Orbital product

Because of the indistingiiishability of the electrons, the antisynnnetric component of any such orbital product must be fonned to obtain the proper mean-field wavefunction. To do so, one applies the so-called antisynnnetrizer operator [24] A= Y.p -lf p, where the pemuitation operator mns over all A pemuitations of the N electrons. Application of 4 to a product fiinction does not alter the occupancy of the fiinctions ( ). ] in it simply scrambles the order which the electrons occupy the ( ). ] and it causes the resultant fiinction... 

By expressing the mean-field interaction of an electron at r with the N- 1 other electrons in temis of a probability density pyy r ) that is independent of the fact that another electron resides at r, the mean-field models ignore spatial correlations among the electrons. In reality, as shown in figure B3.T5 the conditional probability density for finding one ofA - 1 electrons at r, given that one electron is at r depends on r. The absence of a spatial correlation is a direct consequence of the spin-orbital product nature of the mean-field wavefiinctions... 

Wlien considering the ground state of the Be atom, the following four antisyimnetrized spin-orbital products are found to have the largest amplitudes ... 

The following types of innltipole distributions are used to represent the four types of atomic orbital products. 

X spin orbital (product of spatial orbital and a spin function)... 

To understand why integrals over GTOs can be carried out when analogous STO-based integrals are much more difficult, one must only consider the orbital products (XaXc (ri) and XbXd (J l)) which arise in such integrals. For orbitals of the GTO form, such products involve exp(-tta (r-Ra) ) exp(-ac (r-Rc) ). By completing the square in the exponent, this product can be rewritten as follows ... 

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... 

QUANTUM CHEMISTRY COMPUTATIONS IN MOMENTUM SPACE Fourier transform of orbital products ... 

As was mentioned previously, simple orbital products (electron configurations) must be converted into antisymmetrized orbital products (Slater determinants) in order to satisfy the Pauli principle. Thus, proper many-electron wavefunctions satisfy constraints of exchange antisymmetry that have no counterpart in pre-quantum theories. 

In Eq. (1.16a), A is the antisymmetrizer operator that converts the orbital product into a Slater determinant, insuring satisfaction of the Pauli exclusion principle. In this equation (alone), the same spatial orbital might appear twice, with different indices to indicate the change in spin. For example, / i (0,(7 ypf HA) might be the same as i<0)(F K/>,0,0" 2). a doubly occupied spatial orbital (n]m> = 2), with a bar denoting opposite spin in the second spin-orbital. 

The pairwise overlap, symmetrical orthogonalisation and Mulliken approximation together validate the NDO approximation — the orbital product in the orthogonalised GHO basis vanishes to the extent that the Mulliken approximation is realistic. This conclusion obviously has enormous consequences for any NDO approximation schemes. 

The coefficients n, have to obey the condition n, f, imposed by Poisson s electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). 

Similarly, octupoles and hexadecapoles can be thought of as arising from 2p3d and 3d3d atomic orbital products, which leads to n, = 3 and 4, respectively. The... 

Here, Y(j is the 15-element column vector of the angular part of the < (df)0(dj) orbital products, Py is the row vector of the 15 unique elements of the symmetric 5x5 matrix of the coefficients in Eq. (10.3), and Pimp is the row vector containing the coefficients of the 15 spherical harmonic density functions d,mp with / = 0, 2, or 4. Density functions with other / values do not contribute to the d-orbital density. 

The populations Pkl of the symmetry-adapted orbital products follow, in analogy to Eq. (10.7), from... 

Equation (11.7) implies that orbital products can well be approximated in the average by some Mulliken-type expansion [206,218]. 

We have so far said little about the nature ofthe space function, S. Earlier we implied that it might be an orbital product, but this was not really necessary in our general work analyzing the effects of the antisymmetrizer and the spin eigenfunction. We shall now be specific and assume that S is a product of orbitals. There are many ways that a product of orbitals could be arranged, and, indeed, there are many of these for which the application of the would produce zero. The partition corresponding to the spin eigenfunction had at most two rows, and we have seen that the appropriate ones for the spatial functions have at most two columns. Let us illustrate these considerations with a system of five electrons in a doublet state, and assume that we have five different (linearly independent) orbitals, which we label a, b,c,d, and e. We can draw two tableaux, one with the particle labels and one with the orbital labels. 

We saw in Chapter 4 that the number of independent functions is reduced to one if two of the three electrons are in the same orbital. A similar reduction occurs in general. In our five-electron example, if b is set equal to a and c f d, there are only two linearly independent functions, illustrating a specific case of the general result that the number of linearly independent functions arising from any orbital product is determined only by the orbitals outside the doubly occupied set. This is an important point, for which now we take up the general rules. 

The same sort of considerations allow one to determine matrix elements. Eet ui(l) v n) = T be another orbital product. There is a joint overlap matrix between the v- and m-functions ... 

Alternatively, if results of ab initio theory at the single-configuration orbital-product wavefunction level are used to define the parameters of a semi-empirical model, it would then be proper to use the semi-empirical orbitals in a subsequent higher-level treatment of electronic structure as done in Section 6. 

In particular, within the orbital model of electronic structure (which is developed more systematically in Section 6), one can not construct trial wavefunctions which are simple spin-orbital products (i.e., an orbital multiplied by an a or (3 spin function for each electron) such as 1 sa 1 s(32sa2s(32pia2poa. Such spin-orbital product functions must be made permutationally antisymmetric if the N-electron trial function is to be properly antisymmetric. This can be accomplished for any such product wavefunction by applying the following antisymmetrizer operator ... 

It should be noted that the effect of A on any spin-orbital product is to produce a function that is a sum of N terms. In each of these terms the same spin-orbitals appear, but the order in which they appear differs from term to term. Thus antisymmetrization does not alter the overall orbital occupancy it simply "scrambles" any knowledge of which electron is in which spin-orbital. 

The antisymmetrized orbital product A i( )2ct>3 is represented by the short hand I 4> l4>24>3 I and is referred to as a Slater determinant. The origin of this notation can be made clear by noting that (1/a/N ) times the determinant of a matrix whose rows are labeled by the index i of the spin-orbital (f>i and whose columns are labeled by the index j of the electron at rj is equal to the above function A (f>i< )2<1>3 = (1/V3 ) det(cf>i (rj)). The general structure of such Slater determinants is illustrated below ... 

The antisymmetry of many-electron spin-orbital products places constraints on any acceptable model wavefunction, which give rise to important physical consequences. For example, it is antisymmetry that makes a function of the form I Isa Isa I vanish (thereby enforcing the Pauli exclusion principle) while I lsa2sa I does not vanish, except at points ri and 1 2 where ls(ri) = 2s(r2), and hence is acceptable. The Pauli principle is embodied in the fact that if any two or more columns (or rows) of a determinant are identical, the determinant vanishes. Antisymmetry also enforces indistinguishability of the electrons in that Ilsals(32sa2sp I =... 

One needs to learn how to tell which term symbols will be Pauli excluded, and to learn how to write the spin-orbit product wavefunctions corresponding to each term symbol and to evaluate the corresponding term symbols energies. 

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