Equilibrium positions anharmonic crystals

The sensitivity of the phonon frequencies to temperature shows quite clearly the importance of their anharmonicity.42 The width of the Raman peaks, very small at low temperature ( 1cm-1), evolves in parallel with the frequency shift with temperature, which is still a consequence of the phonon-phonon interactions due to the anharmonicity. The fundamental reason for this strong anharmonicity, as well as the importance of the equilibrium-position shifts between 4 and 300 K,45 resides in the weakness of the van der Waals cohesive forces in the molecular crystal. 

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). 

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 ... 

The simulation of molecular crystals can be addressed with atomistic MD, fully accounting for finite temperature and anharmonic effects. Here, a typical simulation is set up by considering a sample built as an 1 x m x n replica of the unit cell (superceU) with 3D periodic boundary conditions applied. The dynamics below the melting point is in most cases hmited to intramolecular vibrations, and oscillations of molecular positions and orientations around their equilibrium values. From the point of view of the supramolecular organization these simulations may not add further information to that of the equilibrium crystal strucmre, but they can be very useful for other purposes. The simulation of crystal supercells in the NpT ensemble, in which the simulation box is free to rearrange under the effect of molecular forces, can be used to benchmark the FF employed [2,119, 127]. The explicit verification that the FF is able to maintain (within a tolerance of a few percent) the crystal cell parameters measured at the same temperature and pressure than in experiments is a necessary test of the acctuacy of the model potential. 

---