Ground state wavefunctions helium

Apart from the demands of the Pauli principle, the motion of electrons described by the wavefunction P° attached to the Hamiltonian H° is independent. This situation is called the independent particle or single-particle picture. Examples of single-particle wavefunctions are the hydrogenic functions (pfr,ms) introduced above, and also wavefunctions from a Hartree-Fock (HF) approach (see Section 7.3). HF wavefunctions follow from a self-consistent procedure, i.e., they are derived from an ab initio calculation without any adjustable parameters. Therefore, they represent the best wavefunctions within the independent particle model. As mentioned above, the description of the Z-electron system by independent particle functions then leads to the shell model. However, if the Coulomb interaction between the electrons is taken more accurately into account (not by a mean-field approach), this simplified picture changes and the electrons are subject to a correlated motion which is not described by the shell model. This correlated motion will be explained for the simplest correlated system, the ground state of helium. 

In the independent particle picture, the ground state of helium is given by Is2 xSo. For this two-electron system it is always possible to write the Slater determinantal wavefunction as a product of space- and spin-functions with certain symmetries. In the present case of a singlet state, the spin function has to be 

There are many ways to improve this independent-particle model by incorporating electron correlation in the spatial part h°(r1, r2) of the wavefunction. Here the Hylleraas function [Hyl29] and the method of configuration interaction (Cl) will be used as illustrations. 

The fundamental idea of Hylleraas was that the attractive force between the nuclear charge and each of the electrons is well accounted for in the single-particle orbitals pls0(ri) and pls0(r2) by their exponential functions with negative exponents in the coordinates rf (see Section 7.1.1)  

one expects that the mutual repulsion of the electrons at the positions rt and r2 can be described by an exponential function with a positive exponent in the relative coordinate r12  

---

For the ground configuration Is2 of helium, only one wavefunction can be written [eq. (2.2.36)]. Hence the ground state of helium is a singlet. 

This way of writing the wavefunction was developed by Slater in 1929, and is therefore known as a Slater determinant. Each term in the determinant consists of a hydrogen-like orbital multiplied by a spin function, and is referred to as a spin orbital. The first row in the determinant contains the two spin orbitals available to an electron in the ground state of helium, both occupied by electron 1. The second row contains the same terms, this time occupied by electron 2. The method can easily... 

Let the ground state of helium be our exanple. We take the ordinary independent-electron wavefunction as our initial approximation ... 

We mentioned in Chapter 5 that the ground-state wavefunction ls(l)ls(2) for helium was much too contracted if the Is functions were taken from the He+ ion without modification. Physically, this arises because, in He+, the single electron sees only a... 

In the 35 years since Pekeris work was reported, even more extensive calculations have been reported. For example, Drake et al. [9] have calculated an upper bound for the ground state of helium having 22 significant figures using a wavefunction having 2358 terms. 

Problem Determine the Slater-type orbital wavefunction and for an electron in a) the ground-state of helium, and b) the 2p orbital of oxygea... 

In order to demonstrate the method, the simplest case, the ground state of the helium atom, will be used. Since the two-electron wavefunction is given by % = (lsO+, IsO-, one has to find the optimized orbitals Ru(r) which are part of starting point is the energy eigenvalue Eg... 

Recasting of correlated wavefunctions in helium (ground state)... 

As implied by the name, a correlated wavefunction takes into account at least some essential parts of the correlated motion between the electrons which results from their mutual Coulomb interaction. As analysed in Section 1.1.2 for the simplest correlated wavefunction, the helium ground-state function, this correlation imposes a certain spatial structure on the correlated function. In the discussion given there, two correlated functions were selected a three-parameter Hylleraas function, and a simple Cl function. In this section, these two functions will be represented in slightly different forms in order to make their similarities and differences more transparent. 

Table 2.2 2 Summary of four trial wavefunctions of helium atom in its ground state...

---