Studying Potential Energy Surfaces

Theoretical predictions of potential energy surfaces and reaction paths can sometimes yield quite surprising results. In this section, we ll consider an example which illustrates the general approach toward and usefulness of studying potential energy surfaces in detail. 

These results suggest another route for isomerization, involving three steps  

A second hydrogen migration from the methyl group back to the central carbon (IV, which is equivalent to T) 

Exploring Chemistry with Electronic Structure Methods 169 

Subsequent calculations at the MP2 level locate the two transition structures like those suggested. In this case, Intrinsic Reaction Coordinate (IRC) calculations were used to confirm that these transition structures do in fact connect the minima in question we ll look at this kind of calculation in detail later in this chapter. 

---

In order to study potential energy surfaces it is necessary to use theoretical approaches for the solution of the Schrodinger equation which give a balanced description for all states involved at various nuclear conformations, in particular for geometries in the neighborhood of the equilibrium as well as close to the dissociation limits. It is clear that only methods that account for correlation are at all acceptable for this purpose the SCF solution would lead to an unrealistic ordering of states and distorted surfaces. Presently, all such methods applicable to larger systems are based on expansion techniques. 

In this section we present results for the correlation energy and equilibrium geometry of H2O obtained with a near-HF one-electron basis (the 39-STO basis described in Chapter 4) via DCI, lEPA, L-CCA, and CCA. Meyer and his coworkers have studied potential energy surfaces, ionization potentials, dipole moments, and polarizabilities of a variety of molecules using CEPA. Some of their impressive results are contained in a review by Meyer. ... 

The above discussion represents a necessarily brief simnnary of the aspects of chemical reaction dynamics. The theoretical focus of tliis field is concerned with the development of accurate potential energy surfaces and the calculation of scattering dynamics on these surfaces. Experimentally, much effort has been devoted to developing complementary asymptotic techniques for product characterization and frequency- and time-resolved teclmiques to study transition-state spectroscopy and dynamics. It is instructive to see what can be accomplished with all of these capabilities. Of all the benclunark reactions mentioned in section A3.7.2. the reaction F + H2 —> HE + H represents the best example of how theory and experiment can converge to yield a fairly complete picture of the dynamics of a chemical reaction. Thus, the remainder of this chapter focuses on this reaction as a case study in reaction dynamics. 

In addition to the dependence of the intennolecular potential energy surface on monomer vibrational level, the red-shifting of the monomer absorption as a fiinction of the number of rare gas atoms in the cluster has been studied. The band origin for the Vppp = 1 -t— 0 vibration in a series of clusters Ar -HF, with 0 < n < 5, was measured and compared to the HF vibrational frequency in an Ar matrix (n = oo). The monomer vibrational frequency Vp p red shifts monotonically, but highly nonlinearly, towards the matrix value as sequential Ar atoms are added. Indeed, roughly 50% of the shift is already accounted for by n = 3. 

Tunable visible and ultraviolet lasers were available well before tunable infrared and far-infrared lasers. There are many complexes that contain monomers with visible and near-UV spectra. The earliest experiments to give detailed dynamical infonnation on complexes were in fact those of Smalley et al [22], who observed laser-induced fluorescence (LIF) spectra of He-l2 complexes. They excited the complex in the I2 B <—A band, and were able to produce excited-state complexes containing 5-state I2 in a wide range of vibrational states. From line w idths and dispersed fluorescence spectra, they were able to study the rates and pathways of dissociation. Such work was subsequently extended to many other systems, including the rare gas-Cl2 systems, and has given quite detailed infonnation on potential energy surfaces [231. 

The Ar-HCl and Ar-HF complexes became prototypes for the study of intennolecular forces. Holmgren et al [30] produced an empirical potential energy surface for Ar-HCl fitted to the microwave and radiofrequency spectra,... 

The stoi7 begins with studies of the molecular Jahn-Teller effect in the late 1950s [1-3]. The Jahn-Teller theorems themselves [4,5] are 20 years older and static Jahn-Teller distortions of elecbonically degenerate species were well known and understood. Geomebic phase is, however, a dynamic phenomenon, associated with nuclear motions in the vicinity of a so-called conical intersection between potential energy surfaces. 

Static properties of some molecules ([193,277-280]). More recently, pairs of ci s have been studied [281,282] in greater detail. These studies arose originally in connection with a ci between the l A and 2 A states found earlier in computed potential energy surfaces for C2H in symmetry [278]. Similar ci s appear between the potential surfaces of the two lowest excited states A2 and B2 iit H2S or of 82 and A in Al—H2 within C2v symmetry [283]. A further, closely spaced pair of ci s has also been found between the 3 A and 4 A states of the molecule C2H. Here the separation between the twins varies with the assumed C—C separation, and they can be brought into coincidence at some separation [282]. 

The full dynamical treatment of electrons and nuclei together in a laboratory system of coordinates is computationally intensive and difficult. However, the availability of multiprocessor computers and detailed attention to the development of efficient software, such as ENDyne, which can be maintained and debugged continually when new features are added, make END a viable alternative among methods for the study of molecular processes. Eurthemiore, when the application of END is compared to the total effort of accurate determination of relevant potential energy surfaces and nonadiabatic coupling terms, faithful analytical fitting and interpolation of the common pointwise representation of surfaces and coupling terms, and the solution of the coupled dynamical equations in a suitable internal coordinates, the computational effort of END is competitive. 

In this chapter, we look at the techniques known as direct, or on-the-fly, molecular dynamics and their application to non-adiabatic processes in photochemistry. In contrast to standard techniques that require a predefined potential energy surface (PES) over which the nuclei move, the PES is provided here by explicit evaluation of the electronic wave function for the states of interest. This makes the method very general and powerful, particularly for the study of polyatomic systems where the calculation of a multidimensional potential function is an impossible task. For a recent review of standard non-adiabatic dynamics methods using analytical PES functions see [1]. 

HCR and co-workers carried out a number of studies by employing 3D potential energy surfaces calculated by means of highly sophisticated ab initio approaches [88,91-101]. The results of these computations are in impressive agreement with the corresponding experimental findings. The discrepancies in the order of 100 wavenumbers, as in early ab initio studies [16,17], have been reduced in the HCR studies to only a few wavenumbers. In conclusion of their paper on the ( H ) system of NH2, Gabriel et al. state We believe... 

The full quantum mechanical study of nuclear dynamics in molecules has received considerable attention in recent years. An important example of such developments is the work carried out on the prototypical systems H3 [1-5] and its isotopic variant HD2 [5-8], Li3 [9-12], Na3 [13,14], and HO2 [15-18], In particular, for the alkali metal trimers, the possibility of a conical intersection between the two lowest doublet potential energy surfaces introduces a complication that makes their theoretical study fairly challenging. Thus, alkali metal trimers have recently emerged as ideal systems to study molecular vibronic dynamics, especially the so-called geometric phase (GP) effect [13,19,20] (often referred to as the molecular Aharonov-Bohm effect [19] or Berry s phase effect [21]) for further discussion on this topic see [22-25], and references cited therein. The same features also turn out to be present in the case of HO2, and their exact treatment assumes even further complexity [18],... 

---