Bom-Oppenheimer potential energy

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... 

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. 

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 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... 

FIGURE 2.3 Stationary points on the Bom-Oppenheimer potential energy surface and on the Gibbs free energy surface (gray). Note the difference between transition structure and transition state. 

Due to their direct relation to the spectral overlap integral, see Eq. (9), the emission and absorption spectra of the dye molecules are of interest in the context of EET processes. The simplest way to model excitation spectra employs the calculation of vertical energy separations, i.e., the separation of the Bom-Oppenheimer potential energy surfaces of the initial state and the final state at the equilibrium structure of the initial state. This energy separation is expected to coincide with the absorption maximum, as rationalized by the Franck-Condon principle (see for example [135]). This assumption is not always appropriate, rylene dyes being a prominent example. These dyes feature a strong 0-0 transition and a pronounced vibronic progression that is even visible in solution at room temperature (see for example [137]). A detailed simulation of the vibrational substructure of the absorption and emission bands is necessary to understand the details of the spectram. 

We will now study the more useful case when the nuclei are moving by classical mechanics with a Bom-Oppenheimer potential energy surface (PES). The points passed by the nuclei are called the trajectory and denoted Q(t). Q(t) is the spatial coordinates for the heavy particles (nuclei) in the system ... 

For the case of molecules, the concept of resonance can be also related to the existence of conical intersections in the Bom-Oppenheimer potential energy surfaces. However, in order to grasp the relationship among resonance, conical intersections and chemical structure, it is essential to state the differences between adiabatic and diabatic representations of molecular and atomic wavefiinc-tions... 

VmepC ) is the Bom-Oppenheimer potential energy along the MEP. The standard-state free energy of reactants CR (7) is computed by well known formulas, and C (T,j) is computed similarly, but excluding contributions from the reaction coordinate s. 

The most obvious way to specify a potential energy surface is as an analytic function of the coordinates. Typically such a function is obtained by semiempirical valence bond theory and/or a valence force field with bond breaking terms or by a fit to ob initio electronic structure calculations of the Bom-Oppenheimer potential energy surface as a function of intemuclear distances. This kind of analytic potential has been widely employed for systems with three atoms and less widely so for systems with 4-12 atoms [37]. Some recent examples include surfaces for the reactions H + CH4 [38], Q (H20)2 + CHCl [7a,39], F + H2 [40], and H + HBr [41]. 

Static (PES) and kinetic (RRKM) information can be complemented by chemical dynamics simulations, which are able to fill some aspects of gas phase reactivity not considered by the previous approaches. In particular, chemical dynamics can be used to model explicitly the collision between the ion and the target atom and thus it is possible to obtain the energy transferred in the collision and (eventually) the reactions. The molecular system, represented as an ensemble of atoms each bearing a mass i evolves on the Bom-Oppenheimer potential energy surface through Newton s equation of motirais ... 

The first term accounts for bond stretching and compression (the term here is quadratic but could be a Morse function), the second accounts for angle bends, the third is a truncated Fourier expansion for torsions involving four sequentially connected atoms, the fourth term accounts for space exclusion and dispersion attraction and the final term approximately accounts for Coulomb interactions. A force field of this kind is a function of the positions of the atoms. It serves as an approximation to the quantum mechanical Bom-Oppenheimer potential energy surface (PES). It is parameterized by a combination of spectroscopic and calorimetric measurements, and by accurate quantum mechanical calculations. 

The motion of the nuclei is often determined by a single Bom-Oppenheimer potential energy surface. This situation was implicitly assumed when discussing the above formulation. However, applications such as photodissociation involve electronically excited states, and there one must frequently account for the nonadiabatic coupling to another, or even to several other, electronic states. 

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