The prediction of molecular equilibrium structures by the standard electronic wave functions

The prediction of molecular equilibrium structures by the standardelectronic wave functions

Trygve HelgakerDepartment of Chemistry, University of Oslo, N-0315 Oslo, Norway

Jurgen GaussInstitut fur Physikalische Chemie, Universita¨t Mainz, D-55099 Mainz, Germany

Poul Jo”rgensenDepartment of Chemistry, Aarhus University, DK-8000 Aarhus C, Denmark

Jeppe OlsenChemistry Center, University of Lund, S-22100 Lund, Sweden

~Received 20 September 1996; accepted 13 January 1997!

A systematic investigation has been carried out of the accuracy of molecular equilibrium structuresof 19 small closed-shell molecules containing first-row atoms as predicted by the following standardelectronicab initiomodels: Hartree–Fock~HF! theory, Mo” ller–Plesset theory to second, third, andfourth orders~MP2, MP3, and MP4!, coupled-cluster singles and doubles~CCSD! theory; CCSDtheory with perturbational triples corrections@CCSD~T!#, and the configuration-interaction singlesand doubles ~CISD! model. For all models, calculations were carried out using thecorrelation-consistent polarized valence double-zeta~cc-pVDZ! basis, the correlation-consistentpolarized valence triple-zeta~cc-pVTZ! basis, and the correlation-consistent polarized valencequadruple-zeta~cc-pVQZ! basis. Improvements in the basis sets shorten the bond distances at alllevels. Going from cc-pVDZ to cc-pVTZ, bond distances are on the average reduced by 0.8 pm atthe Hartree–Fock level and by 1.6 pm at the correlated levels. From cc-pVTZ to cc-pVQZ, thecontractions are about ten times smaller and the cc-pVTZ basis set appears to yield bond distancesclose to the basis-set limit for all models. The models HF, MP2, and CCSD~T! give improvedaccuracy at increased computational cost. The accuracy of the Mo” ller–Plesset series oscillates, withMP3 being considerably less accurate than MP2 and MP4. The MP2 geometries are remarkablyaccurate, being only very slightly improved upon at the MP4 level for the cc-pVQZ basis. TheCCSD equilibrium structures are only moderately accurate, being intermediate between MP2 andMP3. The accuracy of the CCSD~T! model, in contrast, is high and comparable to that observed inmost experimental studies and it has been used to challenge the experimentally determinedequilibrium structure of HNO. The CISD wave function provides structures of low quality.© 1997 American Institute of Physics.@S0021-9606~97!01215-4#

neaisfo

thvethhethcaquda

o

-i-Fored-ofrowingdyra-gu-heulen-

di-t is,the

he

byst

I. INTRODUCTION

In ab initio electronic-structure theory, the Schro¨dingerequation is solved by introducing approximations in the oand N-electron spaces: In the one-electron space, theproximations are introduced by the truncation of the atomorbital basis set; in theN-electron space, the approximationare introduced by the adoption of some particular modelthe representation of theN-electron wave function in Fockspace. These approximations in the one- andN-electronspaces should be introduced in a systematic fashion—establishing hierarchies of models for the wafunctions—so that the errors can be controlled and sothe solutions may be improved upon until in principle texact solution is recovered. Systematic comparisons ofresults obtained at the different levels of the hierarchiesthen be used to make judgements about the usefulness,ity, and reliability of a particular molecular calculation analso to extrapolate the results towards the exact solutionthus estimate probable errors.1,2

In this paper, we present a systematic investigation

6430 J. Chem. Phys. 106 (15), 15 April 1997 0021-9606/97/

Downloaded 07 Jun 2013 to 193.157.137.211. This article is copyrighted as indicated in the abstract

-p-c

r

us

at

enal-

nd

f

the accuracy obtainable inab initio electronic-structure predictions of molecular equilibrium geometries, applying a herarchy of basis sets and a hierarchy of wave functions.statistical significance, we have considered 19 small closshell molecules containing first-row atoms and a varietychemical bonds; bonds between hydrogen and a first-atom as well as single, double, and triple bonds involvtwo first-row atoms. All molecules considered in this stuare dominated by a single closed-shell electronic configution. However, the dominance of the Hartree–Fock confiration in the wave function differs considerably among tmolecules in this study, which includes the ozone molecwith a significant contribution from a second electronic cofiguration.

Over the last decade, a considerable effort has beenrected towards developing hierarchical basis sets—thasequences of basis sets that allow the user to approachbasis-set limit by going to higher levels in the hierarchy. Tatomic natural-orbital~ANO! sets of Almlof and Taylor3

provide the first example of such basis sets and the ANOsWidmark et al.4 are another example. However, the mo

106(15)/6430/11/$10.00 © 1997 American Institute of Physics

. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

haningte

vadavas

--k

nstcunn

n

haorlerafo

tinn

thp,ninl-

suencthsa-oh

esin

rie

ndd inad-cc-

laritalsnsrre-

ardwe

tenttentcc-cc-sisbeenep-

t ba-s ofela-

al-iane

ol-theere

the

elcu-asisn-

6431Helgaker et al.: Molecular equilibrium structures

successful of these new hierarchical basis sets are perthe correlation-consistent sets developed by Dunningco-workers.5–9 Thus, for all wave functions consideredthis study, we carry out calculations using the followinthree correlation-consistent sets: the correlation-consispolarized valence double-zeta~cc-pVDZ! basis, thecorrelation-consistent polarized valence triple-zeta~cc-pVTZ! basis, and the correlation-consistent polarizedlence quadruple-zeta~cc-pVQZ! basis. For a few selectemolecules and wave functions, additional calculations hbeen carried out for even larger correlation-consistent bsets.

Perhaps the simplestN-electron hierarchy of wavefunction models is that provided by Mo” ller–Plesset perturbation theory ~MPPT!, containing the models Hartree–Foc~HF!, second-order Mo” ller–Plesset ~MP2!, third-orderMo” ller–Plesset ~MP3!, and fourth-order Mo” ller–Plesset~MP4!, all of which have been included in this study. Aalternative hierarchy is that based on the coupled-clu~CC! representation of the electronic structure. This partilar hierarchy contains the models HF, MP2, CCSD, aCCSD~T!, where CCSD is the coupled-cluster singles adoubles model10 and CCSD~T! corresponds to CCSD withperturbative triples corrections added.11 In addition to thesemodels, we have included in our study the configuratiointeraction singles and doubles~CISD! model. Although thismodel is considerably less important and less useful tthose belonging to the MPPT and CC hierarchies, its histcal importance makes its inclusion in this study worthwhi

Many investigations have appeared where the accuof molecular equilibrium geometries has been examinedthe standard wave function models.12–18 Our investigationdiffers from the previous studies in being more systemawith regard to the approximations made in the one- aN-electron spaces, thereby making it easier to identify aseparate the errors introduced at the different levelstheory. Previous studies have been less systematic inrespect and have in our opinion not always correctly serated the errors introduced in the one- andN-electron spacesleading in some cases to incorrect conclusions concerthe quality of theN-electron models. The number of moecules considered in this investigation is also larger thanprevious studies and in a few cases new experimental rehave been found, more recent and accurate than those usprevious investigations, increasing the statistical significaof the present study. It should be noted, however, thatpresent investigation concerns onlyclosed-shell moleculecontainingfirst-row atoms. The results presented in this pper therefore do not necessarily carry over to open-shell mecules or to molecules containing heavier elements suctransition-metal compounds.

II. COMPUTATIONAL DETAILS

Calculations of the molecular equilibrium geometrihave been carried out for the 19 molecules in Table I usthe HF, MP2, MP3, MP4, CCSD, CCSD~T!, and CISD wavefunctions. For all models, the calculations have been car

J. Chem. Phys., Vol. 106


apsd

nt

-

eis

er-dd

-

ni-.cyr

cddofisa-

g

inltsd inee

l-as

g

d

out using the correlation-consistent cc-pVDZ, cc-pVTZ, acc-pVQZ basis sets—the primary basis sets considerethis investigation. To explore basis-set saturation further,ditional calculations have been carried out in the largerpV5Z basis for the three molecules N2, H2O, and N2H2. Thecorrelation-consistent sets provide a hierarchy of molecubasis sets, where the occupied Hartree–Fock atomic orbare systematically supplemented with correlating functiodesigned for an accurate and balanced description of colation effects in the atomic valence region.

In addition to exploring the convergence of the standcorrelation-consistent hierarchy of basis sets cc-pVXZ,have for the three molecules N2, H2O, and N2H2 also con-sidered the performance of two related correlation-consisbasis-set hierarchies: the augmented correlation-consissets aug-cc-pVDZ, aug-cc-pVTZ, aug-cc-pVQZ, and aug-pV5Z; and the correlation-consistent core-valence setspCVDZ, cc-pCVTZ, and cc-pCVQZ. In the augmented basets, the standard correlation-consistent basis sets haveaugmented with diffuse functions so as to improve the rresentation of the outer regions of the electronic system.8 Inthe core-valence sets, the standard correlation-consistensis sets have been augmented with correlating functionlarge exponents, as appropriate for a description of corrtion effects in the inner-valence and core regions.9

The CISD calculations and the largest HF and MP2 cculations have been carried out using the Gaussprogram.19 For the remaining calculations in this study, whave used theACESII program.20 The calculated electronicenergies and equilibrium structures for the 19 sample mecules in Table I may be obtained upon request fromauthors. We note that, in all calculations, all electrons wcorrelated~i.e., the 1s orbitals were not kept frozen!. Theexperimental bond distances for the 28 distinct bonds in19 sample molecules are listed in Table II.

III. RESULTS

A. Measures of errors

In order to quantify the errors in the calculations, whave considered several statistical measures. Let the calated bond distances for a given method and for a given bset be denoted byRi

calc and let the corresponding experimetal numbers beRi

exp. The error is then given by

D i5Ricalc2Ri

exp. ~1!

TABLE I. The 19 molecules, on which the statistical analysis is based.

HF, H2O, NH3, CH4, N2, CH2, CO, HCN, CO2, HNC, C2H2, CH2O, HNO,N2H2, O3, C2H4, F2, HOF, H2O2

, No. 15, 15 April 1997


d

rffisusTh

of.

reerten

TZ8 pmtedis,0.1thethe

se-

andnt.ctsthe

to d

s-

6432 Helgaker et al.: Molecular equilibrium structures

We have, for each basis set and each method, calculatemean errorD, the standard deviation in the errorsDstd, themean absolute errorDabs, and the maximum errorDmax forthen528 bond distances,

D51

n (i51

n

D i , ~2!

Dstd5A 1

n21 (i51

n

~D i2D!2, ~3!

Dabs51

n (i51

n

uD i u, ~4!

Dmax5maxi

uD i u. ~5!

Each measure characterizes a specific aspect of the pemance of the methods and the basis sets. Thus, the twomeasuresD andDstd characterize the distribution of errorabout a mean valueD for a given method and basis set, thquantifying both systematic and nonsystematic errors.

TABLE II. Bond length of the molecules in Table I ordered accordingincreasing experimental values.

Molecule Bond Experiment~pm!

1 HF RFH 91.7a

2 H2O RHO 95.7b

3 HOF RHO 96.57c

4 H2O2 RHO 96.7d

5 HNC RHN 99.4e

6 NH3 RHN 101.2f

7 N2H2 RHN 102.8g

8 C2H2 RCH 106.2h

9 HNO RHN 106.3i

10 HCN RCH 106.5j

11 C2H4 RCH 108.1k

12 CH4 RCH 108.6l

13 N2 RNN 109.77a

14 CH2O RCH 109.9m

15 CH2 RCH 110.7n

16 CO RCO 112.8a

17 HCN RCN 115.3j

18 CO2 RCO 116.0o

19 HNC RCN 116.9e

20 C2H2 RCC 120.3h

21 CH2O RCO 120.3m

22 HNO RNO 121.2i

23 N2H2 RNN 125.2g

24 O3 ROO 127.2p

25 C2H4 RCC 133.4k

26 F2 RFF 141.2a

27 HOF RFO 143.5c

28 H2O2 ROO 145.56d

aReference 32. iReference 26.bReference 33. jReference 38.cReference 28. kReference 39.dReference 31. lReference 40.eReference 34. mReference 41.fReference 35. nReference 42.gReference 36. oReference 43.hReference 37. pReference 44.



the

or-rst

e

mean absolute errorDabs represents the typical magnitudethe errors in the calculations andDmax gives the largest error

B. Mean errors

We begin by considering the mean errors, which alisted in Table III and plotted in Fig. 1. From Fig. 1, wconclude that improvements in the one-electron basis shothe bond lengths whereas improvements in theN-electrondescription usually~but not invariably! increase the bondlengths. Thus, going from the cc-pVDZ basis to the cc-pVbasis, the bond lengths are on the average reduced by 0.at the Hartree–Fock level and by 1.6 pm at the correlalevels. Going from the cc-pVTZ basis to the cc-pVQZ basthis contraction is much less pronounced—of the order ofpm. Clearly, for most methods and most applications,cc-pVTZ basis should provide results sufficiently close tobasis-set limit.

For all basis sets, the bond distances increase in thequence HF, CISD, MP3, CCSD, MP2, CCSD~T!, and MP4.Moreover, the Hartree–Fock bond lengths are too shortthe MP4 bond lengths are too long relative to experimeThus, improvements in the description of correlation effetend to increase the bond lengths. We note, however,

TABLE III. The mean deviationsD relative to experiment in the calculatebond distances~pm!.

cc-pVDZ cc-pVTZ cc-pVQZ

HF 22.01 22.80 22.91MP2 1.29 20.15 20.26MP3 0.40 21.16 21.30MP4 1.77 0.30 0.24CCSD 0.96 20.72 20.89CCSD~T! 1.59 20.05 20.19CISD 0.11 21.57 21.80

FIG. 1. Mean errorsD relative to experiment in the calculated bond ditances~pm!.

, No. 15, 15 April 1997


ein

nthD

itde.imQthr

fi-genenddeISla

ut

mcra,

,tly

f

ondthiariare-ane

bles.tednor-for-

ac-ndt insatedheentthehate-lyost2

hatthat

e

-


oscillatory behavior of the Mo” ller–Plesset sequence—thMP2 bond lengths are intermediate between those obtaat the MP3 and MP4 levels.

Since improvements in the one-electron andN-electrondescriptions affect the bond lengths in opposite directiothere is considerable scope for cancellation of errors incalculation of bond distances. For example, at the cc-pVlevel, the CISD bond lengths arein the meanextremely ac-curate with a mean deviation of only 0.1 pm, compared wthe CCSD~T! error of 1.6 pm, almost as large in magnituas the Hartree–Fock error of22.0 pm for this basis setHowever, as the description of the one-electron space isproved, the CISD bond distances shorten. At the cc-pVlevel, the CISD bond distances are much less accuratethe other correlated wave functions, with an average erro21.8 pm compared with the CCSD~T! error of20.2 pm andthe Hartree–Fock error of22.9 pm. A similar behavior isobserved for the MP3 bond distances, which are accurate~inthe mean! at the cc-pVDZ level~error 0.4 pm! but inaccurateat the more complete cc-pVQZ level~error21.3 pm!.

Clearly, the CISD and MP3 models are not of sufciently high quality to yield accurate bond lengths for larbasis sets. The CISD and MP3 models should thereforebe used for the calculation of molecular structures. Thgood performance at the cc-pVDZ level is fortuitous adoes not allow for an improvement of the one-electronscription. It does explain, however, the success of the Cwave function in the 1970s for the calculation of molecustructures for basis sets of polarized double-zeta quality.

At the cc-pVTZ level, two approximations stand ofrom the others: MP2 with a mean error of20.15 pm andCCSD~T! with a mean error of20.05 pm. At this level, theMP4 distances~with a mean error of10.30 pm! are alsoquite accurate, but less so than the simpler MP2 approxition. Again, there appears to be a certain element of canlation of error in these numbers. Thus, at the more elabocc-pVQZ level, the MP2 and CCSD~T! bond distances arewith mean errors of20.26 and20.19 pm, respectively, onthe average less accurate than at the cc-pVTZ level. Alsothe cc-pVQZ level, the MP4 distances are finally slighmore accurate than the MP2 distances~errors 10.24 and20.26 pm, respectively!. We shall return to a discussion othe possible sources of errors in these numbers later.

C. Standard deviations

Having discussed the mean errors in the calculated bdistances, it is appropriate also to consider the standardviations in the errors and thus more fully characterizedistribution of errors in the calculations. The standard devtions are listed in Table IV and plotted in Fig. 2. Only fothree models does the standard deviation decrease withprovements in the basis set: for MP2, for MP4, and in pticular for CCSD~T!. For MP3 and CCSD, the standard dviation decreases from cc-pVDZ to cc-pVTZ but increaseswe go to cc-pVQZ. For the CISD wave function, the stadard deviation increases monotonically and for HartreFock it is always large.



ed

s,eZ

h

-Zanof

otir

-Dr

a-el-te

at

de-e-

m--

s-–

D. Normal distributions

In Fig. 3, we have, for each basis set and eachN-particleapproximation, plotted the normal distributions

r~R!5Nc expF21

2 SR2D

DstdD 2G ~6!

based on the mean values and standard deviations in TaIII and IV. In this expression,Nc is a normalization constantAlthough we make no claim that the errors in the calculabond distances are indeed distributed according to themal distributions, these plots neatly summarize the permance of the various levels of theory.

We note that the Hartree–Fock wave function is charterized by broad distributions centered off the origin, athat its performance does not improve upon improvementhe basis set. In contrast, the Mo” ller–Plesset bond distanceare characterized by distributions that are sharper and loccloser to the origin. The relatively poor performance of tMP3 bond distances compared with MP2 and MP4 is evidfrom these plots. We also note that the progression ofMP4 distribution as the basis set is improved is somewmore satisfactory than that of MP2 theory—both with rspect to the position of the peak and its width—but onslightly so. Indeed, considering the significantly higher cof the MP4 calculations, the improvement of MP4 over MPis rather disappointing.

The performance of the CCSD model is also somewdisappointing; its performance is intermediate between

FIG. 2. Standard deviationsDstd in the errors relative to experiment in thcalculated bond distances~pm!.

TABLE IV. Standard deviationsDstd in the calculated bond distances relative to experiment~pm!.



, No. 15, 15 April 1997


le III andvertical


FIG. 3. Normal distributionsr(R) for the errors in the calculated bond distances. The distributions have been calculated from the mean errors in Tabthe standard deviations in Table IV~pm!. For easy comparison, all distributions have been normalized to one and plotted on the same horizontal andscales.

os

rehed,oth

e-ro

n

usiso-

ob-t

of MP2 and MP3. Clearly, the CCSD wave function is nparticularly well suited for the calculation of bond distanceOnly with the addition of triples corrections at the CCSD~T!level does the coupled-cluster model yield satisfactorysults. Indeed, at the cc-pVTZ and cc-pVQZ levels, tCCSD~T! model performs excellently, with sharply peakedistributions close to the origin. From these investigationsappears that the inclusion of doubles amplitudes to secorder at the MP2 level yields satisfactory results, but thatinclusion of doubles to higher orders~as in MP3, CISD, andCCSD! without the simultaneous incorporation of triples@asin MP4 and CCSD~T!# yields bond distances in worse agrement with the exact solution. We finally note that CISD peforms less satisfactorily than any other correlated methwith the possible exception of MP3.

E. Mean absolute deviations

We now consider the mean absolute deviationsDabslisted in Table V and plotted in Fig. 4. In Table VI, the mea



t.

-

itnde

-d,

absolute deviationsDabs are scaled such that the CCSD~T!error in the cc-pVQZ basis is equal to one. With the obvioexceptions of MP3 and CISD at the cc-pVDZ level, Fig. 4very similar to what we would obtain by plotting the abslute values of the mean valuesD ~compare with Fig. 1!,confirming the systematic nature of the errors usuallytained in ab initio calculations. From Fig. 4, the differen

TABLE V. The mean absolute deviationsDabsrelative to experiment for thecalculated bond distances~pm!.



, No. 15, 15 April 1997


nidSve

ver-Zatanimw

thlanor%toenlao

ulshfohulu

dantn, t8rgtio

el.

ec-

rs.le ofondla-oner-heby,

lv-ionlestherstheWe

enfer-in

pleeri-

d

QZ


behavior of the wave functions at the cc-pVDZ level on oside and the cc-pVTZ and cc-pVQZ levels on the other sis quite evident. Among the correlated methods, the CIand MP3 approximations perform best at the cc-pVDZ leand worst at the cc-pVTZ and cc-pVQZ levels.

At this point, it is appropriate to comment on the relatiperformance of the Mo” ller–Plesset approximations. Compaing with Hartree–Fock theory, we note that, for the cc-pVDbasis, the absolute mean errors relative to the uncorreldescription are 61%, 42%, and 84% at the MP2, MP3,MP4 levels, respectively. Thus, for this basis set, theprovements on the uncorrelated description are small andnote that MP3 performs better than MP2 and MP4. Atcc-pVTZ level, the situation is reversed and the errors retive to the Hartree–Fock description are 21%, 41%, a18%, respectively. Finally, for the cc-pVQZ basis, the errat the MP2, MP3, and MP4 levels are 19%, 45%, and 14

These examples demonstrate quite clearly the oscillabehavior of the Mo” ller–Plesset sequence and the inherinadequacy of the cc-pVDZ basis set in recovering molecuelectronic correlation effects, indicating that any comparisof the performance of correlated methodsrelative to experi-mentbased on experience with the cc-pVDZ basis set shobe treated with caution as it may give a completely faindication of the performance of the various models. Tsmall cc-pVDZ basis does not have the flexibility neededa satisfactory description of the true correlation effects. Tcc-pVTZ basis, on the other hand, yields satisfactory resfor the bond distances and should be sufficient for most pposes.

F. Maximum errors

Finally, in Table VII and Fig. 5, we have listed anplotted the maximum errors for the various basis setsN-electron approximations. These numbers are importanproviding worst-case errors for the different wave functioand basis sets. Thus, we see that, for the cc-pVQZ basisHartree–Fock wave function may give errors as large aspm, and that the maximum CISD and MP3 errors are as laas 5.7 and 4.2 pm, respectively. The CCSD wave func

FIG. 4. Mean absolute errorsDabs relative to experiment in the calculatebond distances~pm!.



eeDl

edd-e

e-ds.rytrn

ldeeretsr-

dinshe.5en

may give errors as large as 3.1 pm at the cc-pVQZ levAgain, the best methods are MP2, MP4, and CCSD~T!,whose maximum errors are 1.7, 1.5, and 1.2 pm, resptively.

Some comments are in order for the maximum erroFirst, these numbers are based on a rather small sampmolecules, containing elements from the first and secrows only. Clearly, larger errors may be obtained in calcutions on other systems and in particular in calculationsmolecules involving heavier atoms. For example, for the vtical cyclopentadienyl–iron distance in ferrocene, tHartree–Fock wave function overestimates the distance21 pm,21 MP2 underestimates the same distance by 19 pm21

whereas CCSD and CCSD~T! give distances within 1–2 pmof the experimental bond length.22 This particular exampleillustrates that, although less accurate for molecules invoing first- and second-row atoms, the CCSD wave functappears to be more robust than MP2 theory for molecucontaining heavier atoms. It should also be noted thatmaximum errors in Table VII may be associated with erroin the experimental measurements rather than errors incalculations, in particular for the most accurate methods.shall return to this point shortly.

G. Detailed plots

In Figs. 6–9, we have plotted the differences betwethe calculated and experimental bond lengths for the difent wave functions: HF in Fig. 6; MP2, MP3, and MP4Fig. 7; CCSD and CCSD~T! in Fig. 8; and CISD in Fig. 9. Inthese plots, the 28 distinct bonds found in the 19 sammolecules have been arranged in order of increasing exp

TABLE VI. The mean absolute deviationsDabs in the calculated bond dis-tances relative to experiment in units of the deviation at the cc-pVCCSD~T! level.



TABLE VII. The maximum absolute deviationsDmax in the calculated bonddistances relative to experiment~pm!.



, No. 15, 15 April 1997


inc-

vehsthoothdnZve

n--

Pmt bc-

8th

.5

nc-ingbe-

cu-O.ing

d

lev

the

t the


mental bond length as given in Table II. Each figure contathree plots—one for each of cc-pVDZ, cc-pVTZ, and cpVQZ basis sets.

From Fig. 6, we note that the Hartree–Fock wafunction—almost without exception—gives bond lengtthat are too short compared with experiment. In contrast,MP4 bond lengths in Fig. 7 are with very few exceptions tlong. The other models may give bond lengths that are eitoo short or too long, with a predominance of too long bonat the cc-pVDZ level and too short bonds at the cc-pVTZ acc-pVQZ levels. In particular, at the cc-pVTZ and cc-pVQlevels, the MP3, CCSD, and CISD models invariably gitoo short bond distances, whereas the cc-pVDZ CCSD~T!model gives bond distances that are too long.

For the Hartree–Fock function, the largest deviatiofrom experiment are found in O3 ~where two electronic configurations are important!, and for the electron-rich nonhydrogen bonds in F2, HOF, H2O2, and N2H2. The MP2 modeldescribes these bonds surprisingly well, whereas the Mmodel still has problems for these bonds. Similar probleare experienced by the CISD model and to some extenthe CCSD model. CCSD~T! describes these bonds quite acurately.

The CCSD~T! results with the cc-pVQZ basis in Fig.have a mean absolute deviation of 0.22 pm, smaller thanestimated error in many experimental investigations. Tmaximum deviation for CCSD~T! in the cc-pVQZ basis oc-curs for the NH bond length in HNO, which is a factor of 5

FIG. 5. Maximum errorsDmax relative to experiment in the calculated bondistances~pm!.

FIG. 6. The errors in the calculated bond lengths at the Hartree–Fock~pm!.



s

e

ersd

s

3sy

hee

larger than the mean error for this basis set and wave fution. In the other wave-function models, the correspondfactors between the maximum and mean deviations aretween 2.9 and 3.6. The large maximum error in CCSD~T!compared with the mean error is probably due to an inacrately determined experimental bond length for NH in HNThis conjecture is substantiated by the fact that the remain

el

FIG. 7. Errors relative to experiment in the calculated bond lengths forMo” ller–Plesset models~pm!.

FIG. 8. Errors relative to experiment in the calculated bond distances aCCSD and CCSD~T! levels ~pm!.

, No. 15, 15 April 1997


atro

eletia

ee

.1th

iful

ughele

ar

e

c-inthhenconle

s-vebs

as-theZ-ort-dif-

wforeth.oran-orre

t t

nd

nd


bond distances between hydrogen and first-row atomsreproduced with an accuracy of a few tenths of a picomein CCSD~T!, whereas the NH distance in HNO has an erof 1.2 pm.

H. Basis-set convergence

We now examine in greater detail the basis-set convgence for the calculation of equilibrium structures. In TabVIII, we give the average and absolute average differenshifts in the bond lengths from cc-pVDZ to cc-pVTZ~DDT!and from cc-pVTZ to cc-pVQZ~DTQ!. The absolute averagdifferential shifts decrease by a factor of 6–8 for all wavfunction models going fromDDT to DTQ. Extrapolationsuggests shifts from cc-pVQZ to cc-pV5Z of the order of 0pm or less. This result is confirmed by calculations onselected molecules N2, H2O, and N2H2.

Basis-set saturation with respect to the addition of dfuse functions has been investigated by carrying out calctions for the molecules N2, H2O, and N2H2 at the HF andMP2 levels using the aug-cc-pVXZ~X5D,T,Q,5! basis sets.The results show that the equilibrium geometries in the amented and nonaugmented basis sets approach each othigher levels and differ by less than 0.1 pm at the quintupzeta level.

Flexibility in the core region has been examined by crying out calculations for the selected molecules N2, H2O,and N2H2 at the HF, MP2, CCSD, and CCSD~T! levels usingthe core-valence basis sets cc-pCVXZ~X5D,T,Q!. The ef-fect of including the core orbitals is very small at thquadruple-zeta level—for example, the N2 equilibrium bondlength is reduced by 0.03 pm going from cc-pVQZ to cpCVQZ at the CCSD~T! level. Thus, the basis-set limit appears to be obtained within a few tenths of a picometer usthe cc-pVQZ basis. It should be understood, however,these results apply only to molecules containing no higthan first-row atoms and that the importance of core-valeand core correlation is considerably larger in systems ctaining heavier atoms. Also, we recall that all calculatiopresented in this paper have been carried out with all etrons correlated.

The differential changes in the bond lengthsDDT andDTQ given in Table VIII indicate that it becomes increaingly more important to use larger basis sets with improments in the correlation description. The average and alute average differential shifts differ in sign only at theDDT

FIG. 9. Errors relative to experiment in the calculated bond distances aCISD level ~pm!.



reerr

r-

l

-

e

-a-

-r at-

-

-

gatren-sc-

-o-

level. All bond distances thus become shorter when increing the basis from double-zeta to triple-zeta quality, sincebonding region is only crudely described at the cc-pVDlevel. The increased flexibility at the cc-pVTZ level increases the electron density in the bonding region and shens the bond length. The average and absolute averageferential shifts differ at theDTQ level for the correlated wavefunctions since the additional flexibility in the basis can nobe used to adjust the finer details in the bonds and therelead to either an increase or a decrease in the bond leng

The differential shifts in the bond lengths are similar fall bond lengths at the correlated levels, differing substtially from the shifts obtained at the HF level. This behaviis clearly seen from Fig. 10, where the differential shifts agiven for the HF, MP2, and CCSD~T! models.

he

FIG. 10. Differential changes in the bond lengths for HF, MP2, aCCSD~T! ~pm!.

TABLE VIII. The average and absolute average differential shifts in bolength from cc-pVDZ to cc-pVTZ~DDT! and from cc-pVTZ to cc-pVQZ~DTQ! for the standard wave-function models~pm!.

uDDTu uDTQu DDT DTQ

HF 0.79 0.11 20.79 20.11MP2 1.44 0.16 21.44 20.12MP3 1.56 0.20 21.56 20.14MP4 1.47 0.16 21.47 20.06CCSD 1.68 0.23 21.68 20.17CCSD~T! 1.64 0.20 21.64 20.14CISD 1.68 0.27 21.68 20.24

, No. 15, 15 April 1997


ecthruigas

thtioo

t iesmyonoethheuve

rv2tiomt

ng

leavndth

Ph

e

th

helaa

ng4l,ve

on-DZ,

uleZ

m

Z

nto

wesis-

is a–5

hatn-cedofn ofhatthendsetnm-

top-elcu-arene-thetro-erba-ed-

oicalmow

cester


I. Systematic trends in the convergence towards theFCI limit

From the material presented in this paper, some gentrends in the convergence of the calculated bond distantowards the FCI treatment may be discerned. Althoughtrue nature of the convergence of the bond lengths withspect to the correlation treatment must remain somewhatcertain as long as we cannot treat correlation effects to horders, it is interesting to speculate on the convergence bon the data presently available to us.

In general, bonds are contracted by improvements inbasis sets and stretched by improvements in the correlatreatment. These generalizations, however, gloss over sinteresting details in the dependency of the bond lengthsthe correlation treatment. Thus, among the methods thatroduce correlation through the inclusion of doublexcitations—that is, MP2, MP3, CCSD, and CISD—the siplest treatment~MP2! gives the longest bond distances. Anfurther improvement in the treatment of the doubles ctracts the bonds back towards the HF limit, the magnitudethis contraction depending on the nature of the improvemin the correlation description. For the cc-pVQZ basis,largest ‘‘back contraction’’ relative to MP2 is observed at tCISD level and amounts to as much as 58%, which shobe compared with the contractions of 39% at the MP3 leand 24% at the CCSD level.

In very general terms, we may rationalize these obsetions as follows:~1! The inclusion of doubles at the MPlevel stretches the bonds since a new type of interacamong the electrons~parametrized by means of doubles aplitudes! has been introduced. The bonds are stretched byrepulsive nature of this interaction.~2! Any further refine-ment in the treatment of the interaction~by relaxation of theamplitudes! reduces its overall repulsive character, allowithe bonds to recontract somewhat.

By this argument, we expect a simple treatment of tripto stretch the bonds further, and any relaxation of the wfunction in the presence of the triples to contract the boagain. However, since the triples are less important thandoubles, we expect these effects to be much smaller thanthe doubles.

The simplest treatment of the triples occurs at the Mand CCSD~T! levels, the two methods differing from eacother by the fact that, at the CCSD~T! level, the doubles havebeen fully relaxed~in the absence of the triples! whereas nosuch relaxation is carried out at the MP4 level. In agreemwith this observation, we find that MP4 and CCSD~T! bothincrease the bond lengths~compared with MP3 and CCSD!but that the MP4 bonds are the longest since neitherdoubles nor the triples have been fully relaxed.

We also expect that a full relaxation of the triples at tCCSDT level23–25should contract the bonds somewhat retive to CCSD~T!. This conjecture has been confirmed bypreliminary calculation on the nitrogen molecule. Goifrom CCSD~T! to CCSDT, the bond contracts from 111.8to 111.80 pm at the cc-pVDZ level. At the cc-pVTZ levethe bond distances are 110.06 and 110.00 pm, respecti



ralesee-n-hed

eonmenn-

-

-fnte

ldl

a-

n-he

sesefor

4

nt

e

-

ly,

and 109.81 and 109.75 pm at the cc-pVQZ level. The ctractions are thus 0.04, 0.06, and 0.06 pm at the cc-pVcc-pVTZ, and cc-pVQZ levels, respectively.

The experimental bond length in the nitrogen molecis 109.77 pm. Correcting the calculated CCSDT cc-pVQbond distance of 109.75 pm for basis sets effects~whichcontract the bond by 0.13 pm at the MP2 level going frocc-pVQZ to cc-pV5Z! and the full inclusion of core andcore-valence correlation effects@which, at the CCSD~T!level, contracts the bond by 0.003 pm going from cc-pVQto cc-pCVQZ#, we obtain a CCSDT limit of 109.62 pm, inerror by20.15 pm relative to experiment. For the nitrogemolecule, we therefore obtain the following errors relativeFCI in the CC hierarchy:23.21 pm ~HF!, 20.81 pm~CCSD!, 20.15 pm~CCSDT!. For the HF wave function, wehave here used the cc-pVQZ value whereas for CCSDhave used the cc-pVQZ value, corrected for the same baset incompleteness error of 0.13 pm as in the CCSD~T! cal-culation. From these numbers, we conclude that therereduction in the error in the bond distance by a factor of 4with each order in the coupled-cluster amplitudes.

We thus find that the smallness of the CCSD~T! errorarises from a cancellation of errors—the contraction twould occur upon relaxation of the triples and upon extesion of the basis beyond cc-pVQZ is approximately balanby the stretching that would occur upon the introductionquadruples and higher amplitudes. The same cancellatioerrors is observed in MP2 theory, where the contraction toccurs upon relaxation of the doubles is balanced bystretching that occurs upon the introduction of triples ahigher amplitudes. At the MP2 level, the cc-pVQZ basis-error is small relative to the total error in MP2. It is unknowwhether this cancellation of errors is fortuitous or a systeatic one@and would occur also for CCSDT~Q! and higher-order wave functions#.

IV. SUMMARY

In ab initio electronic-structure calculations, solutionsthe Schro¨dinger equation are obtained by introducing aproximations in the one- andN-electron spaces. We havinvestigated the accuracy that may be expected in the calated equilibrium structures when such approximationsintroduced in a hierarchical, systematic fashion. The oelectron space is spanned in a systematic fashion usingsequence of correlation-consistent polarized basis sets induced by Dunninget al.5–9 In theN-electron space, we havconsidered the hierarchy of models defined by the pertution series HF, MP2, MP3, and MP4 as well as the couplcluster-based hierarchy HF, MP2, CCSD, and CCSD~T!. Inboth hierarchies, the computational cost scales asn4, n5, n6,and n7, wheren is the number of orbitals. We have alsconsidered calculations using the CISD model. For statistsignificance, we have carried out calculations of equilibriustructures for 19 closed-shell molecules containing first-ratoms and a variety of chemical bonds.

The correlation-consistent basis sets give bond distanthat converge smoothly to within a few tenths of a picome

, No. 15, 15 April 1997


ndthen

ce

h

boern

this

rii-uthhe

e

tsttei

set

O.0e,

sti

l-

hiuromISnth

teorde

TZrese–oreaticandateuf-

ral

ys.

um

em.

d J.

e

G.

.. S., C.art,

ia,

rtlett,e,Lau-

ys.


at the cc-pVQZ level. For the perturbation models, we fithat the MP2 model gives remarkably accurate bond lengwhich are only marginally improved upon by the MP4 modat the cc-pVQZ level. MP3 has significantly larger deviatiofrom experiment than does MP2.

In the coupled-cluster hierarchy, the CCSD wave funtion gives bond lengths that are less accurate than thosthe MP2 level, whereas the accuracy at the CCSD~T! level iscomparable to that of most experimental investigations. Tmean absolute deviation between cc-pVQZ CCSD~T! bondlengths and experimental bond lengths is 0.22 pm, tocompared with the experimental uncertainties, which areten of the order of a few tenths of a picometer. The largdeviation between CCSD~T! and experiment is observed fothe NH bond length in HNO, where the experimental bolength is 1.2 pm longer than the CCSD~T! value. Based onthe documented high accuracy of the CCSD~T! model for theother bond distances in this investigation, we concludethe experimentally determined NH bond length in HNOincorrect.26

It may be of some interest to note that, in our compason of the CCSD~T! model with experiment, we have invarably found that experimental reinvestigations of previomeasurements have improved the agreement withCCSD~T! model. The most striking examples concern tbond distances in H2O2 and HOF, for which the initial com-parisons with old experimental work showed poor agreembetween experiment and theory~as for the NH bond lengthin HNO discussed above!. More recent experimental resulwere then searched for and found to give significantly beagreement with theory. Thus, experimental work on HOF1972 determined the OF distance at 144.2 pm.27 In 1988, twonew experimentally derived structures were published, baon reinvestigations of high-resolution spectra using theorcally determined anharmonic force fields.28,29 These investi-gations put the OF distance at 143.5 pm~Ref. 28! and 143.6pm,29 in better agreement with the cc-pVQZ CCSD~T! dis-tance of 143.2 pm. Similarly, in early measurements, theand OH distances in H2O2 were determined at 147.5 and 95pm, respectively.30 In 1993, an analysis of recent microwavmeasurements put these distances at 145.6 and 96.7 pm31 inconsiderably better agreement with the cc-pVQZ CCSD~T!distances of 145.0 and 96.1 pm. Clearly, differencespersist between theory and experiment for H2O2 and an ex-perimental reinvestigation may be worthwhile for this moecule.

The performance of the CISD model is so poor that tmodel cannot be recommended for the calculation of eqlibrium structures. Indeed, the accuracy of the geometry pdictions of this model deteriorates markedly as we go frsmall to large basis sets. Thus, the initial success of the Cmodel in the early days of correlated calculations rests ostrong cancellation of basis-set and correlation errors atcc-pVDZ level, in particular for small molecular systems.

Concerning basis sets, we find that, for all correlamodels, basis sets of at least cc-pVTZ quality are mandatOn the other hand, the improvements observed in bondtances when going to the larger cc-pVQZ basis sets ar



s,ls

-at

e

ef-st

d

at

-

se

nt

rn

edi-

O

ll

si-e-

Dae

dy.is-so

small that there is usually no need to go beyond the cc-pVlevel. In general, therefore, molecular equilibrium structushould be computed at the cc-pVTZ level. For the HartreFock wave function, the basis-set requirements are mmodest. Thus, although there are noticeable systemchanges in the bond distances between the cc-pVDZcc-pVTZ levels, the Hartree–Fock model is so inaccurthat for most applications the cc-pVDZ level should be sficient.

ACKNOWLEDGMENTS

This work has been supported by the Danish NatuResearch Council~Grant No. 11-0924! and the SwedishNatural Research Council~NFR!.

1C. W. Bauschlicher, S. R. Langhoff, and P. R. Taylor, Adv. Chem. Ph77, 103 ~1990!.

2P. R. Taylor, inLecture Notes in Chemistry 58, Lecture Notes in QuantChemistry, edited by B. O. Roos~Springer, Berlin, 1992!, p. 325.

3J. Almlof and P. R. Taylor, J. Chem. Phys.86, 4070~1987!.4P.-O. Widmark, P.-A˚ . Malmqvist, and B. O. Roos, Theor. Chim. Acta77,291 ~1990!.

5T. H. Dunning, J. Chem. Phys.90, 1007~1989!.6R. A. Kendall, T. H. Dunning, and R. J. Harrison, J. Chem. Phys.96, 6796~1992!.

7D. E. Woon and T. H. Dunning, J. Chem. Phys.98, 1358~1993!.8D. E. Woon and T. H. Dunning, J. Chem. Phys.100, 2975~1994!.9D. E. Woon and T. H. Dunning, J. Chem. Phys.103, 4572~1995!.10R. J. Bartlett, inModern Electronic Structure Theory, Part II, edited by D.R. Yarkony~World Scientific, Singapore, 1995!.

11K. Raghavachari, G. W. Trucks, J. A. Pople, and M. Head-Gordon, ChPhys. Lett.157, 479 ~1989!.

12P. J. DeFrees, B. A. Levi, S. K. Pollack, W. J. Hehre, J. S. Binkley, anA. Pople, J. Am. Chem. Soc.113, 1507~1991!.

13B. G. Johnson, P. M. W. Gill, and J. A. Pople, J. Chem. Phys.97, 7846~1992!.

14G. Scuseria and T. J. Lee, inQuantum Mechanical Electronic StructurCalculations with Chemical Accuracy, edited by S. R. Langhoff~KluwerAcademics, Dordrecht, 1995!, p. 47.

15H. F. Schaefer III, J. R. Thomas, Y. Yamaguchi, B. J. Deleeuw, andVacek, inModern Electronic Structure Theory, edited by D. R. Yarkony~World Scientific, Singapore, 1995!.

16K. Andersson and B. O. Roos, Int. J. Quantum Chem.45, 591 ~1993!.17J. Gauss and D. Cremer, Adv. Quantum Chem.23, 205 ~1992!.18M. Mladenovic, S. Schmatz, and P. Botschwina, J. Chem. Phys.101, 5891

~1994!.19M. J. Frisch, G. W. Trucks, M. Head-Gordon, P. M. W. Gill, M. WWong, J. B. Foresman, B. G. Johnson, H. B. Schlegel, M. A. Robb, EReplogle, R. Gomperts, J. L. Andres, K. Raghavachari, J. S. BinkleyGonzalez, R. L. Martin, D. J. Fox, D. J. DeFrees, J. Baker, J. J. P. Stewand J. A. Pople,GAUSSIAN 92, Gaussian Inc., Pittsburgh, Pennsylvan1992.

20J. F. Stanton, J. Gauss, J. D. Watts, W. J. Lauderdale, and R. J. BaACES II, Quantum Theory Project, University of Florida, GainesvillFlorida, 1992. See also, J. F. Stanton, J. Gauss, J. D. Watts, W. J.derdale, and R. J. Bartlett, Int. J. Quantum Chem.S26, 879 ~1992!.

21W. Klopper and H. P. Lu¨thi, Chem. Phys. Lett.262, 546 ~1996!.22H. Koch, P. Jo”rgensen, and T. Helgaker, J. Chem. Phys.104, 9528~1996!.23J. Noga and R. J. Bartlett, J. Chem. Phys.86, 7041 ~1987!; 89, 3401

~1988!.24G. Scuseria and H. F. Schaefer, Chem. Phys. Lett.152, 382 ~1988!.25J. D. Watts and R. J. Bartlett, J. Chem. Phys.93, 6104~1990!.26F. W. Dalby, Can. J. Phys.36, 1336~1958!.27H. Kim, E. F. Pearson, and E. H. Appelman, J. Chem. Phys.56, 1 ~1972!;E. F. Pearson and H. Kim,ibid. 57, 4230~1972!.

28L. Halonen and T.-K. Ha, J. Chem. Phys.89, 4885~1988!.29W. Thiel, G. Scuseria, H. F. Schaefer III, and W. D. Allen, J. Chem. Ph89, 4965~1988!.

, No. 15, 15 April 1997


-

sc.


30R. L. Redington, W. B. Olson, and P. C. Cross, J. Chem. Phys.36, 1311~1962!.

31G. Pelz, K. M. T. Yamada, and G. Winnewisser, J. Mol. Spectrosc.159,507 ~1993!.

32K. P. Huber and G. H. Herzberg,Molecular Spectra and Molecular Structure, Constants of Diatomic Molecules~Van Nostrand-Reinhold, NewYork, 1979!.

33A. R. Hoy, I. M. Mills, and G. Strey, Mol. Phys.24, 1265~1972!.34R. A. Creswell and A. G. Robiette, Mol. Phys.36, 869 ~1978!.35W. S. Benedict and E. K. Plyler, Can. J. Phys.35, 1235~1957!.36M. Carlotti, J. W. C. Johns, and A. Trombetti, Can. J. Phys.52, 340



~1974!.37A. Baldacci, S. Ghersetti, S. C. Hurlock, and K. N. Rao, J. Mol. Spectro59, 116 ~1976!.

38G. Winnewisser, A. G. Maki, and D. R. Johnson, J. Mol. Spectrosc.39,149 ~1971!.

39H. C. Allen and E. K. Plyler, J. Am. Chem. Soc.80, 2673~1958!.40D. L. Gray and A. G. Robiette, Mol. Phys.37, 1901~1979!.41J. L. Duncan, Mol. Phys.28, 1177~1974!.42P. Jensen and P. R. Bunker, J. Chem. Phys.89, 1327~1988!.43G. Cramer, C. Rosetti, and D. Baily, Mol. Phys.58, 627 ~1986!.44T. Tanaka and Y. Morino, J. Mol. Spectrosc.33, 538 ~1978!.

, No. 15, 15 April 1997


The prediction of molecular equilibrium structures by the standard electronic wave functions

Documents