Second-order vibrational perturbation theory

In the view of applying second-order vibrational perturbation theory, the best option for the coordinate system is the normal-coordinate representation. The cubic and (semidiagonal) quartic force constants are then derived by numerical differentiation of analytically evaluated second derivatives along the normal coordinates [64—66] ... 

Finally, we focus on the derivation of highly accurate structural information, by mixed experimental-theoretical analysis. As already mentioned, the equilibrium structure is directly related to the instead of experimentally measured Bq rotational constants. As a consequence, its elucidation requires explicit consideration of vibrational effects, which, within a pure experimental approach, would require the knowledge of experimental vibrational corrections to rotational constants for all isotopic species considered. A viable alternative is provided by the QM computations of the corresponding vibrational corrections [29], which can be obtained very effectively by second-order vibrational perturbation theory (VPT2) [210, 214] applied to a cubic force field [214-216] (see Section 10.3.2 for an extended account on VPT2). The combination of experimental ground-state rotational constants with computed vibrational corrections see Eq. 10.6) allows... 

In the following, we will give a short account on the approach we recently developed within second-order vibrational perturbative theory (VPT2) [210,214,... 

To determine the H2O semiclassical vibrational eigenstates using the vibrational part of the molecular Hamiltonian, a second order classical perturbation theory method was used. This relates the vibrational energy to the three good constants of the motion Jjj2> the vibrational good actions. These actions... 

Lattice vibrations are calculated by applying the second order perturbation theory approach of Varma and Weber , thereby combining first principles short range force constants with the electron-phonon coupling matrix arising from a tight-binding theory. 

Using second order perturbation theory [3], the mean and mean square values of the mass weighted coordinate x in the vibrational state Ij) with quantum number j are explicitely given by ... 

Second-order perturbation theory gives rise to two additional mechanisms involving an intermediate state that is vibrationally coupled to one and spin-orbit coupled to the other manifold (Fig. 11). 

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 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. 

Once the reliability of CCSD(T) had been established, we could proceed with confidence to use it to predict vibrational frequencies for Be3 and Be. In order to obtain the best possible prediction to aid experimentalists, a full quartic force field was generated for each molecule [76], using finite differences of computed energies, and fundamental frequencies were obtained via second-order perturbation theory. In Table 5.7 we list the CCSD(T) fundamental frequndes and, for comparison, the CCSD, CCSD(T) and MRCI harmonic frequencies. 

The ab initio calculated energies were obtained at the SCF level, followed by the evaluation of the second-order electronic correlation contribution with the many-body perturbation theory [SCF+MBPT(2)]. These calculations were performed on HF/3-21G(d) optimized geometries and include the zero-point vibrational energy corrections. 

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