Harmonic oscillator, potential energy

Calculation of Morse and Harmonic Oscillator Potential Energy Curves... 

Figure Bl.2.3. Comparison of the harmonic oscillator potential energy curve and energy levels (dashed lines) with those for an anharmonic oscillator. The harmonic oscillator is a fair representation of the true potential energy curve at the bottom of the well. Note that the energy levels become closer together with increasing vibrational energy for the anharmonic oscillator. The anharmonicity has been greatly exaggerated.

Figure 13 Potential energy surfaces for electron transfer reactions. Harmonic oscillator potential energy functions for reactants and product are shown, including the nuclear wave functions, which are shaded. The dark shaded region indicates the magnitude of overlap of the nuclear wave functions, which is the Franck-Condon factor, (a) is the normal region, (b) is the activationless region and (c) is the inverted region as defined in the text. (Ref. 72. Reproduced by permission of Nature Publishing Group, www.nature.com)...

The parabola —D + kx best approximates V (x) close to x = 0 and represents the harmonic oscillator potential energy (with the force constant k). The Morse oscillator is hard to... 

Our goal with this derivation is to find a simple and reasonably accurate solution to the energies and wavefunctions of atoms vibrating in a chemical bond. We will first show that the potential energy curve of a chemical bond can be crudely approximated by the harmonic oscillator potential energy jkx . Then we will turn once more to our friends at the 19th-century French Academy for the solution to the differential equation. The final step is relating the mathematical solution to the physical parameters of the molecule. A DERIVATION SUMMARY appears after Eq. 8.24. 

The force constant is the second derivative of the harmonic oscillator potential energy. 

FIGURE 14.29 The Morse potential is a better fit to the potential energy curve of a real molecule than is the harmonic oscillator potential energy surface, superimposed... 

The above results are based upon a harmonic oscillator potential energy curve, which of course does not give a particularly good representation of the potential energy curves for actual molecules. We could include additional terms of 14-18 in the expression for the potential energy a more satisfactory procedure is to assume some appropriate... 

Figure 9.7 Vibrational energy levels determined from solution of the one-dimensional Schrodinger equation for some arbitrary variable 6 (some higher levels not shown). In addition to the energy levels (horizontal lines across the potential curve), the vibrational wave functions are shown for levels 0 and 3. Conventionally, the wave functions are plotted in units of (probability) with the same abscissa as the potential curve and an individual ordinate having its zero at the same height as the location of the vibrational level on the energy ordinate - those coordinate systems are explicitly represented here. Note that the absorption frequency typically measured by infrared spectroscopy is associated with the 0 —> 1 transition, as indicated on the plot. For the harmonic oscillator potential, all energy levels are separated by the same amount, but this is not necessarily the case for a more general potential...

Consider the physical significance of the additional terms in (4.67) as compared to (4.39). The fourth term on the right side of (4.67) represents a shift in the vibrational levels. The constant involves the third and fourth derivatives of V evaluated at Re, and is therefore a consequence of the deviation of the potential energy function from the (quadratic) harmonic-oscillator potential ... 

The addition of the spin-orbit term to the nuclear harmonic oscillator potential causes a separation or removal of the degeneracy of the energy levels according to their total angular momentum (j = l + s). In the nuclear case, the states with... 

This is equivalent to the harmonic oscillator potential, eqn 3.13, if x represents the distance from the equilibrium separation and if the zero of energy is taken as the minimum potential Vt. The approximation is not accurate for vibrations of large amplitude, but is good enough for the lowest allowed energy levels, three of which are shown in Fig. 3.8. 

The product J Eg)coQ is equal to Eg for a harmonic oscillator potential truncated at = Eg, and to 2Eg for a Morse potential with dissociation energy equal to Eg. Equation (2.41) is the low-friction limit result of Kramers. There are other methods to derive the results obtained in the previous section. One is to look for the eigenvalue with smallest positive real part of the ojjerator L defined so that dP/dt = — LP is the relevant Fokker-Planck or Smoluchowski equation. Under the usual condition of time scale separation this smallest real part is the escapie rate for a single well potential. Another way uses the concept of mean passage time. For the one-dimensional Fokker-Planck equation of the form... 

Fig. 3.2. Schematic illustrating the harmonic oscillator potential and the associated wave functions. The wave functions are plotted such that the zeroes of (x) intersect the energy axis at that energy...

This is an aromaticity index based on the energy deformation derived from a simple harmonic oscillator potential and defined as [Krygowski and Wieckowski, 1981 Krygowski, Anulewicz et al, 1983, 1995 Bird, 1997] ... 

We postulate the potential energy curve of a dissociating harmonic oscillator reactant as that shown in Fig. 1. It is a truncated harmonic oscillator potential with a finite number of equally spaced energy levels such that the level N is the last... 

Figure 21.5 The Bell tunnel model (a) Quantum mechanical harmonic oscillator with its ground state wavefunctions. (b) Inverted harmonic oscillator potential, (c) A stream of particles with a Boltzmann distribution of energies hits the barrier. Classical only those particles with W>Vq can pass the barrier.

We will now develop the equations used to compute the local mode energies. After defining the local mode Hamiltonian, we will convert it into the normal coordinates that were used to define the basis sets used for the dynamical calculations. The local mode Hamiltonian, for mode /, is defined in terms of the harmonic oscillator potential (except for the initially excited stretch, as described later)... 

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