Schrodinger equation lowest-energy wavefunction

If we are interested in the ground-state electronic properties of a molecule or solid with a given set of nuclear coordinates we should seek the solution to the Schrodinger equation which corresponds to the lowest electronic energy of the system. However, the inter-electronic interactions in eq. (2.2) are such that this differential equation is non-separable. It is therefore impossible to obtain the exact solution to the full many-body problem. In order to proceed, it is necessary to introduce approximation in this equation. Two types of approximations can be separated, namely, approximations of the wavefunction, VF, from a true many-particle wavefunction to, in most... 

Notice that [] is an energy functional where the magnitude to be varied is the wavefunction Cn- Thus, the lowest eigenvalue Ea of the JV-particle Schrodinger equation H0 — Eo0 is reached at the extremum of this functional ... 

The discussion in the previous sections assumed that the electron d5mamics is adiabatic, i.e. the electronic wavefunction follows the nuclear d5mamics and at every nuclear configuration only the lowest energy (or more generally, for excited states, a single) electronic wavefunction is relevant. This is the Bom-Oppenheimer approximation which allows the separation of nuclear and electronic coordinates in the Schrodinger equation. 

Using proper boundary conditions and approximations, each possible solution of the time-independent Schrodinger equation corresponds to a stationary state of the system the one with the lowest energy is considered the ground state. Also, the spatial wavefunction is independent of time and function of the positions of the electrons, and the nuclei should be normalized and anti-symmetrized. Thus,... 

To go beyond the Hartree-Fock limit and obtain the full solution to the Schrodinger equation (in the non-relativistic and Bom-Oppenheimer limit), one would have to combine various solutions of the product type. In any calculation one obtains more molecular orbitals than needed to accommodate all the electrons in the system. In a system with 2n electrons, the n molecular orbitals with the lowest molecular orbital energies are used in the Hartree-Fock solution for the ground state (this assumes a closed shell system, where two electrons are paired up in each molecular orbital). The rest of the molecular orbitals obtained will be excited molecular orbitals. Of course, other possible wavefunctions of the product type can be formed by using excited molecular orbitals in the product. The set of all such possible products can be used as a basis set to solve the full Schrodinger equation. The solution now looks like ... 

Provided the interaction potential is sufficiently attractive (which it almost always is), the multichannel Schrodinger equation supports bound states at energies below the lowest threshold. The corresponding wavefunction for state n is... 

The lowest six energy levels are graphed and selected wavefunctions sketched for the solution to a one-dimensional Schrodinger equation. On this graph, draw in the potential energy curve that could account for these solutions. 

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