Wave functions for the helium atom

In many places elsewhere in this book we describe the analysis of spectra, the definition and determination of molecular parameters from the spectra, and the relationships between these parameters and the wave functions for the molecules in question. Later in this chapter we will outline the principles and practice of calculating accurate wave functions for diatomic molecules. Before we can do that, however, we must discuss the calculation of atomic wave functions the methods originally developed for atoms were subsequently extended to deal with molecules. This is not the book for an exhaustive discussion of these topics, and so many accounts exist elsewhere that such a discussion is not necessary. Nevertheless we must pay some attention to this topic because the interpretation of spectroscopic data in terms of molecular wave functions is one of the primary motivations for obtaining the data in the first place. 

We have already dealt with the calculation of the wave functions of the hydrogen atom. We now proceed to consider many-electron atoms, first dealing with the simplest such example, the helium atom which possesses two electrons. The Hamiltonian for a helium-like atom with an infinitely heavy nucleus can be obtained by selecting the appropriate terms from the master equation in chapter 3. The Hamiltonian we use is 

Almost all approaches to many-electron wave functions, for both atoms and molecules, involve their formulation as products of one-electron orbitals. If the interelectronic repulsion term in (6.23) is small compared with the other terms, the Hamiltonian is approximately separable into independent operators for each electron and the two-electron wave function, f (1, 2), can be written as a simple product of one-electron functions, 

There are several different ways of proceeding from here, but following others we use the variation method. Multiplication of both sides of the Schrodinger equation by i/o and integration over all coordinates gives an expression for the energy, 

Any approximate wave function // will give a calculated energy Wo from (6.26) which is greater than the true energy Eo, and the objective is to find a wave function which minimises the value of Wo- Ultimately a trial wave function will contain one or more variable parameters, and we will look to find parameter values which minimise the calculated energy however we first develop (6.26). 

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Exercise 1.4. Generation of the numerical radial wave function for the helium atom Is atomic orbital. 

This exercise involves the calculation of the numerical radial wave function for the helium atom in the form rR(r) output by the Herman-Skillman program for the helium atom and its processing into the equivalent of a radial wave function. 

We now include spin in the He zeroth-order ground-state wave function. The function 15(1)15(2) is symmetric with respect to exchange. The overall electronic wave function including spin must be antisymmetric. Hence we must multiply the symmetric space function 15 (1) 15 (2) by an antisymmetric spin function. There is only one antisymmetric two-electron spin function, so the ground-state zeroth-order wave function for the helium atom including spin is... 

The Xk will also be natural orbitals but if is degenerate, they may be a unitary transformation of the Xk corresponding to the same eigenvalue n. For example, the simplest two-electron wave function for the helium atom is... 

Morrison, R. C., Q. Zhao, R. C. Morrison, and R. G. Parr. 1995. Solution of the Kohn-Sham equations using reference densities from accurate, correlated wave functions for the neutral atoms helium through argon. Phys. Rev. A51, 1980. 

We have just explained that the wave equation for the helium atom cannot be solved exacdy because of the term involving l/r12. If the repulsion between two electrons prevents a wave equation from being solved, it should be clear that when there are more than two electrons the situation is worse. If there are three electrons present (as in the lithium atom) there will be repulsion terms involving l/r12, l/r13, and l/r23. Although there are a number of types of calculations that can be performed (particularly the self-consistent field calculations), they will not be described here. Fortunately, for some situations, it is not necessary to have an exact wave function that is obtained from the exact solution of a wave equation. In many cases, an approximate wave function is sufficient. The most commonly used approximate wave functions for one electron are those given by J. C. Slater, and they are known as Slater wave functions or Slater-type orbitals (usually referred to as STO orbitals). 

Morrison, R. C., Zhao, Q., 1995, Solution to the Kohn-Sham Equations Using Reference Densities from Accurate, Correlated Wave Functions for the Neutral Atoms Helium Through Argon , Phys. Rev. A, 51, 1980. 

Consider the wave function for the S ground state of the helium atom. In the Hartree-Fock approximation, the two electrons move independently of each other, each in the field generated by the average motion of the electrons. The motion of one electron is therefore unaffected by the instantaneous position of the other, as illustrated in Figure 7.1. In this figure, we have plotted the Hartree-Fock wave function for the helium ground state with one electron fixed at a position 0.5 o from the nucleus, in a plane that contains the nucleus and the fixed electron. For clarity, combined surface and contour plots are presented, with the positions of the fixed electron and the nucleus marked by vertical bars. As expected, the Hartree-Fock contour plot consists of a set of concentric circles that are undistorted by the fixed electron - that is, the wave-function amplitude for one electron depends only on its distance to the nucleus, not on the distance to the other electron. 

To see the implications of the Hamiltonian singularities for the wave function, it is convenient to express the Hamiltonian in a different set of coordinates. Since the ground-state wave function of the helium atom is totally synunetric, it can be expressed in terms of the three ratlial coordinates... 

If the structure of the helium atom were exactly described by the symbol la9 and that of neon by 1 a22a 2p these atoms would have spherically symmetrical electron distributions.24 However, the mutual repulsion of the two electrons in the atom causes them to avoid one another the wave function for the atom corresponds to a larger probability for the two electrons to be on opposite sides of the nucleus than on the same side (for the same values of the distances of the two electrons from the nucleus, there is greater probability that the angle described at the nucleus by the vectors to the electrons is greater than 90° than that it is less than 90°). This effect, which is called correla-... 

In a few cases, the wave-function F of a monatomic entity can be used for calculating a, e.g. 4.5 bohr3 for the hydrogen atom, or 0.205 A3 for the helium atom in agreement with the experimental value. Gaseous H does not have a Hartree-Fock function stable relative to spontaneous loss of an electron, and it is necessary to introduce correlation effects in order to calculate a which is said to be 31 A3. The value 1.8 A3 for H(-I) in Table 2 derives from NaCl-type LiH, NaH and KH. The anion B2Hg2 has a = 6.3 A3 to be compared with the isoelectronic C2H6 4.47 A3. Since CH4 has a =... 

Abstract. With an eye on the high accuracy ( 10 MHz) evaluation of the ionization energy from the helium atom ground state, a complete set of order ma6 operators is built. This set is gauge and regularization scheme independent and can be used for an immediate calculation with a wave function of the helium ground state. 

DuPre and McTague have used a Hirschfelder-Iinnett wave function for the triplet state of the hydrogen molecule to approximate the wave function of a pair of helium atoms. On evaluating an approximate expres-aon for the polarizability they find Icqs < oo at short range and > o at long range. 

The stationary states of the helium atom, represented on the right side of Figure 29-2, are conveniently divided into two sets, shown by open and closed circles, respectively. The wave functions for the former, called singlet states, are obtained by multiplying the symmetric orbital wave functions by the single... 

So far we have not taken into consideration the spins of the electrons. On doing this we find, exactly as for the helium atom, that in order to make the complete wave functions antisymmetric in the electrons, as required by Pauli s principle, the orbital wave functions must be multiplied by suitably chosen spin functions, becoming... 

There are no analytical forms for the radial functions, / ni(r), as solutions of the radial wave equation. Hartree, in 1928, developed the standard solution procedure, the self-consistent field method for the helium atom by using the simple product forms of equation 1.10 to represent the two-electron wave function. Herman and Skillman (4) programmed a very useful approximate form of the Hartree method in the early 1960s for atomic structure calculations on all the atoms in the Periodic Table. An executable version of this program, based on their FORTRAN code, modified to output data for use on a spreadsheet is included with the material on the CDROM as hs.exe. 

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. 

We now reconsider the helium atom from the standpoint of electron spin and the Pauli principle. In the perturbation treatment of helium in Section 9.3, we found the zeroth-order wave function for the ground state to be 15(1)15(2). To take spin into account, we must multiply this spatial function by a spin eigenfunction. We therefore consider the possible spin eigenfunctions for two electrons. We shall use the notation a(l)a(2) to indicate a state where electron 1 has spin up and electron 2 has spin up a(l) stands for a(Wji). Since each electron has two possible spin states, we have at first sight the four possible spin functions ... 

The two-electron spin eigenfunctions consist of the symmetric functions a(l)o (2), /3(l)/3(2), and [a(l)/3(2) -I- j8(l)a(2)]/ V and the antisymmetric function [a(l)/3(2) —/3(l)a(2)]/ V. For the helium atom, each stationary state wave function is the product of a symmetric spatial function and an antisymmetric spin function or an antisymmetric spatial function and a symmetric spin function. Some approximate helium-atom wave functions are Eqs. (10.26) to (10.30). 

The spatial wavefunction, P(1>2), for the helium atom can be written approximately as the product of two hydrogen-like wave-functions ... 

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