Schrodinger equation adiabatic 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... 

Schrodinger amplitude relation to Klein-Gordon amplitude, 500 Schrodinger equation, 439 adiabatic solutions, 414 as a unitary transformation, 481 for relativistic spin % particle, 538 for the component a, 410 in Fock representation, 459 in the q representation, 492 Schrodinger form of one-photon equation, 548... 

Discrete Fourier transform (DFT), non-adiabatic coupling, Longuet-Higgins phase-based treatment, two-dimensional two-surface system, scattering calculation, 153-155 Discrete variable representation (DVR) direct molecular dynamics, nuclear motion Schrodinger equation, 364-373 non-adiabatic coupling, quantum dressed classical mechanics, 177-183 formulation, 181-183... 

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)... 

The surface-hopping trajectories obtained in the adiabatic representation of the QCLE contain nonadiabatic transitions between potential surfaces including both single adiabatic potential surfaces and the mean of two adiabatic surfaces. This picture is qualitatively different from surface-hopping schemes [2,56] which make the ansatz that classical coordinates follow some trajectory, R(t), while the quantum subsystem wave function, expanded in the adiabatic basis, is evolved according to the time dependent Schrodinger equation. The potential surfaces that the classical trajectories evolve along correspond to one of the adiabatic surfaces used in the expansion of the subsystem wavefunction, while the subsystem evolution is carried out coherently and may develop into linear combinations of these states. In such schemes, the environment does not experience the force associated with the true quantum state of the subsystem and decoherence by the environment is not automatically taken into account. Nonetheless, these methods have provided com-... 

According to Equation (2.29), in the adiabatic representation (index a) one expands the total molecular wavefunction F(R, r, q) in terms of the Born-Oppenheimer states Ej (q R, r) which solve the electronic Schrodinger equation (2.30) for fixed nuclear configuration (R,r). In this representation, the electronic Hamiltonian is diagonal,... 

The END theory was proposed in 1988 [11] as a general approach to deal with time-dependent non-adiabatic processes in quantum chemistry. We have applied the END method to the study of time-dependent processes in energy loss [12-16]. The END method takes advantage of a coherent state representation of the molecular wave function. A quantum mechanical Lagrangian formulation is employed to approximate the Schrodinger equation, via the time-dependent variational principle, by a set of coupled first-order differential equations in time to describe the END. 

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 ... 

Thus, we are forced to stick to the adiabatic representation, which raises other problems. As the complete nuclear Schrodinger equation is solved for both coupled states, all quantum effects like interferences or phase effects are included (see Sec. 7), but one needs to keep track of the phases of the electronic wavehmctions while computing the nonadiabatic coupling elements (NAC). Additionally, we are faced with the strong localization of the NACs, which requires many grid points for the wavepacket propagation and makes the calculations quite time consuming. 

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