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In this work the induced van der Waals interaction between a pair of neutralatoms or molecules is considered by use of a statistical mechanical method.Commonly this interaction is obtained by standard quantum mechanicalperturbation theory to second order. However, the latter is restricted toelectrostatic interactions between charges and dipole moments. So withradiating dipole-dipole interaction where retardation effects are important forlarge separations of the particles, other methods are needed, and the resultinginduced interaction is the Casimir-Polder interaction usually obtained by fieldtheory. It can also be evaluated, however, by a statistical mechanical methodthat utilizes the path integral representation. We here show explicitly by useof the statistical mechanical method the equivalence of the Casimir-Polder andvan der Waals interactions to leading order for short separations whereretardation effects can be neglected. Physically this is well known, but in ouropinion the mathematics of this transition process is not so obvious. Theevaluations needed mean a transform of the statistical mechanical free energyexpression to a form that can be identified with second order perturbationtheory. In recent works [H{\o}ye 2010] the Casimir-Polder or Casimir energy hasbeen added as a correction to calculations of systems like the electron cloudsof molecules.  The equivalence to van der Waals interactions to leading order indicates thatthe added Casimir energy will improve the accuracy of calculated molecularenergies. We here also give numerical estimates of this energy includinganalysis and estimates for the uniform electron gas.

Van der Waals interactions: Evaluations by use of a statistical mechanical method