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Chemical Physics 311 (2005) 63–69

Benchmark four-component relativistic densityfunctional calculations on Cu2, Ag2, and Au2

Fan Wang, Wenjian Liu *

Institute of Theoretical and Computational Chemistry, State Key Laboratory of Rare Earth Materials Chemistry and Applications, College of

Chemistry and Molecular Engineering, Peking University, Beijing 100871, PR China

Received 7 February 2004; accepted 12 October 2004Available online 6 November 2004

Abstract

Optimization of basis functions with the analytical gradient method is implemented at the level of four-component relativisticdensity functional theory (DFT). The basis set requirements for the molecular spectroscopic constants (i.e., bond length, dissocia-tion energy, vibrational frequency, and parallel component of the dipole polarizability tensor) of Cu2, Ag2 and Au2 are investigated,indicating that including up to g-type functions in the basis set is sufficient, whereas the effects of higher angular momentum func-tions are negligibly small. The basis set limit results are estimated and should be taken as benchmark for calibration of other densityfunctional calculations, in particular, those using the same exchange–correlation functionals.� 2004 Elsevier B.V. All rights reserved.

1. Introduction

Density functional theory (DFT) [1] has become themost popular approach for quantum chemical calcula-tions of large systems due to its low computational costand good accuracy for the calculated results. Within theKohn–Sham (KS) [2] implementation of DFT, there aretwo factors that determine the actual accuracy, i.e., thechosen exchange–correlation (XC) functional and one-particle basis set to expand the KS orbitals. The keyproblem is that there is no systematic way to improvethe quality of the XC functionals, although a numberof XC functionals with impressive accuracy have beendeveloped in the last years (for a recent review see [3]).Some of these functionals were determined mainly bythe consideration of known physical properties, andthus can be termed as �physicist�s functionals�. However,the majority of the existent functionals contain a num-ber of parameters that were fixed by fitting the com-

0301-0104/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2004.10.019

* Corresponding author.E-mail address: lwj@chem.pku.edu.cn (W. Liu).

puted results of a training set of systems toexperimental values. Noticeably, such functionals areoften combined with a particular finite basis set, typi-cally of triple-f (TZ) or even double-f (DZ) quality.Such functionals can best be viewed as �basis set func-tionals� or �chemist�s functionals�, since little physics istaken into account in the fitting procedure. In this case,the basis set is an integral part of the XC functionaldevelopment, and the final error is then a mixture of er-rors in the XC functional and the basis set. Recently,Boese et al. [3] investigated extensively the effects of dif-ferent basis sets on the parameters and the transferabil-ity of the errors of the functionals to different basis sets.They concluded that ‘‘the functionals obtained by fittingto one basis set are transferable to other basis sets’’, and‘‘it is probably not important to reach basis set limitwhen developing new density functionals’’. Their con-clusions were drawn upon the fact that the errors ofthe existent XC functionals are still quite large with re-spect to chemical accuracy. However, when more accu-rate XC functionals are to be developed, such an errortransferability is questionable. In addition, as pointed

64 F. Wang, W. Liu / Chemical Physics 311 (2005) 63–69

out by Ahlrichs et al. [4], different implementations ofthe same functional may sometimes lead to different re-sults. If basis set errors are buried in the functionals, thesituation may even be more confusing. Therefore, it isworthwhile studying the basis set limit results in orderto uncover the inherent errors embedded in the XCfunctionals [5,6].

When systems containing heavy elements are studied,relativistic effects have to be taken into account for reli-able descriptions. Relativistic DFT [7–9] was developedshortly after the birth of DFT. Later, it was realized thatrelativistic corrections to the XC functionals only affectinnermost core shells [10] but have no discernible effectson valence electron properties of heavy [11] and evensuperheavy elements [12]. This alleviates greatly the needfor genuine relativistic XC functionals, since nonrelativ-istic XC functionals can safely be used in relativistic DFTcalculations of heavy elements. However, none of theavailable functionals has been �trained� particularly forheavy elements. The good accuracy of the approximateXC functionals obtainable for light elements cannot beexpected in general for heavy elements. This raises theneed for reparametrization of the functionals. However,before this is done, one has to be aware of the inherenterrors of the functionals. Namely, one has to go to basisset limit and had better use the Dirac operator itself toavoid possible errors hidden in approximate relativisticHamiltonians derived from some ad hoc transformationsof the Dirac operator. Although a wide variety of basissets have been developed for light elements, systematicconstructions of decent basis sets for use in relativisticcalculations turn out to be much more difficult [13] andstill deserve ongoing efforts. Caution has to be taken ofthe results derived from four-component Hamiltonianbut with small basis sets: such results [14,15] are by nomeans benchmark for calibration of more approximateapproaches [16]! A four-component relativistic densityfunctional program package, Beijing density functional(BDF) [17,18], was developed in our group a few yearsago. In this package, numerical atomic orbitals (spinors)(NAO) augmented with additional kinetically balancedSlater-type functions (STF) are used as basis sets. Sucha basis set is most compact and yet efficient accordingto our experience. BDF can therefore provide bench-mark results (for a specific XC functional) at relativelylow computational efforts.

In this paper, we implement a scheme for analyticaloptimization of the exponents of the STFs, and investi-gate basis set effects on the spectroscopic constants ofCu2, Ag2 and Au2. These systems are chosen becausethey have simple electronic structure and have been ta-ken as prototypical systems for testing various relativis-tic methods. Although there exist in the literature somefour-component DFT calculations on these systems withextended basis sets (e.g., [19,20]), the basis set limit re-sults still have not been available. BDF accommodates

a number of XC functionals. However, for the presentpurpose, only the local density approximation (LDA)[21] and BP86 [22,23] functionals are considered. Sys-tematic investigations on more XC functionals and sys-tems are of course highly desirable. Here, we mentionthat Swart and Snijders [24] recently investigated theperformance of several XC functionals on the geome-tries of some metallocenes with focus on both basis setand relativistic effects. However, only scalar relativisticeffects have been taken into account and no results onthe energetics have been reported. An extensive investi-gation on the basis set requirements in the ZORA (zer-oth order regular approximation [25–28]) calculationswas also made by van Lenthe and Baerends [29].

2. Optimization of basis set

As mentioned above, combinations of NAOs and va-lence-type STFs are used as basis set in BDF. The NAOsare obtained by solving the Dirac–Kohn–Sham (DKS)equation for a free atom (central field) with finite-differ-ence method, while the STFs are normally chosen bymodifying those nonrelativistic ones [30–32], keepingin mind that s and p1/2 orbitals will be relativisticallycontracted, while p3/2, d, and f orbitals will be relativis-tically expanded. One may also employ the STFs thatwere recently obtained by fitting to numerical scalar rel-ativistic ZORA atomic orbitals [29]. The use of NAOsfor inner shells of heavy elements avoids the difficultiesassociated with optimization of steep functions thatare necessary to describe the weak singularities of inner-most shells such as 1s and 2p1/2. It has been shown thatpoor representation of these shells may have sizeable ef-fects on valence properties, especially of heavy p-blockelements [12]. The (valence) STFs mainly take care ofthe deformation and polarization of atoms when form-ing a molecule. Since the NAOs serve as the backbone(strongly occupied) in the molecular expansions, theadditional STFs usually do not require further optimiza-tion. To confirm this and provide more accurate results,we now implement the optimization of the STF expo-nents with the analytical gradient method. The deriva-tive of the total energy with respect to the exponent atakes the following form:

oEoa

¼Xlm

P lm

ovloa

j F j vm� �

þ vl j F j ovmoa

� �� �

�Xlm

W lmovloa

j vm� �

þ vl jovmoa

� �� �; ð1Þ

F ¼ T þ V N þ V H þ V XC; ð2Þ

P lm ¼Xi

niCliC�mi; ð3Þ

F. Wang, W. Liu / Chemical Physics 311 (2005) 63–69 65

W lm ¼Xi

�iniCliC�mi; ð4Þ

where {vl} are STFs and {ei} are orbital energies. Pand W are the density and energy-weighted densitymatrices, respectively. In Eq. (2), T ; V N; V H; and V XC

are the kinetic operator, nuclear potential, Coulombpotential, and exchange–correlation potential, respec-tively. It has been assumed in the above that the energyhas been SCF converged, i.e., the following stationaryconditions hold

Fui ¼ �iui; ð5Þ

ui ¼Xl

vlCli: ð6Þ

Since the exponents may change in a relatively largerange, we use oE

o ln a ¼ a oEoa instead of oE

oa to make the optimi-zation more stable [33]. The evaluation of the derivativescan be carried out at nonrelativistic, and scalar relativis-tic ZORA, two-component ZORA, and four-compo-nent DFT levels with or without frozen coreapproximation, since they differ only in the kinetic oper-ator. The implementation closely follows that of theanalytical energy gradient already available in BDF[34]. The BFGS [35] scheme is used to update the Hes-sian matrix. In the optimization process the exponentsmay sometimes meet each other and cause lineardependency of the basis set. To avoid this, we haveadopted the same auxiliary term to the total energy asproposed by Porezag and Pederson [33]

Eaux ¼Xa 6¼a0

0:003675x ð1� xÞ3 ðx 6 1Þ

0 ðx P 1Þ

(; ð7Þ

where x ¼ lnðaÞ�lnða0Þln 1:5

� �2

.

To avoid possible linear dependence between theNAOs and any STF to be optimized, an exponent foreach radial NAO should be determined for use in Eq.(7). For this, a normalized STF multiplied with r2 isleast-squares fitted to the large component of a NAOmultiplied with r2 beyond the outmost node. In fact, thisauxiliary term is not needed in most of our calculations.With NAOs in the basis set, the derivatives of the totalenergy with respect to the exponents of the STFs areusually very small and the exponents may change in arelatively large range without significant effects on themolecular properties. This is consistent with our obser-vation that the NAOs usually play the role of backbonein the basis set.

Variational collapse has been an issue in four-compo-nent calculations [36]. The remedy is to use the kinetic-balance condition [37], i.e., the small components of theSTFs should be calculated as

/S ¼ 1

2c~r �~p/L: ð8Þ

However, we have recently demonstrated [38] thatsuch a condition is unnecessary in our case, again dueto use of NAOs. Nonetheless, the condition has been ap-plied in the present calculations.

3. Computational details

The technicalities adopted in BDF have been de-scribed in great detail elsewhere [17,18]. In the presentcalculations, double point group D�

1h symmetry is usedfor the DKS wavefunction, while cylindrical symmetryis used for the numerical integration, which consists of100 radial and 80 angular grid points. The energy differ-ence between the molecule and superposition of spheri-cal atoms is directly evaluated with the generalizedtransition-state method [39] to further suppress numeri-cal noises due to numerical integration. As for basis setsin the following section, single-f (SZ) denotes the NAOs,and double-f denotes that an additional STF is used foreach spinor of all the (n�1) and n-shells (n = 4, 5, 6 forCu, Ag, and Au, respectively). Still larger basis sets (e.g.,triple-f, quadruple-f (QZ)) indicate that more STFs areaugmented for the valence (n�1)d, ns, and np shells. Allthe STFs are optimized in the molecular, instead ofatomic, calculations, with the interatomic distance cho-sen as the equilibrium bond length obtained from LDAcalculations with the TZP basis set. Since the shape oforbitals does not depend much on the chosen XC func-tionals [6,29], the optimization is done only at the LDAlevel. No frozen-core approximation is used in all thecalculations. Nonrelativistic XC functionals, LDA andBP86, are used as they stand, since relativistic correc-tions to the functionals have no significant effects onthe systems studied here (<1 pm in bond lengths, <1kcal/mol in dissociation energies, and about 2 cm�1 invibrational frequencies [11]). The equilibrium bondlength and vibrational frequency are obtained by suita-ble polynomial fitting of point-wise calculations. Themoment polarized atomic reference [40] is used to derivethe dissociation energy. The parallel component (ajj) ofthe dipole polarizability tensor is calculated with the fi-nite-field method at the corresponding equilibrium bondlength.

4. Results and discussion

The gold dimer is perhaps the most important bench-mark molecule to test relativistic electronic structuretheories. Recently, Hess and Kaldor [41] did all-electroncoupled-cluster calculations on the molecular spectro-scopic constants of Au2 using second order scalar

66 F. Wang, W. Liu / Chemical Physics 311 (2005) 63–69

relativistic one-electron Douglas–Kroll–Hess approach.They observed that it is necessary to include at least h-type functions in the basis set to obtain reliable resultsand the basis set superposition error (BSSE) [42] has sig-nificant effects on the results. Since most previous calcu-lations on Au2 did not take the BSSE into account, theysuspected that the satisfactory results of relativistic DFTon Au2 may also rely on the BSSE. To check this, wecalculate the properties of Au2 at different levels ofoptimized basis sets using four-component DFT. Forcomparison, the lighter homologues Cu2 and Ag2 arealso investigated. The calculated bond lengths, dissocia-tion energies, harmonic vibrational frequencies, and di-pole polarizabilities are given in Tables 1–3.

The SZ basis set (i.e., a single set of NAOs) isobviously insufficient for molecular calculations, incontrast to the statements made in [14,15]. The DZbasis set improves quite a lot, and the DZ resultsare actually rather close to those of TZ, the differ-ences being only 0.5–1 pm in the bond lengths, 0.2–0.8 kcal/mol in the dissociation energies, and justabout 0.5 cm�1 in the vibrational frequencies for allthe three molecules at both the LDA and BP86 levels.This indicates that, when extended basis sets cannotbe afforded for large systems, this particular kind of

Table 1Bond length (Re in A), dissociation energy (De in kcal/mol), vibrational fcalculated with four-component DFT

Basis LDA

Re De xe ajj

SZ 2.2024 57.29 276.15 17DZ 2.1602 63.33 300.21 17TZ 2.1586 63.33 300.56 18QZ 2.1552 63.37 300.30 185Z 2.1548 63.40 300.32 18QZ-1f 2.1502 64.10 299.78 18QZ-2f 2.1500 64.12 299.37 18QZ-3f 2.1498 64.13 299.33 18QZ-4f 2.1498 64.21 299.90 18QZ-4f 1g 2.1473 64.41 300.33 18QZ-4f 2g 2.1471 64.41 300.25 18QZ-4f 3g 2.1471 64.41 300.25 18QZ-4f 4g 2.1471 64.41 300.26 18QZ-4f 4gb 2.1476 64.34 300.03 18QZ-4f 4g 1hb 2.1474 64.37 300.11 18Dfc �0.0054 0.84 �0.40Dgd �0.0027 0.20 0.36 �Dhb �0.0002 0.03 0.08 �Limite 2.147 64.4 300.3 18Experimentalf 2.219 47.9 266.4Errorg �0.072 16.5 33.4

a Calculated at the corresponding equilibrium bond length.b Scalar relativistic ZORA results.c Difference between the QZ-4f and QZ results.d Difference between the QZ-4f 4g and QZ-4f results.e Estimated basis set limit.f Refs. [45,46].g Deviation of the calculated result from the experimental value.

DZ basis set augmented with one set of polarizationfunctions may be used to achieve rather good accu-racy (see also [29]). However, the DZ dipole polariza-bilities are still quite off. The TZ results, including thedipole polarizabilities, are very close to those of QZ.The latter are essentially converged as compared tothose of 5Z.

Next, we investigate the effects of polarization func-tions on top of the QZ basis sets. For clarity, we havedocumented the increments from QZ to QZ-4f (Df),and from QZ-4f to QZ-4f 4g (Dg) in the tables. Itcan be seen that including f-functions in the basis sethas important effects for Cu2, and it is necessary toget reliable results for Ag2, while it is crucial for Au2.The effects of g-functions are much smaller, but stillsignificant for Au2 (Table 3; see also [20]). Therefore,higher angular momentum functions should be furtherincluded to really show the basis set convergence.However, due to technical reasons, only up to g-func-tions can be used in BDF for two- and four-componentcalculations at the present time. Nevertheless, we canperform scalar relativistic ZORA calculations withand without h-functions to estimate the effects ofhigher angular momentum functions. This is justifiedbecause the spin–orbit coupling effects in these systems

requency (xe in cm�1), and dipole polarizability (ajj in a.u.)a of Cu2

BP86

Re De xe ajj

5.30 2.2620 45.46 249.94 187.436.10 2.2186 50.60 271.48 188.647.20 2.2170 50.39 271.36 201.428.52 2.2155 50.19 270.86 202.729.81 2.2152 50.22 270.98 203.836.85 2.2106 50.84 270.96 200.639.46 2.2110 50.89 270.49 203.459.88 2.2108 50.91 270.47 203.839.85 2.2101 50.97 271.22 203.789.43 2.2078 51.14 271.78 203.389.47 2.2080 51.16 271.54 203.509.47 2.2080 51.16 271.54 203.519.48 2.2079 51.17 271.56 203.529.50 2.2084 51.21 271.11 203.649.47 2.2080 51.26 271.29 203.611.33 �0.0054 0.78 0.36 1.060.37 �0.0022 0.20 0.34 �0.240.03 �0.0004 0.05 0.18 �0.039.4 2.208 51.2 271.7 203.2

2.219 47.9 266.4�0.011 3.3 5.3

Table 2Bond length (Re in A), dissociation energy (De in kcal/mol), vibrational frequency (xe in cm�1), and dipole polarizability (ajj in a.u.)a of Ag2calculated with four-component DFT

Basis LDA BP86

Re De xe ajj Re De xe ajj

SZ 2.5363 47.88 194.38 234.90 2.6099 36.27 172.66 252.84DZ 2.5049 51.35 206.82 235.27 2.5797 38.66 181.23 254.70TZ 2.4990 51.53 206.67 245.88 2.5748 38.68 181.16 267.87QZ 2.4976 51.60 206.92 246.90 2.5742 38.60 180.94 268.615Z 2.4980 51.58 206.80 247.68 2.5750 38.70 180.90 269.59QZ-1f 2.4853 52.79 207.08 242.72 2.5631 39.61 181.43 263.70QZ-2f 2.4855 52.84 206.89 244.38 2.5632 39.68 181.50 265.40QZ-3f 2.4850 52.86 207.08 245.48 2.5628 39.67 181.49 266.69QZ-4f 2.4847 52.89 207.14 245.53 2.5625 39.69 181.53 266.80QZ-4f 1g 2.4795 53.20 207.93 244.67 2.5576 39.91 182.16 265.82QZ-4f 2g 2.4794 53.21 207.88 244.72 2.5576 39.92 182.11 265.91QZ-4f 3g 2.4795 53.20 207.90 244.76 2.5576 39.92 182.13 266.00QZ-4f 4g 2.4793 53.21 207.89 244.82 2.5574 39.93 182.10 266.16QZ-4f 4gb 2.4810 53.02 207.30 245.13 2.5590 39.79 181.69 266.47QZ-4f 4g 1hb 2.4802 53.07 207.43 245.04 2.5583 39.84 181.83 266.36Dfc �0.0129 1.29 0.22 �1.37 �0.0117 1.09 0.59 �1.81Dgd �0.0054 0.32 0.74 �0.71 �0.0051 0.24 0.57 �0.64Dhb �0.0008 0.05 0.13 �0.09 �0.0007 0.05 0.14 �0.11Limit e 2.478 53.3 208.0 244.7 2.556 40.0 182.3 266.0Experimentalf 2.531 38.6 192.4 2.531 38.6 192.4Errorg �0.053 14.7 15.6 0.025 1.4 �10.1

a Calculated at the corresponding equilibrium bond length.b Scalar relativistic ZORA results.c Difference between the QZ-4f and QZ results.d Difference between the QZ-4f 4g and QZ-4f results.e Estimated basis set limit.f Refs. [47,48].g Deviation of the calculated result from the experimental value.

F. Wang, W. Liu / Chemical Physics 311 (2005) 63–69 67

are pretty small, as can be seen from the differences be-tween the scalar relativistic ZORA and DKS results.On the other hand, an accurate description of thespin–orbit coupling normally does not require exceed-ingly high angular momentum functions, since itmainly results from inner shells. The effects of h-func-tions are indeed negligibly small for all the properties.The effects of higher angular momentum functionsshould be even smaller due to the exponential decaybehavior [43]. This means that for the systems studiedhere, basis functions with angular momenta higherthan g-type are much less important in relativisticDFT than in ab initio (wavefunction based) correlatedcalculations [41]. With g-functions included, the errorof the basis set is already smaller than chemical accu-racy. This is due to the fact that only occupied orbitalsenter the KS formalism. There are strong evidencesshowing that the basis set requirements in DFT andHartree–Fock are very similar [43]. The requirementof high angular momentum functions in ab initio cor-relation calculations is however related to the inefficienttreatment of the Coulomb cusps [44].

To estimate the BSSE, we make counterpoise correc-tions [42] for the DZ, QZ-4f, and QZ-4f 4g basis set re-sults of Au2. The DZ basis set is chosen since BSSE

tends to be large for a small basis set. The results arelisted in Table 4. It can be seen that the BSSE has no dis-cernible effects in our calculations even for the DZ basisset, which is primarily owing to the special feature of thebasis set. The BSSE is also very small for the BP86 func-tional, although the NAOs and STFs are actually ob-tained from LDA calculations. Although notdocumented, the secondary BSSE for the dipole polariz-ability is also very small. In contrast, much larger BSSEswere observed in ab initio calculations even with basissets including i-type functions [41]. This is hardly sur-prising since in ab initio correlation calculations virtualorbitals are also populated, which are more diffuse andthus more affected by the BSSE than the occupied ones.

According to the convergence tendency of the basisset, the basis set limit results can be estimated, the devi-ations of which from experimental values directly un-screen the inherent errors in the functionals (cf. Tables1–3). It can be seen that the performance of LDA ismore systematic than that of BP86, although BP86yields more accurate dissociation energies. As is wellknown, the dipole polarizabilities converge more slowlywith respect to basis set. Nonrelativistically, the dipolepolarizabilities of Au2 and Ag2 are very similar [49].However, due to large relativistic contraction of the

Table 3Bond length (Re in A), dissociation energy (De in kcal/mol), vibrational frequency (xe in cm�1), and dipole polarizability (ajj in a.u.)a of Au2calculated with four-component DFT

Basis LDA BP86

Re De xe ajj Re De xe ajj

SZ 2.5171 60.46 183.30 190.79 2.5756 46.49 164.32 201.75DZ 2.4858 65.74 195.04 192.88 2.5422 51.09 174.81 204.02TZ 2.4740 66.50 195.72 205.41 2.5313 51.57 175.34 217.71QZ 2.4729 66.59 195.56 206.56 2.5303 51.63 175.19 218.825Z 2.4727 66.64 195.66 206.21 2.5299 51.68 175.40 218.50QZ-1f 2.4520 69.10 197.72 204.19 2.5106 53.71 177.49 215.56QZ-2f 2.4509 69.20 197.08 206.52 2.5100 53.78 177.02 218.23QZ-3f 2.4504 69.33 197.64 207.45 2.5092 53.89 177.54 219.18QZ-4f 2.4502 69.35 197.72 207.48 2.5090 53.94 177.58 219.28QZ-4f 1g 2.4418 70.34 199.11 205.60 2.5009 54.68 178.92 217.45QZ-4f 2g 2.4417 70.35 199.04 205.56 2.5008 54.69 178.89 217.40QZ-4f 3g 2.4415 70.37 199.00 205.76 2.5007 54.72 178.83 217.59QZ-4f 4g 2.4415 70.38 199.04 205.82 2.5006 54.74 178.89 217.63QZ-4f 4gb 2.4482 68.47 196.41 204.85 2.5082 52.98 176.34 217.22QZ-4f 4g 1hb 2.4474 68.58 196.55 204.68 2.5074 53.06 176.48 217.00Dfc �0.0227 2.76 2.16 0.92 �0.0213 2.31 2.39 0.46Dgd �0.0087 1.03 1.32 �1.66 �0.0084 0.80 1.31 �1.65Dhb �0.0008 0.11 0.14 �0.17 �0.0008 0.08 0.14 �0.22Limite 2.441 70.5 199.2 205.6 2.500 54.8 179.1 217.4Experimentalf 2.472 53.3 190.9 2.472 53.3 190.9Errorg �0.031 17.2 8.3 0.028 1.5 �11.8

a Calculated at the corresponding equilibrium bond length.b Scalar relativistic ZORA results.c Difference between the QZ-4f and QZ results.d Difference between the QZ-4f 4g and QZ-4f results.e Estimated basis set limit.f Ref.[48].g Deviation of the calculated result from the experimental value.

Table 4BSSE correction for Au2 (Re in A, De in kcal/mol, xe in cm�1)

Basis LDA BP86

Re De xe Re De xe

DZ 0.00007 �0.011 0.005 0.00015 �0.030 0.010QZ-4f 0.00006 �0.010 �0.034 0.00013 �0.038 0.013QZ-4f 4g 0.00006 �0.018 0.059 0.00018 �0.061 �0.025

68 F. Wang, W. Liu / Chemical Physics 311 (2005) 63–69

Au 6s orbital, the dipole polarizability of Au2 is signifi-cantly smaller than that of Ag2. This is in line with thefact that the bond length and dissociation energy ofAu2 are shorter and larger, respectively, than those ofAg2. Primarily due to the incorrect asymptotic behaviorof the XC potential, the present DFT ajj values for Cu2,Ag2, and Au2 are much larger than those from Dirac–Hartree–Fock calculations [49]. The present resultsshould be used as benchmark for calibration of otherDFT calculations, in particular, those using the sameXC functionals. It has been a common practice to usebasis sets of TZP quality in routine calculations. TheTZ-1f results of Au2, e.g., from our own calculations[16], differ from the present values by 1.3 pm in the bondlength, �1.1 kcal/mol in the dissociation energy, and 4

cm�1 in the vibrational frequency, which are small butnot insignificant compared to the errors in thefunctionals.

5. Conclusions

At present, DFT offers the only real possibility ofmodeling large and complex systems containing heavyelements, at a level that purports to include relativistic,exchange and correlation effects in the electronic struc-ture. It has been popular (and more or less justified) todirectly use available nonrelativistic XC functionals inrelativistic DFT calculations. However, the availablefunctionals have never been parametrized by taking

F. Wang, W. Liu / Chemical Physics 311 (2005) 63–69 69

heavy elements in the training set. Therefore, systematicreparametrizations are necessary for better accuracy.Yet, before this is done, one has first to be aware ofthe inherent errors of the functional in mind. In otherwords, one has to know the basis set requirement in or-der that the basis set error would not be an integral partof the functional development. For this purpose, wehave implemented the optimization of basis functionswith the analytical gradient method in BDF. As a firststep toward that goal, and in part to response to thesuspection due to Hess and Kaldor, we have carriedout benchmark four-component DFT calculations forCu2, Ag2, and Au2 with the standard functionals. Itcan be deduced from the present investigation that basissets with up to g-type functions are normally needed inorder to reveal the inherent errors of the functionals.Still higher angular momentum functions are hardlynecessary. Systematic investigations on more XC func-tionals and more challenging systems should be made,and are under progress in our group.

Acknowledgements

The research of this work was sponsored by the Na-tional Natural Science Foundation of China (ProjectNos. 20243001, 20273004 and 20333020). Support wasalso provided by the Scientific Research Foundationfor the Returned Overseas Chinese Scholars, State Edu-cation Ministry of China.

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