Computational and Theoretical Chemistry 1056 (2015) 24–36

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Computational and Theoretical Chemistry

journal homepage: www.elsevier .com/locate /comptc

Study of structures, energies and vibrational frequencies of (O2)n+

(n = 2–5) clusters by GGA and meta-GGA density functional methods

http://dx.doi.org/10.1016/j.comptc.2014.12.0222210-271X/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding authors. Tel.: +977 984 117 9328 (R. Parajuli). Tel.: +81 055 2208681; fax: +81 055 220 8682 (E. Torikai).

E-mail addresses: rparajuli@yahoo.com (R. Parajuli), et@yamanashi.ac.jp(E. Torikai).

Govinda Prasad Khanal a,f, Rajendra Parajuli b,⇑, Elangannan Arunan c, Shinichi Yamabe d, Kenzo Hiraoka e,Eiko Torikai f,⇑a Central Department of Physics, Tribhuvan University, Kathmandu, Nepalb Department of Physics, Amrit Campus, Tribhuvan University, Kathmandu, Nepalc Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, Indiad Institute for Fundamental Chemistry, Kyoto University, Takanao-Nishihiraki-cho-34-4, Kyoto 606-8103, Japane Clean Energy Research Center, University of Yamanashi, 4-3-11 Takeda, Kofu 400-8511, Japanf Interdisciplinary Graduate School of Medicine and Engineering, University of Yamanashi, 4-3-11 Takeda, Kofu 400-8511, Japan

a r t i c l e i n f o a b s t r a c t

Article history:Received 6 November 2014Received in revised form 24 December 2014Accepted 24 December 2014Available online 3 January 2015

Keywords:Oxygen cluster cationsM05-2XM06-2XDissociation energySpin multiplicityHarmonic vibrational frequency

Using Generalized Gradient Approximation (GGA) and meta-GGA density functional methods, structures,binding energies and harmonic vibrational frequencies for the clusters O4

+, O6+, O8

+ and O10+ have been

calculated. The stable structures of O4+, O6

+, O8+ and O10

+ have point groups D2h, D3h, D4h and D5h optimizedon the quartet, sextet, octet and dectet potential energy surfaces, respectively. Rectangular (D2h) O4

+ hasbeen found to be more stable compared to trans-planar (C2h) on the quartet potential energy surface.Cyclic structure (D3h) of O6

+ cluster ion has been calculated to be more stable than other structures.Binding energy (B.E.) of the cyclic O6

+ is in good agreement with experimental measurement. Thezero-point corrected B.E. of O8

+ with D4h symmetry on the octet potential energy surface and zero-pointcorrected B.E. of O10

+ with D5h symmetry on the dectet potential energy surface are also in goodagreement with experimental values. The B.E. value for O4

+ is close to the experimental value when singlepoint energy is calculated by Brueckner coupled-cluster method, BD(T).

� 2014 Elsevier B.V. All rights reserved.

1. Introduction The ion at m/z at 64 detected by rocket-borne mass spectrometer

There is a wide range of experimental and theoretical method-ologies for the understanding of properties and nature of the clus-ter systems. The ultimate goal of such studies is to bridge the gapbetween the gaseous and condensed phases, i.e., the fifth state ofmatter [1]. Study on the cluster system is of paramount importancebecause clusters play very important roles in nature as reviewed byCastleman and Jena [2].

The system we choose here is oxygen cluster cations (O2)n+

which have been studied for fifty years since the pioneering workby Yang and Conway [3,4]. Since then, numerous experimental andtheoretical works on this system have been made. The study of O4

+

is of special interest for a variety of reasons. This ion plays the keyrole in the atmospheric chemistry. O4

+ was found to be the reactionintermediate for the formation of protonated water clusters,H+(H2O)n as a final product in the D-layer of ionosphere [5,6].

was assigned to O4+ [7].

There are a limited number of reports regarding theoreticalstudies of (O2)n

+ for n greater than 2 [8,9]. Zhou et al. [9] measuredinfrared spectra of O6

+ in solid argon and neon. They assigned theabsorption at 1435.0 and 1429.5 cm�1 in solid neon to cyclic(D3h) and trans-planar structures of O6

+, respectively. They carriedout computational calculations for both cyclic and trans-planarstructures of O6

+ by employing B3LYP/6-311+G⁄ theoretical methodand found that the trans-planar structure is more stable than thecyclic structure by 3.3 kcal/mol. Experimentally, they found thatcyclic (D3h) O6

+ is more stable than the trans-planar (C2h) one onannealing to 12 K in solid neon. Moreover, they found the presenceof only trans-planar O6

+ in solid argon. The authors pointed out thepossibility of ‘‘inverse-symmetry breaking phenomenon’’ in largerO6

+ ion like that already pointed out for smaller ions by others[10,11] for H2

+ and (NO)2+. Ricks et al. [12] studied O4

+, O6+ and O8

+

clusters ions by time of flight mass spectrometry and infrared(IR) laser photodissociation spectroscopy. Electron spin resonancespectroscopy (ESR) [13] and matrix IR measurements [14]predicted that the structure of O4

+ has a trans-planar C2h geometry.Thompson and Jacox [14] reluctantly assigned the structure for O4

+

G.P. Khanal et al. / Computational and Theoretical Chemistry 1056 (2015) 24–36 25

as trans-planar due to the lack of a better candidate (the authors[14] observed two prominent absorptions at 1164 and 1320 cm�1

and weaker absorptions at 2808 and 2949 cm�1. They assignedthe absorptions at 1164 and 2808 cm�1 to trans-planar, C2h struc-ture of O4

+). Later on, after the landmark work by Lindh and Barnes[15] for O4

+, Jacox and Thompson [16] reassigned the two absorp-tion spectra at 1320.0 and 2949.0 cm�1 to rectangular, D2h struc-ture. Ricks et al. [12] also, assigned their photodissociation IRspectra at 1323.0 and 2950 cm�1 to anti-symmetric O–O stretchvibration and combination of this with symmetric O–O stretchvibration of rectangular (D2h) structure of O4

+, respectively. More-over, Lindh and Barnes [15] found from their CASSCF (completeactive space self-consistent field) calculations that the rectangularD2h structure and trans-planar C2h structure of O4

+ are the most sta-ble structures. Jacox and Thompson [16] provided arguments onhow rectangular (D2h) structure could be stabilized in solid neon.Recently, Jacox and Thompson [17] studied combination bandsfor the rectangular O4

+ in neon-matrix experiments. There havebeen reports on computational works focused on the study of rect-angular O4

+ as well [18–20]. On the other hand, computationalstudy due to Peel [21] concluded the ground state structure of O4

+

in gas phase is trans-planar at QCISD(T)/6-31G⁄ level of theory.Immediately after its publication, Barnes and Lindh [22] concludedthe symmetry breaking should be considered even at the QCISD(T)level of theory, and concluded that O4

+ has the rectangular structurein its ground state using Brueckner coupled cluster method.

Several experimental studies have been made to determine thebinding energies of (O2)n

+ with n P 2 [3,4,23–26], undergoing clus-ter decay (O2)n

+ ? (O2)n�1+ + O2. Conway and Janik [23] measured

the binding energies for O4+, O6

+, O8+ and O10

+ to be �10.53, �6.51,�2.56, �2.43 kcal/mol, respectively. By the measurement of gasphase equilibria for the solvation reaction of O2

+ with O2 molecules,Hiraoka [25] measured the binding energies of O4

+, O6+, O8

+ and O10+ as

�9.15, �5.89, �2.48, �2.14 kcal/mol, respectively, in good agree-ment with those measured by Conway and Janik [23]. Hiraoka pre-dicted that the positive charge is delocalized in O4

+ and also in O6+

and suggested the rectangular geometry for O4+. After the Hiraoka’s

work [25], Matt et al. [26] studied (O2)n+, n = 2–10 by high resolu-

tion two sector-field mass spectrometer of reversed geometry andcalculated binding energy from the average kinetic energy releasemeasurement taken, in the centre of mass utilizing finite heat baththeory. As the finite heat bath theory is based on statisticalassumption, they could not estimate the binding energies for thecluster with n less than 5. They could only estimate the bindingenergy for O10

+ to be �3.69 kcal/mol because the cluster size shouldbe at least 6 or more to use the finite heat bath theory [26].

By observing gas phase photodissociation spectra of O4+ and that

of O6+, Ricks et al. [12] concluded that O6

+ does not have the O2-sol-vated O4

+, O4+� � �O2, but the positive charge is delocalized in three O2

molecules, i.e., (O2)3+, in agreement with the prediction made by

Hiraoka [25]. Further, by the observation of strong band for O6+

and O8+, they predicted that O8

+ has the O2-solvated O6+, i.e.,

(O2)3+� � �O2. This is also in agreement with the prediction made by

Hiraoka [25]. Ricks et al. also observed the strong harmonic vibra-tional frequency of 1323 cm�1 in the Ar-tagged O4

+ (i.e. O4+Ar) in

infrared photodissociation spectra and the spectrum was assignedto asymmetric O–O stretch frequency for O4

+Ar. The other promi-nent peaks were 2950 cm�1, 3103 cm�1, 3260 cm�1, 3363 cm�1,3519 cm�1, and 3612 cm�1, and 3723 cm�1, 3858 cm�1,4079 cm�1 and 4280 cm�1 with weak intensities. Also, in the caseof O6

+, they obtained strong harmonic vibrational frequency (here‘‘strong’’ refers to IR intensity) at 1425 cm�1 and other weakerbands at 2846 cm�1, 3038 cm�1. Moreover, for O8

+, they observedstrong harmonic vibrational frequency at 1426 cm�1 and otherweaker bands observed at 1488 cm�1, 2865 cm�1, 2916 cm�1 and3025 cm�1.

In this work, we have calculated structures, binding energiesand harmonic vibrational frequencies for the clusters O4

+, O6+, O8

+

and O10+ . The main objective of this work was to check whether

Generalized Gradient Approximation (GGA) and meta-GGA densityfunctional methods can be used for (O2)n

+ systems or not. It wasfound that quartet (S = 3/2) electronic ground state is favorablefor O4

+ [7], whereas O6+ has a sextet (S = 5/2) electronic ground state

[5]. Moreover, we have calculated that O8+ has octet (S = 7/2) elec-

tronic ground state and that of O10+ is dectet (S = 9/2).

After completing this work, we note that Sokolov et al. [27]applied Density cumulant functional theory [28] and showed thatthis method is capable of providing an accurate description ofstructural parameters and vibrational frequencies for O4

+. To thebest of our knowledge, the work presented here will be the firstreport on complete electronic structure calculations of structures,energies and vibrational frequencies of higher size cluster of(O2)n

+ (n = 3, 4, 5). We expect that this study will inspire moreworks to elucidate the structure and energetics of open shell clus-ter (O2)n

+ with cluster size n greater than 2.

2. Computational methods

The geometrical structures of (O2)n+ (n = 2–5) are fully opti-

mized using density functional theory (DFT) employing meta Gen-eralized Gradient Approximation (meta-GGA) with hybrid metaexchange correlation functionals M05-2X and M06-2X developedby Truhlar’s group [29,30]. The B3LYP [31,32] functional cannotperform satisfactorily in non-covalent interactions because it doesnot accurately describe medium-range exchange–correlation (XC)energy [33]. Hence, we have employed meta-GGA DFT methodbecause meta-GGAs include kinetic energy density as an input,and can yield accurate ground-state energies for molecules, solids,and surfaces [34]. We have also performed calculations with B3LYPand BHandH [35] for comparative study. The Brueckner orbitalbased on coupled-cluster single-point calculation was also carriedout [36–38] on the M06-2X O2, O2

+ and O4+ geometries. Brueckner

doubles with a triples contribution, BD(T), gave reliable single-point energies [39]. At the fifth order of perturbation, BD(T) incor-porates more terms than CCSD and QCISD.

The basis sets used in all calculations are Pople’s triple-f (zeta)basis sets 6-311G(3df,2pd) and 6-311++G(3df,2pd). Diffuse basissets are used to take into account the dispersion interactions inweakly bound molecular clusters like (O2)n

+. The GAUSSIAN 09[40], quantum chemistry software suite was used to carry out allcalculations, whereas ChemCraft software [41] was used for visual-ization of optimized structures. We have performed harmonicvibrational frequency calculations not only to confirm that theoptimized geometries are minima on the potential energy hyper-surfaces but also to find the zero-point vibrational energy. The dis-persion effect on the O2

+. . .O2 interaction was examined byinclusion of the D3 version of Grimme’s dispersion with the origi-nal D3 damping function [42] to B3LYP, M05-2X and M06-2X func-tionals. For BHandH, the D3 dispersion correction is not availablein GAUSSIAN 09. Furthermore, the latest functional from theHead-Gordon group, wB97XD [43], which includes dispersioneffect explicitly, was applied to the O2

+. . .O2 system. We have usedthe basis set with diffuse function 6-311++G(3df,2pd) for all ofthese dispersion-corrected density functional theory calculations.

3. Results and discussion

� O2, O2+ and O4

+

Fig. 1 shows the optimized structures of O2, O2+ as well as rect-

angular (D2h point group) and trans-planar structures (C2h point

26 G.P. Khanal et al. / Computational and Theoretical Chemistry 1056 (2015) 24–36

group) of O4+ calculated at M06-2X/6-311++G(3df,2pd) level of

theory. Table 1 shows total energies, bond lengths as well as vibra-tional frequencies for O2 and O2

+ at the M05-2X, M06-2X, BHandHand B3LYP levels of theory using basis sets 6-311G(3df,2pd) as wellas 6-311++G(3df,2pd) at each level of theory. From Table 1, we cansee that the bond lengths of 1.203 Å and 1.105 Å for O2 and O2

+ cal-culated at B3LYP/6-311++G(3df,2pd) levels of theory are very closeto the experimental values of 1.208 Å and 1.116 Å [44], respec-tively. On the other hand, the dissociation energy for O2 calculatedat this level of theory (289.6 kcal/mol without zero-point correc-tion and 290.0 kcal/mol with zero-point correction) is greater thanthe experimental value of 278.3 kcal/mol [45]. In the case of M06-2X/6-311++G(3df,2pd) level of theory, the dissociation energy ofO2 is 2.0 kcal/mol (with zero-point correction as well as withoutit) less than that obtained at B3LYP/6-311++G(3df,2pd). On theother hand, even though the dissociation energy of O2 seems tobe smallest at BHandH level of theory compared to all other levelsof theory, the vibrational frequencies for both O2 and O2

+ at thislevel of theory is very high compared to experimental values [44].

Table 2 represents equilibrium geometries, binding energies(B.E.s) as well as asymmetric O–O stretch frequencies i.e. IR activevibration for rectangular and trans-planar O4

+ calculated at differ-ent levels of theory. At M06-2X/6-311G(3df,2pd) and M06-2X/6-311++G(3df,2pd) levels of theory, the rectangular structure hasall real frequencies, whereas the trans-planar structure seems tohave single imaginary frequency at these levels of theory indicat-ing that it is at the transition-state rather than at true minimum.The zero-point corrected B.E.s for the rectangular structure are�19.4 kcal/mol and �17.6 kcal/mol, respectively at M06-2X/6-311G(3df,2pd) and M06-2X/6-311++G(3df,2pd) levels of theory

Fig. 1. Optimized structures of (a) O2, (b) O2+, (c) rectangular (D2h) O4

+ and (d) trans-planar(S = 1), doublet (S = 1/2) and quartet (S = 3/2) potential energy surfaces, respectively. Bnumber in red near each O-atom represents Mulliken atomic spin density in the atomreferred to the web version of this article.)

Table 1Equilibrium geometries, harmonic vibrational frequencies, total energies of O2 and O2

+ cadifferent methods and basis sets: a ? 6-311G(3df,2pd) and b ? 6-311++G(3df,2pd). E1 ? Ein Å, frequencies in cm�1 and energy in a.u. (or hartrees). Ionization energy, I.E. (kcal/mol

Methods For O2 For O2+

R (O–O)i Freq.iii E1 E2 R (O

M05-2X a 1.1847 1787.05 �150.3494572 �150.345386 a 1.08b 1.1852 1778.08 �150.3528092 �150.348758 b 1.08

M06-2X a 1.1870 1789.07 �150.3151581 �150.311082 a 1.09b 1.1877 1778.63 �150.3188927 �150.314841 b 1.09

BHandH a 1.1688 1876.65 �149.5030138 �149.498739 a 1.07b 1.1695 1863.52 �149.5068616 �149.502616 b 1.07

B3LYP a 1.2026 1657.35 �150.3745877 �150.370812 a 1.10b 1.2033 1645.02 �150.3794885 �150.375741 b 1.10

i Experimental value for bond length of O2 molecules in their ground electronic stateii Experimental value for bond length of O2

+ molecules in their ground electronic stateiii Experimental value of harmonic vibrational frequency = 1580.0 cm�1 for O2 and 190iv Experimental value of ionization energy of O2 = 278.3 kcal/mol [45].

which are not in agreement with experiments [23–25]. Table 3shows the relative energies for the rectangular and trans-planarstructures employing all four methods M05-2X, M06-2X, BHandHand B3LYP with the both basis sets 6-311G(3df,2pd) and6-311++G(3df,2pd).

From Table 2, we see that none of the DFT methods used couldreproduce the experimental binding energies for O4

+, about�10.0 kcal/mol [23–25], and ionization energy of O2, 278.3 kcal/mol [45]. So, we have also carried out study to find the effect of dis-persion on O2

+. . .O2 interaction. We will discuss here the effect ofGrimme’s D3 dispersion (GD3) on different properties of O2, O2

+

as well as that on the both of the structures of O4+, rectangular

and trans-planar. Equilibrium geometry, harmonic vibrational fre-quency and total energy of O2 and O2

+ are summarized in Table S6(see supplementary material). First, consider the effect of disper-sion on equilibrium bond lengths of O2. Equilibrium bond lengthsfor O2 and O2

+, calculated at B3LYP, M05-2X and M06-2X densityfunctionals with basis set 6-311++G(3df,2pd) are 1.203, 1.185,1.188 Å and 1.105, 1.089, 1.092 Å, respectively, whereas those cal-culated by using the dispersion-corrected density functionals onthe same potential energy surfaces, i.e. triplet for O2 and doubletfor O2

+, calculated at B3LYPGD3, M05-2XGD3 and M06-2XGD3functionals with basis set 6-311++G(3df,2pd) are 1.204, 1.186,1.189 Å, respectively for O2 and 1.106, 1.090. 1.092 Å, respectivelyfor O2

+. Moreover, the BHandH functional produces bond lengths forO2 and O2

+ as 1.170 and 1.077 Å, respectively, employing the samebasis set 6-311++G(3df,2pd). Compared to experimental bondlengths of 1.208 Å for O2 and 1.116 Å for O2

+, it seems that the dis-persion-corrected functional B3LYP-GD3 performs slightly betterthan the B3LYP functional among all of the methods used. More-

(C2h) O4+ calculated at the theoretical level M06-2X/6-311++G (3df,2pd) on the triplet

ond lengths are given in angstrom, whereas angles are measured in degrees. The. (For interpretation of the references to colour in this figure legend, the reader is

lculated on triplet and doublet potential energy surfaces, respectively, by using fournergy without ZPE (zero-point energy) and E2 ? Energy with ZPE; distances measured) for O2: 1 ? I.E. without zero-point energy (ZPE) and 2 ? I.E. with ZPE.

ivIonization energy of O2

–O)ii Freq.iii E1 E2 1 2

96 2230.75 �149.8898289 �149.884747 288.4 289.194 2231.45 �149.8906081 �149.885524 290.0 290.7

20 2223.12 �149.8599916 �149.854927 285.6 286.218 2223.39 �149.8606898 �149.855625 287.5 288.2

73 2317.03 �149.0508359 �149.045557 283.7 284.471 2317.32 �149.0515387 �149.046259 285.7 286.4

55 2061.65 �149.9173996 �149.912703 286.9 287.553 2061.53 �149.9179663 �149.913270 289.6 290.2

(3Pg�) = 1.208 Å [44].

(2Qg) = 1.116 Å [44].

5.0 cm�1 for O2+ [44].

Table 2Equilibrium geometries, total energies (a.u.) and strong harmonic vibrational frequencies (cm�1) for rectangular and trans-planar structures of quartet O4

+ calculated at differentlevels of theory. ROO ? distance between O and O in each O2 subunit, RCM ? distance between centre of mass of each O2 subunit (distances in Å), h ? angle between the linejoining two centres of masses of each O2 subunit and one O–O bond axis (in degrees), E1 ? energy without zero-point correction and E2 ? energy with zero-point correction.D ? Binding energies (B.E.s) in kcal/mol. (D1 ? B.E. without zero-point energy and D2 ? zero-point corrected B.E.). Basis sets used: a ? 6-311G(3df,2pd) and b ? 6-311++G(3df,2pd).

Point group Method State Geometry Energy Frequency Binding energy

ROO RCM h E1 E2 Asymm. O–O stretching Imaginary freq. D1 D2

D2h M05-2X/a 4B3g 1.134 2.439 90.0 �300.2701656 �300.260187 1633.88 �51.22 �19.37 �18.86M05-2X/b 4B3g 1.134 2.446 90.0 �300.271507 �300.261588 1613.93 �45.74 �17.63 �17.13M06-2X/a 4B3g 1.137 2.433 90.0 �300.2071845 �300.196935 1618.16 No �20.10 �19.40M06-2X/b 4B3g 1.137 2.440 90.0 �300.2086383 �300.198464 1596.57 No �18.23 �17.56BHandH/a 4B3g 1.120 2.343 90.0 �298.5885471 �298.578013 1689.06 �52.00 �21.77 �21.16BHandH/b 4B3g 1.120 2.349 90.0 �298.590001 �298.579541 1666.19 �61.63 �19.82 �19.24B3LYP/a 4B3g 1.151 2.637 90.0 �300.3415193 �300.332249 1667.16 �103.14 �31.08 �30.58B3LYP/b 4B3g 1.151 2.663 90.0 �300.3433612 �300.334173 1656.45 �109.98 �28.81 �28.33

C2h M05-2X/a 4Bu 1.136 2.895 44.9 �300.2701948 �300.260381 1596.88 No �19.40 �18.98M05-2X/b 4Bu 1.136 2.905 44.7 �300.2718154 �300.262082 1573.48 No �17.82 �17.44M06-2X/a 4Bu 1.139 2.888 44.8 �300.2067594 �300.197133 1600.86 �76.07 �19.84 �19.53M06-2X/b 4Bu 1.138 2.906 44.7 �300.2085169 �300.199009 1574.21 �72.81 �18.16 �17.91BHandH/a 4Bu 1.124 2.767 44.6 �298.589189 �298.578886 1653.57 No �22.18 �21.71BHandH/b 4Bu 1.123 2.773 44.4 �298.5909618 �298.580741 1629.47 No �20.43 �20.00B3LYP/a 4Bu 1.151 3.187 42.8 �300.3451177 �300.335992 1681.59 No �33.34 �32.93B3LYP/b 4Bu 1.151 3.216 42.6 �300.3472947 �300.338223 1671.45 No �31.28 �30.88

Note: For rectangular structure, ROO = 1.165 Å, RCM = 2.38 Å; asymmetric O–O stretching frequency = 1271.0 cm�1: Lindh and Barnes [15].Experimental value of asymmetric O–O stretching frequency = 1323.0 cm�1 [12].

G.P. Khanal et al. / Computational and Theoretical Chemistry 1056 (2015) 24–36 27

over, the quality of simulating vibrational frequency of O2 and O2+ is

slightly improved by adding the GD3 dispersion correction toB3LYP functional. Also, for the functionals M05-2X and M06-2X,the quality of frequency calculation improves by adding this dis-persion correction (see Tables 1 and S6 for comparison). FromTables 1 and S6, we can see that the dissociation energies of O2 cal-culated at none of the DFT methods are in good agreement withexperimental ones. Even addition of the GD3 dispersion correctionto DFTs, B3LYP, M05-2X and M06-2X, could not yield ionizationenergy of O2 molecule in close comparison to the experimentalresults. The GD3 dispersion correction to B3LYP functional resultsin decrease in zero-point corrected ionization energy as well asthat without zero-point correction of this molecule by 0.1 kcal/mol. On the other hand, the dispersion corrected M05-2X DG3and M06-2X DG3 have slightly increased those values from thatwithout DG3 dispersion correction (see Tables 1 and S6). Moreover,deviation in ionization energy of O2 from experimental value isreduced by the use of wB97XD functional compared to other den-sity functionals used.

For the study of effect of dispersion-correction on O2+. . ..O2 inter-

action, we have summarized equilibrium geometries, total energiesand asymmetric O–O stretch frequencies for rectangular and trans-planar O4

+ geometries in Table S7 (see supplementary material).Zero-point corrected B.E.s of the rectangular O4

+ calculated atB3LYP-GD3, M05-2X GD3, M06-2X GD3, wB97XD using basis set6-311++G(3df,2pd) are �29.19, �17.12, �17.47, �22.6 kcal/mol,respectively, whereas those respective values calculated withoutzero-point correction are �29.67, �17.62, �18.13, �23.18 kcal/mol (see Table S7 in supporting information). Comparing these val-ues with those from Table 2, it seems that zero-point correctedbinding energy of the rectangular O4

+ as well as that without thecorrection slightly decreases (by 0.01 kcal/mol with and withoutzero-point correction for M05-2X and by 0.09 kcal/mol withzero-point correction and 0.1 kcal/mol without zero-point correc-tion for M06-2X) after adding the GD3 dispersion correction tothe M05-2X and M06-2X density functional. On the other hand,the binding energy of this cluster ion increases by 0.86 kcal/molwith and without zero-point correction in the case of B3LYP func-tional with GD3 correction. Moreover, close inspection of Table S7in supporting material reveals that B.E. for this cluster ion

calculated by using wB97XD functional is more deviated fromexperimental value than that obtained even at BHandH functional.Regarding vibrational frequency, even addition of the GD3 disper-sion to the B3LYP and M05-2X functionals, one imaginary fre-quency of �100.75 and �59.59 cm�1, respectively is still present(without the dispersion correction, the corresponding imaginaryfrequencies present were �109.98 and �45.74 cm�1). Even thoughaddition of the dispersion correction to B3LYP results in the abso-lute value decrease in imaginary frequency by 9.23 cm�1, there isincrease in the absolute value in imaginary frequency by13.85 cm�1 calculated at the GD3-corrected M05-2X functional.Moreover, in the case of wB97XD dispersion-corrected functional,the imaginary frequency produced is �92.87 cm�1, of which theabsolute value is smaller than that produced by B3LYP-GD3 butgreater than that produced at M05-2X GD3 density functional,using the same basis set 6-311++G(3df,2pd) and on the same quar-tet potential energy-hypersurface.

Zero-point corrected binding energy of trans-planar structure ofO4

+ calculated at GD3 corrected density functionals B3LYP-GD3,M05-2X GD3, M06-2X GD3, wB97XD are �31.33, �17.47,�17.77, �23.58 kcal/mol respectively, whereas the correspondingvalues without zero-point correction are �31.73, �17.82, �18.03,�24.06 kcal/mol. Comparing these values with those from Table 2,we can see that binding energy of the trans-planar structureincreases by 0.45 kcal/mol, including zero-point energy as well asthat without it, after dispersion correction to B3LYP, whereas thereis increase in zero-point corrected B.E. of this structure by0.03 kcal/mol (no change without zero-point energy) by GD3 dis-persion correction to M05-2X. On the other hand, after the disper-sion correction to M06-2X functional, zero-point corrected B.E. ofC2h geometry decreases by 0.14 kcal/mol (without zero-point cor-rection, it is 0.13 kcal/mol) compared to M06-2X functional. More-over, B.E. of the trans-planar O4

+ is larger than that of therectangular one by 0.88 kcal/mol with zero-point energy (it is0.98 kcal/mol without zero-point energy correction). After addingthe GD3 dispersion correction to M06-2X, imaginary frequencyincreases by 7.89 cm�1 for this structure (�72.81 cm�1 byM06-2X method and �80.70 cm�1 by M06-2X GD3 method).

Thus, we have found that even dispersion-correction to thedensity functionals could not give significant improvement to

Table 3Relative energies (kcal/mol) for O4

+ calculated at M05-2X, M06-2X, BHandH and B3LYP levels of theory with basis sets a ? 6-311G(3df,2pd) and b ? 6-311++G(3df,2pd)(1 ? without zero-point correction and 2 ? with zero-point correction). Positive energies (relative to 0.0) mean less stable structures.

Point group M05-2X/a M05-2X/b M06-2X/a M06-2X/b BHandH/a BHandH/b B3LYP/a B3LYP/b

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

D2h 0.18 0.12 0.19 0.31 0.0 0.12 0.0 0.34 0.40 0.55 0.6 0.75 2.26 2.35 2.47 2.54C2h 0.0 0.0 0.0 0.0 0.27 0.0 0.08 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

28 G.P. Khanal et al. / Computational and Theoretical Chemistry 1056 (2015) 24–36

reproduce experimental binding energy of O4+. So, we have

employed BD(T), a high-level ab initio methods to determine theionization energy of O2 and binding energy of O4

+.To improve the accuracy, we have performed single point

energy calculations by the Brueckner coupled-cluster method,BD(T), using the optimum structure we obtained at M06-2Xmethod. We have obtained ionization energy for O2 to be 274.6and 277.14 kcal/mol respectively at BD(T)/6-311G(3df,2pd) andBD(T)/6-311++G(3df,2pd) levels of theory, respectively. Also, thebinding energies for rectangular (D2h) O4

+ has been calculated tobe �11.0 and �9.61 kcal/mol at BD(T)/6-311G(3df,2pd) andBD(T)/6-311++G(3df,2pd) levels of theory, respectively (seeTable S1 in supplementary material).

� O6+

Fig. 2 shows three optimized structures of O6+ with point group

symmetries D3h, C2v and C2h, calculated on the sextet (S = 5/2)potential energy surface using M06-2X/6-311++G(3df,2pd) levelof theory. Tables 4 and 5 summarize the total energies, both withand without zero-point correction, strong harmonic vibrationalfrequencies (m) for all those clusters using all the four theoreticallevels M05-2X, M06-2X, BHandH and B3LYP with the basis sets6-311G(3df,2pd) and 6-311++G(3df,2pd).

Table 6 shows that the D3h symmetric structure seems to bemore stable than other two structures at M05-2X, M06-2X andBHandH levels of theories, on the sextet potential energy hyper-surface, when we take zero-point energy correction into accountas well. Even though the C2h symmetric structure seems to be morestable at B3LYP/6-311G(3df,2pd) level of theory, the addition of

Fig. 2. Optimized structures of cyclic (D3h) O6+ and trans-planar (C2h) O6

+, calculated at thesurface. With zero-point energy correction, the structure (a) is the lowest-energy struclengths are given in angstrom, whereas angles are measured in degrees. Mulliken atomicstructure is also shown below the corresponding structure. (For interpretation of the refearticle.)

diffuse functions ‘‘++’’ in the basis set 6-311G(3df,2pd) producesthe D3h structure(cyclic structure) to be more stable than othertwo isomers in our consideration. On the other hand, withoutzero-point correction, C2v symmetric structure seems to be morestable than D3h symmetric structure by 0.1, 0.1, 0.03, 0.02 kcal/mol at M05-2X/6311G(3df,2pd), M05-2X/6-311++G(3df,2pd),M06-2X/6-311G(3df,2pd), M06-2X/6-311++G(3df,2pd) respec-tively, showing that that the D3h and C2v structures of O6

+ are nearlyisoenergetic. As already pointed out in introduction section, Zhouet al. [9] utilized B3LYP/6-311+G⁄ method and obtained that theenergy of C2h symmetry is lower than D3h symmetric structureby 3.3 kcal/mol. Recently, from the gas-phase IR photodissociationstudy, Ricks et al. [12] has suggested that the stable structure forthe O6

+ cluster ion is the cyclic (D3h) by observing the vibrationalspectrum of this cluster ion in gas-phase.

� O8+

Fig. 3 shows five different optimized structures with geometri-cal parameters, point group symmetries and relative energies forO8

+ cluster calculated at M06-2X/6-311++G(3df,2pd) level of theoryon the octet potential energy surface. All these structures are cal-culated to have all real frequencies at this level of theory. Totalenergies and strong harmonic vibrational frequencies (cm�1) withIR intensity (km/mol) calculated at four levels of theory M05-2X,M06-2X, BHandH and B3LYP using both basis sets 6-311G(3df,2pd)and 6-311++G(3df,2pd) are shown in Tables 7 and 8. For the confir-mation of spin multiplicity (2S + 1) of more stable O8

+ cluster ion tobe 8, we have calculated the total energies along with strong har-monic vibrational frequencies for the cluster ion employing spin

M06-2X/6-311++G(3df,2pd) level of theory on the sextet (S = 5/2) potential energyture. The figure (b) C2v geometry has electric dipole moment 0.0554 Debye. Bondspin density in each O-atom is printed in red near the atom. Electronic state of each

rences to colour in this figure legend, the reader is referred to the web version of this

Table 4Total energies, E (hartree) and corresponding strong harmonic vibrational frequencies, m (cm�1) for three different isomers of O6

+ calculated by using M05-2X and M06-2X meta-GGA functionals. E1 ? Energy without zero-point energy (ZPE), E2 ? Energy with ZPE, a ? 6-311G(3df,2pd), b ? 6-311++G(3df,2pd), Int. ? IR intensity (km/mol). Mode ofvibration is given in parenthesis of the corresponding m.

Point group State E & m M05-2X/a M05-2X/b M06-2X/a M06-2X/b

D3h6A1

00E1 �450.6294405 �450.6327318 �450.5323358 �450.5359502E2 �450.615144 �450.618612 �450.517931 �450.521834m Two 1582.01 (E0) Two 1545.11 (E0) Two 1579.30 (E0) Two 1539.35 (E0)Int. 4271.56 & 5052.13 & 4131.75 & 4898.24 &

4283.13 5023.97 4171.34 4945.13

C2v6A2 E1 �450.6295395 �450.632814 �450.5323833 �450.5359825

E2 �450.615140 �450.618600 �450.517800 �450.521728m 1580.97 (A1) & 1584.12 (B2) 1544.42 (A1) & 1548.70 (B2) 1580.44 (A1) & 1586.00(B2) 1541.63 (A1) & 1546.69 (B2)Int. 4289.31 (A1) & 4289.75 (B2) 5053.342 (A1) & 5037.02 (B2) 4144.90 (A1) & 4127.55(B2) 4933.07 (A1) & 4907.00 (B2)

C2h6Bg E1 �450.6280117 �450.631757 �450.5309802 �450.5351463

E2 �450.614080 �450.618021 �450.516740 �450.521065m 1592.82 (Bu) 1557.77 (Bu) 1599.72 (Bu) 1563.72 (Bu)Int. 17113.68 20268.06 16496.02 19330.65

Observed value of m (cm�1) for O6+ by Ricks et al.: 1425.00 cm�1 [12].

Table 5Total energies, E (hartree) and corresponding strong IR vibrational frequencies, m (cm�1), for three different isomers of O6

+ calculated by using BHandH and B3LYP functionalemploying basis sets a ? 6-311G(3df,2pd), b ? 6-311++G(3df,2pd). E1 ? Energy without zero-point energy (ZPE), E2 ? Energy with ZPE, Int. ? IR intensity (km/mol). Mode ofvibration is given in parenthesis of the corresponding m.

Point group State E & m BHandH/a BHandH/b B3LYP/a B3LYP/b

D3h6A1

00E1 �448.1058554 �448.1093538 �450.7392241 �450.7438569E2 �448.090446 �448.094197 �450.725530 �450.730399m Two 1660.06 (E0) Two 1622.03 (E0) Two 1636.69 (E0) Two 1620.61 (E0)Int. 3921.83 & 3921.61 4570.42 & 4570.30 1192.38 & 1192.27 1387.89 & 1387.73

C2v6A2 E1 �448.1058465 �448.1093415 �450.7392106 �450.7438419

E2 �448.090433 �448.094189 �450.725518 �450.730391m 1659.94 (B2) & 1660.63 (A1) 1621.67 (B2) & 1622.51 (A1) 1636.58 (A1) & 1636.65 (B2) 1620.50 (A1) & 1620.58 (B2)Int. 3924.06 (B2) & 3877.99 (A1) 4582.26 (B2) & 4532.73 (A1) 1186.65 (A1) & 1190.80 (B2) 1384.59 (A1) & 1390.43 (B2)

C2h6Bg E1 �448.1027689 �448.1069716 �450.7427454

E2 �448.088094 �448.092490 �450.729911 #

m 1681.91 (Bu) 1649.42 (Bu) 1674.77 (Bu)Int. 14570.43 16748.30 4624.78

# Geometry got totally distorted.

Table 6Relative energies (kcal/mol) for different structures of O6

+ calculated at different levels of theory with basis sets a ? 6-311G(3df,2pd) and b ? 6-311++G(3df,2pd) [1 ? withoutzero-point correction and 2 ? with zero-point correction]. Positive energies (relative to 0.0) mean less stable structures.

Point group M05-2X/a M05-2X/b M06-2X/a M06-2X/b BHandH/a BHandH/b B3LYP/a B3LYP/b

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

D3h 0.1 0.0 0.1 0.0 0.03 0.0 0.02 0.0 0.0 0.0 0.0 0.0 2.21 2.75 0.0 0.0C2v 0.0 0.003 0.0 0.01 0.0 0.1 0.0 0.1 0.006 0.008 0.008 0.005 2.22 2.76 0.01 0.01C2h 1.0 0.7 0.7 0.4 0.9 0.7 0.5 0.5 1.9 1.5 1.5 1.1 0.0 0.0 #

# The geometry totally distorted during optimization.

G.P. Khanal et al. / Computational and Theoretical Chemistry 1056 (2015) 24–36 29

multiplicities 6, 4 and 2, using the same level of theory M06-2X/6-311++G(3df,2pd) (see Supplementary material Table S2). FromTable S2, we see that octet spin state i.e. spin multiplicity 8 is morefavorable for O8

+ compared to all of the lower spin statesmentioned. We have shown all harmonic vibrational frequenciesas well as the coordinates for all of the optimized structures inSupplementary material.

From Table 9, we can see that, O8+ with D4h point group symme-

try has the lowest energy (without zero-point energy correction)amongst the mentioned five structures at M05-2X/6-311++G(3df,2pd), M06-2X/6-311++G(3df,2pd) and B3LYP/6-311G(3df,2pd) as well as B3LYP/6-311++G(3df,2pd) levels of theory. AtM05-2X/6-311G(3df,2pd) and M06-2X/6-311G(3df,2pd) levels oftheory, C1 symmetric structure of this cluster is more stable withand without zero-point energy correction. At M05-2X/6-311G

(3df,2pd) level of theory, the strong harmonic vibrational fre-quency of this structure is calculated to be 1568.4 cm�1, whereasthat for O6

+ having D3h point group symmetry is 1582 cm�1 (dou-bly-degenerate). Also, at M06-2X/6-311G(3df,2pd) level of theory,this structure has strong harmonic vibrational frequency (here‘‘strong’’ refers to IR intensity) of 1571.3 cm�1, whereas that forO6

+ with D3h symmetry is doubly-degenerate frequency of1579 cm�1 at the same level of theory. This shows that the infraredactive frequency of O8

+ with C1 symmetry is lower than that of theO6

+ with D3h symmetry. The experiment by Ricks et al. [12] foundthat O6

+ has infrared active frequency at 1425.0 cm�1, whereas thatfor O8

+ was 1426.0 cm�1. On the other hand, D4h structure of O8+ has

been calculated to have doubly-degenerate infrared active fre-quency of 1588.9 cm�1 at M05-2X/6-311G(3df,2pd) and 1596.9cm�1 at M06-2X/6-311G(3df,2pd) levels of theory on the same

Fig. 3. Optimized structures of O8+ at M06-2X/6-311++G(3df,2pd) level of theory on the octet (S = 7/2) potential energy surface. Distances are measured in angstrom, angles in

degrees. Relative energies (kcal/mol) without zero-point energy correction are given in braces. Planar structures: (a), (c), (d) and (e); (b) is non-planar. Electric dipolemoments of (b)–(d) are 1.5609, 1.5607 and 0.0535 Debye, respectively. The direction of electric dipole moments are shown by blue pointers. Mulliken atomic spin density ineach O-atom is shown by red printed number near the atom. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version ofthis article.)

30 G.P. Khanal et al. / Computational and Theoretical Chemistry 1056 (2015) 24–36

octet potential energy surface. These frequencies are greater thanthat of the stable O6

+ (i.e. the D3h symmetric).In the case of M05-2X/6-311++G(3df,2pd) and M06-2X/6-

311++G(3df,2pd) levels of theory as well as on the same octetpotential energy surface, D4h symmetric structure for O8

+ was foundto be more stable compared to the other structures considered(with and without zero-point correction at the first method andwithout zero-point correction in the later one). The doubly-degen-erate frequencies calculated for this cluster are 1549.7 cm�1 and1555.2 cm�1 at M05-2X/6-311++G(3df,2pd) and M06-2X/6-311++G(3df,2pd) levels of theory, respectively on the same octetpotential energy surface, whereas those for D3h symmetric O6

+ aredoubly-degenerate 1545.1 cm�1 and 1539.4 cm�1, respectively. Itseems that, while experimental trend due to Ricks et al. [12] is pre-served, simply harmonic vibrational frequencies which are notmass-selected are not sufficient to make conclusion. With zero-point energy correction, C2h symmetric O8

+ seems to be more stablethan D4h symmetric isomer calculated on the same octet potentialenergy surface using the M06-2X/6-311++G(3df,2pd) level of the-ory. But the calculated infrared active frequency of 1590.1 cm�1

for C1 symmetric O8+ is not much closer to experimental value of

1426 cm�1 due to Ricks et al. [12]. On the other hand, the infraredactive frequency calculated at this level of theory for the D4h sym-metric structure, 1555.2 cm�1, is much closer to the experimentalvalue. Thus, we can confirm that D4h symmetric O8

+ cluster is morestable than other structures.

The D4h symmetric structure of O8+ cluster ion has been calcu-

lated to be more stable at B3LYP/6-311G(3df,2pd) and B3LYP/6-311G(3df,2pd) levels of theory (see Table 9). We have already con-cluded that B3LYP functional is not suitable for open-shell calcula-tions such as the cluster of our interest (O2)n

+. [Refer: Govinda P.Khanal, Master thesis, First Principles and density-functional Studyof (O2)n

+, n = 2–10; Central Department of Physics, Kathmandu,Nepal, Sept. 2013].

The binding energy for the cluster O8+, undergoing dissociation

O8+ ? O6

+ + O2 + D, where D is binding energy (B.E.), calculated atdifferent levels of theory has been displayed in Table 13. We havecompared graphically zero-point corrected dissociation energiesfor this and other clusters calculated at four density functionalsB3LYP, BHandH, M05-2X and M06-2X using basis sets 6-311G(3df,2pd) and 6-311++G(3df,2pd) with experimental resultsdue to Hiraoka [25]. The comparisons are shown in Fig. 5. Fromthe plot, we can see that the zero-point corrected binding energyof the cluster with D4h geometry is nearly equal to the experimen-tal results due to Hiraoka [25] at the levels M05-2X/6-311G(3df,2pd) and M05-2X/6-311++G(3df,2pd). Also, the calcu-lated binding energies for the cluster are in good agreement withthe experimental result [25] at the levels M06-2X/6-311G(3df,2pd)and M06-2X/6-311++G(3df,2pd).

To see the distribution of spins of O2 and O2+ in different struc-

tures of O8+ cluster ion, we have shown Mulliken atomic spin den-

sities distributed over each O–O subunit in all of the optimized

Table 7Total energies, E (in hartree), and corresponding strong IR harmonic vibrational frequencies, m (cm�1) for different structures of O8

+ calculated at M05-2X and M06-2X levels oftheory using basis sets a ? 6-311G(3df,2pd), b ? 6-311++G(3df,2pd). E1 ? Energy without zero-point energy (ZPE), E2 ? Energy with zero-point energy (ZPE), Int. ? IR intensity(km/mol). Mode of vibration is given in parenthesis of the corresponding m.

Point group State E & m M05-2X/a M05-2X/b M06-2X/a M06-2X/b

D4h8A2g E1 �600.9830139 �600.9897539 �600.8518575 �600.8592517

E2 �600.964474 �600.971341 �600.833299 �600.840801m Two 1588.88 (Eu) Two 1549.72 (Eu) Two 1596.85 (Eu) Two 1555.24 (Eu)Int. 9215.12 & 9215.12 11192.07 & 11192.07 9165.20 & 9165.20 11229.82 & 11229.82

C18A E1 �600.9837455 �600.9895697 �600.8532665 �600.8587986

E2 �600.965181 �600.971213 �600.834183 �600.840096m 1568.39 (A) 1533.27 (A) 1571.29 (A) 1547.61 (A)Int. 10884.49 12625.95 10347.91 12330.73

C2v8A1 E1 �600.9831552 �600.9890563 �600.8522411 �600.8587989

E2 �600.964537 �600.971020i �600.833332 �600.840096m 1585.70 (B2) 1545.50 (B2) 1584.73 (B2) 1547.60 (B2)Int. 11655.03 13244.36 10296.12 12332.20

C2v8A2 E1 �600.9821145 �600.9883109 �600.8519497 �600.8587885

E2 �600.963639 �600.970143 �600.833185 �600.840249m 1625.01 (B2) 1595.84 (B2) 1637.17 (B2) 1607.40 (B2)Int. 25848.99 28626.38 25087.41 28353.62

C2h8Bu E1 �600.9825152 �600.988819 �600.851566 �600.8585486

E2 �600.964650ii �600.971122 �600.833829 �600.841153m 1618.39 (Bu) 1580.71 (Bu) 1626.34 (Bu) 1590.13 (Bu)Int. 37573.37 44747.33 35101.50 42983.56

i Imaginary frequency (in cm�1): �14.78 (A1);ii Imaginary frequency (cm�1): �6.34 (Bu).

Table 8Total energies, E (in hartree), and corresponding strong IR harmonic vibrational frequencies, m (cm�1) for different structures of O8

+ calculated at BHandH and B3LYP levels oftheory. E1 ? Energy without zero-point energy (ZPE), E2 ? Energy with zero-point energy(ZPE), a ? 6-311G(3df,2pd), b ? 6-311++G(3df,2pd)). Int. ? IR intensity (km/mol).Mode of vibration is given in parenthesis of each m.

Point group State E & m BHandH/a BHandH/b B3LYP/a B3LYP/b

D4h8A2g E1 �597.6145270 �597.6219114 �601.1273007 �601.1362869

E2 �597.595047 �597.602498 �601.110016 �601.119131m Two 1691.03 (Eu) Two 1651.91 (Eu) Two 1631.32 (Eu) Two 1614.44 (Eu)Int. 7681.12 & 7681.12 9350.81 & 9350.81 2284.84 & 2284.84 2712.12 & 2712.12

C18A E1 �597.6166582 �597.6229118 �601.1265603 �601.1352513

E2 �597.596869 �597.603356 �601.109292 �601.118506m 1665.13 (A) 1637.28 (A) 1628.02 & 1631.51(A) 1611.93 & 1613.62 (A)Int. 8795.82 9617.12 2273.20 & 2353.77 2944.74 & 2892.83

C2v8A1 E1 �597.6158041 �597.6220863 �601.1265603 #

E2 �597.596188i �597.602651ii �601.109292m 1678.64 (B2) 1652.74 (B2) 1631.54 (B2)Int. 9396.27 10390.95 2355.12

C2v8A2 E1 �597.6135443 �597.6204755 # #

E2 �597.593922iii �597.600984iv

m 1742.09 (B2) 1652.68(B2) & 720.41(B2)Int. 17831.38 2244.99 & 17530.18

C2h8Bu E1 �597.6127719 �597.61982 # #

E2 �597.593960 �597.601195m 1712.85 (Bu) 1683.76 (Bu)Int. 27977.87 28376.69

Imaginary frequencies (cm�1).i �31.50 (B1).

ii �28.7 (B1).iii �29.26 (A2), �16.09 (B1).iv �26.80 (A2), �15.30 (B1).# Geometry got totally distorted.

G.P. Khanal et al. / Computational and Theoretical Chemistry 1056 (2015) 24–36 31

structures of the cluster ion, calculated at M06-2X/6-311++G(3df,2pd) level of theory (see Fig. 3). Mulliken atomic spin densi-ties in each O-atom in O2 is 1, whereas those in the case of O2

+ is 0.5(see Fig. 1). From Fig. 3, we can see that in D4h symmetric O8

+, spindensities are delocalized identically in each of the four O–O sub-units i.e. no subunit is equivalent to neither O2 nor O2

+. In the caseof C1 and C2v symmetric structures of O8

+ shown as (b) and (c) inFig. 3, one of the O–O subunits has Mulliken atomic spin densitiesof 0.924 in each O-atom. It is slightly less than 1 in the case of O2.

Thus, these two structures seem to be O2-solvated cluster cations.Other remaining two structures, shown in (d) and (e), do not haveO–O subunit nearly equivalent to neither O2 nor O2

+.

� O10+

The two optimized structures of O10+ along with geometrical

parameters calculated at M06-2X/6-311++G(3df,2pd) level of the-ory, on the dectet potential energy surface, are shown in Fig. 4.

Table 9Relative energies (kcal/mol) for different structures of O8

+ calculated at different levels of theory with basis sets a ? 6-311G(3df,2pd) and b ? 6-311++G(3df,2pd). P.G ? Pointgroup symmetry, S ? Electronic state, 1 ? without zero-point correction and 2 ? with zero-point correction.

P.G. S M05-2X/a M05-2X/b M06-2X/a M06-2X/b BHandH/a BHandH/b B3LYP/a B3LYP/b

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

D4h8A2g 0.5 0.4 0.0 0.0 0.88 0.55 0.0 0.22 1.34 1.14 0.63 0.54 0.0 0.0 0.0 0.0

C18A 0.0 0.0 0.12 0.10 0.0 0.0 0.28 0.66 0.0 0.0 0.0 0.0 0.46 0.45 0.65 0.39

C2v8A1 0.3 0.4 0.44 0.20 0.64 0.53 0.28 0.66 0.54 0.43 0.52 0.44 0.46 0.45 #

C2v8A2 1.0 0.9 0.91 0.75 0.83 0.63 0.29 0.57 1.95 1.85 1.53 1.49 # #

C2h8Bu 0.8 0.3 0.59 0.14 1.07 0.22 0.44 0.0 2.44 1.83 1.94 1.36 # #

# The geometry totally distorted during optimization. Positive energies (relative to 0.0) mean less stable structures.

Fig. 4. Optimized structures of cyclic O10+ at M06-2X/6-311++G (3df,2pd) level of theory on the dectet (S = 9/2) potential energy surface. The structures (a) and (b) both are

planar. Relative energies (kcal/mol) without zero-point correction are given in braces. The number printed in red near each O-atom represents Mulliken atomic spin density.Positive energy (relative to 0.0) means less stable structure. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of thisarticle.)

32 G.P. Khanal et al. / Computational and Theoretical Chemistry 1056 (2015) 24–36

Fig. 5. Zero-point corrected dissociation energies calculated at different levels of theory using basis set 6-311G(3df,2pd) are shown in the (a) graph and that using 6-311++G(3df,2pd) in the (b) graph. For comparison, experimental results due to Hiraoka [25] are shown. Point group symmetries considered for O4

+, O6+, O8

+ and O10+ are D2h, D3h,

D4h and D5h, respectively.

Total energies and strong harmonic vibrational frequencies for thecorresponding structures are summarized in Tables 10 and 11. Allof the harmonic vibrational frequencies as well as coordinates ofthe optimized structures calculated at all levels of theory usedare also shown in supplementary material. Also, we have opti-mized both of these isomers of O10

+ using other lower spin multi-plicities 8, 6, 4 and 2. The total energies along with infraredactive vibrational frequencies i.e. strong harmonic vibrational fre-quencies, calculated by employing these different spin multiplici-ties, using M06-2X/6-311++G(3df,2pd) level of theory are shownin Table S3 in supplementary material. From Table S3, we can

see that the dectet state of O10+ is more stable than other lower spin

states mentioned.From Table 12, we see that D5h symmetric O10

+ , calculated on thedectet potential energy surface, seems to be more stable thananother structure with C2h symmetry at theoretical levels usedexcept for M06-2X/6-311G(3df,2pd) and B3LYP/6-311G(3df,2pd)levels of theory. This Table shows that at the M06-2x/6-311++G(3df,2p) level of theory, D5h symmetric O10

+ seems to be morestable compared to the C2h structure and the zero-point correctedB.E. of the cluster is calculated to be �2.14 kcal/mol (see Table 13).This value is exactly equal to the experimental one by Hiraoka [25]

Table 10Total energies, E (in hartree), and corresponding strong IR harmonic vibrational frequencies, m (cm�1) for two different structures of O10

+ calculated at M05-2X and M06-2X levelsof theory with basis sets a ? 6-311G(3df,2pd), b ? 6-311++G(3df,2pd)). E1 ? Energy without zero-point energy (ZPE), E2 ? Energy with ZPE, Int. ? IR intensity (km/mol). Modeof vibration is given in parenthesis of each m.

Point group State E & m M05-2X/a M05-2X/b M06-2X/a M06-2X/b

D5h10A2

00E1 �751.3361041 �751.3453222 �751.1710623 �751.1814753E2 �751.314293 �751.324187 �751.148397 �751.159046m Two 1596.11 (E01) Two 1564.07 (E01) Two 1608.91 (E01) Two 1568.19 (E01)Int. 14846.81 and 14840.43 17600.53 and 17599.12 13166.45 and 13166.54 16398.90 and 16397.30

C2h10Bg E1 �751.3359316 �751.3448571 �751.1712679 �751.1812109

E2 �751.314191 �751.323640 �751.148592 �751.158976m 1636.83 (Bu) 1601.45 (Bu) 1648.39 (Bu) 1615.81 (Bu)Int. 64192.48 78450.09 59392.80 73160.50

Table 11Total energies, E (in hartree), and corresponding strong IR harmonic vibrational frequencies, m (cm�1) for two different structures of O10

+ calculated at BHandH and B3LYP levels oftheory with basis sets a ? 6-311G(3df,2pd), b ? 6-311++G(3df,2pd)). E1 ? Energy without zero-point energy (ZPE), E2 ? Energy with ZPE, Int. ? IR intensity (km/mol). Mode ofvibration is given in parenthesis of each m.

Point group State E & m BHandH/a BHandH/b B3LYP/a B3LYP/b

D5h10A002 E1 �747.1233598 �747.1338 �751.5105475 �751.5233939

E2 �747.099467 �747.110011 �751.489348 �751.502637m Two 1708.15 (E01) Two 1668.38 (E01) Two 1626.80 (E01) Two 1610.42 (E01)Int. 10904.15 and 10904.44 13522.64 and 13522.14 3212.12 and 3212.26 3890.50 and 3890.54

C2h10Bg E1 �747.1215282 �747.1315194 �751.5158223 #

E2 �747.098568 �747.108883 �751.495444m 1733.69 (Bu) 1701.43 (Bu) 1655.71 (Bu)Int. 51833.95 60916.44 15453.52

# Geometry totally distorted during optimization.

Table 12Relative energies (kcal/mol) [1 ? without zero-point energy (ZPE) correction and 2 ? with (ZPE) correction] for two structures of O10

+ calculated at different levels of theory withbasis sets a ? 6-311G(3df,2pd) and b ? 6-311++G(3df,2pd), P.G ? Point group symmetry, e ? Electronic state. Positive energies (relative to 0.0) mean less stable isomers.

P.G S M05-2X/a M05-2X/b M06-2X/a M06-2X/b BHandH/a BHandH/b B3LYP/a B3LYP/b

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

D5h10A2

000.0 0.0 0.0 0.0 0.13 0.12 0.0 0.0 0.0 0.0 0.0 0.0 3.31 3.83

C2h10Bg 0.11 0.06 0.29 0.34 0.0 0.0 0.17 0.04 1.15 0.56 1.43 0.71 0.0 0.0 #

# Geometry totally distorted during optimization.

G.P. Khanal et al. / Computational and Theoretical Chemistry 1056 (2015) 24–36 33

(see Fig. 5). Moreover, by adding diffuse functions to B3LYP func-tional, the C2h symmetric isomer of the cluster did not convergeon the mentioned dectet potential energy surface. The experimentalresults due to other groups have been shown in Table 13.

To get more information about how spins of O2 and O2+ are dis-

tributed over O–O subunit in different structures of O10+ cluster cat-

ion, we have shown Mulliken atomic spin densities distributedover all O-atoms in the optimized structures, calculated at M06-2X/6-311++G(3df,2pd) level of theory (see Fig. 4). From Fig. 4, wesee that all O–O subunits are identical in the D5h symmetric struc-ture and none of the subunits are similar to either O2 or O2

+ showingthat this structure is not O2-solvated structure. Also, the C2h struc-ture of O10

+ calculated on the same dectet potential energy surface,utilizing the same level of theory (see Fig. 4), does not seem to con-tain O–O subunit identical to either of the O2 or O2

+, also showingthat the structure is not O2-solvated structure.

From the Table 13 and Fig. 5, we can see that M05-2X and M06-2X methods produce binding energies for (O2)n

+ clusters in goodagreement with experiments for the cluster size n greater than 2but are bad for the cluster size n = 2. We have already pointedout in O4

+ section that none of DFT methods reproduce experimen-tal binding energies (as well as anti-symmetric O–O stretching fre-quency) for O4

+ cluster ion. Recently, Sokolov et al. [27] havesuccessfully applied new approach to density cumulant functionaltheory [28] (DCT) to study equilibrium structure, harmonic vibra-tional frequencies, and dissociation energy of the rectangular O4

+.

Thus, regarding the performance of all four DFTs, B3LYP, M05-2X, M06-2X and BHandH in determining ionization energy of O2,BHandH/6-311G(3df,2pd) performs slightly better than M06-2X/6-311G(3df,2pd),whereas M05-2X/6-311++G(3df,2pd) performspoorest of all the methods used (M05-2X/6-311++G(3df,2pd) per-forms slightly poorer than the B3LYP/6-311++G(3df,2pd)) (seeTable 1). In the case of binding energy of O4

+, deviation of dissocia-tion energy from experimental value for the rectangular structurecalculated at M05-2X/6-311++G(3df,2pd) is the least among allmethods (and even slightly less deviated than that for the trans-planar one calculated at the same level of theory) (see Table 2).Among them, the B3LYP seems to perform the worst, as alreadypointed out in determining B.E. of O4

+ and it seems from Table 2and Fig. 5 that the BHandH performs better than this functionalbut worse than M06-2X. The performance of the M05-2X and thatof M06-2X are comparable in determining the binding energy ofO4

+. Moreover, in the case of determining B.E. of O6+ M05-2X per-

forms slightly better than M06-2X, with basis set 6-311G(3df,2pd)but adding diffuse functions in the basis set i.e. 6-311++G(3df,2pd)results in slightly better performance of M06-2X (see Table 13 andFig. 5). It seems that the BHandH performs better than B3LYP. Fordetermining the binding energy of O8

+, M05-2X performs slightlybetter than M06-2X comparing with data of Hiraoka [25] (bothof these methods result in B.E. close to experimental values men-tioned in Refs. [23,25]) (see Table 13 and Fig. 5), whereas the per-formance of BHandH is better than B3LYP for the task. In the case

Table 13Binding energies i.e. B.E.s (in kcal/mol) of (O2)n

+ clusters (n = 2–5) undergoing decay (O2)n+ ? (O2)n�1

+ + O2, using different methods. D1 ? B.E. without zero-point energy (ZPE)correction, D2 ? B.E. with ZPE correction, a ? 6-311G(3df,2pd) and b ? 6-311++G(3df,2pd).

Cluster size n Basis sets M05-2X M06-2X BHandH B3LYP B.E.s due to experimental groups

D1 D2 D1 D2 D1 D2 D1 D2

2 a �19.37 �18.86 �20.10 �19.40 �21.77 �21.16 �31.08 �30.58 �10.53l, �9.69m,b �17.63 �17.13 �18.23 �17.56 �19.82 �19.24 �28.81 �28.33 �9.15n

3 a �6.16 �6.01 �6.27 �6.22 �8.97 �8.59 �14.51 �14.10 �6.51l, �5.89n

b �5.28 �5.19 �5.28 �5.35 �7.84 �7.56 �13.18 �12.854 a �2.58 �2.47 �2.74 �2.69 �3.55 �3.68 �8.46 �8.58 �2.56l, �2.48n

b �2.64 �2.49 �2.77 �2.59 �3.57 �3.56 �8.12 �8.155 a �2.28u �2.78u �2.54 �2.52 �3.65 �3.56 �5.43 �5.35 �2.43l, �2.14n,

b �1.73 �2.56 �2.09 �2.14 �3.15 �3.07 �4.78 �4.87 �3.69o

Note: During above calculations, we have considered the structures of O4+, O6

+, O8+, and O10

+ having point group symmetries D2h, D3h, D4h and D5h and corresponding electronicstates 4B3g, 6A1’’, 8A2g, and 10A2’’, respectively.

u With ‘‘integral = ultrafine’’: without the constraint the structure has imaginary frequencies.l Conway and Janik [23].

m Linn, Ono and Ng [24].n Hiraoka [25].o Matt et al. [26].

34 G.P. Khanal et al. / Computational and Theoretical Chemistry 1056 (2015) 24–36

of O10+ , B.E.s determined by M06-2X functional, using both basis

sets 6-311G(3df,2pd) and 6-311++G(3df,2pd) are closer to experi-mental values. Addition of diffuse functions to the basis set311G(3df,2pd) improves the performance of M05-2X in predictingB.E. of O10

+ (see Table 13 and Fig. 5).

4. Conclusions

At M06-2X/6-311G(3df,2pd) and M06-2X/6-311++G(3df,2pd)levels of theory, rectangular (D2h) O4

+ has been found to be morestable compared to trans-planar (C2h) on the quartet potentialenergy surface. At these levels of theory, the latter structure hasbeen found to have an imaginary frequency showing that it is atthe saddle point. Also, zero-point corrected binding energies ofthe rectangular structure at the mentioned levels of theory are�19.4 and �17.6 kcal/mol, respectively. The anti-symmetric O–Ostretch frequencies of the rectangular O4

+ are 1618.2 cm�1 and1596.6 cm�1 at M06-2X/6-311G(3df,2pd) and M06-2X/6-311++G(3df,2pd) levels of theory, respectively. Four different dis-persion corrected calculations on O2, O2

+ and O4+ have been per-

formed. The functional wB97XD seems to be the best among thefour ones used to determine ionization energy of O2. However,the ionization energy obtained by this functional is also highlydeviated from experimental finding. B.E.s of the rectangular (D2h)and trans-planar (C2h) geometry of O4

+ ion calculated at thewB97XD/6-311++G(3df,2pd) theoretical level are �22.60 and�23.58 kcal/mol, with zero-point correction, respectively. Amongthe four dispersion-corrected density functionals, the deviation ofB.E. from experimental value for the O4

+ cluster ion is the least atM05-2XGD3/6-311++G(3df,2pd) level of theory. However, thereexists an imaginary frequency of �59.59 cm�1 for the rectangularstructure of O4

+ calculated at this level of theory. Except the M06-2XGD3 functional, other two functionals, wB97XD and B3LYP-GD3, also produce imaginary frequencies for the rectangular O4

+

but in the case of trans-planar structure of this cluster ion, onlythe M06-2XGD3 produces an imaginary frequency. We concludethat none of the dispersion corrected density functionals accu-rately reproduce experimental ionization energy of O2 as well asbinding energy of O4

+. By using optimized structures of O2, O2+ and

O4+ (rectangular structure) at the level of theory M06-2X/6-

311G(3df,2pd) and M06-2X/6-311++G(3df,2pd), BD(T) single-pointenergy calculations were performed. We have obtained ionizationenergy of O2 equal to 274.6 and 277.14 kcal/mol using BD(T)/

6-311G(3df,2pd) and BD(T)/6-311++G(3df,2pd), respectively. Thebinding energies for the rectangular O4

+ are �11.0 and �9.61 kcal/mol at two levels of theory. These BD(T) results are in good agree-ment with experimental values of ionization energy of O2 as wellas experimental binding energy of O4

+. With zero-point correction,cyclic structure (D3h) of O6

+ cluster ion has been calculated to bemore stable than other structures at M05-2X/6-311G(3df,2pd)and M05-2X/6-311++G(3df,2pd) as well as at M06-2X/6-311G(3df,2pd) and M06-2X/6-311++G(3df,2pd) levels of theoryon the sextet potential energy surface. The zero-point correctedbinding energies of the cyclic O6

+ calculated at M05-2X/6-311G(3df,2pd) and M05-2X/6-311++G(3df,2pd) levels of theoryare �6.01 and �5.19 kcal/mol, respectively. On the other hand,those values predicted by using M06-2X/6-311G(3df,2pd) andM06-2X/6-311++G(3df,2pd) methods are �6.22 and �5.35 kcal/mol, respectively. These values are in good agreement with exper-imental measurements. Moreover, the zero-point corrected bind-ing energies of O8

+ with D4h symmetry on the octet potentialenergy surface are �2.47 and �2.49 kcal/mol, at theoretical levelsM05-2X/6-311G(3df,2pd) and M05-2X/6-311++G(3df,2pd), respec-tively. Those at M06-2X/6-311G(3df,2pd) and M06-2X/6-311++G(3df,2pd) theoretical levels are�2.69 and�2.59 kcal/mol, respec-tively. These values are in good agreement with experiments. Also,zero-point corrected B.E.s of O10

+ with D5h symmetry on the dectetpotential energy surface are �2.78 and �2.56 kcal/mol calculatedat M05-2X/6-311G(3df,2pd) and M05-2X/6-311++G(3df,2pd) theo-retical levels, respectively. Those values calculated for this clusterion on the same potential energy surface at M06-2X/6-311G(3df,2pd) and M06-2X/6-311++G(3df,2pd) levels are �2.52 and�2.14 kcal/mol, respectively. The latter one is exactly equal tothe experimental result due to Hiraoka [25]. This agreementsuggests that the performance of M05-2X functional is better forprediction of B.E. of O10

+ using basis set with diffuse function,6-311++G(3df,2pd), compared to that without diffuse function,6-311G(3df,2pd). From the study, performance of M05-2X seemsto be slightly better than M06-2X functional in the case of O6

+

and O8+, although both methods produce B.E.s for these clusters in

good agreement with experimental values. In the case of O10+

cluster ion also, M05-2X seems to perform better for the estima-tion of B.E. of O10

+ by using basis set with diffuse function,6-311G++(3df,2pd) than using 6-311G(3df,2pd). On the otherhand, the B3LYP functional performs worst among the four DFTmethods in determining B.E.s of (O2)n

+ (n = 2 � 5). In addition, theBHandH functional has been found to perform better than B3LYP

G.P. Khanal et al. / Computational and Theoretical Chemistry 1056 (2015) 24–36 35

but not better than meta-GGA DFTs like M05-2X and M06-2X inthe mentioned task.

In conclusion, the stable structures of O4+, O6

+, O8+ and O10

+ havepoint groups D2h, D3h, D4h and D5h optimized on the quartet, sextet,octet and dectet potential energy surfaces, respectively.

Acknowledgements

One of the authors (GPK) thanks the Indian Academy of Sciences(IAS) Bangalore for summer fellowship in 2012, Prof. G. Rangarajan,the then Chairman International relation cell, Indian Instituteof Science Bangalore, India. Authors acknowledge CentralDepartment of Physics, Kathmandu, Nepal and the SupercomputerEducational Research Centre (SERC) of the Indian Institute ofScience for computational facilities. In addition, GPK would liketo thank ‘‘special doctoral program for green energy conversionscience and technology’’, University of Yamanashi, Japan for finan-cial support to complete the work. RP would like to acknowledgeAcademy of Science for developing world for TWAS associateship(2009–2011) and Department of Inorganic and Physical Chemistry,Indian Institute of Science for visiting positions (2012 October and2014 October).

Appendix A. Supplementary material

In Table S1, BD(T) single point energies calculated for O4+ using

the optimized geometries obtained at M06-2X/6-311G(3df,2pd)and M06-2X/6-311++G(3df,2pd) level of theories are given. Totalenergies, corresponding strong harmonic vibrational frequenciesfor different structures of O8

+ calculated on sextet, quartet, doubletas well as octet potential energy surfaces, using M06-2X/6-311++G(3df,2pd) level of theory, are given in Table S2. Table S3gives information for O10

+ , on the dectet, octet, sextet, quartet anddoublet potential energy surfaces using the same level of theory.Harmonic vibrational frequencies calculated for O4

+, O6+, O8

+, O10+

are shown in Tables S4, S5, S8–S13. Tables S6 and S7 present equi-librium geometries, strong harmonic vibrational frequency, totalenergies, ionization energy of O2 and binding energy of O4

+, respec-tively, calculated at Grimme’s dispersion, GD3, corrected function-als B3LYP-GD3, M05-2XGD3, M06-2XGD3 as well as those atdispersion corrected recent density functionals of Martin Head-Gordan’s group, wB97XD, which account for dispersion effectexplicitly. The optimized coordinates obtained at all of the fourdensity functional methods, B3LYP, M05-2X, M06-2X and BHandH,are shown in Tables S1a–S1d for O2 and O2

+; in Tables S2a–S2d forO4

+; in Tables S3a, S3b, S4a, S4b for O6+; in Tables S5a, S5b, S6a, S6b

for O8+; and in Tables S7a, S7b, S8a, S8b for O8

+. Supplementary dataassociated with this article can be found, in the online version, athttp://dx.doi.org/10.1016/j.comptc.2014.12.022.

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