Self-consistent phonon method

Depending on the character of the molecular motions, one can distinguish several physical situations. In most cases, the molecules are trapped in relatively deep potential wells. Then, they perform small translational and orientational oscillations around well-defined equilibrium positions and orientations. Such motions are reasonably well described by the harmonic approximation. The collective vibrational excitations of the crystal, which are considered as a set of harmonic oscillators, are called phonons. Those phonons that represent pure angular oscillations, or libra-tions, are called librons. For some properties it turns out to be necessary to look at the effects of anharmonicities. Anharmonic corrections to the harmonic model can be made by perturbation theory or by the self-consistent phonon method. These methods, which are summarized in Section III under the name quasi-harmonic theories, can be considered to be the standard tools in lattice dynamics calculations, in addition to the harmonic model. They are only applicable in the case of fairly small amplitude motions. Only the simple harmonic approximation is widely used the calculation of anharmonic corrections is often hard in practice. For detailed descriptions of these methods, we refer the reader to the books and reviews by Maradudin et al. (1968, 1971, 1974), Cochran and Cowley (1967), Barron and Klein (1974), Birman (1974), Wallace (1972), and Cali-fano et al. (1981). 

Just as the self-consistent phonon method, the mean field approximation (Kirkwood, 1940 James and Keenan, 1959) is based on the thermodynamic variation principle for the Helmholtz free energy ... 

This potential was subsequently used in self-consistent phonon lattice dynamics calculations [115] for a and y nitrogen crystals. And although the potential—and its fit— were crude by present day standards, lattice constants, cohesion energy and frequencies of translational phonon modes agreed well with experimental values. The frequencies of the librational modes were less well reproduced, but this turned out to be a shortcoming of the self-consistent phonon method. When, later [ 116,117], a method was developed to deal properly with the large amplitude librational motions, also the librational frequencies agreed well with experiment. 

The Harmonic Method and the Self-Consistent Phonon Method.406... 

Just as the perturbation theory described in the previous section, the self-consistent phonon (SCP) method applies only in the case of small oscillations around some equilibrium configuration. The SCP method was originally formulated (Werthamer, 1976) for atomic, rare gas, crystals. It can be directly applied to the translational vibrations in molecular crystals and, with some modification, to the librations. The essential idea is to look for an effective harmonic Hamiltonian H0, which approximates the exact crystal Hamiltonian as closely as possible, in the sense that it minimizes the free energy Avar. This minimization rests on the thermodynamic variation principle ... 

Such decoupling in the liquid may be strictly justified only in the long-wave approximation.In this sense, such a procedure is justified for the macroscopic description. However, one should remember that this is the correct method in a number of cases also for short wavelengths. For example, this is the case for phonons in solids. In other cases, such as the electron gas in metals (plasmons), acoustic phonons in quantum liquids and so on, this decoupling may be considered as the self-consistent field method or the random phase approximation (the analog of the superposition approximation in the classical theory of liquids). 

This expression still involves processes of electron scattering with respect to each other in the self-consistent approximation (see Figure 6). If these processes are neglected, retaining only the interaction of the electrons with the phonon particles (Figure 7), i.e., if the random phase approximation is used in the self-consistent field method, we obtain for Wif... 

A proof of Eq. (14) is given in several places [27, 66-68]. The advantage of this equation is that it can be used to systematically improve the approximate Hamiltonian and the free energy F by optimizing a set of variable parameters or functions contained in Hq. Specific forms chosen for lead to the Self-Consistent Phonon (SCP) method, the Mean-Field (MF) method and the Time-Dependent Hartree (TDH) or Random-Phase Approximation (RPA). 

Another technique to obtain the effects of the anharmonic terms on the excitation frequencies and the properties of molecular crystals is the Self-Consistent Phonon (SCP) method [71]. This method is based on the thermodynamic variation principle, Eq. (14), for the exact Hamiltonian given in Eq. (10), with the internal coordinates not explicitly considered. As the approximate Hamiltonian one takes the harmonic Hamiltonian of Eq. (18). The force constants in Eq. (18) are not calculated at the equilibrium positions and orientations of the molecules as in Eq. (19), however. Instead, they are considered as variational parameters, to be optimized by minimization of the Helmholtz free energy according to Eq. (14). The optimized force constants are found to be the thermodynamic (and thus temperature dependent) averages of the second derivatives of the potential over the (harmonic) lattice vibrations ... 

In Table 7.4, the differences between the experimental values for A/ and Ga for the first lines compared to B and In can be attributed to the above-mentioned phonon resonances. There are also non-negligible differences between the measured and calculated spacings as well as between the calculated spacings for the deepest levels. As already mentioned, it is possible to use a self-consistent method to obtain experimental acceptor energy levels in... 

This is the fundamental result of the SCF-phonon method. It replaces the classical force constants which are the second derivatives of the potential at equilibrium by the quantum mechanical average of the second derivative of the given potential function, or, alternatively by the second derivative of the average potential with respect to the effective equilibrium position. In the classical limit of narrowly peaked wavefunctions the result reduces to the classical result. Since the wavefunctions depend on the force constant through (3.4), (3.7) must be solved self-consistently. In practice, X is found by minimizing the ground-state energy numerically. 

The application of this method requires knowledge of the explicit form of at least one of the two funetions (F(/)F(0)> or y t), in order to find a solution of the equation. Variants of the approaeh, developed up to the present time are based on different ways of modeling of the dissipation term y(t), conneeted with the secondary zone of atoms [ 18-20]. Adelman and Garrison use Debye s model for phonons of the solid and obtain an equation for the dissipation term whieh, ean be solved numerically. Doll and Dion propose y t) as a linear combination of conveniently chosen functions, where the coefficients are determined by numerieal self-consistency. Another possibility is to model the microscopic interactions in the lattice of the solid in order to derive a dissipation term. Tully presents the friction as a white noise or positionally autocorrelated function of a Brownian oscillator, including both oscillation and dissipation terms. 

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