Principal anharmonicity

The mean and mean square values of the LA coordinate s represent the principal anharmonic and harmonic vibrational contributions, respectively [3]. 

Hence, to first order in the and retaining only the principal anharmonic and harmonic contributions... 

The majority of VTST calculations performed to date have been for atom-diatom collisions.For that kind of collision, reasonably accurate calculations of the vibrational energy levels are possible without excessive labor. For example, for a collinear minimum-energy path the vibrations orthogonal to the path consist of one stretch and a twofold degenerate bend. Use of a curvilinear bend coordinated 57,65 reduces the bend-stretch coupling, and principal anharmonicity can be included accurately in the bend by the harmonic-quartic approximation described above or by the WKB approximation. The stretch can also be treated accurately by the WKB approximation. 5 xt is also possible to estimate the effect of bend-rotational coupling,57 and in particularly... 

A further advantage of representing the potential energy in the internal coordinates is that if the principal anharmonic internal coordinate force constants and cannot be calculated directly from... 

Morse Approximation I and Other Corrections for Principal Anharmonicity... 

Spectroscopists call the force constants that have all indices the same the principal force constants, while the anharmonicity associated with the principal force constants is called principal anharmonicity. The Morse and quadratic-quartic approximations treat only principal anharmonicity. However, as mentioned in Eq. [95], neglecting the cross terms between modes is a much more serious approximation in rectilinear coordinates.Explicitly including cross terms in rectilinear coordinates is expensive and cumbersome... 

Hayward et al. 1994] Hayward, S., Kitao, A., Go, N. Harmonic and anharmonic aspects in the dynamics of BPTI A normal mode analysis and principal component analysis. Prot. Sci. 3 (1994) 936-943 [Head-Gordon and Brooks 1991] Head-Gordon, T., Brooks, C.L. Virtual rigid body dynamics. Biopol. 31 (1991) 77-100... 

Steven Hayward, Akio Kitao, and Nobuhiro Go. Harmonic and anharmonic aspects in the dynamics of BPTI A normal mode analysis and principal component analysis. Physica Scripta, 3 936-943, 1994. 

Even when the harmonic approximation is not quantitatively justified it provides a convenient starting point for exact treatments. Thus, even if the potential energy surface is anharmonic in the bottleneck, it is often smooth enough for there to be a principal saddle point that can be found by minimizing IVU 2. 

The principal progression-forming mode, Vi, is almost harmonic up to vibrational quantum number 6, with the anharmonicity constant Xi, 0.2cm . The overtone band wavenumbers derived from the calculation differ from the experimental values by a few wavenumbers as the vibrational quantum number increases. Other modes show only one or two harmonics. 

The principal progression-forming mode (vi) in the resonance Raman spectrum is harmonic within experimental error with the anharmonicity constant Xu = 0.0cm . For the stibine complex, vj shows a long overtone progression in the resonance Raman spectrum. This mode is also almost harmonic, the anharmonicity constant Xj2 being <0.1cm. Other modes show only one or two harmonics. Again harmonicity is a reasonable assumption in this calculation. 

Of interest also are the results concerning deviations of the atomic fluctuations from simple isotropic and harmonic motion. As discussed in Chapt. XI, most X-ray refinements of proteins assume (out of necessity, because of the limited data set) that the motions are isotropic and harmonic. Simulations have shown that the fluctuations of protein atoms are highly anisotropic and for some atoms, strongly anharmonic. The anisotropy and anharmonicity of the atomic distribution functions in molecular dynamics simulations of proteins have been studied in considerable detail.193"197 To illustrate these aspects of the motions, we present some results for lysozyme196 and myoglobin.197 If Ux, Uy, and Uz are the fluctuations from the mean positions along the principal X, Y, and Z axes for the motion of a given atom and the mean-square fluctuations are... 

The relationship of physical properties to the chemical instability in these materials should provide the vital link which would, in principal, explain the explosivity of these and related materials in the condensed phase. However, this connection is difficult to make and has to be approached from all possible points of view. The crucial mechanism here for thermal behavior may be the transfer of energy via mode-mode coupling processes. A detailed study of the anharmonic dynamics of such crystals coupled with diffraction work under high pressure and variable temperature, therefore, forms the next logical step in this research. 

In the previous section, we utilized local PCA to represent the moving normal modes, which depicted the locally harmonic but globally anharmonic dynamics of proteins. The major difficulty we met within the local PCA was the fact that two principal modes determined in adjacent time-windows, e,variance-covariance matrix C, or the quasidegeneracy in C. From a statistical viewpoint, this difficulty can be attributed to statistical fluctuation in the estimation of the principal modes due to the small sampling size in the determination of local PCA. 

A valence force field includes only valence coordinates and principal force constants a general force field also includes interactions. Thus the harmonic valence terms are the principal ones in the first sum in eq. (35), the harmonic general field consists of the whole first sum, and the anharmonic valence terms comorise the second sum. 

A typical resonance Raman spectrum of iodine consists of a long progression of the vibrational mode showing anharmonicity and rotational structure (Kiefer and Bernstein, 1973 Rousseau and Williams, 1976). This structure is readily understood on the basis of the calculations in Section III,A,B. The very simple Eq. (33) gives a qualitative picture of such a progression. The corresponding REPs are readily derived from the absorption spectrum (Berjot et ai, 1971). Thus the principal features of resonance Raman scattering by I2 are well understood. 

The cubic force field of pyramidal XYg-type molecules contains 14 independent parameters. Usually, however, the number of spectroscopic constants dependent on the enharmonic force field, such as rotation-vibration constants, l-type doubling constants, or anharmonicity constants, is smaller than the number of parameters to be determined. Their number thus has to be reduced by introducing model potentials and imposing certain constraints. Some of the possible routes were presented by Morino et al. [50]. Cubic force constants for NF3 pertinent to model potentials (mostly Morse potentials) have been calculated by several groups of workers [11, 12, 19, 51, 52], Some of the principal quartic constants have been estimated as well [12, 51]. 

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