Hylleraas function compared

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... 

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 ground state energy values compare very well with those obtained by Aquino et al. through a 40-term expansion of generalized Hylleraas functions (see Tables 10 and 11). 

It is observed that nanohartree accuracy can be achieved with Nmsa 13, which should be compared with the Cl expansion where one-electron basis functions with angular momentum quantum numbers up to Lmax 300 ought to be included to obtain this precision. (One should perhaps not compare Afmai directly with L ax as done in Figure 1 since a Hylleraas function for a given (Vmax restricted to even powers of ri2 can be represented by a Cl function with L ax = max/2. It is the terms with odd powers of r 2 which cannot be described by the Cl expansion.)... 

Comparing with the exact second-order energy correction of —0.00723732 obtained from the exact first-order state in Exercise 14.3.3, we find that, whereas the usual expression gives an error of —0.00003088, the Hylleraas fimctional gives an error of only 0.00000013. Note that the Hylleraas functional gives a positive error. 

Fig. 7.11. The helium ground-state wave function with one electron fixed at a position 0.5oo from the nucleus for different principal expansions 2

This contribution examines current approaches to Coulomb few-body problems mainly from a methodological perspective, in contrast to recent reviews which have focused on the results obtained for benchmark problems. The methods under discussion here employ wavefunctions which explicitly involve all the interparticle coordinates and use functional forms appropriate to nonadiabatic systems in which all the particles are of comparable mass. The use of such wavefunctions for states of arbitrary angular symmetry is reviewed, and the kinetic-energy operator, written in the interparticle coordinates, is presented in a convenient form. Evaluation of the resultant angular matrix elements is discussed in detail. For exponentially correlated wavefunctions, problems of integral evaluation are surveyed, the relatively new analytical procedures are summarized, and relations among matrix elements are presented. The current status of Gaussian-orbital and Hylleraas methods is also reviewed. 

Notice that O5 and Og are two-electron functions, which cannot be factorized into one-electron functions. By calculating all matrix elements and solving the 6 x 6 eigenvalue problem, Hylleraas, in 1928, obtained, without comparison, the best description of the helium atom with the energy -2.903329 H, compared to the earlier best value of -2.86 H. With the help of modern computers, it was recently possible to determine the ground state energy with more than accurate 20 decimal places (-2.903724 H) using essentially the Hylleraas method. 

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