The molecular mechanics (MM) [] is widely used in calculations of potential energy surfaces (PES) of organic molecules, both per se and in molecular dynamics simulations. The conventional MM scheme faces, however, significant problems when applied to the metal ion complexes. The main problem is an adequate modeling of coordination sphere, which, in fact, presents two closely related sub-problems - accounting for coordination polyhedron flexibility and accounting for electronic structure details.
A conventional MM considers metal center (as well as all the other atoms) as having a definite ideal (strain-free) stereochemistry, which is allowed to be distorted only slightly. Electronic structure is accounted for only implicitly, through the choice of ideal polyhedron and associated with it deformation potentials/constants. The typical paradigm, valence force field with harmonic potentials (VFF-H) requires definition of a unique set of ideal bond lengths and valence angles around the metal ion and associated stretching/bending constants. For the most metal ions the VFF-H model prevents analysis of real-world, significantly distorted structures, as well as structural rearrangements (which are numerous in coordination chemistry).
The obvious solution of the problem is to reject the VFF-H formalism for metal ions, in parts or at all. The most straightforward way is to describe (angular) deformations in coordination sphere with specially dedicated potential functions rather than with harmonic potentials. In this respect, a moderately successfull use of trigonometric or double-well potentials has been reported[,,,]. A more radical and more popular approach is so-called ''points-on-a-sphere'' (POS) scheme Hambley,Reichert,Cundari,Johnson. It suggests that the shape of coordination polyhedron is ultimately dictated by the ligand-ligand van derWaals interactions thus eliminating the need in definition of idealized polyhedron and associated angular parameters. Recently, it has been shown [,] that this approach may be further improved by considering not van derWaals inter-ligand interactions but ''coordination bonds repulsion'', as it is suggested by well-known and extremely successfull qualitative theories by Gillespie [] and Kepert [].To our mind, this last version of MM, so-called Gillespie-Kepert MM (MMGK) [,] is a maximum of what may be reached within the pure MM framework. It allows a proper description of many cases of significant distortion in coordination geometry (for a discussion and examples see [,]).
However, even this last MMGK version of MM suffers from a general problem of being not capable to account for important consequences of details of electronic structure of transition metal ions (TMI). This is not a technical issue which may be solved by use of more intricated versions of potential functions. As it is mentioned in Ref. [], the physical pre-condition of successfull use of MM for common organic molecules is that their electronic excited states are well separated from the respective ground states on the energy scale. Only one quantum state of electronic system is experimentally observed and the MM (a sort of classical) description becomes valid. By contrast, the behavior of the metal valence d-shell is sufficiently quantum: several electronic states may appear in a narrow energy range close to its ground state, so that sometimes the PES's corresponding to different electronic terms of the d-shell intersect resulting in spin transitions [,].
Consequently, it seems necessary to directly incorporate quantum mechanical (QM) description of the central atom and its closest surrounding into computational scheme while the rest of molecule may be still described within MM formalism. In other words, we need in a hybrid quantum mechanics / molecular mechanics (QM/MM) methods.
Recently a general approach to description of the molecular electronic structure and potential energy has been suggested [], which makes it possible to apply quantum chemical description to that part of molecule in which electronic terms are close on the energy scale, and to use the MM description for the part where the closest electronic states remain distant in energy. Analogous approach has been also applied in papers TchDarh,TchNATOARW for the PES evaluation of the spin-crossover in Fe(II) complexes. It was suggested in Ref. [] to calculate the energy of the d-shell as that of the ground state of the dn-configuration in the effective crystal field (ECF - see below) induced by the ligands. The energy of the d-shell in the ECF approximation depends both on the molecular geometry (position and orientation of the ligands around the central ion) and on the electronic structure of the ligands. However, the approach [] is computationally intensive as it requires calculation of electronic structure of the whole ligand sphere at each step of geometry optimization, which reduces its practical significance[,].
Another approach to incorporating TMI in the general MM scheme was proposed in [,]. It uses the angular overlap model (AOM) in the Cellular Ligand Field (CLF) formulation [] and represents the energy of the d-shell in the crystal field as a sum of one-electron energies of the occupied d-levels in a one-electron approximation. The crystal field acting upon the d-states, in turn, is represented as a sum of contributions from each ligand, described by the AOM parameters e[`(s)] and ep . Dependence of the AOM parameters on metal-ligand distance in Refs. [,] is interpolated by fitting empirical values of the former for different ligands. This approach (CLFSE/MM, Deeth1) has been used for calculation of the structures of nickel (II) complexes with amine and p-type ligands. However, CLFSE neglects the d-electron correlations - which, in fact, are responsible for the form of the energy low-excitation spectrum of the d-shell. Consequently, ECF scheme seems to be of more general applicability (see Discussion).
In the present paper, we report a computationally efficient combination of the local [] version of the ECF method - LECF with the MMGK procedure and the application of the hybrid method to the calculations of molecular geometry for a series of Fe(II) complexes (having different ground state total spin) with nitrogen containing ligands.
The paper is organized as follows: in the next Section we briefly review the basic features of the ECF method [] and of its local incarnation []. Next we describe a new approach allowing to determine the effective crystal field as a sum of contributions from the separate ligands taking into account the chemical nature of the ligands and (with use of perturbation technique) the Coulomb interaction of the ligand with the rest of the complex. The last section provides the examples of application of ECF/MMGK to some real-world problems.
As it is mentioned in the Introduction the key point for incorporation of TMI into MM is to estimate the energy of the d-shell as a function of the ligand sphere geometry and composition. In this section we develop the working formulae based on the ECF theory performing this task.
The ECF method is based on a general concept of separating electron
variables. In the context of the original ECF method [] the
motivation of such a separation is that the electron correlations are much
more significant in the d-shell rather than in the ligands and for that
reason different levels of approximation must be used for different parts of
the TMC. The same concept applies when a hybrid QM/MM method is to be
developed []: electrons have to be divided into groups, some of
the groups whose excited electronic states are available in the experiment
are treated quantum mechanically whereas the behavior of other groups whose
excited electronic states lay high in energy (and are not available in
experiment) are modeled with use of MM. In a TMC containing one transition
metal ion and ligands around it the separation of electrons into groups is
performed as follows: the basis of valence atomic orbitals (AO's) containing
the 4s-, 4p-, and 3d-AO's of the metal atom (for a first transition
row element) and those of the ligand atoms, is according to Ref.
SouTchMis divided into two parts. The first part contains only 3d-orbitals of the transition metal atom (d-system). The rest contains 4s-, 4p-AO's of the metal atom and the valence AO's of the ligand atoms. All
these orbitals together are termed as ligand (l-) system. In the ECF
method [] it is shown that the d-shell can be described by
the effective QM Hamiltonian Hdeff:
| (0) |
| (0) |
| (0) |
| (0) |
| (0) |
| (0) |
|
As one can see the ECF method [] involves the covalent term eq. (5) to the crystal field which is strongly related to the canonical MO's (CMO's) of the l-system. According to calculations SouTchMis,Souexp1,Souexp2,Souexp3,Souexp4 namely the covalent term dominates (up to 80% - 90%) the d-level splitting. The Coulomb interaction with the effective charges on the ligand atoms gives the rest 10%-20% of the splitting and incidentally has local character. The AOM method employed in Refs. [,], by contrast, operates with some local quantities, which are, however, not clearly defined and can not be obtained in other way rather than fitting experimental data Gerloch,Shaffer. The interpolation for the AOM parameters proposed and used in Refs. [,] in the framework of the CLFSE approach is too much simplified in two respects: (i) It does not reflect the chemical identity of the ligand since the single formula (see below) is used ofr all ligands of similar structure and composition, though the AOM theory itself [,] insists on the chemical specificity of its parameters; (ii) The dependence of the AOM parameters on molecular geometry adopted in Refs. [,] does not cover that on orientation of the ligands with respect to the metal atom.
In our recent paper [] we have derived and tested a local
version of the ECF method and calculated the AOM parameters on its basis.
Here we review briefly the derivarion of the local form for the covalent
part of the ECF Wmncov which allows us to recast it into the
form compatible with the MM approach and to reformulate the calculation of
the d-shell energy in terms of the AOM parameters which now become
calculable for arbitrary geometry of the ligand sphere and absorb the
information on the electronic structure of the ligands (see below). For this
end we notice that eq.(5) can be rewritten in terms of the
retarded (ret) and advanced (adv) Green's functions for the l-system
GreenMonster,GreenMonster2:
| (0) |
|
| (0) |
The Green's functions allow to express the result of eq.(5) in the
arbitrary basis of the one-electron states of the l-system. We perform a
unitary transformation of the l-system canonical MO's (CMO's) |i
ñ which are the eigenstates of the respective Fock operator
to the localized MO's (LMO's) | L
ñ for the occupied and
empty MO's separately:
| (0) |
| (0) |
In the basis of the LMO's the resonance integrals entering eq. (5)
acquire the form:
| (0) |
| (0) |
Clearly, the unitary transformation from the CMO's basis to the LMO's basis
of the l-system does not affect the covalent part of the ECF. It turned
out, however (see Ref. []), that the sum over diagonal pairs
with L¢ = L in (12) is the most efficient and convenient
approximation for 10Dq with the error not exceeding 0.1 eV. This accuracy
is also comparable to that of the ECF method itself. The formal expression
for this approximate summation is:
| (0) |
In the previous Sections we reviewed the ECF theory and its local reformulation (LECF) which allowed us to derive and calculate the crystal field. It is particularly important that by this method this can be done for arbitrary geometry of the complex, which is prerequisite for developing a hybrid QM/MM method. It is, however, only a half way to the goal. Though the summation scheme over the local states in eq. (13) is approximate the Green's function of the l-system employed in it is still an exact one. It represents formally in the local basis the result of the complete SCF procedure applied to the whole l-system. It implicitly involves all the orbitals of the l-system namely the metal 4s,4p-AO's and the valence orbitals of the ligands. It would be, however, more computationaly feasible to express the ECF matrix (and thus the energy of the d-shell) in terms of the electronic structure parameters of the free ligands. Toward this end the Green's function of the l-system Gl has to be expressed in terms of free ligands (or even of their fragments - in the case of polydentate ligands with several donor atoms).
The natural way to do so is to apply the perturbation technique taking the
Green's function of a system consisting of noninteracting metal ion and
ligands as an unperturbed one. In general two kinds of interaction between
the central metal ion and the ligands must be turned on along a way from the
state of isolated ligands and TMI to this bound state in the complex. The
first one is the Coulomb interaction of the ligands with the field of the
central ion having the charge (ZM-nd) e. The second one is the
resonance responsible for one-electron transfers between the ligands and the
4s-,4p-AO's of the metal atom. The bare Green's function for the
unperturbed l-system G00l has the following simple block
diagonal form:
| (0) |
Nonvanishing blocks G0L i correspond to separate ligands
(fragments) L i containing the diagonal GLLL (e)
Green's function matrix elements of the lone pair L at L i:
| (0) |
The Green's function matrix block G0M has the form:
| (0) |
| (0) |
The two sorts of interaction mentioned above play, however, different roles.
The Coulomb interaction does not break the block diagonal structure of the
bare Green's function G00l. By contrast the resonance
interaction smashes the block structure. As it is shown in
ECFSpintran,SouTchMis,Souexp1,Souexp2,Souexp3,Souexp4,DarhTchRJPC the
total electron transfer from each ligand to the metal ion 4s-,4p-AO's
amounts to 0.1-0.15 of unit charge per ligand at the physically
reasonable metal-ligand distances (at longer separations it is even
smaller). This makes it possible to take the resonance into account with use
of perturbation theory in the metal-ligand one-electron hopping (i.e. with respect to TML and TLM). In the present paper we restrict
ourselves to the effect of the Coulomb field only i.e. to the zero
order with respect to the resonance. In this setting the LMO's of the l-system can be expanded in terms of the CMO's not of the whole l-system,
but in terms of the MO's of the respective ligand (fragment) only. Thus the
summation index i in the second row of eq.(12) enumerates the
MO's for each ligand (fragment) separately. The CMO orbital energies
(forming narrow quasi-bands) which appear as the Green's function poles can
be replaced by the close values of the orbital energies of the separate
ligands (fragments) within the complex. That allows to conserve the
block-diagonal form also for the approximate Green's function for the l-system:
| (0) |
In the following Subsection we consider an implementation of this approach allowing to express the Green's function of the l-system in terms of the Green's functions of separate ligands taking into account the Coulomb field effects.
The accepted model of electronic structure of TMC considers the metal ion as
a point charge (equal to its oxidation degree or formal charge). The
electron transfers from the ligand orbitals to the metal ion valence AO's
are neglected. Within such an approach the influence of the central ion on
the surrounding ligands reduces to that of the Coulomb field. The Coulomb
field affects the positions of the poles of the Green's function (orbital
energies) of the free ligands. The form of the CMO's of each ligand remain
unchanges which results in a picture of the rigid ligands' MO's. According
to [], the effect of the Coulomb field upon the orbital
energies within the first order perturbation theory can be represented as:
| (0) |
| (0) |
| (0) |
| (0) |
The basis for integration of the ECF with the MM is proposed in Ref.
TchDarh. According to Ref. [] the total energy of a TMC in its
n-th electronic state in the ECF approximation is:
| (0) |
| (0) |
In eq. (24) the term Edeff(n) corresponds to the effective d-shell energy calculated as the n-th eigenstate of the effective d-shell Hamiltonian eq. (1). The crystal field part of this Hamiltonian is estimated in the framework of the local ECF approach eq. (13). In the previous Sections we proposed the rigid ligand's MO model that represents the Green's function G0l eqs. ( 18) - (19) including only the ligand MO energy shifts with Sii(f) of eq. (22) calculated with use of the partial charges calculated for free ligands.
The hybrid scheme involving MM component requires extensive parameterization. The entire set of parameters consists of three subsets. These are the subsets relevant for description of the free d-shell, the parameters of the MM part and those relevant to the junction between the MM and ECF parts.
The d-shell electronic parameters are taken from the ECF method without changes. They are specific exponents of atomic orbitals, d-electron core attraction parameter Udd, the Racah parameters of electron-electron Coulomb repulsion in the d-shell B and C specific either for the complex (if known) or their standard values for the free ion tabulated say in Ref. []. They are described in details in Ref. []
Organic part of a molecule and metal ion coordination sphere, leaving out
effects of the d-shell, is described in the present hybrid procedure in
terms of the MMGK method. According to the MMGK, the arrangement of the
donor atoms around the metal is dictated by mutual repulsion between all
metal-ligand (M-L) bonds. It is assumed that the repulsion occurs between
the effective centers of repulsion lying on the M-L bonds on the distance reff from the metal ion. This term implicitly acccounts for the
electronic effects in the coordination sphere which could not be described
within bare ECF formalism (which considers only d-electron splitting).
Energy of the ''bond repulsion'' in the coordination sphere is:
|
| (0) |
The total MMGK conformation energy of a molecule is:
| (0) |
|
- the energy of bond stretching;
|
- Morse function to allow large variations in metal - donor atom bond lengths in different TMC spin states:
|
|
|
|
|
MM parameters for organic part of molecule were primarily taken from the CHARMM force field [,], while specific metal-dependent parameters are fitted within MMECF method. Full set of the MMGK force field parameters is available on request. The same parameters are used for any spin state.
Since within the LECF energy part the entire system is divided into two parts, i.e. the d-shell and the ligands l-system, their junction requires separate attention. There are two kinds of parameters corresponding,respectively, to the d-l interaction and to the l-system electronic structure characterized below.
d-l interaction parameters
In the original ECF theory the specific parameters describing the interaction between the subsystems were fitted in order to reproduce the d-level splitting for octahedral complexes with a specific donor atom. The set of the intersystem interaction parameters includes the gsd and [`(g)]pd parameters of the Coulomb interaction between d-shell and transition metal valence s- and p-electrons taken from the Oleari's work [], the valence state ionization potentials for the d-shell and the donor atoms taken from [], and the dimensionless factors bML characteristic for a metal - donor atom pair, scaling the resonance interaction. These parameters are transferred from the original ECF [] to the MMECF package without change.
Electronic structure parameters of the l-system
The original ECF [] method employs the CNDO approximation
Pople in order to estimate the parameters of the electronic structure of
the l-system required for the calculation of the effective Hamiltonian eq.
(1). These quantities (one-electron densities, orbital
energies, and MO-LCAO expansion coefficients) had to be calculated for each
molecular geometry by the approximate SCF procedure extended to the entire l-system. In the RLMO approximation these quantities are estimated
petrubatively which is more econimical from the computational point of but
requires a larger number of parametrs. Within the RLMO model the electronic
structure of the free ligand prototype is supposed to be unchanged during
the complex formation. Thus, for the LECF calculations we used the charge
distribution calculated for the free ligand itself i.e. the effective
point charges are found from the CNDO calculation on the free ligand and
considered as electronic structure parameters for the l-system. Also the
orbital energies of the ligand MO's having non-zero contribution to the lone
pair of the donor atom calculated for the free ligand are to be fed to the
MMECF procedure. They are used to estimate the positions of the poles of the
Green's function in the Coulomb field of the charges within the complex
according to the formula eq. (21) with use of the partial densities
of the i-th MO's on the atom A of the ligand (fragment) L :
|
In our present study, the basic procedure for treating PES of TMC within MM approach is constructed. To do so we, first, reformulate in the local form, i.e. in terms of the field increments induced by the lone pairs of the ligands, the semi-empirical ECF theory (which previously allowed to calculate with appropriate accuracy the crystal field induced by the ligand's on the TMC's d-shells). This allowed us to find explicit formulae for the crystal field matrix expressed through the electronic characteristics of the free ligands lone pairs and the procedure to calculate them. In the framework of our approach the crystal field matrix is calculated for arbitrary arrangement and orientation of the ligands around the central TMI. This puts our approach into a close relation with the empirical angular overlap model (AOM) theory [,]. The latter despite its purely empirical character is widely used for interpretation of experimental data on electronic structure of the d-shell in TMC's. Since the AOM based approaches to description of TMI's in the MM framework are present in the literature [,], let us discuss briefly the relation and difference between the AOM and the local ECF.
The formulae eqs. (12), (13) of the local version of the ECF theory are close to formal expressions used to substantiate the AOM Gerloch. As it is mentioned in the Introduction, the AOM is used as a basis for constructing the CLFSE approach to incorporation of TM into MM [,]. Here we establish a relation between the AOM and the local ECF methods.
According to the AOM, the crystal field is given by the sum (cellular
expansion []):
| (0) |
| (0) |
| (0) |
| (0) |
| (0) |
| (0) |
| (0) |
| (0) |
| (0) |
The above explicit formulae for the ell¢L
parameters allow to perform analysis of various approximations employed in
the literature. For example, in Ref. [] the dependence of the AOM
parameters es on separation between the ligand donor and metal
atoms is parameterized in the form:
| (0) |
| (0) |
| (0) |
The main goal was to construct a procedure comprising the LECF approach and general MM by the eq. (24). It is implemented in the MMECF package [] with the RLMO version of the electronic structure of the ligand system. The package includes both gradient minimization and minimization without derivatives [,] procedures. The package allows to consider ligands or its fragments as rigid bodies. As a consequence, the number of geometry variables considerably decreases that speeds up the minimization procedure. Technically, the rigid bodies first pre-optimized with use of the MM potentials only (excluding metal-dependent ones), and in the further calculations their internal geometry is fixed.
A series of complexes of Fe2+ were chosen to check the relevance of the proposed scheme and to parameterize it. This set contains complexes of both monodentate and polydentate ligands. We studied the [Fe(Py)6]2+, [Fe(BiPy)3]2+, [Fe(TerPy)2]2+, [Fe(mBiPy)3]2+ complexes (the ligand are shown in Fig. 1). Both the low-spin and the high-spin complexes are present in the list. In case of the RLMO model, however, the d-shell splitting is too strong as compared to experiment so we limit ourselves to optimal geometry calculation for different spin states.
Initial geometries were taken from the Cambridge Crystal Structure Data Bank (CCSDB). Hydrogen atoms were added where necessary. Corresponding CCSDB code names of the complexes are PYFEFE [], VEWVEY vewvey, NUZKOI [], ZIMBUS []. We first preoptimized geometry of the ligands using the pure MM procedure, and treated the six-membered aromatic rings as rigid fragments (having only six degrees of freedom each). Minimization without derivatives was used with convergence criterion (energy change per iteration) 10-6 kcal/mol.
Since the original ECF method was shown previously [] to provide an accurate description for the crystal field in TMC's itself and also for its dependence on tiny geometry variations Tchheme,ECFSpintran, we first fitted the MM parameters describing the Fe-N interactions (since namely this part of the MM model is affected by introducing the LECF procedure for estimating the d-shell energy). The parameters fit was performed by trial-and-error method with the condition that the total spin for the global energy minimum should coincide with the experiment and the geometry structure at this minimum should be close to the experimental one. Within such a setting we, first, studied the complexes listed above with use of the standard values (B0 = 917 cm-1, C0 = 4040 cm-1) of the Racah parameters for the free Fe2+ion Lever.
This yielded the results present in Table column h = 1 (see below). The general conclusion which could be drawn from it is that the general outline of the potential energy profiles is in agreement with experimental data, though the important details are still wrong. Indeed, with the free ion values of the Racah parameters the high-spin molecules have the total spin 2 in their deepest minima and the low spin ones the total spin 0 in agreement with experiment []. However, the energy differences between the minima corresponding to the correct total spin and other values of the total spin are too small (ca. 1 kcal/mole) for all the molecules considered. On the other hand the geometry for the high spin complexes is systematically shifted towards longer Fe-N separations. In fact it looked like the intersection of the high-spin and low-spin terms in the complexes of iron(II) took place in our calculation at about 2.2 Å rather than at 2.1 Å as it is known to be from the experiment and from previous ECF calculation []. In order to clarify this situation we calculated the 10Dq splitting parameter as a function of the metal- donor atom distance both with the use of the LECF combined with the rigid ligand MO's model (LECF-RLMO) and by the original ECF method for the octahedral configuration of the complex [Fe(Py)6]2+. The result is shown in Fig. 2. It can be characterized in a twofold manner: first, one can notice that the LECF-RLMO curve can be approximately obtained from the exact ECF one by 0.2 Å shift to the larger interatomic separations; second, it can be checked that for the sensitive range of the interatomic separations about 2 Å the ratio 10Dq(LECF-RLMO)/10Dq(ECF) falls in the interval 1.5 ¸2.2.
These observations can be analyzed with use of the Tanabe-Sugano diagrams (see e.g. []). Indeed, as one remembers the relative position of the electronic terms of different total spin depends on the 10Dq/B ratio, with its large values corresponding to the low-spin ground states and the small ration values corresponding to the high-spin ground states. Since the LECF-RLMO model regularly overestimates the splitting parameter due to neglect of the one-electron transfers between the ligands and the metal 4s- and 4p-AO¢s the correct position of the crossection point of the high- and low-spin terms can be obtained by multiplying the free ion B0 and C0 values by a factor h. We studied the complex geometries for four values of the h parameters equal respectively to 1, 1.5, 2, and 2.4. For each value of h the MM parameters for the Fe-N interaction were fitted. Obtained calculation results are summarized in Tables , . The calculation detalis characterizing the quality of the results for corresponding four series of the Racah B and C parameters are presented in Table , while in Table the information for the best fit geometries with the Racah B and C parameters equal to 1430 and 6240 cm-1, respectively (which corresponds to h = 1.5), is presented.
The nearly octahedral [Fe(Py)6]2+ complex has high-spin ground state. Our calculations reproduce the same spin of the ground state. The coordination bond lengths are in an acceptable agreement with experimental data. Valence angles at the metal are in good agreement with experiment as well. However, the torsional angles between pyridine planes differ significantly from the experimental ones. In our opinion, this may be explained by nonbonded interactions in crystal environment. Electronic terms for [Fe(Py)6]2+complex for different h values are presented in Fig. 3. They illustrate the validity of our conjecture concerning the position of the spin-crossection point as a function of h. As one can see for stronger Coulomb interaction in the d-shell the transition to the low-spin ground state requires stronger crystal filed. This results in the shift of the spin-crossection point to the experimentally substantiated value of 2.1 Å .
Tris-(2-methyl-bipyridine)-iron complex [Fe(mBiPy)3]2+ also has the high-spin ground state, in agreement with the experiment. The calculated lengths of the Fe-N bonds for the methylated ring are somewhat shorter than the experimental ones. This disagreement can be possibly cured by adjusting the1,4-van-der-Waals parameters which were not fitted in the present study.
Geometry of both low-spin complexes, [Fe(BiPy)3]2+and [Fe(TerPy)2]2+, are calculated with good accuracy. Even the details of geometry in the case the terpyridyl complex are reproduced: the bond length for the central ring is somewhat longer than that for the side rings. Despite the difference in bond distances one can see a good quality of experimental geometry reproduction by the LECF-RLMO - MMGK calculations. Moreover, the hybrid method predict the minimum energy for each complex to the spin state which observed experimentally. The energy differences between the minima for the best fit Racah parameters are large enough to explain the fact that for the compounds under consideration only one ground spin state is observed.
Finally, we can say that by concert usage of the LECF-RLMO procedure as a QM model for describing the geometry dependence of the d-shell energy and the MMGK procedure as a MM model for the ligand energy, a unified MM-like description for the PES of different spin states of the iron(II) complexes is achieved with use of the single spin-independent parameterization (the full set of parameters are available from the authors upon request).
Similar result has been reached in Ref. [] for a series of nickel(II) complexes with use of the CLFSE-MM methodology. However, the description of the d-shell state in the CLFSE method [] is based on the one-electron approximation and neglects the d-electron correlations - which, in fact, are responsible for the form of the energy low-excitation spectrum of the d-shell. For that reason, the success of the CLFSE method in describing molecules with different total spin is reached for the case of nickel(II) complexes when the one-electron approximation suffice to describe the difference between the d-level splitting and filling for the high-spin (triplet) six-coordinate and the low-spin (singlet) square planar four-coordinate complexes. By contrast, the (L)ECF method explicitly takes into account the electron correlation in the d-shell by employing the full configuration interaction wave function for the latter. It makes it possible to describe correctly the low-lying excited states of the d-shell and their crossovers that are required for description of the spin transitions [,]. That allowed us to trace the effect of very tiny geometry variations on the ground state total spin in the complexes which maintain their octahedral geometry and the whole difference in geometry does not exceed 10% of the metal-ligand bond length.
This work is performed with partial financial support of the Federal Target Program of Russia ''Integracia'' through the grant No A0078. The usage of the CCSDB is supported by the RFBR grant No 99-07-90133, usage of the supercomputer facilities at the Keldysh Institute of Applied Mathematics is supported by the RFBR grant No 01-07-90383.
| Complex | h | 1 | 1.5 | ||||||
| S | E, | Rms | E, | Rms | |||||
| kcal/mol | bonds | angles | tors. | kcal/mol | bonds | angles | tors. | ||
| Fe(bipy)32+ | 0 | -9367.42 | 0.03 | 1.2 | 1.2 | -9336.11 | 0.01 | 1.4 | 0.8 |
| 0.08 | 0.7 | 0.00 | 1.9 | ||||||
| 1 | -9340.31 | 0.09 | 2.3 | 1.6 | -9297.27 | 0.06 | 1.7 | 1.3 | |
| 0.24 | 3.7 | 0.16 | 2.3 | ||||||
| 2 | -9354.38 | 0.12 | 4.2 | 5.0 | -9322.18 | 0.08 | 2.9 | 3.4 | |
| 0.32 | 8.9 | 0.23 | 6.2 | ||||||
| Fe(terpy)22+ | 0 | -9337.06 | 0.05 | 1.5 | 0.8 | -9301.69 | 0.02 | 1.5 | 0.9 |
| 0.12 | 0.6 | 0.03 | 1.9 | ||||||
| 1 | -9315.77 | 0.10 | 2.8 | 1.9 | -9269.24 | 0.07 | 2.1 | 2.1 | |
| 0.26 | 2.9 | 0.18 | 1.2 | ||||||
| 2 | -9326.85 | 0.14 | 5.3 | 3.6 | -9288.03 | 0.11 | 4.4 | 3.4 | |
| 0.39 | 11.3 | 0.30 | 9.4 | ||||||
| Fe(py)62+ | 0 | -9382.08 | 0.06 | 0.8 | 16.3 | -9336.25 | 0.09 | 0.7 | 21.5 |
| 0.15 | 1.4 | 0.23 | 1.2 | ||||||
| 1 | -9365.24 | 0.05 | 0.7 | 30.5 | -9316.22 | 0.07 | 0.7 | 24.6 | |
| 0.13 | 1.1 | 0.18 | 1.2 | ||||||
| 2 | -9382.49 | 0.02 | 0.7 | 28.8 | -9345.72 | 0.02 | 0.7 | 24.9 | |
| 0.05 | 1.2 | 0.06 | 1.2 | ||||||
| Fe(m-bipy)32+ | 0 | -9356.69 | 0.04 | 4.3 | 4.4 | -9313.41 | 0.07 | 4.9 | 5.1 |
| 0.12 | 8.9 | 0.20 | 9.9 | ||||||
| 1 | -9337.31 | 0.03 | 3.0 | 4.1 | -9289.08 | 0.04 | 3.3 | 4.7 | |
| 0.08 | 6.1 | 0.12 | 6.6 | ||||||
| 2 | -9358.23 | 0.04 | 1.7 | 2.5 | -9324.60 | 0.02 | 1.6 | 2.5 | |
| 0.10 | 3.7 | 0.03 | 3.3 | ||||||
| Complex | h | 2 | 2.4 | ||||||
| S | E, | Rms | E, | Rms | |||||
| kcal/mol | bonds | angles | tors. | kcal/mol | bonds | angles | tors. | ||
| Fe(bipy)32+ | 0 | -9235.59 | 0.01 | 1.7 | 1.3 | -9209.11 | 0.02 | 1.9 | 1.2 |
| 0.02 | 2.4 | 0.05 | 3.1 | ||||||
| 1 | -9197.34 | 0.04 | 1.6 | 0.8 | -9165.73 | 0.03 | 1.5 | 1.0 | |
| 0.11 | 1.3 | 0.08 | 1.6 | ||||||
| 2 | -9233.85 | 0.06 | 1.9 | 2.3 | -9209.02 | 0.04 | 1.5 | 1.8 | |
| 0.16 | 3.8 | 0.11 | 2.7 | ||||||
| Fe(terpy)22+ | 0 | -9198.44 | 0.01 | 1.6 | 1.3 | -9172.85 | 0.02 | 1.8 | 1.2 |
| 0.01 | 2.2 | 0.03 | 3.0 | ||||||
| 1 | -9163.55 | 0.05 | 1.9 | 1.3 | -9127.79 | 0.04 | 1.7 | 1.8 | |
| 0.14 | 0.7 | 0.10 | 1.4 | ||||||
| 2 | -9187.09 | 0.08 | 3.2 | 2.3 | -9156.94 | 0.06 | 2.7 | 2.1 | |
| 0.23 | 6.8 | 0.18 | 5.7 | ||||||
| Fe(py)62+ | 0 | -9233.06 | 0.10 | 0.7 | 21.2 | -9197.95 | 0.11 | 0.7 | 21.0 |
| 0.26 | 1.1 | 0.30 | 1.1 | ||||||
| 1 | -9212.45 | 0.08 | 0.7 | 20.3 | -9173.28 | 0.09 | 0.7 | 20.5 | |
| 0.21 | 1.1 | 0.24 | 1.1 | ||||||
| 2 | -9257.70 | 0.04 | 0.7 | 23.6 | -9227.56 | 0.06 | 0.7 | 22.8 | |
| 0.11 | 1.2 | 0.16 | 1.1 | ||||||
| Fe(m-bipy)32+ | 0 | -9208.88 | 0.08 | 5.4 | 5.6 | -9177.23 | 0.09 | 5.7 | 6.4 |
| 0.22 | 10.7 | 0.26 | 11.1 | ||||||
| 1 | -9192.56 | 0.06 | 4.5 | 6.1 | -9156.17 | 0.07 | 4.7 | 6.4 | |
| 0.18 | 9.1 | 0.21 | 9.4 | ||||||
| 2 | -9231.67 | 0.03 | 2.4 | 3.1 | -9204.35 | 0.04 | 2.7 | 4.1 | |
| 0.07 | 4.8 | 0.12 | 5.2 | ||||||
| [Fe(bipy)3]2+ | |||||||
| Ground state S=0 | |||||||
| Exp. | Calc. | ||||||
| S=2 | S=1 | S=0 | |||||
| h | 1 | 1.5 | 1 | 1.5 | 1 | 1.5 | |
| Energy, kcal/mol | -9354.38 | -9209.02 | -9367.42 | -9209.11 | -9340.31 | -9165.73 | |
| Fe-N, Å | 1.97 | 2.29 | 2.19 | 2.05 | 1.97 | 2.24 | 1.94 |
| 1.97 | 2.29 | 2.19 | 2.05 | 1.97 | 2.00 | 2.15 | |
| 1.97 | 2.29 | 2.19 | 2.04 | 1.97 | 2.00 | 2.17 | |
| 1.97 | 2.29 | 2.19 | 2.04 | 1.97 | 2.24 | 2.17 | |
| 1.97 | 2.29 | 2.20 | 2.05 | 1.96 | 2.27 | 2.15 | |
| 1.97 | 2.29 | 2.20 | 2.04 | 1.96 | 2.27 | 1.93 | |
| N-Fe-N, o | 81.9 | 73.0 | 75.7 | 82.3 | 84.7 | 79.4 | 82.0 |
| 94.3 | 94.5 | 94.7 | 93.5 | 93.1 | 97.2 | 94.2 | |
| 89.8 | 100.3 | 98.6 | 90.5 | 88.9 | 93.7 | 87.9 | |
| 94.3 | 94.7 | 94.3 | 94.5 | 93.4 | 97.9 | 96.1 | |
| 174.6 | 161.7 | 166.2 | 175.6 | 177.1 | 169.9 | 177.1 | |
| 174.6 | 159.7 | 164.9 | 174.1 | 176.3 | 175.3 | 172.2 | |
| 94.3 | 93.2 | 94.0 | 93.2 | 92.3 | 97.5 | 96.6 | |
| 89.8 | 102.3 | 98.2 | 90.7 | 89.9 | 89.5 | 92.2 | |
| 94.3 | 95.9 | 96.0 | 94.6 | 92.9 | 94.9 | 96.0 | |
| 81.9 | 73.0 | 75.8 | 82.5 | 84.6 | 79.4 | 76.3 | |
| 94.3 | 94.6 | 94.0 | 93.9 | 93.2 | 94.3 | 95.1 | |
| 89.8 | 99.9 | 95.5 | 89.7 | 89.4 | 89.0 | 88.02 | |
| 174.6 | 161.1 | 164.1 | 174.0 | 176.9 | 167.4 | 170.8 | |
| 94.3 | 94.7 | 93.0 | 92.9 | 92.9 | 95.3 | 94.4 | |
| 81.9 | 73.1 | 75.7 | 82.3 | 84.9 | 73.6 | 82.0 | |
| [Fe(terpy)2]2+ | |||||||
| Ground state S=0 | |||||||
| Exp. | Calc. | ||||||
| S=2 | S=1 | S=0 | |||||
| h | 1 | 1.5 | 1 | 1.5 | 1 | 1.5 | |
| Energy, kcal/mol | -9315.77 | -9156.94 | -9337.07 | -9172.85 | -9326.85 | -9127.79 | |
| Fe-N, Å | 1.88 | 2.32 | 2.23 | 1.92 | 1.86 | 1.96 | 1.90 |
| 1.97 | 2.33 | 2.23 | 2.11 | 2.01 | 2.27 | 2.18 | |
| 1.98 | 2.35 | 2.25 | 2.12 | 2.01 | 2.29 | 2.20 | |
| 1.88 | 2.32 | 2.23 | 1.92 | 1.87 | 1.97 | 1.90 | |
| 1.98 | 2.35 | 2.25 | 2.14 | 2.02 | 2.31 | 2.21 | |
| 1.98 | 2.33 | 2.23 | 2.13 | 2.02 | 2.29 | 2.20 | |
| N-Fe-N, o | 81.7 | 70.9 | 72.8 | 81.2 | 82.9 | 79.2 | 81.1 |
| 81.1 | 70.4 | 72.3 | 81.2 | 83.1 | 78.5 | 80.3 | |
| 179.4 | 178.1 | 175.3 | 180.0 | 179.9 | 179.8 | 179.7 | |
| 98.3 | 107.7 | 111.8 | 99.4 | 97.3 | 101.9 | 100.0 | |
| 99.9 | 111.3 | 103.2 | 99.3 | 97.4 | 101.5 | 99.4 | |
| 162.8 | 141.3 | 145.0 | 162.4 | 166.0 | 157.7 | 161.4 | |
| 98.7 | 108.7 | 109.3 | 98.8 | 97.2 | 100.6 | 98.7 | |
| 90.0 | 98.0 | 95.5 | 91.3 | 90.8 | 93.9 | 92.8 | |
| 91.6 | 94.5 | 95.2 | 91.4 | 90.7 | 91.3 | 91.6 | |
| 98.5 | 110.0 | 105.7 | 98.8 | 96.8 | 101.7 | 99.9 | |
| 92.3 | 93.9 | 95.5 | 91.5 | 91.0 | 91.2 | 90.9 | |
| 91.6 | 99.1 | 94.6 | 91.6 | 91.1 | 92.7 | 91.0 | |
| 81.3 | 70.4 | 72.4 | 80.6 | 82.7 | 78.3 | 80.3 | |
| 80.5 | 70.6 | 72.6 | 80.7 | 82.6 | 78.3 | 80.4 | |
| 161.8 | 141.0 | 145.0 | 161.4 | 165.4 | 156.6 | 160.6 | |
| [Fe(py)6]2+ | |||||||
| Ground state S=2 | |||||||
| Exp. | Calc. | ||||||
| S=2 | S=1 | S=0 | |||||
| h | 1 | 1.5 | 1 | 1.5 | 1 | 1.5 | |
| Energy, kcal/mol | -9382.49 | -9227.56 | -9365.24 | -9173.28 | -9382.08 | -9197.95 | |
| Fe-N, Å | 2.25 | 2.30 | 2.20 | 2.29 | 1.95 | 2.09 | 2.00 |
| 2.28 | 2.30 | 2.21 | 2.04 | 2.20 | 2.14 | 2.05 | |
| 2.29 | 2.30 | 2.20 | 2.29 | 2.19 | 2.11 | 2.02 | |
| 2.26 | 2.30 | 2.20 | 2.29 | 1.95 | 2.09 | 2.01 | |
| 2.22 | 2.30 | 2.21 | 2.04 | 2.20 | 2.13 | 2.05 | |
| 2.25 | 2.30 | 2.20 | 2.29 | 2.19 | 2.10 | 2.02 | |
| N-Fe-N, o | 90.1 | 90.2 | 90.2 | 90.0 | 90.0 | 90.1 | 90.1 |
| 89.4 | 89.0 | 89.9 | 87.8 | 90.0 | 90.0 | 90.0 | |
| 178.5 | 179.0 | 179.9 | 179.8 | 180.0 | 180.0 | 180.0 | |
| 90.0 | 90.4 | 90.1 | 90.0 | 90.1 | 90.0 | 90.0 | |
| 90.1 | 90.2 | 89.9 | 92.0 | 90.0 | 89.9 | 89.9 | |
| 90.2 | 90.8 | 90.2 | 90.3 | 90.6 | 90.0 | 90.0 | |
| 89.6 | 89.6 | 89.9 | 89.9 | 89.9 | 90.0 | 89.9 | |
| 178.5 | 178.0 | 179.5 | 179.9 | 179.3 | 179.7 | 179.7 | |
| 89.6 | 89.3 | 90.1 | 90.1 | 90.0 | 90.2 | 90.0 | |
| 92.1 | 90.0 | 90.0 | 92.1 | 90.2 | 90.1 | 90.0 | |
| 88.3 | 91.2 | 90.3 | 89.8 | 90.2 | 90.3 | 90.2 | |
| 179.4 | 179.2 | 179.6 | 179.6 | 179.4 | 179.8 | 179.8 | |
| 90.3 | 89.9 | 89.9 | 90.1 | 90.0 | 89.9 | 90.0 | |
| 88.5 | 90.8 | 90.2 | 88.1 | 90.1 | 90.1 | 90.2 | |
| 91.9 | 88.7 | 89.4 | 89.8 | 89.3 | 89.5 | 89.8 | |
| [Fe(m-bipy)3]2+ | |||||||
| Ground state S=2 | |||||||
| Exp. | Calc. | ||||||
| S=2 | S=1 | S=0 | |||||
| h | 1 | 1.5 | 1 | 1.5 | 1 | 1.5 | |
| Energy, kcal/mol | -9358.23 | -9204.35 | -9337.31 | -9156.17 | -9356.69 | -9177.23 | |
| Fe-N, Å | 2.20 | 2.29 | 2.20 | 2.21 | 2.12 | 2.04 | 1.96 |
| 2.16 | 2.30 | 2.20 | 2.02 | 1.95 | 2.04 | 1.97 | |
| 2.20 | 2.30 | 2.21 | 2.28 | 2.18 | 2.02 | 1.96 | |
| 2.19 | 2.31 | 2.21 | 2.11 | 2.01 | 2.16 | 2.06 | |
| 2.26 | 2.30 | 2.21 | 2.29 | 2.20 | 2.14 | 2.05 | |
| 2.24 | 2.31 | 2.21 | 2.30 | 2.20 | 2.16 | 2.07 | |
| N-Fe-N, o | 88.3 | 93.6 | 92.8 | 93.1 | 90.9 | 91.2 | 90.1 |
| 170.1 | 177.2 | 173.9 | 176.0 | 171.2 | 178.3 | 175.6 | |
| 74.5 | 73.0 | 75.5 | 79.2 | 82.7 | 81.1 | 84.3 | |
| 86.9 | 88.7 | 86.5 | 87.0 | 84.7 | 86.9 | 86.2 | |
| 113.8 | 109.5 | 110.2 | 109.1 | 110.3 | 99.8 | 98.8 | |
| 86.7 | 87.3 | 85.6 | 83.7 | 83.0 | 88.1 | 87.0 | |
| 162.0 | 166.2 | 166.8 | 172.2 | 173.4 | 171.8 | 172.6 | |
| 74.5 | 73.2 | 75.9 | 78.4 | 81.2 | 81.8 | 84.8 | |
| 86.7 | 89.1 | 87.7 | 92.5 | 91.3 | 90.7 | 89.2 | |
| 111.1 | 106.3 | 106.8 | 104.0 | 103.5 | 99.7 | 98.9 | |
| 83.5 | 89.0 | 87.5 | 90.0 | 88.1 | 91.5 | 90.3 | |
| 74.5 | 73.1 | 75.6 | 73.5 | 76.3 | 81.8 | 84.4 | |
| 109.0 | 109.0 | 108.7 | 100.6 | 99.3 | 100.5 | 99.6 | |
| 95.5 | 92.3 | 90.8 | 90.5 | 89.9 | 87.9 | 86.9 | |
| 151.8 | 155.6 | 157.3 | 162.1 | 163.4 | 170.1 | 172.2 | |
1Corresponding author. Address: L.Y. Karpov Institute of Physical Chemistry, Vorontsovo Pole 10, 103064 Moscow Russia