Separation of Nuclear Motions

We will first consider in detail the collision dynamics of adiabatic processes The nonadiabatic transitions in molecular collisions will be treated later in a more concise form  

The collisions between atoms and molecules may be reactive or non-reactive , depending on whether or not, as a result of the collision, new atoms or molecules are formed The non-reactive collisions are elastic if after the collision the internal states of the colliding particles remain unchanged or inelastic when the internal states change  

The concept of a potential energy surface, arising from the adiabatic approximation, is the basis of both the classical and quantum-mechanical treatment of the dynamics of elastic, inelastic and reactive collisions. The adiabatic potential energy V(x) governs the internal motions of atoms in an isolated system and determines the solutions of the nuclear wave equation (6.1) However, the results of a collision process will be entirely determined by the interaction potential V(x) only if the translation and rotation motion of the overall system do not influence its internal motions  

The translation motion of the whole system of interacting particles can be described by the motion of its center-of-mass in respect to a body-fixed coordinate system This will be a free (inertial) motion with a constant velocity as far as the collision complex can be considered as an isolated system Such is approximately the situation during a collision in a dilute gas where, because of the large intermolecular distances, the interactions of the collision complex with the other molecules may be neglected. As is known from classical mechanics, the free center-of-mass motion can be completely separated from the internal motions, which can then be described in a coordinate system having its origin in the center-of-mass. In quantum mechanics a similar separation is possible by a product representation of the wave function 

The above dynamic separation of motions reduces to the equations 

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If there are large-amplitude nuclear motions, the MD problem should be considered for the system as a whole, and only the potential energy for fixed nuclei (the APE) is calculated by means of QM/MM separation. If an adiabatic separation of nuclear motions in the two regions is possible, the dynamic problem may be considered in the QM region only, the MM region influence being included in the APE as described in Section II.4. 

The separation of nuclear and electronic motion may be accomplished by expanding the total wave function in functions of the election coordinates, r, parametrically dependent on the nuclear coordinates... 

Assumption of a rigorous separation of nuclear and electronic motions (Bom-Oppenheimer approximation). In most cases this is a quite good approximation, and there is a good understanding of when it will fail. There are, however, very few general techniques for going beyond the Bom-Oppenheimer approximation. 

As a point of departure we assume, within a conventional separation of nuclear and electronic motions, an effective Hamiltonian for the motion of two atomic nuclei and their associated electrons both along and perpendicular to the internuclear vector, directly applicable to a molecule of symmetry class for which magnetic effects are absent or negligible [25] ... 

Figure 8. The molecular structure of the [Cr2(CO)io(M2-tf)] anion for the bis(triphenylphosphine)-iminium salt showing (a) a view normal to the Cr-Cr axis (b) a view looking down the Cr-Cr axis. The Cr-Cr intemuclear separation is 3.349(13) A. The thermal ellipsoids of nuclear motion for all atoms are scaled to enclosed 50% probability.

We have already seen in Sec. 3.1 that before the Born-Oppenheimer separation of nuclear and electronic motion is made, the Coulomb Hamiltonian has very high symmetry, but that the clamped-nucleus Hamiltonian has only the spatial symmetry of the nuclear framework. That is, the Hamiltonian... 

In molecular orbital (MO) theory, which is the most common implementation of QM used by chemists, electrons are distributed around the atomic nuclei until they reach a so-called self-consistent field (SCF), that is, until the attractive and repulsive forces between all the particles (electrons and nuclei) are in a steady state, and the energy is at a minimum. An SCF calculation yields the electronic wave function 4C (the electronic motion being separable from nuclear motion thanks to the Born-Oppenheimer approximation). This is the type of wave function usually referred to in the literature and in the rest of this chapter. 

Equation (1-13) or its body-fixed equivalent is of little use for Van der Waals complexes, as it discriminates one nuclear coordinate, e.g. y = 1. Specific mathematical forms of Hamiltonians describing the nuclear motions in Van der Waals dimers have been developed (7). This point will be discussed in more details in Section 12.4. Here we only want to stress that whatever the mathematical form of the Hamiltonian is used to solve the problem of nuclear motions, the results will be the same, if the Schrodinger equation is solved exactly. However, in weakly bound complexes there is a hierarchy of motions due to the strong intramolecular forces which determine the internal vibrations of the molecules, and to much weaker intermolecular forces which determine their relative translations and rotations. This hierarchy allows to make a separation between the intramolecular vibrations with high frequencies and the intermolecular modes with much lower frequencies. Such a separation of the fast intramolecular vibrations and slow rotation-vibration-tunneling motions can be performed if a suitable form of the Hamiltonian for the nuclear motions in Van der Waals molecules is used. 

We wish to divide XT into a part describing the nuclear motion and a part describing the electronic motion in a fixed nuclear configuration, as far as possible. Equations (2.36) and (2.37) do not themselves represent such a separation because 3 is still a function of R,

partial differential operators with respect to these coordinates. The obvious way to remove the effects of nuclear motion from. >iel is by transforming from space-fixed axes to molecule-fixed axes gyrating with the nuclei. 

We could return to the exact equation (2.131) and examine the matrix elements of the Cn, n terms in the same manner as we dealt with the diagonal f,, terms. It is, however, easier to turn to the form of the exact Hamiltonian given in equations (2.113) and (2.120). The terms in this Hamiltonian which cause a breakdown of the adiabatic separation of nuclear and electronic motion are... 

In chapter 2 we discussed at length the separation of nuclear and electronic coordinates in the solution of the Schrodinger equation. We described the Born-Oppenheimer approximation which allows us to solve the Schrodinger equation for the motion of the electrons in the electrostatic field produced by fixed nuclear charges. There are certain situations, particularly with polyatomic molecules, when the separation of nuclear and electronic motions cannot be made satisfactorily, but with most diatomic molecules the Born-Oppenheimer separation is acceptable. The discussion of molecular electronic wave functions presented in this chapter is therefore based upon the Born-Oppenheimer approximation. 

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