Long-range interactions of metastable helium atoms

Polarizabilities, dispersion coefficients, and long-range atom-surface interaction potentials are calculated for the n = 2 triplet and singlet states of helium using highly accurate, variationally determined, wave functions. PACS numbers: 34.20.Cf, 32.10.Dk, 34.50.Dy Typeset using REVTEX 1 The advent of doubled basis sets has made it possible to calculate precisely many properties of two-electron atomic systems [1–4]. We apply variational methods developed previously and demonstrated for the helium atom [5] to calculate nonrelativistic values of the electric dipole, quadrupole, and octupole polarizabilities and corresponding dispersion coefficients for the metastable n = 2 singlet and triplet states, respectively, He(2 S) and He(2 S). Additionally, potentials for the atom-wall interaction of a He(2 S) or a He(2 S) atom and a single perfectly conducting wall or a dielectric wall are calculated with the inclusion of retardation effects due to the finite speed of light. Our results for atom-wall interactions are germane to experiments involving atom-evanescent wave mirrors [6]. In this paper the notation of Ref. [7] is followed very closely; references to equations of Ref. [7] will be preceded by the symbol I. Atomic units are used throughout. The dispersion interaction of two like atoms can be written U(R) = −C6R −6 − C8R −8 − C10R , (1) where the coefficients C6, C8, and C10 are the van der Waals coefficients, R is the interatomic distance, and C6 = (3/π)G(1, 1) , (2) C8 = (15/π)G(1, 2) , (3) C10 = (28/π)G(1, 3) + (35/π)G(2, 2) , (4) with G(l,m) = ∫ ∞ 0 αl(iω)αm(iω)dω , (5) where αl(iω) is the 2 -pole dynamic polarizability function evaluated at imaginary frequency defined by Eqs. (6)–(9) of Ref. [5], and similarly for αm(iω). When the effects of retardation due to the finite speed of light are considered the potential U(R), Eq. (1), can be replaced by [8,9] 2 V (R) = −C6f6(R)R −6 − C8f8(R)R −8 − C10f10(R)R , (6) The coefficient f10(R) will not be considered in this paper as the C10 term is usually negligible. Expressions for the retardation coefficients, f6(R) and f8(R), as integrals involving the dynamic electric dipole polarizabilities, are given in Eqs. I-(5) and I-(7). The form (6) intrinsically includes certain relativistic effects, so that when f6(R) and f8(R) are expanded in powers of the fine structure constant αfs = 1/137.035 989 5 for small distances V (R) ∼ −R[C6 − α 2 fsR W4]− R [C8 − α 2 fsR W6], (7) where W4 = 1 π ∫ ∞ 0 dω ωα 1(iω) (8) and W6 = 3 π ∫ ∞ 0 dω ωα1(iω)α2(iω). (9) The relativistic origin of the coefficient W4 has been discussed by Power and Zienau [10], see also [11]. The coefficient W6 of the factor α 2 fs/R 6 in (7) corresponds to the theory of Power and Thirunamachandran [9] and is equal to the coefficient WLL;4,2 in the theory of Meath and Hirschfelder based on the Breit-Pauli Hamiltonian [11]. As the distance increases retardation arising from the finite speed of light becomes important and the potential approaches its asymptotic form, see Eqs. I-(13) and I-(14), V (R) ∼ −K7R −7 −K9R , (10) with K7 = 23 4π α 1(0) αfs = 250.81 α 1(0), K9 = 531 16π α1(0)α2(0) αfs = 1 447.6 α1(0)α2(0). (11) An expression for the potential VAtD(R, ǫ) for the interaction [12,13] of an atom and a dielectric wall was presented in Eq. I-(15), where R is the atom-wall distance and ǫ is the 3 dielectric constant of the wall. The expression is a double integral that can be evaluated with knowledge of the function α1(iω). For small distances VAtD(R, ǫ) has the limiting form VAtD(R, ǫ) ∼ − C3 R3 ǫ− 1 ǫ+ 1 , (12)