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Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B

On the evaluation of second-order matrix elements of perturbation theory by variational methods

Michele Battezzati

Published Oct 1, 1979• 5 citations
Publisher
DOI
10.1007/bf02739899

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In this paper, we present an efficient alternative method for the evaluation of the three-photon transition matrix element in the dipole approximation from the ground state to bound states in atomic hydrogen. This method is a variation of the Dalgarno–Lewis method for the treatment of the second-order Stark effect in the hydrogen atom. In this approach, the infinite double sum over the complete set of states including the continuum states present in the third-order perturbation theory result is treated exactly. The closed analytical expression obtained for the matrix element, as a function of incident photon energy, clearly displays all singularities present in the original third-order perturbation theory result.
The derivation of the dispersion coefficients for various symmetries of the lower nP\mathrm{nP}nP-nP\mathrm{nP}nP asymptote of the homonuclear alkali-metal dimers is presented. Using the perturbation theory the dispersion coefficients C5{C}_{5}C5​ are obtained in the first order and the C6{C}_{6}C6​ and C8{C}_{8}C8​ coefficients are obtained in the second order in terms of double sums involving atomic radial matrix elements. The sums are separated in individual contributions of each atom by using a complex integral representation. The atomic matrix elements involving Green's functions are evaluated using the Dalgarno-Lewis method. Numerical results for the C5{C}_{5}C5​, C6{C}_{6}C6​, and C8{C}_{8}C8​ dispersion coefficients for Li, Na, K, Rb, and Cs dimers are presented.