Variational energy calculations

M is an n x n matrix with 1/2/i, on the diagonal and 1 /2M for off diagonal elements. This is the Hamiltonian we use for variational energy calculations (cases 1 and 2). We make no further transformations or approximations to this Hamiltonian. More information on the center of mass separation and form of the Hamiltonian, eqn.(6), can be found in the references[9,10,12]. 

Variational RRKM calculations, as described above, show that a imimolecular dissociation reaction may have two variational transition states [32, 31, 34, 31 and 36], i.e. one that is a tight vibrator type and another that is a loose rotator type. Wliether a particular reaction has both of these variational transition states, at a particular energy, depends on the properties of the reaction s potential energy surface [33, 34 and 31]- For many dissociation reactions there is only one variational transition state, which smoothly changes from a loose rotator type to a tight vibrator type as the energy is increased [26],... 

The simultaneous optimization of the LCAO-MO and Cl coefficients performed within an MCSCF calculation is a quite formidable task. The variational energy functional is a quadratic function of the Cl coefficients, and so one can express the stationary conditions for these variables in the secular form ... 

Coupled cluster calculations give variational energies as long as the excitations are included successively. Thus, CCSD is variational, but CCD is not. CCD still tends to be a bit more accurate than CID. 

In figure 2 we show the difference between electrostatic energy calculated by the new method, and E s - This shows that the variation of E g with the choice of... 

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

Miller, M. A. Reinhardt, W. P., Efficient free energy calculations by variationally optimized metric scaling concepts and applications to the volume dependence of cluster free energies and to solid-solid phase transitions, J. Chem. Phys. 2000,113, 7035-7046... 

De Koning, M., Optimizing the driving function for nonequilibrium free-energy calculations in the linear regime. A variational approach, J. Chem. Phys. 2005, 122,... 

In ab initio methods (which, by definiton, should not contain empirical parameters), the dynamic correlation energy must be recovered by a true extension of the (single configuration or small Cl) model. This can be done by using a very large basis of configurations, but there are more economical methods based on many-body perturbation theory which allow one to circumvent the expensive (and often impracticable) large variational Cl calculation. Due to their importance in calculations of polyene radical ion excited states, these will be briefly described in Section 4. 

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