Asymptotically exact calculation of the exchange energies of one-active-electron diatomic ions with the surface integral method

Abstract

We present a general procedure, based on the Holstein–Herring method, for calculating exactly the leading term in the exponentially small exchange energy splitting between two asymptotically degenerate states of a diatomic molecule or molecular ion. The general formulae we have derived are shown to reduce correctly to the previously known exact results for the specific cases of the lowest Σ and Π states of H+2. We then apply our general formulae to calculate the exchange energy splittings between the lowest states of the diatomic alkali cations K+2, Rb+2 and Cs+2, which are isovalent to H+2. Our results are found to be in very good agreement with the best available experimental data and ab initio calculations.

Exchange splitting of the interaction energy and the multipole expansion of the wave function

“…(5) and throughout the paper. The surface-integral method has been extended to hydrogen molecule [19][20][21] alkali-metal dimer cations [22][23][24][25][26][27][28] , neutral homo-and heterodimers [29][30][31][32][33] , excited states of H + 2 ion 34 , and interactions of diatomic molecules with atomic ions 35 . For the discussion of other extensions of this theory see Ref.…”

“…For the interaction of two hydrogen atoms only the first term in the analogous expansion is known 19,20 , although even the functional form of this leading term has been debated recently 21 . For larger systems only approximate form of the leading term is known [22][23][24][25][26][27][28][29][30][31][32][33] . Its accuracy is hard to ascertain since no reference data sufficiently accurate at large R are available.…”

Asymptotics of the exchange-splitting energy for a diatomic molecular ion from a volume-integral formula of symmetry-adapted perturbation theory

“…The technique of Holstein and Herring was extended to the neutral H 2 molecule in the independent works of Gor'kov and Pitaevskii [22] and Herring and Flicker [23]. Extensions to many-electron systems were also provided [6,10,[24][25][26].…”

Efficient algorithm for two-center Coulomb and exchange integrals of electronic prolate spheroidal orbitals

“…An interesting alternative approach to diatomic molecules is the Holstein-Herring method [16,14,33,30] for calculating exchange energies of H + 2 -like molecular ions. This method has recently been extended to two-active-electron systems [28].…”

“…Our approach employs Neumann's expansion of the Coulomb repulsion 1/ |x − y|, solves the resulting integrals symbolically in closed form and subsequently performs a numeric Taylor expansion for efficiency. Thanks to the general form of the integrals, the obtained coefficients are independent of the particular wavefunctions and can thus be reused later.Key features of our algorithm include complete avoidance of numeric integration, drafting of the individual steps as fast matrix operations and high accuracy due to the exponential convergence of the expansions.Application to the diatomic molecules O2 and CO exemplifies the developed methods, which can be relevant for a quantitative understanding of chemical bonds in general.An interesting alternative approach to diatomic molecules is the Holstein-Herring method [16,14,33,30] for calculating exchange energies of H + 2 -like molecular ions. This method has recently been extended to two-active-electron systems [28].…”