Vibrational methods harmonic approximation

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]. 

Vibrational Spectra Many of the papers quoted below deal with the determination of vibrational spectra. The method of choice is B3-LYP density functional theory. In most cases, MP2 vibrational spectra are less accurate. In order to allow for a comparison between computed frequencies within the harmonic approximation and anharmonic experimental fundamentals, calculated frequencies should be scaled by an empirical factor. This procedure accounts for systematic errors and improves the results considerably. The easiest procedure is to scale all frequencies by the same factor, e.g., 0.963 for B3-LYP/6-31G computed frequencies [95JPC3093]. A more sophisticated but still pragmatic approach is the SQM method [83JA7073], in which the underlying force constants (in internal coordinates) are scaled by different scaling factors. 

Regardless of the force field chosen, the calculation of vibrational frequencies by the method outlined above is based on the harmonic approximation. Tabulated values of force constants can be used to calculate vibrational frequencies, for example, of molecules whose vibrational spectra have not been observed. However, as anharmonicities have been neglected in the above analysis, the resulting frequency values are often no better than 5% with respect to those observed. 

A productive exploitation of the synergy between experiment and theory requires that practitioners familiarize themselves with the scope and limitations of the methods they use, so they can avoid pitfalls due to artifacts that may occur both in experiment and in theory. It is, for example, disturbingly easy to create or annihilate bands by formation of suitably scaled difference spectra. On the other hand, the harmonic approximation that is at the basis of all practicable modeling calculations of vibrational spectra may lead to predictions that have no relation to experiment (as demonstrated above for the case of phenylcarbene). 

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. 

The quantities of interest in vibrational spectra are frequencies and intensities. Within the double harmonic approximation, vibrational frequencies and normal modes for solvated molecules are related, within the continuum approach, to free energy second derivatives with respect to nuclear coordinates calculated at the equilibrium nuclear configuration. The QM analogues for vibrational intensities , depend on the spectroscopy under study, but in any case derivative methods are needed. 

Due to the harmonic approximation, most methods will overestimate the vibrational frequencies. Listed in Table 2.4 are the mean absolute deviations of the vibrational freqnencies for a set of 32 simple molecnles with different compnta-tional methods. A clear trend is that as the method improves in acconnting for electron correlation, the predicted vibrational freqnencies are in better accord with experiment. 

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). 

Normal coordinate analysis has been used for many years in the interpretation of vibrational spectra for small molecules.88 It provided the motivation for the application of the harmonic approximation to proteins and their constituent elements (e.g., an a-helix).35 133-136 In this alternative to conventional dynamical methods, it is assumed that the displacement of an atom from its equilibrium position is small and that the potential energy (as obtained from Eq. 6) in the vicinity of the equilibrium position can be approximated as a sum of terms that are quadratic in the atomic displacements i.e., making use of Cartesian coordinates, which are simplest to employ for large molecules, we have... 

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. 

The connecting link between ab initio calculations and vibrational spectra is the concept of the energy surface. In harmonic approximation, usually adopted for large systems, the second derivatives of the energy with respect to the nuclear positions at the equilibrium geometry give the harmonic force constants. For many QM methods such as Hartree-Fock theory (HF), density functional methods (DFT) or second-order Moller-Plesset pertiubation theory (MP2), analytical formulas for the computation of the second derivatives are available. However, a common practice is to compute the force constants numerically as finite differences of the analytically obtained gradients for small atomic displacements. Due to recent advances in both software and computer hardware, the theoretical determination of force field parameters by ab initio methods has become one of the most common and successful applications of quantum chemistry. Nowadays, analysis of vibrational spectra of wide classes of molecules by means of ab initio methods is a routine method [85]. 

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