Harmonic calculation

The ° mn coefficients are the mean values of the generalized spherical harmonics calculated over the distribution of orientation and are called order parameters. These are the quantities that are measurable experimentally and their determination allows the evaluation of the degree of molecular orientation. Since the different characterization techniques are sensitive to specific energy transitions and/or involve different physical processes, each technique allows the determination of certain D mn parameters as described in the following sections. These techniques often provide information about the orientation of a certain physical quantity (a vector or a tensor) linked to the molecules and not directly to that of the structural unit itself. To convert the distribution of orientation of the measured physical quantity into that of the structural unit, the Legendre addition theorem should be used [1,2]. An example of its application is given for IR spectroscopy in Section 4. 

Fig. 10.5 Distribution of exponents, Equation 10.21, for exact harmonic calculated equilibrium and TST kinetic isotope effects (Hirschi, J. and Singleton, D. A., J. Am. Chem. Soc. 127, 3294 (2005))...

The expansion coefficients V5/14 and V10/7 are determined entirely by the Oh symmetry and the normalized forms of spherical harmonics. Calculations of energy involve only one parameter, Dq, and this may be summarized by the relationship... 

Figure 8. A variety of approximations to the classical Lennard-Jones phase diagram. The data points show the results of ESPS studies (discussed in Section IV.D), denoted here by LS. The dashed and solid lines are the results of harmonic calculations (for the two system sizes). The dash-dotted line is a phenomenological parameterization of the anharmonic effects. The scale at the top of the figure shows the pressures at selected points on the (LS N = 123, NPT) coexistence curve. Tieline structure is unresolvable on the scale of the figure. (Taken from Fig. 11 of Ref. 57.)...

It is nice to have a distinctive notation for the curvilinear co-ordinates, which emphasizes their difference from and yet their one-to-one correlation with the Rt co-ordinates. Most authors reporting anharmonic calculations do not in fact make any distinction they denote the curvilinear co-ordinates by the same symbols customarily used to denote the corresponding rectilinear coordinates in harmonic calculations. For many purposes this is satisfactory, particularly since the harmonic force constants are not altered by the change from rectilinear to curvilinear co-ordinates. However, in a general discussion it is important to distinguish the two sets, and so for the remainder of this section we shall follow Hoy et al.12 and write the curvilinear co-ordinates with the symbol Hi. 

The first derivative L tensor elements, are determined as the elements of the L matrix from the preliminary harmonic calculation. The second and third derivative L tensor elements have been determined in one of two ways in the literature of anharmonic calculations. The first method involves setting up an... 

Symmetry, and the Number of Independent Force Constants.—As in harmonic calculations, the rather general discussion of the preceding section can be simplified in particular cases by making use of symmetry, as discussed by Hoy et a/.12 Thus we may choose the curvilinear co-ordinates Jfin linear combinations that span the irreducible representations of the point group we denote such symmetrized curvilinear co-ordinates by the symbol S, and we define them by means of a U matrix exactly analogous to that used for rectilinear coordinates ... 

If the force constants are to be regarded as derivatives of the potential energy V with respect to the coordinates R, it is clearly important to include die coefficients 1/n and to regard the summations as unrestricted as in Eq. (1). Although this is commonly done in harmonic calculations, anharmonic force constants are often defined in a different form in which the numerical factors are omitted and the summations are restricted. 

The free energy of the system also includes entropic contributions arising from the internal fluctuations, which are expected to be different for the separate species and for the liganded complex. These can be estimated from normal-mode analyses by standard techniques,136,164 or by quasi-harmonic calculations that introduce approximate corrections for anharmonic effects 140,141 such approaches have been described in Chapt. IV.F. From the vibrational frequencies, the harmonic contribution to the thermodynamic properties can be calculated by using the multimode harmonic oscillator partition function and its derivatives. The expressions for the Helmholtz free energy, A, the energy, E, the heat capacity at constant volume, C , and the entropy are (without the zero-point correction)164... 

Figure 19. Dispersed fluorescence of ultracold anthracene in a supersonic beam for several values of the excess vibrational energy, which is indicated (in cm 1) on each plot. The right column is experimental,60 and the left column is our 17-mode harmonic calculation (Table II).61 The agreement is excellent at low excitation energies (<766 cm 1), but fails at higher energies owing to the onset of IVR processes.

This conclusion does not apply to the C-H stretching region, where Fermi resonance is apparent [19] and harmonic calculations are of insufficient accuracy to account for the observed spectra, either qualitatively or quantitatively. Our methodology also ignores solvent effects. This is a good approximation when solute-solvent interactions are minimal, as in the case of CC14 solutions of propylene oxide, but not when large, specific interactions exist. 

HARMONIC CALCULATIONS Process Rpritirai Relative Intensity... 

Fignre 6. Comparison of eigenvalue spectra for Met-enkephlin at T = 2.7 kcal/mol for (a) quasi-harmonic estimate of fluctuation kernel and (b) MWG estimate of fluctuation kernel. The MWG approximation produces a much more compact distribution than the linear quasi-harmonic calculation. (Adapted from Church and Shalloway [38].)... 

This simplified equilibration process holds strictly for a cubic crystal in a general non-cubic case, the volume, static pressure and bulk modulus have to be replaced by the strain components (related to unit-cell parameters), stress components and elastic constants, respectively. The computer program PARAPOCS[17] performs the lattice-dynamical, thermodynamical and quasi-harmonic calculations in the general tensorial formalism, and has been used to obtain all results reported below for calcite and aragonite. 

A better insight into the quality of the quasi-harmonic calculation and corresponding equation of state is obtained by considering the other ther-... 

Catti, M., Pavese, A., and Price, G.D. (1993) Thermodynamic properties of CaCOs calcite and aragonite a quasi-harmonic calculation, Phys. Chem. Minerals 19,472-479. 

Even more crucial in this discussion, the calculation of IR spectra with MD is related only to the time-dependent dipole moment of the molecular system, requiring neither any harmonic expansion of the transition dipole moment nor the knowledge of normal modes, in contrast to harmonic calculations. Therefore, if the dipole moments and their fluctuations are accurately calculated along the trajectory, the resulting IR spectmm should be reliable. The vibrations therefore do not rely directly on the curvature of the potential energy surface at the minima... 

The double band around 3,000-3,100 cm is produced by the combined Co,-H and C-H stretch modes of the methyl groups of the peptide. These bands are up-shifted by 40 cm from experiment, and the band spacing ( 50 cm ) is slightly bigger than the experimental one ( 30 cm ). It is nonetheless remarkable that our calculation predicts an intensity of the band so close to experiment. The harmonic calculation [105] does not give such intensity to this mode. 

Fig. 6 Infrared spectrum of gas phase AlaalF" for the 2,500-4,000 cm . Taken from [39]. Left conformational families of AlaslF" taken into account for the dynamics. Right (lop) harmonic calculations from [104] and (bottom) comparison of IR-MPD experiment (exp) and dynamical spectrum from DFT-MD simulations (calc) [39]...

Figure 4. Comparison between the experimental (solid bold line) and theoretical v spectra of 1-methyluracil crystal (anharmonic calculation—Dirac 5 functions and thin solid line harmonic calculation—dashed line) for the 1-methyluracil crystal (a) and the deuterated crystal (b). Figure reprinted with permission from Ref. 58. Copyright American Institute of Physics Publishing LLC.

Ghien, G. H. and R. W. G. Bucknall. 2009. Harmonic calculation of proximity effect on impedance characteristics in subsea power transmission cables. IEEE Trans. Power Deliv. 24(2) 2150-2158. 

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