Anharmonic contribution 849

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 ... 

In the Heitler-London approximation, with allowance made only for biquadratic anharmonic coupling between collectivized high-frequency and low-frequency modes of a lattice of adsorbed molecules (admolecular lattice), the total Hamiltonian (4.3.1) can be written as a sum of harmonic and anharmonic contributions ... 

The thermal expansivity of a solid is in general low at low temperatures and the anharmonic contribution to the heat capacity is therefore small in this temperature region and Cv m Cpm. At high temperatures the difference between the heat capacity at constant pressure and at constant volume must be taken into consideration. 

The ZPVE may be partitioned into harmonic and anharmonic contributions... 

The harmonic frequencies and the anharmonic constants may be obtained from experimental vibrational spectra, although their determination becomes difficult as the size of the system increases. In Table 1.10, we have listed experimental harmonic and anharmonic contributions to the AEs. These contributions may also be obtained from electronic-structure calculations of quadratic force fields (for harmonic frequencies) and cubic and quartic force fields (for anharmonic constants). For some of the larger molecules in Table 1.11, we have used ZPVEs calculated at the CCSD(T)/cc-pVTZ level or higher, see Ref. 12. In some cases, both experimental and theoretical ZPVEs are available and agree to within 0.3 kJ/mol [12, 57],... 

Although the harmonic ZPVE must always be taken into account in the calculation of AEs, the anharmonic contribution is much smaller (but oppositely directed) and may sometimes be neglected. However, for molecules such as H2O, NH3, and CH4, the anharmonic corrections to the AEs amount to 0.9, 1.5, and 2.3 kJ/mol and thus cannot be neglected in high-precision calculations of thermochemical data. Comparing the harmonic and anharmonic contributions, it is clear that a treatment that goes beyond second order in perturbation theory is not necessary as it would give contributions that are small compared with the errors in the electronic-structure calculations. 

To illustrate the anharmonic contribution to RPFR from a particular high frequency mode treated in the ZPE approximation, for example a CH/CD stretch, we recall the oscillator energy neglecting Go is expressed... 

The procedure of Ottonello et al. (1996a) is operationally different from that indicated by Kieffer (1980), which suggested that the calculated Cy be fitted to the calorimetric Cp at 100 K after subtraction of the anharmonicity contribution... 

The amplitude of the periodic orbits is therefore determined by the linear stability with respect to perturbations transverse to the orbit. In this sense, the leading term in expression (2.13), obtained by setting C = 0, treats the dynamics transverse to the orbit at the level of the harmonic approximation. The nonlinear stability properties appear thus as anharmonic corrections to the dynamics transverse to the orbit. These anharmonicities contribute to the trace formula by corrections given in terms of series in powers of the Planck constant involving the coefficients C , which can be obtained as Feynman diagrams [14, 31]. 

The nonlinear polarizabilities in the classical spring problem arise from anharmonic contributions to the spring constant. Resolution of eq. 3 into harmonics of frequency nu using trigonometric identities provides an understanding of how specific orders of anharmonicity in V(x) lead to anharmonic polarizations at frequencies different from that of the applied field S(t). In the classical problem, the coefficients an are determined by the anharmonicity constants in V(x) [10]. 

As a rule the quantum-mechanical force-fields and the corresponding normal frequencies are calculated in a harmonic approximation, while the experimentally accessible frequencies are influenced by anharmonic contributions. The Puley s scaling factors are also found to incorporate the relevant empirical corrections for the vibrational anharmonicity. 

Thus electrical anharmonicity is the non-linear part of the variation of the dipole moment with normal coordinates. There is no reason to expect it to be negligibly small. It can give intensity to overtones and combination tones. In the general case both mechanical and electrical anharmonicities contribute to the intensity. As will be mentioned later they cannot be disregarded when spectra of H-bond complexes are dealt with. 

The harmonic oscillator provides equally spaced eigenenergies in molecular vibrations, additional anharmonic contributions (V = bx3 + cx4 +...), computed numerically, spread the higher-energy vibrational eigenvalues further apart if b > 0, or closer together if b < 0. 

TABLE 6.11 Harmonic and Anharmonic Contributions to hcc of Vinyl Radical Calculated at the PBE0/EPR-I1//PBE/6-311 + G(d,p) Level... 

Finally, it is to be noted that configurational mixing-related anharmonicities of the PE surfaces have the effect of mixing the nuclear and electronic coordinates, even in the two state model (see equations 35-40). These anharmonicities must result in changes in the frequencies of some vibrational modes that are correlated with the reaction coordinate. The size of the anharmonic contribution increases with electron delocalization. 

In the present study, we examine the magnitude of the anharmonic contribution from the host water molecules and also the anharmonic free energy arising from the rotational degrees of freedom of guest molecule by Monte Carlo (MC) simulation with the Gaussian statistics. 

The anharmonic free energy is evaluated for empty hydrate and cubic ice (ice Ic). The calculated free energy due to the anharmonic potential energy surface is given in Table 1. The anharmonic contribution to the free energy of empty... 

The anharmonic contribution to the free energy is evaluated most simply by a thermodynamic integration method with a reference system of a collection of harmonic oscillators. This free energy difference between the real and the reference system, 4 — is given by... 

Table 6. Free energy due to the anharmonic contributions to the ice, empty and filled hydrates. Free energy is in kJ mol The free energy differences between the real and reference systems are denoted by A (kJ per mole of guest). The reference systems for the hydrate encaging spherical guest molecules are corresponding harmonic oscillators. The reference systems for the occupied hydrates by nonspherical guests are harmonic oscillators with spherical guests.

Table 7. Free energy of cage occupation by nonspherical propane and ethane evaluated by the direct calculation of anharmonic contributions and by the mean occupation according to equation (26). The free energy of the corresponding free rotor is omitted. Free energy is in kJ mol and pressure is in 0.1 MPa.

More accurate characterization of the energy levels—beyond the Morse potential— might include additional anharmonic contribution of the form (v- -1) (Ogyg. Vibrational tran.sitions of diatomic molecules occur in the infrared, in the broad range of 50—15, 000 cm A molecule will absorb or emit radiation only if it has a nonzero dipole moment. Thus heteronuclear diatomic molecules such as HCl are infrared (IR) active, while homonuclear diatomics such as H2 and CI2 are not. 

The largest uncertainty in C and S probably results from the neglect of anharmonlclty. Barho (15) and Albright (16) have given estimates for the anharmonic contributions. These vary from 0.1 to 0.3 cal K mol" in both C and S over the range from 300 to 1000 K, respectively. Somewhat larger contributions are possible ased on the anharmonic functions calculated by McBride (17) for CF (g). We estimate the overall uncertainty in the entropy of this JANAF table to be 0.1 cal... 

The most significant uncertainty in the calculated thermodynamic functtions is due to neglect of anharmonic contributions. 

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