Calculation of Positron Binding Energies and ElectronâPositron

Article pubs.acs.org/JPCA

Calculation of Positron Binding Energies and Electron−Positron Annihilation Rates for Atomic Systems with the Reduced Explicitly Correlated Hartree−Fock Method in the Nuclear−Electronic Orbital Framework Published as part of The Journal of Physical Chemistry virtual special issue “Mark S. Gordon Festschrift”. Kurt R. Brorsen, Michael V. Pak, and Sharon Hammes-Schiffer* Department of Chemistry, 600 South Mathews Avenue, University of Illinois at Urbana−Champaign, Urbana, Illinois 61801, United States ABSTRACT: Although the binding of a positron to a neutral atom has not been directly observed experimentally, high-level theoretical methods have predicted that a positron will bind to a neutral atom. In the present study, the binding energies of a positron to lithium, sodium, beryllium, and magnesium, as well as the electron−positron annihilation rates for these systems, are calculated using the reduced explicitly correlated Hartree−Fock (RXCHF) method within the nuclear−electronic orbital (NEO) framework. Due to the lack of explicit electron−positron correlation, NEO Hartree−Fock and full configuration interaction calculations with reasonable electronic and positronic basis sets do not predict positron binding to any of these atoms. In contrast, the RXCHF calculations predict positron binding energies and electron-positron annihilation rates in qualitative agreement with previous highly accurate but computationally expensive stochastic variational method calculations. These results illustrate that the RXCHF method can successfully describe the binding of a positron to a neutral species with no dipole moment. Moreover, the RXCHF method will be computationally tractable for calculating positron binding to molecular systems. The RXCHF approach offers a balance of accuracy and computational tractability for studying these types of positronic systems. constant, has been used.19 Due to the substantial computational cost of SVM and SVMFC calculations, positron binding calculations have not been performed with these highly accurate methods for systems with more than five quantum particles (i.e., four electrons and one positron),19 where e+LiH is the largest molecular system studied.20,21 For larger molecules, Hartree−Fock (HF), second-order Møller−Plesset (MP2), and moderately sized configuration interaction (CI) methods have been employed to study positron binding. These methods are all based on a meanfield reference wave function and therefore do not include as much electron−positron correlation as the SVM methods. Inspired by Crawford’s prediction in 196722,23 that a molecule with a dipole moment greater than 1.625 D would bind a positron, many of these studies using mean-field based selfconsistent-field (SCF) methods have focused on molecular systems with large dipole moments, such as acetone, aldehydes, and nitriles,24−26 yielding binding energies that are typically within 25−50% of the experimental values. Such calculations

1. INTRODUCTION Positron (e+) binding to various species has been investigated computationally over many years. Initially, these calculations were limited to small two- to four-body systems,1−4 but in 1997, two groups demonstrated that a positron would bind to a lithium atom.5,6 This result motivated two decades of computational studies indicating that a positron would bind to a variety of neutral atoms.7−9 Positron binding to a neutral atom has never been directly observed experimentally, although various experiments for detection have been proposed.10−12 Due to the predictions of high-level calculations, it is presumed that a positron will bind to various neutral atoms. Furthermore, positron binding to neutral molecules, including alkanes, has been observed experimentally.13−17 Electron−positron correlation is thought to play an important role in positron binding to neutral atoms and molecules. The stochastic variational method (SVM), which is an approach that accurately includes electron−positron correlation by optimizing a wave function containing antisymmetric products of thousands of explicitly correlated functions,18 has been used for calculations on small atomic systems. For larger atomic systems, the SVM with a frozen core (SVMFC), where the density of the core orbitals is kept © 2016 American Chemical Society

Received: October 6, 2016 Revised: December 20, 2016 Published: December 21, 2016 515

DOI: 10.1021/acs.jpca.6b10124 J. Phys. Chem. A 2017, 121, 515−522

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The Journal of Physical Chemistry A

ally expensive SVM methods predict positron binding to these neutral atoms, whereas the traditional mean-field based SCF methods do not predict such binding. The RXCHF approach offers a balance of accuracy and computational tractability for studying these types of positronic systems.

for molecules with large dipole moments are dominated by direct electrostatic interactions and thus do not appear to require the accurate inclusion of electron−positron correlation. The experimental measurement of the binding energy of a positron to carbon disulfide15 (CS2) has also motivated additional CI based calculations on triatomic molecules without a large dipole moment,27,28 although these calculations use an unconventional definition of the positron binding energy that invokes motions along the normal modes. The mean-field based SCF methods have yet to predict positron binding to neutral atoms or molecules such as alkanes, where electron− positron correlation is likely to play an essential role in the positron binding. The reduced explicitly correlated Hartree−Fock (RXCHF) method29−31 in the nuclear−electronic orbital (NEO) framework32 is an SCF method that includes explicit correlation between different types of quantum particles, such as electrons and positrons. RXCHF differs from the more general explicitly correlated Hartree−Fock (XCHF) method33,34 in that it only includes explicit correlation between the positron and select electronic spin orbitals rather than all electronic spin orbitals as in the XCHF method. The reduction in the number of explicitly correlated spin orbitals significantly improves the scaling properties and therefore decreases the computational cost. The RXCHF method has previously been used to study e+Li and e+LiH.35 In this previous work, the RXCHF calculations agreed well with the SVM calculations for the electronic and positronic densities, the two-photon annihilation rates, and the electron−positron contact densities of these systems. The RXCHF method has better computational scaling properties than the SVM methods and therefore can be extended to larger molecular systems. Moreover, the RXCHF method includes explicit electron−positron correlation, which is lacking in mean-field based SCF methods such as NEO-HF, NEO-MP2, and NEO-CI with limited basis sets. The objective of this study is to benchmark the RXCHF method for computing the binding energy of a positron to Li, Be, Na, and Mg, as well as the electron−positron annihilation rates for these systems. The highly accurate but computation-

2. THEORY AND COMPUTATIONAL METHODS 2.A. RXCHF Method. In this section, we briefly review the equations underlying the RXCHF method. More details about the RXCHF method can be found elsewhere.29−31 For a system of N electrons, Nc classical nuclei, and one positron, the Hamiltonian in atomic units is H=−

1 2

Nc

∑

+

A=1

N

∑ ∇i2 − i=1

1 2 ∇p − 2

ZA + |r p − r cA|

N

Nc

N

∑∑ i=1 A=1

N

∑∑ i=1 j>i

ZA |r ie − r cA|

1 − |r ie − r ej|

N

∑ i=1

1 + Vnn |r ie − r p| (1)

e

c

p

where r , r , and r are the collective spatial coordinates of the electrons, classical nuclei, and positron, respectively, ZA is the charge on the Ath classical nucleus, and Vnn is the Coulombic interaction energy between classical nuclei. In the NEO-HF method, the wave function is the product of an electronic determinant and a positronic orbital ΨNEO‐HF(x1e, ... x eN , x p) = Φe(x1e, ... x eN )χ p (x p) e

(2)

p

where x and x are the collective spin coordinates of the electrons and positron, respectively, Φe is a Slater determinant of the electronic spin orbitals, and χp is the positron orbital. The NEO-HF method is not adequate to describe positronic systems due to the lack of electron−positron correlation. In the RXCHF method, the occupied electronic orbitals are partitioned into two subsystems: a subsystem of Ns special orbitals, which are explicitly correlated to the positron, and a subsystem of Nr regular orbitals, which are not explicitly correlated to the positron, such that Nr + Ns = N. The RXCHFne wave function ansatz is then defined as

ΨRXCHF‐ne(x1r, ... x rNr , x1s, ... x sNs, x p) = Φr (x1r, ... x rNr)Φs(x1s, ... x sNs)χ p (x p)G(r s, r p) = ({χ1r (x1r), ..., χNr (x rNr)} r

×

({χ1s (x1s),

where Φr is a Slater determinant of regular electronic spin orbitals, {χri }, Φs is a Slater determinant of special electronic spin orbitals, {χsi }, the coordinates are defined analogously, and ( is the antisymmetrizer. G(rs,rp) is the correlation factor and is defined as a sum over Gaussian-type geminals g(rsi ,rp) s

p

G (r , r ) =

(4)

i=1

where Ngem

g (r is,

p

r)=

s k i

p2

∑ bk e−γ |r − r | k=1

r p)

(3)

consequence of the separate Slater determinants for regular and special electrons in eq 3, the regular and special electrons are treated as distinguishable types of particles, and exchange between the two electronic subsystems is neglected. This neglect of exchange interactions between the two subsystems is denoted by “ne” in the wave function defined in eq 3. However, we emphasize that full exchange interactions between electrons are included within the regular and special electron subsystems, and therefore, RXCHF does not suffer from any self-interaction errors. The original derivation of RXCHF contained a 1 + G form for the correlation factor rather than the G form shown here. The 1 + G correlation factor can be obtained from the G correlation factor by setting bk = 1/Ns and γk ≈ 0 for a single k value. As will be discussed below, the appropriate form of the correlation factor depends on the system studied.

Ns

∑ g(r is, r p)

...,

χNs (x sNs)}χ p (x p)G(r s, s

(5)

Here bk and γk are predetermined constants, and Ngem is the number of Gaussian-type geminals in the summation. As a 516


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and the positronic orbital is expanded in terms of Npbf positronic basis functions, {ϕpμ′}

With the definition of the Hamiltonian in eq 1 and the RXCHF-ne wave function ansatz in eq 3, the RXCHF-ne energy is defined as the expectation value of the Hamiltonian ERXCHF‐ne

⟨ΨRXCHF‐ne|H |ΨRXCHF‐ne⟩ = ⟨ΨRXCHF‐ne|ΨRXCHF‐ne⟩

Npbf

ψ p(r p) =

(7)

i ∈ {1, ..., Nr} (8)

j=1

f s χis = εisχis +

Nr

∑ εijsχjr

i ∈ {1, ..., Ns} (9)

j=1

f p χ p = ε pχ p

(10)

where f is the Fock operator for the corresponding subsystem and is defined elsewhere.30,34 The electronic Fock equations contain off-diagonal Lagrange multipliers {εrij} and {εsij} that maintain orthogonality between the regular and special orbitals and can be eliminated using a variant of the orthogonality constrained basis set expansion (OCBSE) technique.30,36 To obtain the working equations for RXCHF, eqs 8−10 are converted to spatial orbitals, {ψi}, in conjunction with spatial Fock operators. For simplicity, the equations are given for a system with an even number of regular and special electronic spin orbitals, where each spatial orbital is doubly occupied. However, the equations have also been derived for an odd number of regular electronic spin orbitals31 and for a single special electronic spin orbital.34 The spatial electronic orbitals are expanded in terms of a common set of Nebf electronic basis functions, {ϕeμ} ψi r(r1r) =

Nebf

∑ Cμr ,iϕμe(r1r)

i ∈ {1, ..., Nr /2} (11)

μ=1

ψis(r1s) =

Nebf

∑ Cμs ,iϕμe(r1s) μ=1

Fr Cr = SeCr Er

(14)

FsCs = SeCsEs

(15)

FpCp = SpCpEp

(16)

where F is a Fock matrix over the basis set, C is a matrix comprised of the orbital coefficients, E is an eigenvalue matrix containing the orbital energies, and S is an overlap matrix defined over either the electronic or positronic basis sets {ϕeμ} or {ϕpμ′}, respectively. Full definitions of the Fock matrices over the basis sets are provided in refs 29−31. The inclusion of explicit correlation in the RXCHF method requires the calculation of multiple-particle integrals. RXCHF calculations with one, two, or greater than two explicitly correlated spin orbitals require the calculation of three-, four-, or five-particle integrals, respectively, with computational scaling of N2pbfN4ebf, N2pbfN6ebf, or N2pbfN8ebf, respectively, compared to N4ebf for NEO-HF calculations. The calculation of the multiple-particle integrals is the bottleneck in an RXCHF calculation. However, this calculation has recently been sped up by the implementation of a new integral code based on Rys polynomials.37 RXCHF calculations are computationally tractable when only a relatively small number of electronic orbitals are explicitly correlated to the positronic orbital. Moreover, the computational cost is decreased significantly when restricted basis sets are utilized for the special electronic orbitals (i.e., the special electronic orbitals may be expanded in terms of only certain basis functions localized on specified nuclei). In this case, the scaling mentioned above becomes N2pbfN2ebfN2rebf, N2pbfN2ebfN4rebf, or N2pbfN2ebfN6rebf, respectively, where Nrebf is the number of electronic basis functions in the restricted basis set. Such restricted basis sets have been used for NEORXCHF calculations on systems containing quantum mechanical protons31 but have not yet been implemented for positronic systems. Additional calculations on the systems in this study were performed with NEO-HF and NEO-CI. NEO-HF is a meanfield multicomponent method that does not include any explicit correlation between the positron and electrons, as given by eq 2. NEO-CI is a full CI calculation for the electrons and positron using the NEO-HF wave function as the reference determinant. NEO-CI is, in principle, exact for complete electronic and positronic basis sets, but, in practice, even qualitative accuracy with NEO-CI requires larger basis sets for the electron and positron than can be utilized with current computational resources. More details about the NEO-HF and NEO-CI methods can be found elsewhere.32 2.B. Computational Details. The binding energy of a positron to Li, Be, Na, and Mg was calculated using the RXCHF method. Positronic atoms can decay by one of two pathways

Ns

∑ εijrχjs

(13)

These expansions lead to three coupled matrix equations for the regular electronic, special electronic, and positronic orbitals

The additional approximate exchange term includes exact HF exchange, as well as additional contributions to the exchange that arise from explicit electron−positron correlation. This approximate exchange term contains three-particle and fourparticle contributions but neglects five-particle contributions, as discussed in ref 30. A previous study35 compared RXCHF-ae to RXCHF-fe, which includes full exchange between the regular and special electron subsystems in a rigorous manner. For the e+Li and e+LiH systems, the RXCHF-ae and RXCHF-fe methods provide similar electronic and positronic densities, electron−positron contact densities, and annihilation rates. For the remainder of this paper, RXCHF is defined to be RXCHF with approximate exchange (i.e., RXCHF-ae). Taking the derivative of eq 7 with respect to spin orbitals produces three coupled Fock equations for the regular electronic spin orbitals, special electronic spin orbitals, and positron orbital f r χi r = εirχi r +

Cμp′ϕμp′(r p)

μ ′= 1

(6)

As previously stated, the RXCHF-ne wave function neglects exchange between the regular and special electron subsystems, but approximate exchange, denoted by “ae” in the energy expression and explicitly defined in Appendix A, can be included as an additive term to the RXCHF-ne energy ERXCHF‐ae = ERXCHF‐ne + Eae

∑

i ∈ {1, ..., Ns/2} (12) 517

e+ A → Ps + A+

(17)

e+ A → e+ + A

(18) DOI: 10.1021/acs.jpca.6b10124 J. Phys. Chem. A 2017, 121, 515−522

Article

The Journal of Physical Chemistry A Here Ps is positronium,1 a bound electron−positron species, and A is an atom. Previous calculations have shown that e+Li and e+Na decay via the first pathway, while e+Be and e+Mg decay via the second pathway.7,19 Thus, the binding energies are calculated as BE = E(e+ A) − E(Ps) − E(A+)

As will be discussed in the next section, calculations were performed using both the G and 1 + G forms of the correlation factor for each system. The values of the bk and γk parameters for the G form with Ngem = 9 and for the G part of the 1 + G form with Ngem = 6 were obtained from ref 38. For the calculations of dissociation via the Ps and positively charged atom pathway, the energy of Ps was calculated identically to that of the positronic atom systems but without the presence of the classical nucleus and with only a single electron that resides in an explicitly correlated spin orbital. The NEO-HF and NEO-CI calculations were performed using the aug-pcseg-2 electronic basis set and the same positronic basis sets as the RXCHF calculations. A 1s, 2s, and 2p frozen core was used for the NEO-CI calculations on Mg and Na. The NEO-HF and NEO-CI calculations and the neutral and positively charged atomic calculations without a positron were performed using GAMESS.40

(19)

for e+Li and e+Na and as BE = E(e+ A) − E(A)

(20)

for e+Be and e+Mg. In eqs 19 and 20, E(X) is the energy of species X. Note that the quantity in eq 19 should strictly be called the positronium binding energy for a positively charged atom, but following convention7 in the field, the quantity in eq 19 will be called the positron binding energy. For the current study, both pathways were investigated, and the lowest energy decay pathway was found to agree with the previous calculations. To examine the ability of the RXCHF method to predict other properties of positronic atom systems, the electron− positron annihilation rates of e+Li, e+Be, e+Na, and e+Mg, each with a bound positron, were calculated. The two-photon electron−positron annihilation rate is defined as35,38

3. RESULTS AND DISCUSSION 3.A. Determination of the Correlation Factor. To determine the appropriate correlation factor for each atomic system, the absolute RXCHF energies for e+Li, e+Be, e+Na, and e+Mg using the G and 1 + G correlation factors were computed and are compared in Table 1. The calculations on Mg using the

Ne

λ = 4πr02c⟨∑ δ(r p − r ie)Oê ip⟩ i

Table 1. Absolute RXCHF Energies for e+Li, e+Be, e+Na, and e+Mg Using the G and 1 + G Correlation Factors

(21)

where r0 is the classical electron radius, c is the speed of light, and Ô eip is the singlet spin projection operator ⎛ 1 2⎞ Oê ip = ⎜1 − Sê ip⎟ ⎝ 2 ⎠

energy (h) atom

electronic basis

G

1+G

Li

aug-pcseg-0 aug-pcseg-1 aug-pcseg-2 aug-pcseg-0 aug-pcseg-1 aug-pcseg-2 aug-pcseg-0 aug-pcseg-1 aug-pcseg-2 aug-pcseg-0 aug-pcseg-1 aug-pcseg-2

−7.4687 −7.4786 −7.4826 −14.5285 −14.5551 −14.5618 −161.7006 −161.8990 −161.9126 naa naa naa

−7.4397 −7.4397 −7.4436 −14.5400 −14.5682 −14.5746 −161.6509 −161.8491 −161.8633 −199.3540 −199.5972 −199.6167

(22)

For each atom, three different electronic basis sets were used: aug-pcseg-0, aug-pcseg-1, and aug-pcseg-2.39 Previous molecular RXCHF calculations used the pc-0 basis,30,31 but for this study diffuse functions are included because other ab initio studies have indicated the importance of diffuse electronic orbitals for positronic systems.24−26 Compared to previous RXCHF calculations that used the pc basis set family, the current study used the newer segmented contractions.39 For the positronic orbital, an even-tempered basis set of s-type orbitals was used, ζi = αβi, where ζi is the ith positron basis function exponent, with α = 2.1150 and β = 0.42376. The number of positronic basis functions, Npbf, was increased until no change in the binding energy was observed at the 1 meV level. This convergence occurred at Npbf = 9, resulting in a 9s basis set for the positronic orbital. For all e+A calculations, the number of electronic orbitals explicitly correlated to the positron was equal to the number of valence electrons. The justification for this assumption is based on two points. First, the core electrons remain closer to the nucleus relative to the valence electrons and therefore are in a region of space unlikely to be occupied by the positron due to positron−nucleus repulsion. As explicit electron−positron correlation is short ranged, there is no need to explicitly correlate the positron with electrons that are unlikely to be in the relevant regions of space. Second, the previous SVM calculations on Na and Mg used a fixed core and only explicitly correlated valence electrons to the positron. Thus, for consistent comparison to the SVM fixed core calculations, the positron should be explicitly correlated to only the valence electrons.

Be

Na

Mg

a

Not available (na) because the calculation converged to an unphysical result.

G correlation factor did not converge to a physical result. For the Mg calculations with the G correlation factor using the augpcseg-0 and aug-pcseg-1 electronic basis sets, the special electronic spatial orbital converged to an orbital of p-type symmetry, while for the calculations using the aug-pcseg-2 electronic basis set, the positron orbital was in a region with negligible electronic density, indicating that the positron was not bound to the Mg atom. Thus, the Mg results with the G correlation factor are not provided in Table 1. These results demonstrate the importance of choosing the correct form of the correlation factor for the RXCHF wave function. RXCHF calculations with a single electronic spin orbital explicitly correlated to the positron lead to a lower energy with the G correlation factor than with the 1 + G correlation factor. In contrast, RXCHF calculations with two electronic spin orbitals explicitly correlated to the positron lead to a lower energy with the 1 + G correlation factor than with the G correlation factor when the positron is bound. Thus, for 518


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The Journal of Physical Chemistry A the e+Li and e+Na systems, a correlation factor of the form G is appropriate, whereas for the e+Be and e+Mg systems, a correlation factor of the form 1 + G is appropriate. For the e+Li and e+Na systems, where only a single electronic spin orbital is explicitly correlated to the positron, the G form of the correlation factor gives a lower energy because the positron binds tightly to the valence electron as a Ps species. For the e+Be and e+Mg systems, where two electronic spin orbitals are explicitly correlated to the positron, the positron cannot always be tightly bound to both valence electrons. In this case, a 1 + G form for the correlation factor is more appropriate because this correlation factor has the correct limiting behavior as the distance between an electron and the positron becomes large. These findings are supported by previous SVM calculations indicating that e+Li and e+Na are best described as a Ps and a positively charged atom, while e+Be and e+Mg are best described as a positron and a polarized atom.7 3.B. Positron Binding Energies. The RXCHF, NEO-HF, and NEO-CI binding energies of a positron to Li, Be, Na, and Mg are presented in Table 2. The NEO-HF and NEO-CI binding energies are shown in brackets, indicating a negative binding energy, as the positron is not predicted to bind for all NEO-HF and NEO-CI calculations. Note that the NEO-HF and NEO-CI energies are unphysical because the positron orbital is dominated by the most diffuse positron basis function, which is the same for all systems studied. As a result, the calculated binding energies for NEO-HF and NEO-CI are nearly identical for all of these systems and essentially correspond to the kinetic energy of the unbound positron calculated with the most diffuse basis function. For comparison with the RXCHF calculations, SVM calculations for Li and Be and SVMFC calculations for Na and Mg are also presented in Table 2. For all four atoms, the RXCHF and SVM based methods predict that a positron will bind to the neutral atom, demonstrating the importance of explicit electron−positron correlation for these systems. The substantially larger positron binding energy for Mg compared to Be is qualitatively consistent with the greater polarizability for Mg, which is expected to facilitate the binding of a positron. Specifically, the polarizabilities of the Be and Mg atoms calculated using conventional HF theory with the aug-pcseg-2 basis set were estimated to be 37.0 and 73.1, respectively, in atomic units. The difference in the positron binding energy between the RXCHF calculations with the aug-pcseg-2 electronic basis set and the SVM calculations is ∼0.1 eV for Li, Na, and Mg and 0.004 eV for Be. These RXCHF calculations invoke several approximations that could impact the accuracy: neglect of electron−electron correlation, inclusion of only approximate exchange between the regular and special electronic orbitals, inclusion of a relatively small number of electron−positron Gaussian-type geminal functions with fixed parameters, use of moderate electronic and positronic basis sets, and assumption of the single-configurational wave function ansatz given in eq 3. Given these approximations, the differences between the RXCHF and SVM or SVMFC calculations are perceived to be reasonable. A significant difference is observed in the computational timings for the RXCHF calculations with a single explicitly correlated electronic spin orbital versus the RXCHF calculations with two explicitly correlated electronic spin orbitals, resulting in a shorter computational time for e+Na than for e+Be. As mentioned in section 2, this difference is due to the four-particle integrals that must be calculated for

Table 2. RXCHF, NEO-HF, NEO-CI, and SVM Positron Binding Energies for Li, Be, Na, and Mga,b,c atom

method

electronic basis

binding energy (eV)

Li

NEO-HF NEO-CI RXCHF RXCHF RXCHF SVM41 NEO-HF NEO-CI RXCHF RXCHF RXCHF SVM42 NEO-HF NEO-CI RXCHF RXCHF RXCHF SVM43 NEO-HF NEO-CI RXCHF RXCHF RXCHF SVM19

aug-pcseg-2 aug-pcseg-2 aug-pcseg-0 aug-pcseg-1 aug-pcseg-2

[0.0309] [0.0308] 0.2605 0.1925 0.1582 0.0675 [0.0305] [0.0303] 0.1039 0.0839 0.0823 0.0860 [0.0312] [0.0310] 0.1342 0.1274 0.1222 0.0129 [0.0310] [0.0308] 0.3511 0.3433 0.3242 0.4607

Be

Na

Mg

aug-pcseg-2 aug-pcseg-2 aug-pcseg-0 aug-pcseg-1 aug-pcseg-2 aug-pcseg-2 aug-pcseg-2 aug-pcseg-0 aug-pcseg-1 aug-pcseg-2 aug-pcseg-2 aug-pcseg-2 aug-pcseg-0 aug-pcseg-1 aug-pcseg-2

a The positron binding energies of Li and Na are computed using eq 19, associated with the pathway leading to positronium dissociation. The positron binding energies of Be and Mg are computed using eq 20, associated with the pathway leading to positron dissociation. bThe NEO-HF and NEO-CI binding energies are in brackets to indicate a negative binding energy and are similar for all atoms because the positron orbital is dominated by the most diffuse basis function orbital in each case. Therefore, the NEO-HF and NEO-CI binding energies are not meaningful and are shown only for illustrative purposes. c These calculations were run on a cluster with AMD 2.3 GHz processors. The RXCHF calculations on Li, Be, Na, and Mg with the aug-pcseg-2 electronic basis set used 4096, 4096, 4096, and 8192 processors, respectively, and took 0.7, 4.9, 2.0, and 155 min, respectively. The Li, Be, Na, and Mg NEO-HF, and NEO-CI calculations were run on a single processor, and the NEO-CI calculations took 1218, 2949, 1.4, and 3.1 min, respectively. The several orders of magnitude differences among the NEO-CI timings are due to the use of a frozen core for Na and Mg but not for Li and Be.

RXCHF calculations with two explicitly correlated electronic spin orbitals. The most important result of these calculations is the qualitative prediction of binding of a positron to these atoms, particularly given that the mean-field based SCF methods, NEO-HF and NEO-CI, do not predict binding for any of these systems. Both between the RXCHF calculations with a single explicitly correlated electronic spin orbital provide a larger binding energy than predicted by the corresponding SVM calculations, while both of the RXCHF calculations with two explicitly correlated electronic spin orbitals provide a smaller binding energy than predicted by the corresponding SVM calculations. However, with the limited number of calculations in this study, it is difficult to determine whether this trend will be found in general. 3.C. Electron−Positron Annihilation Rates. The RXCHF electron−positron annihilation rates for e+Li, e+Be, 519


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The Journal of Physical Chemistry A e+Na, and e+Mg are presented in Table 3. The results are given for both the G and 1 + G correlation factors. As expected, the

performed with a single explicitly correlated electronic spin orbital in conjunction with a G correlation factor. For systems corresponding to a positron bound to a polarized atom, RXCHF calculations should be performed with two explicitly correlated electronic spin orbitals in conjunction with a 1 + G correlation factor. Calculations that do not include explicit correlation between the electrons and the positron do not predict positron binding for any of the neutral atoms. Even NEO-full CI calculations with reasonable electronic and positronic basis sets do not predict binding at all. Moreover, the NEO-RXCHF method can be extended to molecular positronic systems that will require explicitly correlating the positron to all valence electrons. The maximum scaling of the NEO-RXCHF method is N2pbfN8ebf and is more favorable with the use of restricted basis sets and smaller numbers of explicitly correlated electronic orbitals. The development of pseudopotentials for heavier atoms would also decrease the computational cost. Thus, the RXCHF method will be computationally tractable for the calculation of positron binding to larger molecular systems. In particular, such computations are expected to be more tractable than the corresponding SVM calculations, which at present have been limited to a maximum of five explicitly correlated quantum particles.19 For example, the RXCHF method could potentially be used to predict the binding of a positron to an alkane molecule, which has been observed experimentally.13,14 The theoretical prediction of positron binding to an alkane molecule by an ab initio method would be an important convergence between theory and experiment. Similarly, the study of positron binding to amino acids, as well as the calculation of annihilation rates and contact densities for these systems, could have important biomedical implications in terms of positron emission tomography.44−46

Table 3. RXCHF and SVM Electron−Positron Annihilation Rates for e+Li, e+Be, e+Na, and e+Mga λ (109 s−1) system +

e Li

e+Be

e+Na

e+Mg

method

electronic basis

G

1+G

RXCHF RXCHF RXCHF SVM41 RXCHF RXCHF RXCHF SVM42 RXCHF RXCHF RXCHF SVM43 RXCHF RXCHF RXCHF SVM19

aug-pcseg-0 aug-pcseg-1 aug-pcseg-2

1.394 1.370 1.351

0.260 0.278 0.287


0.695 0.662 0.676

0.460 0.406 0.410


1.391 1.368 1.342

0.306 0.300 0.299


nab nab nab

0.818 0.808 0.795

reference

1.748

0.334

1.898

0.955

a

Values in italics were obtained with the correlation factor that was shown in section 2.A to give a higher energy and therefore are considered to be less accurate. bNot available (na) because the calculation converged to an unphysical result.

annihilation rates are more accurate when using the correlation factor that leads to a lower energy and is physically motivated (i.e., G for Li and Na and 1 + G for Be and Mg). The calculated annihilation rates are in qualitative agreement with annihilation rates calculated with the SVM method. The deviations for RXCHF calculations with the aug-pcseg-2 electronic basis set range from 0.076 for Be to 0.556 for Na in units of 109 s−1. Note that Be has the lowest magnitude of error in both binding energy and electron−positron annihilation rate, and the other three atoms have similar magnitudes of error for both quantities. Electron−positron annihilation rates were not computed with the NEO-HF and NEO-CI methods because these methods do not lead to positron binding. However, previous calculations for molecular systems illustrated that NEO-HF incorrectly predicts the electron−positron annihilation rate by multiple orders of magnitude compared to experiment and SVM calculations.35 The reasonable agreement of the RXCHF and SVM electron−positron annihilation rates further demonstrates the accuracy of the RXCHF method relative to other SCF based methods.

■

APPENDIX A The approximate exchange term Eae in eq 7 is defined as29,30 Eae = − Nr

+

⎛ Nr Ns p s r ⎜⎜∑ ∑ ⟨χ p χar χbs |Ω(ex) 2 (p , 1, 2)|χ χb χa ⟩ S RXCHF‐ne ⎝ a = 1 b = 1 1

Ns

Ns

(p , 1, 2, 3)|χ p χbs χar χcs ⟩ ∑ ∑ ∑ [⟨χ p χar χbs χcs |Ω(ex,1) 3 a=1 b=1 c=1

(p , 1, 2, 3)|χ p χcs χar χbs ⟩ − ⟨χ p χar χbs χcs |Ω(ex,1) 3 (p , 1, 2, 3)|χ p χcs χbs χar ⟩ + ⟨χ p χar χbs χcs |Ω(ex,2) 3

)

(p , 1, 2, 3)|χ p χbs χcs χar ⟩] −⟨χ p χar χbs χcs |Ω(ex,2) 3

(A1)

RXCHF‑ne

where S is the overlap of the RXCHF-ne wave function, χr, χs, and χp are regular, special, and positron spin orbitals, respectively, and Nr and Ns are the number of occupied regular and special electronic orbitals, respectively. The coordinate dependences in eq A1 are

4. CONCLUSIONS The binding energies of a positron to a neutral atom and the electron−positron annihilation rates have been computed using the NEO-RXCHF method for Li, Be, Na, and Mg. The NEORXCHF binding energies agree with previous SVM and SVMFC binding energies to within 0.004 eV for Be and to within ∼0.1 eV for the other atoms. The NEO-RXCHF electron−positron annihilation rates agree qualitatively with previous SVM and SVMFC electron−positron annihilation rates with percentage errors ranging from 17 to 29%. The RXCHF calculations exhibit a strong dependence on the choice of correlation factor. For systems corresponding to a Ps bound to a positively charged atom, RXCHF calculations should be

|χ p (p)χbs (1)χar (2)χcs (3)⟩ = |χ p χbs χar χcs ⟩

(A2)

|χ p (p)χbs (1)χar (2)⟩ = |χ p χbs χar ⟩

(A3)

The operators in eq A1 are defined as

520

Ω(ex) 2 (p , 1, 2) = g (2, p)[Vee(1, 2)]g (1, p)

(A4)

Ω(ex,1) (p , 1, 2, 3) = 2g (2, p)[Vee(1, 2)]g (3, p) 3

(A5)


Article

The Journal of Physical Chemistry A Ω(ex,2) (p , 1, 2, 3) = g (2, p)[Vee(1, 3)]g (2, p) 3

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(A6)

where Vee(i,j) = |ri − rj|−1 is the electron−electron Coulomb operator for electrons i and j, Vep(p,i) = −|rp − ri|−1 is the electron−positron Coulomb operator for electron i and positron p, and g(i,p) is the correlation factor defined in eq 5 correlating electron i and positron p. The subscripts 2 and 3 in eqs A4−A6 indicate that the operator is a two- or threeelectron operator, respectively.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Sharon Hammes-Schiffer: 0000-0002-3782-6995 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation under CHE-13-61293. This research is also part of the Blue Waters sustained-petascale computing project, which is supported by the National Science Foundation (Awards OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana−Champaign and its National Center for Supercomputing Applications.

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