Two-centre harmonic oscillator integrals

Abstract

Two-centre integrals involving displaced and distorted (or squeezed) simple harmonic oscillator number states, the DSN states, are often encountered in the molecular spectroscopy and in the quantum theory of wave packets and light. For example, the Franck–Condon factors are overlap integrals of the DSN states. In this work, finite, closed and algebraic formulas are derived for the overlap, exponential, trigonometric and polynomial integrals.

Acknowledgements

Andrea Beddoni acknowledges the support of EU under the RTN program 5 (contract No. HPRN-CT-1999-00005). Lauri Halonen would like to thank the Academy of Finland for financial support.

Notes

Present address: Loc. San Nicolà 59/a, 06049 Spoleto (PG), Italy.

Present address: Laboratory of Polymer Chemistry, PO BOX 55 (A.I. Virtasen aukio 1), FIN-00014 University of Helsinki, Finland.

Present address: Laboratory of Polymer Chemistry, PO BOX 55 (A.I. Virtasen aukio 1), FIN-00014 University of Helsinki, Finland.

⁰ Now

and

. Taking the adjoint of a product of a scalar α and ladder operators includes reversing the order of the factors in the product, replacing the raising operator by the lowering operator and vice versa, and taking the complex conjugate of the scalar. For example,

.

Additional information

Notes on contributors

Lauri Halonen *

Present address: Loc. San Nicolà 59/a, 06049 Spoleto (PG), Italy. Present address: Laboratory of Polymer Chemistry, PO BOX 55 (A.I. Virtasen aukio 1), FIN-00014 University of Helsinki, Finland.