Operators Empirical one-electron

Matrix element of a semi-empirical one-electron operator, usually... 

Matrix element of a one-electron operator in semi-empirical theory... 

The Lande parameter ni in a system having spherical symmetry is essentially a one-electron operator quantity, as illustrated in Dr. Schmidtke s lecture (49). In a complex, the L. C. A. 0. M. O. description hence suggest that the new value of na (adapted to the lower Z due to central-field covalency) is multiplied by the square b representing the electronic density of the t2g sub-shell in MX6. Empirically, the dependence of ni is roughly a proportionality to Z, and we hence write... 

A Antisymmetrizing operator A Vector potential P First hyperpolarizability P Resonance parameter in semi-empirical theory B Magnetic field (magnetic induction) X, /r, A, cr Basis functions (atomic orbitals), ab initio or semi-empirical methods rraiipp inrliiflinp basis fiinrHon 7] An infinitesimal scalar rj Absolute hardness h Planck s constant H hjl K h Core or other effective one-electron operator hap Matrix element of a one-electron operator in AO basis Matrix element of a one-electron operator in semi-empirical theory... 

The angular overlap model is a relatively crude method which appears to yield results at least as good as those afforded by the crystal field model. As with all simple empirical models, the AOM depends on many approximations and assumptions which cannot be expected to be even approximately correct. Thus, for example, the parameter a is assumed to depend only on the identity of the metal and the ligand, and on the internuclear distance it is independent of the stoichiometry or stereochemistry. The theoretical basis for assuming the proportionality of the AOM matrix elements to overlap integrals is closely related to the Wolfs-berg-Helmholz approximation for the off-diagonal matrix elements of the one-electron operator ... 

The DIRAC package [55,56], devised by Saue and collaborators, rather than exploiting the group theoretical properties of Dirac spherical 4-spinors as in BERTHA, treats each component in terms of a conventional quantum chemical basis of real-valued Cartesian functions. The approach used in DIRAC, building on earlier work by Rosch [80] for semi-empirical models, uses a quaternion matrix representation of one electron operators in a basis of Kramers pairs. The transformation properties of these matrices, analysed in [55], are used to build point group transformation properties into the Fock matrix. 

The matrix elements of the spin-orbit coupling operator have been included in these works using empirically obtained or computed spin-orbit coupling constants for an effective one electron operator. The Breit-Pauli spin-orbit coupling operator (115) with all multi-center terms was employed for the first time by Kiyonaga, Morihashi and Kikuchi [125]. 

The so-called method of orthogonal operators, mentioned in the Introduction, looks fairly promising in semi-empirical calculations [20,34,134]. Its main advantage is that the addition of new parameters practically does not change the former ones. As a rule this approach allows one to reduce the root-mean-deviation of calculated values from the measured ones by an order of magnitude or so in comparison with the conventional semi-empirical method [135]. Unfortunately it requires the calculation of complex matrix elements of many-electron operators. 

Here Bk s stand for the crystal field parameters (CFP), and Ck(m) are one-electron spherical tensor operators acting on the angular coordinates of the mth electron. Here and in what follows the Wyboume notation (Newman and Ng, 2000) is used. Other possible definitions of CFP and operators (e.g. Stevens conventions) and relations between them are dealt with in a series of papers by Rudowicz (1985, 2000,2004 and references therein). Usually, the Bq s are treated as empirical parameters to be determined from fitting of the calculated energy levels to the experimental ones. The number of non-zero CFP depends on the symmetry of the RE3+ environment and increases with lowering the symmetry (up to 27 for the monoclinic symmetry), the determination of which is non-trivial (Cowan, 1981). As a result, in the literature there quite different sets of CFP for the same ion in the same host can be found (Rudowicz and Qin, 2004). 

In the phosphorescence process the spin-orbit interaction between the singlet-triplet manifolds is strongest for 1,3(ti, x") <- 3,1 (x, 7r ). This can be understood by the rotation of the lone-pair electron into the x-electron system by the spin-orbit operator. This type of atomic interaction motivates the approximations of one-electron, one-center spin-orbit operators cherished in semi-empirical work. The benzene-type of strong... 

Empirical rules have been elaborated to account for competition between these pathways, depending on electrolysis conditions [180], The Coventry group chose to examine a system almost at balance where both pathways operate [183] in order to best identify any sonoelectrochemical effect on mechanism [184], Table 4 shows product ratios (by glc) from the electrooxidation of partially neutralized cyclohex-anecarboxylate in methanol at platinum, at a current density of 200 mAmp cm-2. The first column shows a substantial amount (49%) of the dimer bicyclohexyl from the one-electron pathway, together with cyclohexylmethylether, cyclohexanol, and other products from the two-electron pathway (totaling -30%). The methyl cyclo-hexanoate ester (17%) is considered to arise from acid-catalyzed chemical esterification of the starting material with methanol solvent, due to the quantity of protons produced around the anode since at the high current densities needed, the parasitic... 

As described in the previous sections, all semiempiricals contain parameters. They either replace integrals that are calculated analytically in ab initio approaches, or they are part of empirical formulas that describe the chemical bonding, usually in the two-center one-electron part. These parametric formulas are designed to compensate for the neglect of a large part of the interatomic, three-, and four-center terms that have to be taken into account in first-principles methods. The quality of a semiempirical method therefore strongly depends not only on the formulation of the Fock operator, but also on the choice of the parameter sets. 

An even simpler but less well-justified approximation avoids the calculation of the matrix elements of the two-electron part of the operator altogether. Only the matrix elements of the one-electron part of are computed, and in the sum over nuclei a in Equation 3.3, contributions from each atom are not multiplied by Z but by the effective spin-orbit coupling nuclear charge of atom a, which has been optimized empirically to represent the partial compensation of the one-electron part by the two-electron part of the operator. Recommended values of for atoms of main-group and transition metal elements are listed in Table 3.1. This method is generally acceptable in molecules containing heavy atoms but is not very accurate in those composed of light atoms only. 

A one-electron one-centre semi-empirical spin-orbit operator acting on nodeless pseudoorbitals has the form [25] ... 

However, the solution given by Eq. (48) is based on the form of effective independent-electron Hamiltonians that can be empirically constructed as in Extended Hiickel Theory [168]. Such arbitrariness can be nevertheless avoided by the so-called self-consistent field (SCF), in which the one-electron effective Hamiltonian is considered to depend on the solution of Eq. (40) itself, i.e., the matrix of coefficients (C). The resulting Hamiltonian is called the Fock operator, while the associated eigen-problem is the Hartree-Fock equation ... 

The one-electron, non-empirical VEH method first developed by Nicolas and Durand and adapted to CPs by Br6das and others [240, 241], considers only the valence electrons explicitly, with Gaussian functions of appropriate orbital symmetry typically used for the C and H valence electrons. Coulomb interactions are simulated implicitly. In this, effective Fock operators contain the kinetic term and a sum of atomic potentials ... 

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