Schrodinger equation total wave function

The problem of finding the best approximation of this type and the best one-electron set y2t. . ., y>N is handled in the Hartree-Fock scheme. Of course, a total wave function of the same type as Eq. 11.38 can never be an exact solution to the Schrodinger equation, and the error depends on the fact that the two-electron operator (Eq. 11.39) cannot be exactly replaced by a sum of one-particle operators. Physically we have neglected the effect of the "Coulomb hole" around each electron, but the results in Section II.C(2) show that the main error is connected with the neglect of the Coulomb correlation between electrons with opposite spins. 

IS the total wave function. In shorthand notation Schrodinger s equation can be written as... 

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 ... 

Schrodinger equation. Or alternatively, we may say, equally accurately, that the two electrons occupy two localised orbitals and %b, one at each end of the segment. These are just two different ways of interpreting the same total wave function. 

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,... 

The total matter—radiation Hamiltonian is given in Eq. (9.2). We then solve the Schrodinger equation by expanding the total wave function as. ... 

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 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, Schrodinger equation can be expressed as... 

The total wave function ij/(Q, q, t) of the polyatomic system in question satisfies the time-dependent Schrodinger equation ([Pg.248]

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-... 

In order to define orbitals in a many-electron system, two approaches are possible, which we may refer to as constructive and analytic . The first approach is more common one makes the ad hoc postulate that every electron can be associated with one orbital and the total wave function can be constructed from these orbitals. Then, one is led to an effective one-electron Schrodinger equation from one electron in the field of the other electrons. The underlying model is the independent particle model (IPM). When following the constructive way, one does not know a priori whether the model is a good approximation to the actual physical situation one only knows that it cannot be rigorously correct. The merit of this approach is its relative simplicity from both the mathematical and physical points of view. 

Now we enter the quantization of the radiation field. First, we summarize the Hamilton operator, H, and the Schrodinger equation with total wave function, W , and total energy, E , for the radiation field as an ensemble of harmonic oscillators. 

The total wave function of a molecule comprising N electrons and M nuclei depends on all electronic rj and nuclear Rj coordinates. It can be obtained by solving the time-independent Schrodinger equation [14-16],... 

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