Perturbation theory, of nuclei and electrons

THE MANY-BODY PERTURBATION THEORY OF NUCLEI AND ELECTRONS... 

The types of interactions which can arise in the diagrammatic perturbation theory of nuclei and electrons can be classified according to the number and type of particles involved. First we subdivide the interactions into one-particle and two-particle types. 

First-order diagrammatic perturbation theory of nuclei and electrons... 

Figure 9. Second-order diagrams involving a one- and two-particle interactions in the perturbation theory of nuclei and electrons.

In section 2 we define the total molecular Hamiltonian operator describing both nuclear and electronic motion. The Hartree-Fock theory for nuclei and electrons is presented in section 3 and a many-body perturbation theory which uses this as a reference is developed in section 4. The diagrammatic perturbation theory of nuclei and... 

In the original mathematical treatment of nuclear and electronic motion, M. Bom and J. R. Oppenheimer (1927) applied perturbation theory to equation (10.5) using the kinetic energy operator Tq for the nuclei as the perturbation. The proper choice for the expansion parameter is A = (me/M) /", where M is the mean nuclear mass... 

The electronic contributions to the g factors arise in second-order perturbation theory from the perturbation of the electronic motion by the vibrational or rotational motion of the nuclei [19,26]. This non-adiabatic coupling of nuclear and electronic motion, which exemplifies a breakdown of the Born-Oppenheimer approximation, leads to a mixing of the electronic ground state with excited electronic states of appropriate symmetry. The electronic contribution to the vibrational g factor of a diatomic molecule is then given as a sum-over-excited-states expression... 

The Time Dependent Processes Section uses time-dependent perturbation theory, combined with the classical electric and magnetic fields that arise due to the interaction of photons with the nuclei and electrons of a molecule, to derive expressions for the rates of transitions among atomic or molecular electronic, vibrational, and rotational states induced by photon absorption or emission. Sources of line broadening and time correlation function treatments of absorption lineshapes are briefly introduced. Finally, transitions induced by collisions rather than by electromagnetic fields are briefly treated to provide an introduction to the subject of theoretical chemical dynamics. 

Each electron in the system is assigned to either molecule A or B, and Hamiltonian operators and W for each molecule defined in terms of its assigned electrons. The unperturbed Hamiltonian for the system is then + W , and the perturbation XU consists of the Coulomb interactions between the nuclei and electrons of A and those of B. The unperturbed states, eigenfunctions of are simple product functions for closed-shell molecules, non-degenerate, Rayleigh-Schrodinger, perturbation theory gives the... 

In order to discuss solvation theoretically, we should solve the Schro-dinger equation for the substrate and solvent together. This of course is not practicable. However, chemical evidence suggests that the molecules of solvent and solute in a solution are not usually greatly affected by the interactions between them this therefore should be a favorable case for the use of PMO theory. We therefore regard the interactions as a perturbation, the electrons of each molecule now moving in a field due to the nuclei and electrons of solute as well as solvent, instead of those of solvent or solute alone. 

The expression for the force on the nuclei, Eq. (89), has the same form as the BO force Eq. (16), but the wave function here is the time-dependent one. As can be shown by perturbation theory, in the limit that the nuclei move very slowly compared to the electrons, and if only one electronic state is involved, the two expressions for the wave function become equivalent. This can be shown by comparing the time-independent equation for the eigenfunction of H i at time t... 

Most of the techniques described in this Chapter are of the ab initio type. This means that they attempt to compute electronic state energies and other physical properties, as functions of the positions of the nuclei, from first principles without the use or knowledge of experimental input. Although perturbation theory or the variational method may be used to generate the working equations of a particular method, and although finite atomic orbital basis sets are nearly always utilized, these approximations do not involve fitting to known experimental data. They represent approximations that can be systematically improved as the level of treatment is enhanced. 

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