Ground state of the helium atom

In this section we examine the ground-state energy of the helium atom by means of both perturbation theory and the variation method. We may then compare the accuracy of the two procedures. 

The potential energy V for a system consisting of two electrons, each with mass me and charge —e, and a nucleus with atomic number Z and charge +Ze is 

We regard the term e jr 2 in the Hamiltonian operator as a perturbation, so that 

In reality, this term is not small in comparison with the other terms so we should not expect the perturbation technique to give accurate results. With this choice for the perturbation, the Schrodinger equation for the unperturbed Hamiltonian operator may be solved exactly. 

The unperturbed Hamiltonian operator is the sum of two hydrogen-like Hamiltonian operators, one for each electron 

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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... 

TABLE VI. Ground State of the Helium Atom Obtained by Superposition of Configurations ... 

Kinoshita, T., Phys. Rev. 105, 1490, Ground state of the helium atom. ... 

To arrive at the correct formulation of the ground state of the helium atom it is necessary to also take into account the effect of spin, represented by the functions a and / . There are four possibilities, according to the electrons having the same spin, either up or down ... 

Fig. 13.4 Logarithm of error(Eh) in the configuration interaction energy for the ground state of the helium atom as a function of maximum orbital quantum number, L, of the one-electron basis functions. The data were obtained in an...

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... 

The first combination is symmetric, and the second is antisymmetric with respect to permutation. In the ground state of the helium atom both electrons occupy the same spatial orbital (Is), so that they must have the antisymmetric spin function for the total wave function to be antisymmetric they therefore form a singlet spin state. In order to find a helium atom with a triplet spin state (so-called spins parallel), the spatial part of the wave function must be antisymmetric with respect to interchange. 

Whilst we are discussing the Pauli principle, it is worthwhile to introduce a further method of expressing the antisymmetric nature of the electronic wave function, namely, the Slater determinant [1], Since both electrons in the ground state of the helium atom occupy the same space orbital, the wave function may be written in the form... 

Our early work on the hydrogen atom confined inside prolate spheroidal boxes also dealt with the molecular hydrogen H+ and molecular HeH++ ions [18]. The investigation of the hydrogen molecular ion was extended recently for confinement in boxes with the same shape with penetrable walls [40]. On the other hand, more than ten years ago, we investigated the ground state of the helium atom confined in a semi-infinite space [46] and inside boxes [47] with paraboloidal boundaries. 

For helium, the effect of relativity is so small that it can safely be handled by approximate relativistic treatments, so that the results are in excellent agreement with spectroscopic measurements [10]. The earlier calculations for the ground state of the helium atom used basis sets that were functions of the coordinates s, t, and u ... 

Section 9.3 applied perturbation theory to the ground state of the helium atom. We now treat the lowest excited states of helium. The unperturbed energies are given by (9.48). The lowest unperturbed excited states have ] = 1, 2 = 2 or n, = 2, 2 = 1. and substitution in (9.48) gives... 

John C. Slater (1901-1976), American physicist, for 30 years a professor and dean at the Physics Department of the Massachusetts Institute of Technology, then at the University of Florida, Gainesville, where he participated in the Quantum Theory Project. His youth was in the stormy period of the intense development of quantum mechanics, and he participated vividly in it. For example, in 1926-1932, he published articles on the ground state of the helium atom, the screening constants (Slater orbitals), the antisymmetrization of the wave function (Slater determinant), and the algorithm for calculating the integrals (the Slater-Condon rules). 

In 1929, two years after the birth of quantum ehemistry, a paper by Egil Hylleraas appeared, where, for the ground state of the helium atom, a trial variational funetion, eontaining the inter-... 

Next, this problem is redueed to the Ritz method (see Appendiees L, p. el07 and K, p. el05), and subsequently to the seeular equations (H — eS) c = 0. It is worth noting here that, e.g., the Cl wave funetion for the ground state of the helium atom would be linear eombinations of the determinants where the largest c eoefflcient occurs in front of the other determinants eonstrueted from the 2a D and 2s f spinorbitals (one of the doubly exeited determinants). The Cl wave funetions for all states (ground and exeited) are linear eombinations of the same Slater determinants, they differ only in the c eoeffieients. 

In the ground state of the helium atom the spatial part of the wave-function, i/, j(I)v/jj.(2), is symmetric with respect to interchange of electrons, and therefore the total wavefunction will be antisymmetric only if the spin part of the wavefunction is also antisymmetric. Thus, the overall wavefunction must be ... 

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. 

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