Harmonic approximations

The potential energy function U(R) that appears in the nuclear Schrodinger equation is the sum of the electronic energy and the nuclear repulsion. The simplest case is that of a diatomic molecule, which has one internal nuclear coordinate, the separation R of the two nuclei. A typical shape for U(R) is shown in Fig. 19.1. For small separations the nuclear repulsion, which goes like 1 /R, dominates, and liniR o U(R) = oo. For large separations the molecule dissociates, and U(R) tends towards the sum of the energies of the two separated atoms. For a stable molecule in its electronic ground state U(R) has a minimum at a position Re, the equilibrium separation. 

We are interested in more complex molecules such as [Fe(H20)e]2+-As we discussed in Chapter 6, the equilibrium bond distance between 

Even for a diatomic molecule the nuclear Schrodinger equation is generally so complicated that it can only be solved numerically. However, often one is not interested in all the solutions but only in the ground state and a few of the lower excited states. In this case the harmonic approximation can be employed. For this purpose the potential energy function is expanded into a Taylor series about the equilibrium separation, and terms up to second order are kept. For a diatomic molecule this results in  

The first derivative vanishes since U(R) has a minimum at Re. Within this approximation the nuclear Schrodinger equation reduces to that of a harmonic oscillator, whose frequency to is given by  

For a complex molecule with N internal degrees of freedom, the situation is a little more complicated. Let Rk (A = 1,. .., N) denote the components of R, and R j, their equilibrium values. A Taylor expansion up to second order gives now  

As we saw above, what emerges from our detailed analysis of the vibrational spectrum of a solid can be neatly captured in terms of the vibrational density of states, p(co). The point of this exercise will be seen more clearly shortly as we will observe that the thermodynamic functions such as the Helmholtz free energy can be written as integrals over the various allowed frequencies, appropriately weighted by the vibrational density of states. In chap. 3 it was noted that upon consideration of a single oscillator of natural frequency co, the associated Helmholtz free energy is 

In light of our observations from above, namely that the vibrational contribution to the energy of the crystal may be written as a sum of independent harmonic oscillators, this result for the Helmholtz free energy may be immediately generalized. In particular, we note that once the vibrational density of states has been determined 

Note that all reference to the wavevector has been eliminated as a result of the fact that p (co) is obtained by averaging over all wave vectors within the Brillouin zone, and is hence indifferent to the polarization and propagation vector of a given normal mode. 

In sections 5.22 and 5.24, we have made schematic evaluations of the nature of the vibrational spectrum (see fig. 5.5). At this point, it is convenient to construct approximate model representations of the vibrational spectrum with the aim of gleaning some insight into how the vibrational free energy affects material properties such as the specific heat, the thermal expansion coefficient and processes such as structural phase transformations. One useful tool for characterizing a distribution such as the vibrational density of states is through its moments, defined by 

The most naive model we can imagine is to insist that whatever model we select for the density of states faithfully reproduces the first moment of the exact vibrational density of states. That is. 

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A different approach is to represent the wavepacket by one or more Gaussian functions. When using a local harmonic approximation to the trae PES, that is, expanding the PES to second-order around the center of the function, the parameters for the Gaussians are found to evolve using classical equations of motion [22-26], Detailed reviews of Gaussian wavepacket methods are found in [27-29]. 

This algorithm was improved by Chen et al. [78] to take into account the surface anhannonicity. After taking a step from Rq to R[ using the harmonic approximation, the true surface information at R) is then used to fit a (fifth-order) polynomial to fomi a better model of the surface. This polynomial model is then used in a coirector step to give the new R,. 

While it is not essential to the method, frozen Gaussians have been used in all applications to date, that is, the width is kept fixed in the equation for the phase evolution. The widths of the Gaussian functions are then a further parameter to be chosen, although it appears that the method is relatively insensitive to the choice. One possibility is to use the width taken from the harmonic approximation to the ground-state potential surface [221]. 

S(p) is thus the harmonic approximation counterpart of the function/(p). To simplify the orthography, we use symbol instead of the usual For a... 

In his classical paper, Renner [7] first explained the physical background of the vibronic coupling in triatomic molecules. He concluded that the splitting of the bending potential curves at small distortions of linearity has to depend on p, being thus mostly pronounced in H electronic state. Renner developed the system of two coupled Schrbdinger equations and solved it for H states in the harmonic approximation by means of the perturbation theory. 

In the harmonic approximation, V does not involve the cross-term pj-Pe because pj- and are the symmetry coordinates. It is thus of the form... 

First, let us note that the adiabatic potentials and V [Eq. (67)], even in the lowest order (harmonic) approximation, depend on the difference of the angles 4>j- and t >c this is an essential difference with respect to triatomics where the adiabatic potentials depend only on the radial bending coordinate p. The foims of the functions V, Vt, and Vc are determined by the adiabatic potentials via the following relations... 

We can approximate this firaction of states in the reactant well, by expanding the potential in a harmonic approximation and assuming that the tempera ture is low compared with the barrier height. This leads to an estimate for the rate constant... 

The many approaches to the challenging timestep problem in biomolecular dynamics have achieved success with similar final schemes. However, the individual routes taken to produce these methods — via implicit integration, harmonic approximation, other separating frameworks, and/or force splitting into frequency classes — have been quite different. Each path has encountered different problems along the way which only increased our understanding of the numerical, computational, and accuracy issues involved. This contribution reported on our experiences in this quest. LN has its roots in LIN, which... 

The surfaces of large molecules such as proteins cannot be represented effectively with the methods described above (e.g., SAS), However, in order to represent these surfaces, less calculation-intensive, harmonic approximation methods with SES approaches can be used [1S5]. 

The vibraiimial rroqueiicics are tlenved lioin the harmonic approximation, which assiiiines that the potential surface has a quadratic form. 

The centrifugal distortion constant depends on the stifthess of the bond and it is not surprising that it can be related to the vibration wavenumber co, in the harmonic approximation (see Section 1.3.6), by... 

B4) The corresponding estimates are valid only in harmonic approximation therefore, they are inapplicable to nonnal temperature conditions. The harmonic... 

Many thermodynamic quantities can be calculated from the set of normal mode frequencies. In calculating these quantities, one must always be aware that the harmonic approximation may not provide an adequate physical model of a biological molecule under physiological conditions. 

The corresponding level broadening equals half. In fact is the diagonal kinetic coefficient characterizing the rate of phonon-assisted escape from the ground state [Ambegaokar 1987]. In harmonic approximation for the well the only nonzero matrix element is that with /= 1,K0 Q /> = <5o, where is the zero-point spread of the harmonic oscillator. For an anharmonic potential, other matrix elements contribute to (2.52). 

This formula resembles (3.32) and, as we shall show in due course, this similarity is not accidental. Note that at n = 0 the short action 1 2 ( q) taken at the ground state energy Eq is not equal to the kink action (3.68). Since in the harmonic approximation for the well Tq = 2n/o)o, this difference should be compensated by the prefactor in (3.74), but, generally speaking, expressions (3.74) and (3.79) are not identical because eq. (3.79) uses the semiclassical approximation for the ground state, while (3.74) does not. 

P Q-) =p Q-,Q-,p), which in the harmonic approximation is described by (3.16), PhiQ-iQ-,P) exp(— CO Q1 tanh co ). Having reached the point Q, the particle is assumed to suddenly tunnel along the fast coordinate Q+ with probability A id(Q-), which is described in terms of the usual one-dimensional instanton. The rate constant comes from averaging the onedimensional tunneling rate over positions of the slow vibration mode,... 

In accordance with the one-dimensional periodic orbit theory, any orbit contributing to g E) is supposedly constructed from closed classical orbits in the well and subbarrier imaginary-time trajectories. These two classes of trajectories are bordering on the turning points. For the present model the classical motion in the well is separable, and the harmonic approximation for classical motion is quite reasonable for more realistic potentials, if only relatively low energy levels are involved. 

Even for such a simple molecule, which 1 deliberately constrained to be lineiir and where I assumed that the harmonic approximation was applicable, the potential energy function will have cross-terms. 

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