Quantum chemical methods density matrix elements

Several approaches are available in the literature to generate and evaluate Hamiltonian matrix elements with wavefunctions of charge-localized, diabatic states. They differ in the level of theory used in the calculation and in the way localized electronic structures are created [15, 25, 26, 29-31]. When wavefunction-based quantum-chemical methods are employed, the framework of the generalized Mulliken-Hush method (GMH) [29, 32-34], is particularly successful. So far, it has been used in conjunction with accurate electronic structure methods for small and medium sized systems [35-37]. As an alternative to GMH and other derived methods [38, 39], additional methods have been explored for their applicability in larger systems such as constrained density functional method (CDFT) [25, 37, 40, 41], and fragmentation approaches [42-47], which also include the frozen density embedding (FDE) method [48, 49]. 

It is interesting to compare the possibilities and errors of different hybrid QM/MM schemes. The careful examination and comparison of link atom and LSCF techniques was performed in Ref. [128] using the CHARMM force field [114] and the AMI method [143] as a quantum chemical procedure. In the case of the link atom procedure two options were used QQ - the link atom does not interact with the MM subsystem and HQ - link atom interacts with all MM atoms. The main conclusion of this consideration is that the LSCF and the link atom schemes are of similar quality. The error in the proton affinity determination induced by these schemes is several kcal/mol. It is noteworthy that all the schemes work rather badly in description of conformational properties of n-butane. The large charge on the MM atoms in the proximity of the QM subsystem (especially on the boundary atom) cause significant errors in the proton affinity estimates for all methods (especially, in the case of the LSCF approach where the error can be of tens of kcal/mol). This is not surprising since the stability and transferability of intrabond one- and two-electron density matrix elements Eq. (19) is broken here. It proves that the simple electrostatic model is not well appropriate for these schemes and that a detailed analysis of the... 

Solutions to the eigenvalue equation (16) can be obtained by any of the standard quantum chemical methods, such as Hartree-Fock SCF, multiconfiguration SCF (MCSCF), Mpller-Plesset perturbation, coupled cluster, or density functional theories. The matrix elements of Hr, a one-electron operator, are readily computed, thus formally the inclusion of solvent effects in the quantum mechanical description of the solute molecule appears quite simple. Moreover, gradients of the eigenvalue E are readily computed. 

The theoretical tools of quantum chemistry briefly described in the previous chapter are numerously implemented, sometimes explicitly and sometimes implicitly, in ab initio, density functional (DFT), and semi-empirical theories of quantum chemistry and in the computer program suits based upon them. It is usually believed that the difference between the methods stems from different approximations used for the one- and two-electron matrix elements of the molecular Hamiltonian eq. (1.177) employed throughout the calculation. However, this type of classification is not particularly suitable in the context of hybrid methods where attention must be drawn to the way of separating the entire molecular system (eventually - the universe itself) into parts, of which some are treated explicitly on a quantum mechanical/chemical level, while others are considered classically and the rest is not addressed at all. That general formulation allows us to cover both the traditional quantum chemistry methods based on the wave functions and the DFT-based methods, which generally claim... 

Further, it is understood that each matrix element consists of the components originating in the pure QM, the est and the vdW contribution. The est components are conveniently computed by the quantum chemical calculation package. For instance, in GAUSSIAN program [25], several approximate methods of electronic state calculations are available, e.g., the Hartree-Fock (HF), second-order Moller-Plesset perturbation theory (MP2), conhguration interaction field (CIS), complete active space self-consistent field (CASSCF) method, and the density functional theory (DFT) methods. On the other hand, since the vdW components are expressed as such analytical functions of the mw Cartesian coordinate variables involved in the same atom (A = B) as follows. 

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