Vector potential point nuclear

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

As shown in Chapter 6, the solution of the Schrodinger equation in the adiabatic approximation can be divided into two tasks the problem of electronic motion in the field of the clamped nuclei (this will be the subject of the next chapters) and the problem of nuclear motion in the potential energy determined by the electronic energy. The ground-state electronic energy E (R) of eq. (6.8) (where k = 0 means the ground state) will be denoted in short as V(R), where R represents the vector of the nuclear positions. The function V R) has quite a complex structure and exhibits many basins of stable conformations (as well as many maxima and saddle points). 

As an example of the explicit expression we give the vector potential produced by the nuclear spin of a point nucleus... 

The final step in the MM analysis is based on the assumption that, with all force constants and potential functions correctly specified in terms of the electronic configuration of the molecule, the nuclear arrangement that minimizes the steric strain corresponds to the observable gas-phase molecular structure. The objective therefore is to minimize the intramolecular potential energy, or steric energy, as a function of the nuclear coordinates. The most popular procedure is by computerized Newton-Raphson minimization. It works on the basis that the vector V/ with elements dVt/dxn the first partial derivatives with respect to cartesian coordinates, vanishes at a minimum point, i.e. = 0. This condition implies zero net force on each atom... 

A molecule may be viewed as a number N of nuclear charges, Z e, of a certain arrangement given by their position vectors, Rt for i = 1... JV, surrounded by an electronic cloud of charge density p(r) of finite dimensions. The potential of the electrostatic field at the point R outside the electronic cloud is given by... 

As Lefebvre-Brion and Field [61] point out, the only coupling cases for which the electronic and nuclear motions can be separated are cases (a) and (c) consequently only in these cases can potential curves be defined unambiguously and accurately. However, as we have already pointed out, Hund s coupling cases are idealised descriptions and for most molecules the actual coupling corresponds to an intermediate situation. Moreover, the best description of the vector coupling often changes as the molecular rotation increases. In this section we consider the nature of the intermediate coupling schemes in more detail some of these will appear elsewhere in this book in connection with the observed spectra of specific molecules. 

A global property function is usually expressed as the expectation value of an operator or as the derivative of such an expectation value with respect to an internal or external parameter of the system. In the Born-Oppenheimer approximation, the electronic wave function depends parametrically upon the coordinates of the n nuclei, and therefore a set of the 3 -6 linearly independent nuclear coordinates constitutes the natural variables for such a choice of the potential function. However, the manifold M on which the gradient vector field is bound can be defined on a subset of 1R provided q < 3n-6, for example the intrinsic reaction coordinate (unstable manifold of a saddle point of index 1 of the Born-Oppenheimer energy hypersurface) or the reduced reaction coordinate. 

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