Anharmonicity second order vibrational perturbation

In a smaller molecule (HCP), these diagnostically important changes in vibrational resonance structure are manifest in several ways (i) the onset of rapid changes in molecular constants, especially B values and second-order vibrational fine-structure parameters associated with a doubly degenerate bending mode (ii) the abrupt onset of anharmonic and Coriolis spectroscopic perturbations and (iii) the breakup of a persistent polyad structure 15]. 

V. Barone, Anharmonic vibrational properties by a fully automated second order perturbative approach. J. Chem. Phys. 122, 014108 (2005). 

A similar interaction would be observed between all Fermi polyads containing sets of vibrational levels related by the selection rule A tv = 2, A tv = +1, and the hamiltonian matrix should be diagonalized for each Fermi polyad without the use of perturbation theory. If, on the other hand, the interaction (63) were smaller, or the separation between the unperturbed levels were larger, the interaction could be treated by perturbation theory it can be shown that, in second-order perturbation theory, equation (63) would contribute a term to the vibrational anharmonic constants... 

All this applies to weak and medium strong H-bonds like those encountered for alcohols and many other systems up to carboxylic acid dimers or about 32-42 kJ/mol. (8 or 10 kcal/mol.) Unfortunately vibrational spectra of systems with very strong H-bonds could, with a few exceptions, only be measured in condensed phases. Factors that come in when such systems are examined are potential surfaces with two minima with, in certain cases, the possibility of tunnelling, or flat single minima Most of these systems are likely to be so anharmonic that second order perturbation theory breaks down and the concept of normal vibrations becomes itself question-nable. Many such systems are highly polarizable and are strongly influenced by the environment yielding extremely broad bands 92). Bratos and Ratajczak 93) has shown that even such systems can be handled by relaxation theories. 

Amos et al. " considered the same complex using comparable basis sets, and evaluated the anharmonic constants using standard second-order perturbation formulas, based upon third and fourth derivatives of the SCF energy. This treatment evaluates each vibrational frequency, Vj, in terms of a purely harmonic potential cOj, and anharmonic constants Xjj relating the various modes i and j (assuming all modes are nondegenerate). 

Quadratic terms in die property expansions are considered to be first-order in electrical anharmonicity, cubic terms are taken to be second-order, etc. Similarly, cubic terms in the vibrational potential are considered to be first-order in mechanical anharmonicity, quartic terms are second-order, and so forth. The notation (n, m) is used hereafter for the order of electrical (n) and mechanical (m) anharmonicity whereas the total order (n -I- m) is denoted by I, II,. Although our definition of orders is reasonable other choices are possible. Two key questions are (1) How important are anharmonicity contributions to vibrational NLO properties and (2) What is the convergence behavior of the double perturbation series in electrical and mechanical anharmonicity Both questions will be addressed later. Here we note that compact expressions, complete through order II in electrical plus mechanical anharmonicity, have been presented [19]. The formulas of order I contain either cubic force constants or second derivatives of the electrical properties with respect to the normal coordinates. Depending upon the level of calculation at least one order of numerical differentiation is ordinarily required to determine these anharmonicity parameters. For electrical properties, the additional normal coordinate derivative may be replaced by an electric field derivative using relations such as d p./dQidQj = —d E/dldAj.ACd, = —dk,/rjF where F is the field and k j is... 

Under conditions where large amplitude mechanical perturbations create a dense sea of phonons, especially phonons near the zone edge, the mean-free path in Eq. (15) may be decreased due to phonon-phonon anharmonic coupling. In other words there is a second-order correction to Eq. (15) that reduces the thermal conductivity at higher phonon concentrations. For instance terms where two phonons efficiently combine to pump a doorway vibration drastically reduce the thermal conductivity by converting a mobile pair of phonons into an essentially immobile vibration. Similarly, interactions that convert faster acoustic phonons into slower optic phonons also reduce the mean-free path. 

Perturbation theory has been applied to anharmonic calculations of spectroscopy from ab initio potentials in a large number of studies [19-25,115-121]. In nearly all cases so far, second-order perturbation theory was employed. The representation of the anharmonic potential generally used in these studies is a polynomial in the normal modes, most often a quartic force field. A code implementing this vibrational method was recently incorporated by V. Barone in gaussian [24]. Calculations were carried out for relatively large molecules, such as pyrrole and furan [25], uracil and thiouracil [118], and azabenzenes [119]. We note that in addition to spectroscopy, the ab initio perturbation theoretic algorithms were also applied to the calculation of thermodynamic properties... 

Figure 3. Schematic depiction of first three terms in dimensional perturbation expansion of Eq.(12) for hydrogenic atom. For each the effective potential W(r) for the D oo limit is shown (solid curve). At left, the zeroth-order term corresponds to the electron at rest at the minimum of T (r). In the middle, the first-order term, proportional to 1/D, corresponds to harmonic oscillations (as if potential were replaced by the dashed parabola). At right, the second-order term corresponds to anharmonic vibrations (arising from cubic and quartic portions of the potential).

The argument Rp implies structure relaxation in the field, and P" means the nuclear relaxation part of P, while the subscript oc oo invokes the so-called infinite optical frequency (lOF) approximation. In principle, this procedure allows one to obtain most of the major dynamic vibrational NR contributions in addition to the purely static ones of Eqs.4.5. 7. The linear term in the electric field expansion of Eq. (4) gives the dc-Pockels effect the quadratic term gives the optical Kerr Effect and the linear term in the expansion of beta yields dc-second harmonic generation (all in the lOF approximation). For laser frequencies in the optical region it has been demonstrated that the latter approximation is normally quite accurate [29-31]. In fact, this approximation is equivalent to neglecting terms of the order with respect to unity (coy is a vibrational frequency). In terms of Bishop and Kirt-man perturbation theory [32-34] all vibrational contributions through first-order in mechanical and/or electrical anharmonicity, and some of second-order, are included in the NR treatment [35]. 

From a perturbative standpoint, anharmonicity in the stretching potential mixes harmonic wavefunctions. If the anharmonicity were to produce a mixing of the zero-order n = l and n = 2 states, the nonzero transition integral between the harmonic n = 0 and n = l states would be distributed among the perturbed first and second excited states. This would make both the n = 0 to n = 1 and n = 0 to n = 2 transitions allowed in the anhar-monically perturbed states. Anharmonicity in the potential and nonlinearity in the dipole moment function make it possible to observe vibrational transitions other than An = 1. However, such other transitions are usually weaker or less intense because the harmonic nature of the potential and the linearity of the dipole moment function have the major role. 

---