Lambda-Doubling Effect on Molecular Electric-Field Gradient and Nuclear-Quadrupole Coupling

Abstract

Physical reasons and mathematical arguments are given to show that for a rotating linear molecule or radical, between the degenerate [open phi]+Ω and [open phi]−Ω electronic states of opposite senses of angular momentum, there exists a ``cross-term'' matrix element connected by a nonaxial field-gradient-tensor operator (∇E)2Ω=T2,2Ω/re5. The resulting effect is that the electric-field gradients eq for the two Λ-doubling components are different, giving rise to an additional nuclear-quadruple contribution to the Λ-doubling energy. When the nuclear spin is quantized with respect to the space z axis as in a molecule belonging to Case aβ (Sec. III), it is shown that because the nuclear quadrupole tensor operator transforms with a rotation matrix of the second rank, such Λ-doubling effect on field gradient vanishes for the 2Π½(a) state. For a Case bβ2Π state and a 2Π state intermediate between Cases a and b, the dependence of this effect on J is derived using wavefunctions with a consistent phase and inversion symmetry not otherwise obtainable from the solution of a secular equation. For states with Ω=0 (Table I) the effect is shown to arise from electron spin and orbital interaction in a stationary molecule. For 3Π1(a) and 1Π states, the relative sizes of the ``nonaxial field gradient'' eq2, and the axial field gradient, eq0, are estimated using open-shell electron configurations constructed from one-electron molecular orbitals. The general selection rule based on Ω, the total spin and orbital angular momentum, for the matrix elements over the spinless, one-electron field-gradient operator is justified for various open-shell electron configurations by invoking electron spin—spin interaction in addition to the customary electron spin—orbit interaction. A general method is devised for obtaining diagonal and off-diagonal spin—spin interaction matrix elements over Slater determinantal wavefunctions. In terms of these interactions the relative importance of these configurations of equivalent as well as nonequivalent electrons is given. When the nuclear spin is quantized with respect to the molecular figure axis as in a molecule belonging to Case aα (Sec. V), the additional nuclear quadrupole splitting of the Λ-doubling levels is shown to be independent of J and to be present in states with ΩF=0 only, when the molecule is rotating.