Helium atom functions

Hass6 has considered five variation functions of this form, shown with their results in Table 47-2. The success of his similar treatment of the polarizability of helium (function 6 of Table 29-3) makes it probable that the value — 1.413e2aj /i28 for W" is not in error by more than a few per cent. Slater and Kirkwood1 obtained values 1.13, 1.78, and 1.59 for the coefficient of — e2al/R6 by the use of variation functions based on their helium atom functions mentioned in Section 29e. An approximate discussion of dipole-quadrupole and quadrupole-quadrupole interactions has been given by Margenau.1... 

To anticipate some of the problems involved in energy calculations, we use the helium atom functions already derived to obtain the actual energy levels. The 2-electron Hamiltonian (neglecting spin terms) is... 

The tliird part is tire interaction between tire tenninal functionality, which in tire case of simple alkane chains is a metliyl group (-CH ), and tire ambient. These surface groups are disordered at room temperature as was experimentally shown by helium atom diffraction and infrared studies in tire case of metliyl-tenninated monolayers [122]. The energy connected witli tliis confonnational disorder is of tire order of some kT. 

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 ... 

All of our orbitals have disappeared. How do we escape this terrible dilemma We insist that no two elections may have the same wave function. In the case of elections in spatially different orbitals, say. Is and 2s orbitals, there is no problem, but for the two elechons in the 1 s orbital of the helium atom, the space orbital is the same for both. Here we must recognize an extr a dimension of relativistic space-time... 

The analogy is even closer when the situation in oxygen is compared with that in excited configurations of the helium atom summarized in Equations (7.28) and (7.29). According to the Pauli principle for electrons the total wave function must be antisymmetric to electron exchange. 

To make matters worse, the use of a uniform gas model for electron density does not enable one to carry out good calculations. Instead a density gradient must be introduced into the uniform electron gas distribution. The way in which this has been implemented has typically been in a semi-empirical manner by working backwards from the known results on a particular atom, usually the helium atom (Gill, 1998). It has thus been possible to obtain an approximate set of functions which often serve to give successful approximations in other atoms and molecules. As far as I know, there is no known way of yet calculating, in an ab initio manner, the required density gradient which must be introduced into the calculations. 

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... 

Power Series Expansions and Formal Solutions (a) Helium Atom. If the method of superposition of configurations is based on the use of expansions in orthogonal sets, the method of correlated wave functions has so far been founded on power series expansions. The classical example is, of course, Hyl-leraas expansion (Eq. III.4) for the ground state of the He atom, which is a power series in the three variables... 

Mulliken, R. S., Proc. Natl. Acad. Sci. U.S. 38, 160, "A comparative survey of approximate ground sate wave functions of helium atom and hydrogen molecule/ ... 

For example, the helium atom electron wave function can be written as... 

Heitler-London wave function, 15-16 Helium atom, wave function for, 3 Heterolytic bond cleavage, 46, 51, 47,53 Histidine, structure of, 110 Huckel approximation, 8,9,10,13 Hydrocarbons, force field parameters for, 112... 

The two preceding applications showed that our hydrogenic model fits well with the helium atom and the dihydrogen molecule for the determination of the polarization functions except that their exponent ( is different from Co which is the exponent of the genuine basis set It is obvious that the hydrogenic model will fit less and... 

The existence of the first HK theorem is quite surprising since electron-electron repulsion is a two-electron phenomenon and the electron density depends only on one set of electronic coordinates. Unfortunately, the universal functional is unknown and a plethora of different forms have been suggested that have been inspired by model systems such as the uniform or weakly inhomogeneous electron gas, the helium atom, or simply in an ad hoc way. A recent review describes the major classes of presently used density functionals [10]. 

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). 

To understand how electrons with spin interact, it is useful to examine a system consisting of two electrons, such as the helium atom. Let this two-electron system be described by the wave function, in space coordinates, (r), If the electrons are interchanged the wave function will in general be different, ( ), but since the electrons are identical (ignoring spin) the energy of the system will not be affected. The wave functions therefore belong to degenerate levels. 

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 ... 

The linear combination is used instead of the unsyinmetrical states / (l)cv(2) and j3 2)a ). It is reasonable to expect that each of these spin states could occur in combination with the ground-state function ij> r) to yield four different levels at the ground state. However, for the helium atom only one ground-state function can be identified experimentally and it is significant to note that only one of the spin functions is anti-symmetrical, i.e. 

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...

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... 

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... 

To expose the essence of the R12 method of Kutzelnigg [19], consider the simplest two-electron system, the helium atom in its ground state. The exact wave function in the vicinity of an electron-electron coalescence point r can be expressed [13] as... 

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