1 Introduction

Positrons are unique probes of matter, with applications in materials science (ultra-sensitive diagnostic studies of surfaces, defects and porosity) [1], medical imaging (PET) [2,3,4,5,6,7], astrophysics [8,9,10,11], molecular spectroscopy [12,13,14], and creation of positronium and antihydrogen for tests of fundamental physics including gravity (see e.g., [15,16,17,18,19,20,21,22,23,24,25]).

Positron interactions with matter are characterized by strong many-body correlations. They significantly modify scattering, enhance annihilation rates by orders of magnitudes (see [12] for a review), and lead to positron binding in atoms and molecules [12, 26,27,28,29,30,31,32]. They also make the theoretical description of positron-matter interactions very challenging. Whilst positron binding energies have to date been measured for over 90 molecules (via vibrational-Feshbach resonant annihilation spectra) [12, 33,34,35,36,37,38,39,40,41], it was only recently that we developed a many-body theory approach that provided the first ab initio calculated binding energies in agreement with experiment (for a handful of polar and non-polar molecules) [32, 42]. We have since extended the method to successfully describe positron annihilation on the small non-binding molecules H\(_2\), N\(_2\) and CH\(_4\) [43], and positronic bonding in molecular dianions [44].

In this paper, we apply the many-body approach to investigate positron binding to hydrogen cyanide (HCN). This molecule has a substantial dipole moment of about 3 D [45] meaning that positron binding happens even at the mean-field HF level of theory.Footnote 1 Whilst we are not aware of any reported experimental results to date, previous ab initio calculations of the HCN positron binding energy have been carried out using the Hartree-Fock (HF), Configuration Interaction (CI), and diffusion Monte Carlo (DMC) methods. The CI calculations [47,48,49] reported binding energies \(\varepsilon _b\) = 35, 40, and 44 meV, using different Gaussian basis sets with a limited number of positron basis functions, with configurational state functions limited to ground state, single excitations and electron-positron double excitations. The DMC calculations [50] produced the result of \(\varepsilon _b = 38\pm 5\) meV. As CI and DMC are variational methods, their predictions should be considered as lower bounds on the true binding energy. Notably, the calculations of [48,49,50] all employed a single positron basis centre (close to the nitrogen atom) to expand the positron wave function, and it is known that the CI expansion for positronic systems converges slowly [47, 49]. In addition to the ab initio methods, a recent model-potential calculation, which assumed an isotropic dipole polarizability, found a binding energy in the range 31–81 meV (dependent on the ‘cut-off parameter’ used in the model potential), with a recommended value of 47 meV [51].

Using the same positron basis size as the DMC calculation, our many-body approach gives a binding energy of 41 meV, in good agreement with the DMC result of \(38\pm 5\) meV. However, we performed a basis-set sensitivity study and found that by including Gaussians on additional centres, notably the atoms of the molecule, and additional ghost centres, we obtain converged binding energy that is in the range 63–73 meV (depending on the molecular geometry, and the approximation used for the positron-molecule correlation potential). Moreover, we calculate the anisotropic dipole polarizability tensor, and find that it is highly anisotropic: the component along the molecular axis (of most importance for positron binding, see below) is about 1.3 times larger than the isotropic polarizability, explaining our relative larger binding energy compared with the isotropic-polarizability-based model-potential calculations of [51].

We present the essentials of our many-body theory method and the numerical details for the present calculations, and then discuss our binding energy and annihilation rate calculations, including considerations of basis set convergence.

2 Theory and numerical implementation

A comprehensive description of the many-body theory method and its implementation in our EXCITON+ code is provided in [32], so here we give only a brief overview and detail the basis parameters used for the present calculations. We calculate the positron wave function \(\psi _{\epsilon }\) and binding energy \(\epsilon \) by solving the Dyson equation

$$\begin{aligned} \left( H^{(0)}+{\hat{\Sigma }}_{\epsilon }\right) \psi _{\epsilon }({\varvec{r}}) = \epsilon \psi _{\epsilon }({\varvec{r}}), \end{aligned}$$
(1)

where \(H^{(0)}\) is the zeroth-order Hamiltonian for the positron in the field of the ground-state molecule, and \({\hat{\Sigma }}_{\epsilon }\) is the positron-molecule correlation potential (irreducible self-energy, equivalent to an optical potential [52]), which is non-local and energy-dependent. For positron bound states, the latter property means the equation is to be solved self-consistently at the bound state energy \(E=\epsilon _b\).

We construct the self-energy diagrammatically via its expansion in the residual electron–electron and electron–positron interactions (see Fig. 1 of Hofierka et al. [32]). Three infinite classes of diagrams are included. The first is the so-called GW diagram that describes the polarization of the electron cloud by the positron, and corrections to it that describe screening of the electron–positron Coulomb interaction (random phase approximation) and additional electron–hole attraction (time-dependent Hartree–Fock, or, if screened electron–hole interactions are used as done in this work, the so-called Bethe-Salpeter Equation approximation, GW@BSE). For the positron the GW diagram alone is insufficient, as one must take accurate account of strong attraction due to virtual positronium (Ps) formation (where an electron temporarily tunnels from the molecule to the positron), which is described by the diagram \(\Sigma ^{\Gamma }\) that includes the infinite ladder series of electron–positron interactions. Finally, we also consider the (screened) infinite series of positron-hole repulsive interactions \(\Lambda \), which is similar to \(\Gamma \) in structure. The total self-energy we consider here is the sum of the three channels \(\Sigma =\Sigma ^{GW+\Gamma +\Lambda }\). In practice we work with the matrix elements of \(\Sigma \) in the Hartree-Fock molecular orbital (MO) basis, and construct the individual contributions to \(\Sigma \) by solving the respective Bethe-Salpeter equations for the electron–hole polarization propagator \(\Pi \), the two-particle electron-positron propagator \(G^\textrm{ep}_\textrm{II}\) and the positron-hole propagator \(G_\textrm{II}^\textrm{ph}\) [53]. Their general form is \({{\varvec{L}}}({\omega }) = {{\varvec{L}}}^{(0)}(\omega ) + {{\varvec{L}}}^{(0)}(\omega ){\varvec{KL}}(\omega )\) where the \({{\varvec{L}}}^{(0)}\) are non-interacting two-body propagators and \({{\varvec{K}}}\) are the interaction kernels [53, 54]. In the excitation space of pair product HF orbitals \({{\varvec{L}}} = \left( {\varvec{C}}\omega -{\varvec{H}}\right) ^{-1} = {{\varvec{\xi }}} (\omega -{{\varvec{\Omega }}})^{-1}{{\varvec{\xi }}}^{-1}{{\varvec{C}}}^{-1}\), where the pair transition amplitudes \(\xi \) are the solutions of the pseudo-Hermitian linear-response generalized eigenvalue equations [55,56,57] of the form

$$\begin{aligned} {\varvec{H}} {{\varvec{\xi }}} = {{\varvec{C}}}{{\varvec{\xi }}}{{\varvec{\Omega }}}, \end{aligned}$$
(2)

with \({{\varvec{\xi }}}^{\dag }{{\varvec{C}}}{{\varvec{\xi }}}={{\varvec{C}}}\). The explicit form of the individual matrices can be found in [32]. Here the \({{\varvec{H}}}\) matrix depends on the particular two-particle propagator \({{\varvec{L}}}\) under consideration and the approximation used for it (see Extended Table 4 in [32] for the explicit matrix elements), and \({{\varvec{\xi }}}\) is the matrix of (de-)excitation eigenvectors \({{\varvec{X}}_n}\) (\({{\varvec{Y}}_n}\)) with the corresponding (de-)excitation energies \(\Omega ^+_n\) (\(\Omega ^-_n\)). Expanding the positron Dyson wave function (see Eq. 1) in the positron HF MO basis as \(\psi _{\varepsilon }(\textbf{r})= \sum _{\nu } D^{\varepsilon }_{\nu } \varphi ^+_{\nu }(\textbf{r})\) transforms the Dyson equation to the linear matrix equation \({\varvec{FD}}={\varvec{\varepsilon D}}\), where \( \langle \nu _1 | F | \nu _2\rangle = \varepsilon _{\nu _1} \delta _{\nu _1\nu _2} + \langle \nu _1 |\Sigma _{\varepsilon } | {\nu _2}\rangle \).

The electron (−) and positron (+) Hartree-Fock molecular orbitals are themselves expanded in distinct Gaussian basis sets as

$$\begin{aligned} \varphi _{a}^{\pm }({\varvec{r}})=\sum _{A}^{N_c^{\pm }}\sum _{k=1}^{N_A^{\pm }} C_{a Ak}^{\pm } \chi ^{\pm }_{A_k}({\varvec{r}}), \end{aligned}$$
(3)

where A labels the \(N_c^{\pm }\) basis centres, k labels the \(N_A^{\pm }\) different Gaussians on centre A, each taken to be of Cartesian type with angular momentum \(l^x+l^y+l^z\), such that

$$\begin{aligned} \chi _{A_k}({\varvec{r}}) =&{\mathcal {N}}_{A_k}(x-x_A)^{l^x_{Ak}}(y-y_A)^{l^y_{Ak}}\nonumber \\&\times (z-z_A)^{l^z_{Ak}} e^{-\zeta _{Ak} |\mathbf{r-r}_A|^2}, \end{aligned}$$
(4)

where \({\mathcal {N}}_{A_k}\) is a normalization constant, and C are the expansion coefficients determined from the solution of the Roothaan equations.

For the electrons we include aug-cc-pVQZ [58] basis sets centred on the H, C and N atoms. For the positron, we likewise place aug-cc-pVQZ basis sets on the H and C atoms, but, to capture the long-range correlation and full extent of the positron wave function,Footnote 2 We place a diffuse even-tempered Gaussian basis on the N atom of 10s9p8d7f form (unless otherwise stated) with exponents \(\zeta _{j}=\zeta _{1}\beta ^{j-1},\)with \(j = 1, \dots , N^{l}\), for angular momentum l and parameters \(\zeta _{1}>0\) and \(\beta >1\). In practice we performed binding energy calculations for a range of \(\zeta _{1}\) and \(\beta \) finding that there are broad ranges of stability. For Hartree-Fock calculations, the optimal \(\beta =3.0\) and \(\zeta _{1}\) was set to \(10^{-5}\) for s- and p-type Gaussians and \(10^{-4}\) for d- and f-type Gaussians (atomic units are used throughout unless otherwise stated). For many-body theory calculations, \(\beta =2.2\) and \(\zeta _{1}\) was \(10^{-3}\) for s- and p-type Gaussians and \(10^{-2}\) for d- and f-type Gaussians. It is known that the convergence of many-body theory calculations with respect to the maximal orbital angular momentum is quite slow [59] and arises from the need to describe virtual Ps localized outside the atom by an expansion in terms of single-particle orbitals centred on the nuclei. Finally, to more accurately describe virtual-Ps we place (hydrogen) aug-cc-pVQZ electron and positron basis sets on additional ‘ghost’ centres (up to 18) at manually optimized locations near the molecule; they generate effectively higher angular momenta basis functions (see, e.g. Appendix B in [51]).

The most computationally demanding part of our approach is in the calculation of the virtual-Ps self-energy contribution \(\Sigma ^{\Gamma }\). For this, the BSE matrix dimension is \(N_{\nu }\times N_{\mu }\), the product of total number of positron and excited electron MOs. For the largest calculation considered here, which employed 18 additional ghost centres, \(N_{\nu }=600\) and \(N_{\mu }=453\), resulting in the matrix of \(271,800^2\) elements, the diagonalisation of which demanded about 4.8 TB of RAM.Footnote 3

Table 1 Comparison of HCN positron binding energies \(\varepsilon _b\) for optimized and experimental geometries and previously calculated results
Full size table
Table 2 Calculated static dipole polarizabilities and ionization energies of HCN for optimized and experimental geometries along principal Cartesian axes (diagonal elements of the polarizability tensor) at HF and BSE levels of theory (Eq. 7) with \({\bar{\alpha }}\) their average
Full size table
Fig. 1
figure 1

Positron bound state wave function in the xz plane (with the H, C, and N atoms located at −2.00, 0, and 2.12 a.u. along the z axis) for three different approximations to the positron self-energy: \(\Sigma ^\textrm{BSE}\) (left), \(\Sigma ^{\textrm{BSE}+\Gamma }\) (middle) and \(\Sigma ^{\textrm{BSE}+{\tilde{\Gamma }}+{\tilde{\Lambda }}}\) (right)

Full size image

Solution of the Dyson equation yields not only the positron binding energy but also the positron-bound state wave function \(\psi _{\varepsilon }\). Using it, the 2\(\gamma \) annihilation rate in the bound state \(\Gamma = \pi r_0^2c\delta _{ep}\) (\(\Gamma [\textrm{ns}^{-1}]= 50.47\,\delta _{ep}[\mathrm{a.u.}])\), whose inverse is the lifetime of the positron-molecule complex with respect to annihilation, can be calculated. Here \(r_0\) is the classical electron radius, c is the speed of light and \(\delta _{ep}\) is the contact density

$$\begin{aligned} \delta _{ep}=\sum _{n=1}^{N_e} \gamma _{n} \int |\varphi _n({{\varvec{r}}})|^2 |\psi _{\varepsilon }({{\varvec{r}}})|^2d {{\varvec{r}}}, \end{aligned}$$
(5)

where \(\gamma _n\) are orbital dependent enhancement factors that account for the short-range electron-positron attraction [60, 61]. Previous many-body calculations for atoms by one of us determined the enhancement factors to follow a physically motivated scaling with the orbital energy \(\varepsilon _n\) in atomic units [60, 61]

$$\begin{aligned} \gamma _n = 1+ \sqrt{ {1.31}/{|\varepsilon _n|}} + \left( {0.834}/{|\varepsilon _n|} \right) ^{2.15}, \end{aligned}$$
(6)

which we assume to hold here.

Finally, we also calculate the dipole polarizability tensor as [62, 63]

$$\begin{aligned} \alpha _{ij}(E)=2\sum _n \Omega _n \frac{{\varvec{\mu }}_i^T {\varvec{X}}_n {\varvec{X}}_n^T{\varvec{\mu }}_j}{\Omega _n^2-E^2}, \end{aligned}$$
(7)

where \({\varvec{\mu }}_i\) are vectors of transition dipole moment one-electron integrals for \(i=x,y,z\). Note that in the HF approximation, the eigenvector matrix \({{\varvec{X}}_n}\) (see above) is a unit matrix and the eigenvalues \(\Omega _n\) reduce to the HF orbital energy differences. The isotropic static dipole polarizability is

$$\begin{aligned} {\bar{\alpha }}=\frac{1}{3} \left[ \alpha _{xx}(0)+\alpha _{yy}(0)+\alpha _{zz}(0)\right] . \end{aligned}$$
(8)
Fig. 2
figure 2

Convergence of positron binding energy with respect to the total number of positron and electron excited states included in the calculation (with the positron states accounting for about 60% of the total) for a number of different approximations to \(\Sigma \): the GW@BSE (circles); including the virtual-Ps ladder series of screened electron-positron interactions; the \(\Sigma ^{\Gamma }\) (squares); and additionally the ladder series of positron-hole interactions \(\Sigma ^{\Lambda }\) (triangles up). Results obtained using screened Coulomb interactions in the ‘\(\Gamma \)’ and ‘\(\Lambda \)-block’ are shown as triangles down, with the additional use of GW energies instead of HF ones in the diagrams leading to the results shown as diamonds

Full size image
Fig. 3
figure 3

Convergence of the positron bound state wave function for HCN projected along the main symmetry axis (with H,C,N atoms indicated by circles located at −2.0, 0, and 2.12 a.u.) with respect to the basis size (represented by the number of ghost centres) at the \(\Sigma ^{GW+{\tilde{\Gamma }}+{\tilde{\Lambda }}}\) level of theory. The single atom basis calculation (dotted line) used a 15s15p6d2f positron basis on the nitrogen atom (as done in [50]), while the remaining calculations used a 10s9p8d7f positron basis on nitrogen and aug-cc-pVQZ basis sets on hydrogen, carbon, and ghost atoms. The dashed line corresponds to the zero ghost calculation, the dot-dashed line a calculation with 5 ghost centres, and the solid line a calculation with 14 ghost centres

Full size image

3 Results and discussion

Table 1 presents our calculated HCN positron binding energies, as well as contact densities, dipole moments, and static dipole polarizabilities, compared with earlier CI, DMC and model-potential calculations. We consider two molecular geometries: one optimized at the HF/aug-cc-pVQZ level via minimization of the total electronic HF energy using the Molpro [64, 65] package, ensuring an internally consistent ab initio calculation, while the second is the experimental geometry [45]. Comparing the results obtained for the two geometries, we observe 3% increase in the C-N bond length \(r_\textrm{CN}\) going from HF-optimized to experimental geometry, leading to 3% change in HF binding energy but 9% difference in many-body binding energies. Further, the static dipole polarizability increases by about 4% while ionization energies decrease by about 0.3 eV. We note that the dipole moments are reported at Hartree–Fock level only, while the ionization energies and polarizabilities have been computed at both HF and GW-BSE levels of theory, as shown in Table 2. Table 2 lists the main components of the static dipole polarizability tensor at both HF and GW-BSE levels of theory, with the molecule aligned along the z-axis. Regarding basis-set dependency, these values are not appreciably affected by addition of ghost basis centres, whose primary purpose is to enlarge the virtual basis space in the many-body theory calculations; e.g., dipole polarizabilities remain the same within the stated precision, while the GW ionization energies fluctuate between 14.13 and 14.05 eV. As far as the experimental geometry is concerned, excellent agreement with the reference polarizability and ionization energy data is found [45].

Fig. 4
figure 4

Convergence of positron-electron annihilation contact density in the bound state with respect to the total number of positron and electron excited states included in the calculation (with the positron states accounting for about 60% of the total) for the electron valence MOs of HCN (HOMO-3: triangles down, HOMO-2: squares, HOMO-1: triangles up, HOMO: diamonds, and the total sum shown as circles) with enhancement factors (see Eq. 6). Here, the positron Dyson wave function is computed at \(\Sigma ^{GW+{\tilde{\Gamma }}+{\tilde{\Lambda }}}\) level of theory

Full size image
Fig. 5
figure 5

Electron valence MOs in HCN (red and blue show positive and negative electron wave function regions at ±0.006 isovalue, respectively). The highest occupied MO in (a) is doubly degenerate at 13.8 eV and has \(\pi \) character while the others are \(\sigma \)-type with energies 15.8 eV (b), 22.1 eV (c), and 34.0 eV (d), respectively. The corresponding electron-positron contact density amplitude (magenta) is shown at the \(\Sigma ^{GW+{\tilde{\Gamma }}+{\tilde{\Lambda }}}\) level at 0.001 isovalue

Full size image
Table 3 Total annihilation contact density \(\delta _\textrm{ep}\) and fractional contribution of individual molecular orbitals, at the Hartree-Fock (HF) level of the theory and using the positron Dyson wave function either without (“unenh.”) or with (“enh.”) enhancement factors accounting for the short-range annihilation vertex corrections (Eq. 6)
Full size table

Being a strongly polar molecule, HCN binds the positron even at the level of a static-potential approximation, with a binding energy of \(\sim \)1.89 meV and contact density \(\delta _{\textit{ep}}^{(0)} =1.00\times 10^{-5}\) a.u., in excellent agreement with the corresponding earlier HF results 1.94 meV and 0.967\(\times 10^{-5}\) a.u. [51]. Other methods reported HF binding energies of 1.6 meV [47, 48] and 2.0 meV [50] (see Table 1). Differences are due to slightly different bond lengths, basis sets, and the particular HF method (e.g.,“frozen target” versus “relaxed target” [51]).

Starting with the second-order GW bare polarization diagram \(\Sigma ^{(2)}\), the HF binding energy increases substantially to \(\varepsilon _b=27\) meV. Using the RPA polarizability in the GW self-energy decreases the binding energy considerably down to 7 meV. Adding in the exchange diagrams within the TDHF approximation (or RPA with exchange) increases the binding energy to 24 meV.

The inclusion of BSE screening in GW diagrams reduces this slightly to 23 meV. The inclusion of the virtual-positronium block diagram \(\Gamma \) increases the GW binding energy by a factor of 5 (see Fig. 2) and causes the positron wave function to be strongly peaked near the nitrogen atom (see Fig. 3). Moreover, considerable positron density protrudes into the region of the HOMO \(\pi \) bond, as seen in Fig. 1. Overall, the subsequent inclusion of positron-hole ladder series \(\Lambda \) down by about 40% while the inclusion of screening within the rungs of the ladder diagrams is relatively less important, lowering the binding energies by 2–5% depending on the choice of SCF or GW energies used in the BSE screening kernel.

It is important to note that many-body binding energies converge with the basis set size (and maximum angular momentum) significantly slower than HF binding energies, with the \(\Gamma \) contribution being the slowest (see Fig. 2). We found that using up to 18 ghost atoms located 1.0 Å away from the main symmetry axis with 2 more along the axis next to nitrogen (in the region of maximum positron density, see Fig. 1) increased many-body binding energies by as much as 5% in the case of GW approximation and up to 50% for the virtual-Ps level of theory results compared to a calculation with basis centres on the atoms only (see Fig. 2).

Our final converged results are higher than the previous CI [47, 48] and diffusion Monte Carlo (DMC) [50] calculations by 20–30 meV. However, as CI and DMC are variational methods, their predictions should be considered as lower bounds on the true binding energy. The DMC calculation [50] employed a single even-tempered 15s15p6d2f positron basis centre near the nitrogen atom. Using this basis in our method, a binding energy of 41 meV was obtained, in excellent agreement with the DMC calculation (\(38\pm 5\) meV  [50]). However, using this single-centre basis results in the positron wave function repulsion from the nuclei being poorly resolved, as shown in Fig. 3. Adding the even-tempered 10s9p8d7f on nitrogen and aug-cc-pVQZ basis sets on hydrogen, carbon, and ghost atoms (with the ghost atoms employing the basis of hydrogen) substantially improves the description around the atoms, and enhances the peak near the nitrogen, raising the binding energy to 63 meV. The recent model correlation potential approach of Swann and Gribakin [51] produced a range of binding energies: 31, 47, and 82 meV (with corresponding contact densities equal to 2.3, 4.1, 8.6\(\times 10^{-3}\) a.u.) obtained for cut-off radii of 2.25, 2.0, and 1.75 a.u., with smaller cut-off radii meaning a stronger correlation potential. The recommended value was 47 meV. This model potential approach assumed an isotropic molecular dipole polarizability. However, we see from Table 2 that the crucial zz component of the polarizability tensor (along the molecular axis) is \(\sim \)1.3 times larger than the isotropic value. Our many-body theory approach describes the anisotropic positron-molecule potential ab initio. Thus the larger binding energy we find compared to the model-potential recommended value is to be expected.

The convergence of positron-electron annihilation contact density \(\delta _\textrm{ep}\) with the number of virtual states is shown in Fig. 4. The enhancement factors accounting for the short-range annihilation vertex corrections (Eq. 6) average to about 4.7, with the largest value of about 5.3 for the doubly degenerate HOMO decreasing with the orbital energies through 5.1 and 3.4 down to about 2.5 for the valence orbitals shown in Fig. 5. The relative contributions to the contact density of individual MOs depend on their overlaps with the positron wave function, which is affected by their shape and magnitude in the vicinity of the positron. Notably, the HOMO-1 (at 15.8 eV) contributes more to the overall contact density than HOMO, and also the HOMO-3 (at 34.0 eV) has overlap with the positron wave function than HOMO-2 (at 22.1 eV) as seen in Table 3 and Fig. 5. Finally, we note that the contact densities contain the positron Dyson wave function normalization constant

$$\begin{aligned} \int |\psi _{\varepsilon }({{\varvec{r}}})|^2 d\textbf{r}= \left( 1-{\partial \varepsilon }/{\partial E}\right| _{\varepsilon _b}) ^{-1} \equiv a <1, \end{aligned}$$
(9)

which estimates the degree to which the positron-molecule bound state is a single-particle state, with smaller values of a signifying a more strongly-correlated state. Here we find \(a= 0.986\) at the final \(GW+\Gamma +\Lambda \) level of theory, mirroring the binding energies at each level of theory (\(a=0.997\) for the GW level of theory, decreasing to 0.977 at \(GW+\Gamma \) level).

4 Summary and conclusion

Many-body theory calculations of positron binding to HCN were performed using the Gaussian basis code EXCITON+. The effects of correlations were studied: the process of virtual-Ps formation was found to substantially enhance the binding, a near cancellation of screening corrections to the bare polarization was found, as was a non-negligible role of the positron-hole interaction. Our converged results are about 50% larger (20–30 meV in absolute terms) than previous CI or FN-DMC calculations that used a single positron basis centre. Using a similar basis we obtain results in good agreement, but find that including additional basis centres gives an improved description of the positron wave function (cusps) at the nuclei and also is required to obtain convergence of the virtual-positronium contribution to the positron-molecule correlation potential. We hope this work will stimulate further theoretical and experimental work, to shed light on this discrepancy.