Calculations of nuclear quadrupole coupling in noble gas–noble metal fluorides: Interplay of relativistic and electron correlation effects

Calculations of nuclear quadrupole coupling in noble gas–noble metalfluorides: Interplay of relativistic and electron correlation effects

Perttu Lanttoa!

NMR Research Group, Department of Physical Sciences, P.O. Box 3000, FIN-90014 University of Oulu,Finland and Laboratory of Physical Chemistry, Department of Chemistry, P.O. Box 55 (A.I. Virtasenaukio 1), FIN-00014 University of Helsinki, Finland

Juha Vaarab!

Laboratory of Physical Chemistry, Department of Chemistry, P.O. Box 55 (A.I. Virtasen aukio 1),FIN-00014 University of Helsinki, Finland

!Received 21 June 2006; accepted 20 September 2006; published online 7 November 2006"

The nuclear quadrupole coupling constants !NQCCs" of noble gas and noble metal nuclei in therecently found noble gas–noble metal fluorides !NgMF, where Ng=Ar,Kr,Xe and M=Cu,Ag,Au" are obtained theoretically by high-level ab initio calculations, where both relativisticand electron correlation effects are included, and compared to experimental results. Fully relativisticfour-component Dirac-Hartree-Fock !DHF" calculations are carried out at the basis set limit forelectric field gradient that couples with the electric quadrupole moment of the nucleus, anduncorrelated relativistic effects are extracted by comparing DHF results to nonrelativistic !NR" HFcalculations. Electron correlation effects are investigated both at fully relativistic second-orderMøller-Plesset !DMP2" and at NR MP2 levels of theory, as well as at the NR coupled-cluster singlesand doubles with perturbational triples #CCSD!T"$ level. The validity of the approximation whererelativistic effects, on the one hand, and nonrelativistically obtained correlation effects, on the otherhand, are evaluated separately and assumed to be additive, is investigated by comparison with theDMP2 results. Inclusion of relativistic effects is shown to be necessary for obtaining the correctNQCC trends as the nucleus of interest and/or its neighbors become heavier. Electron correlationtreatment is needed for approaching quantitative agreement with the experimental NQCCs. Theassumption of additive electron correlation and relativistic effects, corresponding to the NRcorrelation treatment added on top of relativistic DHF data, gives qualitatively correct noble gasNQCCs. For noble metal NQCCs, correlation treatment at the relativistic level of theory ismandatory for reaching agreement with experimental results. Current work also confirms theexperimental trends of NQCCs, which have been taken as an indication of nearly covalentinteraction between noble gas and noble metal in the heaviest present systems, especially inXeAuF. © 2006 American Institute of Physics. #DOI: 10.1063/1.2363371$

I. INTRODUCTION

Hyperfine properties probe the electron cloud in imme-diate vicinity of the nuclei. This makes them sensitive pa-rameters for theoretical investigation of electronic structureand, in particular, the effect of special relativity on electronsmoving with high velocities.1 In the context of nuclear mag-netic resonance !NMR" observables, relativistic effects onnuclear shielding !K and spin-spin coupling JKL tensors areimportant already for fairly light elements.2–6 This holds alsofor the nuclear quadrupole coupling !NQC" tensor, ",7–11 thatresults from the interaction of the electric quadrupole mo-ment of the nucleus and electric field gradient !EFG" at thenuclear site. The energetics of the innermost electronic shellsare determined mainly by Coulombic interaction with thenucleus and consequently the electron-electron interactionplays a minor role. However, the magnetic and electric hy-

perfine interactions obtain contributions from the core tailsof semicore and valence orbitals that, in turn, are affected byelectron correlation effects. Indications of significant interde-pendence of relativistic and correlation effects on the NQCtensor are visible, e.g., in the computational study reported inRef. 10.

Recent observations of strongly bound noble gas–noblemetal halides !NgMX", such as XeAuF, have led to sugges-tions of a covalent nature of bonding between noble gas andnoble metal atoms in such systems.12–18 One criterion injudging the strength of Ng-M interaction is to observechanges in the experimentally observed nuclear quadrupolecoupling constants !NQCCs", !. Relativistic effects are ex-pected to play an important role in the ab initio calculationsof NQCCs8 in these systems due to the presence of heavyelements, e.g., Xe and Au.1,21 Furthermore, compounds in-cluding metallic elements are also typically sensitive to elec-tron correlation effects.

The present work deals with ab initio calculations ofthe EFG tensors in the NgMF !Ng=Ar, Kr, Xe, M=Cu,

a"Author to whom correspondence should be addressed. Electronic mail:[email protected]

b"Electronic mail: [email protected]

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Ag, Au" systems. We focus on the computational needs ofNQCC calculations for the naturally occurring quadrupolarisotopes 63Cu, 83Kr, 131Xe, and 197Au, as well as the effectsof the selection of basis set, and the treatment of relativisticand electron correlation effects. An important question con-cerns the coupling of relativity and correlation or, in otherwords, is it necessary to treat electron correlation effects atthe relativistic level of theory, or is it sufficient to considerNR correlation and relativistic effects as additive.

We carry out NQCC calculations at the fully relativisticDirac-Hartree-Fock !DHF" level of theory at the basis setlimit. The NR electron correlation effects are estimated atboth the second-order Møller-Plesset !MP2" and coupled-cluster singles and doubles with perturbational triples#CCSD!T"$ levels. Finally, the coupling of relativistic andcorrelation effects is estimated with valence-correlated fullyrelativistic MP2 level of theory !DMP2".22

II. AB INITIO CALCULATIONS

The equilibrium molecular geometries were optimizedwith the GAUSSIAN 03 !Ref. 23" program package at the MP2level using counterpoise !CP" correction for basis set super-position error !BSSE".24 The scalar relativistic Stuttgart 19-valence-electron pseudopotentials !PPs" were used for thefourth-, fifth-, and sixth-row noble metals.25 The eight-valence-electron PPs were used for the noble gases.26 Withthe latter, the quadruple-zeta level correlation-consistent va-lence basis sets !cc-pVQZ", optimized especially for thesePPs, were used.27 For the noble metals, the #6s5p3d$ valencebasis set25 with added #2f1g$ polarization functions27 wereused. For the light fluorine and argon atoms, the all-electroncorrelation-consistent cc-pVQZ basis sets28,29 were used inthe geometry optimization. The optimized MP2!CP"/PP/

VQZ structures have been used in all the present propertycalculations.

The EFG tensors were computed both at the one-component NR and four-component relativistic levels oftheory. The uncorrelated NR calculations at the HF level aswell as at correlated MP2 and CCSD!T" levels were carriedout using the ACES II quantum chemistry code.30 All orbitalswere fully relaxed and correlated in the MP2 and CCSD!T"calculations. The DIRAC quantum chemistry code31 wasused for the fully relativistic DHF8 and DMP222 calculations.In the DMP2 calculation, only the uppermost valence orbitalspace with DHF orbital energies between −15 and +20 a.u.were correlated while keeping the core orbitals inactive. Anuclear model with finite isotropic Gaussian charge distribu-tion was used for all nuclei.32 The two-electron integrals ofthe type SS-SS !S denoting the small component of the wavefunction" were omitted in both the wave function and prop-erty calculations. This has previously been found to be agood approximation.10 The values used for the nuclear quad-rupole moments QK were, in units of fm2, 25.9 for 83Kr!I= 9

2", −11.7 for 131Xe!I= 3

2", −22.0 for 63Cu!I= 3

2", and 54.7

for 197Au!I= 32

".7Both the NR and relativistic basis sets used in the NQCC

calculations for the present quadrupolar nuclei are presentedin Table I. The basis sets in NR calculations, denoted asHIVu/FIVu, are uncontracted Huzinaga/Kutzelnigg33,34 HIVquality basis sets for F and Ar, as well as the primitive setsby Fægri35 !of similar quality" for the heavier elements!X=Cu, Kr, Ag, Xe, and Au". Although not fully converged,these are sufficiently extended basis sets for reasonable esti-mates of electron correlation effects at the NR level. The useof larger, converged “HIVu32+g”-type basis set37 !with sys-tematically saturated tight and diffuse exponent ranges" was

TABLE I. Basis sets used in the nuclear quadrupole coupling calculations.a

Nonrelativistic Relativistic !large component"

Atom HIVu/FIVu Primary Tightb Polarizationb Diffuseb Totalc

F 11s7p4d1fd 11s7p4d1fe – – – !11, 9, 4:8, 1:5"Ar 12s8p4d2fd 12s8p4d2fe – – – !12, 8, 4:9, 2:6"Cu 16s11p8d2f f,g 19s14p9d3fh 3d2f 1g 1p2f !19, 15, 12:6, 7:9, 1:15"Kr 16s13p12d2f f,g,i 19s16p9d2fh – – 2d !19, 16, 11:9, 2:15"Ag 20s14p11d2f f,g 20s16p11d2fh – – – !20, 16, 11:9, 2:14"Xe 20s16p15d2f f,g,i 20s18p11d2fh – – 1d !20, 18, 12:9, 2:17"Au 22s16p13d8f f 23s19p14d9fh 5d 2g 1p1d1f !23, 20, 20:4, 10:10, 2:20"aAll basis sets are of the uncontracted type. Spherical/Cartesian gaussians were used in NR/relativistic calculations.bAdditional functions used on top of the primary basis set in the final calculations. The tight functions extend the exponent range of the l shell in questiontoward higher exponents, selected from the parent s and p sets in the dual-family manner. Correspondingly, the diffuse functions extend the range of exponentstoward small values.cThe family basis set notation !s , p ,d , f ,g"= !n ,m ,x :y ,x :y ,x :y" denotes that there are n and m functions in the parent s and p sets, respectively. x is thenumber of functions of the l-type in question, with the largest exponent coinciding with the function number y of the parent set and the x−1 smaller exponentsfollowing consecutively.dUncontracted Huzinaga/Kutzelnigg basis set from Refs. 33 and 34.eParent s and p sets from the uncontracted Huzinaga/Kutzelnigg basis sets from Refs. 33 and 34. The exponents of the d and f functions are obtained in thedual-family manner from the set of the parent s and p exponents, respectively, with similar range of exponents as in the NR basis set.fPrimitive one-component basis set from Ref. 35.gThe additional diffuse f exponents from Ref. 34 are used collectively for all atoms in each distinct row of the periodic table.hPrimitive four-component basis set from Ref. 36. Diffuse f functions added in the dual-family manner with exponents in the range of those occurring in theNR basis set.iAdded four diffuse d functions with exponents obtained by successively dividing the smallest existing exponent by three.

174315-2 P. Lantto and J. Vaara J. Chem. Phys. 125, 174315 "2006!


not feasible for the present purposes as these sets pose toohigh demands for the code and computation time.

For the relativistic basis sets,36 the convergence of theNQCC was monitored as tight and diffuse primitives wereindividually added for each l value on top of the primarybasis set. The small-component basis functions are generatedfrom the large-component set by the restricted kinetic bal-ance condition.38 For Ar, Ag, and F nuclei, i.e., not the cur-rently interesting quadrupolar ones, we used primitive HIVu/FIVu-type basis sets in production runs for the sake ofcomputational efficiency. According to test calculations !notshown here" with much larger basis sets used for these nu-clei, the NQCCs of the interesting nuclei !63Cu, 83Kr, 131Xe,and 197Au" are not affected by this approximation.

III. RESULTS AND DISCUSSION

A. Geometries

The optimized geometries are compared with earlier ex-perimental and theoretical MP2 results in Table II. Our equi-librium MP2!CP"/PP/VQZ geometries are of accuracy com-parable to or better than the previous MP2 results !withoutCP corrections" in Ref. 20, when compared to the experimen-tally estimated equilibrium geometries in Refs. 12–18. Therelatively modest rm

!1" model19 has been used in these papers.The M-F bond length is slightly underestimated by thepresent calculations.

B. Basis set convergence in the relativisticcalculations

The basis set convergence for the NQCC of 197Au, illus-trated in Fig. 1 at the relativistic DHF level, is typical for thenoble metals in the present systems. One or two additionaltight and diffuse functions are enough to reach a reasonableconvergence of the NQCC, to within a 0.5% relative changebetween two successive sets, apart from the d orbitals forwhich three to five tight primitives are necessary. Also one or

two g-type polarization functions are necessary for con-verged NQCC for the noble metals. The noble gases such asXe only require a couple of d-type diffuse functions on topof Fægri’s set to reach similar convergence !not shown". Theoscillation of the NQCC as a function of the number of ad-ditional tight p- and g-type functions !Fig. 1" has its origin ina slight numerical instability related to the basis-set lineardependence criteria of the code. This oscillation remainsnevertheless well below the required convergence level!0.5%" and is of no concern presently.

C. Electron correlation effects on NQCCsat the nonrelativistic level

The NQCC values and trends at different levels of theoryas well as the corresponding experimental values are pre-sented in Table III and Fig. 2. Both the magnitudes andtrends of the experimental data are poorly reproduced at theNR HF level. MP2-level correlation is a major improvementbut the correlation effects are overestimated as evidenced bycomparison with the CCSD!T" data. This overestimation ismore clear for the noble gases for which the MP2 correlationeffect can be nearly 50% larger than the CCSD!T" correla-tion effect, e.g., in the case of NQCC!131Xe", while in themetal NQCCs it is only a few percent as shown in Figs. 2!a"and 2!c", respectively. Electron correlation effects evaluatedat the NR level increase the magnitude of noble gas NQCCswhile the magnitudes of the noble metal NQCCs decrease.The trends of the NQCC in a given series of molecules arepractically similar at both correlated and uncorrelated NRlevels. These trends are not in agreement of the experimentaldata. The only case of qualitative agreement of NR theorywith the experiment occurs for NQCC!63Cu" in ArCuF. It isobvious that a relativistic treatment is required.

TABLE II. Molecular geometries of noble gas–noble metal fluorides!NgMF".a

rNg-M rM-F

Complex MP2!CP"b Expc MP2d MP2!CP"b Expc MP2d

ArCuF 2.214 2.219 2.238 1.715 1.753 1.727KrCuF 2.311 2.317 2.322 1.718 1.748 1.718XeCuF 2.439 2.430 2.459 1.723 1.750 1.737ArAgF 2.561 2.558 2.590 1.960 1.986 1.972KrAgF 2.591 2.600 2.609 1.958 1.975 1.969XeAgF 2.665 2.662 2.684 1.957 1.971 1.969ArAuF 2.374 2.391 2.396 1.905 1.918 1.909KrAuF 2.439 2.461 2.454 1.910 1.918 1.913XeAuF 2.542 2.543 2.545 1.918 1.918 1.922

aBond lengths in Å.bPresent calculations of the equilibrium geometry with counterpoise !CP"corrections.cExperimental estimates of the equilibrium geometry from Refs. 12–28, bythe rm

!1" model presented in Ref. 19. The rAu-F bond length has been keptfixed to the equilibrium value corresponding to the AuF complex in theexperimental analysis of gold complexes.dMP2/pseudopotential calculations without CP corrections from Ref. 20. FIG. 1. Basis set convergence of the nuclear quadrupole coupling constant

of 197Au at the Dirac-Hartree-Fock level. Functions of tight !t" or diffuse !d"type were individually added in each angular momentum shell. b0 denotesthe unsupplemented primary basis set.

174315-3 Noble gas–noble metal fluorides J. Chem. Phys. 125, 174315 "2006!


D. Relativistic effects at the uncorrelated Hartree-Focklevel

Uncorrelated relativistic effects on the NQCCs are vis-ible as the difference between the DHF and HF results inTable III. This has been verified with four-component Lévy-Leblond calculations !not shown here" with the present rela-tivistic basis, which give NR results similar to the present HFcalculations. Relativity causes an increase in the positiveEFG at the position of the 83Kr and 131Xe nuclei already inNgCuF and NgAgF complexes. However, the most drasticeffect of relativity on the noble gas NQCC is observed in thegold-containing systems, where the EFG is double the size ofthe quantity in the silver complexes.

While underestimating the magnitude of the experimen-tal data, the uncorrelated relativistic calculations at the DHFlevel are noted to result in the correction of the observedtrend when the adjacent noble metal center becomes heavier.Relativity explains the large magnitudes of bothNQCC!83Kr" and NQCC!131Xe" in gold-containing systems#Figs. 2!a" and 2!b", respectively$, when compared to thecomplexes with lighter noble metals.

The relative relativistic effect on the negative noblemetal EFGs is much smaller than on the noble gas EFGs.While the magnitude of NQCC!197Au" increases at most by

one fifth in XeAuF as compared to the NR HF calculation,the systematic change due to relativity recovers the experi-mental trend as seen in Fig. 2!c". At the same time, the agree-ment with the experimental magnitude of the NQCC!197Au"actually deteriorates. There is a clear dependence ofNQCC!197Au" on the relativistic nature of the nearby noblegas center, contrary to the case of NQCC!63Cu" where therelativistic effect is almost independent of the identity of thenoble gas center. This may be an indication of a weaker bondof the noble gas atom with copper than with gold.

E. Nonrelativistic correlation effects added on topof the relativistic DHF data

Due to the fact that correlated fully relativistic calcula-tions are computationally very demanding, applying correla-tion effects as incremental corrections evaluated at the NRlevel, on top of the uncorrelated relativistic DHF data, is anattractive pragmatic solution. This approximation seems tohave some validity for the noble gas nuclei in the presentcomplexes. The NR-level correlation treatment is qualita-tively sufficient for noble gas NQCCs, which are broughtclose to the experimental values when the NR correlationeffect is added on top of the DHF data #DHF+"MP2 and

TABLE III. Calculated nuclear quadrupole coupling constants in noble gas–noble metal fluorides.a

Nucleus Nonrelativistic DHF+

Complex HFb MP2 !"MP2NR "c CCSD!T"d DHFe "MP2

NR f "CCSD!T"NR f DMP2 !"MP2

R "g Exp

131XeXeCuF −49.7 −73.9!−24.2" −65.5 −62.8 −87.0 −78.6 −95.9!−33.1" −87.8h

XeAgF −43.2 −61.0!−17.8" −56.0 −63.8 −80.6 −75.6 −83.6!−20.8" −82.8i

XeAuF −50.8 −66.1!−15.4" −60.9 −112.0 −127.3 −122.1 −144.5!−32.5" −134.5j

83KrKrCuF 83.3 122.0!38.7" 108.7 96.1 134.8 121.5 144.2!48.1" 129.0k

KrAgF 62.0 86.4!24.4" 80.6 82.7 107.1 101.2 110.7!28.0" 105.1l

KrAuF 79.5 103.8!24.3" 96.8 161.3 185.7 178.6 205.0!43.7" 185.9l

63CuArCuF 79.3 37.2!−42.1" 45.7 78.3 36.2 44.7 31.1!−47.2" 38.1m

KrCuF 81.1 40.3!−40.8" 47.6 81.0 40.2 47.5 36.7!−44.3" 41.8k

XeCuF 83.9 47.7!−36.2" 53.5 84.2 48.0 53.9 41.1!−43.1" 47.8h

197AuArAuF −824.8 −590.7!234.1" −608.8 −862.2 −628.1 −646.2 −273.1!589.1" −323.4n

KrAuF −849.1 −610.7!238.4" −627.1 −941.9 −703.5 −719.9 −372.3!569.6" −404.8l

XeAuF −870.8 −646.5!224.4" −660.6 −1033.1 −808.7 −822.9 −492.5!540.6" −527.6j

aIn MHz.bHartree-Fock calculation with the HIVu/FIVu basis set.cSecond-order Møller-Plesset calculation with the HIVu/FIVu basis set. In parentheses the correlation effect at the nonrelativistic level, as the difference"MP2

NR =MP2−HF.dCoupled-cluster singles and doubles with perturbational triples calculation with the HIVu/FIVu basis set.eRelativistic Dirac-Hartree-Fock-calculation.fDHF results corrected with additive correlation effects evaluated at the NR level, based on the "MP2

NR =MP2−HF and "CCSD!T"NR =CCSD!T"−HF differences

!with HIVu/FIVu basis set".gRelativistic MP2 calculations where valence orbitals with DHF orbital energy between −15. . . +20 a.u. are correlated. In parentheses the correlation effect atthe relativistic level, as the difference "MP2

R =DMP2−DHF.hReference 18.iReference 14.jReference 16.kReference 17.lReference 15.mReference 12.nReference 13.



DHF+"CCSD!T" in Figs. 2!a" and 2!b"$. The former levelof approximation underestimates the magnitude of the ob-served NQCCs slightly, but retains the correct trends intro-duced by the DHF level of theory. However, the fact that aslightly worse agreement is obtained when the NR correla-tion effect is adopted from the CCSD!T" level of theory,reveals that this procedure benefits from error cancellation.The situation is different for the noble metals. While the NRcorrelation effect improves the agreement of the NQCCswith the experimental values, the fact that NQCC!197Au" isoverestimated by about 100% at the best DHF+"MP2 level,indicates that adding the NR-level correlation effect on topof the DHF data is not a good approximation in that case.

F. Correlation effects at the fully relativistic level

As correlation effects on both the noble gas and noblemetal NQCCs are significant at the NR level and, for 197Auin particular, the incremental NR correlation is clearly aninadequate approximation, one needs to scrutinize the possi-bility of the coupling of relativistic and correlation effects. Invery rough terms, a large fraction of the EFG at any !nonhy-drogen" nucleus is determined by the properties of the atomiccenter at which the investigated nucleus sits. Typically, elec-tron correlation effects on noble metal atoms are large due tothe nature of the subvalence d shell. Relativistic effects sta-bilize the valence s shell and destabilize the d shell1,21 andhence may be expected to couple strongly to correlation ef-fects. In contrast, at least free noble gas atoms are remark-

ably immune to electron correlation effects due to the largeenergy gap between occupied and unoccupied atomic states.In such systems, much smaller coupling effects between rela-tivity and correlation may be surmised.

Our DMP2 calculations serve to investigate this cou-pling. This appears to be a very good level of theory fortreating electron correlation from the point of view of thenoble metal NQCC since, at the NR level, MP2 overesti-mates correlation effects only by a few percent for 197Au,when compared to CCSD!T" data. The relative MP2–CCSD!T" difference is larger for 63Cu, presumably due tothe nodeless character of the 3d shell of copper that makes itmore susceptible for correlation effects. While the four-component density functional theory with differentexchange-correlation functionals has been shown to be inap-propriate for producing accurate EFGs for the gold halides,DMP2 gives a reasonable !over"estimation of the electroncorrelation effects on the noble metal EFG in thesesystems.10

In the present DMP2 calculations only the occupied va-lence and lowest virtual orbitals are correlated. This meansthat molecular orbitals arising largely from the following oc-cupied atomic orbitals were not included in the correlationtreatment: F, Ar: 1s; Cu, Kr: 2s2p; Ag: 3s3p; Xe: 3s3p3d;Au: 4s4p, and all orbitals with lower energy. The number ofcorrelated electrons versus all electrons equals in ArCuF: 42/56, KrCuF: 52/74, XeCuF: 52/92, ArAuF: 66/106, KrAgF:62/92, XeAgF: 62/110, KrAuF: 76/124, and XeAuF: 76/142.The accuracy of the process of restricting the correlationtreatment to the present orbital energy interval has been ex-plicitly verified for NQCC!63Cu" in the ArCuF complex. Inthis case the results from calculations including more semi-core orbitals !energy between −40 and +20 a.u., e.g., corre-lating also the 2s2p orbitals of copper" and, alternatively,virtual orbitals !in the range −15. . . +100 a.u." amount to30.92 and 30.57 MHz, respectively, which are close to thevalue of 31.13 MHz resulting from our basic type of calcu-lation. Both effects are to the same direction, i.e., they de-crease the magnitude of the EFG, which means that limitingoneself to a valence-only correlation treatment very slightlyunderestimates the total MP2 correlation effect on NQCC.Possessing this experience, we proceed with the limited cor-related MO space.

A very clear finding from the present DMP2 calculationsis that the treatment of electron correlation at the relativisticlevel is mandatory for reliable noble metal NQCCs. Borrow-ing from findings at the NR level and from Ref. 10, a higher-level correlation treatment than MP2 at a relativistic level oftheory is necessary for convergence of NQCC in the presentsystems. On the basis of the NR MP2 and CCSD!T" com-parison, one may expect a major improvement at the fullyrelativistic CCSD!T" level as MP2 exaggerates correlationeffects. The fully converged values may even require up tofull quadruple excitations at the CCSDTQ level of theory.10

The current principal observation is that the correct trend thatwas obtained for the NQCC!197Au" already at the DHF levelis brought to a close agreement with the experimental valuesonly after treating electron correlation at the genuinely rela-tivistic level #Fig. 2!c"$, while adding the NR correlation on

FIG. 2. Calculated nuclear quadrupole coupling constants in NgMF !Ng=Ar,Kr,Xe; M=Cu,Ag,Au" systems at different nonrelativistic correlationlevels #NR HF, NR MP2, and NR CCSD!T"$, at different fully relativisticcorrelation levels !DHF and DMP2", and at the DHF level with nonrelativ-istic estimates of correlation effects used as additive corrections #DHF+"MP2 and DHF+"CCSD!T"$. Panels from the top left corner: !a"NQCC!131Xe", !b" NQCC!83Kr", !c" NQCC!197Au", and !d" NQCC!63Cu".



top of DHF values is only able to provide one half of what isneeded. Fully relativistic DCCSD!T" calculations would beexpected to be very accurate, but they are beyond our currentresources. It is noteworthy that the correlation effect at therelativistic level #measured by the difference of DMP2 andDHF results for NQCC!197Au"$ greatly exceeds its NR coun-terpart !MP2–HF". This indicates a significant interdepen-dence of relativistic and correlation effects, in agreementwith the expectations based on the qualitative argument dis-cussed above.

For copper, the DHF and NR HF data were very similarbut the correlation effect at the relativistic level !DMP2" isalso clearly larger than at the NR level, with a concomitantslight change in the trend of NQCC!63Cu" when the Ng atombecomes heavier #Fig. 2!d"$. The remarkable finding is thatthe cross coupling of relativistic and correlation effects isvery significant while the relativistic effects alone at an un-correlated level of theory are insignificant. We conclude,based on the DMP2 calculations, that the good agreement ofthe DHF+"MP2 approximation with the experiment for thecopper complexes results from a cancellation of errors.

Qualitatively, the NQCCs of the noble gas nuclei, 83Krand 131Xe, are not much affected by the interplay of electroncorrelation and relativity, and semiquantitative results are al-ready obtained by combining DHF calculation with the MP2estimate of the electron correlation effect, with the latter ob-tained from NR calculations. Large relativity-correlationcoupling effects on the Au center in the XeAuF complex, forexample, are absent from NQCC!131Xe". Apparently, also thelarge relativistic effects at the Ng center itself remain im-mune to correlation, as we anticipated above.

However, when comparing DMP2 and DHF+"MP2 re-sults in more detail and taking into account that MP2 over-estimates correlation effects, the clearly larger electron cor-relation effect evidenced by the DMP2 calculation is to theright direction. The DMP2-level correlation treatment alsoslightly changes the trend of the noble gas NQCCs when themetal atom becomes heavier, rendering the data for the com-plexes containing “anomalous” noble metals, copper andgold,21 further removed from the complex of the “normal”silver #Figs. 2!a" and 2!b"$. Therefore, in order to obtainquantitatively accurate NQCCs for the right reason also forthe heavier noble gases, it is necessary to treat electron cor-relation at a relativistic level. There exists some interplay ofrelativity and electron correlation also for the noble gases,but the additive approximation may still be used due to thefavorable error cancellation.

IV. CONCLUSIONS

In the present work, the experimentally observed nuclearquadrupole coupling constants !NQCC" of krypton, xenon,copper, and gold in noble gas–noble metal fluorides !NgMF"were computed theoretically at both nonrelativistic and fullyrelativistic levels of theory. For the first time, the experimen-tal trends of NQCCs are reproduced in these systems andsemiquantitative agreement with experimental values isreached when electron correlation is treated at the second-order Møller-Plesset !MP2" level, at the relativistic level of

theory. The general finding is that while relativity is neces-sary for reproducing the experimental trends as the NQCCnucleus or its neighbor becomes heavier, electron correlationis needed for better agreement with the observed magnitudesof the NQCCs.

The necessity of treating electron correlation at the rela-tivistic level of theory is clearly demonstrated for the noblemetal NQCCs by comparison of relativistic correlated resultswith the relativistic uncorrelated Dirac-Hartree-Fock results,corrected by incrementally adding the electron correlationeffect found in nonrelativistic correlated calculations. Thelatter procedure leads only to one half of the total correlationinfluence on the NQCC of the 197Au nucleus, in the NgAuF!Ng=Ar,Kr,Xe" series. Despite the finding that the relativ-istic effect at uncorrelated Hartree-Fock level of theory isinsignificant for the NQCC of 63Cu in NgCuF !Ng=Ar,Kr,Xe", incorporation of the MP2-level correlation re-veals a large cross-coupling effect between relativity andelectron-electron interaction.

For the noble gas NQCCs in KrMF and XeMF !M=Cu,Ag,Au", the error made when evaluating correlationcontributions at the nonrelativistic level is much smaller, anderror cancellation renders the approximation of combiningDirac-Hartree-Fock data with nonrelativistic MP2 correlationeffects, as one that produces qualitatively correct results.Nevertheless, acquiring quantitative results for the right rea-son demands a genuinely relativistic correlation treatment. Itmay be expected that CCSD!T" calculations at a relativisticlevel would provide fully quantitative NQCCs in the presentsystems.

ACKNOWLEDGMENTS

P.L. and J.V. are with the Finnish Center of Excellence inComputational Molecular Science !CMS". J.V. is AcademyResearch Fellow of the Academy of Finland. Further finan-cial support from the Emil Aaltonen Foundation is gratefullyacknowledged. Computational resources were partially pro-vided by the Center for Scientific Computing, Ltd. !CSC,Espoo, Finland".

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Calculations of nuclear quadrupole coupling in noble gas–noble metal fluorides: Interplay of relativistic and electron correlation effects

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