Ground-state wave function Hamiltonian equations

Because of the interelectronic repulsion term l/ri2, the electronic Hamiltonian is not separable and only approximate solution of the wave equation can be considered. The obvious strategy would be to use Hj wave functions in a variation analysis. Unfortunately, these are not known in functional form and are available only as tables. A successful parameterization, first proposed by James and Coolidge [89] and still the most successful procedure, consists of expressing the Hamiltonian operator in terms of the four elliptical coordinates 1j2 and 771 >2 of the two electrons and the variable p = 2ri2/rab. The elliptical coordinates 4> 1 and 2, as in the case of Hj, do not enter into the ground-state wave function. The starting wave function for the lowest state was therefore taken in the power-series form... 

The problem is to get a computable expression for the ground state wave function without solving the Schrodinger equation for the many body hamiltonian of (1), obviously an impossible task for any non trivial system. As usual in many body problems, we can resort to the variational principle which states that the energy of any proper trial state ) will be greater or equal to... 

Let H and H, be the operators obtained when t>(r,) in (15.106) is replaced by v (r,) and Wi,(r,), respectively, where and (r,) differ by more than a constant. Prove that the ground-state wave functions ij/o and of these Hamiltonians must be different functions. Hint Assume they are the same function, write the Schrodinger equations for H and H/, subtract one equation from the other, and show that this leads to w (r,) — fc(r,) equals a constant, thereby contradicting the given information and proving that the two wave functions cannot be equal. 

For a given Hamiltonian operator there will be an infinite number of solutions to this equation, each indicated by a different value of the index . We wish to find the ground state wavefunction, y/j, which has an energy Normally, equation (8.1) cannot be solved analytically and the wavefunctions that satisfy the equation are unknown. Under these circumstance it is necessary to formulate a trial wavefunction, which is expected to be a good approximation to the true ground state wave-function. 

Since the lORA equation is variational and the ZORA wave function is not the ground state wave function for the lORA Hamiltonian, the quotient must be an upper bound to the lORA energy for the bound states. We can summarize as follows ... 

Once an effective Hamiltonian for the system has been defined, the wave function T and energy can be evaluated by minimizing the Vsb in Eq. (8.6) with respect to the molecular orbital coefhcients of the ground state wave function using a self-consistent (SCF) procedure to solve the effective Schrodinger equation. 

In equation (A.8), is the wave function which describes the distribution of particles in the system. It may be the exact wave function [the solution to equation (A.l)] or a reasonable approximate wave function. For most molecules, the ground electronic state wave function is real, and in writing the expectation value in the form of equation (A.8), we have made this simplifying (though not necessary) assumption. The electronic energy is an observable of the system, and the corresponding operator is the Hamiltonian operator. Therefore, one may obtain an estimate for the energy even if one does not know the exact wave function but only an approximate one, P, that is,... 

Stated in still other words this means that for any trial density p(r) - which satisfies the necessary boundary conditions such as p( ) - 0, J p( ) dr = N, and which is associated with some external potential Vext - the energy obtained from the functional given in equation (4-6) represents an upper bound to the true ground state energy E0. E0 results if and only if the exact ground state density is inserted into equation (4-8). The proof of the inequality (4-11) is simple since it makes use of the variational principle established for wave functions as detailed in Chapter 1. We recall that any trial density p(r) defines its own Hamiltonian H and hence its own wave function. This wave function can now be taken as the trial wave function for the Hamiltonian generated from the true external potential Vext. Thus, we arrive at... 

Consequently, from the density the Hamiltonian can be readily obtained, and then every property of the system can be determined by solving the Schrodinger equation to obtain the wave function. One has to emphasize, however, that this argument holds only for Coulomb systems. By contrast, the density functional theory formulated by Hohenberg and Kohn is valid for any external potential. Kato s theorem is valid not only for the ground state but also for the excited states. Consequently, if the density n, of the f-th excited state is known, the Hamiltonian H is also known in principle and its eigenvalue problem ... 

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