Potential anharmonicity

Contents Lattice Dynamics. - Symmetry. - Inter-molecular Potentials. - Anharmonic Interactions. - Two-Phonon Spectra of Molecular Crystals. -Infrared and Raman Intensities in Molecular Crystals. 

In the previous section, we described the situation that a single WP created by a pump pulse splits into two counter-propagating WPs due to potential anharmonic-ity. Similar interfering WPs can be generated by time-delayed double pump pulses. 

Thus, the ratio is dependent on experimental parameters such as the optical path length, sample number density, and the phase matching conditions for the intermediate third-order processes, as well as the ratio of the third-and fifth-order response functions. The ratio of the response functions is directly related to the magnitude of the nonlinearity in the system, which is reflected by the magnitude of the potential anharmonicity, g(3), and the nonlinearity in the polarizability, a,2>. For example, let us consider only the NP contribution to the direct fifth-order response [Equation (21)]. For simplicity we will consider a system represented by a single mode, in other words the response is isotropic. If we express the third-order response functions in term of the coordinate [Equation (17)] and ignore all higher order terms,... 

Note, that since q is in mass-weighted coordinates, the units of the force constants fij are simply sec 2 rather than the usual kg-sec potential which only includes quadratic terms is called a harmonic potential. Anharmonicity arises from the higher-order terms. 

A strong contribution of Coulomb attraction (in salts) makes the potential anharmonic and flatter on the high-distance side of the equilibrium position this potential means high expansion coefficients. 

Molecular vibration absorptions are typically observed at infrared wavelengths, which correspond to the resonance frequencies for fundamental molecular vibrations. Because of the molecular potential anharmonicity, however, overtone and combination absorption bands also appear at visible and near-infrared wavelengths. The predominant factor for attenuation in POFs has been the stretching overtone absorptions of C-H bonds. 

Hollenstein H, Marquardt R, Quack M and Suhm M A 1994 Dipole moment function and equilibrium structure of methane In an analytical, anharmonic nine-dimenslonal potential surface related to experimental rotational constants and transition moments by quantum Monte Carlo calculations J. Chem. Phys. 101 3588-602... 

After transforming to Cartesian coordinates, the position and velocities must be corrected for anharmonicities in the potential surface so that the desired energy is obtained. This procedure can be used, for example, to include the effects of zero-point energy into a classical calculation. 

R. M. Levy, O. de la Luz Rojas, and R. A. Friesner. Quasi-harmonic method for calculating vibrational spectra from classical simulations on multidimensional anharmonic potential surfaces. J. Phys. Chem., 88 4233-4238, 1984. 

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

Harmonic analysis (normal modes) at given temperature and curvature gives complete time behavior of the system in the harmonic limit [1, 2, 3]. Although the harmonic model may be incomplete because of the contribution of anharmonic terms to the potential energy, it is nevertheless of considerable importance because it serves as a first approximation for which the theory is highly developed. This model is also useful in SISM which uses harmonic analysis. 

The reason that does not change with isotopic substitution is that it refers to the bond length at the minimum of the potential energy curve (see Figure 1.13), and this curve, whether it refers to the harmonic oscillator approximation (Section 1.3.6) or an anharmonic oscillator (to be discussed in Section 6.1.3.2), does not change with isotopic substitution. Flowever, the vibrational energy levels within the potential energy curve, and therefore tq, are affected by isotopic substitution this is illustrated by the mass-dependence of the vibration frequency demonstrated by Equation (1.68). 

Figure 6.4 Potential energy curve and energy levels for a diatomic molecule behaving as an anharmonic oscillator compared with those for a harmonic oscillator (dashed curve)...

Owing to the effects of mechanical anharmonicity - to which we shall refer in future simply as anharmonicity since we encounter electrical anharmonicity much less frequently -the vibrational wave functions are also modified compared wifh fhose of a harmonic oscillator. Figure 6.6 shows some wave functions and probabilify densify functions (iA A ) for an anharmonic oscillator. The asymmefry in and (iA A ) 5 compared wifh fhe harmonic oscillator wave functions in Figure f.i3, increases fheir magnitude on the shallow side of the potential curve compared with the steep side. 

The only types of anharmonic potential function we have encountered so far are the two illustrated in Figure 6.38, both of which show only a single minimum. There are, however, some vibrations whose potential functions do not resemble either of those but show more than one minimum and whose term values are neither harmonic, nor are they given by Equation (6.88) or Equation (6.89). Such vibrations can be separated into various types, which will now be discussed individually. 

In the case where r > r" there is, when anharmonicity is taken into account, a relatively steep part of the excited state potential curve above v" = 0, giving a relatively broad... 

The corresponding level broadening equals half. In fact is the diagonal kinetic coefficient characterizing the rate of phonon-assisted escape from the ground state [Ambegaokar 1987]. In harmonic approximation for the well the only nonzero matrix element is that with /= 1,K0 Q /> = <5o, where is the zero-point spread of the harmonic oscillator. For an anharmonic potential, other matrix elements contribute to (2.52). 

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