Integrals radial

The expressions of the Sections 1.5 and 1.6 are general and apply to any solution of the Schrodinger equation. In the special case of a Morse potential, the radial integrals in Eq. (1.34) can be evaluated, with some approximations, in closed form. The approximation consists in replacing the lower limit of integration by -oo. This approximation is similar to that used in Section 1.3 when obtaining the wave functions. Thus... 

Here / is a nuelear spin, F is the full momentum of the system, and J is the full electron momentum. The hyperfine splitting constants are expressed through the standard radial integrals ... 

Here gi is the Lande factor and Q, the quadruple momentum of the nucleus, and the radial integrals are defined as follows ... 

In the radial integration with the form (B.28), expressions (B.9) have dominating contributions from the largest negative powers of p. Therefore a single term is chosen in (B.28), and not a negative power series. Consequently we can write [13,20]... 

Examination of Eqs. 2.18, 2.19 or 2.21, 2.22 reveals that the radial integral in the expression for Zy is roughly proportional to the volume v so that Z2f varies nearly as v2. The bound states partition function, Zjb, on the other hand, depends roughly linearly on volume. 

This expression factorizes into a radial integral and an angle-dependent part. The imi) are of course given by the spherical harmonics,... 

Substituting this expansion into the radial integral and changing to the variable q— R — Re, we have for the radial integral... 

The second- and fourth-order radial integrals (Cp and Dq, respectively) are designed not to involve the angular coordinate of the ligands and so hopefully may refer directly to features of the M—L bonds or interactions. The angle 8 must be treated as a parameter, even though it should take a value near that determined for the molecular structure, because of the gross approximation... 

The Onk and Q k are operators which are respectively linear combinations of spherical harmonics and expansions in terms of Cartesian coordinates, 1 = 2 for d-orbitals, 3 for /-orbitals. The parameters B k and A k are, of course, specialized forms of the general form given in equation (2), but including the evaluation of the relevant radial integrals. 

If the metal and donor atom wave functions and their separations are the same, the group radial integrals may be unchanged and we can note that the result for the difference between the t2 and e antibonding rf-orbital sets, Et — E ... 

As for ease of calculation, only a small number (say 30) of radial integration points are required, so that not too many evaluations of the potential are necessary. A complete computer program for quantum-mechanical elastic and inelastic scattering is available (14). 

One of the nagging features of these expressions is that the radial integral from the multipole expansion introduces a factor of r21, and thus the dimensions of B(E, l) and Bsp(E, l) depend on l. 

We shall not consider here the more complex cases if necessary, they may be found in [9, 11] or deduced utilizing methods described there. The expressions for matrix elements of the majority of energy operators (1.16) in terms of radial integrals and transformation matrices or 3n/-coefficients for complex electronic configurations may be found in [14]. 

The expression for quantity g(Rs) in (19.52) follows directly from (5.40). Thus, for p- and d-shells we have simple algebraic formulas for the coefficients of radial integrals (actually, for matrix elements) of all relativistic corrections of the order a2 to the Coulomb energy. For the /-shell such a formula exists only in the case of the orbit-orbit interaction. For the almost filled shell we find... 

Here and further on the symbol (abc) means that parameters a, b and c obey the triangular condition with even perimeter. If in the radial integral there is X[ = nff instead of 2 = ntlji, then this indicates that in the corresponding parts of the integral function / and quantum number l must be replaced by g and V, respectively. Coefficient fk has the form... 

The radial integral of the direct part of interaction Fk is defined by (19.31), whereas that of the exchange part is as follows ... 

The coefficients at the radial integrals in (20.8) have the expressions (for brevity we shall skip further the quantum numbers in the left side of the equalities) ... 

Here the radial integrals are defined according to (19.72), whereas their coefficients are equal to... 

It is worth emphasizing that coefficients (20.29) and (20.30) do not depend on orbital quantum numbers. These numbers define only the parity of summation index k, following from the conditions of the nonvanishing of radial integrals in (20.27) and (20.28). For the direct term, parameter k acquires even values, whereas for the exchange part the parity of k equals the parity of sum h + h-If one or two subshells are almost filled, then the following equalities are valid ... 

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