Harmonic approximation anharmonic frequencies

Vibrational Spectra Many of the papers quoted below deal with the determination of vibrational spectra. The method of choice is B3-LYP density functional theory. In most cases, MP2 vibrational spectra are less accurate. In order to allow for a comparison between computed frequencies within the harmonic approximation and anharmonic experimental fundamentals, calculated frequencies should be scaled by an empirical factor. This procedure accounts for systematic errors and improves the results considerably. The easiest procedure is to scale all frequencies by the same factor, e.g., 0.963 for B3-LYP/6-31G computed frequencies [95JPC3093]. A more sophisticated but still pragmatic approach is the SQM method [83JA7073], in which the underlying force constants (in internal coordinates) are scaled by different scaling factors. 

Regardless of the force field chosen, the calculation of vibrational frequencies by the method outlined above is based on the harmonic approximation. Tabulated values of force constants can be used to calculate vibrational frequencies, for example, of molecules whose vibrational spectra have not been observed. However, as anharmonicities have been neglected in the above analysis, the resulting frequency values are often no better than 5% with respect to those observed. 

In order to obtain better agreement between theory and experiment, computed frequencies are usually scaled. Scale factors can be obtained through multiparameter fitting towards experimental frequencies. In addition to limitations on the level of calculation, the discrepancy between computed and experimental frequencies is also due to the fact that experimental frequencies include anharmonicity effects, while theoretical frequencies are computed within the harmonic approximation. These anharmonicity effects are implicitly considered through the scaling procedure. 

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 ... 

Another hmitation is inherent to the harmonic approximation on which standard quantum mechanical force-field calculations are invariably based. Due to a fortui-tious (but surpisingly systematic) cancellation of errors, the harmonic frequencies calculated by modem density functional methods often match very well with the experimental ones, in spite of the fact that the latter involve necessarily more or less anharmonic potentials. Thus one is tempted to forget that the harmonic approx-imaton can become perilous when strong anharmonicity prevails along one or another molecular deformation coordinate. 

The authors made a more exacting comparison for vibrational frequencies, where experimental data were available for the matrix isolated radical anion. Focusing on one fundamental and one combination band, the CCSD(T)/TZP+ predictions of 1527 and 1955 cm compared reasonably well to the experimental values of 1518 and 2042. Again, the flat nature of the PES in the vicinity of the linear form makes things difficult for theory, since this introduces potentially large anharmonicity that is not accounted for in the usual harmonic approximation employed to compute vibrational frequencies (see Section 9.3.2). 

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. 

This is an unusual structure for an acceptor-hydrogen complex in other semiconductors hydrogen assumes the bond-centre position when binding to acceptors [7], As a direct consequence the vibrational frequency of the complex is not representative of a Ga-H bond, but rather of an N-H bond. The calculated vibrational frequency (in the harmonic approximation) is 3360 cm 1. Anharmonic effects may lower this frequency by as much as 170 cm 1 [8]. 

But we will later see a violation of a simple energy gap behavior such as the above. It is critical to appreciate that this frequency effect is modulated by the factors that depend on the intramolecular potential it suffices to recall that in an harmonic approximation only the transition (001) -> (000) would be allowed. To place this issue in perspective, we write the rate constant [Equation (5)] for transitions out of the anharmonic... 

More sensitive to the level of theory is the vibrational component of the interaction energy. In the first place, the harmonic frequencies typically require rather high levels of theory for accurate evaluation. It has become part of conventional wisdom, for example, that these frequencies are routinely overestimated by 10% or so at the Hartree-Fock level, even with excellent basis sets. A second consideration arises from the weak nature of the H-bond-ing interaction itself. Whereas the harmonic approximation may be quite reasonable for the individual monomers, the high-amplitude intermolecular modes are subject to significant anharmonic effects. On the other hand, some of the errors made in the computation of vibrational frequencies in the separate monomers are likely to be canceled by errors of like magnitude in the complex. Errors of up to 1 kcal/mol might be expected in the combination of zero-point vibrational and thermal population energies under normal circumstances. The most effective means to reduce this error would be a more detailed analysis of the vibration-rotational motion of the complex that includes anharmonicity. 

The symbol v is normally used to express the frequency of tmy given vibrational mode, in units of cm . Since most calculations are restricted to the harmonic approximation, the use of this symbol in the computational literature likewise refers generally to harmonic frequencies. In those cases where anharmonicity is added to the computations, the notation can become confusing in that v usually refers to the anharmonic value, with co reserved for the harmonic approximation to this frequency. The reader should therefore exercise some caution in scanning the original hterature. In this text, we will adopt the convention that v will represent the harmonic frequency in those cases where anharmonicity is included, the distinctions and notation will be clearly delineated. 

The results are presented in Table 3.68 which is divided into frequencies derived using the harmonic approximation and anharmonic data taking account of the fourth-order polynomial description of the PES. It appears that the SCF harmonic frequencies are barely altered at all by BSSE. In fact, the only harmonic frequency to be affected is the v, stretch, v(FH), which is increased by 12 cm at the MP2 level when the BSSE is included. When the treatment is expanded to include anharmonicity, there is again virtually no effect on either frequency from BSSE. However, MP2 calculations do show significant changes account of superposition error raises the v frequency by 32 cm and lowers v(F-N) by a... 

The above discussion of Equations 2 and 3 has been predicted on the assumption of harmonic frequencies for all 3N modes. More realistically, these are at best described as slightly anharmonic frequencies which we approximate with an effective harmonic force field. For lattice frequencies in particular, anharmonicity is expected to be important here it arises both from the anharmonic curvature in the potential and from the expansion of the lattice on warming. Consequently, the force constants used to describe the lattice modes become temperature dependent. The approach amounts to a simple extension of the ideas at the basis of the pseudoharmonic theory of solid lattices (2, 3) to the condensed phases which interest us. One phenomenological result of such anharmonicity is that Equation 3 now takes the form ... 

Such unexpected variations of the anharmonicity can be explained by considerable variations of the main electronic state under influence of high frequency excitation. Under these conditions the electronic states in the nanotubes vary in such a way that deviation from the harmonic approximation is decreasing for G mode but increasing for the D mode and the sum D+G tone. 

If the anharmonicity is small, it can be accounted for in the framework of the body of mathematics used in harmonic approximation. Such analysis was made elsewhere [88], and it was shown that the formulas for the rate constant remain unchanged, but the parameters involved—equilibrium coordinates and phonon frequencies—turn out to be temperature-dependent effective values. The applicability criteria for harmonic approximation were also obtained. 

Within the harmonic approximation of the transition state theory (TST), the prefactors are determined as the ratios between the product of all the eigenmode frequencies at the minimum and that of all the real eigenmode frequencies at the saddle points. This approximation is adequate for systems with slowly varying PES s, to the extent that no anharmonicities are involved. Going beyond the harmonic approximation of TST, in Ref [5], the rate prefactor probabilities are determined as... 

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