Schrodinger equation for the total wave function

The total Hamiltonian with the electronic coordinates expressed in the molecule-fixed axis system is given by (2.120). If the total wave function for electronic and nuclear motion is written as Prvc (, st, R, p, 6) the Schrodinger equation can be expressed as 

In order to develop this equation we assume that RIve can be expanded as a complete set of electronic functions multiplied by nuclear functions 

The electronic functions ffir, R) are also eigenfunctions of the operator P-, since P-commutes with SQ, 

This exact form of the molecular wavefunction PnK was first introduced by Born and Huang [9]. The equation to be solved, equation (2.125), may therefore be written 

Multiplication on the left by the complex conjugate function // and expansion yields 

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We consider a closed-shell dimer AB with NA electrons and nA nuclei that can be assigned to the monomer A, and NB electrons and nB nuclei that can be assigned to the monomer B. The set of electronic coordinates r , = l,... NA + NB will be denoted by r, while the nuclear coordinates, Ry, y = 1,. ..nA + nB, will be denoted in short by R. The coordinates of electrons and nuclei are defined in a space-fixed frame. The nuclear masses will be denoted by My, and atomic units me = e = h= 1 will be used throughout this chapter. The Schrodinger equation for the total wave function Tf,ot can be written as,... 

The approximation involved in factorization of the total wave function of a molecule into electronic, vibrational and rotational parts is known as the Bom-Oppenheimer approximation. Furthermore, the Schrodinger equation for the vibrational wave function (which is the only part considered here), transformed to the normal coordinates Qi (which are linear functions of the "infinitesimal displacements q yields equations of the harmonic oscillator t5q>e. For these reasons Lifson and Warshel have stressed that the force-field calculations should not be considered as classical-me-... 

If we substitute the right-hand side of (10.164) into the total Schrodinger equation (10.162) and project onto a single case (e) basis function ir [sj R > obtain a set of coupled differential equations for the channel wave functions Xa isJaR seen above, for each J and e there are three coupled equations corresponding to the allowed values of J and R. These equations may be written in matrix notation. 

The total classical energy E = H. The Schrodinger equation for the wave function. .. rn,t) which describes the dynamical state of the system is obtained by defining E and p as the differential operators... 

Since the exact solution of Schrodinger s equation for multi-electron, multi-nucleus systems turned out to be impossible, efforts have been directed towards the determination of approximate solutions. Most modern approaches rely on the implementation of the Born-Oppenheimer (BO) approximation, which is based on the large difference in the masses of the electrons and the nuclei. Under the BO approximation, the total wave-function can be expressed as the product of the electronic il/) and nuclear (tj) wavefunctions, leading to the following electronic and nuclear Schrodinger s equations ... 

As the positron approaches the target system it interacts with and distorts it, so that the total wave function no longer has the separable form of equation (3.3). Nevertheless, an equivalent Schrodinger equation can be derived for the positron, the solution to which is a function of the positron coordinate rq only, with the correct asymptotic form but at the cost of introducing a non-local optical potential. 

The coupled Schrodinger equations can be projected onto the fa fa subspace by Feshbach partitioning, giving an equation for the coefficient function Xd(q) in the component faxdiq) of the total wave function. The effective Hamiltonian in this equation is tn + Vd(q) + Vopt, which contains an optical potential that is nonlocal in the <7-space. This operator is defined by its kernel in the fa - fa subspace,... 

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