Adiabaticity, electronic

The simplest approach to simulating non-adiabatic dynamics is by surface hopping [175. 176]. In its simplest fomi, the approach is as follows. One carries out classical simulations of the nuclear motion on a specific adiabatic electronic state (ground or excited) and at any given instant checks whether the diabatic potential associated with that electronic state is mtersectmg the diabatic potential on another electronic state. If it is, then a decision is made as to whedier a jump to the other adiabatic electronic state should be perfomied. 

Finally, in brief, we demonstrate the influence of the upper adiabatic electronic state(s) on the ground state due to the presence of a Cl between two or more than two adiabatic potential energy surfaces. Considering the HLH phase, we present the extended BO equations for a quasi-JT model and for an A -1- B2 type reactive system, that is, the geometric phase (GP) effect has been inhoduced either by including a vector potential in the system Hamiltonian or... 

Several years ago Baer proposed the use of a mahix A, that hansforms the adiabatic electronic set to a diabatic one [72], (For a special twofold set this was discussed in [286,287].) Computations performed with the diabatic set are much simpler than those with the adiabatic set. Subject to certain conditions, A is the solution of a set of first order partial diffei ential equations. A is unitary and has the form of a path-ordered phase factor, in which the phase can be formally written as... 

Here, the integrand is the off-diagonal gradient mahix element between adiabatic electronic states,... 

V is the derivative with respect to R.) We stress that in this formalism, I and J denote the complete adiabatic electronic state, and not a component thereof. Both /) and y) contain the nuclear coordinates, designated by R, as parameters. The above line integral was used and elaborated in calculations of nuclear dynamics on potential surfaces by several authors [273,283,288-301]. (For an extended discussion of this and related matters the reviews of Sidis [48] and Pacher et al. [49] are especially infonnative.)... 

The effective potential matrix for nuclear motion, which is a diagonal matrix for the adiabatic electronic set, is given by... 

Under circumstances that this condition holds an ADT matrix, A exists such that the adiabatic electronic set can be transformed to a diabatic one. Working with this diabatic set, at least in some part of the nuclear coordinate space, was the objective aimed at in [72]. 

Now, we recall the remarkable result of [72] that if the adiabatic electronic set in Eq. (90) is complete (N = oo), then the curl condition is satisfied and the YM field is zero, except at points of singularity of the vector potential. (An algebraic proof can be found in Appendix 1 in [72]. An alternative derivation, as well as an extension, is given below.) Suppose now that we have a (pure) gauge g(R), that satisfies the following two conditions ... 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

U(qJ is referred to as an adiabatic-to-diabatic transformation (ADT) matrix. Its mathematical sbucture is discussed in detail in Section in.C. If the electronic wave functions in the adiabatic and diabatic representations are chosen to be real, as is normally the case, U(q ) is orthogonal and therefore has n n — l)/2 independent elements (or degrees of freedom). This transformation mabix U(qO can be chosen so as to yield a diabatic electronic basis set with desired properties, which can then be used to derive the diabatic nuclear motion Schrodinger equation. By using Eqs. (27) and (28) and the orthonormality of the diabatic and adiabatic electronic basis sets, we can relate the adiabatic and diabatic nuclear wave functions through the same n-dimensional unitary transformation matrix U(qx) according to... 

This choice of elements for the u, (qx,) matrix will diabatize the adiabatic electronic states i andj while leaving the remaining states unaltered. 

By using the fact that for a finite number of adiabatic electronic states n, we choose a U(qx) that satisfies Eq. (47) [rather than Eq. (42) that has no solution], Eq. (35) now reduces to... 

As discussed in Section II.A, the adiabatic electronic wave functions and depend on the nuclear coordinates Rx only through the subset... 

In the two-adiabatic-electronic-state Bom-Huang description of the total orbital wave function, we wish to solve the corresponding nuclear motion Schrodinger equation in the diabatic representation... 

Both terms on the right are related to the rate of change of the adiabatic electronic functions with respect to the nuclear coordinates. The first term Gy is given by... 

The matrix of vectors F is thus the defining quantity, and is called the non-adiabatic coupling matrix. It gives the strength (and direction) of the coupling between the nuclear functions associated with the adiabatic electronic states. 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

The remaining combinations vanish for symmetry reasons [the operator transforms according to B (A") hreducible representation]. The nonvanishing of the off-diagonal matrix element fl+ is responsible for the coupling of the adiabatic electronic states. 

The basis consisting of the adiabatic electronic functions (we shall call it bent basis ) has a seiious drawback It leads to appearance of the off-diagonal elements that tend to infinity when the molecule reaches linear geometry (i.e., p 0). Thus it is convenient to introduce new electronic basis functions by the transformation... 

Now, we discuss briefly the situation when one or both of the adiabatic electronic states has/have nonlinear equilibrium geometry. In Figures 6 and 7 we show two characteristic examples, the state of BH2 and NH2, respectively. The BH2 potential curves are the result of ab initio calculations of the present authors [33,34], and those for NH2 are taken from [25]. 

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