Radial wave functions, transition elements

In the central field approximation, when radial wave functions not depending on term are usually employed, the line strengths of any transition may be represented as a product of one radial integral and of a number of 3n./-coefficients, one-electron submatrix elements of standard operators (C(fc) and/or L(1 S(1)), CFP (if the number of electrons in open shells changes) and appropriate algebraic multipliers. It is usually assumed that the radial integral does not depend on the quantum numbers of the vec-... 

The one-electron submatrix element of operator rk 1 stands for the radial integral of this quantity and the appropriate radial wave functions. Indeed, we can easily show that the right side of formula (27.7) equals zero for k = 0. Therefore, Ml-transitions occur only between levels of one and the same configuration. 

In the 3d transition metal series, the electronic state gradually changes with the atomic number, though these elements possess similar properties. Because the valence electronic state of these elements essentially depends upon the nature of the 3d atomic orbital, we depict the radial wave functions, namely the spatial distribution of the 3d atomic orbitals of the elements from Sc to Zn in Fig.5. From the figure. 

Relativistic effects have to be taken into account for compounds containing transition elements with higher atomic numbers the 5d transition elements (Hf, Ta, W) are of particular concern in the present review. A fiiUy relativistic treatment requires the solution of the Dirac equation instead of the Schrodinger equation. However, in many cases, it is sufficient to use a scalar relativistic scheme (48) as an approximation. In this technique, the mass-velocity term and the Darwin 5-shift are considered. The spin-orbit splitting, however, is neglected. In this approximation a different procedure must be used to calculate the radial wave functions, but the nonrelativistic formalism, which is computationally much simpler than solving Dirac s equation, is retained. 

Unfortunately, the Slater-type orbitals become increasingly less reliable for the heavier elements, including to some extent the first transition series these limitations are described in a recent review by Craig and Nyholm (5 ). The most accurate wave functions to use in these calculations would be the SCF functions obtained by the Hartree-Fock procedure outlined above, but this method leads to purely numerical radial functions. However, Craig and Nyholm (5S) have drawn attention to relatively good fits obtained by Richardson (59) to SCF 3d functions by means of two-parameter orbitals of the type... 

For a description of the induced transitions between rotovibrational levels vj) — v f) of H2, radial matrix elements, Eq. 4.17, are needed. Since the H-H interaction potential is well known, the vibrational wave-functions, (pVj(r), which depend on the rotational excitation j, can be computed with the help of digital computers [217], At fixed intermolecu-lar separations R, the AXt(r,R) are known at three vibrational spacings,... 

In the third transition series the 4/ shell is complete and has spherical symmetry and angular effect. Any effect must be due to the radial part of the appropriate wave functions. Thus so far as the effects on the third series of transition elements are concerned the second implication that the contraction is due to the angular part of the 4/ wave functions does not appear to be convincing. 

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