Long-Range Interactions for Hydrogen: 6P-1S and 6P-2S

The collisional shift of a transition constitutes an important systematic effect in high-precision spectroscopy. Accurate values for van der Waalsinteraction coefficients are required in order to evaluate the distance-dependent frequency shift. We here consider the interaction of excited hydrogen 6P atoms with metastable atoms (in the 2S state), in order to explore the influence of quasi-degenerate 2P, and 6S states on the dipole-dipole interaction. The motivation for the calculation is given by planned high-precision measurements of the transition. Due to the presence of quasi-degenerate levels, one can use the non-retarded approximation for the interaction terms over wide distance ranges.

where R = R A − R B , R = | R|R = R/R. Furthermore, d A = e r A is the electric dipole moment operator for atom A and d B is the same for atom B ( r i with i = A, B is the electron coordinate relative to the atomic nucleus). States of the form, e.g., |6P A |1S B are energetically degenerate with respect to states of the form |1S A |6P B , and are coupled by the van der Waals Hamiltonian. For the 2S-2P system, this problem has been analyzed in Ref. [12], on the basis of nonrelativistic Schrödinger theory. However, in order to evaluate the distance-dependent frequency shift of hyperfine-resolved transitions, one has to invoke a more sophisticated analysis, which has recently been performed in Ref. [11]. A complete rediagonalization of the total Hamiltonian, comprising Lamb shift, fine-structure, and hyperfine effects, becomes necessary. In order to analyze the problem, one has to define a quantization axis, which we choose as the line of separation of the two atoms. This brings the van der Waals Hamiltonian into the form We should add that the interaction remains non-retarded over very wide distance ranges, commensurate with the fine-structure and Lamb shift transition wavelengths in the quasi-degenerate system. Here, we engage in the endeavor of evaluating long-range interactions for the hydrogen 6P -1S and 6P -2S systems. In the latter case, we find it necessary to include, in our basis of states, all 2S, 2P , 6S as well as 6P 1/2 and 6P 3/2 hyperfine-resolved atomic levels. The calculations are motivated, in part, by the prospect of a future high-precision measurement of the 2S-6P transition in atomic hydrogen [13], to supplement ongoing efforts for a resolution of the proton charge radius puzzle [14][15][16] (see the recent work [17] for a discussion of systematic effects in 4P -2S hydrogen systems, which are closely related to the systems under investigation here).
where H LS stands for the Lamb shift, H HFS describes hyperfine effects, and H FS describes fine-structure splittings. These Hamiltonians have to be added for atoms A and B, Here, i = A, B denotes either atom, while α is the fine-structure constant and m e is the electron mass. The momentum operators for the atomic electrons are denoted as p i , while L i is the orbital angular momentum operator. The (dimensionless) spin operator for electron i is S i , while I i is the spin operator for proton i. The CODATA (Committee on Data for Science and Technology) values (see Ref. [18]) for the electronic and protonic g factors are g s ≃ 2.002 319 and g p ≃ 5.585 695. The Bohr magneton is µ B ≃ 9.274 010 × 10 −24 Am 2 , while µ N ≃ 5.050 784 × 10 −27 Am 2 is the nuclear magneton. The expression for H LS in Eq. (4b) follows the Welton approximation [19], which is sufficient for our purposes of calculating long-range interaction coefficients. For the 6P -1S and 6P -2S systems, we define the zero of the energy as the sum of the Dirac energies of the 1S and 6P 1/2 states, and to the sum of the 2S and 6P 1/2 states, respectively. The zero point of the energy excludes both Lamb shift as well as hyperfine effects; in the following, we add the Lamb shift energy to the S states but leave the P states untouched by Lamb shift effects. So, our definition of the zero point of the energy corresponds to the hyperfine centroid of the |(6P 1/2 ) A (1S) B states, and to the hyperfine centroid of the |(6P 1/2 ) A (2P 1/2 ) B system, respectively. The fine-structure energy is added for the 6P 3/2 states.
The matrix elements of the van der Waals Hamiltonian have to be calculated in a hyperfine-resolved basis. Let us take the 6P 1/2 -1S interaction as an example, and exclude the 6P 3/2 states for the time being. The unperturbed states carry the quantum numbers Here, the quantum numbers have their usual meaning, i.e., n is the principal quantum number, while ℓ, J, and F , respectively, are the electronic orbital angular momentum, the total (orbital+spin) electronic angular momentum, and the total (electronic+protonic) atomic angular momentum. The multiplicity of the hyperfine-resolved state is g F = 2F + 1.
After adding the electron orbital and spin angular momenta, and the nuclear (proton) spin |± p , the four 1S states within the hyperfine manifold are given by I. Multiplicities in the 6P 1/2 -6P 3/2 -1S system. One might wonder why Fz = ±3 is possible for F = 2. The answer is that F = 2 here refers to the total angular momentum (electron orbital plus electron spin plus nuclear spin) of one of the atoms, while Fz = ±3 refers to the angular momentum projection of the sum of the total angular momenta of both electrons.
while the hyperfine singlet 6P 1/2 (F = 0) state is given by The hyperfine triplet states in the 6P manifold read as follows, and Here and in the following, we use the notation |n, ℓ, J, F, F z for the unperturbed states with the specified quantum numbers.

A. Orientation
We need to diagonalize the Hamiltonian given in Eq. (3), in a quasi-degenerate basis, composed of two atoms, the first being in a 6P state, the second being in a substate of the 1S hyperfine manifold. Retardation does not need to be considered. For the manifolds composed of the (6P ) A (1S) B and (1S) A (6P ) B states, with all hyperfine levels resolved, we obtain the following total multiplicities when all 6P 1/2 and 6P 3/2 states are added into the basis (see also Table I): g(F z = ±3) = 2, g(F z = ±2) = 10, g(F z = ±1) = 22, g(F z = 0) = 28. The multiplicities are the sums of the multiplicities in the 6P 3/2 -1S and the 6P 1/2 -1S system.
We now turn to the computation of the matrix elements of the total Hamiltonian (3) in the space spanned by the two-atom states which are product states built from any two states of the types given in Eqs. (6) (for the 1S states) and (9) (for the 6P 1/2 states), as well as the 6P 3/2 states. Matrix elements of the van der Waals interaction Hamiltonian are determined with the help of a computer symbolic program [20]. We define the parameters where R = a 0 ρ, and a 0 is the Bohr radius. E h = α 2 m e c 2 is the Hartree energy. Our scale H is equal to one-third of the hyperfine splitting of the 2S state [21], while L 2 is the 2S-2P 1/2 Lamb shift [22], and L 6 approximates the 6S-6P 1/2 Lamb shift. The natural scale for the constants H and L is an energy of order α 3 E h . Hence, we write where we set C H = 0.0231596, C F = 0.158606 and C L,2 = 27 C L,6 = 0.413734. Typical second-order energy shifts encountered in our calculations may be expressed as where T 1 , T 2 and T 3 are numerical coefficients of order unity. Particular attention should be devoted to mixing terms of the Hamiltonian, in the space of the 6P 1/2 and 6P 3/2 states, within the F = 1 manifolds, with the mixing matrix element being given by [23] for an outline of the calculation).
In order to illustrate the mixing term, we temporarily restrict the discussion here to one atom only, say, atom A, omitting the subscript on H HFS ≡ H HFS,A . For the two states, with the same magnetic projection F z = µ, we consider the basis of states The matrix of the Hamiltonian H HFS + H FS in the basis {|a , |b } is evaluated as Here, g p is the proton g factor, while D is a diagonal matrix element, and X is the off-diagonal element given above. The eigenvalues of H Fz=1 HFS+FS are given by The second-order shift in the eigenvalues, ∆ = X 2 /(F − 2D), is numerically equal to 1.412 129 × 10 −13 E h , where E h = α 2 m e c 2 is the Hartree energy. For simplicity, we thus define the parameter The hyperfine splitting energy between 6P 1/2 (F = 1) and 6P 1/2 (F = 0) states thus amounts to H/27, while between 6P 3/2 (F = 2) and 6P 3/2 (F = 1) states, it is 2H/135. The 1S-state hyperfine splitting is 24H. For the product state of atoms A and B, we shall use the notation , which summarizes the quantum numbers of both atoms.

B. States with Fz = 3
The atomic states can be classified according to the quantum number F z ; the z component of the total angular momentum commutes [11] with the total Hamiltonian given in Eq. (3). Within the 6P 1/2 -6P 3/2 -1S 1/2 system, the states in the manifold F z = 3 are given as follows, Here, we have ordered the basis vectors in ascending order of the quantum numbers, starting from the last member in the list. The Hamiltonian matrix evaluates to We have subtracted the sum of the Dirac energies of the 1S and 6P 1/2 hyperfine centroids, and the 1S Lamb shift is absorbed in the definition of the 1S hyperfine centroid energy, as outlined in Sec. II. The energy eigenvalues and eigenvectors corresponding to H Fz =3 are given as follows, The average of the first-order shifts (linear in V) vanishes; there are no second-order shifts (quadratic in V).
We order the 10 states in this manifold in order of ascending quantum numbers, States |ψ 3 and |ψ 6 are 6P 1/2 states, the rest are 6P 3/2 states (see also the multiplicities indicated in Table I). Among the 6P 3/2 states, |ψ 4 and |ψ 7 have F = 1, the rest have F = 2. The Hamiltonian matrix H = H Fz=2 is 10-dimensional, (21) An adjacency graph [11,24] shows that the matrix is irreducible, which in particular implies that there are no hidden symmetries in the Hamiltonian matrix H Fz=2 which would otherwise lead to a further decomposition into irreducible submatrices.
Of particular interest are second-order van der Waals shifts, within the (F z = 2) manifold. We have the quantum number J, F , and F z at our disposal. A weighted average over the possible values of F , namely, F = 1 and F = 2, TABLE II. Second-order van der Waals shifts for 6PJ hydrogen atoms interacting with ground-state hydrogen atoms. Entries marked with a long hyphen (-) indicate unphysical combinations of F and Fz values. We denote the scaled interatomic distance by ρ = R/a0 and give all energy shifts in atomic units, i.e., in units of the Hartree energy E h = α 2 mec 2 . Recall that Fz = Fz,A + Fz,B. The results still involve an averaging over the quantum numbers of the spectator atom; e.g., the result for J = 3/2, F = 2, and Fz = 2 involves and averaging over the second-order shifts of the states |Ψ9 and |Ψ10 given in Eq. (20e). We define ∆ in Eq. (16).
The weighted average vanishes, Observe that the ∆ term, which is the HFS-FS (hyperfine-structurefinestructure) mixing term, only occurs for the F = 1 states, and vanishes for the F = 2 states (see also the entries in Table II). Furthermore, observe that there are no 6P 1/2 states with F = 0 in the manifold F z = 2, because of angular momentum selection rules (we have F z = 2 and hence F ≥ 1 for all states in the manifold).  Table II for the averaged second-order van der Waals shifts in the F z = 0, F z = +1, F z = +2, and F z = +3 manifolds. The Hamiltonian matrix for F z = −3 manifold is identical to that of F z = +3. The F z = −2 manifold has identical diagonal entries to that of F z = +2 while some off-diagonal entries are different. The same is true of the F z = ±1 manifolds. Yet, we have checked that the distance-dependent Born-Oppenheimer energy curves for F z = ±2 and F z = ±1 are alike.

E. Second-Order Energy Shifts
As a function of J and F , within the 6P -1S system, a global averaging over all the states with different individual magnetic quantum numbers leads to the results Comparing to Table II, this average would correspond to an average over the entries in the different rows, for given F (for the multiplicities, see Table I).
One can also average over the possible orientations of F , namely, F = J ± 1 2 , for given J and F z . This amounts to an averaging over the first two entries in the columns, and the third and fourth entry in every column, of Table II. The results are and E(6P 3/2 , F z = 0) F = 1 2 ∆ + 6.025 91 × 10 2 As a function of J, averaging over F and F z leads to the results Without hyperfine resolution, there are four J = 3 2 states and two J = 1 2 states. Hence, the fine-structure average of the latter two results vanishes.

A. Selection of the States
We are now turning our attention to the interaction of excited 6P states with metastable 2S atoms. If we could restrict the basis of states to the 6P 1/2 , 6P 3/2 , and 2S states, just replacing the 1S state from the previous calculation with the metastable 2S states, then the calculation would be relatively easy. However, as it turns out, there is TABLE III. Multiplicities in the 6P 1/2 -6P 3/2 -(6S;2P 1/2 )-2S-1S system. The entries in the first seven rows refer to the 6P 1/2 -6P 3/2 -2S system, and are the same as those for the 6P 1/2 -6P 3/2 -1S system given in Table I. The eighth row gives the number of added (6S, 2P 1/2 ) states which complete the basis of quasi-degenerate basis. Finally, we end up with multiplicities of 40, 30, 12 and 2 for Fz = 0, ±1, ±2, ±3, respectively (ninth row). an additional complication: namely, the |(6P 1/2 ) A (2S) B states are energetically quasi-degenerate with respect to |(6S) A (2P 1/2 ) B states, and removed from each other only by the classic 2S-2P 1/2 Lamb shift (the n = 6 Lamb shift is much smaller). It is thus necessary to augment the basis of states with the 6S-2P 1/2 states. This turns the interaction into a 6P 1/2 -6P 3/2 -(6S;2P 1/2 )-2S system, where the notation indicates that the 6S-2P 1/2 states are merely added as virtual states, for the calculation of second-order energy shifts. Among the basis states, the reference states of interest are the 6P 1/2 and 6P 3/2 states.
A priori, we have 4 + 8 + 4 + 4 + 4 = 24 one-atom states in the 6P 1/2 -6P 3/2 -(6S;2P 1/2 )-2S system, which amounts to 24 2 = 576 states for the two atoms, and the Hamiltonian matrix would have 576 2 = 331776 entries. In order to remain within the quasi-degenerate basis, we should select only those two-atom states composed of an S and a P state, and only those where the principal quantum numbers add up to 2 + 6 = 8. These selection rules drastically reduce the number of states in the basis, according to Table III. Furthermore, the total Hamiltonian (3) commutes with the total angular momentum F . From the 6P 1/2 -6P 3/2 -2S system, we obtain multiplicities of 28, 22, 10 and 2, for the manifolds with F z = 0, F z = ±1, F z = ±2, and F z = ±3. By the addition of the (6S, 2P 1/2 ) states, we end up with multiplicities of 40, 30, 12 and 2, for the manifolds with F z = 0, F z = ±1, F z = ±2, and F z = ±3.
Roughly, one proceeds as in the 6P 1/2 -6P 3/2 -1S system, sets up the matrices of the total Hamiltonian (3) for every F z manifold, and then, convinces oneself that the averages of the first-order energy shifts vanish, both for the entirety of states within every F z manifold as well as for every fine-structure submanifold individually. Then, one takes as reference states the 6P -2S system and evaluates the second-order shifts for every state in a given (J, F, F z ) manifold. The individual multiplicities are given in Table III. Observe that even for given (J, F, F z ), the 6P J -2S reference levels need not all be energetically degenerate in view of a freedom of choice for the hyperfine-resolved complementary 2S state in the two-atom basis.

B. Second-Order Energy Shifts
For the states within the individual (J, F, F z ) manifolds, the averaged second-order energy shifts of the 6P J -2S reference levels are given in Table IV. As a function of J and F , an averaging over the magnetic quantum numbers IV. Average second-order van der Waals shifts for 6PJ hydrogen atoms interacting with 2S metastable atoms. Entries marked with a long hyphen (-) indicate unphysical combinations of F and Fz values. We denote the scaled interatomic distance by ρ = R/a0 and give all energy shifts in atomic units, i.e., in units of the Hartree energy E h = α 2 mec 2 . The notation ∆ is defined in Eq. (16).
These results are obtained from the entries in Table IV, using the weighting factors from Table III for an averaging over the rows. One can also average over the possible orientations of F , namely, F = J ± 1 2 , for given J and F z . This amounts to an averaging over the first two entries in the columns (for J = 1/2), and the third and fourth entry (for J = 3/2) in every column of Table IV. The results of the latter procedure are and As a function of J, averaging over both F and F z leads to the results Without hyperfine resolution, there are four J = 3/2 states and two J = 1/2 states. Hence, an additional average over the fine-structure levels leads to a cancellation of the term proportional to ∆, but the 1/ρ 6 energy shift remains as an overall repulsive interaction among 6P -2S atoms. We had seen in Eqs. (27a) and Eqs. (27b) that the fine-structure average of the van der Waals interaction vanishes for the 6P -1S interaction, in the quasi-degenerate basis which we are using. For the 6P 1/2 -2S and 6P 3/2 -2S systems, the van der Waals interactions are found to be repulsive and the van der Waals coefficients are of order 10 10 in atomic units [see Eqs. (31a) and (31b)]. The latter, comparatively large van der Waals coefficients mainly are due to the virtual (6S; 2P 1/2 ) states which have to be added to the quasi-degenerate basis if the spectator atom is in the metastable 2S state.

V. TIME DEPENDENCE AND OSCILLATORY TERMS
A few remarks on the role of the time dependence of the interaction and the no-retardation approximation are in order, in view of recent works [5,[7][8][9][10]25], part of which discuss time-dependent effects in van der Waals interactions. First of all, let us emphasize that the Hamiltonian (3) is manifestly time-independent. As such, it cannot generate oscillatory terms in energy shifts, for reasons of principle. In order to see this, let us consider the time evolution of a matrix element M = φ|H|φ ′ under the action of H, where |φ and |φ ′ are arbitrary basis states. The matrix element transforms into M (t) = φ| exp(iHt/ ) H exp(−iHt/ )|φ ′ = M (0) = M and therefore is time-independent. The time-independent Hamiltonian matrix therefore has stationary eigenvalues, which describe the time-independent energy eigenvalues of the system. This approach is canonically taken in the analysis of the van der Waals interaction within manifolds of quasi-degenerate states (see also Ref. [12]).
The time-independence of our Hamiltonian (2) corresponds to the non-retardation approximation for the van der Waals approximation. Just like in our recent paper [11], the validity of the non-retardation approximation, is tied to the energetic (quasi-)degeneracy of the levels in our hyperfine-resolved basis. In general, retardation sets in when the phase of the virtual oscillation of an atom changes appreciably over the time that it takes light to travel from one atom to the other, and back [see the discussion surrounding Eqs. (8) and (9) of Ref. [26]]. For our case, the transition wavelengths correspond to fine-structure, Lamb shift and hyperfine-structure transitions. The largest of these is the fine-structure interval E FS ∼ α 4 m e c 2 . The validity of the non-retardation approximation is thus equivalent to the condition Thus, in the distance range where retardation becomes relevant, the overall magnitude of the van der Waals interaction is completely negligible. The interaction, within the manifold of quasi-degenerate states, is thus "instantaneous" from the point of view of virtual transitions, and the no-retardation approximation is justified. In recent works [9,10], a somewhat related but different situation is considered: An atomic interaction is treated where it is assumed that there are optical transitions available, to energetically lower states as compared to the excited reference state, which act as "virtual resonant transitions" and lead to long-range oscillatory tails in the van der Waals interaction, where the oscillations are functions of the interatomic distance (not of time). This is the case, e.g., for an excited 3S-1S system, with the virtual 2P states in the 3S atom acting as virtual resonant states [see also Ref. [3]].
Additionally, in Refs. [5,7,8,25], time-dependent effects have been studied in the context of retarded van der Waals interactions for non-identical atoms. A comparison to formulas given in Refs. [7] and [8], and [5,25], is in order. First, one should observe that the second-order shifts investigated in Ref. [7] diverge in the limit of identical atoms, i.e., for the case of the "interatomic detuning" ∆ AB → 0. Also, the energy shifts given in Eqs. (6) and (7) of Ref. [8] diverge, because (in the notation of Ref. [8]) one has U ijpq ∝ µ A i µ A j µ B p µ B q /∆ AB where U ijpq is a tensor that enters the calculation of the energy shift, and the µ i are the dipole moment operators of non-identical atoms A and B. For the case of perfect degeneracy, we thus have to calculate one-photon rather than two-photon exchange (in the non-retardation approximation). This has been done here and in our recent work [11]. We find, in full analogy to the discussion in Ref. [11], that the average first-order shifts of the hyperfine-resolved levels, in first order of the van der Waals Hamiltonian, vanish.
The interaction with the quantized modes of the radiation field (which mediate the van der Waals interaction) is "switched on" at distance R and time t = 0 in Refs. [7,8]. We here refrain from a discussion on the relevance of this approximation. Otherwise, the interatomic interaction energy otherwise increases as the atoms approach each other from infinity to a finite distance R. We only explore the consequences of the results reported in Refs. [7,8]. Let |Ψ = |6P A |1S B denote a specific state within the 6P -1S hyperfine manifold, and let |Ψ ′ = |1S ′ A |6P ′ B denote a different state, displaced by an energy shift of order We concentrate on the second-order shifts due to quasi-degenerate levels |Ψ ′ = |1S ′ A |6P ′ B , in which case the formalism of [8] and [5] becomes applicable. In Eqs. (20) and (6) of Ref. [8], it is claimed that the "usual" result for the van der Waals energy shift holds only in the limit where Γ is the natural line width of the state (expressed in units of energy), Ω is the Rabi frequency of the excitation, and ∆ AB is the detuning E Ψ ′ − E Ψ . This condition cannot be met in our setting for any Rabi frequency Ω, because we evidently have Γ ∆ HFS ∼ ∆ AB (the natural line width of the 6P states is greater than the hyperfine splitting). Otherwise, time-dependent oscillatory terms are obtained in Ref. [8].
Likewise, it is claimed in Ref. [7] that upon a sudden excitation of atom A at t = 0, at a later time T , there are oscillatory terms in the energy which for our case would be proportional to and thus oscillatory in both space and time. These terms are claimed to influence the energy shift dynamically, after an observation time T . Note that the first term on the right-hand side of Eq. (4) of Ref. [7] reproduces the usual second-order van der Waals contribution (proportional to R −6 ) to an energy shift in the limit of a vanishing wave vector for the transition of atom A, namely, k A → 0 (in the notation of Ref. [7]), which is relevant for our quasi-degenerate manifolds. This term is non-oscillatory in T and gives the leading contribution to the interaction energy for quasi-degenerate systems in the non-retardation limit, as also remarked in Ref. [9]. In any case, the additional oscillatory terms obtained in Eqs. (20) and (6) of Ref. [8] and in Eq. (4) of Ref.
[8] average out to zero over the observation time T . If we are to evaluate position-dependent energy shifts (pressure shifts) within an atomic beam, then we do not know the time T at which an atomic collision occurs, within the beam, after excitation. The oscillatory terms in the energy shifts thus average out to zero, in the calculation of the pressure shifts due to atomic collisions within the beam.
In order to avoid "switching on" the interaction with the quantized modes of the radiation field, suddenly at time t = 0, one generally assumes the excited state to be an asymptotic state (in the context of the S-matrix formalism, see Refs. [19,27]). For a didactic presentation of the application of this formalism to the ac ("alternating-current", oscillatory-field) Stark energy shift due to an oscillatory external laser field, see Ref. [28]. Specifically, for manifestly oscillatory terms in the interaction Hamiltonian such as a laser field, or the quantized electromagnetic field, one damps the interaction infinitesimally at infinity and uses an infinitesimally damped time-evolution operator [see Eq. (21) of Ref. [28]] in order to formulate the energy shift within the Gell-Mann-Low theorem. Or, one matches the S-matrix amplitude with the energy shift generated by the interatomic interaction [9,10,27]. Finally, the calculation of the pressure shift within an atomic beam, using the time-independent van der Waals potentials as input, is discussed in Chaps. 36 and 37 of Ref. [29] (within the impact approximation).

VI. CONCLUSIONS
In this paper, we have studied the van der Waals interaction of excited 6P hydrogen atoms with ground-state 1S and metastable 2S atoms. Within our hyperfine-resolved basis, in order to obtain reliable estimates of the van der Waals interaction coefficients, one needs to consider all off-diagonal matrix elements of the van der Waals interaction Hamiltonian. Specifically, for hydrogen, the nuclear spin I = 1 2 needs to be added to the total electron angular momentum J, resulting in states with the total angular momentum F = J ± 1 2 of electron+nucleus. The explicit construction of the hyperfine-resolved states is discussed in Sec. II. For the 6P -1S system, one needs to include both the 6P 1/2 as well as the 6P 3/2 states in the quasi-degenerate basis, because the 6P fine-structure frequency is commensurate with the 1S hyperfine transition splitting (see Sec. III). The matrix elements of the total Hamiltonian involve the so-called hyperfine-fine-structure mixing term (see Sec. III), which couples the 6P 1/2 (F = 1) to the 6P 3/2 (F = 1) levels [see Eq. (14)].
The explicit matrices of the total Hamiltonian (3) in the manifolds with F z = 3, 2 are described in Secs. III B and III C. Due to mixing terms of first order in the van der Waals interaction between degenerate states in the two-atom system, the leading term in the van der Waals energy, upon rediagonalization of the Hamiltonian matrix, is of order 1/R 3 for the 6P -1S interaction, but it averages out to zero over the magnetic projections. The phenomenologically important second-order shifts of the energy levels are given in Sec. III E, with various averaging procedures illustrating the dependence of the shifts on the quantum numbers, and the dependence of the repulsive or attractive character of the interaction on the hyperfine-resolved levels.
The same procedure is applied to the 6P -2S interaction in Sec. IV, with the additional complication that virtual quasi-degenerate (6S; 2P 1/2 ) also need to be included in the basis. The treatment of the 6P -1S and 6P -2S longrange interactions reveals the presence of numerically large coefficients multiplying the 1/ρ 6 interaction terms, due to the presence of quasi-degenerate levels. The interaction remains nonretarded over all phenomenologically relevant distance scales. The repulsive character of the 6P -2S interaction due to the quasi-degenerate virtual (6S; 2P 1/2 ) levels is obtained as a surprise conclusion from the current investigation.