Ab Initio and Density Functional Theory Reinvestigation of Gas-Phase Sulfuric Acid Monohydrate and Ammonium Hydrogen Sulfate

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1 7178 J. Phys. Chem. A 2006, 110, Ab Initio nd Density Functionl Theory Reinvestigtion of Gs-Phse Sulfuric Acid Monohydrte nd Ammonium Hydrogen Sulfte Theo Kurtén,*, Mrkku R. Sundberg, Hnn Vehkm1ki, Mdis Noppel, Johnn Blomqvist, # nd Mrkku Kulml Deprtment of Physicl Sciences, UniVersity of Helsinki, P.O. Box 64, FIN UniVersity of Helsinki, Finlnd, Lbortory of Inorgnic Chemistry, Deprtment of Chemistry, UniVersity of Helsinki, P.O. Box 55, FIN UniVersity of Helsinki, Finlnd, Institute of EnVironmentl Physics, UniVersity of Trtu, 18 Ülikooli Str., Trtu, Estoni, nd CSCsScientific Computing Ltd., P.O. Box 405 (Keilrnt 14), FIN Espoo, Finlnd ReceiVed: Mrch 2, 2006; In Finl Form: April 10, 2006 We hve clculted the thermochemicl prmeters for the rections H 2 SO 4 + H 2 O T H 2 SO 4 H 2 O nd H 2 - SO 4 + NH 3 T H 2 SO 4 NH 3 using the B3LYP nd PW91 functionls, MP2 perturbtion theory nd four different bsis sets. Different methods nd bsis sets yield very different results with respect to, for exmple, the rection free energies. A lrge prt, but not ll, of these differences re cused by bsis set superposition error (BSSE), which is on the order of 1-3 for most bsis set combintions used in previous studies. Complete bsis set extrpoltion (CBS) clcultions using the cc-pv(x+d)z nd ug-ccpv(x+d)z bsis sets (with X ) D, T, Q) t the B3LYP level indicte tht if BSSE errors of less thn 0.2 re desired in uncorrected clcultions, bsis sets of t lest ug-cc-pv(t+d)z qulity should be used. The use of dditionl ugmented bsis functions is lso shown to be importnt, s the BSSE error is significnt for the nonugmented bsis sets even t the qudruple-ζ level. The effect of nhrmonic corrections to the zero-point energies nd therml contributions to the free energy re shown to be round 0.4 for the H 2 SO 4 H 2 O cluster t 298 K. Single-point CCSD(T) clcultions for the H 2 SO 4 H 2 O cluster lso indicte tht B3LYP nd MP2 clcultions reproduce the CCSD(T) energies well, wheres the PW91 results re significntly overbinding. However, bsis-set limit extrpoltions t the CCSD(T) level indicte tht the B3LYP binding energies re too low by c. 1-2 kcl/mol. This probbly explins the difference of bout 2 for the free energy of the H 2 SO 4 + H 2 O T H 2 SO 4 H 2 O rection between the counterpoise-corrected B3LYP clcultions with lrge bsis sets nd the diffusion-bsed experimentl vlues of S. M. Bll, D. R. Hnson, F. L Eisele nd P. H. McMurry (J. Phys. Chem. A. 2000, 104, 1715). Topologicl nlysis of the electronic chrge density bsed on the quntum theory of toms in molecules (QTAIM) shows tht different bsis set combintions led to qulittively different bonding ptterns for the H 2 SO 4 NH 3 cluster. Using QTAIM nlysis, we hve lso defined proton trnsfer degree prmeter which my be useful in further studies. Introduction In tmospheric conditions, new prticle formtion is thought to involve sulfuric cid nd wter molecules, with possible contributions from mmoni or some orgnic species. 1,2 To obtin structurl nd thermochemicl prmeters for, e.g., nucletion simultions, b initio nd density functionl theory clcultions hve been crried out on sulfuric cid hydrtes 3-7 nd sulfuric cid-mmoni-wter clusters. 8,9 However, the results obtined in different studies vry strongly. To understnd the discrepncies between erlier studies, nd to fcilitte further, more relible tmospheric clcultions, we hve crried out systemtic nlysis of the effects of the computtionl method nd the bsis set on the geometricl, thermochemicl nd bonding topologicl properties of sulfuric cid monohydrte nd mmonium hydrogen sulfte clusters. * Corresponding uthor. E-mil: theo.kurten@helsinki.fi. Deprtment of Physicl Sciences, University of Helsinki. Deprtment of Chemistry, University of Helsinki. University of Trtu. # CSC - Scientific Computing Ltd.. Previous Results Bndy nd Inni 4 clculted the free energy of the rection H 2 SO 4 + H 2 O T H 2 SO 4 H 2 Otobe-0.6 (t 298 K nd 1 tm) t the B3LYP/ G(2d,2p) level, wheres Re et l. 5 gve vlues of -4.3 nd -2.4 t the B3LYP/ D95(d,p) nd B3LYP/D95++(d,p) levels, respectively. (We hve chosen to report ll energies in, s most erlier studies use this unit. 1 ) kj mol -1.) The PW91/DNP computtions of Ding et l. 6 re in greement with Re s results, yielding vlue of -2.5, but Al Ntsheh et l. 7 recently reported vlue of t the PW91/TZP level. Beichert nd Schrems 3 did not clculte the free energy of hydrtion t 298 K. However, the vlue of -6.3 reported by them t 230 K nd t the MP2/ G(2d,2p)//MP2/6-31G** level is significntly lower thn the vlue of -2.9 given by Bndy nd Inni t 223 K. The experimentl vlue given by Hnson nd Eisele 10 (t 298 K) is -3.6 ( 1. The reported vlues of hydrtion entropies lso vry considerbly; rnging from cl/k mol (Bndy nd Inni) to cl/k mol (Al Ntsheh et l.). The /jp CCC: $ Americn Chemicl Society Published on Web 05/16/2006

2 Gs-Phse H 2 SO 4 H 2 O nd Ammonium Hydrogen Sulfte J. Phys. Chem. A, Vol. 110, No. 22, TABLE 1: Thermochemicl Prmeters for the Rection H 2 SO 4 + H 2 O T H 2 SO 4 H 2 O bsis set E 0/ ZPE/ H/ S/ cl K -1 mol -1 G/ HF/ G(2d,2p) b -9.7 MP2/ G(2d,2p) b c -6.3 (230 K) d B3LYP/ G(2d,2p) e B3LYP/D95(d,p) f B3LYP/D95++(d,p) f PW91/DNP g PW91/TZP h experimentl i -3.6 ( 1 E 0, ZPE, H, S nd G re the chnges in electronic energy, zero-point energy, enthlpy, entropy nd Gibbs free energy, respectively. Note tht - E 0 is often lso referred to s the binding energy. All vlues correspond to 1 tm nd 298 K, unless otherwise stted. b Beichert nd Schrems. 3 c Clculted using the 6-31G** bsis set. d Clculted using the 6-31G** bsis set, nd t 230 K. e Bndy nd Inni. 4 f Re et l. 5 Note tht their reported vlue of S corresponded to -1 times the S vlue given here, s they used different sign convention. g Ding et l. 6 h Al Ntsheh et l. 7 Note tht the temperture used ws K, not 298 K. i Hnson nd Eisele. 10 TABLE 2: Thermochemicl Prmeters for the Rection H 2 SO 4 + NH 3 T H 2 SO 4 NH 3 bsis set E 0/ ZPE/ H/ S/ cl K -1 mol -1 G/ B3LYP/ G(2d,2p) b B3LYP/ G(d,p) c MP2/ G(d,p) c experimentl inference d -8 The definitions of the prmeters re the sme s in Tble 1. All vlues correspond to 1 tm nd 298 K. b Inni nd Bndy. 9 c Lrson et l. 8 d Eisele nd Hnson. 11 thermochemicl prmeters reported by different uthors for the rection H 2 SO 4 + H 2 O T H 2 SO 4 H 2 O re presented in Tble 1. Similr discrepncies exist in the cse of mmonium hydrogen sulfte, though there re fewer studies to compre. The only reported vlue for the free energy chnge of the rection H 2 - SO 4 + NH 3 T H 2 SO 4 NH 3 is (t 298 K nd 1 tm) by Inni nd Bndy, 9 clculted t the B3LYP/ G(2d,2p) level. However, the electronic energy chnges given by Lrson et l. 8 t the B3LYP/ G(d,p) nd MP2/ G(d,p) levels re bout 2 lower, s cn be seen from Tble 2. Also, by comprison to the H 2 SO 4 H 2 O results it might lso be expected tht computtions using other bsis set combintions could yield significntly more negtive free energies. There re no direct experimentl mesurements of the free energy of this rection, but on the bsis of mesured concentrtions of lrger (H 2 SO 4 ) n (NH 3 ) m (with n g 2) clusters, Eisele nd Hnson 11 inferred tht the G vlue should be round -8. All the erlier studies computed the zero-point energies nd vibrtionl contributions to the free energy using the hrmonic pproximtion. The only study to ccount for bsis-set superposition ws tht of Beichert nd Schrems, 3 who found tht the bsis set superposition error (BSSE; estimted t the uncorrected equilibrium geometry by the counterpoise method 12 ) ws significnt (up to 1.4 ) for the 6-31G** bsis set but smller thn 0.25 for the lrger G(2d,2p) bsis. Computtionl Detils All our b initio nd density functionl theory (DFT) clcultions hve been performed using the Gussin 03 progrm suite. 13 All the geometries were converged to rootmen-squre (RMS) nd mximum force of less thn nd u, respectively. For the nhrmonic vibrtionl frequency clcultions, the criteri were 10-5 nd u, respectively. The convergence with respect to the electronic energy in the self-consistent field (SCF) step ws u. For the DFT clcultions, the stndrd integrtion grid ws used, except for the nhrmonic vibrtionl clcultions, in which the ultrfine grid ws used. For estimtion of the bsis set superposition energy, the counterpoise correction 12 (CP) ws pplied to both the energy nd geometry. The perturbtive method by which the nhrmonic vibrtionl frequencies were clculted is described in ref 14. The methods we used were the B3LYP hybrid functionl, 15,16 the PW91 density functionl 17 nd the second-order Møller- Plesset perturbtion theory 18 MP2. Further clcultions were lso crried out using the coupled cluster method CCSD(T). 19 The bsis sets included were the D95++(d,p) set, 20 the somewht lrger G(2d,2p) set nd the new correltionconsistent polrized split-vlence sets cc-pv(t+d)z nd ugcc-pv(t+d)z sets, 24 which re improved version of the stndrd cc-pvtz nd ug-cc-pvtz sets 25,26 nd hve been shown to produce more ccurte results for sulfur-contining molecules. 27,28 At the B3LYP level, the cc-pv(d+d)z, ug-cc-pv- (D+d)Z, cc-pv(q+d)z nd ug-cc-pv(q+d)z bsis sets were lso tested. Results nd Discussion Thermochemicl Prmeters. The thermochemicl prmeters for the rections H 2 SO 4 + H 2 O T H 2 SO 4 H 2 O nd H 2 - SO 4 + NH 3 T H 2 SO 4 NH 3, clculted using three different methods nd four different bsis sets, nd both with nd without counterpoise corrections, re presented in Tbles 3 nd 4, respectively. (For the lrgest bsis set ug-cc-pv(t+d)z, nd methods other thn B3LYP, only electronic energies were clculted due to computtionl limittions.) It should be noted tht in test clcultions, only smll (on the order of 0.05 kcl mol -1 with respect to the rection energies) differences were observed between the cc-pvtz nd cc-pv(t+d)z bsis sets. This is unsurprising, s the bonding pttern of the sulfur tom itself does not chnge during cluster formtion. However, s the ddition of the extr d-orbitl on sulfur ws not computtionlly very costly, nd in principle should improve the ccurcy of the clcultion, we hve used the revised bsis sets throughout this study.

3 7180 J. Phys. Chem. A, Vol. 110, No. 22, 2006 Kurtén et l. TABLE 3: Thermochemicl Prmeters for the Rection H 2 SO 4 + H 2 O T H 2 SO 4 H 2 O t 298 K nd 1 Atm, Using Different Methods nd Bsis Sets, with nd without Counterpoise Corrections to the Energy nd Geometry bsis set E 0/ ZPE/ Clerly, the choice of method nd bsis set hs n enormous influence on the formtion energies of tmosphericlly relevnt cluster structures. The lrge difference for exmple between the results obtined by Bndy nd Inni 4 nd Al Ntsheh et l. 7 re thus explined t lest prtilly by the fct tht Bndy nd Inni used bsis set combintion which produces exceptionlly low stbilities, wheres Al Ntsheh et l. employed functionl yielding exceptionlly high stbilities. The Sltertype orbitl-bsed TZP bsis set used in their clcultions ws unfortuntely not vilble for the Gussin 03 progrm, nd H/ S/ cl K -1 mol -1 G/ B3LYP/D95++(d,p) b b b b b c c c c c B3LYP/ G(2d,2p) b b b b b c c c c c B3LYP/cc-pV(T+d)Z b b b b b c c c c c B3LYP/ b b b b b ug-cc-pv(t+d)z c PW91/D95++(d,p) b b b b b c c c c c PW91/ G(2d,2p) b b b b b c c c c c PW91/cc-pV(T+d)Z b b b b b c c c c c PW91/ b ug-cc-pv(t+d)z c MP2/D95++(d,p) b b b b b c c c c c MP2/ G(2d,2p) b b b b b c c c c c MP2/ cc-pv(t+d)z b b b b b c c c c c MP2/ b ug-cc-pv(t+d)z c The definitions of the prmeters re the sme s in Tble 1. b Without counterpoise corrections. c With counterpoise corrections to both the geometry nd energy. TABLE 4: Thermochemicl Prmeters for the Rection H 2 SO 4 + NH 3 T H 2 SO 4 NH 3 t 298 K nd 1 tm (in ) Using Different Methods nd Bsis Sets, with nd without Counterpoise Corrections to the Energy nd Geometry bsis set E 0/ ZPE/ H/ S/ cl K -1 mol -1 G/ B3LYP/D95++(d,p) b b b b b c c c c c B3LYP/ b b b b b G(2d,2p) c c c c c B3LYP/ cc-pv(t+d)z b b b b b c c c c c B3LYP/ b b b b b ug-cc-pv(t+d)z c PW91/D95++(d,p) b b b b b c c c c c PW91/ G(2d,2p) b b b b b c c c c c PW91/ cc-pv(t+d)z b b b b b c c c c c PW91/ b ug-cc-pv(t+d)z c MP2/D95++(d,p) b b b b b c c c c c MP2/ G(2d,2p) b b b b b c c c c c MP2/ cc-pv(t+d)z b b b b b c c c c c MP2/ b ug-cc-pv(t+d)z c The definitions of the prmeters re the sme s in Tble 1. b Without counterpoise corrections. c With counterpoise corrections to both the geometry nd energy. we could not investigte further resons for why their bsolute vlues for the free energies of rection re still twice s lrge s our lrgest PW91 vlues. The effect of bsis set superposition error on the electronic energies, enthlpies nd free energies ws found to be very lrge (severl ) for lmost ll the bsis set combintions studied, the sole exceptions being the DFT/ugcc-pV(T+d)Z clcultions. This is in contrst to the erlier findings of Beichert nd Schrems, 3 who found the BSSE to be insignificnt for the medium-sized G(2d,2p) bsis set.

4 Gs-Phse H 2 SO 4 H 2 O nd Ammonium Hydrogen Sulfte J. Phys. Chem. A, Vol. 110, No. 22, For exmple, t the MP2/D95++(d,p) level, the bsolute vlue of the free energy of hydrtion of sulfuric cid decreses by fctor of 4 (from 3.40 to 0.79 ) when the counterpoise correction is pplied. Clerly, BSSE should be tken into ccount in ny future studies if even semiquntittive results re desired. This will probbly present considerbly difficulties for studies on lrger clusters with, e.g., 5-10 molecules, s the computtionl effort of pplying the counterpoise correction (CP) increses rpidly with the number of frgments. In such cses, it my be necessry to use simplified versions of the full CP method, nd to pply the CP correction only to the energy, not the geometry. Our studies indicte tht for our two-molecule clusters, neglecting the CP correction to the geometry cuses errors on the order of 0.1 with respect to both the binding energies nd the therml contributions to the rection free energy. Thus, lthough energy-only correction schemes do not remove ll of the BSSE, they still decrese it by n order of mgnitude compred to uncorrected clcultions. Appliction of the counterpoise correction decreses the differences between results obtined using different methods nd bsis sets. For exmple, the counterpoise-corrected B3LYP nd MP2 results re quite close to ech other for ech bsis set, s well s the counterpoise-corrected cc-pv(t+d)z nd ugcc-pv(t+d)z results for ech method. The differences between correltion-consistent bsis sets nd the D95++(d,p) nd G(2d,2p) sets re still lrge, nd the counterpoisecorrected PW91 results differ significntly from the corresponding B3LYP nd MP2 results. Adding dditionl ugmented functions to the cc-pv(t+d)z bsis set decreses the BSSE by n order of mgnitude for the DFT clcultions, nd by bout hlf for the MP2 clcultions. The ug-cc-pv(t+d)z energies, both with nd without counterpoise corrections, obtined using different methods re lso slightly more comptible thn the cc-pv(t+d)z energies. The ugmented bsis functions clerly seem to improve the description of the intermoleculr interction. However, it should be noted tht the ug-cc-pv(t+d)z potentil energy surfces, especilly the counterpoise-corrected ones, re very flt ner the energy minimum. This resulted in long nd computtionlly expensive geometry optimiztion runs. The choice of method nd bsis set, s well s the counterpoise correction, hs reltively smll effect on the vibrtionl zero-point energy (ZPE), entropy nd lso the therml vibrtionl contributions to the vrious properties (not shown here). The difference between the totl vibrtionl contributions to the rection free energy between the computtionlly chepest (B3LYP/D95++(d,p) nd PW91/D95++(d,p)) nd most expensive (MP2/cc-pV(T+d)Z)) clcultions is less thn 0.2 kcl mol -1 for both species studied. The pproch of clculting the electronic energies t higher level thn the vibrtionl properties (employed for exmple by Beichert nd Schrems) thus seems to be justified. This will be discussed further below. For mmonium hydrogen sulfte t the PW91/cc-pV(T+d)Z level nd with ll MP2 clcultions, the ZPE chnge increses upon the ppliction of counterpoise corrections. This is probbly indictive of the low ccurcy of the ZPEs computed using the hrmonic pproximtion rther thn ny rel physicl effect. It should be noted tht for mny tmospheric pplictions (such s nucletion studies investigting the role of mmoni in new-prticle formtion 29,30 ), the difference between the energies of the H 2 SO 4 + H 2 O T H 2 SO 4 H 2 O nd H 2 SO 4 + NH 3 T H 2 SO 4 NH 3 rections my be eqully or more relevnt thn the energy vlues themselves. It is therefore fortunte tht, even though the rection energies obtined t different methods vry by severl, the vlues of G(H 2 SO 4 + H 2 O T H 2 SO 4 H 2 O) - G(H 2 SO 4 + NH 3 T H 2 SO 4 NH 3 )or E 0 (H 2 - SO 4 + H 2 O T H 2 SO 4 H 2 O) - E 0 (H 2 SO 4 + NH 3 T H 2 SO 4 NH 3 ) vry considerbly less. The difference in the free energies vries between 4.0 nd 4.9 nd the difference in the electronic energies between 3.8 nd 4.6, for the counterpoise-corrected results obtined using the correltionconsistent bsis sets cc-pv(t+d)z nd (for the electronic energies) ug-cc-pv(t+d)z. Lrger vritions re observed if the smller bsis sets re tken into ccount. Agin, the B3LYP nd MP2 results gree even more closely (to within c. 0.2 kcl mol -1 ), wheres the PW91 method predicts systemticlly bigger differences between the rection energies. Geometricl Prmeters nd QTAIM Anlysis. The electronic chrge densities generted by the b initio nd DFT clcultions were nlyzed with the AIMPAC 31 nd AIM 2000 progrms, 32 using the quntum theory of toms in molecules 33 (QTAIM) to determine the bond criticl points nd bond pths. The AIM2000 pckge ws used to identify the criticl points (CP) of the electron densities F(x,y,z) in ech cluster studied, nd to trce the bond pths present. The AIMPAC progrm ws then used to clculte the vlues of vrious prmeters t the criticl points. As discussed, e.g., in ref 34, the existence of (3, -1) 1 CP (lso known s bond criticl point, BCP) between two toms is necessry nd sufficient criterion for the existence of chemicl bond between them. The line of mximum density pssing through the BCP nd linking the nuclei of the two toms is then clled bond pth. Similrly, (3, +1). CPs correspond to ring structures nd (3, +3) CPs to cge structures. [Criticl points re points where the grdient of function is zero. The type of criticl point is indicted by the nottion (r, s), where the rnk (r) is the number of nonzero eigenvlues of the second derivtive mtrix nd the signture (s) is the sum of the signs of the eigenvlues. Thus, (3, -1) criticl point hs three nonzero eigenvlues, two of which re negtive.] The vlues of the electron density F nd its Lplcin 2 F, s well s the electronic kinetic, potentil nd totl energies (G, V nd E, respectively; see ref 34 for definitions) t the BCP cn be used to chrcterize the nture of the bonding interction. For exmple, closed-shell interctions correspond to positive vlues of 2 F BCP nd E BCP, wheres covlent bonds correspond to negtive vlues. (Vlues of vrious prmeters t the bond criticl point re denoted with the subscript BCP.) Also, the strength of the bonding interction correltes with F BCP, though the functionl form of the correltion is not generlly known. For set of H-bonded complexes, Koch nd Popelier 35 found the correltion between the H-bond energy nd F BCP to be liner s long s the cceptor tom remined unchnged. In chrcterizing the H-bonds of our cluster structures, we hve dopted the clssifiction used by Rozs et l: 36 H-bond is defined s wek if 2 F BCP > 0 nd E BCP > 0, medium-strength if 2 F BCP > 0 but E BCP < 0, nd strong if 2 F BCP < 0 nd E BCP < 0. Both the H 2 SO 4 H 2 O nd H 2 SO 4 NH 3 clusters contin two possible hydrogen bonds: SOH X bond nd XH OdS bond (where X ) O or N). The structures of the H 2 SO 4 H 2 O nd H 2 SO 4 NH 3 clusters re shown schemticlly in Figures 1 nd 2, long with the loctions of the BCPs corresponding to hydrogen bonds. (Figures 1 nd 2 were creted using the MOLEKEL 37,38 progrm.) The hydrogen bond lengths for the H 2 SO 4 H 2 O nd H 2 SO 4 NH 3 clusters re shown in Tbles 5 nd 6 for both uncorrected nd counterpoise-corrected optimum geometries. The ssocited vlues for the electron density, its Lplcin nd the totl energy density t the bond criticl points,

5 7182 J. Phys. Chem. A, Vol. 110, No. 22, 2006 Kurtén et l. Figure 1. Structure of the H 2SO 4 H 2O cluster. The prmeter r 1 is the distnce between the SOH hydrogen nd wter oxygen toms, nd r 2 is the distnce between the wter hydrogen nd SdO oxygen. The bond criticl points corresponding to the two hydrogen bonds (lbeled BCP 1 nd BCP 2) re shown schemticlly. Note tht the ctul positions of the toms nd BCPs vry depending on the method used, s shown in Tble 5. Figure 2. Structure of the H 2SO 4 NH 3 cluster. The prmeter r 1 is the distnce between the SOH hydrogen nd mmoni nitrogen toms, nd r 2 is the distnce between the mmoni hydrogen nd SdO oxygen. The bond criticl points corresponding to the two hydrogen bonds (lbeled BCP 1 nd BCP 2) re shown schemticlly. Note tht the ctul positions of the toms nd BCPs vry depending on the method used, s shown in Tble 6, nd tht BCP 2 my be missing for some bsis set combintions. computed using the AIMPAC progrm, re lso shown. As the counterpoise correction is only pplied to the energy nd geometry nd does not yield BSSE-free wve function, no counterpoise-corrected AIM prmeters re given. (The wve functions cn, of course, be clculted lso t the CP optimum geometry, but they will show little difference to the uncorrected ones, s the wve function is still contminted by bsis set superposition.) We hve lso clculted proton trnsfer rtio, defined s the rtio of the electron density of the SOH X (where X ) O or N) H-bond BCP to tht of the SO-H BCP, F BCP (H X)/F BCP (SO-H). The lrger the trnsfer rtio, the more evenly shred the proton is between the donor nd cceptor groups. For trnsferred proton the rtio should be lrger thn one. H 2 SO 4 H 2 O Clusters. For H 2 SO 4 H 2 O, ring structure with ring criticl point (not shown in Figure 1) nd two hydrogen bonds (the corresponding BCPs re indicted in Figure 1) ws found with ll bsis set combintions used in this study. The SOH O bond ws lwys of medium strength, with electron density vlues rnging from to u. The OH OdS bond ws lwys wek, with electron density vlues between nd u. It cn be seen by compring Tbles 3 nd 5 tht the higher stbilities (lower rection free energies) given by the PW91 clcultions re ssocited with shorter hydrogen bonds, higher chrge densities t the bond criticl points corresponding to the hydrogen bonds nd greter degrees of proton trnsfer. Similrly, the lower stbilities given by clcultions using the G(2d,2p) bsis set correspond to longer hydrogen bond pths, lower chrge densities nd smller degree of proton trnsfer. The B3LYP nd MP2 bond lengths re in resonble greement with ech other for the lrger bsis sets, wheres the MP2 chrge densities re consistently lower thn the corresponding DFT ones. This is quite generl phenomenon, s demonstrted, e.g., by the clcultions in ref 39. Appliction of the counterpoise correction increses the length of the stronger H-bond by Å for the DFT clcultions nd Å for MP2 clcultions. As expected, the smllest increses were observed when the ug-cc-pv(t+d)z bsis set ws used, for which the BSSE is reltively smll. For the DFT/ug-cc-pV(T+d)Z computtions, in which the BSSE is only bout 0.2, the counterpoise correction to the H-bond lengths ws insignificnt. The increse in length of the weker H-bond vried more strongly, but the corrections t the MP2 level were gin lrger thn t the DFT levels. In contrst to the electronic energies, neither the ppliction of the counterpoise correction nor the use of dditionl ugmented bsis functions led to systemtic decrese in the differences between the bond lengths obtined with different methods. All the intermoleculr SOH O distnces clculted with the lrger cc-pv(t+d)z nd ug-cc-pv(t+d)z bsis sets re in resonble greement with the experimentl vlue of ( Å reported in the rottionl spectroscopic study of Ficcio et l. 40 The B3LYP nd MP2 vlues re slightly lrger nd the PW91 vlues slightly smller thn the experimentl vlue, but the differences re less thn 0.05 Å. However, the intermoleculr OH OdS distnces clculted with the B3LYP nd MP2 methods re considerbly ( Å for the uncorrected nd for the counterpoise-corrected vlues) longer thn the experimentl vlue of 2.05 ( 0.01 Å, wheres the PW91 vlues re in good greement with it. This would seem to support the ssertion by l Nthseh et l. tht the B3LYP functionl predicts too long bond lengths nd too low binding energies for wekly bound complexes such s sulfuric cid monohydrte. H 2 SO 4 NH 3 Clusters. For the H 2 SO 4 NH 3 cluster, the sitution is more complicted. In the PW91/D95++(d,p), PW91/ G(2d,2p) nd PW91/cc-pV(T+d)Z clcultions, ring structure with two H-bonds ws found. Like in the H 2 SO 4 H 2 O cluster, the intermoleculr NH OdS interction ws lwys wek, nd the SOH N interction of medium strength, except for the PW91/cc-pV(T+d)Z cse in which it ws strong. However, in the PW91/ug-cc-pV(T+d)Z clcultions s well s in ll the B3LYP nd MP2 clcultions, the hydrogen bonding pttern ws different. In the B3LYP/ G(2d,2p) nd B3LYP/ug-cc-pV(T+d)Z clcultions, the second H-bond ws missing. In the other B3LYP clcultions, the PW91/ug-ccpV(T+d)Z clcultion nd lso in ll MP2 runs, ring structure contining highly curved bond pth between the SdO oxygen nd the nitrogen tom of mmoni ws seen insted. See Figure 3 for representtive illustrtion of the three different bonding ptterns. This curved bonding interction either my be n rtifct of the Gussin 03 or AIM 2000 progrms or my reflect genuine bonding interction. A comprison of, for exmple, the B3LYP/D95++(d,p) nd B3LYP/ G(2d,2p) results seems to indicte tht this bonding interction certinly seems to be stbilizing: the (counterpoise-corrected) rection free energy is over 1.5 higher t the D95++(d,p) level, where the interction is found, thn t the G(2d,2p) level, where it is bsent. A comprison of Tbles 4 nd 6 lso seems to indicte tht there is correltion between BSSE nd the nomlous-looking bonding. At the B3LYP level, the curved

6 Gs-Phse H 2 SO 4 H 2 O nd Ammonium Hydrogen Sulfte J. Phys. Chem. A, Vol. 110, No. 22, TABLE 5: Hydrogen Bond Lengths nd Topologicl Prmeters t the Corresponding Bond Criticl Points for the H 2 SO 4 H 2 O Cluster, Using Different Methods nd Bsis Sets, with nd without Counterpoise Corrections to the Energy nd Geometry bsis set r 1/ Å F 1/ 100 u 2 F 1/ 100 u E 1/ u r 2/ Å F 2/ 100 u 2 F 2/ 100 u bond pth is bsent in the computtions with low BSSE vlues, nd present in those with high BSSE vlues. This is not very surprising; fter ll, the rtificil orbitl overlp cused by bsis set superposition might well be mnifested s n nomlouslooking bond pth. However, the presence of the nomlouslooking bond pth in the PW91/ug-cc-pV(T+d)Z nd MP2/ ug-cc-pv(t+d)z clcultions, both of which hve low BSSE vlues, does not support this hypothesis. Accordingly, plusible E 2/ u H-trnfer rtio B3LYP/ b b D95++(d,p) c c B3LYP/ b b G(2d,2p) c c B3LYP/ b b cc-pv(t+d)z c c B3LYP/ b b ug-cc-pv(t+d)z c c PW91/ b b D95++(d,p) c c PW91/ b b G(2d,2p) c c PW91/ b b cc-pv(t+d)z c c PW91/ b b ug-cc-pv(t+d)z c b MP2/ b b D95++(d,p) c c MP2/ b b G(2d,2p) c c MP2/ b b cc-pv(t+d)z c c MP2/ b b ug-pv(t+d)z c c experimentl d ( ( 0.01 r n is the H-bond length, F n is the electron density, 2 F n is its Lplcin nd E n is the totl electronic energy computed t the bond criticl point corresponding to hydrogen bond n. Index 1 corresponds to the SOH O H-bond, nd index 2 to the OH SdO H-bond. The H trnsfer rtio corresponds to the S-OH O H-bond nd is defined bove. For detils, see the text, Figure 1 nd refs 34 nd 35. b Without counterpoise corrections. c With counterpoise corrections d Reference 40. TABLE 6: Intermoleculr Distnces Relted to Hydrogen Bonding nd Topologicl Prmeters t the Corresponding Bond Criticl Points for the H 2 SO 4 NH 3 Cluster, Using Different Methods nd Bsis Sets, with nd without Counterpoise Corrections to the Energy nd Geometry bsis set r 1/ Å F 1/ 100 u 2 F 1/ 100 u E 1/ u r 2/ Å F 2/ 100 u 2 F 2/ 100 u E 2/ u H-trnfer rtio B3LYP/ b b d d d D95++(d,p) c c B3LYP/ b b G(2d,2p) c c B3LYP/ b b d d d cc-pv(t+d)z c c B3LYP/ b b ug-cc-pv(t+d)z b c PW91/ b b e e e D95++(d,p) c c PW91/ b b e e e G(2d,2p) c c PW91/ b b e e e cc-pv(t+d)z c c PW91/ b b d d d ug-cc-pv(t+d)z c c MP2/ b b d d d D95++(d,p) c c MP2/ b b d d d G(2d,2p) c c MP2/ b b d d d cc-pv(t+d)z c c MP2/ b b d d d ug-cc-pv(t+d)z c c r n is the H-bond length, F n is the electron density, 2 F n is its Lplcin nd E n is the totl electronic energy computed t the bond criticl point corresponding to hydrogen bond n. Index 1 corresponds to the S-OH N interction, nd index 2 to the other intermoleculr bond criticl point, if present. The H trnsfer rtio corresponds to the S-OH N H-bond nd is defined bove. For detils, see the text, Figure 2 nd refs 34 nd 35. b Without counterpoise corrections. c With counterpoise corrections. d Corresponds to N OdS bond pth. e Corresponds to N-H OdS bond pth.

7 7184 J. Phys. Chem. A, Vol. 110, No. 22, 2006 Kurtén et l. TABLE 7: Thermochemicl Prmeters ( ) for the Hydrtion of Sulfuric Acid with One nd Two Wter Molecules, t 298 K nd 1 tm, Using the Hrmonic Approximtion nd Anhrmonic Corrections species ZPE H S G H 2SO 4 H 2O b b b b c c c c H 2SO 4 (H 2O) b b b b c c c c Note tht tighter convergence criteri were used thn for the vlues given in Tbles 3 nd 4. The definitions of the prmeters re the sme s in Tbles 1-4. b Hrmonic frequencies. c Anhrmonic frequencies. Figure 3. Three different bonding ptterns observed for the H 2SO 4 NH 3 cluster. At the B3LYP/ G(2d,2p) level, only one hydrogen bond is found. At the PW91/ G(2d,2p) level, both SOH N nd NH OdS bond re found. At the MP2/ G(2d,2p), highly curved N OdS bond pth is observed. This figure ws creted using the AIM2000 progrm. 32 explntion is tht there is n interction between the oxygen tom nd the N-H ntibonding moleculr orbitl. Unfortuntely, further nlysis of the bonding interctions ws hindered by the fct tht the Gussin 03 progrm is incpble of writing wve function files contining g-type functions or higher. Thus, for exmple cc-pv(q+d)z or ug-cc-pv(q+d) wve functions could not be generted for the H 2 SO 4 NH 3 cluster. In the future, the connection between bsis set superposition nd this kind of curved bond pth could be tested by generting inherently BSSE-free wve functions using the chemicl Hmiltonin pproch of Myer et l. 41 However, this is beyond the scope of this study. Proton Trnsfer Rtios. The hydrogen trnsfer rtios clculted here help to explin some differences in the structurl trends observed in erlier studies of H 2 SO 4 (H 2 O) n nd H 2 SO 4 (NH 3 ) (H 2 O) n clusters. For exmple, Bndy nd Inni 4 predicted tht the most stble configurtions of H 2 SO 4 (H 2 O) n re neutrl for n ) 0-7, wheres Re et l. 5 computed the ionic nd neutrl configurtions to be eqully stble for n ) 4 nd the ionic configurtions more stble for n g 5. Similrly, Inni nd Bndy predicted tht the switch from neutrl to ionic structures occurs t n ) 4 for H 2 SO 4 (NH 3 ) (H 2 O) n clusters, wheres our recent B3LYP/D95++(d,p) study 30 indictes tht ionic structures re more stble thn neutrl ones lredy for n ) 1. Correspondingly, the proton trnsfer rtio clculted for both H 2 SO 4 H 2 O nd H 2 SO 4 NH 3 with the B3LYP/ G(2d,2p) method used by Bndy nd Inni is smller thn tht clculted with the B3LYP/D95++(d,p) method used by Re et l. nd in our study. On the other hnd, Lrson et l. 8 predicted proton trnsfer for the H 2 SO 4 (NH 3 ) (H 2 O) cluster using the MP2/ G(d,p) method, even though the MP2 trnsfer rtios clculted here re consistently lower thn the corresponding B3LYP vlues. It should, however, be noted tht Lrson et l. clculted only binding energies, not free energies, nd tht they predicted proton trnsfer for the H 2 SO 4 (NH 3 ) (H 2 O) cluster lso using the B3LYP/ G(d,p) method. Compring the results of Bndy nd Inni with those of Lrson et l., one cn see tht smll chnge in the bsis set (in this cse, dding p-type polriztion function to ech hydrogen tom nd d-type polriztion function to ll other toms) cn hve huge effects on the structurl results. This might lso indicte tht the lrge differences observed between the G(2d,2p) bsis set nd the cc-pv(t+d)z or ug-cc-pv(t+d)z sets, ll of triple split-vlence qulity, is relted primrily to the presence of f-type polriztion functions in the correltion-consistent sets. Effect of Anhrmonic Corrections. We hve checked the vlidity of the hrmonic pproximtion by clculting the nhrmonic frequencies nd vibrtionl contributions to the free energy of hydrtion of sulfuric cid mono- nd dihydrte t the B3LYP/cc-pV(T+d)Z level using the nhrmonic corrections implemented in the Gussin 03 progrm. The relevnt thermochemicl prmeters obtined using hrmonic nd nhrmonic frequencies re given in Tble 7. On the bsis of ref 14, tighter convergence criteri were used for this computtion, s mentioned in the section Computtionl Detils. This ffected the optimized bond lengths by less thn Å. A comprison of Tbles 7 nd 3 shows tht for exmple the free energy of hydrtion of sulfuric cid chnges by bout 0.06 when tighter convergence criteri nd the ultrfine integrtion grid re used insted of the defult settings. However, given the mgnitudes of the other error sources reveled by this study, it is clerly not cost-effective to routinely use convergence nd grid settings much tighter thn the defult ones. The nhrmonic frequencies computed using the Gussin 03 progrm re in resonble greement with those computed recently by Miller et l. 42 using the GAMESS pckge nd the CC-VSCF method t the MP2/TZP level. The sum of the differences between the nine experimentlly observed vibrtionl wvenumbers 43 nd the corresponding computed wvenumbers is 219 cm -1 for our results nd 104 cm -1 for those of Miller et l. This is probbly due to the higher ccurcy of the CC-VSCF nd MP2 methods. However, for the purposes of thermochemicl nlysis, lso our nhrmonic computtions re significnt improvement over the hrmonic pproximtion. It cn be seen from Tble 7 tht the effect of nhrmonic corrections is fr from insignificnt. The bsolute vlues of the rection free energies, for exmple, increse by c. 15% when nhrmonicity is ccounted for. However, compred to the errors

8 Gs-Phse H 2 SO 4 H 2 O nd Ammonium Hydrogen Sulfte J. Phys. Chem. A, Vol. 110, No. 22, TABLE 8: Rtio of Anhrmonic to Hrmonic Zero-Point Energies for Four Different Moleculr Structures species H 2SO 4 (H 2O) 2 H 2SO 4 H 2O H 2SO 4 H 2O ZPE nhrm/zpe hrm cused by bsis-set superposition nd to the differences in results obtined t different bsis set combintions, the errors cused by the hrmonic pproximtion re reltively smll. One possibility is to ccount for nhrmonicity through the use of scling fctors, which cn be pplied directly to either the vibrtionl frequencies or the zero-point energies nd therml vibrtionl contributions. When the use of scling fctors is considered, it should be noted tht the nhrmonicity (defined, e.g., s the rtio of nhrmonic nd hrmonic vibrtionl frequencies or zero-point energies) is different for clusters nd for isolted molecules. Intrmoleculr bonds re usully rigid, nd thus close to hrmonic, wheres intermoleculr bonds my disply significnt nhrmonicity. This is illustrted by Tble 8, in which the rtio of nhrmonic to hrmonic zero-point energies of the four species studied re compred. Thus, using the sme scling fctor both for isolted molecules nd for cluster structures will inevitbly led to errors, which my be of the sme order of mgnitude s the originl error cused by the hrmonic pproximtion. It might be possible to ttin greter ccurcy by using different scling fctors for frequencies corresponding to intr- nd intermoleculr vibrtions, but such n pproch is beyond the scope of this study. It should be noted tht the lrge nhrmonicity of the intermoleculr bonds leds to significnt devition of the vibrtionlly verged geometry from the optimized minimum energy geometry. For the H 2 SO 4 H 2 O cluster, the vibrtionlly verged nhrmonic zero-point SOH O H-bond length ws greter by Å nd the OH OdS H-bond greter by Å thn the minimum energy bond length. This mkes the lrge difference between the clculted minimum-energy bond lengths in Tble 5 nd the experimentl vlues by Ficco et l. 40 even more surprising, s the experimentl vlues correspond to the vibrtionlly verged structure nd could therefore be expected to be lrger, not smller, thn the clculted ones. Higher-Level Computtions. As mentioned bove, the ZPE nd therml contributions to the rection free energies vry reltively little between different methods, wheres the differences in electronic energies re significnt. Thus, it might be fesible to compute the minimum geometry nd thermochemicl prmeters using lower level of theory nd the electronic energy t higher level. Severl such schemes exist, e.g., the G1 nd G2 methods. 44 However, these do not ccount explicitly for bsis set superposition (which we hve found to be significnt even for reltively lrge bsis sets), nd often employ configurtion interction methods, which re not size consistent. As future tmospheric b initio studies will certinly include compring properties of clusters of vrying size, this is clerly undesirble. Furthermore, most such schemes use constnt scling fctors to ccount for nhrmonicity, which is not very good pproch for cluster structures, s discussed in the previous section. We hve chosen to clculte the single-point binding energy of H 2 SO 4 H 2 O using the CCSD(T) method, which is both sizeconsistent nd known to give very ccurte results. 45 Becuse it is counterproductive to employ high-level correlted theories with less thn triple-vlence bsis sets, 45 we hve used the ccpv(t+d)z bsis set for these clcultions. Due to computtionl considertions (nd lso due to the fct tht there re no direct experimentl results to compre with) the H 2 SO 4 NH 3 cluster ws not studied t the CCSD(T) level. TABLE 9: Binding Energies ( )ofh 2 SO 4 H 2 O Clculted t the CCSD(T) - Level, Including Counterpoise Corrections (CP), with Geometries Optimized t Lower Levels of Theory method of geometry E 0,CCSD(T) - optimiztion E 0,CCSD(T) E 0,X CP correction G X + E 0,CCSD(T) - E 0,X B3LYP/cc-pV(T+d)Z PW91/cc-pV(T+d)Z MP2/cc-pV(T+d)Z B3LYP/cc-pV(D+d)Z The CCSD(T) clcultions use the sme bsis set s the corresponding lower-level clcultions. - E 0,X is the (counterpoise-corrected) binding energy clculted t the corresponding lower level of theory. The vlue of G X + E 0,CCSD(T) - E 0,X corresponds to Gibbs free energy obtined by clculting the electronic energy t the CCSD(T) level nd the zero-point nd therml contributions t the lower level of theory. As the optimum geometries obtined using different methods re somewht different (see Tbles 5 nd 6), we hve run CCSD(T) clcultion t the (counterpoise-corrected) optimum geometry corresponding to ech of the three lower-level clcultions: B3LYP/cc-pV(T+d)Z, PW91/cc-pV(T+d)Z nd MP2/cc-pV(T+d)Z. The energies of the free sulfuric cid nd wter molecules were lso clculted t the optimum geometries corresponding to these methods, s relxing the rectnts but not the product t the CCSD(T) level would led to n rtificilly low binding energy. (A full geometry optimiztion of H 2 SO 4 H 2 O t the CCSD(T) level, on the other hnd, would be prohibitively expensive.) Counterpoise corrections to the electronic energy were included in the CCSD(T) clcultions to remove the bsis set superposition error. For comprison, we hve lso clculted the CCSD(T) binding energy using the ccpv(d+d)z bsis set, t the B3LYP/cc-pV(D+d)Z geometry. The results re presented in Tble 9. It cn be seen from Tble 9 tht the CCSD(T)/cc-pV(T+d)Z electronic energies clculted t the B3LYP nd MP2 minimum geometries differ little from the corresponding B3LYP nd MP2 energies, wheres the CCSD(T) binding energy clculted t the PW91 geometry is significntly lower thn the originl PW91 vlue. This indictes (though unfortuntely does not prove) tht the PW91/cc-pV(T+d)Z potentil energy surfce is overbinding compred to the CCSD(T)/cc-pV(T+d)Z surfce, wheres the B3LYP/cc-pV(T+d)Z nd MP2/cc-pV(T+d)Z re quite close to it. Becuse the CCSD(T) results re lmost certinly more ccurte thn ny of the other methods, it would thus be resonble to conclude tht the B3LYP nd MP2 rection energies nd geometries should be considered more relible thn the PW91 ones. (This conclusion is lso supported by the fct tht the B3LYP nd MP2 results re, when lrger bsis sets re used, very similr to ech other.) However, the fct remins tht both the PW91 rection energies nd the corresponding H-bond lengths re closer to the experimentl vlues. A full resolution of the problem would probbly require geometry optimiztion t the CCSD(T)/ug-cc-pV(T+d)Z level or higher, which is computtionlly unfesible t the moment. We hve lso ttempted to clculte the bsis-set limits of the binding energies of H 2 SO 4 H 2 O nd H 2 SO 4 NH 3 t the B3LYP level. This ws done by clculting the counterpoisecorrected binding energies using the cc-pv(d+d)z, cc-pv- (T+d)Z, cc-pv(q+d)z, ug-cc-pv(d+d)z, ug-cc-pv(t+d)z nd ug-cc-pv(q+d)z bsis sets. (Due to computtionl limittions, the ug-cc-pv(q+d)z energy ws computed t the ugcc-pv(t+d)z geometry.) The bsis-set limits could then be estimted using the extrpoltion formuls 46,47

9 7186 J. Phys. Chem. A, Vol. 110, No. 22, 2006 Kurtén et l. nd E(X) ) E( ) + A 1 e -BX (1) TABLE 10: Bsis-Set Limit Binding Energies of H 2 SO 4 H 2 O nd H 2 SO 4 NH 3 Clculted t the B3LYP Level nd Bsis-Set Limit Binding Energies of H 2 SO 4 H 2 Otthe CCSD(T) Level ( ) E(X) ) E( ) + A 2 X -3 (2) H 2SO 4 H 2O H 2SO 4 NH 3 bsis set E 0 correction E 0 correction CP CP B3LYP/cc-pV(D+d)Z B3LYP/cc-pV(T+d)Z B3LYP/cc-pV(Q+d)Z B3LYP/cc-pV( +d)z, eq B3LYP/cc-pV( +d)z, eq B3LYP/cc-pV( +d)z, eq 2 b B3LYP/ug-cc-pV(D+d)Z B3LYP/ug-cc-pV(T+d)Z B3LYP/ug-cc-pV(Q+d)Z c B3LYP/ug-cc-pV( +d)z, eq B3LYP/ug-cc-pV( +d)z, eq B3LYP/ug-cc-pV( +d)z, eq 2 b CCSD(T)/cc-pV( +d)z, eq 2, CP d CCSD(T)/cc-pV( +d)z, eq 2, no CP e Fitted using ll dt points. b Fitted using the X ) 3 nd X ) 4 dt points only. c Computed t the ug-cc-pv(t+d)z geometry. d Computed t the corresponding B3LYP geometries, using the counterpoise-corrected X ) 2 nd X ) 3 dt given in Tble 9. e Computed t the corresponding B3LYP geometries, using the X ) 2 nd X ) 3 dt given in Tble 9, without counterpoise corrections. where E( ) is the bsis-set limit energy, A 1, A 2 nd B re constnts, X ) 2 for the cc-pv(d+d) nd ug-cc-pv(d+d)z sets, X ) 3 for the cc-pv(t+d) nd ug-cc-pv(t+d)z sets nd X ) 4 for the cc-pv(q+d) nd ug-cc-pv(q+d)z sets. For the totl electronic energy, eq 1 holds for Hrtree-Fock methods nd eq 2 for correlted b initio methods. 45 Though not rigorously proven, both formuls hve been successfully employed for mny other moleculr properties, nd lso for DFT methods. (See for exmple ref 27, where both formuls re pplied to B3LYP results.) Equtions 1 nd 2 were fitted seprtely to the dt corresponding to the ugmented nd nonugmented bsis sets. Though eqs 1 nd 2 re usully fitted to uncorrected energies, we hve followed the recommendtions in ref 48 nd used the counterpoise-corrected energies for the fitting, though this led to nonmonotonic behvior for the ugmented bsis set energies. (However, becuse the differences between the ug-cc-pv(t+d) nd ug-cc-pv(q+d) energies were very smll, this hd negligible effect on the results.) Eqution 2 ws lso fitted using only the X ) 3 nd X ) 4 dt points, on the bsis of recommendtions from ref 49. Furthermore, we hve ttempted to estimte the mgnitude of bsisset effects beyond the triple-ζ level for the CCSD(T) method by clculting the CCSD(T)/cc-pV( +d)z binding energy using eq 2 nd the two dt points corresponding to the B3LYP geometries given in Tble 9. (This vlue must be considered s n order-of-mgnitude estimte only, s the very lrge BSSE error indictes tht the CCSD(T)/cc-pV(D+d)Z results re quite unrelible.) For comprison, the CCSD(T)/cc-pV( +d)z energy hs been clculted both with nd without counterpoise corrections. The results re presented in Tble 10. The counterpoise-corrected qudruple-ζ energies re very close to the B3LYP bsis set limit, irrespective of the extrpoltion formul used. However, it should be noted tht without counterpoise corrections, the difference between the cc-pv- (Q+d)Z nd cc-pv( +d)z energies is round 1. The ugmented bsis sets, on the other hnd, converge much fster, nd the effect of the counterpoise correction t the ugcc-pv(q+d)z level is smller thn the differences between the different extrpoltion procedures. Even the counterpoisecorrected ug-cc-pv(d+d)z energies re within 0.5 of the bsis-set limit energies. This indictes tht the ug-ccpv(d+d)z bsis set might represent resonble compromise between computtionl requirements nd ccurcy. It cn be seen from Tble 10 tht the bsis-set limit correction to the B3LYP/cc-pV(T+d)Z electronic energy chnge of the H 2 SO 4 + H 2 O T H 2 SO 4 H 2 O rection is c to The correction thus cts to further increse the difference between computtionl nd experimentl results. This could be tken s n indiction tht the counterpoise correction overcompenstes for the bsis-set superposition error nd leds to rtificilly low binding energies. The mgnitude of this possible overcompenstion cn be estimted by compring the bsis-set limit energies clculted from dt with nd without counterpoise corrections. If no overcompenstion is present, the vlues should be identicl. For the ugmented bsis sets, the difference between the two results (using eq 2 nd the full dt set) is less thn 0.1 for both clusters studied, indicting (though not proving) tht the degree of overcompenstion t the B3LYP level is smll. For the nonugmented bsis sets, the counterpoise-uncorrected bsis-set limit binding energies re round 0.4 higher thn the counterpoisecorrected ones, indicting the possible presence of some degree of overcompenstion, though not enough to explin the difference between experimentl nd clculted B3LYP energies. (It should be noted tht the differences my lso be cused by the incompleteness of our dt set, which includes only double-, triple-, nd qudruple-ζ bsis sets.) Compring the CCSD(T)/cc-pV( +d)z vlues given in Tble 10 to the corresponding cc-pv(d+d)z nd cc-pv(t+d)z vlues in Tble 9, one cn see tht the CCSD(T)/cc-pV(T+d)Z binding energy is significntly lower thn the estimted bsis-set limit: c. 0.6 for the counterpoise-corrected nd 0.8 kcl mol -1 for the uncorrected vlues. Also, the difference between the bsis-set limit binding energies clculted from counterpoisecorrected nd uncorrected energies is over 1, indicting tht overcompenstion my be n importnt fctor t the CCSD(T) level. The fitted CCSD(T)/cc-pV( +d)z vlues, though not quntittively ccurte due to the unrelibility of the CCSD(T)/ccpV(D+d)Z dt, strongly indicte tht the B3LYP binding energies re too low by bout 1-2 (depending on whether the counterpoise-corrected or uncorrected CCSD(T) dt is used for comprison). This is lmost certinly the min reson for the lrge differences between erlier studies nd experimentl vlues. From Tble 7, we cn estimte the mgnitude of nhrmonic corrections to the hydrtion free energy to be round Combining the free energy of hydrtion clculted t the B3LYP/cc-pV(T+d)Z level with this nhrmonic correction nd the difference between the B3LYP/cc-pV( +d)z nd CCSD(T)/cc-pV( +d)z binding energies, we obtin rnge of -2.2 to -2.9 s best-guess vlue of the free energy of hydrtion t 298 K. (The higher vlue corresponds to counterpoise-corrected nd the lower vlue to uncorrected bsis-set limit energies.) This is in modertely good greement with the upper limit of the experimentl rnge of -3.6 ( 1. Further clcultions t the CCSD(T)/ug-cc-pV(T+d)Z or CCSD(T)/cc-pV(Q+d)Z level would be required to better determine the mgnitude of