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A.V. Sinitsky**, M.B. Darhovskii**, A.L. Tchougréeff*,**, and I.A. Misurkin*
* Karpov Institute of Physical Chemistry
Vorontsovo Pole 10, 103064, Moscow, Russia;
** Center for Computational Chemistry
at the M.V. Keldysh
Institute for Applied Mathematics RAS

Effective crystal field for trivalent \\ first transition row ions

Effective crystal field for trivalent
first transition row ions

Abstract

In the present work the semi-empirical effective crystal field (ECF) method previously designed for electronic structure calculations of transition metal complexes (TMC) and utilizing non-Hartree-Fock trial wave function and parameterized for complexes of doubly-charged Cr2+, V2+, Mn2+, Fe2+, Co2+, and Ni2+ cations is extended to complexes of triply-charged cations of 3d-elements. With the parameters adjusted the ECF method is applied to calculations of ground states and low-energy spectra of the d-shells of fluoro-, chloro-, aqua-, amino- and cyano-complexes of the triply-charged cations. Obtained total spin and symmetry of the ground states match with the experimentally observed ones. Satisfactory agreement between the calculated and experimental d-shell electronic transition energies is achieved as well.

Introduction

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 10Dq 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.

Effective Crystal Field method

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 YM, and the single determinant wave function YL for remaining electrons of the TMC:
F = YM(nd)YL(nl).
(0)
Such a function is a special case of the group function [] approximation where the groups are formed by electrons in the d-shell and those in other orbitals, respectively. This form of the wave function allows to use difference of the characteristic values of interaction parameters in the d-shell and in the ligands for taking into account electron correlations at different levels.

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 nd 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:
H = Hd+Hl+Hc+Hr,
(0)
where Hd is the Hamiltonian for d-electrons in the field of the TMC atomic cores, Hl is the Hamiltonian for the ligand subsystem electrons, Hc and Hr are, respectively, the operators of Coulomb and resonance interactions between two subsystems of the TMC.

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 nd 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:


Heff
=
PH0P+Hrr
H0
=
Hd+Hl+Hc
Hrr
=
PHrQ(EQ-QH0Q)-1QHrP
(0)

where P is the operator projecting to the subspace of the functions with the fixed number nd of the d-shell electrons and nl = N-nd is that of the ligand subsystem electrons; and Q = 1-P.

The TMC eigenstate energies must be obtained from the relation:
En = < Fn | Heff(En) | Fn > .
(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:
Heff(En) Heff(E0),
(0)
where E0 is the ground state energy of the Hamiltonian H0. Thus obtained effective Hamiltonian corresponds to the second order of the Raleigh-Schrödinger perturbation theory in Hr. Variation principle applied to the effective Hamiltonian with the trial function of the above form leads to the self-consistent system of equations:


HdeffFdn
=
EdnFdn
HleffFl
=
ElFl.
Hdeff
=
Hd+ < Fl | Hc + Hrr | Fl >
Hleff
=
Hl+ < Fdn | Hc +Hrr | Fdn > .
(0)

In the above system the effective Hamiltonian Hdeff for the d-electron subsystem depends on the wave function of the ligand subsystem Fl, and in its turn the effective Hamiltonian Hleff for the ligand susbsytem depends on the d-electrons' wave functions Fnd. These equations must be solved self-consistently as well. In the ECF method Ref.[] the Slater determinant Fl, 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 Hleff for the ligand subsystem to the corresponding effective (dressed) Hamiltonian reduces to renormalization of one-electron parameters related to the transition metal ion:
Uiieff = Uii+ 1
5
nd

m 
gmi
ZMeff = ZM-nd.
(0)
where Uii is the parameter of the interaction of 4s and 4p-electrons (i = 4s, 4px, 4py, 4pz) with the metal core, ZM is metal core charge, gmi are the Oleari parameters of intraatomic Coulomb interactions. The Fl function thus obtained is used further for constructing the effective Hamiltonian for the d-shell. Diagonalization of the latter gives both the wave functions of the d-shell and the d-electron state energies. The effective Hamiltonian for the d-shell after averaging eq. (6) intersubsystem interaction operators Hc and Hrr over the ground state of the ligand system takes the form:
Hdeff =

mns 
Umneffdms+dns+ 1
2


mnrh 


st 
(mn | rh)dms+drt+dhtdns,
(0)
where dms+ (dns) are operators of creation (annihilation) of electron with the spin projection s on the m-th d-AO; (mn | rh) are the two-electron integrals of the Coulomb interaction in the d-shell. Effective one-electron parameters Umneff of the d-shell contain contributions from the Coulomb and from the projected (eq.(5)) resonance interaction with the ligand subsystem:
Umneff = dmnUdd+Wmnatom+Wmnion+Wmncov,
(0)
where
Wmnatom = dmn(

i s,p 
gmiPii)
Wmnion =

l 
(Pll-Zl)Vmnl.
(0)
Here Pii is one-electron density matrix element for the ligand subsystem, Pll = i lPii, Zl is the l-th atom core charge, Vmnl is the matrix element of the d-electron potential energy in the electrostatic field of a unit point charge placed on the l-th ligand atom. The covalence contribution to the ECF is given by
Wmncov = - (MO)

i 
bmibni

(1-ni)2
DEdi
- ni2
DEid


(0)
where bmi is the resonance integral between the m-th d-orbital and the j-th ligand MO, nj (=0,1) is the occupation number of the j-th MO, DEdi (DEid) are excitation energies required to transfer an electron from the d-shell (i-th MO) to the i-th MO (d-shell). The intersubsystems resonance integrals are calculated by the equation:
bmk = (Id+Ik)SmkbM-L
(0)
where Id and Ik are valence ionization potentials of the d-shell and of the k-th AO in the ligand subsystem respectively, Smk is the AO's overlap integral, bM-L is a dimensionless parameter scaling the resonance interaction between the d-shell and a ligand atom. Charge transfer energies DEdi and DEid are estimated according to:
DEdi
=
Id+ei-Gdi,
DEid
=
-Ad-ei-Gdi ,
(0)
where ei is the i-th MO energy, Gdi is the Coulomb interaction energy between an electron and a hole localized in the d-shell and on the i-th MO respectively; Id and Ad are, respectively, the ionization potential and electron affinity of the d-shell.

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 10Dq 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.

Results and discussion

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 3d-shell of the corresponding isolated cation, e.g. the d-electron subsystem of the Fe3+ complexes has 3d5 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 Udd, zd, 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 Udd for each metal ion and the bM-L resonance parameters for each pair metal (M)-donor atom (L) have been fitted. The bM-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: V3+, Cr3+, Mn3+, Fe3+, and Co3+. (The data on Ni3+ and Cu3+ 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 Udd and bM-L parameters determining the value of one-electron splitting parameter 10Dq 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 10Dq and of the excitation energies is achieved. The values of Udd for each metal of the considered V, Cr, Fe, and Co series were modified so that the positive values of DEdj and DEjd Eq.(13) are guaranteed in order to ensure the stability of the system with respect to electron transfer between the subsystems. After that the bM-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 10Dq values, the ionic and covalent contributions to them and the respective experimental values are given in Table 1. The fitted values of the parameters Udd and bM-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 4s-,4p-, and 3d-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 Udd 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 bM-L for the trivalent cations are as a rule smaller than those for the divalent ones with exceptions of hexahydrate complexes of V3+ and Co3+. One also may notice that the values bM-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 Udd 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 [VF6]3+, [V(H2O)6]3+, and [CoF6]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 bM-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 10Dq 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 bM-L parameter variation along the row is broken. Right in the case of the cobalt hexaaquacomplexes pair the difference of bM-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 bM-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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