Hylleraas wavefunction

Due to its importance, the Hylleraas wavefunction introduced in equ. (1.20) will also be discussed in the context of Cl. For simplicity, only the three-parameter Hylleraas function is chosen. The essential quantity, responsible for the correlated motion of the electrons, in this function is the distance rl2 between the electrons. Usually, rl2 is expressed as 

If this expansion is applied to the three-parameter Hylleraas function, one obtains [GMM53] 

The F rur2) are rather cumbersome, but completely known, functions which depend on r , r , and f (compare the related case of the functions f/ru r2) defined in equ. (7.109b)). They are normalized in such a way that the coefficients c( 2 describe the weight given to the individual (f-components. For the three-parameter Hylleraas function, one has [GMM53]  

The role of these /-components will now be made more explicit. Using equs. (7.5) 

This result states that the angular momenta ( in the expansion of the Hylleraas function impose the following symmetry properties on the individual angular momenta and (2 attached to the two electrons with spatial directions and r2, respectively  

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Alternative methods are based on the pioneering work of Hylleraas ([1928], [1964]). In these cases orbitals do not form the starting point, not even in zero order. Instead, the troublesome inter-electronic terms appear explicitly in the expression for the atomic wavefunction. However the Hylleraas methods become mathematically very cumbersome as the number of electrons in the atom increases, and they have not been very successfully applied in atoms beyond beryllium, which has only four electrons. Interestingly, one recent survey of ab initio calculations on the beryllium atom showed that the Hylleraas method in fact produced the closest agreement with the experimentally determined ground state atomic energy (Froese-Fischer [1977]). 

The wavefunction is expanded in terms of Hylleraas basis sets incorporating explicitly the electron correlation effects... 

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

The Hylleraas function, with its improved properties as compared to a Hartree-Fock function, is called a correlated wavefunction, because it takes into account the mutual electron-electron interaction much better, and the motion of electrons beyond a mean-field average is termed correlated motion or the effect of electron correlations. (The definition of electron correlation is used here in the strict terminology. The mean-field average of electron-electron interactions is frequently also called electron correlation.) Comparing equ. (1.20) with equ. (1.16b) one has... 

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. 

A few people are working on Hylleraas type and James-Coolidge type wavefunctions of many-electron systems. Their present calculations are preparatory, but will expand enormously in the production steps. We should not forget that these calculations are most important from the basic point of view. 

Figure 18. Electron pair density, h(u), its radial derivative, h (u ), and their difference, for H, calculated using a 204 term Hylleraas-type wavefunction, as described in Ref. [71],...

To improve on a single determinant reference we must develop a superior treatment of the two-electron interaaion. Logically, this would involve explicit use of the two-particle operator in a trial wavefunaion, and such Hylleraas-type trial wavefunctions have been used to obtain the best results (for, e.g.. He, Li, Be, H2, and LiH). Analysis of certain other analytic properties of the correlation cusp l/( r,- — r, ) have also been exploited to develop better descriptions without having to use wavefunctions that are explicitly dependent on r j, but such methods also have many computational restriaions. 

Since explicit many-body terms are critical for a compact description of the wavefunction, let us review some early work along these lines. Hylleraas and Pekeris had great success for He with wavefunctions of the form... 

Note that in the present case the matrix elements depend on the final density p . Moreover, because this density is obtained from the transformed wavefunction, they also depend on the expansion coefficients. For this reason, Eq. (177) must be solved iteratively. Such a procedure has been applied - in a sample calculation - to the 2 S excited state of the helium atom. The upper-bound character of the energy corresponding to the energy functional for the transformed wavefunction B,p( r,- ) with respect to the exact energy is guaranteed by the Hylleraas-Undheim-MacDonald theorem. 

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