Concept harmonic approximation

In the simplest adiabatic case with an orbital singlet term, potential energy of the crystal lattice is parabolic with one minimum point. At low temperatures, vibrations of the lattice are localized at the bottom of this well, and as a rule, the so-called harmonic approximation applies. This corresponds to the so-called polaron effect and brings us to the concept of electrons coated with phonons. 

The following first section of this appendix describes quantities that are measured when registering spectra obtained using various experimental set-ups and their relations with molecular quantities. These relations form the basis of the interpretations of molecular spectra. The second section describes some general properties of a distribution that are used in various chapters of this book when this distribution is the band of a spectram. The third section deals with such concepts as normal modes in the harmonic approximation, while the fourth section deals with force constants, reduced masses, etc., and offers comparisons of these various quantities. The last section provides a more specific calculation of the first and second moment of a band such as which corresponds to a normal mode characterized by a strong anharmonic coupling with a much slower mode. 

This chapter has shown how the zero-temperature analyses presented earlier in the book may be extended to incorporate finite-temperature effects. By advancing the harmonic approximation we have been able to construct classical and quantum mechanical models of thermal vibrations that are tractable. These models have been used in the present chapter to examine simple models of both the specific heat and thermal expansion. In later chapters we will see how these same concepts emerge in the setting of diffusion in solids and the description of the vibrational entropies that lead to an important class of structural phase transformations. 

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]. 

The concept of the vibrational force field was initially taken from vibrational spectroscopy where the potential energy of a molecule, upon deformation, could be described by a Taylor series with respect to the potential energy at the equilibrium structure. In this case it turns out that the second derivative gives the force constant, and if the harmonic approximation is assumed, then all higher terms may be neglected. 

To the extent that we can trust the harmonic approximation, each level of vibrational excitation (each increment in vibrational quantum number v) costs one vibrational constant in energy. As Table 8.2 shows, the vibrational constants of typical stretching motions place vibrational excitation energies in the infrared region of the spectrum. We can measure vibrational transitions that occur by absorption or emission or by scattering. Section 6.3 introduced the concept of Raman scattering, which in principle can be applied to the spectroscopy of any degree of freedom, but which is most commonly used for spectroscopy of vibrational states. 

In either case (i.e., rigid or flexible), entropic contributions can be calculated by employing an harmonic approximation [85]. The fundamental concept is to characterize the basin of attraction (y) by the properties of its corresponding local minimum (0p, and not by a random sampling of conformations. These properties include the local minimum energy value, E, and the convexity around the local minimum. Essentially, the convexity measme is used to approximate the basin of attraction region as a hyperparabola centered at the local minimum. Therefore, the anharmonic nature of the tme basin, which defines the deviation from approximated harmonic behavior, controls the error associated with this assumption. 

The chapter starts with a brief review of thermodynamic principles as they apply to the concept of the chemical equilibrium. That section is followed by a short review of the use of statistical thermodynamics for the numerical calculation of thermodynamic equilibrium constants in terms of the chemical potential (often designated as (i). Lastly, this statistical mechanical development is applied to the calculation of isotope effects on equilibrium constants, and then extended to treat kinetic isotope effects using the transition state model. These applications will concentrate on equilibrium constants in the ideal gas phase with the molecules considered in the rigid rotor, harmonic oscillator approximation. 

The statistical mechanical form of transition theory makes considerable use of the concepts of translational, rotational and vibrational degrees of freedom. For a system made up of N atoms, 3N coordinates are required to completely specify the positions of all the atoms. Three coordinates are required to specify translational motion in space. For a linear molecule, two coordinates are required to specify rotation of the molecule as a whole, while for a non-linear molecule three coordinates are required. These correspond to two and three rotational modes respectively. This leaves 3N — 5 (linear) or 3N — 6 (non-linear) coordinates to specify vibrations in the molecule. If the vibrations can be approximated to harmonic oscillators, these will correspond to normal modes of vibration. 

Force Fields. The basic assumption underlying molecular mechanics is that classical physical concepts can be used to represent the forces between atoms. In other words, one can approximate the potential energy surface by the summation of a set of equations representing pairwise and multibody interactions. These equations represent forces between atoms related to bonded and nonbonded interactions. Pairwise interactions are often represented by a harmonic potential - 6q) ]... 

Still, these concepts are found to be useful and their usefulness stems from timescale considerations. We will repeatedly see that for many chemical processes the relevant timescales for environmental interactions are short. This does not mean that the system sees its environment for just a short time, but that the dynamics is determined by a succession of short time interactions. If subsequent interactions are uncorrelated with each other, each can be treated separately and for this treatment a harmonic bath picture might suffice. Two conditions need to be satisfied for this to be a good approximation ... 

This approximation parallels the adiabatic approximation made in Chapter 6, where the fast motion of electrons was separated from the slow motion of the nuclei. There the total electronic energy became the potential energy for the motion of nuclei, here the total vibrational energy (the energy of the corresponding harmonic oscillators in their quantum states) becomes the potential energy for the slow motion along s. This concept is called the vibrationally adiabatic approximation. 

In recent decades the thinking of physicists has largely been dominated by attempts to describe systems in terms of linear differential equations and their solutions. Deviations fi om their harmonic behaviour, which lead to non-linear terms in the differential equations, have been treated as perturbations by introducing interactions between the quasi-particles, correspond to the harmonic solutions (electron-electron and electron-phonon collisions, etc.). The idea of the soliton concept is to solve the non-linear differential equations, not by numerical approximations but analytically and to associate new quasi-particles wifli exact solutions, the solitons. 

This chapter begins with a classical treatment of vibrational motion, because most of the important concepts that are specific to vibrations in polyatomics carry over naturally from the classical to the quantum mechanical description. In molecules with harmonic potential energy functions, vibrational motion occurs in normal modes that are mutually uncoupled. Coupling between vibrational modes inevitably occurs in the presence of anharmonic potentials (potentials exhibiting cubic and/or higher order terms in the nuclear coordinates). In molecules with sufficient symmetry, the use of group theory simplifies the procedure of obtaining the normal mode frequencies and coordinates. We obtain El selection rules for vibrational transitions in polyatomics, and consider the rotational fine structure of vibrational bands. We finally treat breakdown of the normal mode approximation in real molecules, and discuss the local mode formulation of vibrational motion in polyatomics. 

In the previous chapter, we solved the problem of the quantized harmonic oscillator and derived key concepts such as the reduced mass and the isotope shift. We were on the verge of treating rotation but you will soon see it is a two-dimensional problem, which needs to be split into two onedimensional problems. Basically the motion of a gas-phase molecule is translation and free rotation and it takes two coordinates (9, c])) to describe such rotational motion even when we assume constant bond lengths within the molecule. We know from the previous chapter that molecules do vibrate but the motion of the vibrations is much smaller than rotations described by (0, < )). Therefore it is a good approximation to assume constant bond lengths. Thus, we have to solve the Schrbdinger equation for a problem in more than one dimension. 

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