Quantum chemical approach for positron annihilation ...

Quantum chemical approach for positron annihilation spectra of atoms andmolecules beyond plane-wave approximationYasuhiro Ikabata, Risa Aiba, Toru Iwanade, Hiroaki Nishizawa, Feng Wang, and Hiromi Nakai

Citation: The Journal of Chemical Physics 148, 184110 (2018); doi: 10.1063/1.5019805View online: https://doi.org/10.1063/1.5019805View Table of Contents: http://aip.scitation.org/toc/jcp/148/18Published by the American Institute of Physics

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THE JOURNAL OF CHEMICAL PHYSICS 148, 184110 (2018)

Quantum chemical approach for positron annihilation spectraof atoms and molecules beyond plane-wave approximation

Yasuhiro Ikabata,1 Risa Aiba,2 Toru Iwanade,2 Hiroaki Nishizawa,3 Feng Wang,4and Hiromi Nakai1,2,5,6,a)1Waseda Research Institute for Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku,Tokyo 169-8555, Japan2Department of Chemistry and Biochemistry, School of Advanced Science and Engineering, Waseda University,3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan3Department of Theoretical and Computational Molecular Science, Division of Computational MolecularScience, Institute for Molecular Science, Myodaiji, Okazaki, Japan4Department of Chemistry and Biotechnology, Faculty of Science, Engineering and Technology,Swinburne University of Technology, P.O. Box 218, Hawthorn, Victoria 3122, Australia5CREST, Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan6Elements Strategy Initiative for Catalysts and Batteries (ESICB), Kyoto University, Katsura,Kyoto 615-8520, Japan

(Received 17 December 2017; accepted 26 April 2018; published online 11 May 2018)

We report theoretical calculations of positron-electron annihilation spectra of noble gas atoms andsmall molecules using the nuclear orbital plus molecular orbital method. Instead of a nuclear wave-function, the positronic wavefunction is obtained as the solution of the coupled Hartree-Fock orKohn-Sham equation for a positron and the electrons. The molecular field is included in the positronicFock operator, which allows an appropriate treatment of the positron-molecule repulsion. The presenttreatment succeeds in reproducing the Doppler shift, i.e., full width at half maximum (FWHM) ofexperimentally measured annihilation (γ-ray) spectra for molecules with a mean absolute error lessthan 10%. The numerical results indicate that the interpretation of the FWHM in terms of a specificmolecular orbital is not appropriate. Published by AIP Publishing. https://doi.org/10.1063/1.5019805

I. INTRODUCTION

The positron, known as the antiparticle of the electron,can collide with and annihilate an electron, emitting γ-rays.Although the number of γ-rays depends on the direction ofthe spins at the moment of annihilation, two γ-rays with oppo-site directions are emitted in most cases. The energy of theγ-ray is 511 keV if the total momentum of the two particlesis zero, and a non-zero total momentum shifts the energy dueto the Doppler effect. A number of measurements of energyshifts caused by the electron-positron annihilation generate aspectrum with a peak at 511 keV. The shape of the spectrumreflects the electronic and positronic states, which are utilizedto analyze defects in semiconductors. This is called Dopplerbroadening spectroscopy, which is reviewed in Ref. 1 togetherwith positron lifetime spectroscopy.

Positron annihilation spectra have also been measured forisolated systems such as noble gas atoms2–7 and molecules.7,8

Various theoretical approaches have also been applied tonoble gas atoms.5,6,9–17 In addition, the annihilation spectra ofmolecules and noble gas atoms have been analyzed using theplane-wave positron (PWP) method.18–30 The PWP methodapproximates the positronic wavefunction as a plane wave,while the electronic state is determined by a quantum chemicalcalculation. However, the PWP method severely overestimates

a)Author to whom correspondence should be addressed: [email protected].

the full width at half maximum (FWHM) of the spectrum evenif the lowest energy plane-wave positron (LEPWP) approxi-mation is adopted, which sets the wavenumber of the planewave to zero. The overestimation originates in neglectingpositron-nucleus repulsion, which is discussed in Refs. 19,20, and 31. As such, the determination of the positronicwavefunction under the potential of the nuclei and elec-tronic wavefunction of a molecule, i.e., the molecular field, isdesirable.

In this report, the nuclear orbital plus molecular orbital(NOMO) method32–35 is adopted as a theoretical approach forcalculating positron annihilation spectra. The NOMO methodintroduces the concept of the orbital into nuclei as the nuclearorbital (NO), describing the total nuclear wavefunction usinga one particle wavefunction for each nucleus. Solving theHartree-Fock (HF) equation of the NOMO method33 deter-mines NOs and MOs simultaneously. Density functional the-ory (DFT) can also be formulated within the NOMO methodframework.35 Instead of a nuclear wavefunction, the NOMOand related methods can compute the positronic wavefunctionas a positronic orbital (PO), as explained in Sec. II.

After the first report of the NOMO method,32 severalgroups have proposed theoretically similar or equivalent meth-ods,36–43 which rely on the introduction of the NO andexpansion of MOs and NOs using Gaussian basis functions.The representatives are the multicomponent molecular orbitalmethod,37,38 nuclear and electronic orbital method,39,40 andany particle molecular orbital method.42,43 In addition to

0021-9606/2018/148(18)/184110/9/$30.00 148, 184110-1 Published by AIP Publishing.

184110-2 Ikabata et al. J. Chem. Phys. 148, 184110 (2018)

treating nuclear quantum effects, these multicomponent quan-tum chemical methods were applied to calculate the boundstate of positron-atom and positron-molecule interacting sys-tems.44–51 The point of the present work is the first appli-cation of such an approach to compute positron annihilationspectra.

In Sec. II, the theory of the NOMO method for a positron-containing system and the formulation of a positron annihila-tion spectrum are presented. The accuracy of our approach isdiscussed on the basis of the numerical results for noble gasatoms, diatomic molecules, and fluoro derivatives of methaneand benzene.

II. THEORETICAL ASPECTSA. Positron annihilation spectrum

The annihilation amplitude for one electron in the ithMO and one positron is given under the independent particleapproximation by52

Ai(P) =∫

exp(−iP · r) ϕei (r) ϕp(r) dr, (1)

where ϕei (r) is the ith MO, ϕp(r) is the positronic wavefunction,

and P is the total momentum of the two photons. In the case ofatoms, the MO designates the atomic orbital (AO). The proba-bility distribution function with respect to total momentum intwo-photon annihilation is given by

Wi(P) = |Ai(P)|2. (2)

When P = 0, the photon energy is Eγ1 = Eγ2 = mc2 = 511 keV,where m is the mass of the electron (positron), and c is the speedof light. When P , 0, the Doppler effect shifts the γ-ray energyby ±ε, which is given by

ε =Pc2

cos θ, (3)

where θ is the angle between the direction of the photon andthe center-of-mass velocity of the electron-positron pair. Thephoton energy spectrum for the ith MO is written similar tothe Compton scattering form,

wi(ε, c) =∫

Wi(P)

(2π)3δ(ε −

12

P · c) dP. (4)

Averaging wi(ε,c) over the direction of the emission andintroducing polar coordinates lead to

wi(ε) =1c

∫ ∞2 |ε |/c

∫Wi(P)

(2π)3P dΩPdP, (5)

where ∫Wi(P)(2π)3

dΩP4π is the spherically averaged annihilation

momentum density.20 The total annihilation spectrum w(ε) isobtained as the summation over MOs,

w(ε) =∑

i

wi(ε). (6)

To discuss the contribution of each MO to the total spectrum,we define the following factor pi:

pi =wi(0)w(0)

. (7)

We also discuss the ratio of the integrated values of wi andw, which is equal to the ratio of annihilation rates within theindependent particle model,

ζi =∫∞−∞ wi(ε)dε

∫∞−∞ w(ε)dε

. (8)

MOs and a positronic wavefunction are required for thetheoretical calculation of the annihilation spectra. The PWPmethod treats the wavefunction of a positron ϕp as a planewave with a wavenumber vector k,

ϕp(r) = exp(−ik · r). (9)

The treatment using a wavenumber of zero (k = 0) is called theLEPWP approximation. To compute the annihilation spectrausing the PWP method, conventional HF or DFT calculationsare performed to obtain MOs.

B. The NOMO method for positron-moleculeinteractions

The author’s group proposed the NOMO method32–34

to describe nuclear wavefunctions by introducing the NO.Instead, a PO for the system of positron-molecule interac-tions can be obtained within the NOMO framework. Hereafter,all nuclei are assumed to be treated classically, i.e., as pointcharges. For the system consisting of Nn nuclei, Ne electrons,and one positron, the total Hamiltonian is

HNOMO = −

Ne∑µ

12∇µ

2 −12∇p

2 +Ne∑µ<ν

1rµν−

Ne∑µ

1rµp

−

Ne∑µ

Nn∑P

ZP

rµP+

Nn∑P

ZP

rpP+

Nn∑P<Q

ZPZQ

rPQ, (10)

where the summation with respect to µ and ν runs over elec-trons and that of P and Q runs over nuclei. Moreover, r is theinterparticle distance, and ZP is the charge of the Pth nucleus.The total wavefunction of the NOMO method at the HF level,ΦNOMO/HF, is given by

ΦNOMO/HF = ψ

ei (xe

1)ψej (xe

2) · · · ψek(xe

Ne ) × ψ

p(xp), (11)

where i, j, and k are the indices of the occupied MOs. Thesuperscripts e and p mean the electron and positron, respec-tively. The electronic part of the wavefunction is a single Slaterdeterminant consisting of spin orbitals. The spin orbital for theelectron ψe

i is given as a product of the spatial orbital ϕej and

spin function,

ψei (x) =

ϕej (re)α(ω)ϕe

j (re)β(ω). (12)

The spin orbital for the positron is represented in the samemanner.

Equations (10) and (11) derive the total energy of theNOMO/HF method. The variational method is applied to thetotal energy with the orthonormalization condition for all theorbitals, leading to the following NOMO/HF equation:

184110-3 Ikabata et al. J. Chem. Phys. 148, 184110 (2018)

f eψei (xe

1) =

he +

∑j

(Je

j − Kej

)+ Jp

ψe

i (xe1) = εe

iψei (xe

1),

(13)

f pψp(xp) =

hp +

∑j

Jej

ψp(xp) = εpψp(xp), (14)

where f and ε are the Fock operator and orbital energy, respec-tively. he

i and hp are the one-particle Hamiltonians for the ithelectron and positron, respectively. J is the Coulomb operator,

Jej ψ

ei (xe

1) =

∫dxe

2

ψe∗j (xe

2)ψej (xe

2)

|re1 − re

2 |

ψe

i (xe1), (15)

Jpψei (xe

1) =

[∫dxpψ

p∗ (xp)ψp(xp)|re

1 − rp |

]ψe

i (xe1), (16)

and K is the HF exchange operator,

Kej ψ

ei (xe

1) =

∫dxe

2

ψe∗j (xe

2)ψei (xe

2)

|re1 − re

2 |

ψe

j (xe1). (17)

The positronic Fock operator f p contains the molecular field,i.e., the mean field from nuclei and electrons. After integrationwith respect to the spin variable, the MOs and PO are expandedusing the basis function χ,

ϕej (r) =

∑µ

Ceµi χ

eµ(r), (18)

ϕp(r) =∑µ

Cpµ χ

pµ(r). (19)

This expansion transforms the NOMO/HF equation into a cou-pled matrix equation, which is an extension of the Roothaanequation.53 MOs, PO, and corresponding orbital energies arethen obtained simultaneously as a self-consistent field solutionof the coupled matrix equation.

The NOMO/HF method for positron-atom or positron-molecule systems has a similarity with the static HF methodreported in the literature,6,15 where both schemes assumethe independent particle model and do not incorporate thepositron-electron correlation. The static HF method calculatesthe positronic wavefunction by solving the Schrodinger equa-tion using the potential of the HF solution for an atom. Onthe other hand, the NOMO/HF method simultaneously deter-mines both the positronic and electronic wavefunctions bysolving the coupled HF equations. In addition, the NOMO/HFmethod has employed the expansion of the orbitals using Gaus-sian basis functions, which makes it readily applicable topolyatomic systems because well-established techniques forquantum chemical calculations are available.

NOMO/DFT is derived as an extension of the Kohn-Sham formalism.54 The Kohn-Sham orbitals for electronsand positrons are introduced through the connection with theelectronic density ρe and positronic density ρp,

ρe(r) =∑

i

ϕei (r)

2, (20)

ρp(r) = ϕp(r)2. (21)

In the present report, the Kohn-Sham orbitals in Eqs. (20) and(21) are called MO and NO, as in the case of the NOMO/HFmethod, and are used to compute positron annihilation spectra.The Kohn-Sham equation of NOMO/DFT is given by

f eψei (xe

1) =

he +

∑j

Jej + V e

XC + Jp

ψe

i (xe1) = εe

iψei (xe

1),

(22)

f pψp(xp) =

hp +

∑j

Jej

ψp(xp) = εpψp(xp), (23)

where V eXC is the electron-electron exchange-correlation

potential. In this study, the numerical calculation of theelectron-positron correlation is neglected. The electronicexchange-correlation terms are numerically computed usingthe fuzzy cell method.55

III. COMPUTATIONAL DETAILS

All calculations were performed using the modified ver-sion of the General Atomic and Molecular Electronic Struc-ture System (GAMESS) program,56 and the 6-311++G(d,p)Gaussian basis set was used as the electronic basis set through-out this study. The calculations using the PWP method wereperformed with the wavenumber of the positron wavefunctionset to k = 0.05 a.u.

The positronic wavefunction of the NOMO method wasexpanded using basis functions centered on all nuclei. Thepositronic basis sets for noble gas atoms, diatomic molecules,methane derivatives, and benzene derivatives are the (8s8d),(8s8p8d), (8s6p4d), and (6s3p1d) Gaussian functions, respec-tively. Note that the contribution of the p function vanishes forthe atomic case due to spherical symmetry. The even-temperedscheme33,57 was introduced to determine the exponents of thepositronic basis functions. The mth exponent ξm is

ξm = αβm−1. (24)

α = 0.0001 and β =√

10 were employed except for benzeneand its derivatives, for which α was changed to 0.0001 ×

√10

and 0.001 for the s and d functions, respectively.The NOMO/DFT calculations adopted the following den-

sity functionals for electron-electron exchange-correlationterms: a local density approximation (LDA) based on theSlater-Dirac exchange58,59 and Vosko-Wilk-Nusair correla-tion,60 Becke 88 exchange61 and Lee-Yang-Parr correla-tion62 (BLYP), Becke’s three parameter exchange and LYPcorrelation (B3LYP),63 and long-range exchange correctedBLYP (LC-BLYP).64 The range-separation parameter of theLC-BLYP functional was set to 0.47.

IV. RESULTS AND DISCUSSIONA. Annihilation in noble gas atoms

The annihilation spectra of noble gas atoms computedusing the NOMO/HF method are assessed in this subsection.Figure 1 shows the radial distribution of electron, positron,and contact densities of a neon atom interacting with apositron. The electron and positron densities were calculated

184110-4 Ikabata et al. J. Chem. Phys. 148, 184110 (2018)

FIG. 1. (a) Radial distribution of the electron, positron, and contact densityof a neon atom obtained by the NOMO/HF calculation. (b) Percentages ofMOs for the total contact density.

using the MOs and PO determined by the NOMO/HF equa-tion [Eqs. (13) and (14)]. The contact density ρep(r) of theNOMO/HF wavefunction is given by

ρep(r) = 〈ΦNOMO/HF |

Ne∑i

δ(ri − r)δ(rp − r)|ΦNOMO/HF〉

=

Ne∑i

|ϕei (r)|2 |ϕp(r)|2. (25)

The radial distribution of the electron density shown inFig. 1(a) has two peaks corresponding to the shell structure.The positron density monotonically increases as the distancefrom the nucleus becomes large. The positron-nucleus repul-sion causes a significant diffuse distribution of the positron. Asa result, the contact density is distributed in the intermediateregion, where 2s and 2p orbitals are dominant components, asshown in Fig. 1(b). Considering that the covalent and van derWaals radii of neon are 0.58 Å65 and 1.54 Å,66 respectively,valence electrons in the chemical bonding region should beessential for annihilation spectra.

The result of Fig. 1 was obtained using significant dif-fuse (8s8d) positronic basis functions. Here, we discuss thepositronic basis-set dependence of the NOMO/HF calcula-tion of the positron-neon interacting system in terms of thetotal energy and contact density. The total energy decreasesexponentially with respect to the addition of the basis func-tion with a smaller exponent (Fig. S1 in the supplementary

material). The (8s8d) PO basis function resulted in the totalenergy higher than the conventional MO/HF calculation,i.e., a neon atom without a positron, only by 73.7 µhartree= 0.193 kJ/mol, implying that the energy of the positron issignificantly lower than the threshold of the positronium for-mation. We denote that a similar energy difference betweenMO/HF and NOMO/HF is confirmed in the case of heliumand argon atoms.

The radial distribution of the positron and contact densi-ties is shown in Fig. S2 in the supplementary material. Thepositron density is more delocalized as the more diffuse func-tion is added. While the absolute value of the contact densitymonotonically decreases with additional diffuse functions, thecontact density has a peak in the region of valence electronsfor any basis set. The distribution nature of the contact den-sity, which is important for the discussion of the shape, width,and orbital decomposition of the annihilation spectrum givenby Eqs. (6)–(8), is shown in Fig. S3 of the supplementarymaterial. This figure clearly shows that the positronic basis-set dependence is smaller than the methodological difference,i.e., PWP and NOMO. We also denote that the contact densityobtained by the NOMO/HF method distributes in the valenceregion. The low-energy positron and reasonable distributionof the contact density validate the calculation of the positronannihilation spectrum using the NOMO method.

The examination of total energy and density distributionsgives us the physical picture of the annihilation within theNOMO/HF and NOMO/DFT framework without the electron-positron correlation. The positronic wavefunction is signifi-cantly delocalized around the atom or molecule and has largerand smaller overlaps with valence and core electrons, respec-tively. The annihilation occurs mainly with an electron at thevalence region. The physical picture of the PWP method israther different: an incident wave of a positron into a molecule.Because of no interaction between the positron and molec-ular field, the location of annihilation only depends on thedistribution of electrons.

The γ-ray spectra of three noble gas atoms on a loga-rithmic scale are shown in Fig. 2. The total spectra [w(ε)from Eq. (6)] are normalized to unity at ε = 0. For all atoms,log[w(ε)] is quadratic around ε = 0, which is consistent withthe fact that a Gaussian function is a good approximation toexperimental annihilation spectra.2–8 In the case of the heliumatom, log[w(ε)] is linear at large ε, indicating that the spectrumdecays like a Slater function. As for the neon atom, log[w(ε)]is nonlinear in the large ε region. While the 2p orbital is dom-inant in the whole range, the 2s orbital has a non-negligiblecontribution at small and large ε. The inner-shell 1s orbital isalso important for the large ε region. As for the argon atom,an inflection point appears at ε ≈ 3.5 keV. All the 2p, 3s, and3p orbitals affect the total spectrum in the large ε region. Thetotal and orbital spectra of the argon atom in Fig. 2 agree wellwith the spectra obtained by the many-body theory based onthe Dyson equation.16,17

The FWHMs of the annihilation spectra of the noble gasatoms calculated using the PWP and NOMO methods arelisted in Table I. For comparison, previously reported val-ues obtained by the static HF method6,15 and experiment7 arealso listed. The percent contributions of individual MOs to the

184110-5 Ikabata et al. J. Chem. Phys. 148, 184110 (2018)

FIG. 2. Logarithmic representation of positron annihilation spectra of (a)helium, (b) neon, and (c) argon atoms obtained by the NOMO/HF method.

spectrum, i.e., pi [Eq. (7)] and ζ i [Eq. (8)], are also shown.The FWHM values of the whole spectra are in the order of Ne> He > Ar, regardless of the methodology. The PWP methodoverestimated the FWHMs for Ne and Ar atoms by more than1.5 keV. By contrast, the results of the NOMO method arein good agreement with those of the static HF calculations,in which the positron wavefunction is calculated by solvingthe Schrodinger equation with the potential from the nucleusand HF wavefunction of the electrons. According to the pre-vious study based on a diagrammatic many-body theory,17

the good agreement of the FWHM based on the HF level ofwavefunction theory and independent particle approximationis originated in the cancellation due to the positron-electroncorrelation potential and many-body effect in the annihilationamplitude. This fact supports the usefulness of our scheme.

As for the FWHM of MOs (i.e., AOs for the noble gasatoms), both the PWP and NOMO methods showed the same

trend: 1s > 2s > 3s, 2s < 2p, and 3s < 3p. The first relation-ship occurs because the inner-shell electrons move at a higherspeed. The other relationships are ascribed to the orbital angu-lar momentum of the p orbitals. Except for the 1s orbital of Ar,the NOMO method provides smaller FWHM for each orbitalcompared to the PWP method. In addition, the contributionof inner-shell MOs to the spectrum calculated by the NOMOmethod is smaller than those calculated by the PWP method.By contrast, the contribution of the outermost shells is largewhile using the NOMO method. These changes cause a reduc-tion in the overbroadening of the annihilation spectra. Thus,the annihilation spectra obtained by the NOMO/HF methodhave reasonable shapes and are consistent with the previousexperimental and theoretical studies.

B. Annihilation in molecules

The annihilation spectra of 17 molecules, namely, threediatomic molecules, methane and four fluoro derivatives, andbenzene and eight fluoro derivatives, were computed using thePWP and NOMO methods. We confirmed that the positron-molecule interaction energy is between zero and +1.0 kJ/molfor all molecules. This means the low-energy positron isdescribed as in the case of the noble gas atoms. In addition tothe HF calculation, NOMO/DFT calculations were carried outas well. Table II summarizes the total FWHM. The PWP/HFmethod overestimates the total FWHMs with a mean abso-lute error (MAE) and mean absolute percentage error (MAPE)of 1.23 keV and 47.6%, respectively, while the NOMO/HFmethod suppresses the overestimation to only 0.38 keV and14.3%, respectively. NOMO/DFT further reduces the errorwith a MAPE less than 10%. The B3LYP functional, whichmixes 20% HF exchange, demonstrates slightly larger errorscompared to the other functionals. By contrast, the LC-BLYPfunctional results in the smallest error. As a result, the inclusionof the positron wavefunction in the NOMO method signifi-cantly improves the accuracy of the calculated Doppler shiftthan the PWP method for the same molecules. The presentresult implies that the basis-set expansion of the positronicwavefunction for molecules, i.e., the linear combination ofGaussian functions set at each nucleus, works well.

Table II also indicates that a more accurate molecule-positron wavefunction could more accurately reflect the exper-imental measurements. Although the Doppler shift calculatedusing the PWP method correctly indicates the general trendof increasing Doppler shift, as the hydrogen atoms of themethane (CH4) molecule is substituted by fluorine (F) atomsone by one, the present NOMO/HF method more accurately(and precisely) calculates the Doppler shifts (ε) of CH2F2 andCHF3. For example, the FWHM ∆ε was measured as 2.86and 2.85 keV, respectively,7 which is calculated as 4.21 and4.52 keV using the PWP method, respectively, and as 3.30 and3.30 keV using the present NOMO/HF method, respectively.

In addition, for the double fluoro-phenyl derivatives(C6H4F2), the measurements exhibit apparently larger FWHMfor o-C6H4F2 (∆ε = 2.66 keV) than its m-C6H4F2 (∆ε= 2.52 keV) and p-C6H4F2 (∆ε = 2.53 keV). The trendwas reflected using the NOMO method, of which the HF cal-culations produced 3.34 keV for the ortho-isomer and 2.98and 2.91 keV for the meta- and para-isomers, respectively.

184110-6 Ikabata et al. J. Chem. Phys. 148, 184110 (2018)

TABLE I. FWHM ∆ε (in keV) and the percent contribution of MOs (p and ζ ) of noble-gas atoms. Previouslyreported ∆ε and ζ values are also listed for comparison.

PWP/HF NOMO/HF Static HF6,16Expt.7

∆ε p ζ ∆ε p ζ ∆ε ζ ∆ε

He 1s 2.79 100.0 100.0 2.60 100.0 100.0Total 2.79 2.60 2.53 2.50

Ne 1s 17.80 6.6 20.0 14.74 0.5 2.82s 3.54 33.1 20.0 2.83 29.3 13.82p 5.83 60.2 60.0 4.29 70.2 83.4Total 5.06 3.87 3.82 3.36

Ar 1s 32.84 1.9 11.1 43.73 0.0 0.02s 8.04 7.8 11.1 5.00 0.1 1.0 5.18 0.32p 16.19 12.3 33.3 8.99 0.2 4.3 9.41 0.83s 2.58 25.5 11.1 1.88 24.7 18.7 1.86 18.13p 3.75 52.4 33.3 2.89 75.0 76.0 2.89 80.8Total 3.89 2.66 2.65 2.30

The PWP method is not sensitive to the positions of the fluo-rine atoms in this case and produces 3.79, 3.79, and 3.80 keV,respectively. Experimentally, the FWHMs of fluorobenzenedepend on the number of adjacent fluorine atoms. Namely, theFWHMs of C6H5F, m-C6H4F2, and p-C6H4F2, which haveno adjacent fluorine atoms, are around 2.5 keV. In the caseof two adjacent fluorine atoms (o-C6H4F2, 1,2,4-C6H3F3, and1,2,4,5-C6H2F4), the FWHMs are around 2.7 keV. The NOMOmethod demonstrates a similar trend: about 2.75 and 3.05 keVfor zero and two adjacent fluorine atoms, respectively, whenthe LC-BLYP functional is employed.

Table III shows the FWHMs and contributions at ε = 0for individual MOs for N2 and CO molecules. Two trendsregarding the FWHMs of individual MOs of the N2 molecule,namely, 1σg < 1σu and 2σg < 3σg < 2σu, are confirmed. In

the case of the CO molecule, the common trend in FWHM is1σ > 2σ and 5σ < 3σ < 4σ. In addition, π orbitals are con-firmed to have larger FWHMs than valence σ orbitals becauseof the p-orbital character in π orbitals, where p orbitals havelarger FWHM values than s orbitals with the same principalquantum number (Table I). Compared to the PWP method,the FWHMs of MOs calculated using the NOMO methodare smaller. In comparing the PWP method to the NOMOmethod, the contribution of the inner-shell and lower valenceMOs become less dominant. By contrast, the contribution ofhigher valence MOs related to chemical bonds increases inpercentage terms. The results of these two diatomic moleculesare ascribed to the uniform and non-uniform distributionsof a positron given by the PWP and NOMO methods,respectively.

TABLE II. Calculated and experimental values of ∆ε for 17 molecules (in keV).

NOMO/DFT

PWP/HF NOMO/HF LDA BLYP B3LYP LC-BLYP Expt.7

H2 2.08 1.82 1.75 1.75 1.77 1.73 1.71N2 3.63 2.44 2.33 2.31 2.35 2.29 2.32CO 3.56 2.37 2.13 2.12 2.18 2.13 2.23CH4 2.94 2.32 2.27 2.27 2.29 2.24 2.09CH3F 3.72 3.15 2.99 2.97 3.02 2.96 2.77CH2F2 4.21 3.30 3.11 3.08 3.15 3.08 2.86CHF3 4.52 3.30 3.15 3.16 3.19 3.11 2.85CF4 4.74 3.35 3.31 3.28 3.34 3.26 3.04C6H6 3.37 2.36 2.31 2.28 2.31 2.27 2.23C6H5F 3.59 2.99 2.77 2.76 2.83 2.78 2.43o-C6H4F2 3.79 3.34 3.09 3.08 3.15 3.09 2.66m-C6H4F2 3.79 2.98 2.80 2.79 2.84 2.79 2.52p-C6H4F2 3.80 2.91 2.74 2.72 2.78 2.73 2.531,2,4-C6H3F3 3.98 3.30 3.04 3.02 3.10 3.03 2.711,2,4,5-C6H2F4 4.13 3.24 3.03 3.02 3.08 3.01 2.77C6HF5 4.28 3.40 3.18 3.17 3.23 3.16 2.89C6F6 4.41 3.40 3.19 3.18 3.24 3.16 2.95

MAE 1.23 0.38 0.22 0.21 0.26 0.21MAPE (%) 47.6 14.3 8.5 8.1 9.7 7.9

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TABLE III. FWHM ∆ε (in keV) and percent contribution of MOs p for two diatomic molecules obtained by thePWP and NOMO methods.

PWP/HF NOMO/HF

NOMO/DFT

LDA BLYP B3LYP LC-BLYP

∆ε p ∆ε p ∆ε p ∆ε p ∆ε p ∆ε p

N2 1σg 12.00 5.1 8.67 0.4 8.63 0.2 8.72 0.2 8.71 0.2 8.67 0.21σu 12.27 4.7 9.37 0.4 9.25 0.2 9.31 0.2 9.32 0.2 9.29 0.22σg 2.80 20.8 2.03 9.1 2.06 9.8 2.05 9.7 2.05 9.9 2.00 10.12σu 3.42 18.2 2.78 23.1 2.72 23.0 2.72 22.7 2.74 22.8 2.69 22.91πu 3.97 31.2 3.00 32.9 3.00 30.7 2.97 31.0 2.99 31.6 2.96 30.83σg 2.23 20.0 1.56 34.2 1.45 36.2 1.44 36.2 1.47 35.4 1.43 35.8Total 3.63 2.44 2.33 2.31 2.35 2.29

CO 1σ 12.71 4.3 10.92 0.3 10.76 0.1 10.84 0.1 10.85 0.1 10.81 0.12σ 10.41 5.8 7.45 0.4 7.34 0.3 7.40 0.2 7.40 0.2 7.37 0.23σ 2.89 20.6 2.13 8.6 2.09 9.1 2.09 9.1 2.11 9.6 2.04 9.64σ 4.02 14.9 2.96 16.2 2.89 14.9 2.87 15.1 2.90 15.9 2.83 15.91π 4.14 29.4 3.11 38.7 3.02 26.4 2.98 26.8 3.02 27.7 2.99 27.15σ 2.06 25.1 1.53 35.9 1.47 49.2 1.45 48.6 1.48 46.3 1.45 47.0Total 3.56 2.37 2.13 2.12 2.18 2.13

The FWHMs and contributions at ε = 0 of the MOs ofmethane, benzene, and their fluoro derivatives are summa-rized in the supplementary material. Overall, the inner-shellorbitals have large FWHMs and small percent contributions.Conversely, the valence MOs have small FWHMs and largecontributions. Here, we mention the lowest occupied valenceorbitals (LOVO), which have attracted attention in previousPWP studies25–29,31 because of the agreement with the FWHMof the experimental spectra. Figure 3 plots the FWHMs of theLOVO and the highest occupied molecular orbital (HOMO)to clarify the relationship with experimental FWHMs. TheLOVOs have lower FWHMs than the whole spectra mea-sured experimentally, while the HOMOs generally have higher

FIG. 3. FWHMs of the annihilation spectra ∆ε for the LOVO, HOMO, andtotal spectra of 17 molecules. The data points are taken from the results ofNOMO/DFT calculations with the LC-BLYP functional.

FWHMs. There is no linear relationship between the FWHMsof MOs and the experimental values.

Table III and Table SI of the supplementary materialindicate additional common trends between the diatomicmolecules and methane derivatives; the inclusion of thepositron wavefunction in the molecule-positron system sig-nificantly reduces the contributions from inner valence MOsbut increases the percent contribution of the outer valenceelectrons. Moreover, including the electron correlation effectin the wavefunction also increases the percent contribution.It is particularly apparent in the case of CO. For example,the percentage contribution of the HOMO is 25.1% usingthe PWP/HF method, which increases to 35.9% using theNOMO/HF method, but to 49.2% using the NOMO/DFT withthe LDA functional.

Before closing this section, we remark on the positron-electron correlation, which is definitely important for the cal-culation of positron binding energies in molecules.49,67–69 Thisstudy does not evaluate the positron-electron correlation func-tional of NOMO/DFT, which is theoretically desirable formore accurate evaluations of positron-molecule interactingsystems. There are a number of reports that evaluated the bind-ing energy with the positron-electron correlation (for example,see Refs. 49 and 67–71). The electron-nucleus correlation inthe NOMO framework35,72–75 may be applied or extended todescribe the positron-electron correlation.

Here, we emphasize that the main topic of this study is notthe binding energy but the annihilation spectrum. The anni-hilation occurs whether the positron forms a complex witha molecule or is not bound due to the repulsive interaction.According to the experiments based on the vibrational Fes-hbach resonance,52,76 N2, CH4, CHF3, and CF4 do not havea positive binding energy with a positron, while benzene andits fluoro derivatives have.52,77 The numerical accuracy in theFWHM of the annihilation spectrum is good in both cases,suggesting that the positron-electron correlation is an impor-tant factor but is not as essential as in the case of the binding

184110-8 Ikabata et al. J. Chem. Phys. 148, 184110 (2018)

energy. This fact implies that the description of a low-energypositron and reasonable probability distribution of annihilationis important, whether the calculated positron wavefunction isbound or not.

V. CONCLUSION

We applied the NOMO method to the calculation ofpositron annihilation spectra for atoms and small molecules.The wavefunction of a positron was determined with respectto the molecular field. Qualitatively reasonable spectra ofnoble gas atoms were obtained under the independent particleapproximation for the annihilation amplitude, and the over-broadening observed in the PWP method was reduced. Theresults of the NOMO calculations correspond to the phys-ical picture that the positron annihilation processes occurthrough a diffused positronic wavefunction around an atom or amolecule, which is rather different from the direct annihilationof an incident wave in the PWP method.

To the best of our knowledge, this is the first report ofthe theoretical calculation of positron annihilation spectra forpolyatomic molecules beyond the PWP method. The NOMOcalculation with an electronic exchange-correlation densityfunctional provided more accurate FWHMs of spectra within aMAE of less than 10%. The present study also reveals that theinclusion of the positron wavefunction and electron correla-tion effects in positron-atom or positron-molecule interactingsystems significantly increases the percent contribution of theouter valence electrons of the molecule. We found that no spe-cific MO such as LOVO and HOMO gives the FWHM closeto the total or experimental FWHM, which is contrary to theprevious interpretation based on the PWP method.

SUPPLEMENTARY MATERIAL

See supplementary material for the detailed results ofpolyatomic molecules in conjunction with the basis-set depen-dency for the neon atom.

ACKNOWLEDGMENTS

Some of the present calculations were performed at theResearch Center for Computational Science (RCCS), OkazakiResearch Facilities, Institutes of Natural Sciences (NINS).This study was supported in part by a Grant-in-Aid forScientific Research “KAKENHI No. 26248009” from theJapanese Ministry of Education, Culture, Sports, Scienceand Technology (MEXT), Japan, and by the Core Researchfor Evolutional Science and Technology (CREST) Program,“Theoretical Design of Materials with Innovative FunctionsBased on Relativistic Electronic Theory” of the Japan Scienceand Technology Agency (JST). F.W. acknowledges the Aus-tralian Research Council (ARC) Discovery Project (DP) No.DP110101371.

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