Nuclear motion Schrodinger equation electronic states

Let us define x (R>.) as an n-dimensional nuclear motion column vector, whose components are Xi (R i) through X (R )- The n-electronic-state nuclear motion Schrodinger equation satisfied by (Rl) can be obtained by inserting Eqs. (12)... 

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 n-electronic-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 referred 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. 

This can be used to rewrite the diabatic nuclear motion Schrodinger equation for an incomplete set of n electronic states as... 

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... 

Nuclear motion Schrodinger equation direct molecular dynamics, 363-373 vibronic coupling, adiabatic effects, 382-384 electronic states ... 

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 ... 

In one quantum mechanical approach based on the diabatic approximation , the electron is assumed to be confined initially at one of the reactant sites and electron transfer is treated as a transition between the vibrational levels of the reactants to those of the products. The quantum mechanical treatment begins with the time dependent Schrodinger equation, Hip = -ihSiplSt, where the wavefunction tj/ is written as a sum of the initial (reactant) and final (product) states. In the limit that the Bom-Oppenheimer approximation for the separation of electronic and nuclear motion is valid, the time dependent Schrodinger equation eventually leads to the Golden Rule result in equation (25). 

The fundamental approximation used for describing the electron and nuclear motion in molecules and in condensed media is the well-known adiabatic approximation. Let us recall its essence. It is based upon the large difference in the masses of electrons and nuclei. Due to this difference the electron motion is fast in comparison with the nuclear motion, and thus electrons have time to adjust themselves to the nuclear motion and at every moment they can be in a state very close to the one they would be in if nuclei were immobile. Within this picture, as the first step in the construction of the complete wave function of the system, it proves useful to find wave functions describing electron motion with fixed positions of the nuclei, i.e. to resolve the Schrodinger equation... 

The Born-Oppenheimer approximation allows us to decouple the electronic and nuclear motions of the free molecule of the Hamiltonian Hq. Solving the Schrodinger equation //o l = with respect to the electron coordinates r = r[, O, gives rise to the electronic states (r, R) = (r n(R)), n = 0,..., Ne, of respective energies En (R) as functions of the nuclear coordinates R, with the electronic scalar product defined as (n(R) n (R))r = j dr rj( r. R) T,-(r, R). We assume Ne bound electronic states. The Floquet Hamiltonian of the molecule perturbed by a field (of frequency co, of amplitude 8, and of linear polarization e), in the dipole coupling approximation, and in a coordinate system of origin at the center of mass of the molecule can be written as... 

The Bom-Oppenheimer separation of the electronic and nuclear motions is a cornerstone in computational chemistry. Once the electronic Schrodinger equation has been solved for a large number of nuclear geometries (and possibly also for several electronic states), the potential energy surface (PES) is known. The motion of the nuclei on the PES can then be solved either classically (Newton) or by quantum (Schrodinger) methods. If there are N nuclei, the dimensionality of the PES is 3N, i.e. there are 3N nuclear coordinates that define the geometry. Of these coordinates, three describe the overall translation of the molecule, and three describe the overall rotation of the molecule with respect to three axes. For a linear molecule, only two coordinates are necessary for describing the rotation. This leaves 3N - 6(5) coordinates to describe the internal movement of the nuclei, which for small displacements may be chosen as vibrational normal coordinates . 

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