Adiabatic electronic basis sets

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... 

The coupled equations have also been solved using the adiabatic electronic basis set for the H2 E+ states (Glass-Maujean, et al, 1983). In this basis set, the d/dR derivative, acting on the unknown md vibrational functions, appears in the second term on the right-hand side of the coupled equations, namely,... 

The total orbital wave function for this system is given by an electronically adiabatic n-state Bom-Huang expansion [2,3] in terms of this electronic basis set as... 

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. 

At this stage, we would like to emphasize that the same transformation has to be applied to the electronic adiabatic basis set in order not to affect the total wave function of both the elecbons and the nuclei. Thus if is the electronic basis set that is attached to 4> then and are related to each other as... 

Just like the adiabatic framework, expressed in terms of equation (4), the diabatic one which avoids the NACTs altogether follows from the same basic observation, namely, that molecular systems contain fast and slow moving particles. However, instead of applying an electronic basis set which varies with the nuclear coordinates one may employ a single basis set as calculated at one point in configuration space. Following this idea it can be shown that the relevant SE is [19] ... 

It is important to mention that the relation between the phases of the D-matrix and Berry s phase of each of the electronic adiabatic states is discussed by R. Baer [23]. In particular, for a real electronic basis set where the two phases are shown to be identical. 

In addition to the electronically adiabatic representation described by (4) and (5) or, equivalently (57) and (58), other representations can be defined in which the adiabatic electronic wave function basis set used in expansions (4) or (58) is replaced by some other set of functions of the electronic coordinates rel or r. Let us in what follows assume that we have separated the motion of the center of mass G of the system and adopted the Jacobi mass-scaled vectors R and r defined after (52), and in terms of which the adiabatic electronic wave functions are i] l,ad(r q) and the corresponding nuclear wave function coefficients are Xnd (R). The symbol q(R) refers to the set of scalar nuclear position coordinates defined after (56). Let iKil d(r q) label that alternate electronic basis set, which is allowed to be parametrically dependent on q, and for which we will use the designation diabatic. We now proceed to define such a set. LetXn(R) be the nuclear wave function coefficients associated with those diabatic electronic wave functions. As a result, we may rewrite (58) as... 

There is, however, a serious shortcoming associated with such a q-independent electronic basis set l d(r). If we consider, for example, a two-state adiabatic expansion involving only iK ad(r q) and ijjjl ad(r q), the corresponding diabatic expansion in thei]i, -d(r q) must contain a sufficiently large number of terms to represent those two adiabatic electronic wave functions well for all values of the q sampled by xad(R)- This is, in general, an unacceptably large number. In addition, it can be shown [25] that in general no other choice of 1,d(r q) makes W(1)d(R) vanish for all R. 

An alternative approach to the selection of A(q) is to consider an electronically adiabatic expansion truncated at a small number X of terms and require A(q) to be an (X X >T)-dimensional matrix. In this case, neither the adiabatic nor the diabatic electronic basis set is complete, but we assume that the adiabatic expansion in X terms is sufficiently accurate for our purposes. We now wish to select this A(q) so as to minimize the effect of the term in (75) containing W(1)ad(R). Ideally, we would like to force this matrix to vanish identically. Unfortunately, this is not always possible, as we shall now show. 

In this expression, W 1)ad(R) (k = i, j) is the X X N matrix whose row n and column n element is the k element of the W(n1, 1ad(R) vector, i.e., [W ad(R)], and the brackets in its right-hand side denote the commutator of the two matrices within. When n and n are allowed to span the complete infinite set of adiabatic electronic quantum numbers, condition (102) is satisfied [24,26], (99) has a solution, and the resulting A(q) leads to the q-independent diabatic electronic basis set mentioned in connection with (83). For the small values of X case being considered here, (102) is in general not satisfied and (98) does not have a solution. On the other hand, the equation obtained by replacing in (99) W(1)ad(R) by its longitudinal part VRd><1)ad(q) [see remark after (98)], namely... 

The gauge invariance of the group-Born-Oppenheimer approximation provides a good starting point to discuss diabatic states. In contrast to Eq. (21a), where this approximation is formulated in the adiabatic electronic basis, Eq. (26) is expressed in an arbitrary basis. Elimination of the derivative couplings appearing in the latter equation amounts to setting to zero the left hand side of Eq. (27b) ... 

Note that is the vector matrix of derivative couplings in the adiabatic electronic basis and the gauge transformation (R) is the unitary transformation matrix connecting the adiabatic and diabatic basis sets. In the above example of two real electronic states, Eq. (32) is identical to Eq. (30a) where x is set to zero ... 

This is often referred to as the Born-Huang expansion [58]. We assume that the electronic basis set [ /), either adiabatic or diabatic, are orthonormal at each nuclear configuration as... 

The nuclear derivative couplings emerge from a connection matrix for the orthogonal electronic basis states adiabatic parameter vector R. Suppose that R depends smoothly on time t and the electron basis set is determined at each point in R-space, namely, R(t). The wavefunction... 

If more than one electronic state is involved, then the electronic wave function is free to contain components from all states. For example, for non-adiabatic systems the elecbonic wave function can be expanded in the adiabatic basis set at the nuclear geometry R t)... 

In the lower electronic manifolds, we can use the B-O adiabatic approximation as the basis set, that is,... 

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