Helium wavefunctions

In eq. (2.2.35) we see that the helium wavefunction can be factorized into spatial (or orbital) and spin parts. It should be noted that such a factorization is possible only for two-electron systems. Yet another way of writing (1,2) of He as given in eq. (2.2.35) is... 

Assume that the helium wavefunction is a product of two hydrogen-like wavefunctions (that is, neglect the term for the repulsion between the electrons) in the n = 1 principal quantum shell. Determine the electronic energy of the helium atom and compare it to the experimentally determined energy of —1.265 X 10 J. (Total energies are determined experimentally by measuring how much energy it takes to remove all of the electrons from an atom.)... 

VV e now wish to establish the general functional form of possible wavefunctions for the two electrons in this pseudo helium atom. We will do so by considering first the spatial part of the u a efunction. We will show how to derive functional forms for the wavefunction in which the i change of electrons is independent of the electron labels and does not affect the electron density. The simplest approach is to assume that each wavefunction for the helium atom is the product of the individual one-electron solutions. As we have just seen, this implies that the total energy is equal to the sum of the one-electron orbital energies, which is not correct as ii ignores electron-electron repulsion. Nevertheless, it is a useful illustrative model. The wavefunction of the lowest energy state then has each of the two electrons in a Is orbital ... 

The final wavefunction stUl contains a large proportion of the Is orbital on the helium atom, but less than was obtained without the two-electron integrals. 

In order to exemplify these ideas, we construct three initial or generating wavefunctions for the S state of the helium atom, and generate two types of... 

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 [ p( r,- ) with respect to the exact energy is guaranteed by... 

Calculations of IIq(O) are very sensitive to the basis set. The venerable Clementi-Roetti wavefunctions [234], often considered to be of Hartree-Fock quality, get the sign of IIq(O) wrong for the sihcon atom. Purely numerical, basis-set-free, calculations [232,235] have been performed to establish Hartree-Fock limits for the MacLaurin expansion coefficients of IIo(p). The effects of electron correlation on IIo(O), and in a few cases IIq(O), have been examined for the helium atom [236], the hydride anion [236], the isoelectronic series of the lithium [237], beryllium [238], and neon [239] atoms, the second-period atoms from boron to fluorine [127], the atoms from helium to neon [240], and the neon and argon atoms [241]. Electron correlation has only moderate effects on IIo(O). 

Part of the proof of equation 13 acknowledges this is not so for a Hartree Product. To remedy this, first consider a two-electron system, such as helium. Two equivalent Hartree Product wavefunctions for this system are... 

As detailed in Section 5.1.2, the photoelectron distribution in a channel, identified by a parent ion a = Is, 2s, 2p is obtained by projecting the propagating wavefunction l (t) onto the helium scattering states, which satisfy incoming boundary conditions... 

A. J. Thakkar and V. H. Smith Jr., Phys. Rev. A, 15,1 (1977). Compact and Accurate Integral-Transform Wavefunctions. I. The 11S State of the Helium-like Ions from H to through Mg10. ... 

Apart from the demands of the Pauli principle, the motion of electrons described by the wavefunction P° attached to the Hamiltonian H° is independent. This situation is called the independent particle or single-particle picture. Examples of single-particle wavefunctions are the hydrogenic functions (pfr,ms) introduced above, and also wavefunctions from a Hartree-Fock (HF) approach (see Section 7.3). HF wavefunctions follow from a self-consistent procedure, i.e., they are derived from an ab initio calculation without any adjustable parameters. Therefore, they represent the best wavefunctions within the independent particle model. As mentioned above, the description of the Z-electron system by independent particle functions then leads to the shell model. However, if the Coulomb interaction between the electrons is taken more accurately into account (not by a mean-field approach), this simplified picture changes and the electrons are subject to a correlated motion which is not described by the shell model. This correlated motion will be explained for the simplest correlated system, the ground state of helium. 

In the independent particle picture, the ground state of helium is given by Is2 xSo. For this two-electron system it is always possible to write the Slater determinantal wavefunction as a product of space- and spin-functions with certain symmetries. In the present case of a singlet state, the spin function has to be... 

Depending on the individual orbital angular momenta /a and tb involved, the summation can go up to infinity. Such a situation occurs for double photoionization in helium where the 1P° state of the continuum pair wavefunction can be obtained by an unlimited coupling of individual orbital momenta (esep, sped, edef,...). However, in the case of photon-induced two-step double ionization the formulation... 

Starting in a manner similar to the treatment of single photoionization described in Section 2.1, double photoionization in helium caused by linearly polarized light will be treated first with uncorrelated wavefunctions. A calculation of the differential cross section for double photoionization then requires the evaluation... 

The matrix element Mfi derived so far for the differential cross section of double photoionization in helium is based on uncorrelated wavefunctions in the initial and final states. For simplicity the initial state will be left uncorrelated, but electron correlations in the final state will now be included. The significance of final state correlations can be inferred from Fig. 4.43 without these correlations an intensity... 

Starting from a different treatment of double photoionization in helium, based on properties of the wavefunctions in the threshold region, and special coordinates (hyperspherical coordinates) for the description of the correlated motion of the electrons, different predictions for this 0 parameter have been obtained (see [HSW91, KOs92] with references therein).)... 

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... 

Figure 7.6 Radial wavefunctions Pls(r) = rRls(r) of helium. HYDR is the hydrogenic wavefunction with Z = 2 HF is the Hartree-Fock wavefunction. From [BJ066].

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