Diabatic electronic basis

Mead C A and Truhlar D G 1982 Conditions for the definition of a strictly diabatic electronic basis for molecular systems J. Chem. Rhys. 77 6090... 

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... 

W (Rj.) is an n X n diabatic first-derivative coupling matrix with elements defined using the diabatic electronic basis set as... 

In what follows we introduce the model Hamiltonian using both diabatic and adiabatic representations. Adopting diabatic electronic basis states /j ), the molecular model Hamiltonian can be written as [162, 163]... 

These elements are slowly varying function of nuclear coordinates and generally treated as constants in accordance with the applicability of the Condon approximation in a diabatic electronic basis [7,78]. The quantity C/(f) = ( I /(0) I /(0)> is the time autocorrelation function of the wave packet (WP) initially prepared on the /th electronic state and, I /(0 = I /(0). 

To analyze the vibronic structures of the X, A and B electronic states Ph we constructed a vibronic Hamiltonian in a diabatic electronic basis which treats the nuclear motion in the X state adiabatically, and includes the nonadiabatic coupling between the A and B electronic states. The Hamiltonian terms of the dimensionless normal coordinates of the electronic ground state (XMi) of phenide anion is given by [19]... 

In our theoretical formulation for PCET [26, 27], the electronic structure of the solute is described in the framework of a four-state valence bond (VB) model [41]. The most basic PCET reaction involving the transfer of one electron and one proton may be described in terms of the following four diabatic electronic basis... 

Therefore, before solving the coupled equations, it was necessary to transform to the diabatic electronic basis in order to eliminate this d/dR derivative. To date there has been only one example for bound states, the Hj three-particle system (Hunter and Pritchard, 1967), where the coupled equations were solved directly without such a transformation. 

Two coordinate systems and basis sets describing to the collision of two identical atoms in states S and P are employed. Following Wataiiabe [27], first we introduce the standard space-fixed coordinate system with axis along the relative nuclear angular momentum 1, i.e., axis perpendicular to the collision plane, axis (, in the direction opposite to the asymptotic velocity vector v, and axis ij normal to and (. The following notation is used for the diabatic electronic basis states ... 

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... 

Two classes of diabatic electronic basis functions are of interest here ... 

It is often cumbersome to solve the group-Born-Oppenheimer equation (21a) because of the terms V. Furthermore, these terms describe the coupling between electronic states in our g manifold via the momenta of the nuclei, and we commonly have more experience in understanding the impact of couplings via potentials than via momenta. It has, therefore, been popular and desirable, starting already many decades ago, to formulate the nuclear equations of motion in a so called diabatic electronic basis instead of the adiabatic one, which we have used above in Secs. 2... 

The most straightforward numerical technique for the solution of Eq. (1) is based on the expansion of the state vector (t)) in a complete set of time-independent basis functions. Such a complete basis can be constructed as the direct product of diabatic electronic basis states l n) and suitable orthonormal states xiyj) for each nuclear degree of freedom (see Chapter 7)... 

A very useful starting point for the study of non-adiabatic processes, which are common in photochemistry and photophysics, is the vibronic coupling model Hamiltonian. The model is based on a Taylor expansion of the potential surfaces in a diabatic electronic basis, and it is able to correctly describe the dominant feature resulting from vibronic coupling in polyatomic molecules a conical intersection. The importance of such intersections is that they provide efficient non-radiative pathways for electronic transitions. Not only is the position and shape of the intersection described by the model, but it also predicts which nuclear modes of motion are coupled to the electronic transition which takes place as the system evolves through the intersection. 

Adopting diabatic electronic basis states > ), the molecular model Hamiltonian can be written as d22... 

Figure 8.4 Alexander-Stark-Werner (ASW) diabatic potential matrix used in the quantum calculation for computing rate constants of the nonadiabatic F(P3/2> P1/2) + D2 reaction. Ucr > (2 = 0, 1, ff = 1/2) represents the six diabatic electronic basis fxmctions, A and B are the two spin-orbit coupling terms, Vj ,n,i,2 are the four diabatic potential surfaces.

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