Recent data have determined that the structure of the high pressure
$\epsilon$ phase of solid oxygen consists of clusters composed of four O$_2$
molecules. This finding has opened the question about the nature of the
intermolecular interactions within the molecular oxygen tetramer. We use
multiconfigurational ab initio calculations to obtain an adequate
characterization of the ground singlet state of
  (O$_2$)$_4$ which is compatible with the non magnetic character of the
$\epsilon$ phase. In contrast to previous suggestions implying chemical
bonding, we show that (O$_2$)$_4$ is a van der Waals like cluster where
exchange interactions preferentially stabilize the singlet state. However, as
the cluster shrinks, there is an extra stabilization due to many-body
interactions that yields a significant softening of the repulsive wall. We show
that this short range behavior is a key issue for the understanding of the
structure of $\epsilon$-oxygen

Bartolomei, Massimiliano

Campos-Martínez, José

Carmona-Novillo, Estela

Hernández, Marta I.

Hernández-Lamoneda, Ramón

Pérez-Ríos, Jesús

English

arXiv

Recent data have determined that the structure of the high-pressure ε phase of solid oxygen consists of clusters composed of four O2 molecules. This finding has opened the question about the nature of the intermolecular interactions within the molecular oxygen tetramer. We use multiconfigurational ab initio calculations to obtain an adequate characterization of the ground singlet state of (O2)4, which is compatible with the nonmagnetic character of the ε phase. In contrast to previous suggestions implying covalent bonding, we show that (O 2)4 is a van der Waals-like cluster where exchange interactions preferentially stabilize the singlet state. Nevertheless, as the cluster shrinks, there is an extra stabilization due to many-body interactions, i.e., an incipient chemical bonding that just yields a softening of the repulsive wall. We show that these findings can be used to model the intra- and intercluster distances of ε-O2 observed in the lower-pressure range and are consistent with inelastic x-ray measurements of O2 K-edge excitations. © 2011 American Physical Society.funding by MICINN (Spain, FIS2010-22064- C02-02). R.H.L. was supported by a sabbatical grant by MEC (Spain, SAB2009-0010) and Conacyt (Mexico, Ref. 126608). J.P.-R. is a pre-JAE CSIC fellow.Peer Reviewe

Pérez Ríos, Jesús

Hernández Lamoneda, Ramón

Digital.CSIC

PHYSICAL REVIEW B 84, 092105 (2011)Molecular oxygen tetramer (O2)4: Intermolecular interactions and implications for the  solid phaseMassimiliano Bartolomei,* Estela Carmona-Novillo, Marta I. Herna´ndez, Jesu´s Pe´rez-Rı´os, and Jose´ Campos-Martı´nezInstituto de Fı´sica Fundamental, Consejo Superior de Investigaciones Cientı´ficas (IFF-CSIC), Serrano 123, ES-28006 Madrid, SpainRamo´n Herna´ndez-LamonedaCentro de Investigaciones Quı´micas, Universidad Auto´noma del Estado de Morelos, 62210 Cuernavaca, Mor. Me´xico(Received 13 July 2011; published 22 September 2011)Recent data have determined that the structure of the high-pressure  phase of solid oxygen consists of clusterscomposed of four O2 molecules. This finding has opened the question about the nature of the intermolecularinteractions within the molecular oxygen tetramer. We use multiconfigurational ab initio calculations to obtainan adequate characterization of the ground singlet state of (O2)4, which is compatible with the nonmagneticcharacter of the  phase. In contrast to previous suggestions implying covalent bonding, we show that (O2)4 isa van der Waals-like cluster where exchange interactions preferentially stabilize the singlet state. Nevertheless,as the cluster shrinks, there is an extra stabilization due to many-body interactions, i.e., an incipient chemicalbonding that just yields a softening of the repulsive wall. We show that these findings can be used to model theintra- and intercluster distances of -O2 observed in the lower-pressure range and are consistent with inelasticx-ray measurements of O2 K-edge excitations.DOI: 10.1103/PhysRevB.84.092105 PACS number(s): 36.40.−c, 31.15.ae, 34.20.Gj, 62.50.−pRecently, the determination of the structure of the highpressure  phase of oxygen1,2 has raised interest in the studyof molecular oxygen oligomers. In contrast to previouslyproposed structures based on the dimer3 and herringbonechains,4 two independent x-ray diffraction experiments1,2definitively concluded that -O2 consists of layers of welldefined (O2)4 aggregates. They were found to form prismswith the O2 axes perpendicular1 or nearly perpendicular2 to arhombic, nearly squared, base. There is a hierarchy of distancesin this phase, the O2 bond length, which nearly keeps the gasphase value (≈1.21 A˚) at all pressures, and the intra- andintercluster distances (2.34 and 2.66 A˚ at 11.4 GPa), whichdecrease monotonically with pressure up to the boundarywith the metallic ζ phase.2 Other key properties of thisphase, suggesting increasing intermolecular interactions, area dark-red color, a strong infrared absorption, and a magneticcollapse.3,5–7 Further evidence for a different intermolecularbonding has come from inelastic x-ray scattering8 where,at the lower pressure boundary of this phase (10 GPa), adiscontinuous shift of about 1.1 eV in the electronic transitionsfrom 1s to 1πg orbitals was found.A few works1,9–11 have attempted to rationalize the stabilityand the bonding of the (O2)4 species in the frameworkof the density functional theory (DFT), but with unclearresults. Thus, authors of Ref. 9 found that the D4h cuboidstructure corresponds to a local energy minimum that theyrecognized as unstable when higher levels of theory wereapplied. Furthermore, DFT calculations in Ref. 10 failedto show that the experimental (O2)4 geometry is the moststable one compared with other chain structures,4 showingthe need for additional studies. Then, it is apparent thatdespite recent progress12–14 made in DFT methodologies fortreating dispersion forces, the multiconfigurational characterof molecular oxygen clusters in low-spin multiplicity statesstill remains a serious problem for such techniques.As a matter of fact the nature of the bonding in suchmolecular clusters has been a subject of debate for severaldecades. A chemically bound dimer was expected consideringthe open shell 3−g character of O2 where two unpairedelectrons occupy degenerate πg orbitals. However, a numberof experimental15–17 and theoretical18–20 works clarified that inthe gas phase, (O2)2 has the typical features of a van der Waalscomplex: a well depth of tens of meV and retention of themolecular properties within the complex. In fact, it has beenshown18,19 that a singlet species of D2h symmetry is stabilizeddue to exchange interactions but not in a sufficient extent tolead to chemical bonding.We report here high level supermolecular ab initio calcula-tions of the (O2)4 cluster. Our goal is a reliable characterizationof the singlet ground state, which is consistent with themagnetic collapse7 and spectroscopy3,5,6 of the  phase. Tothis end, we use a multiconfigurational ansatz, which isunavoidable for spin multiplicities of (O2)4 lower than themaximum one (nonet). We proceed in analogy to our previouswork on the dimer19,20 and treat the highest spin complexby means of a restricted coupled cluster theory with singles,doubles, and perturbative triple excitations [RCCSD(T)]. Inaddition, the singlet-nonet splitting can be well described atthe multiconfigurational complete active space second-orderperturbation (CASPT2) theory. Finally, the (O2)4 singletenergy is obtained by adding to the RCCSD(T) nonet potentialthe singlet-nonet CASPT2 splitting. The aug-cc-pVQZ21 basisset has been used at all levels of theory. For the CASPT2calculations, the active space is defined by distributing eightelectrons in eight molecular orbitals correlating asymptot-ically with the O2 πg shell. As customary, these orbitalshave been previously optimized with the complete active spaceself-consistent field (CASSCF) method. The counterpoisemethod22 was applied to correct interaction energies forthe basis set superposition error. As for the (O2)4 geometry,we use a cuboid structure with D4h symmetry. The centersof mass of O2 form a square whose side is changed in therange 1.5–25 A˚, while the intramolecular distance is keptfixed to 1.2065 A˚. Calculations have been performed withthe MOLPRO2006.1 package.23 To study the role of many-bodyinteractions, we compare the supermolecular calculations with092105-11098-0121/2011/84(9)/092105(4) ©2011 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW B 84, 092105 (2011)-120-80-40                                 V(d) (meV)2 3 4 5 6d (Å)-1200-800-4000ΔV(d) (meV)1101001000supermolecularpairwisenonetsinglet(O2)4  many-body interaction energydFIG. 1. (Color online) Upper panel: interaction energies (in meV)for the nonet and ground singlet states of (O2)4 as functions of thesquare side d (in A˚). Supermolecular approach is represented in solidlines while the pairwise approach in dashed lines. Lower panel: many-body interaction energy for the singlet state V (d) obtained as thedifference between supermolecular and pairwise energies. The largevalues of V at small d’s might be a key feature to explain theclustering of O2 molecules in the  phase.estimations based on the summation of pure pair interactionsobtained at the same level of theory as those of (O2)4. In thepairwise approach, we have obtained expressions for the (O2)4energy that are compatible with a well-defined total spin of thecomplex. It must be noted that there are three singlet states24asymptotically correlating with four O2(3−g ), and that here,we are reporting the calculations of the ground singlet state.The interaction energies of (O2)4 in the singlet and nonetstates as functions of the square side d are reported in theupper panel of Fig. 1 together with the pairwise estimationsof the corresponding interactions. Equilibrium parameters ofthese potentials are given in Table I. As can be seen from theparameters of the singlet and nonet potential wells, (O2)4 is avan der Waals-like complex in the gas phase, mainly stabilizedby dispersion interactions. The exchange interaction, however,plays a role making the potential well of the singlet state deeperand shifted at shorter intermolecular distances than that ofthe nonet state. As in the dimer,19,20 the exchange interactionfavors the states of lowest spin multiplicity.More insight is gained when comparing the supermolecularcalculations with the pairwise estimations. For the nonetstate, the pairwise approximation reproduces very well theTABLE I. Binding energy De (in meV) and equilibrium distanceRe (in A˚) of the (O2)4 potentials reported in Fig. 1.nonet singletDe Re De Resupermolecular 56.6 3.37 122.6 2.99pairwise 57.1 3.37 108.4 3.08supermolecular energies indicating that many-body effects arenot particularly relevant. However, for the singlet state, ananalogous agreement is only achieved for the largest d sizesof the cluster. Around the minimum of the singlet well, thesupermolecular energies are already lower than the pairwiseones (see Table I), but it is for shorter distances where aremarkable softening of the repulsive wall is found. This is dueto a many-body effect, as shown in the lower panel of Fig. 1where it can be noticed that the many-body interaction energyfor the singlet state increases dramatically as d decreases,being about 1.2 eV at d = 2 A˚. The origin of this effectmust be in the exchange interactions,25 since polarizationcontributions to many-body interaction energies, if important,should be shown also for the nonet state, which is notthe case.In a molecular crystal, the first response to pressure isa “squeezing out of van der Waals space,” in other words,the penetration to the repulsive region of the intermolecularpotentials.26 Therefore the peculiar behavior at short clus-ter sizes is relevant indeed for understanding the structureof the high-pressure  phase,1,2 as is discussed in thefollowing.Some clues for the softening of the repulsive wall of theground singlet state are given in Fig. 2, where we showCASSCF molecular orbitals arising from the interaction ofthe eight half-occupied πg orbitals of O2. An analogous cal-culation for singlet (O2)2 is shown for the sake of comparison.The properties of these optimized orbitals barely change fromthe asymptote up to the equilibrium distance (d ≈ 3 A˚), butfor shorter d’s the interaction does give rise to bonding andantibonding orbitals. Interestingly, the four bonding orbitals in(O2)4 are more stable than the bonding orbitals of (O2)2 and theassociated occupation numbers increase faster as d decreases,as a result of many-body exchange effects. However, a doubleoccupancy only occurs for very short distances (d < 1.8 A˚).Moreover, the (O2)4 orbital stabilization is much smaller thanthe electron-electron Coulomb repulsion contribution to thetotal interaction. Thus the result is an incipient chemicalbonding leading not to a minimum but just to a softeningof the repulsive region of the potential, as shown in Fig. 1.This multiconfigurational analysis differs from those of Refs. 9and 8 where, based on a simpler monoconfigurational picture,it was suggested that all the bonding orbitals were doublyoccupied leading to large binding energies with respect tothe isolated O2 molecules. The present analysis also gives aqualitative insight into the observation of a ≈1-eV shift inthe πg ← 1 s transitions at the boundary of the  phase,8since the splitting between the (O2)4 antibonding energies andthe isolated πg orbitals is of the same order of magnitude(≈1.5 eV) in the relevant range (d ≈ 2.4 A˚).092105-2BRIEF REPORTS PHYSICAL REVIEW B 84, 092105 (2011)-20-15-10-50510Energy (eV)1.5 2 2.5 3 3.5 4 4.5 5d (Å)00.511.52mean occupation number(O2)4(O2)2bonding orbitalsHOMO molecular orbitalsbonding orbitalsantibonding orbitalsantibonding orbitalsπg*πg*FIG. 2. (Color online) Upper panel: energies of the HOMOsoriginated from the O2 πg orbitals vs intermolecular distance d , for theground singlet states of (O2)4 and (O2)2. Note that for the tetramer,each line corresponds to a couple of almost degenerate HOMOs.Lower panel: as in the upper panel but for the occupation numbers.In order to study whether the present results are adequatefor a description of the  phase, we consider a very simplemodel for the energy of a layer of (O2)4 clusters, with the O2interatomic distance kept fixed.27 A basic unit cell is shownin Fig. 3 where the tetramers form rhombuses of length d andangle α. It is assumed that, at a given pressure, the centers ofmass of the clusters are fixed and determined by the latticeparameters a and b. Values of these parameters as functionsof pressure were taken from Ref. 2. In addition, the angle α isfixed to 81.4◦ as derived from data reported at 11.4 GPa.2 Weassume that all clusters increase/decrease their size d at a timeand study the subsequent modification of the cell energy. Thisenergy is given as a sum of intra and intercluster contributionsE(d) = V intra(d) + 12∑i,jV interij (rij ), (1)where V intra is the supermolecular (O2)4 singlet potential andV interij is a pair potential between molecules i and j belonging todifferent clusters. The corresponding intermolecular distancerij is determined by d, a, b, and α. A spin-averaged(O2)2 potential was used for V interij because the nonmagneticcharacter7 of the  phase suggests that the dependence on spinmust be washed out. In Fig. 3 (upper panel), it is shown that,1.9 2.1 2.3 2.5d (Å)010002000300040005000Energy (meV)P=18.3 GPaEVintra   Vinter10 12 14 16 18 20 22 24P (GPa)22.22.42.62.83Distance (Å)aΣbmDdDmdmdFIG. 3. (Color online) Upper panel: energy of a layer of -O2[Eq. (1)] as a function of the cluster size, d , for a pressure of18.3 GPa (solid line). In the inset, the model unit cell is shown wherethe O2 axes are perpendicular to the a-b plane and the interclusterdistances, rij , are displayed by dashed lines, D being the shortest oneamong them. A minimum of the energy is obtained for the intraclusterdistance dm. Lower panel: Pressure dependence of dm and Dm (lines)compared with data of Ref. 2 (circles). See text for details.within the cell, the optimum size of the cluster is considerablyreduced with respect to the gas phase equilibrium distance. For18.3 GPa, a minimum in the total energy is obtained at aboutdm = 2.17 A˚, in agreement with the observed intraclusterdistance of 2.185 A˚2 and with the value obtained at 17.6 GPain another independent experiment.1 In the lower panel ofFig. 3, we show that the pressure dependence of the optimizedintra and intercluster distances dm and Dm agrees fairly wellwith the observations2 (Dm is the shortest intercluster distance,obtained as a function of dm, a, b, and α) for pressures lowerthan 20–25 GPa.We would like to stress that, despite the simplicity ofthe model, the key element is the behavior of the repulsivewall of the ground singlet state, adequately calculated at amulticonfigurational level of theory. Indeed, substitution ofthe ground singlet energy V intra(d) with that obtained at alower level of theory28 would lead to an optimum intraclustersize far less compatible with the measurements. As discussedin Ref. 1, the fact that both intra and intercluster distancescompress at nearly the same rate is a clear indication of a ratherweak interaction between the O2 molecules. We believe thatthe present findings of a van der Waals cluster with an incipientchemical bond for short sizes will be a key ingredient in futurerationalization and modeling of the structure of -oxygen as092105-3BRIEF REPORTS PHYSICAL REVIEW B 84, 092105 (2011)well as the pressure dependence of its vibrational modes. As anexample, the pressure dependence3 of the IR vibron frequencydisplays a minimum at about 20 GPa suggesting a change inthe O2 interatomic distance. This behavior, already observedin other molecular crystals,29 should be interpreted and takeninto account in more detailed analyses.We thank Yuichi Akahama for sending us structural dataof Ref. 2 and funding by MICINN (Spain, FIS2010-22064-C02-02). R.H.L. was supported by a sabbatical grant by MEC(Spain, SAB2009-0010) and Conacyt (Mexico, Ref. 126608).J.P.-R. is a pre-JAE CSIC fellow. We thank Nestor A. Aguirrefor assistance in the preparation of the figures.*maxbart@iff.csic.es1L. F. Lundegaard, G. Weck, M. I. McMahon, S. Desgreniers, andP. Loubeyre, Nature (London) 443, 201 (2006).2H. Fujihisa, Y. Akahama, H. Kawamura, Y. Ohishi, O. Shimomura,H. Yamawaki, M. Sakashita, Y. Gotoh, S. Takeya, and K. Honda,Phys. Rev. Lett. 97, 085503 (2006).3F. A. Gorelli, L. Ulivi, M. Santoro, and R. Bini, Phys. Rev. B 63,104110 (2001).4J. B. Neaton and N. W. Ashcroft, Phys. Rev. Lett. 88, 205503 (2002).5Y. A. Freiman and H. J. Jodl, Phys. Rep. 401, 1 (2004).6Y. Akahama and H. Kawamura, Phys. Rev. B 61, 8801 (2000).7I. N. Goncharenko, Phys. Rev. Lett. 94, 205701 (2005).8Y. Meng, P. J. Eng, J. S. Tse, D. M. Shaw, M. Y. Hu, J. Shu,S. A. Gramcsh, C. Kao, R. J. Hemley, and H. K. Mao, Proc. Natl.Acad. Sci. USA 105, 11640 (2008).9R. Steudel and M. W. Wong, Angew. Chem. Int. Ed. 46, 1768(2007).10B. Militzer and R. J. Hemley, Nature (London) 443, 150 (2006).11Y. Ma, A. R. Oganov, and C. W. Glass, Phys. Rev. B 76, 064101(2007).12A. Becke and E. Johnson, J. Chem. Phys. 127, 124108 (2007).13E. Johnson, S. Keinan, P. Mori-Sa´nchez, J. Contreras-Garcı´a,A. Cohen, and W. Yang, J. Am. Chem. Soc. 132, 6498 (2010).14J. Toulouse, I. C. Gerber, G. Jansen, A. Savin, and J. G. Angyan,Phys. Rev. Lett. 102, 096404 (2009).15C. Long and G. Ewing, J. Chem. Phys. 58, 4824 (1973).16L. Biennier, D. Romanini, A. Kachanov, A. Campargue, B. Bussery-Honvault, and R. Bacis, J. Chem. Phys. 112, 6309 (2000).17V. Aquilanti, D. Ascenzi, M. Bartolomei, D. Cappelletti, S. Cavalli,M. de Castro Vitores, and F. Pirani, Phys. Rev. Lett. 82, 69 (1999).18P. Wormer and A. van der Avoird, J. Chem. Phys. 81, 1929 (1984).19M. Bartolomei, M. I. Herna´ndez, J. Campos-Martı´nez, E. Carmona-Novillo, and R. Herna´ndez-Lamoneda, Phys. Chem. Chem. Phys.10, 5374 (2008).20M. 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Molecular oxygen tetramer (O2)4: Intermolecular interactions and implications for the ε solid phase

The Molecular Oxygen Tetramer: Intermolecular Interactions and
  Implications for the $\epsilon$ Solid Phase

The Molecular Oxygen Tetramer: Intermolecular Interactions and Implications for the $\epsilon$ Solid Phase

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The Molecular Oxygen Tetramer: Intermolecular Interactions and Implications for the $\epsilon$ Solid Phase