Second quantization

Second quantization (SQ) concepts were introduced in chapter 2 as a general tool to treat excitations in molecular collisions for which the dynamics were described in cartesian coordinates. This SQ-formulation, which was derived from the TDGH representation of the wave function, could be introduced if the potential was expanded locally to second order around the position defined by a trajectory. It is, however, possible to use the SQ approach in a number of other dynamical situations, as for instance when dealing with the vibrational excitation of diatomic and polyatomic molecules, or with energy transfer to solids and chemical reactions in the socalled reaction path formulation. Since the formal expressions in the operators are the same, irrespective of the system or dynamical situation, the algebraic manipulations are also identical, and, hence, the formal solution the same. But the dynamical input to the scheme is of course different from case to case. In the second quantization formulation of the dynamical problems, one solves the operator algebraic equations formally. Once the formal solution is obtained, we can compute the dynamical quantities which enter the expressions. The advantage over state or grid expansion methods is significant since (at least for bosons) the number of dynamical operators is much less than the number of states. In order to solve the problem to infinite order, that is, also the TDSE for the system, the operators have to form a closed set with respect to commutations. This makes it necessary to drop some two-quantum operators. Historically, the M = 1 quantum problem, namely that of a linearly forced harmonic oscillator, was solved using the operator algebraic approach by Pechukas and Light in 1966 [131]. In 1972, Kelley [128] solved the two-oscillator (M = 2) problem and the author solved the M = 3 and the general problem in 1978 [129] and 1980 [147], respectively. The general case was solved using graph theory designed for the problem and it will not be repeated here. But the formulas are given in this chapter and in the appendices B and C.

Introduction

Molecular dynamics deals with the motion of and the reaction between atoms and molecules. The fundamental theory for the description of essentially all aspects of the area has been known and defined through the non-relativistic Schrdinger equation since 1926. The “only” problem, therefore, is the solution of this fundamental equation. Unfortunately, this solution is not straightforward and, as early as 1929, prompted the following remark by Dirac (1929). . . The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the application of these laws leads to equations much too complicated to be soluble. . . . Dirac could, for that matter, have added the area of molecular biochemistry. But here the systems become even bigger and therefore the above statement is even more correct. What neither Dirac nor anybody else at that time could foresee was the invention of the computer. With that, a whole new area, namely that of computational chemistry, was created. The recent five-volume work Encyclopedia of Computational Chemistry (1998[1]), with several hundred entries, bears witness to the tremendous evolution in this particular area over the last fifty years or so. The success of computational chemistry has to do not only with computers and the increase in computational speed but also with the development of new methods. Here again it should be emphasized that the availability of computers makes the construction of approximate methods a very rich and diverse field with many possibilities. Thus, this combination of computer power and the invention of theoretical and computational methods has changed the pessimistic point of view into an optimistic one. To quote Clementi (1972), “We can calculate everything.” Although this statement, at least in 1972, was somewhat optimistic, development since then has shown that the attitude should be quite optimistic. The purpose of approximate methods should be, and always is, to try to circumvent the bad scaling relations of quantum mechanics.

Approximate theories

Many calculations of rates or cross sections will have to be carried out using reliable approximate theories, simply because the number of systems and the number of detailed rate constants needed for interpreting experimental data or performing chemical kinetic simulations is too large for an attempt to produce these numbers completely from first principles. Aside from this, the information on potential energy surfaces is also a problem, which makes it necessary, in order to estimate whether the surfaces are reliable, to perform easy-to-do calculations before deciding whether more elaborate calculations can be justified using the surface at hand. However, this situation sets up some requirements for approximate theories, before they can be used for large-scale calculations: • They should “reproduce” as closely as possible exact benchmark calculations on simpler systems or in lower dimensions. • They should be extendable to larger systems and dimensions. • They should be able to treat inelastic, as well as reactive, processes. • They should be extendable to multi-surface problems. • They should be easier to do than the exact quantum calculations. • They should be able to take advantage of an increase in computer speed. A number of benchmark calculations, often performed with model potential surfaces, are now available. The first of such, and still popular for comparison, are the calculations on colinear inelastic collisions performed in 1966 by Secrest and Johnson [73]. Since then, calculations on colinear reactive processes for A+BC [74] and inelastic AB+CD systems appeared [76]. By the late eighties, calculations on atom-diatom reactions in full dimensionality (4 dimensions) have been possible, and benchmark calculations on systems as D+H2 [100] and F+H2 [77] are available. The “exact” calculations are performed by expanding the wave function in basis sets consisting of eigenfunctions to part of the hamiltonian. These “channels” are coupled by the remaining part of the hamiltonian, and the calculations are, therefore, often referred to as coupled channel (CC) calculations. The approximate methods are developed to treat the many degrees of freedom in larger systems.

Conclusion

In this book, we have discussed the problems concerning mixing of classical and quantum mechanics, and we have given several possible solutions to the problem and a number of suggestions for the setup of working computational schemes. In the present chapter, we give some recommendations as to which methods one should use for a given type of system and problem. As can be seen from the tables and what is apparent from the discussion in the previous chapters, the quantum-classical method has been and is used for solving many different molecular dynamics problems. Recommendations, as far as molecule surface or processes in solution are concerned, have not been incorporated, the reason being that the methods here are still to some extent under development. We have seen that the quantum-classical approach can be derived in two different fashions. In one method the classical limit ħ→ 0 is taken in some degrees of freedom. In the other approach the quantum mechanical equations are parameterized in such a fashion that classical equations of motions are either pulled out of or injected into the quantum mechanical. Thus the first method involves and introduces the classical picture in certain particular degrees of freedom—in the second method the classical picture is in principle not introduced—it is just a reformulation of quantum mechanics. This reformulation has the exact dynamics as the limit. However, if exact calculations are to be performed, the reformulation may not be advantageous from a computational point of view, and, hence, standard methods are often more conveniently applied. We prefer the second approach for introducing the quantum-classical scheme because, as mentioned, it automatically has the exact formulation as the limit. The approach is most conveniently implemented through the trajectory driven DVR, or the so-called TDGH-DVR method, which gives the systematic way of approaching the quantum mechanical limit from the classical one. Thus, the method interpolates continuously between the classical and the quantum limit—a property it shares with, for instance, the FMS method and the Bohm formulation.

More complex systems

By more complex systems we mean systems containing on the order of hundreds or thousands of atoms, or molecules with less atoms but with “complicated” motions, the latter being the case when considering collisions between polyatomic molecules. In the present chapter we deal with quantum-classical methods for treating energy transfer in collisions involving polyatomic molecules, molecule surface scattering, reactions in polyatomic systems and solution. We will assume that it is possible to construct realistical potential energy surfaces for the systems. Obviously, these surfaces will be of empirical or semi-empirical nature. In some of the methods, as for instance the reaction path method, one tries to minimize the information needed on potential energy surfaces. Chemical reactions and energy transfer processes in the gas phase are often studied using just a single adiabatic Born-Oppenheimer potential energy surface. However non-adiabatic effects, that is, coupling between different electronic states, is an important aspect in chemistry. If the coupling between the various electronic states can be neglected, the “electronic” effect reduces to that of a statistical degeneracy factor ge [180].

Rigorous theories

In this chapter we discuss theories which are rigorous in their formulation but which in order to be useful need to be modified by introducing approximations of some kind. The approximations we are interested in are those which involve introduction of classical mechanical concepts, that is, the classical picture and/or classical mechanical equations of motion in part of the system. At this point, we wish to distinguish between “the classical picture,” which is obtained by taking the classical limit ħ → 0 and the appearance of “classical equations of motion.” The latter may be extracted from the quantum mechanical formulation without taking the classical limit—but, as we shall see later by introducing a certain parametrization of quantum mechanics. Thus there are two ways of introducing classical mechanical concepts in quantum mechanics. In the first method, the classical limit is defined by taking the limit ħ → 0 either in all degrees of freedom (complete classical limit) or in some degrees of freedom (semi-classical theories). We note in passing that the word semi-classical has been used to cover a wide variety of approaches which have also been referred to as classical S-matrix theories, quantum-classical theories, classical path theory, hemi-quantal theory, Wentzel Kramer-Brillouin (WKB) theories, and so on. It is not the purpose of this book to define precisely what is behind these various acronyms. We shall rather focus on methods which we think have been successful as far as practical applications are concerned and discuss the approximations and philosophy behind these. In the other approach, the ħ-limit is not taken—at least not explicitly— but here one introduces “classical” quantities, such as, trajectories and momenta as parameters, and derives equations of motion for these parameters. The latter method is therefore one particular way of parameterizing quantum mechanics. We discuss both of these approaches in this chapter. The Feynman path-integral formulation is one way of formulating quantum mechanics such that the classical limit is immediately visible [3]. Formally, the approach involves the introduction of a quantity S, which has a definition resembling that of an action integral [101].