Helium atom Hylleraas function

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. 

To test the accuracy and convenience of the method of superposition of configurations, the problem of the ground state of the helium atom has recently been reexamined by several authors. According to Hylleraas (1928), the total wave function may be expressed in the form... 

Hylleraas was first to use a wave function with explicit r, -dependence in his 1929 breakthrough study of the helium atom [16,17], Applications of these... 

In 1952 ten Seldam and de Groot [99] reported the first study of the confined helium atom. They performed a variational calculation via a trial wave function based on an ansatz proposed by Hylleraas in 1929, which is... 

Table 2 Correlation energy (CE) estimated as the difference between the wave function expanded with 40 Hylleraas functions (H-WF) and the HF wave function obtained with optimized exponents, for the lowest singlet state of confined helium atom. All quantities are in hartrees...

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. 

Explicitly correlated wave function fheory [14] is anofher imporfanf approach in quantum chemistry. One introduces inter-electron distances together with the nuclear-electron distances and set up some presumably accurate wave function and applies the variation principle. The Hylleraas wave function reported in 1929 [15] was the first of this theory and gave accurate results for the helium atom. Many important studies have been published since then even when we limit ourselves to the helium atom [16-28]. They clarified the natures and important aspects of very accurate wave functions. However, the explicitly correlated wave function theory has not been very popularly used in the studies of chemical problems in comparison with the Hartree-Fock and electron correlation approach. One reason was that it was generally difficult to formulate very accurate wave functions of general molecules with intuitions alone and another reason was that this approach was rather computationally demanding. 

Key words Helium atom - Electron correlation -Explicitly correlated wave functions - Hylleraas expansion... 

Figure 2 The wave function of helium atom in its electronic ground state. The upper part (a) represents the difference between the Hartree-Fock and Hylleraas wave functions in a plane that contains the nucleus and fixed electron, called in the literature the Coulomb hole [5]. In the middle part (b) the F12 geminal function with 7 = 1.0 is plotted. The bottom part (c) represents the difference between (a) and (b).

In 1929, two years after the birth of quantum chemistry, a paper by Hylleraas appeared, where, for the ground state of the helium atom, a trial variational function, containing the interelectronic distance explicitly, was applied. This was a brilliant idea, since it showed that already a small number of terms provide very good results. Even though no fundamental difficulties were encountered for larger atoms, the enormous numerical problems were prohibitive for atoms with larger numbers of electrons. In this case, the progress made from the nineteen twenties to the end of the twentieth century is exemplified by transition from two- to ten-electron systems. 

Hylleraas function (p. 506) harmonic helium atom (p. 507) James-Coolidge function (p. 508) Kolos-Wolniewicz function (p. 508) geminal (p. 513)... 

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. 

Having examined the behaviour of the exact wave function in the ground-state helium atom, let us now consider the description of this atom by approximate electronic wave functions. We begin by examining the standard expansion of Cl theory and then go on to investigate how this model may be amended to allow for a better description of the short-range electronic interactions. We conclude this section with a discussion of the Hylleraas expansion. 

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

The Bohr model gave the correct energies for the hydrogen atom but failed when applied to helium. Hence, in the early days of quantum mechanics, it was important to show that the new theory could give an accurate treatment of helium. The pioneering work on the helium ground state was done by Hylleraas in the years 192 1930. To allow for the effect of one electron on the motion of the other, Hylleraas used variational functions that contained the interelectronic distance ri2- One function he used is... 

For two-electron atoms, many approaches have been applied a review made by Aquino reported the techniques used up to 2009 [20], To date, the expansion of the wave function in terms of Hylleraas-type functions is the technique that gives the lowest energies for several confinement radii [21-23], which can be used as reference when other techniques are proposed for the study of these systems. However, such a technique has not been used for atoms with several electrons, for example, beryllium. In this sense, in this chapter we test the many-body perturbation theory to second order, as a technique to estimate the CE for confined many-electron atoms. In the next section, we discuss the theory behind of the HF method, and the basis set proposed for its implementation for confined atoms. In the same section, the many-body perturbation theory to second order proposed by Moller and Plesset (MP2) [24] also is discussed, and we give some details about the implementation of our code implemented in GPUs [25]. Finally, we contrast our results for helium-like atoms with more sophisticated techniques in order to know the percent of correlation energy recovered by the MP2 method. 

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