Helium atom Hamiltonian

In the last section, we wrote the helium-atom Hamiltonian H = + H, where... 

There is more than one way to divide a Hamiltonian H into an unperturbed part H° and a perturbation H, Instead of the division (9.40) and (9.41), consider the following way of dividing up the helium-atom Hamiltonian ... 

Write the Hamiltonian for the helium atom, which has two electrons, one at a distance r and the other at a distance r2-... 

The helium atom serves as a simple example for the application of this construction. If the nucleus (for which Z = 2) is considered to be fixed in space, the Hamiltonian operator H for the two electrons is... 

Nonrelativistic Hamiltonian of the helium atom in the non-recoil limit (m/mN 0) is... 

We have already dealt with the calculation of the wave functions of the hydrogen atom. We now proceed to consider many-electron atoms, first dealing with the simplest such example, the helium atom which possesses two electrons. The Hamiltonian for a helium-like atom with an infinitely heavy nucleus can be obtained by selecting the appropriate terms from the master equation in chapter 3. The Hamiltonian we use is... 

The energy of the helium atom calculated above is the first-order energy, which differs from the true energy by an amount called the correlation energy this is a measure of the tendency of the electrons to avoid each other. The simplest improvement to the trial wave function is to allow Z in (6.29) to be a variable parameter, which we call (not to be confused with the spin-orbit coupling parameter in equation (6.20)) Z in the Hamiltonian (6.23) remains the same. The expression for the calculated energy,... 

In a first report, Marin and Cruz [16] studied the helium atom confined in an impenetrable spherical box where they used the direct variational method to optimize the energy value. The Hamiltonian for a spherically confined helium atom within a hard box is given by... 

The wave function equation yn = Is (1). Is (2) (Lowest energy level of the helium atom) would be an eigen function of the Hamiltonian for a two electron system, under the following conditions ... 

We find the same result for the other one-electron Hamiltonian, so the contribution to the energy of the helium atom from the one-electron terms in equation 5.32 is... 

Using the result in equation 5.38 we can recast the calculation of the energy of the helium atom as the problem to determine the eigenfunctions of the one-electron Hartree-Fock-Slater Hamiltonian. This identifies universally the Fock operator in the equation... 

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