The SLG approximation

To simplify the interpretation of the energy in the SLG approximation, we further regroup the individual terms and rewrite them as ... 

The derivation is based on the variational principle for energy and it naturally starts from writing it down. The analysis of the properties of the ESVs pertinent to the SLG approximation performed in Section 3.2.2 allows us to rewrite the energy eq. (2.88) using our current knowledge of the transferability of the density related ESVs as follows ... 

The geminal wave functions in eq. (2.60) in the SLG approximation are by definition obtained by diagonalizing the effective Hamiltonian for the m-th bond. These latter are as previously given by eq. (3.1) which can be recast to the form ... 

The latter expression flows to the same value as in the SLG approximation for the free ligand, as for the LPs the following (HFR-type) relation naturally holds ... 

Accuracy of the SLG approximation can be improved by perturbation theory. Second quantization provides us a powerful tool in developing a many-body theory suitable to derive interbond delocalization and correlation effects. The first question concerns the partitioning of the Hamiltonian to a zeroth-order part and perturbation. LFsing a straightforward generalization of the Moller-Plesset (1934) partitioning, the zeroth-order Hamiltonian is chosen as the sum of the effective intrabond Hamiltonians ... 

Due to the fact that the SLG wave function belongs to the GF approximation (Section 1.7), it is subject to numerous selection rules characteristic of GF. Their explicit form can be easily obtained using the second quantization formalism. Since the operators of electron creation on the right and left HOs satisfy usual anticommutation relations for orthogonal basis and the number of particle operators have the usual form ... 

Electronic energy in the semiempirical SLG approximation is thus represented as a sum of five contributions ... 

The SLG-MINDO/3 and SCF-MINDO/3 methods have approximately the same accuracy while calculating the heats of formation of organic compounds. Significant deviations from the experiment (for both wave functions) are observed for the branched compounds. The heats of formation are significantly overestimated for both types of wave functions. It is clear that the intrabond correlation has not much to do with this defect. The NDDO parametrization partially rectifies this by a more detailed account of two-center integrals. 

To demonstrate the computational capacities of the SLG-MINDO/3 method we carried out calculations (for the fixed geometry) for a series of normal, saturated hydrocarbons ranging from CH4 to C20H42 by the SLG-MINDO/3 and SCF-MINDO/3 methods. It has been shown that the dependence of computation time on the system size is essentially non-linear in the case of the HFR approximation and is practically linear for the geminal approach. The SLG-MINDO/3 procedure is faster than the SCF-MINDO/3 one even for the simplest hydrocarbons. In the case of the normal hydrocarbon C20H42 (its semiempirical calculation uses 122 basis functions) the computation time for two methods differs 30 times in favor of the SLG approach. 

The general scheme of the derivation of mechanistic models of PES from a QC description of molecular electronic structure reduces to the following moves. In the SLG based semiempirical QC there are the variables of two classes (electronic structure variables - ESV) geminal amplitudes and variables describing the HOs. ESVs of these classes must be (approximately) estimated, although the ESVs depend on molecular composition and geometry. 

Perturbative estimate of ESVs with respect to noncorrelated bare Hamiltonian. The specificity of each bond and molecule in the approach based on the SLG expressions for the wave function is taken into account perturbatively by using the linear response approximation [25]. We need perturbative estimates of the expectation values of the pseudospin operators which, in their turn, give values of the density matrix elements according to eq. (3.5). According to the general theory (Section 1.3.3.2) the linear response 5(A) of an expectation value of the operator A to the time independent perturbation AB of the Hamiltonian (A is the parameter characterizing the intensity of the perturbation) has the form ... 

Numerical experiments concerning the density ESVs transferability. The above analytical results have been supplied by numerical estimates done to get a feeling of the real sense of the first and second order approximations. Numerical results on the ESVs (Tzm), (f2m), and ( +m) obtained by the SLG method eq. (3.1) using the MINDO/3 parameterization and by the approximate formulae of eqs. (3.9), (3.12),... 

The constructions of different approximations will be done in the sections that follow on the basis of the variational principle for molecular electronic energy in the SLG-based approximation. We shall demonstrate that this treatment leads to a mechanistic model which can in a sense be considered a generic or deductive form of MM. It means that although the simple balls-and-springs model can hardly be justified from any general point of view, it does not mean that any other mechanistic model cannot be justified at all. And that is what we shall provide. 

We are going to deduce a mechanistic model for molecular energy from the SLG-based QM method described in Section 2.4. We shall perform transformations and approximations, following me line mentioned above and arrive naturally at a tetrahedral representation of heavy atoms consistent with facts known from stereochemistry and usually interpreted in the VSEPR framework. The mechanistic model of PES will be derived in terms of these objects. 

Multiplying the resonance integral by the quadrupled transferable spin bond order PoL = CQ- (3.14) results in the resonance energy of the m-th bond which is the only nontrivial contribution to the molecular energy at this (FAFO) level of approximate treatment of the MINDO/3 Hamiltonian using the SLG trial wave function. Within this picture the hybridization tetrahedra interact and the interaction energy depends on separations between centers of the tetrahedra, their mutual orientation, with respect to the bond axis. 

In this section we apply our methodology of constructing hybrid models of molecular electronic structure to the case of coordination compounds. Our main tool will be the SLG/HFR hybrid scheme described below. It will be used to formalize the difference between the organic and inorganic parts of the coordination compound molecule. After it is done the ESVs relevant for the most problematic inorganic part will be selected and reasonably approximated. 

The most precise approximations are given by eqs. (3.16) and (3.17) yielding results which perfectly coincide with the exact (SLG-MINDO/3) ones even for very polar O-H and F-H bonds. This may be qualified as using estimates of the second order in /r, provided the bond polarity (or equivalently (r2TO)) are linear in /. if the orders up to the second are considered. Estimates obtained in the limit Cm > 1 by the formulae eq. (3.12) give reasonable results for the ESVs of the bonds in not too polar molecules at their equilibrium geometries. The bond- and atom-specific corrections of the first and second order in Cm1 and //,m acquire the form ... 

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