21 August 1998
.Chemical Physics Letters 293 1998 7280
A systematic ab initio investigation of the open and ringstructures of ozone
Thomas Muller a, Sotiris S. Xantheas b, Holger Dachsel b, Robert J. Harrison b,Jaroslaw Nieplocha b, Ron Shepard c, Gary S. Kedziora c,1, Hans Lischka a,)
a Institut fur Theoretische Chemie und Strahlenchemie der Uniersitat Wien, A-1090 Vienna, Austria b Pacific Northwest National Laboratory, Richland, WA 99352, USA
c Theoretical Chemistry Group, Chemistry Diision, Argonne National Laboratory, Argonne, IL 60439, USA
Received 11 May 1998
1 .The energy difference between the open and the ring isomer of ozone as well as the dissociation energy O X, A 3 1 3 y. 3 .. .O X, S qO P have been determined at the CCSD T , MR-CISD and MR-AQCC levels of theory. Using correlation2 g
.consistent basis sets up to quintuple-zeta quality, the estimated complete basis set limits for CCSD T and MR-AQCC liewithin 1 kcalrmol of the experimental value of 26.1"0.4 kcalrmol and place the ring isomer by 4.8 and 5.3 kcalrmol,respectively, above the dissociation limit. Zero-point vibrational corrections increase the latter two values by 1.4 kcalrmol.q 1998 Elsevier Science B.V. All rights reserved.
Ozone plays a fundamental role in atmosphericchemistry. Its importance has lead to extensive theo-retical and experimental research efforts focused onboth its ground state and excited state properties.Extensive investigations of the potential energy sur-
.faces PES of O have been performed by3w x w xBanichevich et al. 1 and by Xantheas et al. 2 .
Several previous studies have focused especially on .the characterization of the global open C and2v
1 . .local ring D minima on the ground state X, A3h 1w x.PES 36 . Although the ring isomer has not been
observed experimentally yet, interest in this species
) Corresponding author.1 Present address: Department of Chemistry, Northwestern Uni-
versity, Evanston, Il 60208, USA.
stems from its potential use as a high-energy fuel w x.see, e.g., Refs. 7,8 . In order to assess the suitabil-ity of the ring isomer for such use, it is important to: .1 accurately determine its energy separation withrespect to the global minimum and whether it liesabove or below the lowest dissociation limit O2 3 y. 3 . .X, S qO P ; and 2 investigate accessiblegpathways for its formation, an effort involving thecharacterization of its excited states. The aim of thisstudy is to address the first question. Future studieswill concentrate on the characterization of the ex-cited states for this system.
It has been noted earlier from the analysis of the w x.CI wavefunction see, e.g., Ref. 4 as well as
recently using the T diagnostic in the coupled-clus-1w xter expansion 5 that both the open and the ring
isomers do not represent true single-reference cases.Moreover, the importance of triple excitations in
0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. .PII: S0009-2614 98 00798-2
( )T. Muller et al.rChemical Physics Letters 293 1998 7280 73
coupled-cluster calculations has been stressed by Leew xand Scuseria 5,6 . The recent investigation by Wattsw xand Bartlett 9 even suggests non-negligible contri-
butions from quadruple and higher excitations. Inparticular, we note the strong dependence of theenergy separation D E between the open and theorring structure on the inclusion of triple excitations.D E amounts to y23.3 and y29.1 kcalrmol at theor
. w xCCSD and CCSD T levels, respectively 5 .
1 .The dissociation energy E of O X, A intodiss 3 1 3 y. 3 .O X, S qO P is even more difficult to com-2 g
.pute than D E . A FORS CASSCF calculationorgave only 20% of the experimental E and even andissinternally contracted multi-reference configuration
interaction with single and double excitations ic-.MR-CISD using a cc-pVDZ basis produced only
60% of the experimental value. Only the much largerw xcc-pVQZ basis gave acceptable results 10 .
Consideration of the multi-reference character ofthe ozone molecule at the open, ring and dissociationgeometries is the main focus of our investigation.The most straightforward method to use would beMR-CI. However, truncated MR-CI calculations suf-fer from the well-known size-extensivity problemw x11 . This error can be reduced substantially by usingmethods such as MR averaged coupled-pair func-
. w xtional MR-ACPF 12 or MR average quadratic . w xcoupled cluster MR-AQCC 13 . Results from
MR-AQCC calculations are compared with those .obtained from single-reference CCSD T calcula-
tions.Conceptually, the simplest reference space is the
.complete active space CAS . However, with anincreasing number of electrons andror active or-bitals, its use soon leads to an extremely large com-putational effort. Therefore, another aspect of ourwork is to search for more economical referencespaces which give results comparable to those ob-tained from CAS references.
In addition to studying the correlation require-ments for the accurate calculation of E and D E ,diss orwe study the convergence with respect to the one-electron basis set. The systematic series of correla-
w tion-consistent basis sets cc-pVnZ sets n s.xD, T, Q, 5 developed by Dunning and co-workers
are used in order to determine the complete basis set . w xlimits CBS cf. Refs. 1418 and references.therein of the energies and geometries calculated
using the various methods and reference selectionschemes.
2. Computational details
The calculations were carried out using uncon- . w xtracted and internally contracted ic- MR-CISD 19 ,
w x . w xMR-AQCC 13 and CCSD T 20,21 levels of the-ory. Standard polarized valence cc-pVnZ, aug-cc-
. .pVnZ and core-valence cc-pCVnZ basis sets de-veloped by Dunning et al. nsD, T, Q, 5; for refer-
w x.ences see Ref. 17 have been used. A CASSCFcalculation, used in this study as a reference calcula-tion, with 18 electrons in 12 valence orbitals .. w xCAS 18r12 2 was performed first. The three 1soxygen orbitals were kept doubly occupied. Follow-
w xing the notation used in Ref. 2 the dominant orbitaloccupations and the list of active orbitals as well as
the doubly-occupied orbitals are as follows C2v.symmetry adapted basis sets were used throughout :
open structure: .2 .2 .2 .2 .2 .2 .2 .21a 1b 2a 3a 4a 2b 5a 3b1 1 1 1 1 1 1 1 .2 .2 .2 .2 .0 .0 .06a 4b 1b 1a 2b 7a 5b ;1 1 2 2 2 1 1
ring structure: .2 .2 .2 .2 .2 .2 .2 .21a 1b 2a 3a 4a 2b 5a 3b1 1 1 1 1 1 1 1 .2 .2 .2 .2 .0 .0 .06a 1b 1a 2b 4b 7a 5b .1 2 2 2 1 1 1
These configurations constitute ;83% and ;87%, respectively, of the weight of the CASSCFwavefunctions.
Based on these CASSCF calculations, a series ofreference configuration sets of increasing complexity
.starting from a single-reference SR to a full-va- .lence space CAS 18r12 reference were constructed
for the use in subsequent uncontracted MR-CISDand MR-AQCC calculations. The three oxygen 1sorbitals were kept frozen throughout. The variousreference configuration spaces are characterized bythe number of inactive and active orbitals, the totalnumber of reference configurations and configura-
.tions, respectively see Table 1 . The following ref-erence configuration sets have been used.
fl The single-reference set SR containing only thedominant configuration as given above.
.fl A minimal reference space denoted MIN-REFadditionally includes the next important configura-
.2 .2tion, i.e. the double excitation 1a 2b for2 2the open structure and the pair of degenerate double
( )T. Muller et al.rChemical Physics Letters 293 1998 728074
.2 .2 .2 .2excitations 6a 4b and 3b 4b for1 1 1 1the ring structure.
fl SEL-REF denotes a threshold selected set witha threshold criterion of 0.05 for the absolute value ofthe CI or AQCC CSF expansion coefficients. Theselection is carried out iteratively. Based on an SRcalculation an initial reference space is determined.The corresponding MR calculation yields an im-proved set of expansion coefficients to which theselection criterion is applied. This procedure is re-peated until the reference remains unchanged. Closeto the equilibrium geometries the reference space isinsensitive to geometry relaxation. Therefore the ref-erence space is kept fixed during the optimization ofeach structure.
fl A reference configuration selection schemebased on 5 primary configurations the combined set
of MIN-REF references for the open and ring struc-.ture from which all reference configurations are
generated by single and double excitations within theset of active orbitals. The set PRIM1-REF containsthe 10 active orbitals which are occupied in theprimary references only, i.e. the orbitals 7a and 5b1 1are excluded from the original list of 12 orbitals.PRIM2-REF contains 12 internal orbitals, but re-
.stricts the three 2s-type orbitals 3a , 4a , 2b to be1 1 1inactive. PRIM3-REF is the same as PRIM2-REFbut lifts the restriction on the 2s-type orbitals.PRIM2-REF and PRIM3-REF are approximations to
. .CAS 12r9 and CAS 18r12 , respectively, reducing
the total number of configurations by up to 80% see.Table 1 . The two most important configurations
were selected as primary references for the calcula-tion of the dissociated system.
.fl A CAS 12r9 reference space with 12 elec-trons in the valence space of 9 oxygen p orbitals.
.fl A full valence space CAS 18r12 with 18electrons in the valence space of 12 orbitals.
.Internally-contracted ic- MR-SDCI calculationshave been carried out with a full-valence CAS . .18r12 reference space exclusively. The CCSD Tcalculations are based on the RHF reference configu-rations as given above for the two minima. Uncon-tracted MR-SDCI and MR-AQCC geometry opti-mizations have been limited to the smaller referencespaces SR, MIN-REF and SEL-REF. The calculation
.of the dissociation energy E is based on: 1 thedisssupermolecule approach with a geometry optimized
3 y . 3O molecule S state and an oxygen atom P2 g. .state at a distance of 100 A; and 2 on separate
calculations for the oxygen molecule and oxygenatom, respectively, with appropriately split referenceconfiguration spaces. The differences in E ob-disstained from these two approaches illustrate the size-extensivity error. The CBS limit E was determinedby a fit of the total energies E of a series ofncc-pVnZ results to the exponential form E sE qn BeyC n.
The uncontracted MR-CISD and MR-AQCC cal-culations have been carried out using the COLUMBUS
Table 1Characterization of the various configuration selection schemes in the CI calculations for the open and ring structures of ozone
aReference set Internal orbitals Number of references Number of configurations in MRCIcinactive active cc-pVTZ cc-pVQZ cc-pV5Z
SR 9 0 1 64 242 536d d dcMIN-REF 7 2r3 2r3 184 708 1570d d dc .SEL-REF MRCI 5r4 7r8 7r14 125 4644 10317d d dc .SEL-REF MR-AQCC 4 8 10r14 1205 4644 10317
PRIM1-REF 0 10 20 1127 4383 9768PRIM2-REF 3 9 169 11126 43506 97156PRIM3-REF 0 12 437 24547 97171 215873
.CAS 12r9 3 9 666 38790 152744 341929 .CAS 18r12 0 12 4067 129053 514747 1157095
a In thousands.b h-functions omitted.c Pairs of values separated by slashes refer to the numbers of orbitalsrconfigurations for the open and ring structures, respectively.dRing structure.
( )T. Muller et al.rChemical Physics Letters 293 1998 7280 75
w xprogram package 2224 . CI expansions up to 515million configurations for single-point calculations
are obtained with the larger reference spaces cf..Table 1 . For these cases, the recently developed
w xparallel CI program pciudg 25 of the COLUMBUSsuite has been used. The pciudg module is a mas-sively parallel implementation of the standard ciudg
w xCI program using the global array tools 26 . Thepciudg calculations were performed on a CRAY T3Eusing up to 512 processors with a parallel efficiencyof more than 99%.
The geometry optimizations for the uncontractedCI and AQCC calculations have been performed
w xwith the analytic gradient program 27,28 of COLUM-BUS using the AO integral and AO derivative integral
w xpackages of DALTON 29 . The internally contracted .ic-MR-CISD and the CCSD T calculations were
w xcarried out using the MOLPRO 96 30 suite of pro-grams.
3. Results and discussion
Equilibrium geometries of the open and ringstructures of ozone are listed in Table 2 for selectedcomputational methods and choices of basis sets. Foreach method the optimum values of the internalcoordinates converge smoothly with increasing basis
. .set size. MR-AQCC AQCC SEL-REF , CCSD Tand ic-MR-CISD yield similar results. The ic-MR-CISD exhibits the longest OO bond distancesthroughout and is in best agreement with experimentw x 31 . The addition of diffuse functions i.e., the useof the family of augmented correlation consistent
.basis sets has an almost negligible effect, especiallyfor the QZ and 5Z basis sets. Core correlation i.e.,
. .inclusion of the O 1s orbitals in the active spacewhen using the cc-pCVnZ sets was found to reduce
the OO bond distance by a few thousands of an A.The MIN-REF and SR-AQCC calculations yield re-
Table 2Optimal geometries for the open and ring structures
. .Methodrconfiguration selection Basis set Open structure C Ring structure D2v 3h . . .R A a 8 R A
SR-AQCC cc-pVTZ 1.273 116.0 1.434MR-AQCCrMIN-REF cc-pVTZ 1.271 116.7 1.445MR-AQCCrSEL-REF cc-pVDZ 1.280 116.3 1.458
cc-pVTZ 1.270 116.6 1.440cc-pVQZ 1.264 116.8 1.434
MR-CISDrSEL-REF cc-pVTZ 1.252 116.8 1.438
.CCSD T rRHF cc-pVDZ 1.285 116.6 1.462cc-pVTZ 1.276 116.9 1.445cc-pVQZ 1.269 117.1 1.440cc-pV5Z 1.267 117.2 1.438aug-cc-pVDZ 1.285 116.6 1.468aug-cc-pVTZ 1.276 117.1 1.447aug-cc-pVQZ 1.269 117.2 1.440aug-cc-pV5Z 1.267 117.2 1.438cc-pCVDZ 1.283 116.7 1.461cc-pCVTZ 1.273 117.0 1.442cc-pCVQZ 1.266 117.1 1.437
.ic-MR-CISDrCAS 18r12 cc-pVDZ 1.291 116.2 1.467cc-pVTZ 1.280 116.6 1.446cc-pVQZ 1.272 116.8 1.440cc-pV5Z 1.271 116.9 1.438
w xexp. 31 1.272 116.8
( )T. Muller et al.rChemical Physics Letters 293 1998 728076
sults of quite acceptable quality. The largest devia-tions are found for MR-CISD calculations with thesmaller reference spaces which produced OO bond
distances that are too low by as much as 0.03 A. Atypical example consists of the SEL-REF MR-CISDcalculation. Accurate results are obtained only withthe much larger reference spaces used in the inter-nally contracted CI calculations.
.The CCSD T results for the energy differencebetween the open and the ring structure D E as welloras the dissociation energy E are summarized indissTable 3. The variation of the dissociation energy ofthe open structure is graphically illustrated in Fig. 1.We note a monotonic convergence of the computedE with increasing basis set towards a convergeddissvalue. The corresponding CBS limits for the threefamilies of the correlation-consistent basis sets are
.within 0.2 kcalrmol cf. Table 3 , only 1.3 kcalrmolaway from the experimentally estimated value of
w xy26.1"0.4 kcalrmol 2,32 . Since the results forthe various types of basis sets are quite close to eachother, especially for the larger sets, further...