Quartic anharmonicities

This formula is derived in Appendix 3). With regard to various cubic and quartic anharmonic interactions, the quantity ft is characterized by a certain combination of these anharmonic contributions and becomes dependent on k (see Eq. (4.3.14) for a related quantity and Ref. 140). However, this dependence is insignificant compared to the k-dependence appearing in the denominators of Eqs. (4.3.32) and (4.3.34). Therefore, spectral characteristics defined by formulae (4.3.32) can with good reason be regarded as proportional to certain functions of lateral interaction parameters and of the resonance width 77 ... 

Force fields up to quartic anharmonic terms are now known with reasonably high accuracy for several triatomic molecules and the results shown in Table 3 for H2O are typical. However, even for these there has had to be an assumption that some of the quartic interaction terms are zero in order that the equations from which the constants are derived shall have unique solutions. It can be seen moreover that some of the cubic and quartic terms have uncertainties which are larger than the values of the constants themselves. 

Figure 18. Plot of potential energy vs. distortion coordinate for a material with a harmonic potential and a material with positive and negative quartic anharmonic terms.

The structure/property relationships that govern third-order NLO polarization are not well understood. Like second-order effects, third-order effects seem to scale with the linear polarizability. As a result, most research to date has been on highly polarizable molecules and materials such as polyacetylene, polythiophene and various semiconductors. To optimize third- order NLO response, a quartic, anharmonic term must be introduced into the electronic potential of the material. However, an understanding of the relationship between chemical structure and quartic anharmonicity must also be developed. Tutorials by P. Prasad and D. Eaton discuss some of the issues relating to third-order NLO materials. 

EFF (Empirical force field) [186] has been designed just for modeling hydrocarbons. It uses the quartic anharmonic potential for the bond stretching, and the cubic anharmonic for the valence angle bending. No out of plane or electrostatic terms are involved, although the cross terms, except torsion-torsion and bend-torsion ones, are included. 

The MM3 method [190] is parametrized for as much as 153 atomic types eventually covering almost all chemical elements in common use. The quartic anharmonic potential is used for the bond stretching, sixth power expansion... 

The primary motive for attempting calculations of this kind is simply our desire to determine the potential function V(r) more accurately and over a wider range of co-ordinate space. Even if our immediate ambition is only to determine the equilibrium configuration and the harmonic force field, our ability to withdraw this information from spectroscopic data is limited by the need to make corrections arising from the cubic and quartic anharmonic force field. 

The two terms in the formula (24) for x have a simple physical interpretation the first is a correction to the energy levels owing to quartic anharmonicity in the first order of perturbation, and the second is a correction due to cubic anharmonicity in the second order of perturbation, It is well known that these... 

It is instructive to consider two of the formulae for the spectroscopic constants in more detail, and for this we choose — and xrs for an asymmetric top, these being respectively the coefficients of (vT + )J, the vibrational dependence of the rotational constant, and (vr + i)(fs + i), the vibrational anharmonic constant quadratic in the vibrational quantum numbers. As for diatomic molecules these two types of spectroscopic constant provide the most important source of information on cubic and quartic anharmonicity, respectively. The formulae obtained from the perturbation treatment for these two coefficients in the effective hamiltonian are as follows ... 

Equation (5), VER involves a higher-order anharmonic coupling matrix element, which gives rise to decay via simultaneous emission of several phonons nftjph (multiphonon emission). In the ACN case, three phonons must be emitted simultaneously via quartic anharmonic coupling (or four phonons via fifth-order coupling, etc.). 

The Morse potential provides a useful model for the anharmonic stretching vibrations of a polyatomic molecule. It is superior to a harmonic oscillator perturbed by cubic and quartic anharmonicity terms in terms of both convergence (a V(r) that includes an r3 term cannot be bound, thus cannot have any rigorously bound vibrational levels) and the need for a smaller number of adjustable parameters to describe both the potential energy curve (a and De for Morse frr, frrri and frrrr for the cubic plus quartic perturbed harmonic oscillator) and the energy levels. 

Figure C3.5.2. VER transitions involved in the decay of vibration Q by cubic and quartic anharmonic coupling (from [34]). Transitions involving discrete vibrations are represented by arrows. Transitions involving phonons (continuous energy states) are represented by wiggly arrows. In (a), the transition denoted (i) is the ladder down-conversion process, where Q is annihilated and a lower-energy vibration ra and a phonon p are created.

D. G. Truhlar, Oscillators with quartic anharmonicity Approximate energy levels,/. Molec. Spec. 38 4151 (1971). 

In Section A, we will describe several methods that only take into account poles and are therefore only appropriate at low orders. The low-order terms of dimensional expansions are relatively easy to derive, so these methods are useful for a quick qualitative analysis. This level of approximation corresponds to a simple physical model Eq is the energy of the infinite-P Lewis configuration, E corresponds to the harmonic Langmuir oscillations, and E2 represents cubic and quartic anharmonicities [13]. 

The most complicated anharmonic terms are found in the transverse [111] configuration, where both "forms" mentioned above appear at the same time forces are not parallel to the displacement, and th ir variation is not linear. Two calculations were performed with u = +0.004 (-2,1,1) a and the calculated forces were projected on the 3 perpendicular directions (-2,1,1), (111) and (0,-l,l). Whereas all projections on (0,-1,1) are zero, the "longitudinal" (111) component of the force at the first-neighbor site (<=+l) is, with the above ul, as much as 46 % of the "transverse" (-2,1,1) one at the sites <=-l, 2, the non-parallel components are still respectively 13 % and 20 % of the parallel ones. For obtaining the harmonic force constants, only the (-2,1,1) projection of the force was retained - and (in contrast to the transverse [100] case) a noticeable anharmonic behavior was still found k determined with displacements +u and -u were averaged, in order o eliminate the cubic contributions - because they still differed considerably for k j the cubic term lu in g(5.4.1) represents 7 Z of the ha onic one. It was verified" by a calculation with larger u that the influence of quartic anharmonicity is negligible. 

In order to better understand the structure of the scalet equation formalism, we consider the problem of the quartic anharmonic oscillator potential... 

For the quartic anharmonic oscillator case, the (approximate) form of a depends on whether c b, m) is positive or negative. In the positive case, the term (in the exponential of the second and third modes given above) dominates. If c(6, m) is negative, then the a terms (with a real factor) dominates. As such we have ... 

We examine the behavior of the above asymptotic expansions, at the real turning points (for simplicity), for the ground state of both the quar-tic anharmonic oscillator (e = m = = l), and the double well quartic anharmonic oscillator e = g = 1 and m —5). 

The (convergent) asymptotic series results agree with the direct integration of the scalet equation, where the missing moments and energies can be obtained by MRF analysis. Specifically, for the quartic anharmonic oscillator, Egr = 1.392351642, the nonzero missing moments are /x(0) =. 6426706223, and p(2) =. 3573293777. For the quartic double well potential, Egr = -3.410142761, p(0) = 0.3223013271, and p(2) = 0.6776986729. 

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