Vibrational perturbation theory

A theory should just consider electron propagation with a weak probability of exciting a vibration. Perturbation theory seems to be justified due to the smallness of the electron-vibration coupling (Migdal s... 

At this point the first-principles perturbative (FP) approach becomes valuable. The same kinds of perturbative models are used to describe the vibrational-rotational motions as in the SP approach. However, data from electronic structure theory computations or potential energy functions are used to parameterize the formulas instead of spectroscopically obtained data. The FP approach has for example, been pursued by Martin et al. [16-18] and by Isaacson, Truhlar, and co-workers [19-25]. This avenue is especially valuable when spectroscopic data are not available for a molecule of interest. Codes are available that can carry out vibrational perturbation theory computations, using a grid of ab initio data as input SURVIBTM... 

Effects of vibrations on rotational motion can be conveniently taken into account by means of vibrational perturbation theory. As for centrifugal distortion, we here only recall the relevant issues and refer the reader to the specialistic literature [1,2,27-30]. 

The starting point for the vibrational perturbation theory is the semirigid Hamiltonian due to Watson [28, 29],... 

Vibrational Corrections In order to compare theoretically calculated spectroscopic parameters to experiment, one must consider the effect of molecular vibrations. This is because the properties alluded to above depend upon the structure of the molecule and therefore must be averaged over the vibrational motion of the system under consideration. Force field evaluations in conjimction with vibrational perturbation theory allow the estimation of zero-point vibrational corrections to molecular properties. 

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,... 

The Seetion entitled The BasiC ToolS Of Quantum Mechanics treats the fundamental postulates of quantum meehanies and several applieations to exaetly soluble model problems. These problems inelude the eonventional partiele-in-a-box (in one and more dimensions), rigid-rotor, harmonie oseillator, and one-eleetron hydrogenie atomie orbitals. The eoneept of the Bom-Oppenheimer separation of eleetronie and vibration-rotation motions is introdueed here. Moreover, the vibrational and rotational energies, states, and wavefunetions of diatomie, linear polyatomie and non-linear polyatomie moleeules are diseussed here at an introduetory level. This seetion also introduees the variational method and perturbation theory as tools that are used to deal with problems that ean not be solved exaetly. 

The Time Dependent Processes Seetion uses time-dependent perturbation theory, eombined with the elassieal eleetrie and magnetie fields that arise due to the interaetion of photons with the nuelei and eleetrons of a moleeule, to derive expressions for the rates of transitions among atomie or moleeular eleetronie, vibrational, and rotational states indueed by photon absorption or emission. Sourees of line broadening and time eorrelation funetion treatments of absorption lineshapes are briefly introdueed. Finally, transitions indueed by eollisions rather than by eleetromagnetie fields are briefly treated to provide an introduetion to the subjeet of theoretieal ehemieal dynamies. 

This Introductory Section was intended to provide the reader with an overview of the structure of quantum mechanics and to illustrate its application to several exactly solvable model problems. The model problems analyzed play especially important roles in chemistry because they form the basis upon which more sophisticated descriptions of the electronic structure and rotational-vibrational motions of molecules are built. The variational method and perturbation theory constitute the tools needed to make use of solutions of... 

Treating the full internal nuclear-motion dynamics of a polyatomic molecule is complicated. It is conventional to examine the rotational movement of a hypothetical "rigid" molecule as well as the vibrational motion of a non-rotating molecule, and to then treat the rotation-vibration couplings using perturbation theory. 

The interaction of a molecular species with electromagnetic fields can cause transitions to occur among the available molecular energy levels (electronic, vibrational, rotational, and nuclear spin). Collisions among molecular species likewise can cause transitions to occur. Time-dependent perturbation theory and the methods of molecular dynamics can be employed to treat such transitions. 

The tools of time-dependent perturbation theory can be applied to transitions among electronic, vibrational, and rotational states of molecules. 

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. 

Condition (3.19) is usually satisfied in processes of vibrational dephasing [41, 130, 131, 132], Because of this condition the dephasing is weak and the effect of rotational structure narrowing is pronounced. A much more important constraint is imposed by inequality (3.18). It shows that perturbation theory must be applied to a rather dense medium and even then only the central part of the spectrum (at Aa> < 1/t , 1/tc) is Lorentzian. 

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

Vibrational broadening in [162] was taken into account under the conventional assumption that contributions of vibrational dephasing and rotational relaxation to contour width are additive as in Eq. (3.49). This approximation provides the largest error at low densities, when the contour is significantly asymmetric and the perturbation theory does not work. In the frame of impact theory these relaxation processes may be separated more correctly under assumption of their statistical independence. Inclusion of dephasing causes appearance of a factor... 

One possibility for this was demonstrated in Chapter 3. If impact theory is still valid in a moderately dense fluid where non-model stochastic perturbation theory has been already found applicable, then evidently the continuation of the theory to liquid densities is justified. This simplest opportunity of unified description of nitrogen isotropic Q-branch from rarefied gas to liquid is validated due to the small enough frequency scale of rotation-vibration interaction. The frequency scales corresponding to IR and anisotropic Raman spectra are much larger. So the common applicability region for perturbation and impact theories hardly exists. The analysis of numerous experimental data proves that in simple (non-associated) systems there are three different scenarios of linear rotator spectral transformation. The IR spectrum in rarefied gas is a P-R doublet with either resolved or unresolved rotational structure. In the process of condensation the following may happen. 

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 ... 

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