For the positron–atom system, the solution of the Dyson equation (Eq. 1) in a Gaussian basis yields a discrete set of n continuum pseudostates of energy ɛ _n , which decay exponentially rather than oscillate at large positron–atom separations, and are normalized to unity instead of to an asymptotic plane wave, as required by a true continuum state. Although these are not true continuum states, they can be used to extract information about positron elastic scattering from the target, as outlined in Swann and Gribakin [42]. First, we determine the s-type pseudostates ¹ of a free positron, i.e., eigenstates of the positron kinetic energy Hamiltonian in the Gaussian basis, with energies ε n 0 ( 0 ) . Since these energies increase monotonically with n ₀ (where n ₀ = 1, 2, …), there exists an invertible function f such that f 0 n 0 = ε n 0 0 . (2) Then, we determine the phase shift for the s-type pseudostates of energy ε n 0 for the positron in the dressed field of the atom as follows: δ 0 = n 0 − f 0 − 1 ε n 0 π , (3) where the inverse function f 0 − 1 is constructed by the interpolation of integer n ₀ against ε n 0 ( 0 ) . In practice, for even-tempered Gaussian-basis sets, the energies ε n 0 ( 0 ) and ε n 0 grow approximately exponentially with n ₀. It is, therefore, easier to determine function g by the interpolation of n ₀ vs. ln ( ε n 0 ( 0 ) ) , making g ( ln ( ε n 0 ) ) = f 0 − 1 ( ε n 0 ) , which is nearly linear. The phase shift for positron energy ε n 0 is then given by δ 0 = n 0 − g ln ε n 0 π . (4) The same procedure is used for p- and d-type pseudostates, utilizing p- and d-type free positron pseudostates to form invertible functions f ₁(n ₁) and f ₂(n ₂).

Since Σ ̂ E depends on the energy E of the pseudostate involved and the pseudostate energies are not known a priori, we first calculate Σ _E on a dense energy grid and interpolate to the energy of the pseudostate. For all of our calculations, we use a linear energy mesh for the self-energy, typically using 30 points between 0 and 0.3 a.u. We also tested a denser exponential energy mesh but found negligible improvement in accuracy, owing to the weak energy (E) dependence of the eigenvalues.

In addition to scattering phase shifts, we determined the scattering length a from the effective-range expansion of the s-wave phase shift for momenta k = 2 ε → 0 [43], independently fitting to each of k ⁡ cot δ 0 = − 1 a + π α 3 a 2 k , (5a) k ⁡ cot δ 0 = − 1 a + π α 3 a 2 k + C 1 k 2 ⁡ ln C 2 k , (5b) tan δ 0 = − a k 1 − π α k 3 a − 4 α k 2 3 ln C α k 4 − 1 , (5c) δ 0 = − a k − π α k 2 3 − 4 a α k 3 3 ln α k 4 + C k 3 , (5d)

where α is the static dipole polarizability of the atom determined by EXCITON+ at the BSE level of theory and C, C ₁, and C ₂ are constants. We use the first four or five lowest energy discrete datapoints of δ ₀(k) for fitting. Finally, the elastic scattering cross section is obtained as a sum over the partial waves l = 0, 1, 2 (s, p, d − waves), which dominate at low positron energies [44]: σ el = 4 π k 2 ∑ l = 0 2 2 l + 1 sin 2 δ l k . (6)

For a gas of number density n _g , the positron annihilation rate is parametrized as λ = π r 0 2 c n g Z eff , where r ₀ is the classical electron radius, c is the speed of light, and Z _eff is the effective number of electrons that participate in the annihilation process. Formally, Z _eff is equal to the electron density at the positron, Z eff k = ∫ ∑ i = 1 N e δ r − r i | Ψ k r 1 , … , r N e , r | 2 ⁡ d r 1 … d r N e d r , (7) where Ψ _k is the total wavefunction of the system, with the electron coordinate r _i and positron coordinate r. It describes the scattering of the positron of momentum k by the atom and is normalized asymptotically to the product of the ground-state target atomic wavefunction and positron plane wave. Using the finite basis approach, it can be approximated by [42] Z _eff = 4πδ _ep A ⁻², with the normalization factor A 2 = ( 2 l + 1 ) − 1 2 2 ε π d ε / d n [42], and the annihilation contact density in the independent-particle approximation is as follows: δ ep = 2 ∑ i = 1 N e / 2 γ i ∫ | φ i r | 2 | ψ r | 2 d r . (8) The summation in Eq. 8 runs over all occupied electronic orbitals φ _i , including vertex enhancement factors γ i = 1 + 1.31 / | ε i | + 0.834 / | ε i | 2.15 for orbital i with energy ɛ _i (in a.u.) that account for the effects of short-range electron–positron Coulomb attraction [24, 27]. The integral in Eq. 8 is calculated as a four-centered overlap integral over pairs of electron and positron basis functions χ i ± ( r ) χ j ± ( r ) . To speed up calculations and reduce the memory cost, we employ density fitting (DF), which involves approximating the electronic density using N _aux auxiliary (corresponding aug-cc-pVTZ or QZ type) basis functions { χ ̃ μ − ( r ) } μ = 1 N aux such that χ i − ( r ) χ j − ( r ) ≈ ∑ μ d μ i , j χ ̃ μ − ( r ) with optimal fitting coefficients d μ i , j determined using the Coulomb metric [35, 45–49]. The use of DF reduces four-centered integrals (which for basis size N requires memory ∼ N ⁴) to products of three-centered integrals and matrix elements of the Coulomb operator in the auxiliary basis (of order N ² N _aux, where N _aux ≳ N). We found DF implementation gives results within 0.5% of the exact calculation.

When analyzing the results of the many-body calculations, it is instructive to consider the physically motivated form of the s-wave Z _eff at low momenta k [16, 50] Z eff k = F κ 2 + k 2 + A k 4 + B , (9) where F, B, A and κ are constants. We also compute the Maxwellian average Z _eff at room temperature: Z ̄ eff = ∫ 0 ∞ Z eff k exp − k 2 / 2 k B T 2 π k B T 3 / 2 4 π k 2 d k , (10) where k _B is the Boltzmann constant and k _B T = 9.28 × 10⁻⁴ a.u. at room temperature T = 293 K.

Figure 9 shows Z _eff calculated for s-wave positron on He using the zeroth-order annihilation vertex [setting the enhancement factor γ _i = 1 in Eq. 8] for different approximations of the positron Dyson wave function: calculated at HF, Σ⁽²⁾ and Σ^2+Γ from the present Gaussian-based approach and the previous B-spline MBT approach. The HF results are in excellent agreement, confirming the veracity of the Gaussian basis combined with shifted pseudostate method (including the use of density fitting for the integrals, as described previously). The Σ⁽²⁾ and Σ^2+Γ annihilation rates are in good agreement, although the Gaussian-basis results are slightly smaller than the B-spline results, mirroring what was found previously for the phase shifts. Regardless, we can here assess the relative effect of the higher-order diagrams on Z _eff.

Figures 10–13 show the s-, p-, and d-wave partial-wave contributions to the total momentum-dependent positron annihilation rate Z _eff for the sequence He–Kr. It should be noted that at low-positron momenta k, the s-wave contribution always dominates and the annihilation rates increase as one moves along the noble-gas sequence. The second-order diagram Σ⁽²⁾ provides the largest contribution to the s-wave Z _eff at low momenta for all atoms except for krypton. In all atoms except for helium, the BSE approximation lowers Σ⁽²⁾ Z _eff due to screening of electron-hole interactions. For helium p- and d-waves, the higher-order MBT diagrams modify Z _eff only slightly. The virtual positronium diagram increases the annihilation rates significantly, and it becomes more important as the atom size increases. In argon and krypton, it contributes more to low-energy Z _eff than the second-order Σ⁽²⁾ diagram (see [16] for more details). Finally, the positron-hole ladder series diagram decreases Z _eff in all cases. We found that (static) screening of the ladder diagrams has a negligible effect on the Z _eff results. Specifically, using dressed instead of bare Coulomb interactions in the ladders results in < 1 % decrease in Z ̄ eff for He and Ne, and 6% and 10% decrease for Ar and Kr, respectively. However, using dressed (GW) energies instead of HF energies in the screening kernel mostly cancels out these changes (to within 2%). When compared with the previous B-spline results, our s-wave results are noticeably larger for all of the atoms. The opposite is true for d-wave and p-wave results (with the exception of neon). This is reflected in the total Z _eff results, with the current MBT results being higher at low k, but B-spline being higher as k increases (with the exception of neon, where the difference between the s-wave results are too much for the additional partial waves to overcome). When compared with the semi-empirical results of Ref. [84], He and Ne match the shape well but are noticeably larger (5% for He and 20% for Ne), Ar stays within 15%, and Kr matches the shape well but is about 25% lower.

Table 2 shows the values of Z _eff at room temperature (k = 0.053 a.u.) and thermally averaged values using the fit in Eq. 9 for our best calculation (BSE + Γ + Λ) compared with the previous results. Overall, our thermalized Z _eff results tend to be higher than the previous theoretical data. Notably, the agreement with previous MBT results [16] is worse than in the case of phase shift results. This could be due to the (energy-dependent) enhancement factors that approximate the annihilation vertex correction. We note that a proper ab initio description of the annihilation vertex is beyond current capabilities of our approach. The calculated thermally averaged annihilation rate Z ̄ eff is in excellent accord with a previous measurement of [81] for He (within 2.8%), while for neon, argon, and krypton, we calculate Z ̄ eff values that are 20%, 16%, and 12% larger than measurements of [81, 82], respectively. However, our Z ̄ eff results for Ar and Kr are lower than positron trap measurements of [83] by 8% and 18%, respectively.