Potential matrix element special cases

The case under consideration requires special values of the angular-momentum quantum numbers in the reduced potential matrix elements (7.67), namely = 0, = A = 1, L = X, L = L + 1. The corresponding... 

This reduces to equation (7) for the special case k = k = 0] again, the potential matrix elements are independent of J and diagonal in K, The off-diagonal matrix elements of the operator (J — i) in equation (19) are the same as in the atom-diatom case, and are given by equation (9) with an additional factor of Skk>-... 

In special cases some of these terms may be identically equal to zero, for example, with the electric dipole transition operator (see (4.12) at k = 1) the intrashell terms are zero, and with the kinetic and potential energy operators the intershell terms are zero (at h h) -either case follows directly from the explicit form of relevant one-electron reduced matrix elements. 

We have written the operator Fl(x) as a function of the combined space-spin coordinates X, because while the spin summations can be carried out in Jl(x) before calculating matrix elements, Kl(x) may connect spin-orbitals that are off-diagonal in the spin wavefunctions however in the special case of the density matrix p (xi, Xa) arising from a wavefunction that is a spin singlet (5 = 0) one can show that must also be diagonal. This leads to a useful simplification here since we can usually assume this property for Wlo, and it means that Vl(x) reduces to a (non-local) function of the space variable r only we can therefore consistently parameterize the matrix elements for the whole potential, (/bI Vl(x) j) without having to decompose them into different spin combinations for the Coulomb and exchange potentials. 

Extension of pseudopotential theory to the transition metals preceded the use of the Orbital Correction Method discussed in Appendix E, but transition-metal pseudopotentials are a special case of it. In this method, the stales are expanded as a linear combination of plane waves (or OPW s) plus a linear combination of atomic d states. If the potential in the metal were the same as in the atom, the atomic d states would be eigenstates in the metal and there would be no matrix elements of the Hamiltonian with other slates. However, the potential ix different by an amount we might write F(r), and there arc, correspondingly, matrix elements (k 1 // 1 r/> = hybridizing the d states with the frce-eleclron states. The full analysis (Harrison, 1969) shows that the correct perturbation differs from (5K by a constant. The hybridization potential is... 

Although this formula could hardly be of any practical use in spectroscopy, its importance lies in the fact that the FC overlap has an aniytic closed formnla for the harmonic oscillator potential. It was derived for the first time by Ansbacher with a minor mistake, which has inspired some authors [13,14] to derive analogous formulae for other potentials. The above equation is beautiful and elegant because it represents an exact and closed expression for the FC overlap, with the restriction of being valid only for the special case of the harmonic oscillator. The application of this method to matrix elements of monomial, exponential and Gaussian operators is straightforward and has been published elsewhere [15,16]. 

The quantity k r is equivalent to k r cos 0 = 27r r cos d/X, where 6 is the angle formed between the vectors k and r. The matrix elements ( m W t) k limit r to the molecular dimensions over which the wave functions k and molecular spectroscopy are on the order of 10 A for vacuum-ultraviolet light, and are of course much longer for visible, IR, and microwave spectroscopy. Hence k r is typically much less than 1, and the series expansion of exp(ik r) converges rapidly. In the special geometry we have assumed for our vector potential,... 

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