Harmonic potential curves

Harmonic potential curve (---------) and corresponding vibrational ground state... 

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. 

Comparison of the simple harmonic potential (Hooke s law) with the Morse curve. 

Owing to the effects of mechanical anharmonicity - to which we shall refer in future simply as anharmonicity since we encounter electrical anharmonicity much less frequently -the vibrational wave functions are also modified compared wifh fhose of a harmonic oscillator. Figure 6.6 shows some wave functions and probabilify densify functions (iA A ) for an anharmonic oscillator. The asymmefry in and (iA A ) 5 compared wifh fhe harmonic oscillator wave functions in Figure f.i3, increases fheir magnitude on the shallow side of the potential curve compared with the steep side. 

Provided the potential curves are harmonic, relatively simple relationships can be set up. Thus the curves may be described by ... 

Figure 6. Matching the calculated harmonic potential to the potential curve obtained by Slater with simple MO theory.

Figure 1.3 Potential curve of a molecule (ground state of HC1). The full curve is the Morse potential of Eq. (1.6). The dashed curve is the harmonic approximation. De is the dissociation energy, and re is the equilibrium separation.

The discrete energy levels sketched as horizontal lines on each potential curve of Figure 5.10 are consistent with the quantized energy levels (phonon levels) of a harmonic oscillator. For each harmonic oscillator at frequency 12, the permitted phonon energies are given by... 

For the purpose of numerical computation, we consider a lattice with M = 10 sites. Our goal is to generate the ground-state population of the harmonic potential for R = M - I, represented by the dashed curve in Figure 3.36, from the ground state for R = 0, represented by the solid curve in Figure 3.36. The ground-state wave function satisfies... 

Figure 3.36 Schematic diagram of the lattice system and the harmonic potential. The solid and dashed curves show the harmonic potential at = 0 and t = respectively.

Figure 3.37 The initial (left panel) and the target populations (right panel) for each site in the lattice. The curves show a portion of the harmonic potential for t = 0 and = M - 1 in the left and the right panels, respectively.

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...

Messina et al. [25] test the time-dependent Hartree reduced representation with a simple two-degree-of-freedom model consisting of the h vibration coupled to a one-harmonic-oscillator bath. The objective function is a minimum-uncertainty wavepacket on the B state potential curve of I2. Figure 12, which displays a typical result, shows that this approximate representation gives a rather good account of the short-time dynamics of the system. 

As a second example we analyze the anisotropic charge-permanent dipole potential where SACM and PST differ from each other. Here, for demonstration, we only consider the low-energy perturbation and the high-energy harmonic oscillator limits. In the former limit, the adiabatic channel potential curve for the lowest channel j = m = 0 has the form... 

Figure 8. Lowest adiabatic channel potential curves [33] for the interaction of electronic ground state N2 with ions (q = ionic charge, Q = N2 quadnipole moment N, M = free-rotor quantum numbers k,v = harmonic oscillator quantum numbers for more details, see Ref. 33).

A very important aspect of the results described above for De is that the error obtained at a certain level of approximation is systematic. This fact combined with the fact that the results improve as the method improves are aspects of ab initio methods which are at least as important as the final accuracy of the results. So far the only property discussed is De. It is clear that the most important chemical information, such as reaction pathways and thermochemistry, is obtained from relative energies, but the accuracy of other properties is also of interest. If we look at the equilibrium bond distance Re and the harmonic vibrational frequency we, these properties also display a systematic behaviour depending on the method chosen and this systematic behaviour is easy to understand. Since the RHF method dissociates incorrectly, the potential curves tend to rise too fast as the bond distance is increased. At the RHF level this leads to too short equilibrium bond distances and vibrational frequencies that are too high. When proper dissociation is included at the MCSCF level, the opposite trend appears. Since the dissociation energies are too small at this level the potential curves rise too slowly as the bond distance increases. This leads to too long bond distances and too low frequencies. These systematic trends are nicely illustrated by the results for three of the previously discussed diatomic molecules. For H2 the experimental value for Re is 1.40 ao and for uje it is 4400 cm-1. At the RHF level Re becomes too short, 1.39 ao, and we becomes too high, 4561 cm-1. At the two configuration MCSCF level Re becomes... 

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