L.Y. Karpov Institute of Physical Chemistry,
Vorontsovo Pole 10, 105064 Moscow, Russia
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, highaccuracy methods, like ab initio and DFT, are still computationally too demanding when dealing with coordination compounds. Qualitative approaches, such as pointsonasphere (POS) [] or metalligand sizematch selectivity concept [], still play an important role in theoretical coordination chemistry of both transition and nontransition metals. Thus, effective numerical although maybe semiquantitative 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 twocenter 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 semiempirical 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 semiempirical 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 geminalbased approaches proposed in the literature use somehow predefined carrier spaces of oneelectron states, either empirically constructed or extracted from preceding HartreeFock calculations by some localization procedure []. By contrast, employing of the special form of oneelectron 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 semiempirical methods [,,,] is comparable and in some cases even better than that of the corresponding SCFbased methods. The characteristic features of these methods  linear scalability of required computational resources [], correct asymptotic behaviour of the electronic structure under the sbonds cleavage, variational determination of oneelectron states  make the SLGbased semiempirical quantum chemistry an important alternative to the common SCFbased semiempirical procedures. However, the SLG based methods are nonuniversal, since the SLG trial wave function assumes the electronic structure of a molecule to be well depictable by a combination of twoelectron 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 pbonds 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 twoelectron, twocenter 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 pelectron system is given within this hybrid SLG/SCF method. The calculations show that molecular geometries and heats of formation for both pconjugated doublebonds 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 Zn^{2+} 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 Zn^{2+} 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(H_{2}O)_{n}^{2+} (see [] and references therein). Binding energies of some small complexes of Zn^{2+} with multiple ligands, such as acetone, Nmethylacetamide, 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 Zn^{2+} [,,]. An important feature of the SIBFA approach is the presence of energy contribution E_{ct} 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 Zn^{2+} which is characteristic for the coordination bond formation. However, the expression for E_{ct} in the SIBFA method contains large empirical scaling factor to reproduce the value of E_{ct} 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 Zn^{2+} complexed with monodentate DNAbase 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 fourcoordinated 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.
Semiempirical (AM1, PM3) approaches are also in use as methods of modeling for the latter complexes. In paper [] the accuracy of three semiempirical 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/6311G^{*}) 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 (rootmeansquared) 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 zinccontaining 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 MNDObased parameterization of the electronic Hamiltonian of the SCF part focused on the proper description of closest coordination sphere of the metal and of pelectron subsystems of the ligands when necessary. Then we discuss the results of our modeling.
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 pelectron 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 oneelectron 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:
 (0) 
Each HO is uniquely assigned to some electron group. If the group is to be
described by the SCF method the molecular spinorbitals are constructed as
linear combinations of the hybrid spinorbitals:
 (0) 
 (0) 
Each geminal for the twoelectron twocenter bond
is in its turn a superposition of three singlet twoelectron
configurations:
 (0) 
 (0) 
 (0) 
The wave function for the entire molecule in the SLG/SCF approximation is
represented by the antisymmetrized product of geminals and molecular
orbitals:
 (0) 
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 h^{A} 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 corecore 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 c^{s}_{i p} Eq.(2), the Jacobi angles defining the h^{A} matrices (and by this the shapes and directions of the HOs) and the amplitudes u_{m}, v_{m}, and w_{m} 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 h^{A} []. The geminal amplitudes u_{m}, v_{m} , and w_{m} 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 HartreeFock (RHF), restricted openshell HartreeFock (ROHF), and unrestricted HartreeFock (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 oneelectron matrix elements of the effective operator.
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
4pAOs in the case of zinc). We denote the group function describing
electrons in metal (M) and LPs as F_{MLP}. 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 pelectron systems like in benzene or pyridine rings
or in the SLG approximation if they are local sbonds. With these
notions the overall wave function acquires the form:
 (0) 
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 psystems (such as imidazole, pyridine etc.) the latter are treated as separate SCFgroups without electron transfers between them, i.e. with constant number of electrons.
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 reparameterization of the MNDO Hamiltonian for the Zn atom with AOs included into the SCFbased group F_{MLP} is needed to take into account influence of the SLGbased groups. We expect the change of parameters of the SLGbased groups to be small, as for aromatic molecules with psystems []. 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 twoelectron onecenter parameters fitted in [] are kept fixed, parameter U_{ss} is shifted to 17.989 eV to obtain the heat of formation for the Zn^{2+} ion to be close to the experimental one of 665.1 kcal/mol. (It estimates formally as 2U_{ss}g_{ss}+H_{Zn}, where the parameter H_{Zn} is the atomization energy of Zn and its value accepted in MNDO is 31.7 kcal/mol; we keep it fixed). Remaining parameters: U_{pp}, b_{s}, b_{p}, a and orbital exponents z_{s} and z_{p}, were fitted. Later in Ref. [] the MNDO/d method was introduced specifically for the Zn(II) compounds with another parameterization where U_{ss} as other oneelectron 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(CH_{3})_{2}, Zn(Acac)_{2} and Zn(H_{2}O)^{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 U_{ss} as one giving the correct value for the heat of formation of the Zn^{2+} ion. The resonance parameters b_{p}^{SCF} of the organogenic atoms (C, N, O, H) in the SLG/SCF (MNDO) method that has already been fitted for the SCFtreated pelectron groups in [] also retain their values. The modified MNDO b_{p}^{SCF} parameter for the N atom HO involved into pelectron system was fitted in [] to 21.97 eV. For oxygen atom the value of the analogous parameter b_{p}^{SCF}(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 b_{s} is tuned for the N and O donor atoms in the SCFtreated MLP group independently from other parameters mentioned above.
Then, we performed test calculations using the MNDO [] and MNDO/d [] parameterizations for the complexes Zn(NH_{3})_{4}^{2+} (1) and Zn(Im)_{6}^{2+} (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 (metaldonor 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 z_{s} and z_{p}, mostly controlling the extent of the charge transfer and the electron density distribution over 4s and 4pAO'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 b_{s}(Zn) and b_{p}(Zn). The new values b_{s}(Zn)=1.30 eV and b_{p}(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 U_{pp} and a parameter values for the Zn atom as well as the b_{s}^{SCF} 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 Zndonor 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 b_{s}^{SCF} fitting may be almost zero m value for EDF of Zndonor atom bond errors in selected complexes (less than 0.005 Å).
The results of SLG/SCF (MNDO) calculation with parameters U_{pp} and a for zinc atom given in Table ) and shifted b_{s}^{SCF} 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 zincdonor atom bond length errors. With this set of the parameters the m values for errors in ZnN and ZnO 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 SCFbased 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. Twoelectron onecenter 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 rootmeansquared differences (rmsd) between the calculated and experimental coordination (zincdonor 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(H_{2}O)_{6}^{2+} composition of the complex and another with the Zn(H_{2}O)_{18}^{2+} composition, reflecting the specific solvation of the hydrated cation by further water molecules, as discussed in Ref. []. In the first case the ZnO 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 SCFtreated 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 4porbitals 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 substitution of the ligands in the coordination sphere of Zn^{2+} 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 sixcoordinated 12electron (according
to specification of []) complexes of the form
ZnL_{5}X. In the case of the ZnL_{6} complex the
frontier MOs have the form:

 (0) 
We performed corresponding calculations with use of the SLG/SCF (MNDO) model on several characteristic examples. First, the hexaimidazole zinc complex 2 is considered, where one imidazole ligand is substituted for ammonia resulting in the Zn(Im)_{5}(NH_{3})^{2+} complex. So, L=Im, X=NH_{3}, the value of F_{LL} = 23.95 eV, F_{XX} = 23.02 eV, and the difference dF_{XL} 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 transposition 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 hexaaminozinc complex 3, where one ammonia ligand is substituted for pyridine ligand resulting in the Zn(NH_{3})_{5}(py)^{2+} complex. In this complex L=NH_{3}, X=py, and in the initial configuration corresponding to the unsubstituted complex the matrix element F_{LL} = 24.22 eV, and F_{XX} = 24.76 eV, so the difference between them is dF_{XL} = 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 transammonia 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 nonsymmetrical  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)_{6}^{2+} is substituted to a linear acetonitrile ligand CH_{3}CN to get Zn(py)_{5}(CH_{3}CN)^{2+}. Here L=py, X=CH_{3}CN, F_{LL} = 23.95 eV, F_{XX} = 26.61 eV, dF_{XL} = 2.7 eV, and the sign of the difference dF_{XL} is negative, thus acetonitrile is a weaker donor ligand. The variations of the coordination bond lengths in Zn(py)_{5}(CH_{3}CN)^{2+} as compared to Zn(py)_{6}^{2+} are 0.22 Å and 0.1 Å for acetonitrile and transpyridine, 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 transligands 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 socalled 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 oneelectron 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 metalcontaining 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 NN^{o} 040332146, 040332206, 050333118. The usage of the CCSDB is supported through the RFBR grant No 020790322.
no.  ligand  ligand  CCSD refcode  Ref. 
composition  abbreviation  
1  tetraamine  FUZCUY  []  
2  hexaimidazole  Im  HIMZZN  [] 
3  hexaamine  RAJNOF  []  
4  tris(ethylenediamine)  en  CIBKON  [] 
5  bis(ciscyclohexane1,3,5triamine)  chta  JUNNAH  [] 
6  bis((2pyridylmethyl)amine)  pma  KUTNES  [] 
7  bis(1,4,7Triazacyclononane)  [9]aneN_{3}  KIQPAB  [] 
8  bis(diethylenetriamine))  dien  CEJYEV  [] 
9  tris(3aminopropyl)amine  amp  BIHQOY  [] 
10  hexahydrate  CAXYUV  []  
11  hexamethanol  MeOH  CIRJIW  [] 
12  hexaethanol  EtOH  CIRNIA  [] 
13  bis(12crown4)  [12]aneO_{4}  MEYLEH  [] 
14  tris(1,2ethanediol)  diol  ZNETD001  [] 
Parameter  SLG/SCF  SCF 
z_{s}  1.732  1.732 
z_{p}  1.394  1.394 
U_{ss}  18.023  18.023 
U_{pp}  10.8  12.242 
b_{s}  1.300  5.017 
b_{p}  1.750  0.712 
a  1.58  1.518 
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 2.211 2.182 N2 2.173 2.212 2.166 N3 2.177 2.212 2.173 N4 2.168 2.211 2.182 N5 2.173 2.212 2.166 N6 2.178 2.212 2.173 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 2.080 1.992 N2 2.035 2.080 1.983 N3 2.094 2.110 2.097 N4 2.033 2.070 2.001 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 rmsd 0.002 0.051 13 O1 2.074 2.088 2.086 O2 2.070 2.088 2.098 O3 2.072 2.089 2.089 O4 2.074 2.088 2.086 O5 2.070 2.088 2.098 O6 2.072 2.089 2.089 rmsd 0.020 0.006 14 O1 2.259 2.281 2.244 O2 2.258 2.278 2.220 O3 2.243 2.277 2.279 O4 2.268 2.280 2.214 O5 2.241 2.277 2.282 O6 2.256 2.281 2.249 O7 2.242 2.279 2.269 O8 2.252 2.277 2.349 rmsd 0.036 0.041
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}(NH_{3})^{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)  2.165  2.169  
N2(Y)  2.165  2.169  
Zn(NH_{3})_{5}(Py)^{2+}  4.4  N3(Py)(Z)  2.214  2.170 
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}(CH_{3}CN)^{2+}  16.4  N3(CH_{3}CN)(Z)  2.079  2.285 
N4(X)  2.286  2.288  
N5(Y)  2.321  2.288  
N6(Z)  2.181  2.285 
^{1}Corresponding author. Email: mike@hcc.keldysh.ru