On the performance of correlation consistent basis sets for the

On the performance of total atomization

of correlation consistent basis sets for the calculation energies, geometries, and harmonic frequencies

Jan M. L. Martin Departenient SBG, Limburgs Universitair Centrum. Universitaire Campus, B-3590 Diepenbeek, Belgium and Department of Chemistry, University of Antwerp (UIA), Universiteitsplein I, B-2610 Wilrijk, Belgium

(Received20 December 1993; accepted14 February 1994) The total atomizationenergies(ED, values),geometries,and harmonic frequenciesfor a number of experimentally well-describedmolecules have been calculated at the CCSD(T) (coupled cluster) level using Dunning’s correlation-consistentcc-pVDZ( [ 3 sip 1d]) , cc-pVTZ( [ 4s3p2d lfl) , and cc-pVQZ( [ 5 s4p 3 d2f 1g]) basis sets.Additivity correction are proposedfor binding energiesand geometries.Using a three-termadditive correction of the form proposedby Martin [J. Chem. Phys. 97, 5012 (1992)] mean absoluteerrors in XD, are 0.46 kcal/mol for the cc-pVQZ, 0.93 for the cc-pVTZ, and 2.59 for the c-pVDZ basis sets.The latter figure implies that, althoughunsuitablefor quantitatively accuratework, three-termcorrectedCCSD(T)/cc-pVDZ binding energiescan still be used for a rough estimate when the cost of larger basis set calculations would be prohibitive. CCSD(T)/cc-pVQZ calculationsreproducebond lengths to 0,001 A for single bonds, and 0.003 A for multiple bonds; remaining error is probably partly due to core-core and core-valence correlation.CCSD(T)/cc-pVTZ calculationsresult in additional overestimatesof 0.001 A for single, 0.003 A for double, and 0.004 A for triple bonds.CCSD(T)/cc-pVDZ calculationsresult in further overestimatesof 0.01 A for single bonds, and 0.02 A for multiple bonds. CCSD(T)/cc-pVDZ harmonic frequenciesare in surprisingly good agreementwith experiment,except for pathological caseslike the umbrella mode in NH3 . Both CCSD(T)/cc-pVTZ and CCSD(T)/cc-pVQZ harmonic frequencies generally agree wifh experiment to 10 cm-’ or better; performanceof cc-pVQZ is somewhatsuperior on multiple bonds or the umbrella mode in NH,. Again, a sourceof remaining error appearsto be core correlation. The use of MP2/6-31G* referencegeometriesin the ED, calculation can result in fairly substantialerrors in the uncorrectedXD, values for systemswith cumulatedmultiple bonds. These errors however appearto be largely absorbedby the three-term correction.Use of CCSD(T)/cc-pVDZ referencegeometriesappearsto have no detrimentaleffect on computed ZD, values and is recommendedfor caseswhere only single-point calculations in the cc-pVTZ basis set are possible.

INTRODUCTION

The quality of predicted molecular properties in an calculation is largely defined by two factors: electron correlation and the size of the one-particlebasis set.“2 To the former problem, a satisfactory solution-at least for systems without excessive multireference effects-has emerged in recent years in the shape of coupled cluster the0ry.sThis leavesthe one-particlebasis set as the principal accuracy-determiningfactor, Since computer time in correlated calculations increasesroughly as the fourth power of the number of basis functions, a compromise has to be reachedbetween basis set dimension and the desired accuracy. For very accuratecalculations,two main families of basis sets-both based on general contractions’-are gaining acceptance.The first are the atomic natural orbital (ANO) basis sets of Almlof and Taylor,5*6which are generatedby picking the natural orbitals with the highest occupationnumbers from an atomic configurationinteraction calculation in a very large primitive set. AN0 basis sets are now available for first- and second-rowatomsF6 as well as for transition metals;7a variant called “density-matrix averagedANOs” is available for first-row’ and second-row’atoms. ab initio

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The second such family are the correlation consistent basis sets of Dunning,” which are based on energy optimization of relatively small primitive sets. They are available for first-row”‘” and second-row12atoms. In a previous paper,13the author has comparedthe performanceof AN0 and correlation consistentbasis sets for the calculations of total atomization energies(ZD, values). It was found that AN0 and correlation-consistentbasis sets of equal contracted size perform essentiallyequally well, despitethe fact that the smaller primitive set of the correlation-consistent basis setsresults in significant computer time savings due to much shorter integral evaluationtimes. In Ref. 13, a three-term additivity correction was proposed of the form ~&,lT= a,n,+b,n,+cptilnptir

(1)

where n,, n=, and npair representthe number of cr bonds, r bonds, and electron pairs, respectively, and a, b, and c are constantsspecific for the basis set, electron correlation level, and theoretical level of the reference geometry. It was found13that, using this correction at the CCSD(T) level, a [ 5s4p3d2flg] AN0 or correlationconsistentbasis set predicts BD, values with a mean absolute error of 0.50 kcal/ mol or better. 00(11)/8186/8/$6.00

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The presentpaper investigatesthe relative performance of various correlation consistentbasis sets in further depth. First, the performanceof correlation-consistentbasis sets, as a function of size, for geometriesand harmonic frequencies is addressed.Second,the questionariseswhetherany useful information can be obtained from calculations with a correlation-consistentbasis set of only VDZP (valence doublezeta plus polarization)quality. Finally, the questionas to whetherlow-level referencegeometriesdeterminedat low levels of theory (such as the MP2/6-31G* referencegeometries employedin Gl theory14)have a detrimentaleffect on computedXD, values is addressed. METHODS

All calculations were carried out using the MOLECULE/ program system’5”6 running on the Cray Y-MP 8/464 at San Diego SupercomputerCenter (SDSC), and the ACES II packageI running on an IBM RW6000 model 350 at SDSC. The CCSD(T) method” was used throughout;this is the coupledclustersmethodwith all single and double substitutions’g’20supplementedwith a quasiperturbative estimate for the effect of triple excitations.‘* For systems that are dominatedby a single referenceconfiguration, such as all moleculestreatedin the presentpaper,this method is known to yield correlation energies ., near the basis set n-particle limit.21~22 Three of Dunning’scorrelationconsistentbasis sets” are consideredin this paper.The cc-pVDZ (correlation consistent polarized valencedouble zeta) basis set is a [3s2p ld] general contraction of a (9s4p 1d) primitive set. The ccpVTZ (correlation consistent polarized valence triple zeta) one is a [4s3p2dlfl generalcontractionof a (lOs5p2dlfi primitive set. Finally, its cc-pVQZ (correlationconsistentpolarized valence quadruple zeta) counterpart is a [5s4p3d2flg] general contraction of a (12s6p3d2flg) primitive set. (For hydrogen,the contractedsets are [2slp], [3s2pld], and [4s3p2dlfl, respectively, obtained from (4slp), (5~2~ Id), and (6s3p2dlf) primitive sets.) Since all thesebasis sets are only minimal contractionsin the ( 1s) core orbital, the (Is)-like core orbitals were constrainedto be doubly occupied in all coupled cluster calculations; all results quoted in this paper therefore completely neglect core-core and core-valence correlation. Their rigorous inclusion would require much larger primitive sets, including special “hard” (high-exponent)p, d, andf functions. Spherical harmonics were used throughout. SCF and coupled cluster equation were convergedto essentially machine precision. Geometry optimizations were performed by repeated multivariate parabolic interpolation. Step sizes herein were reducedprogressivelyas convergencewas approached. Harmonic frequencies were calculated using doublesided finite differencesin symmetry-adapiedinternal coordinates,using step sizes of 0.01 A or rad. The coordinatetransformations involved were performed with the aid of the BMA? and INTDER programs.24 Finally, atomic -energiesinvolved in the ED, calculations were taken from Ref. 13 for the cc-pVTZ and cc-pVQZ basissets,and computedusing ACES II for the cc-pVDZ basis SWEDEN/TITAN

J.Chem.Phys.,Vol.

TABLE I. CCSD(T) total energies with different basis sets.

H C N 0 F '3% CH4

co co2 Hz

Hz0 HCN HF NH, N2 H&O F2

NNO

cc-pVDz// cc-pVDZ

cc-pVTZJ/ cc-pVDZ

-0.499278 -37.760324 -54.478433 -74.909 890 -99.521541 -77.110865 -40.387627 -113.054976 -188.148256 -1.163 673 -76.241305 -93.189 560 -100.228 156 -56.402802 -109.276 482 -114.219 034 -199.098 408 -184.233 981

-0.499810 -37.780637 -54.514488 -74.973822 -99.620 299 -77.186683 -40.437572 -113.155412 -188.326959 -1.172 124 -76.332082 -93.274584 -100.338349 -56.472714 -109.373 359 -114.333 466 - 199.295 249 -184.406675

cc-pvTzf cc-pvlz -0.499810 -37.780637 -54.514488 -74.973 822 -99.620 299 -77.187648 -40.438099 -113.155579 -188.327218 -1.172331 -76.332217 -93.275 219 -100.338 356 -56.473 197 -109.373 937 -114.333 872 -199.296 112 -184.407 184

cc-pVQz// cc-pVQZ -0.499 950 -37.786405 -54.524589 -74.993411 -99.650 187 -77.209317 -40.450888 -113.187906 -188.384564 -1.173 796 -76.359798 -93.301303 -100.373180 -56.493053 -109.404 391 -114.369 009 - 199.358 906 -184.461 624

set. The set of moleculesselectedis the same as in Ref. 13, namely, H2, N2, F2, CG, CH,, NH,, H,O, HF, HCN, CO,, C2H,, N,O, and H&O. Besidestheir heat of formation, the geometriesand anharmonicforce fields of these molecules are all well known (with some qualifications: vi& infra), enabling us to obtain the anharmonic zero-point energy (ZPE) and there compare directly with an “experimental” ED,.

For some of the speciesdiscussedin the presentpaper (specifically, NH,, N20, CO,, and H,CO), anharmonicforce fields at the CCSD(T)/cc-pVTZ and harmonic frequenciesat the CCSD(T)/cc-pVQZ levels were previously computedby the presentauthor and co-workers.25-27 Total energiesat the respective equilibrium geometries were obtained as a byproduct of these investigations, and are reported for the first time here. RESULTS AND DISCUSSION

Total energiesare given in Table I. A summary of results with different correction formulas is presentedin Table II, together with the coefficients for the three-term correction. Raw, one-term corrected, three-term corrected, and experimental total atomization energiescan be found in Table III, along with Gl l4 and G2” theory results taken from the cited references.(For NNO, which was not included in those papers, the requiredcalculationswere performedusing GAUSSIAN 9z2’on the Cray Y-MP.) Computedgeometriesand harmonic frequenciesare given in Table IV and V, respectively, together with the correspondingexperimental values. First we will focus on the ED, values.At the CCSD(T)/cc-pVQZ level from its optimal geometry,and with the three-termcorrection, agreementwith experimentcan only be describedas excellent. All errors fall below 1 kcal/mol, which brings us well within the goal of “chemical accuracy”(generally defined as +l-2 kcallmol). The mean absoluteerror amounts to only 0.46 kcal/mol. By contrast, Gl theory14reachesa mean absoluteerror of 1.41 kcal/mol, with a maximum of

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TABLE II. Coefficients for three-term correction formula and error statistics for computed total atomization energies (kcal/mol).

Level of theory Gl theory G2 theory CCSD(T)/cc-pVDZ CCSD(T)/cc-pVTZ CCSD(T)/cc-pVTZ CCSD(T)/cc-pVQZ CCSD(T)/cc-pVQZ

CCSD(T)lcc-pVDZ CCSD(T)/cc-pVTZ CCSD(T)/cc-pVTZ CCSD(T)/cc-pVT2 CCSD(T)/cc-pVQZ CCSD(T)/cc-pVQZ

Reference geometry MP2/6-31G* MP2/6-31G* l-term correction CCSD(T)/cc-pVDZ MP2/6-31G* CCSD(T)/cc-pVlZ MP2/6-3 1G” CCSD(T)/cc-pVQZ 3-term correction CCSD(T)/cc-pVDZ MP2/6-31G* CCSD(T)/cc-pVDZ CCSD(T)/cc-pVTZ MYP2/6-3lG* CCSD(T)/cc-pVQZ

Mean absolute error

Max. absolute error

1.41 1.08

2.97 2.60

2.60 1.58 1.36 1.08 0.69

5.51 4.03 3.23 3.30 1.98

2.59 0.92 0.95 0.93 0.47 0.46

4.96 2.02 1.97 1.99 1.26 0.98

bn

9.407 4.233 3.835 3.798 1.814 1.277

0.346 -1.103 -0.636 -0.706 -0.576 -0.089

0.481 0.857 0.813 0.784 0.781 0.668

TABLE III. Computed and experimental total atomization energies (kcal/mol). Raw Basis geometry

GHz CH4

co co2 H2 H20

HCN HF NH3 N2

H2C0 F2

NNO

Basis geometry

C?HZ C% co co2 H2

Hz0 HF NH3 NZ

H2C0 F2

NNO

ab inirio values

cc-pVDz CCSD(T)/ cc-pVDZ

cc-pVTZ CCSD(T)/ cc-pVDZ

371.27 395.45 241.44 356.52 103.61 208.87 283.34 126.34 267.65 200.56 345.29 27.19 230.44

392.69 412.71 251.60 375.68 108.25 225.05 300.98 136.95 287.90 216.10 363.57 34.29 253.44

cc-pvTZ MPU 6-31G*

cc-pVQZ ME/ 6-31G*

cc-pVDZ CCSD(T)/ cc-pVDz

404.32 421.33 260.48 387.68 111.50 231.28 313.22 139.89 297.56 228.64 373.94 37.96 270.16

404.60 420.23 260.19 388.86 110.35 232.3 1 313.01 140.99 297.74 227.88 374.07 37.97 270.03

401.49 425.06 261.56 386.40 113.37 228.38 313.21 136.09 296.91 230.09 375.04 36.94 270.68

cc-pvTZ CCSD(T)/ cc-pvTZ

l-term correction cc-pVDZ CCSD(T)/ cc-pVDZ

cc-pvTZ CCSD(T)/ cc-pvTZ

cc-pVQZ CCSD(T)/ cc-pVQZ

Expt.’

401.54 425.72 261.62 386.79 113.70 229.05 313.61 136.43 297.92 230.83 375.56 37.28 270.80

403.69 423.44 258.64 386.24 111.85 232.07 311.78 140.42 298.60 226.86 374.22 38.30 267.62

403.57 421.18 258.81 387.71 110.48 232.70 311.79 141.32 298.15 226.99 373.87 38.09 268.40

405.36 420.21 259.31 389.14 109.48 232.55 311.75 141.05 297.90 228.46 373.73 38.20 270.38

cc-pvTZ CCSD(T)/ cc-pVDZ

cc-pvTZ CCSD(T)/ cc-pVT2

cc-pVQZ CCSD(T)/ cc-pVQZ

Gl theoryb

G2 theorf

403.91 421.67 260.26 387.53 111.45 231.45 312.84 140.15 297.50 228.60 373.9K 37.49 270.75

404.14 421.61 260.17 387.39 111.47 231.32 312.93 140.05 297.48 228.72 373.89 37.93 270.67

404.38 420.57 259.88 388.61 110.31’ 232.35 312.69 141.15 297.62 227.98 374.02 37.92 270.55

403.50 417.77 260.71 391.44 109.48 230.17 313.18 140.06 294.93 227.68 374.46 39.09 270.17

403.70 419.97 261.11 391.74 110.33 232.47 312.88 141.86 297.23 226.68 375.66 38.19 269.91

cc-pVQZ CCSD(T)/ cc-pVQZ

393.30 399.48 413.04 417.10 251.71 256.08 375.84 383.62 108.38 109.12 225.13 229.97 301.38 307.71 139.96 136.95 294.06 288.20 216.47 222.90 363.83 369.79 34.84 36.73 253.76 262.95 3-term correction

“See Ref. 13 for references and details. bNNO: this work, others: Ref. 14. ‘TWO: this work, others: Ref. 28. J. Chem. Phys., Vol. 100, No. 11,l

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TABLE IV. Computed and experimental geometries (A. degrees).

C2Hz

rcc rCH

CH4

ICH

co co2

rc0 rc0

H2

rnk4

Hz0

IOH

HCN

rCt.4

e rCH

HF NH3

f-m rm

NZ H&O

rNN

e rc0 rCH

I9 F2

rw

NNO

QN

Mean signed error

Single w/o HF Double Triple

rNO

CCSD(T)/ cc-pVDZ

CCSD(T)/ cc-pvlz

CCSD(T)/ cc-pVQZ

1.2287 1.0790 1.1037 1.1446 1.1745 0.7609 0.9663 101.91 1.1753 1.0826 0.9199 1.0273 103.54" 1.1189 1.2156 1.1199 115.21 1.4577 1.1478 1.1953 0.0182 0.0143 0.0144 0.0215

1.2097 1.0637 1.0890 1.1357 1.1663 0.7426 0.9594 103.58 1.1601 1.0668 0.9172 1.0141 105.64" 1.1038 1.2096 1.1033 116.19 1.4158 1.1329 1.1896 0.0018 0.0016 0.0057 0.0069

1.2065 1.0634 1.0879 1.1314 1.1626 0.7419 0.9579 104.12 1.1564 1.0668 0.9162 1.0124 106.18" 1.1003 1.2066 1.1022 116.44 1.4129 1.1291 1.1870 0.0008 0.0007 0.0024 0.0032

Experiment 1.20241: 1.2033b 1.0625: 1.0605b 1.086c 1.128323d 1.1600e 0.74144d 0.95843,’ 0.95721g 104.44,’ 104.52s 1.1532h 1.0655h 0.916808d 1.0116,' 1.02sj 106.7,' 107.d 1.097685d 1.2033(10)k 1.1005(20)k 116.18(15)k 1.41193d 1.12729’ 1.18509’

‘Reference 34. bReference 35. 3. L. Gray and A. G. Robiette, Mol. Phys. 37, 1901 (1979). dK. R Huber and G. Herzberg, Constants ofDiutomic Molecules (Van Nostrand Reinhold, New York, 1979). 7. L. Teffo, 0. N. Sulakshina, and V. I. Perevalov, J. Mol. Spectrosc. 156, 48 (1992). %?Jensen, J. Mol. Spectrosc. 128, 478 (1988). %. Carter and N. C. Handy, J. Chem. Phys. 87,4294 (1987). hA. M. Smith, S. L. Coy, W. Klemperer, and K. K. Lehmann, J. Mol. Spectrosc. 156,48 (1992). ‘J. L. Duncan and I. M. Mills, Spectrochim. Acta 20, 523 (1964). jA. R. Hoy, I. M. Mills, and G. Strey, Mol. Phys. 24, 1265 (1972). ‘rr. L. Duncan, Mol. Phys. 28, 1177 (1974). ‘J. L. Teffo and A. Chidin, J. Mol. Spectrosc. 135, 389 (1989).

2.97 kcal/mol, for the samesample.(Thesefigures are somewhat improved using G2 theory.28)Admittedly, Gl and G2 theory require much less computer time (with efficient codes) than CCSD(T)/cc-pVQZ calculations. At the CCSD(T)/cc-pVTZ level, which is comparableto Gl theory in computationalcost, mean absoluteand maximum errors amount to 0.93 and 1.99 kcal/mol, respectively. This is a quantitative improvement over Gl theory (less so for G2 theory-),albeit not a qualitative one. Comparisonof the corrected CCSD(T)/cc-pVTZ and the GUG2 results reveals a couple of interestingtrends.For example,it appearsthat Gl and G2 XD, values are on the high side for species like C02, while corrected CCSD(T)/cc-pVTZ results tend to be on the low side. For AH, hydrides, on the other hand, Gl theory tends to be too low (which does not appearto be the case for G2 theory) and correctedCCSD(T)/cc-pVTZ tends to lead to correct results or overestimates.In some cases, when both Gl/G2 theory and CCSD(T)/cc-pVTZ represent the maximum of what is computationally feasible, it might seem appealingto take the averageof three-term corrected CCSD(T)/cc-pVTZ and Gl or G2 theory: this leads to mean absolute errors of 0.87 and 0.74 kcallmol, and maximum errors of 1.81 and 1.44 kcal/mol, respectively. J.Chem.Phys.,Vol.

The rationale for using a three-termcorrection formula rather than a Gl/G2 type three-termcorrection has beendiscussed at .length in Ref. 30. Briefly summarizing, it was found there that, using a one-termcorrection,an spdfg basis set is requiredto reach a mean absoluteerror comparableto that of G2 theory (which employs basis sets of only spdf quality). Introduction of different correction terms for (Tand rr bondsresultedin a small improvementfor spdf basis sets, but a substantialone for spdfg ones.Finally, introduction of different correction terms for pairs and bonds improved results significantly for spdf basis sets, and perceptibly for spdfg ones, especially in terms of the maximum error for the sample. A detailed analysis of the Gl and G2 results revealedthat their relatively low mean absoluteerrors were at least partly due to a fortunate error compensationinvolving a hydrogenbasis set of only sp quality. One could argue that 13 molecules points is not really enough to decide between a one-term correction formula similar to those used in Gl and G2 theory, and a three-term correction such as employed here. However, all 13 points have XD, values known to 0.1 kcal/mol or better, whereas many of the moleculesusedin calibrating Gl and G2 theory have error bars an order of magnitudelarger. Illustration for

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TABLE V. Computed [CCSD(T)] and ex p erimental harmonic frequencies (cm-‘).

CzH2

WI Y w3 m4

05 01 w2

Y 04

co co2

Hz Hz0

HCN

0 WI 0, 0, w *I Y 0, WI 0, *3

I-IF NH3

co 0’ 02 w3 04

N2 HaCO

0 WI 0, 03 *4

F2

IWO

05 % 0 Q-4 0, *3

Mean abserr. w/o problem

cc-pVDz

cc-pvTZ

cc-pVQZ

Experiment

526.9 733.9 1985.7 3409.9 3500.6 1333.8 1550.5 3040.4 3175.0 2143.9 650.2 1338.0 2398.4 4382.5 1690.3 3821.9 3921.9 707.5 2098.3 3446.9 4150.2 3434.2 1182.0 3568.6 1688.1 2338.9 2924.7 1786.8 1534.9 1175.7 2986.7 1265.9 783.8 579.9 1307.7 2283.9 26.8 17.4

577.6 746.3 2001.1 3410.2 3511.2 1344.0 1570.8 3034.7 3153.9 2153.7 660.3 1346.3 2396.5 4409.4 1668.9 3841.2 3945.8 716.0 2111.7 3443.8 4177.4 3471.9 1109.2 3591.5 1687.9 2346.0 2929.2 1780.7 1543.2 1192.2 2995.8 1274.9 920.0 601.1 1297.2 2282.8 11.9 9.0

595.4 146.4 2005.9 3409.9 3501.9 1345.3 1570.4 3036.2 3157.1 2164.4 670.5 1352.0 2396.6 4403.5 1659.2 3844.9 3952.0 721.8 2123.4 3435.9 4162.2 3480.5 1084.1 3608.8 1679.6 2356.2 2933.5 1781.4 1537.6 1190.3 3003.7 1272.9 921.1 602.1 1303.5 2287.9 8.5 6.5

624.0a 746.7= 2007.6” 3415.2= 3495.1= 1367.4b 1582.7b 3025.5b 3156.gb 2169.81358’ 672.gd 1353.gd 2396.5d 4401.213’ 1647.8; 1644.9= 3831.8: 3828.6” 3942.4; 3939.8’ 726.9(2)g 2127.2(4)s 3442.5(1)g 4138.32’ 3478(12),’ 3503h 1030h 3597(g),’ 3591.6h 1684(g),’ 1689.gh 2358.57’ 2937.4: 2977.gk 1777.8j 1778.3k 1544j 1529k 1188.3j 1191k 3012: 2997k 1269.4: 1298.7” 916.64” 596.3’ 1298.3’ 2282.1’

“Reference 35. bD. L. Gray and A. G. Robiette, Mol. Phys. 37, 1901 (1979). of Diatomic Molecules (Van Nostrand Reinhold, New York, 1979). ‘K. P. Huber and G. Herzberg, Constants dJ. L. Teffo, 0. N. Sulakshina, and V. I. Perevalov, J. Mol. Spectrosc. 156, 48 (1992). l? Jensen, J. Mol. Spectrosc. 128, 478 (1988). fS. Carter and N. C. Handy, J. Chem. Phys. 87, 4294 (1987). gA. M. Smith, S. L. Coy, W. Klemperer, and K. K. Lehmann, J. Mol. Spectrosc. 134, 134 (1989). hA. R. Hoy, I. M. Mills, and G. Strey, Mol. Phys. 24, 1265 (1972). ‘K. K. Lehmann and S. L. Coy, J. Chem. Sot. Faraday 2 84, 1389 (1988). jL. B. Harding and W. C. Ermler, J. Comput. Chem. 6, 13 (1985). kD. E. Reisner, R. W. Field, J. L. Kinsey, and H. L. Dai, J. Chem. Phys. SO, 5968 (1984). ‘J. L. Teffo and A. Chedin, J. Mol. Spectrosc. 135, 389 (1989).

the points made above can be found in Table III, where the fairly seriousindividual errors (especiallyfor CH4, COz, N2, and N,O) in the one-term correctedCCSD(T)/cc-pVQZ energiesdrop to less than 1 kcal/mol using the three-termcorrection. It should be noted that the largest error among the corrected CCSD(T)/cc-pVTZ results is for H,. This basically representsthe fact that, since the Hz dissociationenergyconvergessomewhatfaster with respectto the basis set than the other CTbond energies,the correction tends to “overshoot” somewhatfor the Hz case. (Note the presenceof a similar

effect for G2 theory.In Gl theory,where the correction term is chosento reproducean exact Hz dissociationenergy,XD, for the ten-electronhydrides is consistently too low as a result.) Since the cc-pVQZ basis set is much closer to convergencefor cr bonds already,the effect is less prominent here. A refereepointed out that ZD, values for the hydrides AH, appearto be somewhatinferior to those obtained in previous work,31where a mean absoluteerror of 0.12 kcal/ mol per bond was achievedfor the AH,(A=Li-F) hydrides and hydride radicals using a basis set of roughly cc-pVTZ quality supplementedwith sp bond functions. Resultsfor the

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ten-electron hydrides CH,, NH,, H,O, and HF can be compared directly with the presentwork. The mean absoluteerror in Ref. 31 for those four species is 0.14 kcal/mol per bond; the correspondingvalues for three-term corrected ccpVTZ and cc-pVQZ results are 0.53 and 0.10 kcal/mol, respectively. [The issue of a three-term versus a one-term correction (as used in Ref. 31) is largely irrelevant here since n,=O throughout and na=npair except for CH4 (where no=np,ir+l).] It is not surprising that a cc-pVTZ basis set wiIl be outperformed by a cc-pVTZ+(bond function) basis set; however, the cc-pVQZ results appear to be at least as good, and are not restricted to AH, compounds.(The extension of the bond function basis sets3*employed in Ref. 31 to more general compounds without incurring excessivebasis set superposition error (SSS8) is no trivial matter; as discussedin Ref. 32, multiple bonds will probably require spd bond functions, which in turn will require spdfg atomic basis sets to keep BSSE down to an acceptablelevel.) Can any useful result for XD, be obtained from CCSD(T)/cc-pVDZ energies?Needlessto say, the raw CD, values are all but unusable:using the three-term correction, however, one reachesa surprisingly low mean absoluteerror of 2.59 kcal/mol. The maximum error amounts to 4.96 kcalf mol. This means that, although the numbers are not very useful for quantitative work, at least a rough estimateof XD, can be obtained when CCSD(T)/cc-pVDZ represents the limit of what is computationally possible, such as in the case of large cluster molecules. Note that in this case a simple Gl-like correction does only slightly less well than the threeterm formula, since both CTand rr bonds are far away from saturation in the basis set. Summarizing, one can say that three-term corrected CCSD(T) energieswill yield a fair estimate of ED, with the cc-pVDZ basis set, a number of “chemical accuracy” with the cc-pVTZ basis set, and an accuratenumber with the ccpVQZ basis set. Taking C2H, as an example, this would correspondto 48, 116, and 230 basis functions, respectively. Theoretically, computer time would then roughly go up as 1:34:527; in practice the proportions are smaller, especially on a vector machine where the greater vector lengths for larger basis sets result in improved machine performance. For example, on the Cray Y-MP at SDSC, a single-point CCSD(T) energy calculation for CO, took 16 s with the ccpVDZ, 113 s with the cc-pVTZ, and 772 s with the cc-pVQZ basis set. Next, let us addressthe issue of using lower-level reference geometries.Results at the MP2/6-31G* reference geometry (as used in Gl theory) are given in Ref. 13, and will not be repeatedhere. At the CCSD(T)/cc-pVTZ level, they result in an average error relative to the geometry at that level of only 0.55 mhartree. The maximum error however, for N20, is 2.79 mhartree. This is related to the known inability of MP2 to handle cumulated multiple bonds properly (see, e.g., Refs. 26,33). Using CCSD(T)/cc-pVDZ reference geometries,on the other hand, also leads to an averageerror of 0.44 mhartree,but to a maximum error of only 0.96 mhartree. This suggeststhat, especially in outlandish molecules such as clusters, CCSD(T)/cc-pVDZ reference geometries are certainly to be preferred above MP2/6-3 1G” ones. Coef-

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ficients for the three-term correction can be found in Table III. As can be seen,the mean absoluteerror of corrected vs. experimental XD, values is essentially the same as for the full equilibrium geometry. Peculiarly enough, the effect of using MP2/6-31G” geometrieson the mean and maximum errors is negligible at the CCSD(T)/cc-pVTZ level: this is at least partly due to an error compensationin the computed MP2/6-31G* geometries.Effects might be somewhat more perceptible at the CCSD(T)/cc-pVQZ level: presumably, if one can afford to do an energy calculation in a cc-pVQZ basis set, one can afford to do a better geometry optimization than MP2/6-31G” as well. Since there were substantial effects on the uncorrected energies,this means that some of the geometry effects get absorbedin the three-term correction. Some light is shed on this by considering a simple one-term correction such as is used in Gl and G2 theory. As is seenin Table II, the effects of using better geometriesthere are quite measurable.(Note that for the cc-pVDZ basis set, it hardly matters whether the one-term or the three-term correction is used, since the basis set is quite unsaturatedfor both (T and rr bonds. For the cc-pVTZ and cc-pVQZ basis sets, however, the three-term correction representsa real improvement.)This illustrates the fact that errors in the geometry tend to be much larger for -multiple than for single bonds as well, with a concomitant error in the energy that is roughly secondorder: So therefore, a correction that differentiates between single and multiple bonds will tend to iron out thesedifferences.Next comes the issue of the molecular geometries.As can be seenfrom Table IV, these are affected quite substantially by the size of the basis set, but rather systematically so. The cc-pVQZ results are of course in very good agreementwith experiment; for single bonds, errors are in the 0.001 w range except for the CH bond in C,H,. However, the very recent work of Bramley et al.,34 in which the original force field of Strey and Mills35 was refined by fitting variationally computed transition energiesto a large collection of experimentaldata, suggeststhat the geometry should be revised to ycc= 1.20241 A and rHH= 1.0625 A, the latter of which is only 0.0009 A shorter than our computed value. On average then, single bond lengths are overestimatedby 0.0008 A at the CCSD(T)/ cc-pVQZ level of theory. Predictably, somewhatlarger errors are seenfor multiple bonds. To the extent that a partitioning between double and triple bonds can be made on the basis of the relatively limited data here, double and triple bonds are overestimatedby 0.0024 and 0.0032 A, respectively.It is probably more realistic to state that multiple bond lengths are overestimatedby 0.003 A, on average. At the CCSD(T)/cc-pVTZ level, the mean overestimates are 0.0018 w for single bonds, 0.0057 w for double, and 0.0069 A for triple bonds. Note that here the difference between double and triple bonds is somewhatmore conspicuous. Effects on single bond lengths are somewhat spurious; for example,deleting F2 from the samplereducesthe error to 0.0016 A. Oversimplifying, one could say that bringing down the basis set from cc-pVQZ to cc-pVTZ will result in bond lengtheningsof 0.001 A for single, 0.003 A for double, and 0.004 L%for triple bonds. Still the effects are fairly sys-

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tematic, and a reliable estimateof the experimentalr, geometry could probably quite well be obtained by applying the above numbers as additivity corrections. Effects start to become more spurious at the CCSD(T)/ cc-pVDZ level. Here the average overestimate for single bonds actually exceeds that for double bonds, which is an artifact of the grossly overestimatedbond distance in Fs. If F2 is taken out of the sample,the value drops from 0.018 to 0.014 A. The other figures, 0.014 A for double and 0.021 A for triple bonds, are a bit too large to lead to accurateresults when used as additivity corrections. (Nevertheless,the numbers would still be more reliable than the uncorrectedones.) Errors are generally less systematicthan for the larger basis sets. Nevertheless, CCSD(T)/cc-pVDZ geometries would certainly be preferable over MP2 results in big basis sets exceptfor very “well-behaved” molecules,since geometries would almost always be qualitatively correct+ven with substantialmultireferenceeffects (see,e.g., Refs. 33, 36, and 37)-and geometrieswould be consistently biasedto overestimated bond lengths. Finally we turn to the computed frequencies.The most striking result here are the usually good harmonic frequencies predicted at the CCSD(T)/cc-pVDZ level: The mean absolute error is only 27 cm-‘! When two notorious problem cases (the umbrella mode in NH, and the lowest bending mode in acetylene)as well as Fz are eliminated, this drops to 17 cm-‘. What this means is that, even at this relatively inexpensivelevel of theory, harmonic frequenciescan be calculated that will give an infrared spectroscopist an idea “where to look,” and that will be sufficient for a number of purposes,such as computing vibrational partition functions. (The effect of errors in frequencieson computed thermodynamic functions over a wide temperature range was discussed in detail in Ref. 38). Going to a cc-pVTZ basis set brings a real improvement, namely, to a mean absoluteerror of 12 cm-’ with, and of 9 cm-’ without “problem cases.” Note that the frequency for F,, contrary to the cc-pVDZ result, is in quite good agreement with experiment, since there is no problem with the geometry here. In most cases, the harmonic frequenciesare within 10 cm-’ of the experimental ones. It should be noted that in some cases(such as NH,), the experimentalharmonics are not even preciseto 10 cm-‘, and that even this accuracyis the exceptionrather than the rule for most practical cases.A further improvement can be seenwhen moving up to the cc-pVQZ basis set, especially for triple and cumulated double bonds, as well as for the problem cases. However, this improvement-to 8.5 cm-’ with, and 6.5 cm-’ without, problem cases-does not entirely justify the quite considerableadditional computational cost. It is rather doubtful that further basis set extensionwill greatly improve on these figures, since some basis set saturation already appearsto have been achieved. What, then, would be remaining sourcesof error? For the moleculesconsideredhere, and certainly for the AH, species,incompletenessin the electron correlation treatment should have a negligible effect. For first-row compounds, furthermore, relativistic effects on the frequenciescan fairly confidently be dismissed. Inclusion of core-core and especially corevalence correlation is probably the main possible improve-

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ment to the theoreticaltreatment.Lastly, it should be pointed out that at least part of the experimental values are not as precise as they seem, due to problems with the analysis of the data (e.g., due to severeresonancesin formaldehyde). A peculiar phenomenonshould be noted for the stretching frequenciesof CH4. Even with the cc-pVQZ basis set, the totally symmetric stretch os is off by 10.7 cm-‘, while the triply degenerateasymmetric stretch w4 is calculated within 0.3 cm-’ of experiment.A similar phenomenonwas previously pointed out in a study of the force field of H,C0.27 Since the computed frequenciesare evidently near convergencewith respect to the one-particle basis set, and since for molecules like the first-row hydrides, CCSD(T) is very close to an exact n-particle solution, the most important remaining error is probably neglect of core-core and corevalence correlation. From a naive “local mode” perspective then, one would expect this effect to be substantiallyamplified in a vibration that symmetrically stretches four CH bonds, whereasthe effects would partially cancel each other out in an antisymmetric stretching vibration. This is exactly what is observedin both CH, and BH,;3g unfortunately,experimental data for NH3 are not precise enough to find supporting evidencethere.

CONCLUSIONS

In this paper the following points have been established: Using the three-termcorrection in Eq. (l), mean absolute errors in ED, are 0.46 kcallmol for the cc-pVQZ, 0.93 for the cc-pVTZ, and 2.59 for the cc-pVDZ basis sets.The latter figure is not adequatefor quantitative work, but implies a usableestimateof ED, when CCSD(T)/cc-pVDZ represents the computationally feasible maximum; CCSD(T)/cc-pVQZ calculations overestimatesingle and multiple bond lengths by about 0.001 and 0.003 A, respectively; CCSD(T)/cc-pVTZ bond lengths are still longer by about 0.001 A for single, 0.003 A for double, and 0.004 A for triple bonds; CCSD(T)/cc-pVDZ bond lengths are overestimated by an additional 0.01 8, for single, and 0.02 A for multiple bonds; CCSD(T)/cc-pVDZ harmonic frequencies are nevertheless in surprisingly good agreementwith experiment; CCSD(T)/cc-pVTZ harmonic frequencies generally agree with experiment to 10 cm-’ or better. CCSD(T)/ccpVQZ frequenciesare still somewhatbetter, at a disproportionate increasein computationalcost; the use of MP2/6-31G” reference geometries may lead to substantial errors in uncorrected ED, values for some molecules. However, the 3-term correction absorbsthese effects almost completely; the use of CCSD(T)/cc-pVDZ reference geometrieshas no detrimental effect at all on computed ZD, values, and is recommended for cases where MP2 is ill-behaved and CCSD(T)/cc-pVTZ optimizations are prohibitive in cost.

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ACKNOWLEDGMENTS

The author acknowledgesa Senior ResearchAssistant fellowship of the National Science Foundationof Belgium (NFWO/FNRS), and wished to thank Dr. Peter R. Taylor for helpful discussions.The calculation reported in this work were carried out while the author was a researchassociateat the San Diego SupercomputerCenter (SDSC), which is acknowledgedfor generouslyproviding computer time on its Cray Y-MP/864. This paper forms a part of researchresults of a programin Inter-University Attraction Poles,initiated by the Belgian state-Prime Minister’s office-science policy programming. ‘E. R. Davidson and D. Feller, Chem. Rev. 86, 681 (1986). 2J. Ah&f and P. R. Taylor, Adv. Quantum Chem. 22, 301 (1992). 3R. J. Bartlett, J. Phys. Chem. 93, 1697 (1989). 4R. C. Raffenetti, J. Chem Phys. 58,4452 (1973). ‘5. AlmlBf and P. R. Taylor, J. Chem. Phys. 87, 4070 (1987). 6J. Almlof and l? R. Taylor, J. Chem. Phys. 92, 551 (1990). ‘C. W. Bauschlicher, Jr. and P. R. Taylor, Theor. Chim. Acta 86, 13 (1993). *P. 0. Widmark, P. A. Malmqvist, and B. 0. Roos, Theor. Chim. Acta 77, 291 (1990). ‘P. 0. Widmark, B. Joakim Persson, and B. 0. Roos, Theor. Chim. Acta 79, 419 (1991). “T. H. Dunning, J. Chem. Phys. 90, 1007 (1989). “K. A. Peterson, ‘R. A. Kendall, and T. H. Dunning, J. Chem. Phys. 99, 1930 (1993). r2D. E. Woon andT. H. Dunning, J. Chem. Phys. 99, 1914 (1993). t3J. M. L. Martin, J. Chem. Phys. 97, 5012 (1992). t4J. A. Pople, M. Head-Gordon, D. J. Fox, K. Raghavachari, and L. A. Curtiss, J. Chem. Phys. 90, 5622 (1989). “J. Almliif, C. W. Bauschlicbr, Jr., M. R. A. Blomberg, D. P. Chong, A. Heiberg, S. R. Langhoff, P. A. Malmqvist, A. P. Rendell, B. 0. Roos, P. E. M. Siegbahn, and I? R. Taylor, MOLECIJLE~WEDEN, an electronic structure program system. “T .. J Lee 7A. P. Rendell, and J. E. Rice, TITAN, a set of electronic structure programs.

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