Relativistic potential curves

The relativistic Hamiltonian may be defined by adding Hso to Hel. The eigenfunctions of this new Hamiltonian are the relativistic wavefunctions, i,n, which define the relativistic potential curves... 

Le Roy, et al, (2002) have reviewed all of the different types of experimental observations and theoretical calculations for HI. By an empirical analysis, they have shown that, because in HI the spin-orbit interaction is especially important, the adiabatic relativistic potential curves can explain all of the experimental data without introducing residual nonadiabatic coupling. For the lighter halogen hydrides, the J = 1/2 J = 3/2 branching ratio can be obtained from the solution of inhomogeneous coupled equations with a source term representing the initial vibrational wavefunction multiplied by the electronic transition moment (Band, et al., 1981). These calculations are based on adiabatic electronic (or diabatic relativistic) potential curves (see, for example, for HC1, Alexander, et al., 1993 and for HBr, Peoux, et al., 1997). 

Xis the complete non-relativistic Hamiltonian given in (6.130). The Hermitian matrix in (6.141) is diagonalised to obtain the coefficients in the wave hmctions (6.140) and effective potential curves for the coupled states using these wave functions as a function... 

Adiabatic corrections for H2 were first calculated by Kolos and Wolniewicz [27], and much later confirmed by Bishop and Cheung [55]. The best potential curves for H2 and D2, incorporating both adiabatic and relativistic corrections, have been tabulated by Bishop and Shih [56]. Bishop and Cheung [55] have also carried out non-adiabatic calculations for H2, the energy of the lowest level being lowered by 0.42 cm 1 compared with the adiabatic value. A small number of calculations for excited states have also been reported. 

Another method, devised by Cohen et al. to determine oxygen-rate gas collision parameters is to define an effective spin-orbit operator that includes r dependence, Zeff/r3, where the value of Zeff is adjusted to match experimental data (76). Langhoff has compared this technique with all-electron calculations using the full microscopic spin-orbit Hamiltonian for the rare-gas-oxide potential curves and found very good agreement (77). This operator has also been employed in REP calculations on Si (73), UF6 (78), U02+ and Th02 (79), and UF5 (80). The REPs employed in these calculations are based on Cowen-Griffin atomic orbitals, which include the relativistic mass-velocity and Darwin effects but do not include spin-orbit effects. Wadt (73), has made comparisons with calculations on Si by Stevens and Krauss (81), who employed the ab initio REP-based spin-orbit operator of Ermler et al. (35). 

True relativistic ab initio calculations have been done for hydride molecules such as T1H (Pyykko and Desclaux, 1976) and have been extended to heavy nonhydride molecules, such as I2 (de Jong, et ai, 1997). Other types of calculations involve treating the spin-orbit operator as a perturbation and adding its effects to the nonrelativistic potential curves (for example, Ne2, Cohen and Schneider, 1974 LiHg, Gleichmann and Hess, 1994). 

The slope of the repulsive potential at R" (or at the R" values of the two maxima in the v" = 1 probability distribution) may be determined from the width of ct(E). The vertical excitation energy of the repulsive state at JR" is determined by the E at which a E) reaches its maximum value. In this semi-classical approximation, the repulsive potential curve can be determined from a E) provided that /i(i .) varies no more rapidly than linearly in R (Child, et al., 1983). When a sufficient quantity of cr E) data is obtained from free-bound absorption or emission transitions originating from several bound vibrational levels, it is then also possible to determine the shape of the bound potential (Le Roy, et al., 1988). The /(-dependence of /i(JR) 2 can arise from two sources (i) the /(-dependence of the fractional contributions of several different A-S basis states to a single relativistic adiabatic fi-state (ii) /(-variation of the transition moment between A S basis states arising from the molecule to separated atom evolution of the LCAO characters of the occupied orbitals (iii) /(-variation of the configurational character (Configuration Interaction) of either electronic... 

Figure 7.6 Schematic relativistic adiabatic potential curves of HI (from Langford, et al., 1998). The ground state dissociates to the H + I asymptote.

The slight difference in distance is a manifestation of the relativistic contraction in the W2 molecule. The computed EBO is 5.17 for M02 and 5.19 for W2, and much closer to six than for Cr2. Figure 9.1 illustrates the difference in the potential curves (See also Table 9.1). The increased bond energy in M02 and W2 compared to Crj can pardy be explained by the decreased electron repulsion in the more diffuse 4d and 5d shells compared to the compact 3d shell. For heavier elements, spin-orbit coupling, which is often quenched in the molecule but large in the atoms, will decrease the bond energy. For example, it reduces Dg in W2 by more than 1 eV. 

Figure 2 A comparison of valence electron calculations of the potential energy curve of HgH 84 (a) non-relativistic core potential, SCF valence calculation (b) relativistic core potential, SCF valence calculation (c) relativistic core potential, MCSCF valence calculation. In each case the zero of energy is the sum of the appropriate atomic energies calculated in the same manner...

According to Eq. (81) with /S = 6, the three a values for H in Figure 4.14 are —3.708, 2.000, and 9.708. Thus, the asymptotically lowest hyperspherical potential supports an infinite series of Feshbach resonances in the nonrelativistic approximation, although only three lowest members remain as resonances after corrections for the relativistic and radiative effects [80, 82], as was mentioned in Section 3.1.2. Only the lowest member is indicated in the figure by a horizontal line. This resonance is supported by the diabatic potential with A = — 1 connecting from the lowest curve for large p to the middle curve for small p. 

The calculation of term values directly by treating each state as a separate variational problem is also fraught with difficulties, or rather with two difficulties. These are the problems of relativistic and correlation energy. As Figure 1 shows, the potential-energy curve computed by solution of the... 

Fig. 4. All-electron, effective potential, and average relativistic effective core potential configuration-interaction potential-energy curves of Xe2 and Xe2+. Dashed curves are from allelectron calculations and AREP curves are less repulsive than EP.

Fig. 6. Average relativistic effective core potential and relativistic effective core potential energy curves for two states of Bi2. HF, Hartree-Fock GVB(pp), eight-configuration perfect-pairing generalized valence bond FVCI, full-valence Cl based on the GVB(pp) wave functions FV7R, full-valence Cl plus single and double promotions to virtual MOs relative to seven-dominant configurations. (The FVCI and FV7R calculations include the REP-based spin-orbit operator.)...

---