Slater-Condon integrals

Here, p2 and G represent the well known Slater-Condon integrals in terms of whieh the eoulomb and exehange integrals ean be expressed ... 

In other words, we estimate the spectrum of the AF-singlet ground state by applying a shift, to the calculated undectet spectmm. For this case we use an empirical shift of 13500 cm so as to yield maximum coincidence between the calculated and observed spectra, a value easily obtained, for example, if we had used the Slater-Condon integrals from the Fe ion in equation (1) rather than the values for neutral Fe. 

The Slater—Condon integrals Ft(ff), Ft(fd), and Gj-(fd), which represent the static electron correlation within the 4f" and 4f 15d1 configurations. They are obtained from the radial wave functions R, of the 4f and 5d Kohn—Sham orbitals of the lanthanide ions.23,31... 

Table 3. Calculated Slater—Condon Integrals, Spin—Orbit Coupling Constants, and Ligand Field Parameters (in cm-1) for CsMgBr, Eu2+ Considering the Ground (GC) and Excited (EC) Configurations Local Structures of the Eu2+ Impurity...

Additional parameters are required in INDO/S. These are the Slater-Condon integrals, Eq. [12d], and it is necessary to evaluate, for example, (sp ps) and p,py pypP) they are taken from atomic spectroscopy. Predictions from the INDO/S model are similar to those of CNDO/S for n- n" excitations for molecules containing H and the first-row atoms. The INDO/S model is more successful for n >7t excitations (and is capable of distinguishing the n- n singlet and triplet states, separated at first order by terms that are zero under CNDO). The INDO model is much more successful for molecules containing heavier elements, where the Slater-Condon integrals are much larger. 

There have been several attempts to combine information from atomic physics with the Anderson impurity scheme outlined above, so as to calculate multiplet structure consistently with the occurrence of hybridisation. Prom the atomic physics standpoint, the possibility of correcting for the loss of localised 4/-electron density by reducing the Slater-Condon integrals by 10 or 20% respectively, depending on whether one or two 4/ electrons are involved, has been suggested [620]. The precise amount of the correction required is of course borrowed from parametric studies, because atomic physics per se can yield no information whatever on the reduction of charge density due to hybridisation. This is the reverse path to the one outlined above, i.e. it feeds back information from the parar metric studies into atomic calculations, whose ab initio character must of course be sacrificed. 

This is a popular semiempirical method (often referred to as ZINDCVS or ZINDO) for calculation of electronic spectra of both organic molecules and TM species. 75,277,278320 INDO/S parametrization was carried out at the CIS level (see Section 2.38.4.2). The Slater-Condon integrals, which are used to evaluate the TERIs, were taken from atomic spectroscopy data. The calculated transition energies are chosen to match energies of absorption maxima, as opposed to absorption band origins. 

The //yj matrices are, in practice, evaluated in temis of one- and two-electron integrals over the MOs using the Slater-Condon mles [M] or their equivalent. Prior to fomiing the Ffjj matrix elements, the one-and two-electron integrals. 

The one-eenter exchange integrals that INDO adds to the CNDO schcmccan be related to th e-Slater-Condon param eters h", O. and F used to describe atomic spectra. In particular, for a set of s, p,. p,.. t, atom ie orbitals, all the on e-ecn ter in tegrals are given as ... 

In this form, it is elear that E is a quadratie funetion of the Cl amplitudes Cj it is a quartie funetional of the spin-orbitals beeause the Slater-Condon rules express eaeh <

Cl matrix element in terms of one- and two-eleetron integrals < > and... 

These density matriees are themselves quadratie funetions of the CI eoeffieients and they refleet all of the permutational symmetry of the determinental funetions used in eonstrueting F they are a eompaet representation of all of the Slater-Condon rules as applied to the partieular CSFs whieh appear in F. They eontain all information about the spin-orbital oeeupaney of the CSFs in F. The one- and two- eleetron integrals < I f I > and < (l)i(l)j I g I (l)ic(l)i > eontain all of the information about the magnitudes of the kinetie and Coulombie interaetion energies. 

As a result, the exaet CC equations are quartic equations for the ti , ti gte. amplitudes. Although it is a rather formidable task to evaluate all of the eommutator matrix elements appearing in the above CC equations, it ean be and has been done (the referenees given above to Purvis and Bartlett are espeeially relevant in this eontext). The result is to express eaeh sueh matrix element, via the Slater-Condon rules, in terms of one- and two-eleetron integrals over the spin-orbitals used in determining , ineluding those in itself and the Virtual orbitals not in . 

Apart from minor exceptions all other parameters are given the same values as in standard INDO (electroneguivities, Slater-Condon parameters, bonding parameters) or CNDO/S (one-centre repulsion integrals) methods. Two-centre repulsion integrals are usually evaluted by the Ohno-Klopman formula. 

In the Slater-Condon [480] theory the centers of gravity (baricenters) of different terms arising from a given configuration (definite L and S values) can be expressed as multiples of the Fk integrals. 

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... 

At the INDO level Pople et al 27> have expressed the one-center repulsion integrals in terms of Slater-Condon Fft and G parameters. Thus... 

The 16 spin orbitals in this determinant are the Kohn-Sham spin orbitals of the reference system each is the product of a Kohn-Sham spatial orbital y/i and a spin function a or jS. Equation 7.18 can be written in terms of the spatial KS orbitals by invoking a set of rules (the Slater-Condon rules [34]) for simplifying integrals involving Slater determinants ... 

Using the Slater-Condon parameters all symmetry allowable two-electron one-center integrals, which brought about a modification of the matrix elements of the Fock operator, are taken into account. This allows one to write down the matrix elements of the effective Fock operator ... 

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... 

The last ratio is independent of the parameters, a and 3 and may regarded as a purely group theoretical (kinematic) result. The same result is obtained dynamically without group theory through the Slater-Condon parameters which are integrals of e2/rjj over analytic hydrogenic orbitals. The chain has been interpreted as the sum of a p orbital-p orbital and a spin orbital-spin orbital magnetic dipole interaction. This inteipretation is clearly nonphysical. 

Coulomb and exchange integrals between f electrons expressed in Slater-Condon-Shortley parameters2. 

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