Radial factor

For interactions between two quadmpolar molecules which have 0 and 0g of the opposite sign, at a fixed separation r, the angular factor in equation (A1.5.13t leads to a linear stmcture, 0 = 0g = 0, being the most attractive. Linear molecules may also prefer a 2 rectangular or non-planar cross arrangement with 0 = 0g = nil, which allows them to approach closer and increase the radial factor. 

They must be coupled by separate radial factors in a full calculation [2] but, to the extent that non-adiabatic coupling between the upper and lower... 

The radial factor in the total wave function must therefore be an eigenstate of... 

The radial factors of the hydrogen-like atom total wave functions ip r, 0, tp) are related to the functions Sni(p) by equation (6.23). Thus, we have... 

As a function of r the general expression for the radial factors of the wave function becomes... 

Not surprisingly, formalisms with very diffuse density functions tend to yield large electrostatic moments. This appears, in particular, to be true for the Hirshfeld formalism, in which each cos 1 term in the expansion (3.48) includes diffuse spherical harmonic functions with / = n, n — 2, n — 4,... (0, 1) with the radial factor rn. For instance when the refinement includes cos4 terms, monopoles and quadrupoles with radial functions containing a factor r4 are present. For pyridin-ium dicyanomethylide (Fig. 7.3), the dipole moment obtained with the coefficients from the Hirshfeld-type refinement is 62.7-10" 30 Cm (18.8 D), whereas the dipole moments from the spherical harmonic refinement, from integration in direct space, and the solution value (in dioxane), all cluster around 31 10 30 Cm (9.4 D) (Baert et al. 1982). 

X-ray diffraction, 39 160-161, 164-165 Radial factor, 22 214-218 Radiation, see also Irradiation cosmic, see Cosmic radiation terrestial origin, 3 288-293 Radiation chemistry, heterogeneous kinetics of, 3 198-203... 

Co(II) in chloroform (40), In this latter case, the AF difference is about 5 kcal/mole, so the Ni-Co AF difference is smaller. Compare this with the 20 kcal/mole site-preference energy for octahedral Ni(II) over Co(II), according to crystal-field theory [for example, (56)], The fundamental assumption of crystal-field theory, of course, is that the radial factors in cation-coordination-sphere relations are constant, which is tantamount to saying that bonding does not change. As we have seen in deriving Eq. (24), binding factors are very important (8) and, as the above numerical relations confirm, play by far the dominant role. 

These manipulations have brought us to a familiar equation we recognize (4.23) as the Schrodinger equation (1.132) for a one-dimensional harmonic oscillator with force constant ke. Before we can conclude that (4.23) and (1.132) have the same solutions, we must verify that the boundary conditions are the same. For quadratic integrability, we require that S(q) vanish for q = oo. Also, since the radial factor F(R) in the nuclear wave function is... 

This is transfer covariant if all quadratically integrable functions are represented in the same orbital basis. Requiring fps to be orthogonal to all unique representation, but introduces Lagrange multipliers in the close-coupling equations. An alternative is to require... 

We think of this integral as consisting of an angular" factor which we define as +1 and a radial" factor S MXt The angular" factor is called angular because in... 

Another interesting possibility, which will not be discussed here, may consist into expanding the radial factors e and re as a linear combination of primitive gaussian functions [69]. Studies about this simpler approximation is imder way in our Laboratory. 

The radial factor R r) (Table 2-4) is determined by the quantum numbers n and /, the principal and angular momentum quantum numbers. 

It is a characteristic feature of AOM that its matrix elements may be factorized into a parametric factor, which one may call the radial factor, and a factor which makes up the coefficient to this semiempirical parameter 8—11). The latter factor depends only on angular coordinates The angular properties of the I orbitals, and the relative astronomical... 

The eigenfunctions of the free electron confined in the same prolate spheroids are expressed as products of regular radial and angular spheroidal wave functions [16] Chapter 21, in the respective coordinates u and v, and the eigenfunctions of Equations (34) and (35). The radial functions are expressed as infinite series of spherical Bessel functions of order m + s and argument kfu. Its eigenvalues are determined by the boundary condition on the radial factor,... 

Electron-electron interaction is significant in studies where spin-orbital degeneracy, or near degeneracy, is present. The atomic d-orbitals have a common radial factor and their 15 distinct products give rise to 120 density-density integrals. These are expressed in terms of three basic ones, F0, F2, and F4 in the Slater-Condon formulation. The 100 spin-orbital densities are linear combination of the orbital ones and the 100 by 100 interaction integral matrix has a rank of 15 and is expressed by the Slater-Condon parameters. 

The number of electrons in the system is denoted by q. This conclusion depends on the assumption that the radial factor of the atomic orbitals is the same and will be void when a more general model is invoked. 

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