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M.B. Darkhovskii 1, A.M. Tokmachev and
A.L. Tchougréeff

L.Y. Karpov Institute of Physical Chemistry,
Vorontsovo Pole 10, 105064 Moscow, Russia

MNDO parameterized hybrid SLG / SCF method as used \\ for molecular modeling of Zn(II) complexes

MNDO parameterized hybrid SLG / SCF method as used
for molecular modeling of Zn(II) complexes

Dedicated to Prof. A.A. Levin on the occasion of his
75th birthday


The McWeeny's group functions technique is a natural way to introduce local description into quantum chemistry. It can be also a basis for construction of numerically effective computational schemes with almost linear scaling of computational costs with the size of a system. The scheme of strictly local geminals / self-consistent field (SLG/SCF) is based on the general group function approach combining different descriptions for different electron groups: two-electron two-center bonds are described by geminals while those with other numbers of electrons are described by Slater determinants in the one-electron (SCF) approximation. The SLG/SCF approach was previously successfully applied to description of organic molecules. In this paper we apply it to the coordination compounds of Zn(II) containing ligands with nitrogen and oxygen donor atoms. In this class of molecules the group corresponding to the metal and its closest coordination sphere includes vacant 4s- and 4p-AOs of the Zn2+ ion and the lone pairs located on the donor atoms. This group is described by the one-determinant wave function, while the original MNDO Hamiltonian parameterization for Zn atom is adjusted for correct description of geometry and relative stability of the complexes.

Keywords: Zn(II) complexes, hybrid methods, strictly local geminals.


The problem of modeling metal ion complex formation remains actual in computational chemistry of coordination compounds for decades []. The progress in modern quantum chemistry allows to make reliable predictions on the electronic structure and properties of polyatomic molecular systems including metal complexes. However, high-accuracy methods, like ab initio and DFT, are still computationally too demanding when dealing with coordination compounds. Qualitative approaches, such as points-on-a-sphere (POS) [] or metal-ligand size-match selectivity concept [], still play an important role in theoretical coordination chemistry of both transition and non-transition metals. Thus, effective numerical although maybe semi-quantitative tools suitable for modeling coordination bonds and reproducing their qualitative aspects are strongly in demand. The recent studies reviewed in [] are devoted to analysis of chemical bonding in various metal complexes. They are mainly focused on the covalent metal complexes with more or less isolated two-center bonds. By this the electronic structure of compounds considered in [] differs on an intuitive level from the picture of unsaturable and nondirected ``coordination'' bonds characteristic for complexes of metals with organic ligands with donor atoms [].

Recently a new hybrid semi-empirical method called SLG/SCF [] based on the general strategy of describing molecular electronic structure in terms of electron groups residing in orthogonal carrier subspaces and each populated by integer constant numbers of electrons [] has been proposed. This method extends the original semi-empirical method [,,,] based on strictly local geminals (SLG) where the trial wave function is chosen in the form of antisymmetrized product of SLGs. Most of the geminal-based approaches proposed in the literature use somehow predefined carrier spaces of one-electron states, either empirically constructed or extracted from preceding Hartree-Fock calculations by some localization procedure []. By contrast, employing of the special form of one-electron basis states, that is, of strictly local atomic hybrid orbitals which are variationally determined from the energy minimum condition, is the important feature of the SLG approach [,,,]. The quantum chemical scheme [,,,] is implemented with use of semiempirical Hamiltonians. (Some works employing optimized carrier spaces later appeared in the ab initio context [,,].) The MINDO/3 [], MNDO [], AM1 [], and PM3 [] parameterizations for the molecular Hamiltonian were used. Accuracy of predicting molecular characteristics, such as heats of formation, molecular geometries [,,,], and vertical ionization potentials [] for the SLG based semi-empirical methods [,,,] is comparable and in some cases even better than that of the corresponding SCF-based methods. The characteristic features of these methods - linear scalability of required computational resources [], correct asymptotic behaviour of the electronic structure under the s-bonds cleavage, variational determination of one-electron states - make the SLG-based semiempirical quantum chemistry an important alternative to the common SCF-based semiempirical procedures. However, the SLG based methods are non-universal, since the SLG trial wave function assumes the electronic structure of a molecule to be well depictable by a combination of two-electron chemical bonds and lone pairs. It is clear that coordination compounds do not correspond to this picture, along with many other important molecular systems, like those containing conjugated p-bonds etc. The extension of the SLG approach to these classes of molecular systems with essential delocalization of the electron groups is proposed in []. It is done in the framework of the general group function theory [] by treating different electron groups by different procedures: some of the groups (related to the two-electron, two-center bonds and lone pairs) are described by the SLGs and others are described by single determinant build upon molecular orbitals. This extension of the SLG approach puts the SLG/SCF method among other hybrid methods, when different parts of a molecule are treated by different methods of quantum chemistry [].

In [] the analysis of electronic structure, heats of formation and geometries of a test set of organic molecules with p-electron system is given within this hybrid SLG/SCF method. The calculations show that molecular geometries and heats of formation for both p-conjugated double-bonds and aromatic systems formed by carbon atoms only and containing heteroatoms are reproduced with acceptable accuracy.

In the present work we extend the approach [] to the coordination compounds. Among these we chose for consideration the complexes of the Zn2+ ion. This choice is influenced on one hand by the fact that these compounds play a significant role in chemistry and especially in biochemistry, as active sites of many enzymes which contain the Zn2+ ions with different coordination numbers. The Zn(II) complexes are intensively studied by various methods of computational chemistry. There are model ab initio calculations of small systems like hydrates Zn(H2O)n2+ (see [] and references therein). Binding energies of some small complexes of Zn2+ with multiple ligands, such as acetone, N-methylacetamide, imidazole and water are studied within the DFT approximation in a series of papers [,]. In one of the mentioned papers [] the active site of the carbonic anhydrase enzyme is modeled by three imidazole and one water molecules and the free energy of the complex formation is calculated. However, the study was not extended to the complexes with larger ligands which characterizes the common capabilities of ab initio calculations in coordination chemistry of zinc. The proposed SIBFA (``sum of interactions between fragments ab initio computed'') [] molecular mechanics force field has been used to predict the structure of some Zn(II) model complexes with small ligands like ammonia, water, glycine etc., where methodology of empirical calculations of molecular structure and interaction energy is based on the fragmentation of the complex to the metal and ligands (flexible or rigid) and on the modeling of electrostatic interaction by that of transferable multipoles and polarizabilities distributed on the atoms and bonds of the fragments precalculated with use of an ab initio procedure []. The SIBFA force field was tested by comparing its energy components with ab initio ones singled out by the standard energy decomposition procedure based on the ``reduced variation space'' decomposition, and applied to rather polyatomic systems, such as complexes of mercaptocarboxamides with Zn2+ [,,]. An important feature of the SIBFA approach is the presence of energy contribution Ect estimated by the second order perturbation theory on the transfer of electron density from the lone pairs of donor atoms to the vacant orbitals of Zn2+ which is characteristic for the coordination bond formation. However, the expression for Ect in the SIBFA method contains large empirical scaling factor to reproduce the value of Ect taken from ab initio SCF calculations.

The COSMOS force field [] represents another approach to treating the coordination interactions between metal ions and donor atoms. It puts an accent on accounting the polarization of the ligands. The electron transfers are in contrast with SIBFA included implicitly through a scheme [] employing the Pauling's ``bond energy - bond order'' logarithmic relations []. The bond orders used in the COSMOS are set constants, not depending on interatomic separations, so that it is not possible to treat consistently elimination of one of the coordinated ligands. Also, the angular dependence of energy is oversimplified: it is completely switched off for coordination numbers larger than four, i.e., precisely for the cases when it might be expected to be important. The COSMOS procedure was applied to modeling Zn(II) hydrates with different numbers of coordinated water molecules, and to pentahydrated Zn2+ complexed with monodentate DNA-base ligands. The interaction energies of these model complexes appear lower than corresponding ab initio results cited in [], but are nevertheless reasonable. Also the COSMOS method [] has been applied to calculation of some four-coordinated complexes of zinc with experimentally determined molecular structure []. However, results of geometry and energy calculations of these ``real'' complexes are shortly reported in the cited work and are not compared with experiment in details. One of the weak points of these methods (both SIBFA and COSMOS) is that the results of calculations on model compounds are compared to ``precise'' ab initio calculations rather than to experimental data. The proposed force fields are not applied to coordination compounds of larger number of atoms.

Semi-empirical (AM1, PM3) approaches are also in use as methods of modeling for the latter complexes. In paper [] the accuracy of three semi-empirical methods (AM1, PM3 and MNDO/d) when applied to description of Zn(II) complexes is analyzed thoroughly. It is concluded that the MNDO/d method is the most reliable one for modeling zinc - donor atom interactions and gives more accurate results compared to other two methods. The authors of Ref. [] conclude that for coordination bonds of zinc atom with the oxygen and nitrogen donor atoms the MNDO/d method gives results of a satisfactory accuracy. However, the MNDO/d method encounters difficulties when describing complexes with polydentate ligands. This analysis partially inspired development of a new 'ZnB' parameterizaton [] of the PM3 method for Zn(II) compounds in biological systems. In that paper the heats of formation and geometry data both from the set of [] and the authors' [] extensive data set obtained for some zinc metalloenzyme mimics and water complexes by the DFT (B3LYP/6-311G*) calculations are used for parameterization. Indeed, the resulting coordination distances and heats of formation appear to be better described by the new ZnB parameterization than either by the PM3 itself or by AM1 or MNDO. (The authors of Ref. [] did not include the MNDO/d method for the comparison of distances and heats of formation.) On the other hand, the RMS (root-mean-squared) errors in distances for biological mimics and water complexes are still too large: 0.08 and 0.12 Å, respectively.

The MNDO/d method is applied to analysis of catalytic mechanisms of zinc-containing enzyme carboxypeptidase which catalyzes peptide's carbonyl group cleavage []. The enzyme's model active site contains zinc atom pentacoordinated by one water molecule and four amino acid residues as by monodentate ligands. A new possible transition state of the catalytic reaction consistent with some experimental data is found then.

Taking into account that in the previous work [] the SLG/SCF approach was tested mainly with the MNDO parameterization, we also decided to use this version of the hybrid SLG/SCF approach for calculations on Zn(II) complexes. Another reason of our choice is that the methods AM1 and PM3 seem to give artificial minima [] at intermolecular separations characteristic for most of coordination complexes.

In the next Section we briefly review the general structure of the SLG/SCF hybrid method. Then, we apply the proposed procedure to various Zn(II) complexes using an MNDO-based parameterization of the electronic Hamiltonian of the SCF part focused on the proper description of closest coordination sphere of the metal and of p-electron subsystems of the ligands when necessary. Then we discuss the results of our modeling.

Basics of the SLG/SCF hybrid method

The construction of the method is based on the electron group functions formalism []. The electronic structure of entire molecule is represented by a set of electron groups. These correspond to chemical bonds, lone pairs, local unpaired valencies, delocalized p-electron systems, or can be constructed in other way based on chemical intuition. The definition of any group includes the number of electrons in it and the set of one-electron basis functions spanning the corresponding carrier space. The settings for the group in a general case also include its spin state and a method (approximate form of the trial wave function) used for calculation of the group wave function. This general setting imposes restrictions on the molecular wave function. In a general semiempirical setting this weakness must be compensated by relevant parameterization. Nevertheless, the required modifications of the parameters according to our experience [] turn out to be marginal (although necessary). Below we briefly sketch the construction of the SLG/SCF approach which is presented in details in [].

The construction of the hybrid orbitals (HOs) reduces to an orthogonal transformation of the atomic orbitals (AOs) centered on atom A:

tps+ =

i A 
where ais+ is the creation operator of the electron with spin projection s on the ith AO, | tps ñ = tps+| 0 ñ is the HO, and the SO(4) matrix hA determines the transformation from AOs to HOs on the atom A bearing the valence sp-basis. (For the Fermi operator formalism see [,]; for an account of SO(n) groups see []). The orthogonality of matrices hA and that of the basis AOs centered on different atoms (within the semiempirical setting) ensures the orthogonality of the carrier spaces assigned to the groups and, thus, the strong orthogonality of the electronic wave functions for different groups. The transformation Eq. (1) is meaningful only in the case of non-hydrogen atom with orbitals belonging to at least two different electron groups.

Each HO is uniquely assigned to some electron group. If the group is to be described by the SCF method the molecular spin-orbitals are constructed as linear combinations of the hybrid spin-orbitals:
bi s+ =

p {S} 
where p runs over the HO's forming the one-electron basis in the S-th group treated by the SCF approximation and the expansion coefficients satisfy usual orthonormalization conditions:

p {S} 
csi pcsjp = dij .

Each geminal for the two-electron two-center bond is in its turn a superposition of three singlet two-electron configurations:
gm+ = umrma+rmb++vmlma+lmb++wm(rma+lmb++lma+rmb+),
which are two ionic configurations (both electrons are on the same end of a chemical bond) and the covalent (Heitler-London type) one, respectively, representing a two-center two-electron bond (the HOs at the ``right'' and ``left'' ends of the bond are denoted as | rm ñ and | lm ñ ). The normalization condition for the geminal amplitudes um, vm, and wm reads:
á 0| gmgm+| 0 ñ = um2+vm2+2wm2 = 1.
In the case of an electronic lone pair only one configuration survives with the amplitude equal to unity:
gm+ = rma+rmb+
This may be equally considered either as an SCF or as an SLG function.

The wave function for the entire molecule in the SLG/SCF approximation is represented by the antisymmetrized product of geminals and molecular orbitals:
| Y ñ =


[(i { S}) || (s)] 

| 0 ñ
where index S runs over all SCF-treated groups and i runs over the occupied MOs formed according to Eq. (2) from the basis states spanning the S-th SCF treated group. Therefore, the wave function Eq. ( 7) is quite general.

The electronic Hamiltonian in the NDDO approximation is transformed as well to the basis of HOs. This transformation preserves the structure of the Hamiltonian. The expressions for molecular integrals in the HO basis through the transformation matrices hA and molecular integrals in the AO basis can be found in Refs. [,]. The total energy of the molecule is found as usual by averaging the electronic Hamiltonian over the electronic wave function and adding contribution representing the core-core repulsion specific for the MNDO scheme [].

The energy of the molecule is minimized according to variational principle with respect to the electronic structure variables which are the MO LCAO coefficients csi p Eq.(2), the Jacobi angles defining the hA matrices (and by this the shapes and directions of the HOs) and the amplitudes um, vm, and wm defining the geminals. An iteration scheme which alternates the HOs optimizations defining the groups' carrier spaces and determinations of electron structure variables for the groups (the amplitudes of the geminals and MO coefficients respectively) is implemented in Ref. []. The optimal HOs are obtained by gradient minimization of the energy with respect to sextuples of parameters determining transformation matrices hA []. The geminal amplitudes um, vm , and wm are obtained by diagonalization of 3×3 matrices of effective Hamiltonians for each geminal representing a chemical bond. For determination of MOs in the SCF groups three procedures well known in the literature []: restricted Hartree-Fock (RHF), restricted open-shell Hartree-Fock (ROHF), and unrestricted Hartree-Fock (UHF), have been implemented in Ref. []. The effective Fock operators are constructed so that they take into account the presence of other electron groups. This modifies one-electron matrix elements of the effective operator.

Results and discussion

Application of the SLG/SCF method to Zn(II) complexes

The hybrid method [] briefly described in the previous Section allows to calculate electronic structure of different electron groups at different levels effectively taking into account interactions between the groups. In the coordination compounds, as it is shown in [], it is natural to single out the group of electrons in the closest surrounding of the metal ion. In the considered donor complexes the carrier space for this group of electrons is spanned by the lone pairs' (LP) HOs of the donor atoms and by the metal valence AOs (4s- and 4p-AOs in the case of zinc). We denote the group function describing electrons in metal (M) and LPs as FMLP. This group is to be treated by the SCF method. Other groups (pertaining to ligands) are either treated in the SCF approximation if they are p-electron systems like in benzene or pyridine rings or in the SLG approximation if they are local s-bonds. With these notions the overall wave function acquires the form:

In the case of ligands with oxygen donor atoms both LPs of oxygen are included into the MLP group. In molecules with the ligands bearing p-systems (such as imidazole, pyridine etc.) the latter are treated as separate SCF-groups without electron transfers between them, i.e. with constant number of electrons.

Test calculations

Our main purpose was to test the effect of introducing trial wave function given by Eqs. (7), (8) upon the structural and energy characteristics of coordination polyhedra. Thus, the re-parameterization of the MNDO Hamiltonian for the Zn atom with AOs included into the SCF-based group FMLP is needed to take into account influence of the SLG-based groups. We expect the change of parameters of the SLG-based groups to be small, as for aromatic molecules with p-systems []. In the original MNDO parameterization [] for the Zn atom the parameters were fitted on a small set of di- and triatomic molecules containing zinc. A modified MNDO parameterization of zinc atom was proposed in [] where values of two-electron one-center parameters fitted in [] are kept fixed, parameter Uss is shifted to -17.989 eV to obtain the heat of formation for the Zn2+ ion to be close to the experimental one of 665.1 kcal/mol. (It estimates formally as -2Uss-gss+HZn, where the parameter HZn is the atomization energy of Zn and its value accepted in MNDO is -31.7 kcal/mol; we keep it fixed). Remaining parameters: Upp, bs, bp, a and orbital exponents zs and zp, were fitted. Later in Ref. [] the MNDO/d method was introduced specifically for the Zn(II) compounds with another parameterization where Uss as other one-electron parameters differ from those previously mentioned. Atomic parameters for zinc were modified on the basis of parameterization procedure making use of extended data set on compounds of Zn(II), like Zn(CH3)2, Zn(Acac)2 and Zn(H2O)2+, not considered in earlier parameterizations. The MNDO/d method gives rather accurate results for these species.

In general we accept the strategy of parameterization proposed in Ref. [] (see also above) but restrict ourself to making only initial adjustment of the parameters for the Zn atom to reproduce some practically interesting values, such as geometry characteristics (mainly bond lengths between Zn and donor atoms). It is mostly useful because there is a lot of structural information for many zinc coordination complexes; however, the experimental gas phase heats of formation for these complexes are very scarce. For that reason, some authors [,] use ab initio estimates for the heats of formation of other zinc complexes instead of its unknown experimental values. We try to avoid such practice, so our approach is not a parameterization in the strict sense.

Taking this into account and analyzing the results of Ref. [] we started from the MNDO/d parameterization keeping fixed the value of Uss as one giving the correct value for the heat of formation of the Zn2+ ion. The resonance parameters bpSCF of the organogenic atoms (C, N, O, H) in the SLG/SCF (MNDO) method that has already been fitted for the SCF-treated p-electron groups in [] also retain their values. The modified MNDO bpSCF parameter for the N atom HO involved into p-electron system was fitted in [] to 21.97 eV. For oxygen atom the value of the analogous parameter bpSCF(O) fitted in [] is 39.0 eV. (The resonance parameters were changed in the original parameterization of MNDO for the SLG method [] itself). These values are used in further calculations. The resonance parameter bs is tuned for the N and O donor atoms in the SCF-treated MLP group independently from other parameters mentioned above.

Then, we performed test calculations using the MNDO [] and MNDO/d [] parameterizations for the complexes Zn(NH3)42+ (1) and Zn(Im)62+ (2) (where Im = imidazole), with tetrahedral and octahedral coordination, respectively, to get an overall comparison of these two parameterizations of Zn atom within the SLG/SCF context. It turned out that the geometry characteristics (metal-donor atom bond lengths) are better described by the MNDO/d, while the extent of charge transfer to the metal ion obtained with the MNDO parameters seems to conform better with the intuitive picture of charge redistribution in complexes giving resultant (Coulson) charge on the metal close to unity rather than to zero, which is the case for the MNDO/d parameters. The MNDO/d method seems to overestimate the extent of charge transfer which may affect resulting binding energy value. It is important since the major contributions to the binding energy of zinc coordination compounds are electrostatic and charge transfer energies []. However, calculation of these complexes by the SLG/SCF method with the original MNDO/d parameters set for zinc gives a good agreement between extents of the charge transfer to the metal obtained by it and by the original MNDO. Thus, the values of the MNDO/d orbital exponents for the zinc atom zs and zp, mostly controlling the extent of the charge transfer and the electron density distribution over 4s- and 4p-AO's of Zn, are concluded to be acceptable for the SLG/SCF context. By contrast, the geometry characteristics (bond lengths) obtained by the SLG/SCF with the MNDO/d parameterization poorly agree with the experimental ones. The molecular geometry is largely influenced by the resonance parameters bs(Zn) and bp(Zn). The new values bs(Zn)=1.30 eV and bp(Zn)=1.75 eV allow to reproduce the bond lengths in the considered complexes with use of the SLG/SCF method with a better accuracy than the MNDO/d method in its purely SCF version.

To fit Upp and a parameter values for the Zn atom as well as the bsSCF parameter for donor atoms, we use geometry characteristics (bond lengths). For their fitting, we select 14 complexes with ligands containing N and O donor atoms given in Table using the data both from Ref. [] and from the Cambridge Crystal Structure Data Bank (CCSDB). In that Table the ligand composition, abbreviations and corresponding CCSDB reference codes for the selected complexes are given. Some less widely known ligands are depicted in Fig. . In order to integrally characterize the calculated Zn-donor atom bond lenghts, we use the empirical distribution functions of the differences between the calculated and experimental bond lenghts.

In general, the plot of the empirical distribution function (error function, EDF) in the normal scale together with its linear fit is a good statistical test on presence of the systematic error in the data. It characterizes both the range of errors' magnitudes and the probability for that or another value of the error to appear in the test set. It is reasonable to assume that random nonsystematic errors are normally distributed with the zero mean value. Significant difference of the EDF mean deviation (m) from zero is then indicative for certain biases in the measurements or numerical experiments. The variance (or the s value) of this normal law characterize likelihood of facing the error too large by its absolute value. Thus, the condition for parameters bsSCF fitting may be almost zero m value for EDF of Zn-donor atom bond errors in selected complexes (less than 0.005 Å).

The results of SLG/SCF (MNDO) calculation with parameters Upp and a for zinc atom given in Table ) and shifted bsSCF parameters for N and O donor atoms (24.0 and 36.5 eV, respectively) reasonably fulfil the mentioned requirement concerning m value of EDF for zinc-donor atom bond length errors. With this set of the parameters the m values for errors in Zn-N and Zn-O bond lengths are -0.003 Å (s=0.039 Å) and -0.001 Å (s=0.032 Å), respectively. The standard SCF (MNDO) calculations for the same complexes result in the m values of 0.030 Å (s=0.029 Å) and 0.028 Å, respectively (s=0.035 Å). Thus, the systematic error for SLG/SCF (MNDO) is almost zero, which can be considered as an advantage in comparison with the purely SCF-based method yielding much larger systematic error. The distance variance (s) for SLG/SCF (MNDO) and SCF (MNDO/d) are comparable. Corresponding EDF and linear fit plots are given in Figs. , .

It can be seen from Table that the new parameters only slightly differ from their original MNDO/d values. Two-electron one-center parameters of zinc atom are not given here because they are not reparameterized and are taken from [].

Detailed results of the geometry optimization performed by the SLG/SCF (MNDO) (with the obtained MNDO parameterization given in Table ) and the SCF (MNDO/d) methods are present in Table . The improvement of bond length accuracy in case of the SLG/SCF (MNDO) method is evident as compared to the SCF (MNDO/d). The values of root-mean-squared differences (rmsd) between the calculated and experimental coordination (zinc-donor atom) bond lengths are in some cases lower in the SLG/SCF (MNDO) approach. In the case of complex 10 there are two possible ways of estimating the energy, one using formally the Zn(H2O)62+ composition of the complex and another with the Zn(H2O)182+ composition, reflecting the specific solvation of the hydrated cation by further water molecules, as discussed in Ref. []. In the first case the Zn-O distance is obtained to be longer by 0.01 Å than in the second one and closer to the experimental value.

It is worthwhile to note that in the SLG/SCF method the description of coordination bonds is achieved by considering only the MLP region (electron group) around the metal atom within an SCF-treated group which considerably reduces the overall computational costs.

The metal partial atomic charges on zinc and donor atoms calculated by the SLG/SCF (MNDO) are in most cases greater (by magnitude) than obtained by the SCF (MNDO/d) method. The Zn effective atomic charge is as a rule about 0.6 0.9 for SLG/SCF and 0.2 0.4 for SCF calculations. (The valence 4s- and 4p-orbitals populations of the Zn atom are, correspondingly, lower for the SLG/SCF calculations). Thus, the overall charge on the zinc atom obtained in the SLG/SCF (MNDO) calculations is as a rule close to unity, whereas in the MNDO/d one it is rather small (from our point of view - unrealistically).

Effects of ligand substitution : numerical experiment

Effects of substitution of the ligands in the coordination sphere of Zn2+ complexes were also of interest for us. Qualitative theory of metal complex geometry is given in []. Model considerations [] of the substitution in octahedral complexes are based on perturbative treatment of variation of diagonal matrix element of a model Fockian describing the close ligand shell of a complex. Within the SLG/SCF treatment for the complexes introduced in the present paper the corresponding Fock operator for the MLP electron group appears naturally and its matrix elements can be evaluated directly.

We consider substitution in six-coordinated 12-electron (according to specification of []) complexes of the form ZnL5X. In the case of the ZnL6 complex the frontier MOs have the form:
(1/   __
where s stands for the i-th s-LPs included into the MLP group. In this setting the theory [] predicts that by symmetry reasons only the normal mode corresponding to tetragonal distortion of the complex is capable to mix these frontier MOs (in the first order of perturbation theory), so only this distortion can take place as a result of the substitution. According to the theory [], in this case the amplitude Q of the normal mode responding to the substitution is the product of electronic and vibronic factors:

Q = - 1
ca la dFXLb ,   b = b(s,s)

R = R0 
where coefficients are the same as in Eq. (9), dFXL is the difference of diagonal matrix elements of the group MLP Fockian for the lone pair of the substituent X and for that of the predominant ligand L, dFXL = FXX-FLL (electronic factor), and b is the derivative of the off-diagonal Fockian matrix element with respect to the metal-donor atom bond length (vibronic factor). In the theory [] the latter multiplier is assumed to be always positive. It is confirmed by our calculations. Thus, the sign of Q depends only on that of dFXL, since the expansion coefficients ca, la are chosen to be positive. According to [], the effect of the substituent in octahedral 12[`(e)] complexes manifests itself in shortening of the bond in the trans-position with respect to the substituted position if the substituent X is a stronger donor than the substituted ligand L. The donor strength in its turn can be identified with the magnitude of the diagonal matrix element of the Fock operator over its lone pair HO and a stronger one is that one whose matrix element is less negative.

We performed corresponding calculations with use of the SLG/SCF (MNDO) model on several characteristic examples. First, the hexa-imidazole zinc complex 2 is considered, where one imidazole ligand is substituted for ammonia resulting in the Zn(Im)5(NH3)2+ complex. So, L=Im, X=NH3, the value of FLL = -23.95 eV, FXX = -23.02 eV, and the difference dFXL is 0.93 eV, so ammonia is a stronger donor than imidazole. The effect is manifested by the bond shortening for the N6 atom in the trans-position to the incoming ammonia molecule. This is perfectly reproduced in our calculations (see Table ). The extent of the observed substitution effect upon the bond lengths for N3 and N6 atoms is ca. 0.06 Å and the variation of the coordination bond lengths under substitution is almost ideally symmetric.

Then, we consider hexaamino-zinc complex 3, where one ammonia ligand is substituted for pyridine ligand resulting in the Zn(NH3)5(py)2+ complex. In this complex L=NH3, X=py, and in the initial configuration corresponding to the unsubstituted complex the matrix element FLL = -24.22 eV, and FXX = -24.76 eV, so the difference between them is dFXL = -0.54 eV. Thus, the variation of this diagonal matrix element is negative, and the pyridine is a weaker donor than ammonia. As one can see from Table , the theoretically expected effect upon geometry is reproduced, and the trans-ammonia ligand and the substituent pyridine both go away from the zinc atom. The magnitude of elongation of the coordination bond lengths under substitution is smaller than that in the complex 2 and non-symmetrical - 0.044 Å and 0.020 Å for N3 and N6 atoms, respectively.

As the third example, one pyridine ligand in the hypothetic hexapyridine zinc complex Zn(py)62+ is substituted to a linear acetonitrile ligand CH3CN to get Zn(py)5(CH3CN)2+. Here L=py, X=CH3CN, FLL = -23.95 eV, FXX = -26.61 eV, dFXL = -2.7 eV, and the sign of the difference dFXL is negative, thus acetonitrile is a weaker donor ligand. The variations of the coordination bond lengths in Zn(py)5(CH3CN)2+ as compared to Zn(py)62+ are -0.22 Å and -0.1 Å for acetonitrile and trans-pyridine, respectively and become shorter which is in variance with the simplest variant of the theory []. We also see that the two distances change to a different extent. The reasons of the mentioned disagreement may be various. First, we mention that the variance with the precise results of the theory [] occur when somewhat more bulky ligands are involved and at least partially can be attributed to steric effects. On the other hand predictions of the simplest version of the theory [] can be modified when going to its more refined variant involving other orbitals in addition to the frontier ones. For example, another normal mode, involving antisymmetric displacement of the trans-ligands can be populated. Finally we notice that the case of the nitrile substituent results in a much stronger variation of the diagonal Fock matrix element than in other cases so that the perturbative treatment of Ref. [] may become invalid.


In the present work we performed the parameterization of a recently proposed hybrid SLG/SCF method at the MNDO level in order to calculate heats of formation and equilibrium geometries of Zn(II) complexes with ligands containing nitrogen and oxygen donor atoms. It is shown that the method is able to reproduce with remarkable accuracy geometry characteristics of the complexes. The method takes into account the influence of the metal ion upon electronic structure of the ligands both by polarization and by charge transfer. The latter is responsible for chemical bonding and thus for the metal complex formation. The standard ab initio and DFT techniques provide no specialized approach to the description of the so-called coordination bonds [] which are characterized by principal unsaturability and flexibility of geometry orientations and placement of donor atoms around metal cation. As compared to empirical approaches taking into account the charge redistribution throughout the complex formation (like SIBFA and COSMOS) the proposed one uses a sequential quantum chemical procedure to model the electron redistribution. The results of the calculations on substituted complexes are in fair agreement with the qualitative theory of mutual influence of the ligands in complexes. The usage of inherently local description for one-electron states of the complex may be helpful for developing in the future consistent junctions to the surrounding treated by molecular mechanics methods which may be relevant for modeling metal-containing enzymes.


The authors gratefully acknowledge valuable discussions with Dr. I.V. Pletnev and Dr. V.V. Zernov. This work was supported through the RFBR grants NNo 04-03-32146, 04-03-32206, 05-03-33118. The usage of the CCSDB is supported through the RFBR grant No 02-07-90322.

Table 0: Ligand names and CCSD reference codes for the calculated molecules.
no. ligand ligand CCSD refcode Ref.
composition abbreviation
1 tetraamine FUZCUY []
2 hexa-imidazole Im HIMZZN []
3 hexaamine RAJNOF []
4 tris(ethylenediamine) en CIBKON []
5 bis(cis-cyclohexane-1,3,5-triamine) chta JUNNAH []
6 bis((2-pyridylmethyl)amine) pma KUTNES []
7 bis(1,4,7-Triazacyclononane) [9]aneN3 KIQPAB []
8 bis(diethylenetriamine)) dien CEJYEV []
9 tris(3-aminopropyl)amine amp BIHQOY []
10 hexa-hydrate CAXYUV []
11 hexa-methanol MeOH CIRJIW []
12 hexa-ethanol EtOH CIRNIA []
13 bis(12-crown-4) [12]aneO4 MEYLEH []
14 tris(1,2-ethanediol) diol ZNETD001 []

Table 0: Fitted for SLG/SCF method and initial MNDO/d Hamiltonian parameters for Zn atom.
Parameter SLG/SCF SCF
zs 1.732 1.732
zp 1.394 1.394
Uss -18.023 -18.023
Upp -10.8 -12.242
bs 1.300 5.017
bp 1.750 0.712
a 1.58 1.518

Table 0: Distances and rmsd of coordination bonds, calculated from structures of the investigated complexes optimized by the SLG/SCF (MNDO) and SCF (MNDO/d) methods. Experimental geometry references can be found in Table 1.
Complex Distance SLG/SCF SCF exp.
no. to atom
1 N1 2.042 2.085 2.052
N2 2.041 2.083 2.052
N3 2.041 2.084 2.052
N4 2.041 2.084 2.052
rmsd 0.011 0.032
2 N1 2.213 2.232 2.180
N2 2.216 2.263 2.200
N3 2.214 2.227 2.200
N4 2.213 2.236 2.180
N5 2.216 2.238 2.200
N6 2.214 2.242 2.200
rmsd 0.023 0.048
3 N1 2.169 2.215 2.207
N2 2.169 2.216 2.207
N3 2.170 2.217 2.207
N4 2.169 2.217 2.207
N5 2.169 2.216 2.207
N6 2.170 2.216 2.207
rmsd 0.035 0.009
4 N1 2.166 2.217 2.214
N2 2.168 2.215 2.213
N3 2.169 2.220 2.229
N4 2.165 2.220 2.241
N5 2.170 2.220 2.229
N6 2.165 2.220 2.233
rmsd 0.060 0.011

Table 3. Continued.

Complex Distance SLG/SCF SCF exp.
no. to atom
5 N1 2.179 2.228 2.187
N2 2.182 2.227 2.182
N3 2.183 2.228 2.191
N4 2.182 2.228 2.192
N5 2.180 2.227 2.194
N6 2.183 2.228 2.206
rmsd 0.013 0.036
6 N1 2.175 2.191 2.159
N2 2.225 2.193 2.151
N3 2.197 2.220 2.158
N4 2.175 2.191 2.159
N5 2.225 2.193 2.151
N6 2.197 2.220 2.158
rmsd 0.049 0.047
7 N1 2.167 2.2112.182
N2 2.173 2.2122.166
N3 2.177 2.2122.173
N4 2.168 2.2112.182
N5 2.173 2.2122.166
N6 2.178 2.2122.173
rmsd 0.010 0.039
8 N1 2.176 2.215 2.219
N2 2.188 2.226 2.163
N3 2.177 2.224 2.230
N4 2.170 2.224 2.245
N5 2.188 2.227 2.155
N6 2.165 2.215 2.204
rmsd 0.048 0.040
9 N1 2.037 2.0801.992
N2 2.035 2.0801.983
N3 2.094 2.1102.097
N4 2.033 2.0702.001
rmsd 0.038 0.074

Table 3. Continued.

Complex no. Distance to atom SLG/SCF SCF exp.
10 O1 2.057 2.066 2.052
O2 2.067 2.086 2.049
O3 2.067 2.087 2.049
O4 2.067 2.086 2.049
O5 2.066 2.086 2.049
O6 2.066 2.087 2.049
rmsd 0.017 0.037
11 O1 2.097 2.128 2.086
O2 2.097 2.128 2.086
O3 2.097 2.128 2.086
O4 2.097 2.128 2.086
O5 2.097 2.128 2.086
O6 2.097 2.128 2.086
rmsd 0.000 0.042
12 O1 2.077 2.131 2.079
O2 2.077 2.130 2.079
O3 2.078 2.130 2.079
O4 2.077 2.131 2.079
O5 2.077 2.130 2.079
O6 2.078 2.130 2.079
rmsd0.002 0.051
13 O1 2.074 2.088 2.086
O2 2.070 2.0882.098
O3 2.072 2.0892.089
O4 2.074 2.0882.086
O5 2.070 2.0882.098
O6 2.072 2.0892.089
rmsd0.020 0.006
14 O1 2.259 2.281 2.244
O2 2.258 2.2782.220
O3 2.243 2.2772.279
O4 2.268 2.2802.214
O5 2.241 2.2772.282
O6 2.256 2.2812.249
O7 2.242 2.279 2.269
O8 2.252 2.277 2.349
rmsd0.036 0.041

Table 0: The substitution effects in bond lengths and dissociation energy D, DD = D(ZnL5X)-D(ZnL6), for selected complexes calculated by the SLG/SCF (MNDO) method.
Complex DD, Donor Bond length, Å
kcal/mol atom Substituted Initial
N1(X) 2.235 2.213
N2(Y) 2.222 2.216
Zn(Im)5(NH3)2+-4.1 N3(Amm)(Z) 2.156 2.214
N4(-X) 2.231 2.213
N5(-Y) 2.226 2.216
N6(-Z) 2.156 2.214
N1(X) 2.1652.169
N2(Y) 2.1652.169
Zn(NH3)5(Py)2+ 4.4 N3(Py)(Z) 2.2142.170
N4(-X) 2.165 2.169
N5(-Y) 2.166 2.169
N6(-Z) 2.190 2.170
N1(X) 2.282 2.288
N2(Y) 2.323 2.288
Zn(Py)5(CH3CN)2+ -16.4 N3(CH3CN)(Z) 2.079 2.285
N4(-X) 2.286 2.288
N5(-Y) 2.321 2.288
N6(-Z) 2.181 2.285


Figure 4: Ligands used in the calculations.


Figure 4: Empirical distribution function for difference in experimental bond lengths (Å) and those calculated by the SLG/SCF (MNDO) method of Zn(II) complexes with nitrogen and oxygen donor atoms.


Figure 4: Empirical distribution function for difference in experimental bond lengths (Å) and those calculated by the SCF (MNDO/d) method of Zn(II) complexes with nitrogen and oxygen donor atoms.


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