Many-body Theory Calculations of Positron Binding to Halogenated Hydrocarbons

Positron binding energies in halogenated hydrocarbons are calculated \emph{ab initio} using many-body theory. For chlorinated molecules, including planars for which the interaction is highly anisotropic, very good to excellent agreement with experiment and recent DFT-based model-potential calculations is found. Predictions for fluorinated and brominated molecules are presented. The comparative effect of fluorination, chlorination and bromination is elucidated by identifying trends within molecular families including dihaloethylenes and halomethanes based on global molecular properties (dipole moment, polarizability, ionization energy). It is shown that relative to brominated and chlorinated molecules, fluorinated molecules generate a less attractive positron-molecule potential due to larger ionization energies and smaller density of molecular orbitals close to the HOMO, resulting in very weak, or in most cases loss of, positron binding. Overall, however, it is shown that the global molecular properties are not universal predictors of binding energies, exemplified by consideration of CH$_3$Cl vs.~\emph{cis.}-C$_2$H$_2$F$_2$: despite the latter having a larger dipole moment, lower ionization energy and similar polarizability its binding energy is significantly smaller (25 meV vs.~3 meV, respectively), owing to the important contribution of multiple molecular orbitals to, and the anisotropy of, the positron-molecule correlation potential.

Also recently, a model-polarization-potential method [16] was used to calculate binding in chlorinated hydrocarbons, in a joint theory-experimental study [7].Although good agreement was found with experiment for many of the molecules considered, for planar molecules the calculations substantially overestimated the measured binding energies, with the suggestion that this was due to the model assuming an isotropic long-range positron-molecule interaction [23].By contrast, DFT-model calculations for planar chloroethylenes [19] accounted for the anisotropy approximately and found better overall agreement with experiment.The method relied on an adjustable gradient parameter , whose value the authors of Ref. [19] were able to chose to replicate the binding energies of dichloroethylenes to within around 10 meV, but this value led to underestimated binding energies for tri-and tetrachloroethylene, at worst by 30 meV.The anisotropy of the positron-molecule potential, not captured by the global molecular properties, is thus important, and ab initio calculations are demanded for fundamental understanding and description of the body of experimental data.
The purpose of this Letter is twofold.First, we apply our many-body theory approach [24] to study positron binding in the chlorinated hydrocarbons considered in the recent model calculations [7,19] and experiment [7], accounting for the positron-molecule correlations and anisotropic potential ab initio.We find very good (excellent in cases) agreement with experiment and DFT-based model calculations, including for the planar molecules.Secondly, we go beyond the previous chlorinated studies [7,19] and make predictions for fluorinated and brominated molecules, and elucidate the comparative effects of fluorination, chlorination and bromination.We find that compared to their brominated and chlorinated counterparts, fluorinated molecules generate a successively less attractive positron-molecule potential resulting in very weak or loss of binding.We identify trends in binding based on global molecular properties (,  and ) for families including the sequences of cis/ [25], and halomethanes.However, we find the global properties to be poor universal indicators of binding energies, exemplified by CH 3 Cl and cis-C 2 H 2 F 2 which have similar ,  and  but significantly different positron binding energies (25 meV vs 3 meV).We explain this and the overall results, and provide further fundamental insight by considering the individual MO contributions to the positron-molecule correlation potential, showing that e.g., the decrease (or loss of) binding for bromination→chlorination→fluorination is due to successively higher molecular orbital ionization energies and smaller density of states close to the HOMO.Theoretical approach.-Adetailed description of our MBT approach is given in [21].Briefly, we solve the Dyson equation [26,27] ( Ĥ0 + Σ )  (r) =   (r) self-consistently for the positron wave function   (r) with energy .Here Ĥ0 is the zeroth-order Hamiltonian of the positron in the static (Hartree-Fock) field of the molecule and Σ is the positron self energy (an energy-dependent, non-local correlation po-tential) [28].We calculate it using a diagrammatic expansion in electron-electron and electron-positron interactions, see Fig. 1 of [21], involving three main diagram classes: the GW diagram, which describes polarization, screening of the electron-positron Coulomb interaction, and electron-hole interactions; the virtual-positronium (vPs) formation ladder series, which describes the temporary tunnelling of an electron to the positron, denoted Σ Γ ; and the positron-hole repulsion ladder series, denoted Σ Λ The significant enhancement and enabling of binding due to these correlations were delineated in [21].Here we quote results only for our most sophisticated self-energy Σ +Γ+Λ [29].We expand the electron and positron wave functions in Gaussian basis sets, using augcc-pVXZ bases (X=T,Q) [30] on atomic centres as well as additional hydrogen aug-cc-pVXZ bases on "ghost" centres 1 Å from the molecule to resolve regions of maximum positron density.For all of the molecules considered, we placed 5 ghosts around each halogen atom in the molecule in the shape of a square-pyramidal cap, with each ghost 1 Å from the halogen (see Supplemental Material "SM").We also use diffuse eventempered positron bases of the form 10987  6, with exponents  0 ×  −1 ( 0 = 0.00001-0.006and  = 2-3), ensuring the positron is described well at large distances  ∼ 1/, where  = √ 2  .For molecules with > 2 chlorines the positron wave function is delocalized (Fig. 1), and we found that accurate description of the vPs contribution requires a prohibitively large basis set [31] (for our current computing resources), and our ab initio calculations are not converged, though are lower bounds.Thus we also performed MBT-based model calculations approximating Σ ≈ Σ (2) + Σ (Λ) , using the second-order self-energy scaled to approximate the virtual-Ps contribution as introduced and justified in [21]: ab initio calculations give  in the range 1.4 to 1.5 for the HOMOs [see [21] and also Fig. 2  (d).]This approach still calculates the anisotropic polarization potential ab initio, but is much less computationally expensive.
Chlorinated molecules: comparison with experiment and model calculations.-Ourcalculated positron binding energies   for the chlorinated hydrocarbons considered in the recent isotropic-polarization-potential (IPP) [7] and DFT model calculations [19] and experiment [7], and our predictions for their fluorinated and (select) brominated counterparts are presented in Table I. Figure 1 summarizes this for the chlorinated molecules, and also presents the calculated bound-state positron Dyson orbitals for chlorinated and select chlorofluorinated molecules, showing that the positron localizes around the halogens.Overall, very good agreement is found between the ab initio MBT calculations and experiment.For CH 3 Cl, our calculated   = 25 meV is in excellent agreement with both experiment and the IPP model calculations.We find excellent agreement with experiment for CH 2 Cl 2 , and for cis-C 2 H 2 Cl 2 (for which both the IPP and DFT models substantially overestimate) and trans-C 2 H 2 Cl 2 , and reasonable agreement for vinylidene chloride C 2 H 2 Cl 2 .Overall, our ab initio results are in good agreement with the DFT-based calculations [19] (including vinyl chloride, for which there is no measurement).The results of the MBT-based model calculation, which im- portantly augment our unconverged ab initio results for the molecules with > 2 chlorines, are presented in the final column of Table I.The model calculations with  ∼ 1.5 generally give excellent agreement with experiment (with the exception of ethylene).Fluorinated molecules: predictions.-Comparedto the chlorinated molecules, in the fluorinated counterparts we find (see Table I) that positron binding is either lost or greatly reduced (as explained in the next section).We predict bound states for fluoromethane, difluoroethylene, vinyl fluoride (a few tenths of a meV each) and cis-1,2-difluoroethylene (  ∼ 3 meV).Although fluoromethane is known to be VFR active,   was found to be too small to measure [32].However, our prediction of a weak bound-state for fluoromethane of ∼ 0.3 meV is in agreement with that derived from the Z eff fit of the annihilation spectrum of CH 3 F, which until now had not been corroborated with any theoretical calculations [33].This contradicts a recent machine-learning-based prediction that fluoromethane does not bind a positron [20].Our prediction of TABLE I. Calculated MBT positron binding energies (meV) for halogenated hydrocarbons compared with experiment and model-potential calculations.For calculations denoted '< 0' binding was not observed.Where   < 1 meV, we quote values to 1 decimal place.Molecules marked '*' are those for which we believe our ab initio calculations to be unconverged and we recommend the model-MBT result (final column, see text).Also shown are calculated HF dipole moments, isotropic dipole polarizabilities (calculated at the @BSE level) and ionization energies (calculated at the @RPA level and used in the energy denominators of the self-energy analytic expressions [21]).a Model-polarization-potential calculations of Swann and Gribakin, assuming isotropic asymptotic interaction [7].
b DFT is the density-functional theory using the Perdew-Burke-Ernzerhof exchange functional result from Suzuki et al. [19].c Using a scaled self-energy Σ = Σ (2) + Σ (Λ) with  ranging from 1.4 to 1.5 to account for vPs formation [21].d Molecule is VFR active, but   is too small to measure [32].0.3 meV was derived from the Z eff fit of the VFR-based annihilation spectrum [33].e CF 4 is not VFR active [32].f From Ref. [32], where the uncertainty in the  eff peak positions from which   was measured was reported to be between 10 and 15 meV.
a bound state for CH 2 F 2 with   = 0.2 meV concurs with the 0.4 meV prediction by an earlier empirical model [3].Our lack of binding in CF 4 is consistent with experiment; this molecule is known to not be VFR active [32].We also considered 1chloro-1-fluoroethylene and (Z)-chlorofluoroethylene, and report binding energies of 5 meV and 32 meV.These values lie between the fully chlorinated and fluorinated binding energies (see below).
Comparative effect of fluorination, chlorination and bromination; the role of MO energies and density of states.-Figure 2 (a)-(c) show the calculated   as a function of the global molecular properties , , and  for the dihaloethylenes (cis/Z-C 2 H 2 XY and the isomers of C 2 H 2 Cl 2 ) and halomethanes CH 3 X, where X,Y= F, Cl, or Br.These present three distinct cases.Across the cis-dihaloethylenes  and  vary weakly, and the increase in   going from X,Y= F 2 → ClF → ... → Br 2 follows an increase in : in a given family a more polarizable target is more attractive to the positron.Across the halomethanes,  is almost constant, and the increase in   going from F to Br follows both an increasing  and decreasing  (the less tightly bound electrons are more susceptible to perturbation from the positron).For the isomers of C 2 H 2 Cl 2 ,  and  are vary weakly, and the decrease in   from cis-C 2 H 2 Cl 2 to vinylidene chloride to the non-polar trans-C 2 H 2 Cl 2 is due to successively decreasing .These three distinct cases highlight that the global molecular properties can explain trends in   for families of molecules, but they are not reliable universal predictors of binding energies, as exemplified by considering CH 3 Cl and cis-C 2 H 2 F 2 .These have very similar , but whilst cis-C 2 H 2 F 2 has a larger  and lower , it has a lower binding energy (3 meV vs. 25 meV).To explain this, and the reduction or lack of binding in fluorinated molecules in general,  I.
we consider the individual molecular orbital contributions to the correlation potential.We do so via the strength parameter S = −  ⟨| Σ |⟩/  [21,34], where  is an excited positron Hartree-Fock (HF) orbital of energy   , with the self energy taken as Σ ≈ Σ (2+Γ) , i.e., the sum of the bare polarization Σ (2) and the virtual-Ps Σ (Γ) diagrams.Figure 2 (d) shows S (Γ) , S (2+Γ) and the ratio  = S (2+Γ) /S (2) for individual MOs as a function of the MO energy for the sequence of cisdihaloethylenes: the strength parameters mainly decrease with increasing MO ionization energy because more tightly bound orbitals are more difficult for the positron to perturb [21].Additionally, Fig. 2 (e) shows the cumulative S (2+Γ) obtained by summing from the HOMO to the core orbitals.Moving from C 2 H 2 Br 2 through to C 2 H 2 F 2 sees both the total S (2+Γ) and the density of states near the ionization energy decrease: e.g., in C 2 H 2 F 2 there is a ∼ 5 eV gap between the HOMO and the HOMO−1, while this gap is approximately half as wide for C 2 H 2 Cl 2 and C 2 H 2 ClF and half as wide again for C 2 H 2 Br 2 .Further, the contributions to the cumulative S (2+Γ) below the HOMO for C 2 H 2 F 2 are smaller than those for the other three molecules as the MOs have larger .In general the transition from Br to Cl to F either shifts all the energy states to more negative energies, or at least drives the sub-HOMO energies further from the HOMO energy, inhibiting the molecule's ability to bind the positron (SM Fig. S1 shows MO energies of all molecules considered).We now consider CH 3 Cl and cis-C 2 H 2 F 2 [red triangle and blue square in Fig. 2 (a)-(c)].Figure 2 (f) shows their cumulative S (2+Γ) strength parameter.We see that although CH 3 Cl has a larger , its HOMO is doubly degenerate, and contributes relatively more to the strength than the singly-degenerate HOMO of CH 2 F 2 (a second doubly degenerate state of  character also contributes strongly at ∼ 17 eV for CH 3 Cl).Thus, in spite of CH 3 Cl having a smaller dipole moment (which governs the strength of the static potential [35]), its larger correlation potential (which contributes to binding non-linearly; see Extended Data Fig. 3 of [21]) ultimately results in stronger binding.

(g).
Summary.-Many-body theory calculations of positron binding to chlorinated hydrocarbons were found to be in good to excellent agreement with experiment and recent modelpotential-based DFT calculations.Additionally, new predictions elucidated the comparative effects of fluorination, chlorination and bromination: trends within molecular families based on the global molecular properties ,  and  were identified, as was the importance of describing the positronmolecule potential anisotropy, and accounting for the energies and density of electron states (at least near the HOMO).We suggest that any accurate universal formula for positron bind-ing energies should thus include these molecular properties.As well as providing fundamental insight, our results provide benchmarks and can inform other computational approaches to the positron-molecule and many-electron problems.perturbed by the positron and will not tunnel to the positron as readily.
Z eff spectrum of CH 3 Br.-Forvibrational-Feshbach resonant annihilation, the positron-momentum-dependent annihilation spectrum can be estimated by the Gribakin-Lee model [3], viz., where ε is the incident positron's energy (with momentum k = √ 2ε), δ ep is the contact density, ν is a vibrational mode of the  molecule with degeneracy g ν , k ν is related to the energy of the mode as k 2 ν /2 = ω ν − |ε b |, Γ ν and Γ e ν are the total and elastic resonance widths respectively (and their ratio is close to unity), and ∆(E) is related to the energy distribution of the positrons in the experimental beam.Supplemental Figure 2 shows the calculated Z eff (ε) spectrum using our calculated values of ε b = 56 meV and δ ep = 1.139 × 10 −2 a.u. as the free parameters of the model.For comparison we also show the experimental spectrum, and the original result using the Gribakin-Lee model that used ε b = 40 meV and assumed that the contact density was proportional to the square root of the binding energy as δ ep = (0.66/2π) √ 2ε b .As shown in Supplemental Figure 2, our calculated Z eff spectrum is downshifted and slightly enhanced relative to the original Gribakin-Lee model.It was recently reported that some of the earlier measurements of positron binding energies may have contained systematic errors [2]-a new measurement for CH 3 Br would therefore be of interest.

FIG. 2 .
FIG. 2. Dependence of positron binding energies on global molecular properties and individual MOs.(a)-(c): calculated   vs. calculated polarizabilities, dipole moments and ionization energy for the brominated (orange), bromochlorinated (black), chlorinated (red), chlorofluorinated (green) and fluorinated (blue) molecules; symbols denote molecular families: squares are cis-dihaloethylenes C 2 H 2 XY, triangles are halomethanes CH 3 X (X,Y = Br, Cl, F) and circles are isomers of C 2 H 2 Cl 2 .Dashed lines are guides; (d) the positron-molecule correlation strength parameters S Γ  (circles) and S 2+Γ  (squares), and the ratio   ≡ S 2+Γ  /S 2  (crosses) for each MO  against the MO HF ionization energies (vertical lines between panels) for the cis-dihaloethylenes sequence [colours as in (a)-(c)].(e) the corresponding cumulative S 2+Γ obtained by summing from the HOMO to the core orbitals.(f) the cumulative strength S 2+Γ for CH 3 Cl (red; asterisks denote double degeneracy) and cis-C 2 H 2 F 2 (blue).(g) the calculated unenhanced (  = 1) and enhanced contact densities for molecules with a Σ +Γ+Λ bound state.Colours and symbols as in (a)-(c), diamonds are remaining molecules from TableI.

TABLE I .
Calculated positron binding energies (in meV) for chloromethane at the Σ GW +Γ+Λ level of many-body theory compared with experiment using several configurations of G1 ghost atoms (pink atoms in diagrams).
a Using screened Coulomb interactions and GW @RPA energies in the ladder diagrams for Σ Γ and Σ Λ .

TABLE II
. Calculated renormalization factors, a, for the positron Dyson wave functions at the Σ GW +Γ+Λ level of many-body theory a .aUsing screened Coulomb interactions and GW @RPA energies in the ladder diagrams for Σ Γ and Σ Λ .