Harmonic approximation vibrational properties

The thermodynamic properties were computed with the molecular geometry and vibrational frequencies given above assuming an ideal gas at 1 atm pressure and using the harmonic-oscillator rigid-rotor approximation. These properties are given for the range 0-2000°K in the Appendix (Table AI). 

In our discussion the usual Born-Oppenheimer (BO) approximation will be employed. This means that we assume a standard partition of the effective Hamiltonian into an electronic and a nuclear part, as well as the factorization of the solute wavefunction into an electronic and a nuclear component. As will be clear soon, the corresponding electronic problem is the main source of specificities of QM continuum models, due to the nonlinearity of the effective electronic Hamiltonian of the solute. The QM nuclear problem, whose solution gives information on solvent effects on the nuclear structure (geometry) and properties, has less specific aspects, with respect the case of the isolated molecules. In fact, once the proper potential energy surfaces are obtained from the solution of the electronic problem, such a problem can be solved using the standard methods and approximations (mechanical harmonicity, and anharmonicity of various order) used for isolated molecules. The QM nuclear problem is mainly connected with the vibrational properties of the nuclei and the corresponding spectroscopic observables, and it will be considered in more detail in the contributions in the book dedicated to the vibrational spectroscopies (IR/Raman). This contribution will be focused on the QM electronic problem. 

By taking as a reference the calculation in vacuo, the presence of the solvent introduces several complications. In fact, besides the direct effect of the solvent on the solute electronic distribution (which implies changes in the solute properties, i.e. dipole moment, polarizability and higher order responses), it should be taken into account that indirect solvent effects exist, i.e. the solvent reaction field perturbs the molecular potential energy surface (PES). This implies that the molecular geometry of the solute (the PES minima) and vibrational frequencies (the PES curvature around minima in the harmonic approximation) are affected by the presence of a solvating environment. Also, the dynamics of the solvent molecules around the solute (the so-called nonequilibrium effect ) has to be... 

In the limit of static fields, the nuclear relaxation contribution (from now on just vibrational ) to the polarizabilities can be computed in the double harmonic approximation, i.e. assuming that the expansions of both the potential energy and the electronic properties with respect to the normal coordinates can be limited to the quadratic and the linear terms, respectively (i.e. assuming both mechanical and electric harmonicity). 

The harmonic approximation reduces to assuming the PES to be a hyperparaboloid in the vicinity of each of the local minima of the molecular potential energy. Under this assumption the thermodynamical quantities (and some other properties) can be obtained in the close form. Indeed, for the ideal gas of polyatomic molecules the partition function Q is a product of the partition functions corresponding to the translational, rotational, and vibrational motions of the nuclei and to that describing electronic degrees of freedom of an individual molecule ... 

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). 

The zeroth-order approximation in the BK perturbation treatment of pure vibrational NLO is the double harmonic model. As far as electrical properties are concerned this approximation includes just the terms in the instantaneous property expression that are linear in the normal coordinates (there is no vibrational contribution from the constant term). To these are added the quadratic terms in the pure vibrational (or mechanical) potential which constitute the usual harmonic approximation. Then, in zeroth-order roughly half of the square brackets vanish leaving ... 

Omission of dynamics. Minimization identifies the static configuration of lowest energy and there is no representation of the vibrational or other dynamical properties of the system. In formal terms, these are zero Kelvin calculations, with zero point motion omitted. It is, however, relatively straightforward to add a treatment of the vibrational properties of the system within the harmonic (or quasi-harmonic) approximations. Such methods will be discussed in Chapter 3. 

If the thermodynamic properties are calculated within the harmonic approximation, in which the normal modes of vibration are assumed to be independent and harmonic, the cell has no thermal expansion. PARAPOCS (Parker and Price, 1989) extends this to the quasi-harmonic approximation. In this method the vibrations are assumed to be harmonic but their frequencies change with volume. This provides an approach for obtaining the extrinsic anharmonicity which leads to the ability to calculate thermal expansion. 

In this section we report a second extract of the study we have published on the Journal of the American Chemical Society about solvent effects on electronic and vibrational components of linear and nonlinear optical properties of Donor-Acceptor polyenes. In a previous section we have presented the analysis on geometries, here we report the results obtained for the electronic and vibrational (in the double harmonic approximation) static polarizability and hyperpolarizability for the two series of noncentrosym-metric polyenes NH2(CH=CH) R (n=l,2), with R=CHO (series I) and with R=N02 (series II) both in vacuo and in water. 

Vibrational corrections. For energies, this would typically be inclusion of zero point energies while for properties this may correspond to a vibrational averaging. The corrections may again be done at several levels of accuracy, for example using a harmonic approximation or also including anharmonic effects. 

Given that we can calculate the total energy of a molecule as a function of its geometry, we can then calculate the electronic and vibrational states (Wj) associated with the nuclear motions. The energies of the vibrational states are usually calculated in the harmonic approximation. Using statistical mechanics, we can now start to evaluate thermodynamic properties. If we have a set of N particles or molecules distributed over a set of energy levels Wj then the population of level j is... 

An alternative approach to the dynamics of a protein or one of its constituent elements (e.g., an a-helix) is to assume that the harmonic approximation is valid. Early attempts to examine dynamical properties of proteins or their fragments used the harmonic approximation. They were motivated by vibrational spectroscopic studies [24], where the calculation of normal mode frequencies from empirical potential functions has long been a standard step in the assignment of infrared spectra [25]. In calculating the normal vibrational modes of a molecule, one assumes that the vibrational displacements of the atoms from their equilibrium positions are... 

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 ... 

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