Slater sum rules

As in the spin-polarized NR case, the convenience of having only two potentials to represent magnetic interactions is obtained at a price. This price includes some contamination of the SCF solutions with a mixture of multiplets, which can sometimes be resolved by projection techniques, including for example, the Slater Sum Rule of atomic theory. The ease of calculation of an R potential which treats exchange in open-shell heavy atom systems reasonably well, without introducing artificial (and incorrect) spin-polarization is a considerable advantage. 

The BS plus spin projection method discussed here is closely connected to the simple open-shell singlet method for optical excitations based on the Slater sum rule and ASCF (self-consistent-field total energy difference method). The mixed spin excited state is like the BS state, also of mixed spin. The Slater sum rule method" " is also quite effective for multiplet problems for excited states of transition metal complexes as shown in the work of Dahl and Baerends. ... 

The amplitude for the so-ealled referenee CSF used in the SCF proeess is taken as unity and the other CSFs amplitudes are determined, relative to this one, by Rayleigh-Sehrodinger perturbation theory using the full N-eleetron Hamiltonian minus the sum of Foek operators H-H as the perturbation. The Slater-Condon rules are used for evaluating matrix elements of (H-H ) among these CSFs. The essential features of the MPPT/MBPT approaeh are deseribed in the following artieles J. A. Pople, R. Krishnan, H. B. Sehlegel, and J. S. Binkley, Int. J. Quantum Chem. 14, 545 (1978) R. J. Bartlett and D. M. Silver, J. Chem. Phys. 3258 (1975) R. Krishnan and J. A. Pople, Int. J. Quantum Chem. 

The frozen-core (fc) approach is not restricted to spin-independent electronic interactions the spin-orbit (SO) interaction between core and valence electrons can be expressed by a sum of Coulomb- and exchange-type operators. The matrix element formulas can be derived in a similar way as the Sla-ter-Condon rules.27 Here, it is not important whether the Breit-Pauli spin-orbit operators or their no-pair analogs are employed as these are structurally equivalent. Differences with respect to the Slater-Condon rules occur due to the symmetry properties of the angular momentum operators and because of the presence of the spin-other-orbit interaction. It is easily shown by partial integration that the linear momentum operator p is antisymmetric with respect to orbital exchange, and the same applies to t = r x p. Therefore, spin-orbit... 

These rules are a consequence of the fact that the spin-orbit operator for the many-electron states is a sum of one-particle operators according to (5) and the Slater-Condon rules for matrix elements between states of such operators [121]. 

The Hamilton operator consists of a sum of one-electron and two-electron operators, eq. (3.24). If two determinants differ by more than two (spatial) MOs there will always be an overlap integral between two different MOs which is zero (same argument as in eq. (3.28)). Cl matrix elements can therefore only be non-zero if the two determinants differ by 0, 1, or 2 MOs, and they may be expressed in terms of integrals of one- and two-electron operators over MOs. These connections are known as the Slater-Condon rules. If the two determinants are identical, the matrix element is simply the energy of a single determinant wave function, as given by eq. (3.32). For matrix elements between... 

On a most elementary level, the CSFs can be reduced to sums over Slater determinants and then the Slater-Condon rules can be used to calculate the actual matrix elements. Recipes to do this are readily found in the literature. 

In this expression the energies are just sums of orbital energies so that the energy denominators are simply orbital energy differences. The numerators are the squares of related pairs of MO repulsion integrals (by Slater s rules). 

To calculate the electron affinity energy, we need to consider a determinant as large as 3x3, but this proves easy if the useful Slater-Condon rules (see Appendix M available at booksite.elsevier.com/978-0-444-59436-5) are applied. Rule I gives (we write everything using the ROHF spinorbitals, then note that the three spinorbitals are derived from two orbitals, and then sum over the spin variables) ... 

The energy expression can be found using a set of rules known as the Slater-Condon-Shortley rules. These rules are discussed in all the classic texts, the idea being that the energy expression which involves integration over the coordinates of all the electrons can be reduced to a much simpler sum of terms involving the coordinates of one and (at most) two electrons. The variational energy works outas... 

Let us first become familiar with the spin-free (sf) case we have a pair of Slater determinants interacting via (i.e., a sum of spin-independent one-and two-electron operators, h and 12). Their matrix element is given by (Sla-ter-Condon rules) ... 

Since the vibronic coupling is a sum of the one-electron operators Va(ri), the matrix element over Slater determinants can be deduced using the following rule [36] ... 

Before we enter into a more detailed discussion on the determination of the molecular electric quadrupole moments and on additivity rules for atom susceptibilities, we will draw some general conclusions from the theoretical expressions for the g- and -values given in Eqs. (1.2) and (1.4), respectively. We first restate that the perturbation sums are necessarily zero if the total electronic wavefunction (for simplicity we may tliink of a Slater determinant) has cylindrical symmetry with respect to the rotational axis in consideration. To see this, we recall that in cylindrical coordinates with a as the sjmimetry axis ... 

The rules given by Slater must be used when evaluating matrix elements of any operator in a Slater determinantal representation (Slater, 1960, p. 291). These rules reduce the matrix elements of one- and two-electron operators between many-electron Slater determinantal functions to simple sums over spatial orbital... 

The sum-over-states perturbation theory (SOS-PT) requires an evaluation of matrix elements between the ground and monoexcited electron configurations and again the Slater rules are helpful... 

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