Anharmonic potential surface

R. M. Levy, O. de la Luz Rojas, and R. A. Friesner. Quasi-harmonic method for calculating vibrational spectra from classical simulations on multidimensional anharmonic potential surfaces. J. Phys. Chem., 88 4233-4238, 1984. 

Figure 7.1 The simulated temporal motion of a vibrational wave packet created on an anharmonic potential surface. The parameters are evaluated as 300 fs and 37 ps.

In my talk I surveyed recent advances in the methodology and selected 2D-IR spectra of secondary structures. The results promise to provide structurally based kinetic probes for conformational dynamics, sharp tests of anharmonic potential surfaces and novel information regarding the transient and equilibrium vibrational dynamics of peptides. The heterodyned 2D-IR approach has proven useful in determining structures of peptides in solution and the anharmonic nature of the potential surfaces of peptides and secondary structures [1-10], as have polarized photon echo [2,6,10-12] or pump-probe techniques [4,13-16]. 

While this procedure is possible, it involves a very large waste of effort. Furthermore it is practically impossible to find the eigenfunctions for a reasonable (i.e. nonseparable, anharmonic) potential surface for more than a few atoms at moderate energies. Even if the... 

Hollenstein H, Marquardt R, Quack M and Suhm M A 1994 Dipole moment function and equilibrium structure of methane In an analytical, anharmonic nine-dimenslonal potential surface related to experimental rotational constants and transition moments by quantum Monte Carlo calculations J. Chem. Phys. 101 3588-602... 

After transforming to Cartesian coordinates, the position and velocities must be corrected for anharmonicities in the potential surface so that the desired energy is obtained. This procedure can be used, for example, to include the effects of zero-point energy into a classical calculation. 

The anharmonic modes for both the a symmetric and 67 asymmetric CH stretching vibrations have been explored. In order to perform a reasonable anharmonic treatment, we had to take into account the stretching of the bonds to larger elongations than for the harmonic description where displacements can be confined close to the equilibrium geometry. Consequently, correlation effects were included in the determination of the potential surface. The electronic calculations were carried out at the MP2 level, which insures a good description of the CH bond potential towards dissociation. A double zeta... 

How does this happen The answer is easily envisioned if the assumption in Marcus treatments, that the relevant potential surfaces cross only and always in the harmonic parabolic region, is dropped. When the crossing of potential surfaces for reactant and product occurs in the highly anharmonic region near the... 

Near the minimum of the ground electronic surface, the anharmonicities generally play the role of perturbations so that the spectra are regular and the first mechanism is weak. It should become more important when the potential surface deviates significantly from the parabolic shape. The second mechanism, by contrast, may have a very marked effect, as illustrated by the example of N02 [5, 6],... 

The harmonic approximation is unrealistic in a dynamical description of the dissociation dynamics, because anharmonic potential energy terms will play an important role in the large amplitude motion associated with dissociation. An accurate potential energy surface must be used in order to obtain a realistic dynamical description of the dissociation process and, as in the quasi-classical approach for bimolecular collisions, a numerical solution of the classical equations of motion is required [2]. 

Apart from the heat bath mode, the harmonic potential surface model has been used for the molecular vibrations. It is possible to include the generalized harmonic potential surfaces, i.e., displaced-distorted-rotated surfaces. In this case, the mode coupling can be treated within this model. Beyond the generalized harmonic potential surface model, there is no systematic approach in constructing the generalized (multi-mode coupled) master equation that can be numerically solved. The first step to attack this problem would start with anharmonicity corrections to the harmonic potential surface model. Since anharmonicity has been recognized as an important mechanism in the vibrational dynamics in the electronically excited states, urgent realization of this work is needed. 

L. Goodman, M. J. Berman, A. G. Ozkabak, J. Chem. Phys. 90, 2544 (1989). The Benzene Ground State Potential Surface. III. Analysis of b2u Vibrational Mode Anharmonicity Through Two-Photon Intensity. 

Fig. 1.3. Upper Schematic view (dotted line) of cross-section of many-dimensional highly anharmonic potential energy surfaces for reactants plus solution (R) and (dotted line omitted) for products plus solution (P). TS occurs at the intersection. Lower Plot of free energy G for the above R and P systems vs. the reaction coordinate U.

All this applies to weak and medium strong H-bonds like those encountered for alcohols and many other systems up to carboxylic acid dimers or about 32-42 kJ/mol. (8 or 10 kcal/mol.) Unfortunately vibrational spectra of systems with very strong H-bonds could, with a few exceptions, only be measured in condensed phases. Factors that come in when such systems are examined are potential surfaces with two minima with, in certain cases, the possibility of tunnelling, or flat single minima Most of these systems are likely to be so anharmonic that second order perturbation theory breaks down and the concept of normal vibrations becomes itself question-nable. Many such systems are highly polarizable and are strongly influenced by the environment yielding extremely broad bands 92). Bratos and Ratajczak 93) has shown that even such systems can be handled by relaxation theories. 

In order to calculate the response function of the Feynman diagram R3, it is further assumed that the transition frequency co 12 is anharmonically shifted with respect to the ground states transition frequency so that, u>n = >oi -A. Another assumption that can be made (see later for a discussion of these assumptions) is that the fluctuations between both level pairs are strictly correlated <5co12 = <5 j0i. This implies that only the harmonic part of the potential surface is perturbed by the bath fluctuations and the anharmonicity of the vibrator is unaffected. We then obtain for R3 ... 

Figure 12-3. IR-UV double resonance spectrum of GC (structure C) in the mid-IR frequency range (recorded at the FELIX free electron laser facility), compared with three types of ab intio calculations. Harmonic frequencies were obtained at the RI-MP2/cc-pVDZ, RI-MP2/TZVPP, and semiempirical PM3 levels of electronic structure theory. Anharmonic frequencies were obtained by the CC-VSCF method with improved PM3 potential surfaces [30]...

The anharmonic free energy is evaluated for empty hydrate and cubic ice (ice Ic). The calculated free energy due to the anharmonic potential energy surface is given in Table 1. The anharmonic contribution to the free energy of empty... 

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