Positron Binding and Annihilation in Alkane Molecules
PPositron Binding and Annihilation in Alkane Molecules
A. R. Swann ∗ and G. F. Gribakin † School of Mathematics and Physics, Queen’s University Belfast, University Road, Belfast BT7 1NN, United Kingdom (Dated: August 26, 2019)A model-potential approach has been developed to study positron interactions with molecules. Binding energiesand annihilation rates are calculated for positron bound states with a range of alkane molecules, including ringsand isomers. The calculated binding energies are in good agreement with experimental data, and the existenceof a second bound state for n -alkanes (C n H n + ) with n ≥
12 is predicted in accord with experiment. Theannihilation rate for the ground positron bound state scales linearly with the square root of the binding energy.
The ability of positrons to bind to molecules underpins thespectacular phenomenon of resonantly enhanced positron an-nihilation observed in most polyatomic gases [1]. In this pro-cess, the positron is captured by the molecule, its excess energytransferred into molecular vibrations [2, 3]. The correspondingannihilation rates depend strongly on the molecular size anddisplay remarkable chemical sensitivity [4–8]. Observationof energy-resolved resonant annihilation [9] has also enabledmeasurements of the positron binding energy ε b . Binding ener-gies ranging from few to few hundred of meV, have been deter-mined experimentally for over 70, mostly nonpolar, molecularspecies, including alkanes, aromatics, partially halogenatedhydrocarbons, alcohols, formates, and acetates [10–13].This body of data is barely understood from a theoreticalstandpoint, in spite of a long history of the question [14, 15].Nearly all existing calculations of positron binding consid-ered strongly polar molecules, where binding is guaranteed atany level of theory [16]. A variety of methods were used, in-cluding Hartree-Fock [17], configuration interaction [18–20],diffusion Monte Carlo [21–23], explicitly correlated Gaussians[24], and the any-particle molecular-orbital approach [25]. Themajority of calculations examined simple diatomics, such asalkali hydrides [22] and metal oxides [21], or triatomics: hy-drogen cyanide [19, 23] and CXY (X, Y = O, S, Se) [26](see Ref. [27] for more information). In spite of this effort,of all the molecules studied experimentally, theoretical pre-dictions are available only for five strongly polar species (ac-etaldehyde, propanal, acetone, acetonitrile, and propionitrile[28–30]), and the best agreement does not exceed 25% (foracetonitrile, ε b =
136 meV, theory [30], vs. 180 meV, exper-iment [11]). Critically, quantum-chemistry calculations haveso far failed to predict positron binding to nonpolar moleculeswith any degree of accuracy [31].To address this problem, we construct a simple physicalmodel that enables calculations of positron binding to a widerange of polyatomic species and has predictive capability. Weapply the model to a range of alkanes and find good agreementwith experiment, which confirms that the effective positron-molecule potential is largely “additive” and distributed overthe molecule, and that its short-range part is just as importantas the long-range behavior determined by the molecular po-larizability. While this short-range part cannot be described ab initio with the required accuracy, we show that it can beparametrized in a reliable way. This opens the way for cal- culating positron binding energies, annihilation rates, and γ spectra for all molecules that have been studied experimen-tally and for making predictions for other molecules. Under-standing positron binding to molecules also sheds light on itscounterpart—the problem of electron attachment to moleculesand formation of molecular anions. Theoretical approach. —Since accurate predictions ofpositron binding to polyatomic molecules are beyond the ca-pacity of the best ab initio calculations, we use a model-correlation-potential approach [27]. The electrostatic poten-tial V st of the molecule is calculated at the Hartree-Fock levelusing the standard 6-311++G( d , p ) basis, and then a potentialthat describes long-range polarization of the molecular elec-tron cloud by the positron is added. The explicit form of thispotential is V cor ( r ) = − (cid:213) A α A | r − r A | (cid:104) − exp (cid:16) −| r − r A | / ρ A (cid:17)(cid:105) , (1)where the sum is over the atoms A in the molecule, r is theposition of the positron, and r A is the position of nucleus A , relative to an arbitrary origin. (Atomic units (a.u.) are usedthroughout, unless stated otherwise.) This model potential usesthe hybrid polarizabilities α A of the molecule’s constituentatoms [32], which take into account the chemical environmentof the atom within the molecule. The factor in brackets providesa short-range cutoff, characterized by the cutoff radius ρ A ,which is a free parameter of the theory. Its values are expectedto be comparable to the radii of the atoms involved, e.g., in therange of 1–3 a.u. Far from the molecule, the potential takesthe asymptotic form V cor ( r ) (cid:39) − α / r , where α = (cid:205) A α A is the molecular polarizability [33]. The short-range part ofthe potential accounts for other important electron-positroncorrelation effects, such as virtual positronium formation.The Schrödinger equation for the total potential V st + V cor issolved to obtain the positron binding energy ε b and the positronwave function. In practice, this is done using the standardquantum-chemistry package gamess with the neo plugin [34–37], which we have modified to include the model potential V cor [27]. We use an even-tempered Gaussian basis consisting of 12 s -type primitives centered on each C nucleus, with exponents0 . × i − ( i = s -type primitives centeredon each H nucleus, with exponents 0 . × i − ( i = Binding energies for alkanes. —Here we apply the methodto alkanes, which are nonpolar or very weakly polar molecules. a r X i v : . [ phy s i c s . a t o m - ph ] A ug B i nd i ng ene r g y ( m e V ) s t b o u n d s t a t e 2 n d b o u n d s t a t e FIG. 1. Positron binding energies for n -alkane molecules C n H n + .Black crosses, experiment [38]; blue circles, present calculation; or-ange triangles, zero-range potential model [39]. While no quantum-chemistry calculations of positron bindinghave been reported for them before, positron binding energieshave been measured for most of the n -alkanes C n H n + with n = does not support a positron boundstate, and while ethane C H appears to bind a positron, ε b is too small to measure), and also for isopentane C H , cy-clopropane C H , and cyclohexane C H [38]. The bindingenergy for the n -alkanes was found to increase close to linearlywith n , and a second bound state was observed for n ≥ α C = .
096 and α H = .
650 a.u., which pro-vide the best fit, α = n α C + ( n + ) α H , of the polarizabilitiesof alkanes [32]. We use the same cutoff radius for the C andH atoms, and set ρ A = .
25 a.u. to reproduce the measured ε b =
220 meV for dodecane C H . Figure 1 shows the val-ues of ε b obtained for the n -alkanes C n H n + in terms of n .Also shown are the experimental data [38] and the crude zero-range-potential (ZRP) calculations (in which each of the CH or CH groups was replaced by a short-range deltalike poten-tial, whose strength was chosen to fit the binding energy fordodecane) [39]. The present calculations and the experimentaldata are also shown in Table I. We obtain generally very goodagreement with the experimental data. For n = n = n =
12 (dodecane), in agreement with experiment, while theZRP model only shows this for n =
13. For n = n ≈
12 upwards, thecalculated binding energies show signs of saturation and dropbelow the near-linear trend observed for smaller n ; this effect iseven more pronounced in the ZRP data. Indeed, for n =
14 and
TABLE I. Calculated binding energies ε b , independent-particle-approximation contact densities δ ( ) ep , and enhanced and renormalizedcontact densities δ ep for n -alkane molecules C n H n + . Also shownare the experimental (exp.) binding energies [38]. Square bracketsindicate powers of 10. n ε b ε b (exp.) δ ( ) ep δ ep (meV) (meV) (a.u.) (a.u.)2 − . a >0 − − .
302 10 5 . [− ] . [− ] .
81 35 1 . [− ] . [− ] .
75 60 2 . [− ] . [− ] .
23 80 3 . [− ] . [− ] . . [− ] . [− ] . . [− ] . [− ] . . [− ] . [− ]
10 188 . . [− ] . [− ]
11 206 . . [− ] . [− ]
12 221 . . [− ] . [− ] b .
14 0 2 . [− ] . [− ]
13 234 . . [− ] . [− ] b .
12 3 . [− ] . [− ]
14 246 . . [− ] . [− ] b .
56 50 3 . [− ] . [− ]
15 255 . . [− ] . [− ] b .
19 4 . [− ] . [− ]
16 264 . . [− ] . [− ] b .
40 100 4 . [− ] . [− ] a With no binding, this value is determined by the size of the basis. b Second bound state.
16, our ε b for the first bound states underestimate the experi-mental values by 5 and 15%, respectively, although the secondbound state is still very well described. The exact reasons forthis discrepancy are not clear. One possibility is that at roomtemperatures such large chain molecules may favor conforma-tions other than linear [40], for which the calculations wereperformed. At the other end of the scale, our calculations with ρ A = .
25 a.u. fail to predict a bound state for n = γ -ray spectra [41]. This is incontrast to strongly polar molecules, where the bound positronis strongly localized around the negative end of the dipole[12, 27]. The wave function of the second bound state has a p -wave character. It changes sign when crossing a nodal surface(“plane”) near the centre of the molecule.Besides the near-linear increase of the binding energy for n -alkanes, the experiment found that isopentane C H , cyclo-propane C H , and cyclohexane C H have the same bindingenergies as the n -alkanes with the same number of carbon FIG. 2. Contour plots of the first (upper panel) and second (lowerpanel) bound positron states for dodecane C H . The contour forwhich the magnitude of the wave function is largest is indicated. Thechange in the magnitude of the wave function between neighboringcontours is ∆ = . atoms [38]. Using our method, we find that the binding energyfor isopentane is ε b =
59 meV, which is only 5% greater thanthe calculated value of 56 meV for n -pentane. Both values areclose to the experimental value ε b =
60 meV [38]. (The accu-racy of the experimental determination of ε b is likely no betterthan 5 meV, due to uncertainties in the energy of the positronbeam.) For neopentane, our calculations yield ε b =
57 meV,though there are no measurements for this isomer. The similar-ity between the binding energies for the three isomers suggeststhe long-range behavior of V cor (which is the same in all threecases) is more important for positron binding than the effectsof the molecular geometry. The calculated values for cyclo-propane and n -propane are ε b = .
66 and 4 . n -propane.Similarly, the calculated binding energies for cyclohexane and n -hexane are 76 and 87 meV, respectively, which can be at-tributed to the 7% smaller polarizability of cyclohexane. Ex-perimentally, they were reported to have same binding energyof 80 meV [38]. However, updated analysis using a somewhathigher resolution beam indicates ε b =
80 meV for cyclohexane and ε b =
95 meV for n -hexane [42], in close accord with thecalculations. Annihilation rates for alkanes. —The 2 γ annihilation ratefor the positron from the bound state, averaged over the elec-tron and positron spins, is given by Γ = π r c δ ep , where r isthe classical electron radius, c is the speed of light, and δ ep is the electron-positron contact density in the bound state [1].A useful conversion from the contact density to the annihila-tion rate is Γ [ ns − ] = . × δ ep [ a.u. ] . The lifetime of thepositron-molecule complex with respect to annihilation is 1 / Γ .We use the wave functions of the electronic molecular or-bitals along with the positron wave function to calculate theelectron-positron contact density δ ep , viz., δ ep = (cid:213) i γ i ∫ | ϕ i ( r )| | ψ ( r )| d τ, (2)where the sum is over all of the occupied Hartree-Fock elec-tronic spin orbitals with wave functions ϕ i , ψ is the positronwave function, and γ i is an annihilation vertex enhancementfactor , specific to spin orbital i . The enhancement factor isintroduced to improve on the independent-particle approxi-mation by accounting for an increase of the electron densityat the positron due to their Coulomb interaction [43]. Simi-lar enhancement factors are used in calculations of positronannihilation in solids [44, 45]. Recent many-body-theory cal-culations for atoms have shown that the enhancement factorsare, to a good approximation, functions of the spin-orbitalenergy ε i [27, 43]: γ i = + (cid:114) . − ε i + (cid:18) . − ε i (cid:19) . . (3)We also renormalize the positron wave function, to take into ac-count the underlying many-body nature of V cor . The true corre-lation potential that describes the interaction of a positron witha many-electron system is a nonlocal and energy-dependentoperator Σ E ( r , r (cid:48) ) [46, 47]. When using it in the Schrödinger-like Dyson equation, the negative-energy eigenvalue ε = − ε b that corresponds to a bound state becomes a function of E ,i.e., ε = ε ( E ) and has to be found self-consistently. The cor-responding positron wave function is, in fact, a quasiparticle wave function, normalized as [48, 49] ∫ | ψ ( r )| d τ = (cid:18) − ∂ε ∂ E (cid:19) − ≡ a < . (4)By considering the dependence of the binding energy on themolecular polarizability, we have determined values of a foreach molecule. The values range from a = .
992 for C H to0.933 for C H , for the first bound state, and from a = . H to 0.946 for C H , for the second bound state.Figure 3(a) shows the contact density for each of the n -alkanes, for the first and second bound states, when the latterexists. Results are shown for the independent-particle approx-imation ( γ i = a = C on t a c t den s i t y ( a . u . ) ε b (meV )0.000.010.020.03 C on t a c t den s i t y ( a . u . ) ab FIG. 3. Electron-positron contact density for n -alkane moleculesC n H n + . Panel (a) shows the contact density in terms of n , whilepanel (b) shows it as a function of √ ε b . Black symbols, independent-particle approximation; blue symbols, with enhancement factors andrenormalisation. Circles, first bound state; squares, second boundstate. In (b), thin black and blue dashed lines are fits of the respec-tive first-bound-state data, and thick red dashed line is a fit of thecalculated contact densities for positron-atom bound states [1]. also shown in Table I. Including the enhancement factors andrenormalization increases the contact density by a factor of ap-proximately 4.5 compared to the independent-particle approx-imation, irrespective of the size of the molecule. The growthof the contact density with the size of the molecule is relatedto an increase in the positron binding energy. Previous studiesof positron-atom bound states found that the contact densitygrew linearly with √ ε b , specifically, as δ ep ≈ . × − √ ε b ,where δ ep is in a.u. and ε b is in meV [1, 3]. This depen-dence is related to the probability of finding the positron inthe vicinity of the target for weakly bound s -type states. Fig-ure 3(b) shows that the contact density for the n -alkanes alsoscales linearly with √ ε b , with δ ( ) ep ≈ . × − √ ε b in theindependent-particle approximation (thin black dashed line),and δ ep ≈ . × − √ ε b , when the enhancement factorsand renormalization are included (thin blue dashed line). Thuswe see that the contact densities for positron bound stateswith alkanes are about 1.8 times greater than those for thepositron-atom bound states, for the same binding energy. Thisdifference must be related to the fact that in atoms, positronaccess of high-electron-density regions is always impeded by the nuclear repulsion, while in molecules it is easier for thepositron to approach the electrons as they are shared betweenthe constituent atoms. It is also worth noting that the con-tact density for the second bound state remains finite when itsbinding energy goes to zero. Such behavior is characteristic of p -type states that remain localized in the limit ε b → . × − for cyclopropane, 1 . × − for isopentane andneopentane, and 1 . × − a.u. for cyclohexane. With the ex-ception of cyclopropane, the contact densities for the variousisomers and ring forms are very close to those for the corre-sponding n -alkane in Table I. For cyclopropane, the contactdensity is half that of n -propane. This is related to the fact thatthe calculated binding energy for cyclopropane is six timessmaller than that of n -propane. Summary. —We have developed a method for calculatingpositron-molecule binding energies and annihilation rates anddemonstrated its predictive capabilities for the alkanes. Thesequantities are key to understanding positron resonant anni-hilation in molecules. Our method allows one to investigatepositron binding to other molecules that have been studied ex-perimentally. It can also be used to make predictions for othermolecular species, to guide future experimental effort and pro-vide comparisons for more sophisticated quantum-chemistrycalculations. The positron wave function can also be used tocalculate the annihilation γ spectra, where much of the exper-imental data [41] still awaits theoretical analysis [50]. Acknowledgments. —We are very grateful to J. R. Danielson,S. Ghosh, and C. M. Surko for providing recent unpublishedexperimental data. 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