A general formalism is used to express the long-range potential energies in inverse powers of the separation distance between two like atomic or molecular systems with $P$ symmetries. The long-range molecular interaction coefficients are calculated for the molecular symmetries $\Delta$, $\Pi$, and $\Sigma$, arising from the following interactions: He($2 ^1P$)--He($2 ^1P$), He($2 ^1P$)--He($2 ^3P$), and He($2 ^3P$)--He($2 ^3P$). The electric quadrupole-quadrupole term, $C_{5}$, the van der Waals (dispersion) term $C_{6}$, and higher-order terms, $C_{8}$, and $C_{10}$, are calculated \textit{ab initio} using accurate variational wave functions in Hylleraas coordinates with finite nuclear mass effects. A comparison is made with previously published results where available.
Accurate description of the interactions between two excited atoms (or molecules) at longrange is fundamentally important for studies of molecular excited state spectroscopy [1,2], associative ionization [3,4], and other collisional processes [5], and is at the heart of several schemes for quantum computation [6,7]. At sufficiently large separations, the mutual electrostatic interaction energy between the two excited atoms can be accurately described using an expansion of the potential energy in inverse powers of the separation distance R.
The terms describe the electric quadrupole-quadrupole interaction at order R -5 and the instantaneous dipole-dipole (e.g. dispersion) interaction at order R -6 [8] and higher order instantaneous multipole-multipole interactions at orders R -8 and R -10 .
Long-range interactions involving few-electron atoms are the only interactions that presently can be rigorously calculated with high accuracy. Different levels of approximation are needed for the calculations of long-range forces for alkali-metal and alkaline-earth atoms. [5,9,10]. Sizeable discrepancies between various calculations in the literature can occur, as illustrated in the comparisons of C 6 coefficients, for example, given by Zhang et al. [12] for Li (2p)-Li (2p) and given by Yurova [13] for Na(3p)-Na(3p). For helium, it is possible to perform a highly-accurate ab initio calculation of atomic properties and longrange interaction coefficients. Such results could become benchmarks for eventual ab initio calculations of alkaline-earth atomic interactions. Alkaline-earth and other two-electron excited P atoms are currently being studied as the optimal candidates for frequency-based standards and optical clock experiments [14].
We had previously studied the long-range interaction coefficients C n (with n ≤ 10) for all He(n λ S)-He(n ′ λ ′ S) and He(n λ S)-He(n ′ λ ′ P ) systems of the energetically lowest five states:
He(1 1 S), He(2 3 S), He(2 1 S), He(2 3 P ) and He(2 1 P ) and the finite nuclear mass effects for like isotopes [15,16,17]. In this work, we present results for more complicated set of interactions between two like isotope helium atoms with P symmetries. Degenerate perturbation theory is needed to derive the interaction terms for some of the terms. Section II introduces a general formalism for calculating dispersion coefficients between two like atomic or molecular systems of P symmetry. Section III presents numerical results of dispersion coefficients C 5 , C 6 , C 8 , and C 10 for the following three systems He(2 1 P )-He(2 1 P ), He(2 1 P )-He(2 3 P ), and He(2 3 P )-He(2 3 P ).
In this work, atomic units are used throughout. At large distances R between two atoms a and b, the Coulomb interaction [18], treated as a perturbation to the two isolated atoms, is
where
In the above, T (ℓ) µ (σ) and T (L)
-µ (ρ) are the atomic multipole tensor operators defined by
and
and Q i and σ i are the charge and the position vector of the i th particle in atom a, respectively.
Similarly, q j and ρ j are for the j th particle in atom b. The coefficient K µ ℓL in Eq. ( 2) is
and (ℓ, L,
Since the Coulomb interaction V is cylindrically symmetric about the molecular axis R or z axis [6,10], the projection of the total angular momentum of the combined system a-b along the z axis (with magnetic quantum number M), is conserved. Therefore, states with M = ±2, ±1, and 0 are not mixed with each other, corresponding to the ∆, Π, and Σ molecular states, respectively. The ∆ and Π states are degenerate with respect to the sign of M and the degeneracy can not be removed physically in the free combined system a-b.
Therefore, we only study the states with positive M in this work.
For two like isotope atoms a and b in P symmetry, the zeroth-order wave function for the ∆ state of the combined system a-b can be written in the form:
where M a = M b = 1 are the magnetic quantum numbers, α is the normalization factor, and β describes the symmetry due to the exchange of two initial states Ψ na and Ψ n b . If two atoms are both in the same P state, then α = √ 2 and β = 0; if they are in different P states, then α = 1 and β = ±1 [10].
According to the perturbation theory, the first-order energy is
where, after some angular momentum algebra, one gets (see also Ref. [11]),
The second-order energy is
where χ ns (L s M s ; σ)ω nt (L t M t ; ρ) is an allowed intermediate state with the energy eigenvalue E nsnt = E ns + E nt , and the prime in the summation indicates that the terms with E nsnt = E (0) nan b should be excluded. Substituting Eqs. ( 1) and (6) into Eq. ( 11), we obtain
with
After applying the Wigner-Eckart theorem, we have
In Eq. ( 16), G 1 is the angular-momentum part and F 1 is the oscillator strength part. Their expressions are
and
and ∆E nsna = E ns -E na , etc. For the special case where the two initial states Ψ na and Ψ n b are the same and ℓ = ℓ ′ , ḡns;nana reduces to the absolute value of the 2 ℓ -po
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