave function of the complex as an antisymmetrized product of the multiplet (full CI) state for some fixed number of d-electrons and of some closed shell state of the remaining ligand electrons. However, the wave function of this remainder is not considered explicitly and that is the reason why the splitting parameters of the CFT cannot be satisfactorily calculated within its own framework. In the effective Hamiltonian method [14] two most important features of the electronic structure of TMC, namely, the presence of an isolated group of strongly correlated d-electrons on the metal atom and the existence of the closed shell ligands are taken into account explicitly. The distinction between the ligands and the d-shell of TMC is used in order to describe d-electrons and electrons in the ligand orbitals employing different levels of accounting of the electron correlations. The following features were implemented in the method: (1) the multiplet structure of d-electrons in the central ion (to do so the the configuration interaction (CI) is completely taken into account in the d-shell); (2) electronic structure of the ligands must be treated realistically, but the single determinant approximation is sufficient for them; and (3) the weak covalency resulting from the metal-ligand interactions must be included as well.

The method had been parameterized and then applied to the metal fluorine complexes, metal hexahydrate and hexamine complexes, metal hexachloro and tetrachloro complexes, and metallocenes [19-22]. In all the cases studied we observe the perfect agreement between the experimental data concerning the spin and symmetry of the ground state of the complex and our calculation. The spectra of the d − d excitations were reproduced within the accuracy of 1000cm^{−1}.

The ML_{4}Z_{2} complexes of tetragonal symmetry D_{4h} are of specific interest for our theory. The assignment of the absorption bands in the d − d-spectra of these complexes is normally made on the basis of some presumed picture of the d-levels split by the ligand field. This picture is frequently based upon some qualitative reasoning on the basis of the positions of the axial and equatorial ligands in the spectrochemical series [18]. Empirical approaches of that kind clearly are not capable to explain the splitting patterns obtained by such fits. As an example we can mention a controversy lasting for more than a decade concerning the relative position of the b_{2}(xy) and the e_{g}(xz, yz) levels in the fields of tetragonal symmetry. Our method by contrast allows one to calculate directly the splitting parameters and the energies for the d-multiplets in the crystal fields for various ligand arrangements around the central ion. For that reason it seems to be interesting to analyze the results of calculations on the tetragonal complexes ML_{4}Z_{2} and to compare these results with existing experimental data on d − d spectra of these complexes.

II. Account of the method

The formal derivation of the effective Hamiltonian method proceeds as follows [14]. We separate the whole set of the valence atomic orbitals (AO) of the TMC (it includes the 4s−, 4p- and 3d-AOs of the metal and the valence AOs of the ligand atoms) into two parts. The first part contains only 3d-orbitals of the transition metal atom (d-subsystem). The second part contains 4s−, 4p-AOs of the transition metal and the valence orbitals of the ligand atoms (ligand subsystem or l-subsystem). Then the total Hamiltonian for a TMC can be written as follows:

H = H_{d} + H_{l} + H_{c} + H_{r}  (1)

where H_{d} is the Hamiltonian for the d-electrons in the field of the atomic cores of TMC, H_{l} is the Hamiltonian for the electrons of the l-subsystem, H_{c} and H_{r} are respectively the Coulomb and the resonance interaction operators between the two subsystems.

For most of TMC with closed-shell ligands the excitations in the ligand subsystem are very high in energy as compared to the energies of the d − d excitations, and thus their contribution is negligible. The ground state of the ligand subsystem can be described by a Slater determinant Φ_{l}(^{1}A_{1}) with zero total spin. The wave function Φ_{n} for the n-th electronic state of a TMC then takes the form:

Φ_{n} = (∑_{k} C_{k}^{n} | n_{d}k >) ∧ Φ_{l} = Φ_{d}^{n} ∧ Φ_{l}  (2)

where | n_{d}k > are the spin and symmetry adapted n_{d}-electron wave functions constructed on the metal d-orbitals; C_{k}^{n} are variation parameters. Both the spin multiplicity and the point symmetry of the functions of this type coincide with the multiplicity and the symmetry of the functions Φ_{d}^{n} of the d-subsystem.

The wave functions of the type Eq. (2) correspond to the fixed integer number (n_{d}) of electrons in the d-shell of the metal ion. It is not that bad from the point of view of description of electron distribution itself since for most of the complexes the total charge transfer between the d-shell and the ligands usually does not exceed a few hundredth. A more serious problem is that all the matrix elements of the resonance operator H_{r} calculated with the functions of that type are vanishing. That prevents any correct description of the interaction between the ligands and the d-shell within the above class of the trial wave functions with the Hamiltonian Eq. (1). In order to include the effects of the resonance interaction between the subsystems we consider the effective Hamiltonian Heff[14] which operates in the subspace spanned by the functions Eq. (2). Its eigenvalues coincide with those of the exact Hamiltonian Eq. (1):

Heff = P H_{o}P + H_{RR}
H_{o} = H_{d} + H_{l} + H_{c}  (3)
H_{RR} = P H_{r}Q(EQ − QH_{o}Q)^{−1}QH_{r}P

Here P is the projection operator on the subspace of functions with fixed number of d-electrons; Q = 1 - P.

The variational problem for the effective Hamiltonian Heff within the subspace spanned by the wave functions of the type Eq. (2) splits into a pair of interconnected equations for the functions Φ_{d}^{n} and Φ_{l}(see e.g. [14,23]):

Heff_{d}Φ_{d}^{n} = E_{d}^{n}Φ_{d}^{n}  (4)

Heff_{l}Φ_{l} = E_{l}Φ_{l}

with the effective Hamiltonians for the subsystems defined by:

Heff_{d} = H_{d}+ < Φ_{l} | H_{c} + H_{RR} | Φ_{l} >  (5)

Heff_{l} = H_{l}+ < Φ_{d}^{n} | H_{c} + H_{RR} | Φ_{d}^{n} >

Since the l-subsystem is described by a single Slater determinant Φ_{l} the latter must be found from the self-consistent field procedure applied to the Fockian F eff_{l} derived from the Hamiltonian Heff_{l} by the standard method [23,24]. Proceeding semiempirically we apply the standard CNDO parameterization [24] for all ligand atoms. The core attraction parameters U_{ss} and U_{pp} of the metal 4s- and 4p-orbitals and the metal core charge Z_{M} are renormalized according to:

U_{ss} → U_{ss} + n_{d}g_{sd}
U_{pp} → U_{pp} + n_{d}ḡ_{pd}  (6)
Z_{M} → Z_{M} − n_{d}

Here g_{sd} = (ss | dd)−(sd | ds)/2; ḡ_{pd} is the average of the integrals g_{iμ}, where i = 4p_{x}, 4p_{y}, 4p_{z} and μ = 3d_{z^{2}}, 3d_{xz}, 3d_{yz}, 3d_{x^{2}}_{−y^{2}}, 3d_{xy}.

The solution of the Hartree-Fock problem for the l-subsystem with the above approximate CNDO Fockian gives the one electron density matrix P_{kl}, the energies of the molecular orbitals (MO) ϵ_{i}, and MO LCAO coefficients c_{ik}[24]. These quantities completely describe the electronic structure of the l-subsystem within the accepted approximation. They are used to construct the effective Hamiltonian Heff_{d} Eq. (5) in the following way. The operators H_{c} and H_{RR} are averaged over the ground state Φ_{l} of the l-subsystem which leads to the effective Hamiltonian Heff_{d} of the form [14]:

Heff_{d} = ∑_{μνσ} Ueff_{μν}d_{μσ}^{+}d_{νσ} + \frac{1}{2} ∑_{μνρη} ∑_{στ}(μν | ρη)d_{μσ}^{+}d_{νσ}d_{ρτ}^{+}d_{ητ}  (7)

where the Coulomb interaction between d-electrons is taken as in the free ion and the effective core attraction parameters for metal d-electrons Ueff_{μν} contain the corrections originating both from the Coulomb and the resonance interaction of d-electrons with the l-subsystem:

Ueff_{μν} = δ_{μν}U_{dd} + W ion_{μν} + W cov_{μν}  (8)

where

W ion_{μν} = δ_{μν}( ∑_{i∈s,p} g_{μi}Pii) + ∑_{L}(P_{LL} − Z_{L})V_{μν}^{L}  (9)

is an ionic term having the standard CFT form and

W cov_{μν} = − ∑_{i} β_{μi}β_{νi}(\frac{(1 − n_{i}/2)}{ΔEdi} − \frac{(n_{i}/2)}{ΔEid})  (10)

comes from the resonanse. Here Pii is the diagonal matrix element of the one-electron density matrix of the ligand subsystem; P_{LL} = ∑_{l∈L} P_{ll} is the electronic population on the ligand atom L; Z_{L} is the core charge of the ligand atom L; V_{μν}^{L} is the matrix element of the potential energy operator describing the interaction between ad-electron and an electron placed on the ligand atom L; n_{i} is the occupation number of the i-th ligand MO(n_{i} = 0 or 2); ΔEdi(ΔEid) is the energy which is necessary to transfer an electron from the d-shell (from the i-th MO) to the i-th MO(to the d-shell); β_{μi} is the resonance integral between the μ-th d-orbital and the i-th ligand MO which is expressed through the resonance integrals β_{μk} between the μ-th d-orbital and the k-th ligand AO:

β_{μk} = −(I_{d} + I_{k})S_{μk}β^{ML},

where I_{d} and I_{k} are the valence state ionization potentials, S_{μk} is the overlap integral between the μ-th d−AO and the k-th ligand AO, and β^{ML} is the only fitting parameter specific for each ML pair where M stands for a transtion metal atom and L stands for a donor atom in the ligand (like nitrogen, oxygen etc.).

When deriving the expression Eq. (8) for the matrix element Ueff_{μν} between the d-orbitals we lifted some of the limitations of [14]. First, we do not need any more the high (not less than tetragonal) symmetry of the ligand environment and, second, we neglect the splitting of d-levels by the electric field of the charges residing in the ligand system when calculate the energy denominators occurring in the above formula for the covalent contribution

W cov_{μν}.

The covalence term dominating the d-level splitting is apparently analogous by its origin to the ligand field parameters of the angular overlap model (AOM) [25]. The important difference between them is that in the AOM all the ligand field parameters are fit to the experimental spectra for each ligand and are not transferable from one ligand to another even if the donor atom is the same. In our approach the electronic structure of the ligands is taken into account explicitly. That allowed to parameterize (see [14,19-22] and below) only the magnitude of the hopping between the orbitals of the given donor atom and d-orbitals of the given transition metal. The same value can be used for all the ligands containing given donor atom, thus allowing to reproduce the effects of the variations of the ligand structure on the crystal field induced by the ligands and thus on the d − d spectrum.

After the effective Hamiltonian for d-electrons is constructed the states of n_{d} electrons in the d-shell are calculated by diagonalizing the matrix associated with Heff_{d} in the n_{d} electron wave function basis set. The ground state of the whole complex is then obtained by taking the external product of the ligand Slater determinant and that n_{d}-electron wave function obtained by the diagonalization of Heff_{d} which has the lowest eigenvalue. The d − d excited states (they are the low lying excitations of the whole complex) are then obtained by multiplying other n_{d}-electron functions by the same Slater determinant. The d − d excitation energies can be then estimated as the differences between the eigenvalues of the correspondidng eigenstates of the effective Hamiltonian Heff_{d}.

III. Results and discussion

We applied the method described in the previous section to calculate electronic structure of the ligand sphere and d-shells of a series of the ML_{4}Z_{2} compounds where M = V, Co, Ni; L = H_{2}O, NH_{3}, P y; Z = H_{2}O, NCS^{−}, Cl^{−}. The spin multiplicity and the spatial symmetry of the ground states of the considered mixed complexes are correctly reproduced by our method (the parameters β^{ML} were found from the calculations [20,21] on the corresponding pure complexes (ML_{6}^{2+} and ML_{6}^{4−}). For all three considered Ni(II) complexes (Ni(NH_{3})_{2}(NCS)_{2}, Ni(P y)_{4}Cl_{2}, and Ni(P y)_{4}(H_{2}O)_{2}^{2+}) the ^{3}B_{1g} state is the ground state in accord with the experiment [26,27]. The detailed account of these results will be presented elsewhere. In this paper we concentrate mainly on two hydrochloro compounds: V (H_{2}O)_{4}Cl_{2} and Co(H_{2}O)_{4}Cl_{2}.

V (H_{2}O)_{4}Cl_{2}. In this complex water molecules occupy the equatorial positions whereas the axial positions are occupied by the chlorine atoms [28].

The following interatomic distances were used throughout the calculation: R(V − O) = 2.15Å(as in the V (H_{2}O)_{6}^{2+} complex) and R(V − Cl) = 2.50Å(as in the VCl_{6}^{4−} complex).

The 3a_{2u} HOMO of the l-system of this complex is composed mainly by the p_{σ}-orbitals of the chlorine atoms, whereas its 5e_{u} LUMO is formed largely by the p_{x(y)}-orbitals of the vanadium atom. This composition is analogous to that of the frontier orbitals of the l-system in the VCl_{6}^{4−} complex [20]. Comparing the atomic charges in the V (H_{2}O)_{4}Cl_{2} complex with those in the V (H_{2}O)_{6}^{2+} and VCl_{6}^{4−} complexes [20,21], one can see that the substitution of the two water molecules in the axial positions by the Cl^{−} ions diminishes the total charge on the central ion as compared to the hexaquacomplex due to the transfer of the electronic density from the Cl^{−} anions to the 4s- and 4p_{z}- metal orbitals. The atomic charges residing on the oxygen atoms do not change. The charges on the chlorine atoms are slightly smaller than those in the VCl_{6}^{4−} complex.

Now let us turn to the results of our calculation on the d-system of the V (H_{2}O)_{4}Cl_{2} complex. The term ^{4}B_{1g} is the ground state in accord to the experiment [28]. The assignments of the d − d-transitions in [28] were based on the following assumption concerning the splitting of the d-levels:

e_{g}(xz, yz) < b_{2g}(xy) < b_{1g}(x^{2} − y^{2}) < a_{1g}(z^{2})  (11)

Our calculation gives another picture of the d-level splitting:

b_{2g}(xy) < e_{g}(xz, yz) < b_{1g}(x^{2} − y^{2}) < a_{1g}(z^{2})  (12)

For the three electrons placed in the d-shell of the complex the both give the same ground state since irrespective to its sign the parameter dπ of the tetragonal splitting between the b_{2g}(xy) and e_{g}(xz, yz) orbitals is small. The spectral data of [28] do not allow one to distinguish between the pictures given by (11) and (12).

Let us turn to the analysis of the ionic and the covalent contributions to the d-level splitting in the V (H_{2}O)_{4}Cl_{2} complex. According to our data the splitting between the d_{z^{2}}- and d_{x^{2}−y^{2}}-orbitals described by the dσ parameter is equally contributed by the ionic (large difference of the negative charges residing on the oxygen and chlorine atoms: -0.397 and -0.659 respectively) and the covalent contributions. The greater destabilization of the d_{z^{2}}-orbital as compared to the d_{x^{2}−y^{2}}-orbital is appears from the virtual electron transfer from the 3s- and p_{σ}-orbitals of the Cl atoms to the d_{z^{2}}- orbital. The splitting between the d_{xy} and d_{xz}, d_{yz} is controlled largely by the ionic contribution. The positive sign of the dπ parameter (which means that b_{2g}(xy) < e_{g}(xz, yz)) is conditioned by the larger destabilization of the d-orbitals of the e_{g}-symmetry due to the additional contribution from the virtual transfer of an electron to these orbitals from the high occupied 2e_{g} −MO formed by the p_{π}-orbitals of Cl.

Co(H_{2}O)_{4}Cl_{2}. Calculations on this complex present particular interest since it was an object of thorough experimental studies [29-31] which, however, have not brought definitive interpretation of its electronic structure. The results of the measurements were interpreted in [29,30] in terms of the tetragonal crystal field. The measurements of the magnetic momenta [31] and of the polarized electronic absorption spectra performed on the monocrystals [29,30] revealed that the ground state of the complex is a high-spin orbitally nondegenerate state. For the tetragonal geometry assumed in [29,30] it is the ^{4}A_{2g} term appearing from the split octahedral ^{4}T_{1g} term. Irrespective to the details of the assignements of the optical transiitons the both works consistently place the b_{2g}(xy) level above the degenerate e_{g}(xz, yz) level (Eq. (11)). The reason was that within the tetragonal field the ^{4}A_{2g} ground state for seven d-electrons (three holes) is possible only for this relative position of these levels. In terms of the splitting parameters that means that dπ is negative. We performed the calculations for the D_{4h} geometry with the following interatomic distances: R(Co−O) = 2.12Å and R(Co−Cl) = 2.43Å[29] and obtained a positive value of dπ. That automatically led to the ^{4}E ground state in a sharp contrast with the experimental data. However, what one would expect from a molecule with the spatially degenerate (E) ground state is the Jahn - Teller distortion. That is exactly what happens. The real geometry of the complex is not tetragonal, but can be described as an elongated octahedron with an additional distortion placing the equatorial water oxygens in the corners of a rectangle. For the real (distorted) geometry the degenerate ground state is not possible regardless the sign of the dπ parameter. For the experimental D_{2h} geometry where the water oxygens occupy the corners of a planar rectangle with the sides of 3.05 and 2.94Å respectively [29]. The water molecules in the real structure are only slightly shifted from their idealized positions and for that reason the results of the calculation on the ligand subsystem do not differ too much for the real and idealized structures. We performed the calculation and found that the sign of dπ, of course, does not change because of the slight geometry variations accompanying the Jahn - Teller distortion. However, the ground state of the complex is not spatially degenerate any more though it is still spin degenerate. The d − d excitation spectrum is reasonably reproduced within the accuracy of 1000cm^{−1}. For example, in our calculation on the spectrum we do not have spurious low-energy d − d-states with the energies of ca. 1000 − 1500cm^{−1} which were present in the fit of [30] without any explanation. We did not try to reproduce the spectral data in detail together with the assignments given in [29,30] since they differ significantly and are both based on the wrong symmetry of the crystal field. However, it is worthwile to note that we reproduce the general features of the spectra common for the both experimental works (see Table 1). In our calculation performed with the Racah parameters B = 853cm^{−1}C = 3687cm^{−1}[30] we have the spin allowed bands in the region 6000 − 9000cm^{−1}; relatively weak (spin forbidden) transitions in the 12000 − 14000cm^{−1} region; a series of the bands of different intensity in the 18000 − 19000cm^{−1} and 20000 − 22000cm^{−1} regions; and weak bands about 25000−27000cm^{−1} - all in perfect agreement with the available experimental data.

As it was in the case of V (H_{2}O)_{4}Cl_{2}, the transition from the hexahydrocobalt complex to the mixed complex diminishes the charge on the metal due to electron density transfer from the chlorine atoms to the 4s- and 4p_{z}- orbitals of the metal. The charges on the axial and equatorial donor atoms differ significantly (-0.575 and -0.378, respectively).

The analysis of the contributions to the d-level splitting in the Co(H_{2}O)_{4}Cl_{2} complex shows that the positive value of the dσ parameter (b_{1g}(d_{x^{2}−y^{2}}) < a_{1g}(d_{z^{2}})) is mainly due to presence of the high energy occupied 4a_{1g} − MO formed by the p_{σ}-orbitals of oxygens and chlorines. Since this MO is much higher in energy than the occupied 2b_{1g} − MO formed by the 2s- and p_{σ}- oxygen orbitals and the energy of the corresponding virtual electron transfer from this MO to the corresponding d-orbital is smaller as compared to the analogous process for the 4a_{1g} − MO and the destabilization of the a_{1g}(d_{z^{2}}) orbital is larger than that of the b_{1g}(d_{x^{2}−y^{2}})-orbital. If the splitting of the b_{2g}(d_{xy})- and e_{g}(d_{xz}, d_{yz})-orbitals were controlled by the covalent contribution only that would give negative value of dπ. The reason is on one hand the larger resonance integral between the filled 1b_{2g} − MO of the ligand system and d_{xy}-orbital as compared to the integral between the 2e_{g}-HOMO and the d_{xz}−, d_{yz}-orbitals, leading to the larger destabilization of the d_{xy}-orbital and on the other hand the presence of the stabilizing contribution for the d_{xz}−, d_{yz}-orbitals coming from the virtual electron transfer from these orbitals to the vacant 4e_{g} − MO of the ligand subsystem. Nevertheless, the ionic contribution to dπ coming from the difference between the charges on the equatorial and axial ligands overcomes, and the total dπ is positive (like (12)).

IV. Conclusion

In the present paper we describe the effective Hamiltonian method and analyze the results of its application to the calculation of the d-level splitting and d − d spectra in the similar tetrahydrodichloro complexes of divalent vanadium and cobalt. Purely experimental approach to the problem of the relative positions of the d-levels led to contradictive results. The effective Hamiltonian method allows one to perform systematic calculations of the ligand field for various ligand environments. It predicts for both the complexes the same order of the d-levels (12) which seems to be reasonable in view of the similarity of their composition.

The spatially degenerate ground state of seven d-electrons arises within our method in the case of Co(H_{2}O)_{4}Cl_{2} in a visible contradiction to the experimental data. It allows, however, to rationalize the observed distortion of the equatorial square formed by the water molecules to the rectangular configuration. For dπ < 0 as it is adopted in [29,30] the only possible ground state for seven d-electrons in the tetragonal environment is^{4}A_{2g}. It correlates with the experiment only in that respect that the ground state is not spatially degenerate. However, for such a ground state it is difficult to explain the observed distorted geometry. On the other hand for dπ > 0 as it results from our calculation for the tetragonal ligand arrangement the only possible high spin ground state for seven d-electrons is ^{4}E. In this case the required distortion arises naturally due to the Jahn - Teller theorem. Incidentally, the distorted geometry is actually observed for the Co(H_{2}O)_{4}Cl_{2} compound. This distortion would have no reason to appear should the ground state of the tetragonal d^{7}-complex be spatially nondegenerate. We consider this as a rare case when X-ray measurements make it possible to visualize the order of d-levels in the crystal field. Since it is confirmed by the experiment it may serve as an argument in favor of the positive sign of dπ and thus of the capability of our method to reproduce some fine balance of the different contributions to the crystal field. We can thus conclude that the calculations by the method [14] provide important supplementary information to the experimental probes of the electronic structure of TMC.

Acknowledgments

This work has been largely supported by the Russian Foundation for Fundamental Research (RFFR) through the grant 93-03-4276. This work has been completed during a visit of one of us (ALT) to the Max-Planck-Institut für Physik komplexer Systeme, Dresden whose hospitality is gratefully acknowledged as the generous support from the Laboratoire de Chimie Inorganique, Université de Paris-Sud, Orsay (URA CNRS 420), which made possible the visit of ALT to Prague and his participation in the work of the 8-th International Congress on Quantum Chemistry.

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State Calculated energies (cm^{−1}) Observed bands [29,30]
4B_{2g}  0
4B_{3g}  152
4B_{1g}  465
4B_{2g} 6273 6000 - 9000
4B_{3g}  6342
2B_{1g} 8626 broad strong
4A_{g}  9036
2A_{g}  12556  12000 - 14000
2A_{g}  14447  weak
4B_{1g}  15487  15000 - 17000
2B_{2g}  16356
2B_{3g} 16438 broad shoulder
2B_{1g}  17626
2B_{2g}  18102
2B_{3g}  18196
4B_{1g} 18382 18600
2B_{3g}  20634
2B_{2g}  20903
2B_{1g}  21866
4B_{3g}  22179  22250
4B_{2g}  22438  broad
2A_{g}  22480
2B_{2g} 25280 25000 - 27000
2B_{3g}  25295
2A_{g} 26500 weak broad
2B_{3g}  26754
2B_{2g}  26836
2B_{1g}  27488
2B_{1g}  27992