Hamiltonian molecular, modification

Several modifications of EH theory for transition elements have been proposed, including those of Cotton et al. (25), Fenske et al. (26), and Canadine and Hiller (27). The explanation of the use of several different versions of EH theory lies in the use of an effective Hamiltonian and the attempt to identify it with the Hamiltonian used by Roothaan (28) in Hartree-Fock molecular theory. The SCF Hamiltonian is written... 

Appendix. Modification of the Molecular Hamiltonian and Factorization of the Matrix Element... 

Just as the perturbation theory described in the previous section, the self-consistent phonon (SCP) method applies only in the case of small oscillations around some equilibrium configuration. The SCP method was originally formulated (Werthamer, 1976) for atomic, rare gas, crystals. It can be directly applied to the translational vibrations in molecular crystals and, with some modification, to the librations. The essential idea is to look for an effective harmonic Hamiltonian H0, which approximates the exact crystal Hamiltonian as closely as possible, in the sense that it minimizes the free energy Avar. This minimization rests on the thermodynamic variation principle ... 

This classification is useful for molecules which do not have stereoisomers. This point is important in chemistry. The asymptotic hamiltonians of normal molecules are invariant to parity. For stereoisomers, the molecule assumes under inversion a configuration in space which cannot be made to coincide with the original configuration by rotation. For these type of molecules, we will talk of a symmetry-broken molecular hamiltonian. These right and left hand modifications exist as real molecules that can interconvert into each other via transition structures having appropriate symmetry. From the present standpoint, there exists different electronic wave functions for the R- and L-molecules. Thus, each subset cannot be used to expand wave functions of the other. 

The life is relativistic and of the same kind should be the quantum chemistry. The four-component calculations involving large number of dynamic correlation are extremely time-consuming. The important thing, however, is that one is now able to formulate the fully equivalent two-component algorithms for those calculations. Since the final philosophy of the two-component calculations is similar to non-relativistic theory and most of mathematics are simply enough to comprehend the routine molecular relativistic calculations are possible. The standard nonrelativistic codes can be used with simple modification of the core Hamiltonian. The future is still in the development of the true two-component codes which will be able to deal with the spin-orbit interaction effect not in a posterior way as it is done nowadays. 

Using the effective interaction kernel function of Eq. (5.26) in the Hamiltonian (5.19), we can study the modification of molecular properties in disordered phases, like liquids, liquid crystals or crystals with static disorder. A similar approach has been proposed by Sese et al. [174-176] for the study of molecular liquids at CNDO level. 

The all-electron DFT LCAO approach with Gaussian basis sets was extended to scalar-relativistic calculations of periodic systems [561]. The approach is based on a third-order DKH approximation, and similar to the molecular case, requires only a modification of the one-electron Hamiltonian. The effective core Hamiltonian is obtained by applying the DKH transformation to the nuclear-electron potential Vn-Considering that relativistic effects are dominated by the short-range part of the Coulomb interaction, it is proposed to replace the nuclear-electron Coulomb operator used to build the DKH Hamiltonian by a short-range Coulomb operator... 

On the other hand the hamiltonian of the isolated solute Hq allows us to compute the internal energy of this molecule and if one assumes that there is no entropy variation associated with the modification of the molecular charge distribution, the equilibrium properties of the system are reached by minimizing the sum ... 

Under the circumstances, a number of theoretical methods have been already developed to improve the QM/MM-MD method, e.g., the modification of the semi-empirical QM Hamiltonians [7, 52-54], the optimization of the QM/MM empirical parameters [10] and the replacement of the empirical repulsion potential functions [55]. However, these methods need the numerical values of some reasonable reference quantities to optimize the parameters for some specific molecular systems. Moreover, it is usually hard to obtain the reference experimental or computational ones in solution. It is, therefore, reasonable and plausible as a second best strategy that the closer MM solvent molecules around the QM solute should be included into the QM region to avoid the serious problems in the boundary between QM and MM regions. This is because the most serious problem is originating in the quantum-mechanical behaviors. On the basis of such strategy, we have developed the number-adaptive multiscale (NAM) QM/MM-MD [56, 57] and the QM/MM-MD method combined with the fragment molecular orbital (FMO) one, i.e., FMO-QM/MM-MD method [20]. 

Molecular order in the smectic-A phase was probed recently via a proton NMR study of three aromatic solutes in the liquid crystal 8CB [35]. The results were analyzed in the context of a simple modification of K-M theory for dissolved non-uniaxial solutes. The smectic solute Hamiltonian was written in the form ... 

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