Hamiltonians electrostatic component

Most of the variational treatments of spin-orbit interaction utilize one-component MOs as the one-particle basis. The SOC is then introduced at the Cl level. A so-called SOCI can be realized either as a one- or two-step procedure. Evidently, one-step methods determine spin-orbit coupling and electron correlation simultaneously. In two-step procedures, typically different matrix representations of the electrostatic and magnetic Hamiltonians are chosen. 

As an example of application of the method we have considered the case of the acrolein molecule in aqueous solution. We have shown how ASEP/MD permits a unified treatment of the absorption, fluorescence, phosphorescence, internal conversion and intersystem crossing processes. Although, in principle, electrostatic, polarization, dispersion and exchange components of the solute-solvent interaction energy are taken into account, only the firsts two terms are included into the molecular Hamiltonian and, hence, affect the solute wavefunction. Dispersion and exchange components are represented through a Lennard-Jones potential that depends only on the nuclear coordinates. The inclusion of the effect of these components on the solute wavefunction is important in order to understand the solvent effect on the red shift of the bands of absorption spectra of non-polar molecules or the disappearance of... 

The basic equations that rule the mechanics of H-bonds are developed in this appendix. They have been already established in the appendix of Ch. 5 but take here a slightly different form that makes the role of the mass m of the H-atom more apparent, in view of predicting effects of an H/D substitution. The formation of an H-bond is the result of an electrostatic interaction between the electrons and the nuclei of two molecules X-H and Y. Molecules are quantum objects that are ruled by an Hamiltonian H that depends on the coordinates r of electrons, q of the H(D)-atom that establishes an H(D)-bond and Q that defines the relative positions of the two molecular components X-H and Y. r stands for aU coordinates of all electrons e. The relative coordinates q and Q of the nuclei are defined in Figure 2.1. Q stands for all three intermonomer coordinates Q, Qg and defined in this figure. The quantum description is necessary for this H-bond, because a classical description fails to describe any chemical bond. This Hamiltonian H writes ... 

All other interactions between the particles comprising the molecule are, at least for the present, neglected. In particular there is no mention in the Hamiltonian of any spin-dependent (magnetic) interactions both the phenomenological Pauli (two-component) and the relativistic Dirac (four-component) interaction terms amongst electron and nuclei are replaced by the simple electrostatic model. 

In this scheme, the INT is partitioned into electrostatic, exchange, repulsion, polarization, and dispersion. Let us now look at the mathematical derivation of the various components. For a molecular system (AB) with wavefunction O and total energy Hamiltonian operator H, the expectation value is given as... 

Symmetry-adapted perturbation theory is a well-motivated theoretical approach to compute the individual components of intermolecular interactions, namely, the electrostatic, induction, dispersion, and exchange-repulsion terms. The approach is a double-perturbation theory that uses a Hartree-Fock reference, with a Fock operator F written as the sum of Fock operators for the individual molecules. Both the intramolecular correlation potential W) and the intermolecular interactions (V) are treated as perturbations, so that the Hamiltonian is expressed as... 

The intramolecular CHA fonnaUsm received no direct numerical applications. However, the application of the same philosophy to the BSSE problem of intermolecular interactions has been found rather useful [1, 3,4], An eneigy decomposition formalism has also been developed [1], in which the different energy components were defined as the expectation values of the corresponding physical terms of the Hamiltonian the analysis of one of them (that of the diatomic electrostatic interactions in a point-charge qjproximation) had led to the definition of the bond order index [5-8], which has been widely applied in studying different chemical problems. 

In order to use this transformation for the Hamiltonian as represented by Eq. (6.68), microscopic density terms that are quadratic in nature need to be written in the form given on the left-hand side in Eqs. (6.77) and (6.78). Electrostatic terms in He are already in the appropriate form. It is only the terms in Hw that needs to be rewritten. This can be achieved by rewriting in terms of order parameters and total density. For an n component system, all microscopic densities can be described by n—1 independent order parameters (due to the incompressibility constraint serving as the nth relation among the densities). There are many different ways of defining these order parameters. One convenient definition, which makes mathematics simple, is the deviation of densities of solutes from the solvent density, that is, defining c )j(r) = Qj(r)—Qj(r) forj = 1,2,... (n—1), wherej is the index for different solutes (monomers, counterions, and the salt ions). Using the transformation for each quadratic term in the Hamiltonian (cf. Eq. (6.68)), the partition function becomes... 

Similar to the distributed-multipole expansion of molecular electrostatic fields, one can derive a distributed-polarizability expansion of the molecular field response. We can start by including the multipole-expansion in the perturbing Hamiltonian term W = Qf(p, where we again use the Einstein sum convention for both superscripts a, referencing an expansion site, and subscripts t, which summarize the multipole components (/, k) in just one index. Using this approximation for the intermolecular electrostatic interaction, the second-order energy correction now reads ... 

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