Slaters Bond Functions

Fairly soon after the Heitler-London calculation, Slater, using his determi-nantal functions, gave a generalization to the n-electron VB problem[10]. This was a popular approach and several studies followed exploiting it. It was soon called the method of bond eigenfunctions. A little later Rumer[ll] showed how the use of these could be made more efficient by eliminating linear dependencies before matrix elements were calculated. 

Slater s bond eigenfunctions constitute one choice (out of an infinite number) of a particular sort of basis function to use in the evaluation of the Hamiltonian and overlap matrix elements. They have come to be called the Heitler-London-Slater-Pauling (HLSP) functions. Physically, they treat each chemical bond as a singlet-coupled pair of electrons. This is the natural extension of the original Heitler-London approach. In addition to Slater, Pauling[12] and Eyring and Kimbal[13] have contributed to the method. Our following description does not follow exactly the discussions of the early workers, but the final results are the same. 

Consider a singlet molecule with 2n electrons, where we wish to use a different atomic orbital (AO) for each electron. We can construct a singlet eigenfunction of the total spin as the product of n electron pair singlet functions 

the total spin operator may be written as S2 = SftS + SZ(SZ + 1), and, therefore, it is seen that S2 = 0 and is a singlet spin function. 

We now multiply by a product of the orbitals, one for each particle, i(1) 2(2) U2n(2n), where x, 2, 2n is some particular ordering of the orbital set. When we apply the antisymmetrizer to the function of space and spin variables, the result can be written as the sum of 2 SDFs. It is fairly easily seen that there are (2n) /(2 n ) different 2n-electron functions of this sort that can be constructed. Rumer s result, referred[ll] to above, shows how to remove all of the linear dependences in this set and arrive at the minimally required number, (2n) /[n (n + 1)1], of bond functions to use in a quantum mechanical calculation. 

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The calculations are not all at exactly the same bond length R. The basis set is indicated after the slash in the method. R, L, C, and T are basis sets of Slater-type functions. The aug-cc-pVDZ and aug-cc-pVTZ basis sets [360] are composed of Gaussian functions. SCF stands for self-consistent-field MC, for multiconfiguration FO, for first-order Cl, for configuration interaction MR, for multireference MPn, for nth-order Mpller-Plesset perturbation theory and SDQ, for singles, doubles, and quadruples. 

The Slater valence-bond function leads to an energy expression that contains the single exchange integrals between bonded orbitals, such as a and b, with the coefficient +1. These integrals are usually negative,... 

RMS root-mean-square SCVB spin coupled valence bond SDF Slater determinantal functions STF standard tableau function VB valence bond... 

The MODPOT/VRDDO LCA0-M0-SCF programs have been meshed in with the configuration interaction programs we use. In this Cl (71) program each configuration is a spin- and symmetry-adapted linear combination of Slater determinants in the terms of the spin-bonded functions of Boys and Reeves ( 2, 3> 7 0 as formu-... 

The spin functions used by the program are the set of Rumer ° (or bonded) functions. Besides highlighting the formation or disintegration of covalent bonds, this basis has the advantage that the structures in it are very easily decomposed into Slater determinants. The coefficients of the individual determinants are just + 1. The Rumer basis of spin functions is not orthogonal, and a linearly independent set is systematically generated using... 

Before going on to IA2, let us express the Heitler-London valence-bond functions for H2 as Slater determinants. The ground-state Heitler-London function (13.101) can be written as... 

Other calculations tested using this molecule include two-dimensional, fully numerical solutions of the molecular Dirac equation and LCAO Hartree-Fock-Slater wave functions [6,7] local density approximations to the moment of momentum with Hartree-Fock-Roothaan wave functions [8] and the effect on bond formation in momentum space [9]. Also available are the effects of information theory basis set quality on LCAO-SCF-MO calculations [10,11] density function theory applied to Hartree-Fock wave functions [11] higher-order energies in... 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

For four electrons, for example, with only spin degeneracy (the number of occupied orbits equalling the number of electrons), Slater gave the function J(SI i — ipii — iPiii- - I iv) as representing the structure in which orbits a and b are bonded together, and also c and d. Here Pi- Piv... 

Figure4.7 Relativistic bond contractions A re for Au2 calculated in the years from 1989 to 2001 using different quantum chemical methods. Electron correlation effects Acte = te(corn) — /"e(HF) at the relativistic level are shown on the right hand side of each bar if available. From the left to the right in chronological order Hartree-Fock-Slater results from Ziegler et al. [147] AIMP coupled pair functional results from Stbmberg and Wahlgren [148] EC-ARPP results from Schwerdtfeger [5] EDA results from Haberlen and Rdsch [149] Dirac-Fock-Slater...

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