Harmonic potential energy

Quartic terms cannot be neglected relative to cubic. It is true that they represent a higher order of the potential energy expression. However, first order terms of type j tpi P y>t dz, where the ipt are eigenfunctions of the harmonic potential energy and P represents deviations from anharmonicity, vanish when P is a cubic (or any odd powered) term but not when P is a quartic (or any even powered) term. 

We have seen this expression before in Chapter 5, where it was the starting point for describing vibrational modes. We found in Chapter 5 that a natural way to think about this harmonic potential energy surface is to define the normal modes of the system, which have vibrational frequencies v, (/... 

Fig. 2.7. The bonding potential V(r) and its representation by a harmonic potential energy function and an additional cubic term.

Fig. la c. Illustration of the time dependent theory of emission spectroscopy for one-dimensional harmonic potential energy surfaces, a schematic view of the emission transition, b time dependence of the overlap < (f> t) >, e calculated emission spectrum... 

Fig. 12. PET can be studied on the basis of intersecting harmonic potential-energy curves. In the approach of Marcus, the free energy of a reacting system is represented as a function of nuclear geometry on the horizontal axis. During excitation, there is a vertical transition (Franck-Condon) to a point on the excited-state surface, followed by vibrational relaxation. Electron transfer takes place at the crossing of the excited-state and ionic potential-energy curves. The transition-state energy, AGe, corresponds to the energy difference between the minimum on the excited-state surface and the point of intersection...

There are several possible approaches to simplifying the vibrational Hamiltonian given by Eq. (3.13). Some of these will be outlined here. The first approach consists of removing the harmonic potential energy cross terms and the kinetic energy terms by the equivalent of a normal coordinate transformation. We define a new set of coordinates Q... 

If we consider a set of platinum atoms (for example) along a line, so that the 1-dimensional Bravais lattice vector is R na, where a is the platinum interatomic distance and n an integer, the harmonic potential energy has the form [2]... 

Let us consider a set of metal atoms of 1-dimensional configuration so that the Bravais lattice vector is a set of R = na, a is the metal interatomic distance, and n an integer as stated above, the harmonic potential energy, t/harm> is... 

Figure 7 The time auto correlation function and the corresponding spectrum for a Gaussian wave packet propagating on an excited harmonic potential energy surface, (a) The short time decay of C(/) (cf. Eq. (17)) and the broad spectrum (= the Franck Condon envelope (cf. (18)). (b)The longer time dependence of C(r) and the corresponding, vibrationally resolved, spectrum.

Let us consider a model consisting of three bound diabatic basis states Ai(q)) associated with harmonic potential energy functions Ui ). For the sake of simplicity, let us suppose that reactant and product are described by diabatic energy functions in terms of a single... 

The static aspect of the JT theorem, which has been treated in a number of papers 4, 51, 52, 53), results in different energies for both the "Eg components these are expressed in the simplest manner by harmonic potential energy surfaces... 

TRANSITIONS BETWEEN ONE-DIMENSIONAL HARMONIC POTENTIAL ENERGY SURFACES... 

Figure 1 Electronic absorption and luminescence transitions between initial and final states represented by harmonic potential energy curves. The solid and dotted spectra denote calculated luminescence and absorption spectra, respectively. The electronic origin transition, Eqo, and the offset between potential energy...

Figure 2 Time-dependent view of the luminescence transition in Figure 1. Only the potential energy curve of the final state is shown. Time-dependent wave functions are given for times of 0,10, and 50 fs. The bottom panel shows the absolute value of the autocorrelation function, visualized as the overlap between the time-dependent wavefunction (t) and the wavefunction at time zero. The first recurrence at 95 fs occurs after a single vibrational period of the 350 cm vibrational frequency used to define the harmonic potential energy curve.

Figure 11 Autocorrelation functions used to calculate the spectra in Figure 10 from the potential energy surfaces in Figure 9. Dotted and solid lines correspond to autocorrelation functions obtained from the potential surfaces in Figures 9a (harmonic potential energy surface) and 9b (potential energy surface with coupled coordinates), respectively, a) short-time comparison over the time interval for the initial decrease of the autocorrelation function b) comparison over a time interval corresponding to several vibrational periods c) difference trace between the autocorrelation functions in b.

Harmonic potential energy curves may be linearized only within a narrow energy interval therefore, it is to be expected that the relation between their slopes, and hence a, must vary smoothly over a sufficiently great variation in AZ7 . However, the linearization interval is broader if the curves U X) have an inflection typical of, for example, an anharmonic oscillator, but near the minimum of the curves, where they can always be approximated by a parabola, the linearity is inevitably disturbed. In fact, when a certain series of homogeneous reactions of proton transfer were studied over a sufficiently broad range of Af/° (about 80 kJ moF ), a smooth variation in a was obser-ved " however, nobody has so far reliably established a smooth variation in the true value of a with the potential for any electrode reaction moreover, in the case of the hydrogen evolution reaction, a good constancy in a is observed over an interval of overpotentials of about 1.5 This... 

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. 

The correspondence between the continuum elasticity equations developed for the stress and strain tensors and the strain energy density, and the general relations between the force F and time derivative of the position r, or the spatial derivative of the potential energy as given by classical mechanics. The last two equations correspond to the special case of a harmonic potential energy, which is implicit in the linear relation between stress and strain. 

The Born-Oppenheimer energy acts as the vibrational potential energy. We assume a harmonic potential energy. That is, we assume that the potential energy depends on the q variables in the following way ... 

Fig.5.1. Potential energy of a diatomic molecule after MORSE, as defined by (5.1). The parameters are those of the electronic ground state of the HCl molecule D=4.43 eV, X=1.9047 108cm-l, ao=1.2745 A. a is the mean interatomic distance at temperature T. The dashed curve indicates the corresponding harmonic potential energy...

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