Adiabatic electronic basis

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... 

First, we consider the situation where the coherence time, i.e., the time during which Cp(t) is substantially different from zero, is so short (sub-fs) that we can ignore all of the dynamics in the material system during this time. This is a relevant limit for X-rays from incoherent synchrotron radiation and laser plasma sources [4]. The time-dependent material system is created by the pump UV-laser pulse and it, therefore, contains amplitude on the electronic ground state as well as on one (or several) excited electronic state(s). Adopting a density-operator description, we expand the state of the material system in the adiabatic electronic basis,... 

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 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 ... 

The expression (18) has been useful in assessing the relevance of the derivative couplings in the adiabatic electronic basis. What to expect for the residual couplings By differentiating both sides of the identity... 

The two-state flux operator described below is derived from Eq. (10) in a diabatic as well as in an adiabatic electronic basis. We note that the flux operator is Hermitian and its other properties are well documented in the literature. We note that the flux operator F should not be confused with the derivative coupling elements Fij introduced in Eq. (2). 

In order to calculate the reaction probability in the two-state adiabatic electronic basis, the flux operator needs to be represented in this basis. As the quantity in the flux operator depends only on the reaction coordinate... 

Since is non-diagonal, the flux operator will have the same property in the adiabatic electronic basis. On substituting in Eq. (10) one arrives at the same expression as in Eq. (13) for the diagonal elements (/n and 722) of the flux operator in the adiabatic basis, and its off-diagonal elements take the following form7°... 

Defining the energy normalized time-independent reactive scattering wave function in the adiabatic electronic basis as... 

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