Helium ground-state correlations

Figure 4-1 Convergence of the correlation energy for the helium ground state with the number N of pair natural orbitals.

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. 

Structure of the Exchange, Correlation and Correlation-Kinetic-Energy Fields and Potentials for the Helium Ground State... 

Mainly for considerations of space, it has seemed desirable to limit the framework of the present review to the standard methods for treating correlation effects, namely the method of superposition of configurations, the method with correlated wave functions containing rij and the method using different orbitals for different spins. Historically these methods were developed together as different branches of the same tree, and, as useful tools for actual applications, they can all be traced back to the pioneering work of Hylleraas carried out in 1928-30 in connection with his study of the ground state of the helium atom. 

Power Series Expansions and Formal Solutions (a) Helium Atom. If the method of superposition of configurations is based on the use of expansions in orthogonal sets, the method of correlated wave functions has so far been founded on power series expansions. The classical example is, of course, Hyl-leraas expansion (Eq. III.4) for the ground state of the He atom, which is a power series in the three variables... 

A weakness of the development in the literature up to now has been that too much effort has been concentrated on the helium problem, whereas more complicated systems have been only scarce-ly treated. The reason is obvious it is much easier to test a new method for treating correlation on the ground state of helium, and if the method fails on this simple system, it will certainly not work on a more complicated system either. In treating energy differences in many-electron systems, simple methods will often produce results in excellent agreement with experiment owing to a fortuitous cancellation of errors, but a test on helium will then often reveal the faults of the approach. Even in the future, one can therefore expect that the helium problem will be paid a great deal of interest. 

LennarD-Jones, J. E., and Pople, J. A., Phil. Mag. 43, 581, Ser. 7, The spatial correlation of electrons in atoms and molecules. I. Helium and similar two-electron systems in their ground states. Analysis of in-out effect and angular effect. 

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. 

Following these ideas, the ground state of helium which includes electron correlations can then be represented as... 

These preliminary, but important clarifications, lead us to a discussion of the correlated motion within the Cl picture for both electrons in the ground state of helium. For this purpose it is illustrative to analyse the three-parameter Hylleraas function, the correlation properties of which have previously been described, in terms of Cl functions. Looking only for the individual components of orbital angular momenta r = i2 which couple to the desired Se state, one gets [GMM53]... 

The ground state of helium is itself a rather special case as the wave function is relatively compact. It is thus not difficult to get a reasonable representation of this wave function with a rather modest, correlated basis set. Hylleraas[16]... 

To apply these equations, we need the wavefuncdons m> in order to get the dipole moment transition elements and the frequencies spectral series, where only the ground state need be near-exact. This is done by diagonalizing the Hamiltonian matrix formed from a large number of basis functions (which implicitly include the interelectronic coordinate and thus electron correlation). We do this for each symmetry state that is involved. All the ensuing eigenvalues and eigenvectors are then used in the sum-over-states expressions. For helium we require S, P, and D states and for H2 (or D2) E, II, and A states. 

Calculation of the ground state energy of the helium atom is a critical case as well because it is the first example of correlation energy, the difference between the Hartree-Fock energy and the exact value. The energy required to remove one electron from a neutral He atom is the first ionization potential... 

If the interaction between two ground-state He atoms were strictly repulsive (as predicted by MO theory), the atoms in He gas would not attract one another at all and the gas would never liquefy. Of course, helium gas can be liquefied. Configuration-interaction calculations and direct experimental evidence from scattering experiments show that as two He atoms approach each other there is an initial weak attraction, with the potential energy reaching a minimum at 3.0 A of 0.0009 eV below the separated-atoms energy. At distances less than 3.0 A, the force becomes increasingly repulsive because of overlap of the electron probability densities. The initial attraction (called a London or dispersion force) results from instantaneous correlation between the motions of the electrons in one atom and the motions of the electrons in the second atom. Therefore, a calculation that includes electron correlation is needed to deal with dispersion attractions. 

---