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First-Order Relativistic Corrections toMP2 Energy from Standard GradientCodes: Comparison with Results fromDensity Functional Theory

¨ROBERT FRANKE, CHRISTOPH VAN WULLENLehrstuhl fur Theoretische Chemie, Ruhr-Universitat Bochum, D-44780 Bochum, Germany¨ ¨

Received 3 April 1998; accepted 8 June 1998

ABSTRACT: The evaluation of the first-order scalar relativistic corrections toMP2 energy based on either direct perturbation theory or the mass]velocity andDarwin terms is discussed. In a basis set of Levy-Leblond spinors the one- and´two-electron matrix elements of the relativistic Hamiltonian can be decomposedinto a nonrelativistic part and a relativistic perturbation. Thus, a programcapable of calculating nonrelativistic energy gradients can be used to calculatethe cross-term between relativity and correlation. The method has been applied

Ž . Žto selected closed-shell atoms He, Be, Ne, and Ar and molecules CuH, AgH,.and AuH . The calculated equilibrium distances and harmonic frequencies were

compared with results from first-order relativistic density functional calculations.It was found that the cross-term is not the origin of the nonadditivity ofrelativistic and correlation effects. Q 1998 John Wiley & Sons, Inc. J ComputChem 19: 1596]1603, 1998

Keywords: direct perturbation theory; MP2 gradients; density functional theory;relativistic effects; nonadditivity of relativity and correlation

Introduction

olving the Dirac]Hartree]Fock problem isS the initial step in relativistic calculations formany-electron systems. Standard methods to rem-

Correspondence to: R. FrankeContractrgrant sponsor: Deutsche Forschungsgemeinschaft

edy deficiencies of the one-electron approximationconnected to the so-called ‘‘dynamic’’ correlation

Ž .effect are the configuration interaction CI ap-Ž .proach, many-body perturbation theory MBPT

techniques, and, most promising, the coupled clus-Ž .ter CC methods. In general, these methods are

essentially the same as in the case of nonrelativis-tic theory. Fully relativistic four-component calcu-lations at the CI level1 as well as at the MBPT

( )Journal of Computational Chemistry, Vol. 19, No. 14, 1596]1603 1998Q 1998 John Wiley & Sons, Inc. CCC 0192-8651 / 98 / 141596-08

FIRST-ORDER CORRECTIONS TO MP2

level2 have been developed for molecules. A fullyrelativistic Fock-space CCSD method for atoms hasbeen implemented and applied to closed- andopen-shell atoms.3 The major difference from a

Žnonrelativistic treatment from a computational.point of view is that the relativistic functions are

complex four-component spinors. Thus, in spite ofsymmetries between integrals, fully relativistic cal-culations are far more time-consuming than theirnonrelativistic counterparts. To handle molecularsystems with heavy atoms and a large number ofelectrons, which is already a serious task in nonrel-ativistic calculations, much effort has been spenton developing methods that can be consideredas approximations to the complete relativistictreatment. The most widely used approach is based

Ž .on the relativistic effective core potentials RECPsŽ .see, e.g., ref. 4 . The one-electron mass]velocity

Ž .Darwin MVD term has been used to calculate avariety of molecular properties including relativis-

Ž . Žtic correlation contributions up to MBPT 4 see,.e.g., ref. 5 . To take higher order contributions

to relativistic corrections into account, methodsbased on the Douglas]Kroll]Hess approach,6 theZORAr CPD scheme,7 ] 9 and direct perturbation

Ž .10 ] 12theory DPT have been developed. Recently,Klopper gave a simple recipe for the calculation ofthe scalar first-order relativistic corrections in theframework of DPT.13 It will be shown that thisrecipe follows from a representation of the one-and two-electron matrix elements of the Dirac]

Coulomb operator in a basis of Levy-Leblond´spinors. Calculations based on an implementationof this scheme into a nonrelativistic MP2 energygradient code for selected closed-shell atoms andmolecules are reported and compared with datafrom the literature. In addition, for molecular sys-tems, we carried out first-order relativistic density

Ž .functional DFT calculations for comparison.

Method

The key idea of DPT is the different handling ofpositronic and electronic eigenstates of the Diracequation. For the latter, a transformation in four-component spinor space is applied to the Dirac

Ž .equation, which replaces the bispinor C s w, x

ww w˜ Ž .C s ª C s s 1ž / ž /x cxž /x̃

where c is the velocity of light in atomic units, wthe so-called ‘‘large component,’’ and x the ‘‘smallcomponent.’’ This transformation changes the met-ric from 1 to

1 0ˆ Ž .S s 2y2ž /0 c

ând the one-electron Dirac operator, D, after sub-tracting the rest mass, to

ªªV̂ s pˆ Ž .D s 31ªª ˆs p y2 q ? V� 02c

ªˆwhere V is the nuclear potential operator, p is theªmomentum operator, and s the vector of the three

Ž .Pauli matrices s , s , s . From the second line ofx y zthe one-electron Dirac equation in modified metric

ˆ ˆ Ž .DC s ESC 4

a relation is obtained between the small and largecomponent

ªªs pw 1 ªª y2Ž . Ž .x s s s pw q O c 51 2ˆŽ .2 y V y E2c

A special choice of four-component single-particlebasis functions, the Levy-Leblond spinors, shallńow be considered

fw9 Ž .C9 ' s 6ªª1ž /x 9 ž /s pf2

where f is a two-component spinor, g ? h, consist-ing of the spatial function g and a spin function h.

Ž .Functions according to eq. 6 are solutions of theLevy-Leblond equation,14 a linear nonrelativistic´wave equation for spin 1r2 particles. This equa-tion is the limit c ª ` of the Dirac equation with

Ž . 11c-dependent metric 4 for electronic states. Func-Ž .tions of the form defined by eq. 6 limit the

domain of the Levy-Leblond operator, which then´becomes bounded from below.11 In expanding thelarge component w of the Dirac spinor in modified

Ž . y2metric 1 in powers of c

` 12 k Ž .w s w 7Ý 2 kcks0

with the solution of the nonrelativistic Schrodingerëquation, w 0, and performing the nonrelativistic

Ž .limit c ª `, one gets from eq. 5 a relation be-tween large and small component, which holds

JOURNAL OF COMPUTATIONAL CHEMISTRY 1597

¨FRANKE AND VAN WULLEN

Žexactly for the Levy-Leblond spinors if g is cho-´.sen as a solution of the Schrodinger equation .¨

Thus, for sufficiently large values of c—that is,small relativistic perturbation—functions of thistype are suitable basis functions for solving the

Ž .Dirac equation in modified metric 4 in its alge-braic approximation: wave functions for which all

Ž y2 .expectation values are correct up to O c can beexpanded in such a basis set, and, for large valuesof c, the Dirac operator is bounded from below inthe variational space. There is a one-to-one map-

Ž .ping between nonrelativistic two-component andŽ .relativistic four-component basis functions. Once

the one- and two-particle matrix elements of theDirac]Coulomb operator have been obtained, thecalculation proceeds as in the nonrelativistic case.Because the basis set restricted to the Levy-Leblond´

Ž y2 .spinors can give a wave function only up to O cthis method cannot be used to calculate higherorder relativistic corrections.

In the basis of the Levy-Leblond spinors one´gets for the matrix elements of the one-electronDirac operator in c-dependent metric

X ˆ X ˆ ˆ² < < : ² < < : ² :D s C D C s g T q V g ? h N hmn m n m n m n

1 ªª ªªˆ² < < : Ž .q f s pVs p f 8m n24c

ˆ Ž .where T is the nonrelativistic kinetic energy op-erator, and for the metric

² X X: ² : ² :S s C N C s g N g ? h N hmn m n m n m n

12² < < : ² : Ž .q g p g ? h N h 9ˆm n m n24c

For choosing the electron interaction as

1Ž .V i , j s

r12

thus describing the many-electron system by theŽDirac]Coulomb Hamiltonian the argumentation

given here is not restricted to this special choicefor the electron interaction and holds also for the

.Gaunt and Breit interaction one gets, for the two-electron matrix elements in Mulliken notation

Ž X X X X .C C N C Cm n r s

Ž .s f f N f fm n r s

1 ªª ªªq s pf s pf N f f½ m n r sž /24cªª ªª y4Ž . Ž .q f f N s pf s pf q O c 105m n r sž /

Inspection of the relativistic one- and two-electronmatrix elements in the basis of Levy-Leblond´spinors shows a decomposition into a nonrelativis-tic part and a relativistic part with the leading

Ž y2 .order O c . Thus, the scalar first-order relativis-tic correction in the formulation of the DPT holdsthe same as for the quasi-relativistic Pauli opera-tor. For both formulations, scalar first-order rela-tivistic energy corrections can be obtained by us-ing codes for analytical energy gradients where theintegrals describing the nuclear displacement arereplaced by the scalar relativistic parts of the inte-

Ž y2 .grals up to O c given previously. This is therecipe given in a recent publication by Klopper.13

In the course of this work an extension of theMPGRAD program,15 which is part of the TURBO-MOLE package,16 according to the following work-ing equations

1 ª ª 2ˆ² < < : ² < < :E s Y r pVp s q W r p sˆÝ ÝMP2DPT r s r s½24c rs rs

ªª Ž .q G rs N ptpu 11Ž .Ý r st u 5rstu

1 ª3² < < :E s Y 2p Z r d r sŽ .Ý ÝMP2MVD r s A A2 ½4c rs A

14² < < :y r p sˆ

2

ª3² < < : Ž .y2p G rs d r tu 12Ž .Ý r st u 12 5rstu

and the corresponding equations for the calcula-tion of the relativistic corrections of the SCF en-

17 Ž .ergy have been implemented. In eqs. 11 andŽ .12 , Y denotes the relaxed density matrix, W theenergy-weighted density matrix, and G the re-laxed two-particle density matrix, as known fromthe theory of MP2 gradients.18, 19 Z is the nuclearA

charge of atom A. The MP2 results reported in thiswork were obtained from calculations in which allelectrons were correlated. The relativistic densityfunctional calculations were carried out with acode on top of the TURBOMOLE package,20 whichwas augmented by a routine for the calculation ofthe two-electron Darwin term.21 The BP-86 func-tional22, 23 was used throughout. For the velocity oflight, a value of 137.0359895 a.u.24 was used. Nu-clei were considered point particles. The calcula-tions have been carried out in part on IBM RSr6000

VOL. 19, NO. 141598


workstations in our laboratory and partly on theSGI PowerChallenge XL-12 of the Rechenzentrumder Ruhr-Universitat Bochum.¨

Results and Discussion

Calculations for the scalar first-order relativisticcorrection to the MP2 energy by means of DPT, aswell as perturbation theory based on the MVDoperator, were performed for the ground state ofthe closed-shell atoms He, Be, Ne, and Ar. Theenergy data are presented in Table I. The calcula-tions were carried out with basis sets used in thequoted references to make a direct comparisonpossible. From a methodological point of view, ourcalculations are essentially the same as those fromref. 13. They differ only in the manner of imple-mentation. The results of ref. 13 were calculated byapplying finite perturbation theory instead of us-ing analytical gradients. Note that ref. 13 addition-ally contains relativistic corrections to MP3, MP4,

Ž .CCSD, and CCSD T correlation energies. The re-sults obtained in this work replicate those of ref. 13in regard to all figures obtained. For the He and Beatoms, comparison with results from fully rela-

Ž .tivistic calculations ref. 3 shows agreement withinŽ .the limited accuracy one digit given in that work.

In the case of both the He and Be atoms, theDPT and MVD results differ only by a few hun-dredths of mE for the basis sets used. If onehcompares the MVD and DPT results from calcula-tions with smaller basis sets, particularly forsmaller sets of s-functions, the difference increases.For a basis set of TZP size, as typically used inquantum chemical calculations, the difference be-tween them is 62 mE for the Ne atom. The differ-hence between the TZP value and the result of acalculation with the considerably larger 14s10p8d6fbasis amounts to 16 mE for DPT and 76 mE forh hMVD. This reflects the superior convergence be-havior of the DPT compared to MVD approxima-tion with respect to atom-centered Gaussian basisfunctions. This has already been found for one-electron systems.25 As expected, the error of theapproximate relativistic calculations increases withincreasing value of the nuclear charge. Compari-son with the DHF-MP2 values of refs. 26, 27, and28 shows that the absolute error for DPT calcula-tions is 8 mE for Ne and 84 mE for Ar, whichh hcorrespond to a relative error of 3% for Ne and 8%for Ar.

The one- and two-electron contributions for theMVD correction to the MP2 energy are of the sameorder of magnitude but with different signs, whichmeans that one should take two-electron Darwin

TABLE I.MP2 Contributions to Scalar First-Order Relativistic Correction to Ground-State Energy of Selected Closed-Shell

( )Atoms Energies Given in mE .h

One-electron Two-electron RelativisticAtom Basis Method contribution contribution correction Reference

3He 20s15p11d DTP 1.67 This workMVD y10.466 12.122 1.66 This workDHF-MP2 2.0 Ref. 3

3Be 24s10p12d DPT 8.05 This workMVD y56.967 64.982 8.01 This workDHF-MP2 8.0 Ref. 3

30Ne TZP DPT y258.45 This workMVD y520.771 199.873 y320.90 This work

2614s10p8d6f DPT y242.82 This workDPT y243.0 Ref. 13MVD y803.430 558.655 y244.78 This workMVD y245.0 Ref. 13DHF-MP2 y251.0 Refs. 27, 28

28Ar 16s11p9d7f DPT y995.77 This workDPT y996.0 Ref. 13MVD y3146.225 2144.883 y1001.34 This workMVD y1001.0 Ref. 13DHF-MP2 y1080.0 Ref. 28



FIGURE 1. Potential energy curves of AuH. The SCF energies are shifted by y986.85 Hartree, the MP2 energies by( ) ( )y984.00 Hartree, and the SCF-DPT energies by y2.65 Hartree. The MP2-DPT a and MP2-DPT b values are plotted

unchanged. The equilibrium distances are indicated by inverse colored plot symbols and given in angstrom units.( ) ( ) ( ) ( ) ( ) ( ) ( )l SCF, B MP2, ' SCF-DPT, v MP2-DPT a , ^ MP2-DPT b .

term into account if one is interested in energydata calculated from the MVD perturbation. Thisis much less pertinent in the SCF case. For the Neatom and the 14s10p8d6f basis set one gets a valueof y137331 mE for the one-electron contributionh

and 117 mE for the two-electron Darwin contribu-htion at the Hartree]Fock level.

The coinage metal hydrides represent one of themost popular sets of molecules for investigatingthe different approximations in relativistic theory.We report on the study of bond lengths and har-monic frequencies within the DPT and MVD ap-proximation employing MP2 as well as DFT withthe BP-86 functional.22, 23 The equilibrium distance

and the vibrational frequency have been obtainedfrom a polynomial fit to a typical set of seven datapoints in the vincinity of the equilibrium distance.We compared our data mainly with DHF andDHF-MP2 results from a recent study by Collins etal.2 For Cu, we used a 20s14p10d3f primitive basisset, starting from the 20s13p10d set of ref. 29,

Ž .adding a p-set using the next exponent § from18

the well-tempered series, and adding three f-setsŽ .with exponents § ]§ for notation see ref. 29 .15 17

This basis has been contracted to 15s10p7d3f. ForAg, we started from the 23s16p13d basis set of ref.

Ž . Ž .29 and added a p-set § and five f-sets § ]§ .21 17 21

The basis was then contracted to 18s13p10d5f. For

VOL. 19, NO. 141600


Au, the 24s18p15d10f basis set of ref. 29 was aug-Ž .mented by one p-set § and contracted to22

18s13p11d7f. For the hydrogen atom a standardTZP basis set30 was employed.

The results for equilibrium bond distances andharmonic frequencies for CuH are presented inTable II. To investigate the effect of the two-elec-tron Darwin term on molecular properties we ad-ditionally report results for the one-electron MVDperturbation denoted ‘‘1e.’’ The E andMP2DPTE contributions incorporate the nonadditiv-MP2MVDity of relativistic and correlation effects, respec-tively. To examine the importance of this contribu-tion, results are given for adding only the MP2energy to the relativistic SCF results denoted byŽ . Ž .‘‘ a .’’ Variant ‘‘ b ’’ also includes the cross-terms

Ž . Ž .given by eqs. 11 and 12 . The SCF and MP2results of this work are in excellent agreementwith the values of ref. 13. Comparison with theDHF values of ref. 2 shows that the nonrelativisticequilibrium distance on the Hartree]Fock level isreproduced perfectly, while there is a small devia-

˚tion of 0.001 A between DHF and SCF-DPT. Thefrequencies differ by 14 cmy1 for the nonrelativis-

TABLE II.˚( )Bond Lengths r in A and Harmonic Frequenciese

y 1( )v , in cm of CuH Calculated with a 20s14p10d3fePrimitive Basis Contracted to 15s10p7d3f for Cuand a Standard TZP Basis for H Compared with

aValues from Literature.

Nonrelativistic Relativistic

Method r v r ve e e e

SCF-DPT 1.569 1656 1.542 1709SCF-MVd1e 1.542 1708SCF-MVD 1.542 1709

( )MP2-DPT a 1.447 2057 1.426 2133( )MP2-DPT b 1.423 2158

MP2-MVD1e 1.423 2157( )MP2-MVD a 1.426 2134( )MP2-MVD b 1.423 2158

DFT-DPT 1.473 1940 1.452 2008DFT-MVD1e 1.453 2008DFT-MVD 1.452 2009

13SCF-DPT 1.569 1.54213MP2-DPT 1.450 1.423

2DHF 1.569 1642 1.541 16992DHF-MP2 1.454 2024 1.428 2101

33Exp. 1.463 1941

a ( ) (Variant a without the cross-term of E resp.MP2DPT) ( )E between relativity and correlation; variant b in-MP2M VD

cludes this term. ‘‘1e’’ denotes that the two-electron contri-bution is neglected.

tic case and 10 cmy1 for the relativistic calcula-tions. On the MP2 level, the deviations are slightly

˚ ˚Ž .larger. The bond length is 0.007 A 0.005 A shorter,as the DHF-MP2 value from ref. 2 and the har-

y1 Ž y1 .monic frequency 33 cm 57 cm larger for theŽ .nonrelativistic relativistic case. The main reasons

for these deviations probably involve the use ofdifferent basis sets and the different treatment ofcore correlation in both studies. The DFT-DPT re-sults show significantly larger differences from the

˚ ˚Ž w xDHF-MP2 results 0.019 A 0.024 A longer and 84y1 w y1 x wcm 93 cm smaller in the nonrelativistic rela-

x.tivistic case , but match the experimental valuesbetter than those from the MP2-DPT or DHF-MP2calculations. The results from MVD and DPT cal-culations are very similar for all methods. Inclu-sion of the two-electron Darwin term has almostno effect on bond lengths and harmonic frequen-cies.

The effect of the cross-term between relativityand correlation is small but significant. Neglect of

˚this contribution leads to a 0.003-A-longer bondlength for both MVD and DPT and a smaller

Ž y1 y1harmonic frequency 25 cm for DPT, 24 cm.for MVD .

Ž .The results for AgH Table III show the samegeneral behavior as those for CuH. The deviationsbetween the DHF-MP2 values of ref. 2 and MP2-DPT, as well as MP2-MVD results, are similar tothose in the case of CuH. Again, there is a greaterdifference between the full relativistic values fromref. 2 and the DFT results. The DFT values areremarkably close to the experimental data andvery similar to the ZORA-DFT values of ref. 7.Neglect of E or E contributions leadsMP2DPT MP2MVDto an increase in bond length of approximately

˚0.006 A.The relativistic and nonrelativistic potential en-

ergy curves of AuH are shown in Figure 1. Rela-tivistic corrections are essential for the correct pre-diction of the molecular properties of AuH.

The relativistic contraction on the Hartree]Fock˚level of theory is 0.203 A, calculated with first-order

DPT. The corresponding value of ref. 2, calculated˚with DHF, is 0.261 A. The contraction is signifi-

cantly larger due to the importance of higher orderrelativistic contributions. The contraction effectedby correlation determined on the MP2 level of

˚theory calculated in this work is 0.128 A. Collins et˚ 2al. reported a value of 0.120 A. Assuming a simple

additive relationship between the two, one wouldanticipate a combined relativistic and correlation

˚contraction of 0.331 A, based on the values in this˚work, and 0.381 A using the data of ref. 2, but the



TABLE III.˚( )Bond Lengths r in A and Harmonic Frequenciese

y 1( )v in cm of AgH Calculated with a 23s17p13d5fePrimitive Basis Contracted to 18s13p10d5f for Agand a Standard TZP basis for H Compared with




SCF-DPT 1.777 1490 1.706 1601SCF-MVD1e 1.707 1600SCF-MVD 1.706 1601

( )MP2-DPT a 1.651 1732 1.587 1883( )MP2-DPT b 1.580 1907



2DHF 1.779 1473 1.700 16052DHF-MP2 1.663 1699 1.585 18737DFT-ZORA 1.62 1810

34Exp. 1.618 1760



Ž .difference between the MP2-DPT b value for thebond length and the value of the nonrelativistic

˚SCF calculation is only 0.312 A; that is, relativisticcontraction at the MP2 level is much smaller. The

˚corresponding value from ref. 2 is 0.334 A. Inspec-tion of Tables II and III shows that this distinctivenonadditivity is not present in the cases of CuHand AgH. A remarkable consequence of this studyis that this nonadditivity is not connected to thecross-terms E or E —the differenceMP2DPT MP2MVD

Ž .between the MP2-DPT a value for the bond lengthand the value of the nonrelativistic SCF calculation

˚is 0.304 A. Adding independently the relativisticand correlation corrections to the Hartree]Fockpotential energy curve yields bond length andvibrational frequencies not far different from theresults of calculations including the cross-term.However, one may not, for example, simply addthe relativistic and correlation bond contractions toevaluate the equilibrium distance starting from theHartree]Fock value.

In comparison with the DHF-MP2 data, the rel-ativistic equilibrium distance calculated with first-

˚Ž .order MP2-DPT b is 0.018 A too long and the˚DFT-DPT value 0.061 A too long. The MP2-DPT

TABLE IV.˚( )Bond Lengths r in A and Harmonic Frequenciese

y 1( )v in cm of AuH Calculated with a 24s18p15d10fePrimitive Basis Contracted to 18s13p11d7f for Auand a Standard TZP Basis for H Compared with




SCF-DPT 1.824 1499 1.621 1915SCF-MVD1e 1.620 1911SCF-MVD 1.620 1920

( )MP2-DPT a 1.696 1736 1.520 2284( )MP2-DPT b 1.512 2321



2DHF 1.831 1464 1.570 20672DHF-MP2 1.711 1695 1.497 2496

31DKH-CCSD 1.525 22887DFT-ZORA 1.54 2290

4ARPP CEPA-1 1.512 225433Exp. 1.524 2305



Ž .b value is in excellent agreement with the valueof Schwerdtfeger et al.4 calculated with a relativis-tic pseudopotential and the CEPA-1 approxima-tion. Concerning the DKH-CCSD approach, whichgives the best theoretical prediction of propertiesfor AuH documented so far,31 the deviation of the

˚Ž .MP2-DPT b value is 0.013 A for the equilibriumdistance and 33 cmy1 for the harmonic frequency.Although the quality of the MP2 and DFT results

Žare certainly comparable MP2 bond lengths are.slightly too small, DFT bond lengths too long , one

must remember that the DFT calculations are atleast one order of magnitude less demanding.

Conclusions and Outlook

It has been shown that the decomposition of thematrix elements of the DPT approach into a non-relativistic part and a relativistic perturbation fol-lows from a matrix representation in a basis ofLevy-Leblond spinors. The first implementation´for the calculation of relativistic corrections to the

VOL. 19, NO. 141602


MP2 energy by means of DPT and MVD with aMP2 gradient program has been described. Com-parison with relativistic DFT calculations and liter-ature data indicates the reliability of this method.It seems to be very promising to implement thisansatz into an RI-MP2 gradient program such asthe one most recently developed by Weigend andHaser.32¨

Acknowledgment

Ž .We thank E. Eliav Ilyabaev for providing uswith basis sets, W. Klopper for sending a preprintof ref. 13 prior to publication, and W. Kutzelniggfor helpful comments.

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