Schrodinger equation diabatic representation

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... 

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... 

By substituting the expression for the matrix elements in Eq. (B.21), we get the final form of the Schrodinger equation within the diabatic representation... 

In a diabatic representation, the electronic wave functions are no longer eigenfunctions of the electronic Hamiltonian. The aim is instead that the functions are so chosen that the (nonlocal) non-adiabatic coupling operator matrix, A in Eq. (52), vanishes, and the couplings are represented by (local) potential operators. The nuclear Schrodinger equation is then written... 

In complete analogy to the diabatic case, the equations of motion in the adiabatic representation are then obtained by inserting the ansatz (29) into the time-dependent Schrodinger equation for the adiabatic Hamiltonian (7)... 

For the two-state case with real electronic wave functions, the nuclear motion Schrodinger equations are given by (74) and (106) for the adiabatic and diabatic representations, respectively. For this case, all the matrices in those equations have dimensions 2X2 and the xad(R) and xd(R) vectors have dimensions 2X1, whereas those appearing in W(1)ad and W(1)d have the dimensions of R, namely, 3(N — 1) X 1 where N is the number of nuclei in the system. Equation (69) furnishes a more explicit version of (74) and the A(q) appearing in (106) is given by (107) with (3(q) obtained from (115). These versions of (74) and (106) are rigorously equivalent, once the appropriate boundary conditions for xad(R) and xd(R) discussed in Secs. III.B.l and III.B.2 are taken into account. The main differences between and characteristics of those equations are the following ... 

Zener employed the time-dependent Schrodinger equation in the diabatic representation (14) ... 

We now discuss the necessary details of the computational aspects to obtain the reaction probability by the scheme developed above. In the following we proceed with the coupled-surface calculations the uncoupled-surface calculations follow from them in an elementary way. The time-dependent Schrodinger equation (TDSE) is solved numerically in the diabatic electronic representation on a grid in the (i , r, 7) space using the matrix Hamiltonian in Eq. (11). For an explicitly time-independent Hamiltonian the solution reads... 

Then, employing the dipole form of the external field interaction [Eq. (3.43)], the Schrodinger equation for the nuclear wavefunctions in the diabatic representation is given in the matrix form. 

The coupled time-dependent Schrodinger equation for the vibrational wavepackets i R,t) and i 2 R,t) on the electronic states 1 and 2, respectively, in a diabatic representation is given as... 

The following part of this section describes our recent advances in applying accurate quantum wave packet methods to compute rate constants and to understand nonadiabatic effects in tri-atomic and tetra-atomic molecular reactions. The quantum nonadiabatic approaches that we present here are based on solving the time-dependent Schrodinger equation formulated within an electronically diabatic representation. 

Alternatively, a diabatic representation can be used. " In this representation, the electronic wavefunctions used to expand the total wavefunction are not the eigenfunctions of the electronic Hamiltonian, but they are chosen so as to eliminate the derivative coupling. Therefore, the coupling terms do not appear in the Schrodinger equation, but the matrix element Hij = ([Pg.87]

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