Parametric orbital term

The Thouless determinantal electronic wave function z) = det Xi(xj) in Eq. (21) is an example of such proper parametrization. The dynamical spin orbitals are expressed in terms of atomic spin orbitals centered on the various nuclei... 

The exact nature of the axial field is irrelevant within this model — for example, it could be any of the distortions illustrated in Fig. 3 — and its effect is represented parametrically by a splitting between the orbital doublet and orbit singlet arising from the triplet 2 T2 term. The splitting, A in Fig. 4 (defined conventionally as positive if the singlet lies lowest), is thus defined, not in terms of the operator... 

As a parametrization scheme this model is a typical intermediate between the crude spherical parametrization of the B and C parameters and the complete (and therefore impractical) set of the tetragonal repulsion integrals [31]. A similar strategy was used before by Koide and Pryce [32] in their treatment of octahedral d5-complexes they showed that quasi-degenerate terms of the half-filled d-shell configuration could be splitted by introducing a covalency difference between the t2g and eg orbitals. 

In the parametrization of equ. (4.68) the terms associated with the Legendre polynomials Pk(cos ab) represent that part of the angular correlation which is independent of the light beam, while the terms associated with the bipolar harmonics are due to the multipole expansion of the interactions of the electrons with the electric field vector. The link between geometrical angular functions and dynamical parameters is made by the summation indices ku k2 and k. These quantities are related to the orbital angular momenta of the two individual emitted electrons, and they are subject to the following conditions ... 

The function ij/(r, 9, p) (clearly ij/ could also be expressed in Cartesians), depends functionally on r, 6, p and parametrically on n, l and inm for each particular set (n. I, mm ) of these numbers there is a particular function with the spatial coordinates variables r, 6, p (or x, y, z). A function like /rsiiir is a function of x and depends only parametrically on k. This ij/ function is an orbital ( quasi-orbit the term was invented by Mulliken, Section 4.3.4), and you are doubtless familiar with plots of its variation with the spatial coordinates. Plots of the variation of ij/2 with spatial coordinates indicate variation of the electron density (recall the Bom interpretation of the wavefunction) in space due to an electron with quantum numbers n, l and inm. We can think of an orbital as a region of space occupied by an electron with a particular set of quantum numbers, or as a mathematical function ij/ describing the energy and the shape of the spatial domain of an electron. For an atom or molecule with more than one electron, the assignment of electrons to orbitals is an (albeit very useful) approximation, since orbitals follow from solution of the Schrodinger equation for a hydrogen atom. 

An approximate (non-SCF) molecular-orbital method involving extensive parametrization of the required integrals and with the overlap integral represented as a simple product of radial and angular terms... 

For some purposes, a basis set consisting of hybridized atomic orbitals is particularly suitable in LCAO—MO calculations. By taking hybrids directed along the chemical bonds instead of pure atomic orbitals defined in terms of arbitrary axes, one simultaneously retains the essential features of the bond orbital picture and the standard delocalized method. This method has been developed in a parametric form similar to the standard Hiickel method including or not including overlap integrals 41). 

Nowadays, the success of the methods proposed by Hoffmann 50> and by Pople and Segal 51> among the chemists tends to promote the use of pure atomic orbital bases for all-valence treatments. The first method is a straightforward application of the Wolfsberg-Helmholz treatment of complexes to organic compounds and is called the Extended Hiickel Theory (EHT), because its matrix elements are parametrized in the same way as the Hiickel method with overlap for n electrons. The other method, known under the abbreviation Complete Neglect of Differential Overlap (CNDO), includes electron repulsion terms by extending to a orbitals the successful approximation of zero-differential overlap postulated for n electrons. 

---