Second-order perturbation theory approximate

The virtual orbitals / are optimised using a second-order perturbation theory approximation to the energy [31] so that we need only evaluate the diagonal and first row elements of the hamiltonian and overlap matrices ... 

In fact, we have already used a modeling strategy when Po(AU) was approximated as a Gaussian. This led to the second-order perturbation theory, which is only of limited accuracy. A simple extension of this approach is to represent Pq(AU) as a linear combination of n Gaussian functions, p, (AU), with different mean values and variances [40]... 

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... 

Werner, H.-J., Manby, F.R. Explicitly correlated second-order perturbation theory using density fitting and local approximations. J. Chem. Phys. 2006, 124, 054114. 

The spin-orbit mean field (SOMF) operator (56-58) is used to approximate the Breit—Pauli two-electron SOC operator as an effective one-electron operator. Using second-order perturbation theory (59), one can end up with the working equations ... 

The instantaneous OH frequency was calculated at each time step in an adiabatic approximation (fast quantal vibration in a slow classical bath ). We applied second-order perturbation theory, in which the instantaneous solvent-induced frequency shift from the gas-phase value is obtained from the solute-solvent forces and their derivatives acting on a rigid OH bond. This method is both numerically advantageous and allows exploration of sources of various solvent contributions to the frequency shift. 

The onset of sudden variations in vibrational fine structure is one of the most sensitive indicators of a change in resonance structure. The magnitudes of fine-structure parameters are determined by second-order perturbation theory (a Van Vleck or contact transformation) [17]. The energy denominators in these second-order sums over states are approximately independent of vib as long as the <01 <02 - 3/v-6 resonance structure is conserved. 

Then for a large energy gap Az = e(4A2g) - e(6Aig), and with the help of the substitution D = 2/5Az, the second-order perturbation theory yields the approximate roots of the form listed in Table 35. 

Reaction field theory with a spherical cavity, as proposed by Karlstrom [77, 78], has been applied to the calculation of the ECD spectrum of a rigid cyclic diamide, diazabicyclo[2,2,2]octane-3,6-dione, in an aqueous environment [79], In this case, the complete active space self-consistent field (CASSCF) and multiconfigurational second-order perturbation theory (CASPT2) methods were used. The qualitative shape of the solution-phase spectrum was reproduced by these reaction field calculations, although this was also approximately achieved by calculations on an isolated molecule. 

If Hu is much less negative than H22, it is possible to apply second-order perturbation theory to eq. (37). Since the eigenvalues then contain only even powers Sj, S12,. .. and since it is clear that the Wolfsberg-Helmholz method anyhow is a dubious approximation to the M.O. if Sjg is not rather small compared to S1, it may be interesting to find the approximate eigenvalues accurate until second power of the overlap... 

Quantum mechanically, resonance Raman cross-sections can be calculated by the following sum-over-states expression derived from second-order perturbation theory within the adiabatic, Born-Oppenheimer and harmonic approximations... 

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