Wave function interelectronic distances

In other words, the exact wave function behaves asymptotically as a constant 4- l/2ri2 when ri2 is small. It would therefore seem natural that the interelectronic distance would be a necessary variable for describing electron correlation. For two-electron systems, extremely accurate wave functions may be generated by taking a trial wave function consisting of an orbital product times an expansion in electron coordinates such as... 

The outcome was certainly good but, according to Hylleraas opinion, the series (Eq. III.2) converged too slowly. In 1929, Hylleraas tried instead to introduce the interelectronic distance r12 in the wave function itself, which is then called a correlated wave function. In treating the S ground state, he actually used the... 

Because of the success of the r12 method in the applications, one had almost universally in the literature adopted the idea of the necessity of introducing the interelectronic distances r j explicitly in the total wave function (see, e.g., Coulson 1938). It was there-fore essential for the development that Slater,39 Boys, and some other authors at about 1950 started emphasizing the fact that a wave function of any desired accuracy could be obtained by superposition of configurations, i.e., by summing a series of Slater determinants (Eq. 11.38) built up from a complete basic one-electron set. Numerical applications on atoms and molecules were started by means of the new modern electronic computers, and the results have been very encouraging. It is true that a wave function delivered by the machine may be the sum of a very large number of determinants, but the result may afterwards be mathematically simplified and physically interpreted by means of natural orbitals.22,17... 

For systems containing three or more electrons very little is so far known about the foundation for the method of correlated wave functions, and research on this problem would be highly desirable. It seems as if one could expect good energy results by means of a wave function being a product of a properly scaled Hartree-Fock function and a correlation factor" containing the interelectronic distances ru (Krisement 1957), but too little is known about the limits of accuracy of such an approach. 

The method of superposition of configurations as well as the method of different orbitals for different spins belong within the framework of the one-electron scheme, but, as soon as one introduces the interelectronic distance rijt a two-electron element has been accepted in the theory. In treating the covalent chemical bond and other properties related to electron pairs, it may actually seem more natural to consider two-electron functions as the fundamental building stones of the total wave function, and such a two-electron scheme has also been successfully developed (Hurley, Lennard-Jones, and Pople 1953, Schmid 1953). 

An important difference between the BO and non-BO internal Hamiltonians is that the former describes only the motion of electrons in the stationary field of nuclei positioned in fixed points in space (represented by point charges) while the latter describes the coupled motion of both nuclei and electrons. In the conventional molecular BO calculations, one typically uses atom-centered basis functions (in most calculations one-electron atomic orbitals) to expand the electronic wave function. The fermionic nature of the electrons dictates that such a function has to be antisymmetric with respect to the permutation of the labels of the electrons. In some high-precision BO calculations the wave function is expanded in terms of basis functions that explicitly depend on the interelectronic distances (so-called explicitly correlated functions). Such... 

The interelectronic distance is introduced into the wave function through the following two-electron integrals ... 

The essence of the method of incomplete separation of variables consists in introducing the interelectronic distances (usually only in an open shell) in an atom. Then its wave function... 

In order to achieve a high accuracy, it would seem desirable to explicitly include 4.11 Methods Involving Interelectronic Distances terms in the wave functions which are linear in the interelectronic distance. This is the idea in the R12 methods developed by Kutzelnigg and co-workers.-" The first order The necessity for going beyond the HF approximation is the fact that electrons are correction to the HF wave function only involves doubly excited determinants (eqs. further apart than described by the product of their orbitd densities, i.e. their motions (4 35) (4.37)). In R12 methods additional terms are included which essentially are ... 

Therefore, although it is possible in principle to account for electronic correlation without explicit reference to interelectronic distances, it should be emphasized that for the ground state of the simple system of the isolated hydrogen molecule a virtual check with experiment has been olitained only with explicit inclusion of interelectronic distances. - This approach is based on a power series expansion of the wave function, which for the hydrogen molecule may be written as... 

This means that the first derivative of the angular average of the wave function with respect to the interelectronic distance at points where the positions of the two electrons coincide is related to the value of the wave function at the same points. If the latter value does not happen to vanish, then the first derivative is non-zero as well. This means that ip ri, ra) has a cusp i.. a discontinuity in its first derivative) as illustrated in Fig. 3. 

For hydrogen the exact wave function is known. For helium and lithium, very accurate wave functions have been calculated by including interelectronic distances in the variation functions. For atoms of higher atomic number, the best approach to finding a good wave function lies in first calculating an approximate wave function using the Hartree-Fock procedure, which we shall outline in this section. The Hartree-Fock method is the basis for the use of atomic and molecular orbiteils in many-electron systems. 

We have already indicated two of the ways in which we may provide for instantaneous electron correlation. One method is to introduce the interelectronic distances r into the wave function (Section 9.4). This method is only practicable for systems with a few electrons. 

The cusp condition imphes that the exact wave function must be Hnear in the interelectronic distance for small values of ri2. 

---