Bom-Oppenheimer potential

There are many facets to a successful quantum dynamics study. Of course, if comparison with experimental results is a goal, the underlying Bom-Oppenheimer potential energy surface must be known at an appropriately high level of electronic structure theory. For nonadiabatic problems, two or more surfaces and their couplings must be determined. The present chapter, however, focuses on the quantum dynamics of the nuclei once an adequate description of the electronic structure has been achieved. 

As discussed in Section 2, one key assumption of reaction field models is that the polarization field of the solvent is fully equilibrated with the solute. Such a situation is most likely to occur when the solute is a long-lived, stable molecular structure, e g., the electronic ground state for some local minimum on a Bom-Oppenheimer potential energy surface. As a result, continuum solvation models... 

One way to include these local quantum chemical effects is to perform ab initio calculations on an HOD molecule in a cluster of water molecules, possibly in the field of the point charges of the water molecules surrounding the cluster. In 1991 Hermansson generated such clusters from a Monte Carlo simulation of the liquid, and for each one she determined the relevant Bom Oppenheimer potential and the vibrational frequencies. The transition-dipole-weigh ted histogram of frequencies was in rough agreement with the experimental IR spectrum for H0D/D20 [130],... 

Database/3 and the other data used in this paper consist entirely of zero-point-exclusive data, which allows for direct comparisons with calculated Bom-Oppenheimer potential energy surfaces, i.e., the sum of the electronic energies and nuclear repulsion. Although the G3X and CBS families of methods have standard geometry and frequency calculations associated with them, in this paper only the potential energy surfaces are required to compare with Database/3. The geometries used are optimized QCISD/MG3 geometries for all calculations in this paper. 

Given the lack of success with the phase-tailored UV pulse we tried a different approach a two-pulse scheme, where a weak-field ultrashort UV pulse is combined with a strong infrared (IR) field [7]. The strong IR field affects the nuclear dynamics such that the propagator AT, in Eq. (1) in this case describes the nuclear dynamics under the influence of Vt + fijEm (r), where fii is the dipole moment in the electronic state i. The idea that this approach may work is based on the fact that the nuclear dynamics no longer takes place on the pure Bom-Oppenheimer potentials and this may speed up the process. Various applications of IR+UV two-pulse schemes have been discussed recently [9-12]. 

Evaluating the energy e for different values of R gives the effective potential for the nuclei in the presence of the electron. This function is called the Bom-Oppenheimer potential surface or just the potential surface. In order to evaluate e(R) we have to determine HAA, HAB, and SAB. These quantities, which can be evaluated using elliptical coordinates, are given by... 

We shall show how the Bom-Oppenheimer potential energy for the Hj ion can be calculated exactly using series expansion methods, even though an exact analytical solution cannot be obtained. Figure 6.29 shows the coordinate system used for an electron moving in the field of two clamped nuclei. In atomic units the Hamiltonian is... 

The Bom-Oppenheimer approximation is not always correct, especially with light nuclei and/or at finite temperature. Under these circumstances, the electronic distribution might be less well described by the solution of a Schroedinger equation. Non-adiabatic effects can be significant in dynamics and chemical reactions. Usually, however, non-adiabatic corrections are small for equilibrium systems at ordinary temperature. As a consequence, it is generally assumed that nuclear dynamics can be treated classically, with motions driven by Bom-Oppenheimer potential energy functions ... 

In chemical dynamics, one can distinguish two qualitatively different types of processes electron transfer and reactions involving bond rearrangement the latter involve heavy-particle (proton or heavier) motion in the formal reaction coordinate. The zero-order model for the electron transfer case is pre-organization of the nuclear coordinates (often predominantly the solvent nuclear coordinates) followed by pure electronic motion corresponding to a transition between diabatic electronic states. The zero-order model for the second type of process is transition state theory (or, preferably, variational transition state theory ) in the lowest adiabatic electronic state (i.e., on the lowest-energy Bom-Oppenheimer potential energy surface). 

Figure 4.1. Schematic representation of Bom-Oppenheimer potential energy surfaces. Using the photochemical nomenclature, the ground-state surface of a closed-shell system, which is the lowest singlet surface, is labeled Sg, followed by S, Sj, etc. in order of increasing energies. The triplet surfaces are similarly labeled T T], etc.

Figure 13. The (a) ground and (b) excited- electronic-state Bom-Oppenheimer potential surfaces. The n/2 pulse moves half of the initial amplitude, y(O), from a surface a to surface b. After the pulse the nuclear wavefunctions of the ground-state and excited-state surfaces are denoted x Ui) and r (ti (b) Wavepacket evolution of xb. Motion of the wavepacket causes the overlap of the ground-state wavefunction and the excited-state wavefunction to decay, resulting in free induction decay. x remains in place in coordinate space, (c) The n pulse exchanges the amplitude of surface a with surface b. (

Figure 15. Harmonic-excited-state Bom-Oppenheimer potential energy surface. The classical trajectory that originates at rest from the ground-state equilibrium geometry is shown superimposed. 

The result is that, to a very good approximation, as treated elsewhere in this Encyclopedia, the nuclei move in a mechanical potential created by the much more rapid motion of the electrons. The electron cloud itself is described by the quantum mechanical theory of electronic structure. Since the electronic and nuclear motion are approximately separable, the electron cloud can be described mathematically by the quantum mechanical theory of electronic structure, in a framework where the nuclei are fixed. The resulting Bom-Oppenheimer potential energy surface (PES) created by the electrons is the mechanical potential in which the nuclei move. When we speak of the internal motion of molecules, we therefore mean essentially the motion of the nuclei, which contain most of the mass, on the molecular potential energy surface, with the electron cloud rapidly adjusting to the relatively slow nuclear motion. 

What is left to understand about this reaction One key remaining issue is the possible role of other electronic surfaces. The discussion so far has assumed that the entire reaction takes place on a single Bom-Oppenheimer potential energy surface. However, three potential energy surfaces result from the interaction between an F... 

The Bom-Oppenheimer potentials and wave fhnctions, depending on are obtained by solving the problem... 

For the hydrogen molecule, one may currently get a very high accuracy in predicting rovibrational levels. For example, exact anal) tic formulas have been derived that allow one to compute the Bom-Oppenheimer potential with the... 

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