Hamiltonian energy separation

Thus when expressed in normal coordinates, the potential energy becomes a sum of terms, each involving only one coordinate, and the kinetic energy stays as such a sum. (Hence the Hamiltonian is separable in normal coordinates.)... 

The generalization of the pseudopotential method to molecules was done by Boni-facic and Huzinaga[3] and by Goddard, Melius and Kahn[4] some ten years after Phillips and Kleinman s original proposal. In the molecular pseudopotential or Effective Core Potential (ECP) method all core-valence interactions are approximated with l dependent projection operators, and a totally symmetric screening type potential. The new operators, which are parametrized such that the ECP operator should reproduce atomic all electron results, are added to the Hamiltonian and the one electron ECP equations axe obtained variationally in the same way as the usual Hartree Fock equations. Since the total energy is calculated with respect to this approximative Hamiltonian the separability problem becomes obsolete. 

When kT is large with respect to the energy gap, the population of each level is just one over the number of levels or functions. When kT is small with respect to the energy separation, then only the lowest level is occupied. The new energy levels S are linear combinations of the S, Ms > functions of each metal ion. The functions and energies can be calculated by the simple Heisenberg Hamiltonian, that for a dimer is... 

Once we have removed the terms which couple different electronic states (at least to a certain level of accuracy), we can deal with the motion in the other degrees of freedom of the molecule for each electronic state separately. The next step in the process is to consider the vibrational degree of freedom which is usually responsible for the largest energy separations within each electronic state. If we perform a suitable transformation to uncouple the different vibrational states, we obtain an effective Hamiltonian for each vibronic state. Once again, we adopt a perturbation approach. 

Where R, Pj are position and momentum vectors for particle i and Rij = R, — Ry. This Hamiltonian is separable into a Hamiltonian for the kinetic energy of the center-of-mass and a Hamiltonian describing the internal interaction energy. A transformation leading to this separation is given by [9]... 

Even though the one-electron H2+ Hamiltonian has been used to derive all 10 molecular orbitals (only one of which is even half-filled in H2+), a MO diagram built from the expanded basis will enable us to draw quite accurate boundary surfaces and relative energy separations for all the second row homonuclear diatomics. (This has been verified by the more rigorous approaches that are detailed later.) The figures sketched in Figure 9 show the MO diagrams that result from the 10 X 10 secular determinant. 

Eq. (18) can be transformed to a simpler form where the centrifugal energy separates out from other kinetic terms. The resulting Hamiltonian has the following form ... 

The simplest way to include solvation effects is to calculate the reaction path and tunneling paths of the solute in the gas phase and then add the free energy of solvation at every point along the reaction path and tunneling paths. This is equivalent to treating the Hamiltonian as separable in solute coordinates and solvent coordinates, and we call it separable equilibrium solvation (SES) [74]. Adding tunneling in this method requires a new approximation, namely the canonical mean shape (CMS) approximation [75]. 

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