Interelectronic distance

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). 

Fiq. 10. Diagram Showing the Symbols Used for the Various Intehnuclear and Interelectronic Distances... 

Whereas the one-electron exponential form Eq. (5.5) is easily implemented for orbital-based wavefunctions, the explicit inclusion in the wavefunction of the interelectronic distance Eq. (5.6) goes beyond the orbital approximation (the determinant expansion) of standard quantum chemistry since ri2 does not factorize into one-electron functions. Still, the inclusion of a term in the wavefunction containing ri2 linearly has a dramatic impact on the ability of the wavefunction to model the electronic structure as two electrons approach each other closely. 

However, this deficiency of the FCI expansion is easily rectified by including in the wavefunction a single extra term that is linear in the interelectronic distance. The resulting wavefunction may be written as [42]... 

At the other extreme in terms of system size and accuracy stand brute-force approaches such as those based on wavefunctions with explicit interelectronic distances. 

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 symmetry requirements and the need to very effectively describe the correlation effects have been the main motivations that have turned our attention to explicitly correlated Gaussian functions as the choice for the basis set in the atomic and molecular non-BO calculations. These functions have been used previously in Born-Oppenheimer calculations to describe the electron correlation in molecular systems using the perturbation theory approach [35 2], While in those calculations, Gaussian pair functions (geminals), each dependent only on a single interelectron distance in the exponential factor, exp( pr ), were used, in the non-BO calculations each basis function needs to depend on distances between aU pairs of particles forming the system. 

Here r, (with magnitude r,) describes the position of electron i relative to the nucleus, rj2 is the interelectron distance ri —r2l, and V, is with respect to the coordinates of r,. We will later refer to the momentum operators p, = — iV,-. 

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

Gordy and Gordy and Morehouse derived an equation for calculating this average or effective distance based upon the assumption that the location of each spin dipole may be assigned approximately to a point. The result may be stated as in Eq. 1, in which 2D in gauss is the separation between the two outermost of the six (or four) lines, and R is the effective interelectronic distance in angstrom units ... 

The zero-field values for TMM, D = 0.024 cm (2D = 513 G) and E < 0.001 cm , were compatible with the expected triplet ground state and with an effective interelectron distance of 4.8 A. Moreover, when the species was generated in a crystalline host, the lines were visibly further split into a... 

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... 

---