Rovibrational effects on NMR shieldings in a heavy-element system: XeF2

Rovibrational effects on NMR shieldings in a heavy-element system: XeF2Perttu Lantto, Sanna Kangasvieri, and Juha Vaara Citation: J. Chem. Phys. 137, 214309 (2012); doi: 10.1063/1.4768471 View online: http://dx.doi.org/10.1063/1.4768471 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v137/i21 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 137, 214309 (2012)

Rovibrational effects on NMR shieldings in a heavy-element system: XeF2

Perttu Lantto,1,a) Sanna Kangasvieri,2 and Juha Vaara1

1Department of Physics, University of Oulu, P.O. Box 3000, FIN-90014, Finland2Laboratory of Physical Chemistry, Department of Chemistry, University of Helsinki, P.O. Box 55(A.I. Virtasen aukio 1), FIN-00014, Finland

(Received 9 July 2012; accepted 7 November 2012; published online 5 December 2012)

Fully quantum-mechanical treatment of the effects of thermal rovibrational motion in a heavy-element molecule with relativistic effects is carried out for the heavy 129/131Xe and light 19F nuclearshieldings in the linear XeF2 molecule. More importantly, purely quantum-mechanical, intramolecu-lar phenomena, the primary and secondary isotope effect on these shieldings, respectively, are treatedwith including both the zero-point vibrational and finite-temperature effects. While large solvent ef-fects influence the experimental absolute shielding constants and chemical shifts (thereby makingcomparison of experiment and theory very difficult), they are not significant for the isotope shifts.We study the role of electron correlation at both nonrelativistic (NR) and relativistic [Breit-Pauli per-turbational theory (BPPT) as well as 4-component Dirac theory] level. We obtain quantitative agree-ment with the nearly solvent-independent experimental 19F secondary isotope shifts. This implies apromising accuracy for our predictions of the experimentally so far non-existing primary Xe isotopeshift and the temperature dependence of Xe and F chemical shifts corresponding to a low pressuregas phase. To achieve this, a combination of high-level ab initio NR shielding surface is found neces-sary, in the present work supplemented by relativistic corrections by density-functional theory (DFT).Large errors are demonstrated to arise due to DFT in the NR shielding surface, explaining findingsin recent computational studies of heavy-element isotope shifts. Besides a high-quality property hy-persurface, the inclusion of thermal effects (in addition to zero-point motion) is also necessary tocompare with experimental results. The geometry dependence of the different relativistic influenceson the wave function, Zeeman interaction, and hyperfine interaction, as well as their role in the tem-perature dependence of both the Xe and F shielding constants and their isotope shifts, are discussed.The relativistic rovibrational effects arise from the same individual contributions as previously foundfor the chemical shifts and shielding anisotropies. In general, the spin-orbit interactions are moresensitive to rovibrational motion than the scalar relativistic contributions. A previously suggestedthird-order BPPT contribution to shielding anisotropy is shown to be important for a better agree-ment with experiment. © 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4768471]

I. INTRODUCTION

Inclusion of the effects of zero-point vibrational (ZPV)motion of the nuclei on the calculations of the spectroscopic,such as NMR, parameters is quite common nowadays.1–3 Thetreatment of finite-temperature effects is far less common,however. There exist semi-automated methods to treat ZPV inconnection with nonrelativistic (NR) computations, but onlyfew studies have been published on relativistic (REL) heavy-element NMR shieldings.4–8 Comprehensive studies of finite-temperature rovibrational effects for such cases are com-pletely lacking in the literature. The sensitivity of the mainrelativistic spin-orbit (SO) corrections to light-atom NMRshielding on rovibrational motion has been reported.9, 10 Fulltreatment of finite-temperature rovibrational effects on therelativistic shielding has been shown to be necessary for thesecondary isotope effects of 13C shielding in CX2 (X = O, S,Se, Te) molecules.11, 12 To quantitatively account for the ex-perimental normal halogen dependence of 13C and 1H shield-ings in methyl halides, such a treatment was also needed.13

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

The comparison of the computational rovibrational pa-rameters with the experimental temperature dependence iscomplicated for absolute shieldings (or chemical shifts) asthese properties are very sensitive to the environment.11, 12

Instead, experimentally observed isotope shifts of nuclearshielding are pronouncedly intramolecular parameters, i.e.,they are much less sensitive to the solvent effects and are,therefore, more amenable to comparison with theoreticalresults.11, 12 Hence, if low-density gas-phase experiments arenot available, the isotope shifts are preferable for experiment-theory comparison.

The 129Xe nucleus is known to cover a wide chemicalshift range, starting from a couple of hundred of ppm (withrespect to Xe atom) for chemically non-bonded Xe introducedto host media to over 7000 ppm for xenon in molecules suchas xenon fluorides and oxofluorides.14 In Ref. 15 it was shownthat accurate modeling of Xe shift in fluorides necessitatesboth relativistic treatment and a careful inclusion of correla-tion effects. So far these requirements can only be met at abinitio level by using the relativistic Breit-Pauli perturbationaltheory (BPPT),16–18 which treats relativity via corrections toNR shielding. Practical, fully relativistic, correlated ab initio

0021-9606/2012/137(21)/214309/10/$30.00 © 2012 American Institute of Physics137, 214309-1

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214309-2 Lantto, Kangasvieri, and Vaara J. Chem. Phys. 137, 214309 (2012)

methods for shielding do not exist so far. Furthermore, as theNR part has been shown to be very sensitive to the electroncorrelation treatment, quantitative accuracy can only be ob-tained by using an ab initio correlation method for at least theNR part of the xenon shielding tensor, σ , due to the failureof density-functional theory (DFT) with the currently avail-able functionals.15, 19, 20 In addition, the performance of DFTat NR level was shown to deteriorate as a function of the nu-clear mass for group-12 metals.21

DFT was found to provide insufficiently accurate chem-ical shifts in the calibration study of a prototypic, novel Xemolecule HXeCCH.19 This led us to devise a pragmatic piece-wise approximation, in which a benchmark-level ab initio cal-culation is carried out for the NR part and combined with aDFT description of the relativistic BPPT corrections, whichfeature a somewhat smaller susceptibility to DFT errors. Thisapproximation was used to predict an unexplored Xe shiftrange of 500–1600 ppm for a series of novel Xe molecules.20

None of the 14 tested DFT functionals provided results accu-rate enough for quantitative investigations of Xe-molecules.20

It has been shown that in difficult electron correlation cases,the relativistic corrections should also be pursued at a corre-lated ab initio level.15, 20

Within the BPPT, the relativistic heavy-element chemi-cal shift as well as shielding anisotropy arise from five mainrelativistic contributions, the so-called “BPPT-5” terms classi-fied as paramagnetic relativistic (REL-p = p-KE/OZ + p/mv+ p/Dar) and spin-orbit (SO-I = FC-I + SD-I) terms.15, 19–21

These mechanisms are also responsible for the relativisticintermolecular shielding interactions, such as those experi-enced by a Xe atom inside a fullerene cage,22 Xe in benzenesolutions,23 as well as those found important for chemicalshift anisotropy relaxation of Xe gas.24, 25 In addition to theBPPT-5 terms, the recently proposed26 spin-Zeeman-inducedrelativistic SO contribution (SOS) was found very importantfor shielding anisotropy �σ .20 In the special case of a lin-ear molecule, such as presently, the SOS contribution can beobtained from the axial component of the SD-I term in theshielding tensor as �σ SOS = −3 × σ SD−I

|| .We are not aware of any study treating both harmonic

and anharmonic thermal effects on heavy-nucleus shieldingsor isotope shifts at a quantum-mechanical level. Typically,only the ZPV correction is computed by adding harmonic vi-brational corrections to the shielding value at the effectivezero-temperature geometry.2 For example, the ZPV correc-tions to transition metal shieldings have been studied andcombined with classical thermal effects estimated by ab ini-tio molecular dynamics simulations.4, 5 This combination ofquantum and classical treatments was also used in a studyof the isotope shifts of transition metal shieldings.7 In a verydetailed study of chlorine isotope shifts of 125Pt shieldings aquantum mechanical treatment of ZPV only was used.8 Clas-sical treatment of thermal effects is usually rationalized bytheir often smaller magnitude than that of the ZPV contri-bution. In addition to the conceptual difficulties in combin-ing quantum and classical treatments of rovibrational effects,the a priori assumption of small thermal contributions is notnecessarily justified in the case of heavy elements, as shownpresently. Although ZPV effects may provide a qualitative

explanation of experimental shifts or isotope effects, thereare thermal effects significant for quantitative comparison.Furthermore, the thermal contributions cannot be accuratelymodeled by a classical treatment (vide infra). A large part ofthe remaining discrepancies with experiment4, 5, 7, 8 are mostprobably due to the use of DFT for the shielding surfaces,which may produce decisive systematic errors, as shown be-low.

In this paper, we apply controlled approximation for therelativistic finite-temperature rovibrational effects on 129Xeand 19F shieldings in XeF2 at a quantum-mechanical level.A cubic ab initio force field is computed in order to pro-duce a high-level description of the isotope- and temperature-dependent nuclear motion. Averages of the displacement co-ordinates are combined in a Taylor series expansion of theproperty hypersurface around equilibrium geometry. At theNR level, DFT shieldings and their derivatives are comparedwith ab initio ones and the former are found to behave unre-alistically. In contrast, for the relativistic terms, both resultsat the equilibrium geometry and the systematic behavior ofdifferent DFT functionals imply that reasonable data may beobtained by DFT in this case. The relativistic rovibrationalcorrections on both absolute shieldings and isotope shifts byBPPT and four-component Dirac theories are shown to be ingood mutual agreement, despite the fact that absolute shield-ings and (to a lesser extent) shielding anisotropies deviatefrom the fully relativistic results at equilibrium geometry. Weshow that the piece-wise combination methodology leads tounprecedented, quantitative accuracy for the secondary iso-tope shift of 19F [see below for Figure 4 and Table V], there-fore allowing to speculate that high-quality estimates couldalso be obtained for thermal effects on the absolute shield-ings of an unsolvated molecule, as well as the primary Xe iso-tope shifts. There are no experimental findings to verify this,yet. The method possesses significant potential for analysingthe corresponding gas- and liquid-phase experiments,respectively.

II. COMPUTATIONAL DETAILS

Third-order Taylor expansion of the potential energy sur-face in curvilinear internal coordinates (the two bond lengthsr and r′, as well as the bond angle within two perpendicularplanes θ and θ ′),

V = 1

2frr [(�r)2 + (�r ′)2] + frr ′�r�r ′

+ 1

2fθθ [(�θ )2 + (�θ ′)2]

+ 1

6frrr [(�r)3 + (�r ′)3]

+ 1

2frrr ′�r�r ′(�r + �r ′)

+ 1

2frθθ (�r + �r ′)[(�θ )2 + (�θ ′)2], (1)

was fitted to total energies computed at 42 displaced geome-tries in the close vicinity of the equilibrium geometry.11, 12

These calculations were carried out at coupled-cluster

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214309-3 Lantto, Kangasvieri, and Vaara J. Chem. Phys. 137, 214309 (2012)

singles, doubles, and perturbative triples [CCSD(T)] level us-ing a scalar-relativistic pseudopotential (ECP28MDF)27 forthe heavy Xe atom, with the corresponding aug-cc-pVQZ va-lence basis set. F atoms were equipped with the correspondingall-electron aug-cc-pVQZ basis set.28, 29 This combination ishere denoted as AVQZ.

The NR shielding tensors at equilibrium geometry, re, aswell as the shielding constant hypersurfaces up to second-order terms in displacement coordinates (angle degeneracydue to equivalent θ and θ ′ coordinates included in the lastterm),

〈σ 〉T = σe + σr〈�r〉T + σr ′ 〈�r ′〉T

+ 1

2σrr〈(�r)2〉T + 1

2σr ′r ′ 〈(�r ′)2〉T

+ σrr ′ 〈�r�r ′〉T + σθθ 〈(�θ )2〉T

= σe + 〈σr〉T + 〈σr ′ 〉T + etc. (2)

were computed at Hartree-Fock (HF) and second-orderMøller-Plesset perturbation theory (MP2), as well as CCSD,and CCSD(T) levels with the parallel CFOUR code.30 In ad-dition to the DIRAC31 code for four-component relativisticcalculations of shielding tensors at HF (DHF) and DFT withB3LYP33, 34 exchange-correlation functional (D-B3LYP) lev-els, the DALTON quantum chemistry program32 was used tocalculate all the 16 relativistic BPPT contributions16–18 tothe nuclear shielding tensors. The best estimate of the BPPTterms at re was obtained by combining the ab initio CCSD re-sults for the terms involving only singlet operators with mul-ticonfigurational Hartree-Fock (MCHF) data for the tripletterms. In MCHF calculations, the same restricted active space(RAS) wave function was used as in our earlier work on XeF2

in Ref. 15. The DFT calculations for BPPT contributions tothe shielding hypersurface were carried out with BLYP,34, 35

B3LYP, and BHandHLYP34, 36 functionals.In ab initio calculations of NR shieldings and shield-

ing surfaces, the HIV (IGLO-IV) basis sets37, 38 for F andFIV set for the Xe atoms were used. The latter was con-structed in the spirit of Kutzelnigg/Huzinaga basis sets37, 38

from the primitives of the Fægri one-component basis set.39

For BPPT shieldings at the DFT level, the FIVu6 basisset (26s22p21d2f)15 was used for Xe. It was obtained byadding six sets of tight spd functions to the uncontractedFIV set. This has been shown to provide almost fully con-verged BPPT contributions and an even better convergence ofthe Xe chemical shift in XeF2.15 Similarly, the HIVu2 basisset for F (13s9p3d1f) was constructed from the correspond-ing HIV set to produce the FIVu6(Xe)/HIVu2(F) combina-tion for the DFT BPPT calculations. The benchmark-qualityCCSD/RAS results at the re geometry were obtained by thesmaller FIVu2/HIVu2 combination, onto which the basis setcorrection from changing the basis set from FIVu2/HIVu2 toFIVu6/HIVu2, estimated at the DFT/BHandHLYP level, wasthen added. The Dirac calculations, as well as correspond-ing BPPT calculations for comparison, were carried out usingnovel completeness optimized40 (co-r) basis sets both for F(Ref. 41) and Xe (Ref. 42). All relativistic BPPT contribu-tions as well as Dirac calculations at D-B3LYP level were

carried out by placing common gauge origin (CGO) at Xe nu-cleus due to the non-existing gauge-including atomic orbital(GIAO) implementations in the codes we apply. OtherwiseGIAOs were used. Dirac calculations were of the unrestrictedkinetic balance type with the explicit two-electron integralsinvolving only the small-component basis functions omittedboth in wave function and property calculations. In the for-mer, they were replaced by a simple Coulombic repulsivecorrection.43 This approximation has been found to produceonly a small difference as compared to a calculation with afull set of small-component integrals.44

The rovibrational averaging (see details from Refs. 11and 12) was carried out according to Eq. (2) by combiningthe NR shielding surface computed at different levels withthe relativistic DFT BPPT surface. As in the case of poten-tial surface, the property surfaces were obtained by fitting theresults from a sufficient number (61) of displaced geometriesaround the re structure to the second-order shielding surfacein internal coordinates. While the temperature dependent dis-placement coordinate averages, 〈�r〉T, etc, are of high qualitydue to the theoretical level of potential energy surface, thequality of shielding derivatives, σ r, etc of Eq. (2) determinethe overall accuracy.

III. RESULTS AND DISCUSSION

A. Equilibrium geometries and force fields

The current cubic force field at CCSD(T)/AVQZ level forXeF2 (shown in Table I) is the best computational one existingso far. As seen in comparison with the experimental harmonicforce field, it produces harmonic frequencies of good quality.Also the equilibrium bond length is very close to the experi-mental one.

B. Correlation effects and DFT performance

Correlation effects on NR shielding parameters at re

(Table II) appear to be quite easy to describe as alreadyMP2 gives reasonable data as compared to CCSD(T), withCCSD already quite close to the latter. In contrast, the prop-

TABLE I. Equilibrium geometry, harmonic frequencies as well as harmonicand cubic anharmonic force constants for XeF2.a

CCSD(T)b Exp.c

re (Å) 1.9746 1.9744ω1 (cm−1) (asymmetric stretching) 535.35 526.0ω2 (cm−1) (bending) 218.87 214.2ω3 (cm−1) (symmetric stretching) 572.93 566.1frr (aJ Å−2) 3.023 2.950frr ′ (aJ Å−2) 0.185 0.147fθθ (aJ rad−2) 0.807 0.782frrr (aJ Å−3) − 14.216frrr ′ (aJ Å−3) − 0.959frθθ (aJ Å−1 rad−2) − 0.656

aFrequencies for the 129Xe19F2 isotopomer.bForce field computed with aug-cc-pVQZ basis set and corresponding relativistic ECPs.cReference 45.

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214309-4 Lantto, Kangasvieri, and Vaara J. Chem. Phys. 137, 214309 (2012)

TABLE II. Correlation dependence of nonrelativistic (NR) and relativistic(REL) shielding constants σ and anisotropies �σ in XeF2 at equilibriumgeometry.a

Contr. Method σ F �σ F σXe �σXe

NRb HF 450.7 70.0 2172.7 5214.2MP2 434.3 93.8 2479.8 4753.4CCSD 427.0 105.1 2469.1 4769.4CCSD(T) 420.0 115.5 2551.5 4645.7DFT/BLYP 370.3 190.4 2242.0 5111.4DFT/B3LYP 388.5 163.0 2206.3 5164.5DFT/BHandHLYP 416.0 122.2 2158.7 5236.0

RELc HF 4.2 47.6 1008.8 − 599.4HFd 3.2 51.9 1019.3 − 648.2DHFd,e − 8.3 48.7 1330.6 − 875.8DFT/BLYP − 24.9 103.5 1056.8 − 639.9DFT/B3LYP − 19.6 92.7 1045.5 − 633.2DFT/B3LYPd − 21.1 98.4 1054.3 − 674.4DFT/D-B3LYPd,f − 28.9 95.3 1361.8 − 822.7DFT/BHandHLYP − 10.9 76.0 1026.3 − 614.1Ab initiog − 4.1 50.4 1009.6 − 479.1

Basis set correctionh 0.0 0.0 17.6 5.9〈σ 〉 300 K − σ e

i − 9.3 − 2.1( − 7.1) ( − 12.3)

Totalj 406.5 165.9 3576.6 4172.5Exp.k Gas46 379.1 3678

Liquid47 205 4722Solid46 125 4260Solid48 4245

aIn ppm. re(Xe-F) = 1.9746 Å. Anisotropies with respect to the symmetry axis:�σ = σ || − σ⊥.bWith FIV/HIV basis set for Xe/F.cBPPT calculation with FIVu6/HIVu2 basis set for Xe/F if not otherwise noted. BPPTis always calculated with CGO at Xe.dWith co-r basis set for F,41 27s25p21d1f for Xe.42

eRelativistic 4-component Dirac-Hartree-Fock calculation using GIAO.fRelativistic 4-component Dirac calculation at DFT/B3LYP level using CGO at Xe.gCCSD/RAS level with FIVu2/HIVu2 basis set for Xe/F. RAS with the same active spaceas in Ref. 15. FIVu2/HIVu2 basis set used.hChange of REL part from FIVu2(Xe) to FIVu6(Xe) basis set at DFT(BHandHLYP)level.i Rovibrational correction at the CCSD(T)/B3LYP level. NR part in parenthesis.jNR[CCSD(T)]+BPPT[CCSD/RAS] + basis set correction + rovibrational correctionat T = 300 K.kObtained by using the absolute shielding scales of Refs. 14 and 49 for 129Xe and 19F,respectively.

erty derivatives turned out to be much more difficult (videinfra). In fact, all the present DFT functionals are in trou-ble already at re. For 19F, DFT drastically overestimates thecorrelation effects, although the increase of exact exchangeadmixture (EEX) in the hybrid functionals clearly improvesthe situation. BHandHLYP(50% EEX) gives σ F(NR) veryclose to the CCSD(T) result. In contrast, all the DFT func-tionals give too small correlation effects for σ Xe(NR). In thiscase, BHandHLYP produces a correlation effect of wrong(deshielding) sign. DFT errors in the shielding anisotropiesare also significant.

The relative magnitudes of the BPPT shielding terms areoverestimated by all the present DFT functionals as seen inTables I–IV in the supplementary material.51 However, in thenumerically large BPPT contribution to σ Xe, the differencesbetween the DFT functionals are quite small. The DFT dataalso agree reasonably well with the ab initio CCSD/RAS

results. There is a substantial exaggeration in the small,deshielding relativistic contribution to σ F. The DFT errors arelarge for both �σ Xe and �σ F, especially due to the singlemost important contribution, the new relativistic SOS term.26

The errors due to perturbational treatment of relativity inBPPT when compared to Dirac results are in line with theprevious findings,15–21 that is, the systematic BPPT error inabsolute shielding constant is observed, as in here, to mainlyarise from the isotropic core contributions that are relativelyinsensitive to electron correlation and chemical environment.In practice, they almost fully cancel out in the chemical shifts(differences of shielding constants). As shown in the follow-ing, they are not sensitive to the geometry changes of themolecule either. The same BPPT-5 terms as for the chemi-cal shift are left to be the main relativistic contributions to thetemperature averages of shielding constant. As the tempera-ture dependence of isotope shift is obtained from difference ofthe chemical shifts for different isotopomers at different tem-peratures, one can expect an even more complete cancellationof relativistic BPPT errors. This is confirmed by the current,so far unequalled, match with the experimental secondary 19Fisotope shifts and further supported by the good agreementwith the 4-component results for the thermal effects of bothnuclear shieldings and isotope shifts (vide infra). We exploitof error cancellation as usual in modeling. In the case of rovi-brational averaging of XeF2, the errors in the BPPT treatmentof relativity are evidently surpassed by the errors of the DFTcorrelation treatment of the currently available functionals.

The SOS term was shown to bring the perturbation-theoretical, uncorrelated HF estimate of �σ Xe in XeF2 closeto the corresponding, fully relativistic 4-component value.26

In the present correlated treatment, the combination of theprincipal BPPT (mainly BPPT-5) terms with the SOS correc-tion partially cancels with the NR value. By using the presentab initio correlated CCSD(T)/AIV data for the NR part andCCSD/RAS data for the relativistic BPPT terms, combinedwith the basis set correction [FIVu2(Xe)→FIVu6(Xe)] esti-mated at the DFT/BHandHLYP level, as well as the presentrovibrational correction at T = 300 K in the “Total” result inTable II, we obtain only slightly over-/underestimated σ F/σ Xe

when compared with gas-phase experimental results.46 Alsothe estimated non-rovibrationally corrected �σ Xe of ca.4173 ppm is close to the experimental solid state results4260(10) ppm46 and 4245(20) ppm,48 differing slightly morefrom the NMR relaxation result in a solvent.47 The currentab initio result for �σ F of ca. 166 ppm is also reasonablyclose to the solid-state NMR results of 125(5) ppm46 and150(20) ppm.48

Hence, through a combination of ab initio and DFT mod-eling we have achieved theoretical estimates of the shield-ing parameters in XeF2 with a fairly good agreement withthe experiment, especially for shielding anisotropies, reckon-ing with the experimental error limits. We note that an accu-rate correlation treatment is important and especially the NRshielding is sensitive. Therefore, the inaccuracy of the cur-rently available DFT functionals is mostly responsible for theremaining errors in computational studies, regardless of thelevel of relativistic treatment [e.g., BPPT, zeroth-order regularapproximation (ZORA), or 4-component Dirac theory] used.

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214309-5 Lantto, Kangasvieri, and Vaara J. Chem. Phys. 137, 214309 (2012)

FIG. 1. Temperature dependence of and rovibrational contributions to 129Xeshielding constant in 129XeF2. The (a) nonrelativistic and (b) relativistic(BPPT) rovibrational contributions are at CCSD(T) and DFT/B3LYP levels,respectively. In panel (c), the total MP2, CCSD, and CCSD(T) results containthe relativistic contribution at the DFT/B3LYP level.

This prevents in many cases from quantitatively reproducingboth the heavy- and light-atom shielding tensors, as well asinduces errors affecting rovibrational averaging (vide infra).

C. Temperature dependence of 129Xe shielding

Almost full mutual cancellation of the negative NR andpositive REL contributions (Figures 1(a) and 1(b)) results in apractically negligible (less than 0.1%), slightly negative ZPVcorrection to σ Xe, which increases only up to few ppm due tofinite temperature contributions (Figure 1(c)).

The modeling of rovibrational contributions is by nomeans trivial. The high-quality CCSD(T) NR shielding sur-face combined with the accurate CCSD(T)/AVQZ cubic force

TABLE III. Temperature derivative of the shielding constants in XeF2 atroom temperature.a

129Xe shielding 19F shielding

Method NR RELb TOTAL NR RELb TOTAL

BLYP − 38.9 21.7 − 17.2 − 17.9 − 4.0 − 21.9B3LYP − 48.7 21.4 − 27.3 − 15.3 − 3.9 − 19.2B3LYPc − 47.5 21.9 − 25.6 − 15.7 − 4.0 − 19.7D-B3LYPc,d,e − 47.7 18.9 − 28.8 − 15.8 − 3.4 − 19.2BHandHLYP − 64.4 20.6 − 43.8 − 11.5 − 4.0 − 15.5HF − 86.3 19.7 − 66.6 − 6.7 − 2.2 − 8.9HFc − 85.6 20.7 − 64.9 − 6.7 − 2.4 − 9.0DHFc,d − 85.6 14.8 − 70.8 − 6.7 − 2.9 − 9.6MP2 − 30.9 21.4f − 9.5 − 11.6 − 3.9f − 15.5CCSD − 43.6 21.4f − 22.2 − 10.8 − 3.9f − 14.7CCSD(T) − 30.1 21.4f − 8.7 − 12.9 − 3.9f − 16.8Exp.g 471.8 − 4.2

aIn ppm/K. 129XeF2 isotopomer. Linear fit to results in the T = 240–330 K range.bBPPT calculations with FIVu6/HIVu2 basis set if not otherwise noted.cWith co-r basis set for F,41 27s25p21d1f for Xe.42

dRelativistic 4-component Dirac calculation.eCGO placed at Xe.fRelativistic BPPT contribution from DFT/B3LYP calculation.g Ref. 50. Experimental results in acetonitrile-d3 solvent.

field furnishes high-quality NR rovibrational averages. How-ever, the performance of DFT is far from satisfactory with allthe present functionals as seen in Figure 1(a) and the tabu-lated shielding derivatives in Table V in the supplementarymaterial.51 The increase of EEX in the series of functionalsBLYP→B3LYP→BHandHLYP worsens the situation as theresults approach the faulty HF data. It is the pure DFT (BLYP)functional among them that gives the best estimate for boththe overall magnitude and temperature dependence (Table III)of the rovibrational effect on NR 129Xe shielding.

Amongst the ab initio levels of theory, already the low-level MP2 correlation treatment appears sufficient in produc-ing rovibrational effects on NR σ Xe practically in full agree-ment with the best CCSD(T) data. The CCSD results are fur-ther away from the CCSD(T) benchmark, indicating that thegood MP2 performance is coincidental, however.

While the performance of DFT is not satisfactory forthe NR shieldings, relativistic BPPT corrections are estimatedconsistently (Figure 1(b)). The BLYP and B3LYP functionalsproduce practically identical results and even BHandHLYP(with poor NR performance) are close. The total relativisticrovibrational contribution is about 80% of the NR shieldingat room temperature. The relativistic effect is entirely due tothe BPPT-5 terms (see Table VII and Figure 1 in the supple-mentary material51). The non-correlated HF data are acciden-tally in the same region with the DFT results, as the largeerror in the non-correlated SO-I terms is cancelled by the er-rors in the REL-p terms. The HF failure with both BPPT and4-component Dirac methods is evident in relativistic shield-ing derivatives of Table V in the supplementary material.51

The table also displays a reasonable agreement between therelativistic methods at DFT/B3LYP level. The largest differ-ences between relativistic methods occur in the second-orderstretching derivatives. The corresponding thermally averagedquadratic displacements 〈�r�r ′〉T and 〈(�r)2〉T are small

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214309-6 Lantto, Kangasvieri, and Vaara J. Chem. Phys. 137, 214309 (2012)

and only depend a little on temperature. Hence, the differ-ences in these derivatives cause just a small, constant differ-ence both in the average shielding and isotope shift (see be-low).

At DFT level, REL-p is unimportant and almost unaf-fected by rovibrational motion, although the individual terms(especially p/mv) are sensitive to temperature. As the FC-Icontribution remains very small throughout the whole investi-gated T range, the main relativistic rovibrational contribution,as well as the temperature dependence of σ Xe, is mostly dueto the SD-I spin-orbit contribution.

Among the terms in Eq. (2), the first- and second-orderrovibrational contributions, 〈σ Xe

r 〉T and 〈σ Xeθθ 〉T , are the largest

and most sensitive to T. Heavy cancellation between the dif-ferent terms takes place however, rendering full treatmentnecessary.

The temperature derivative of σ Xe at room temperatureis negative (dσ NR

Xe /dT = −30 ppb/K) at the NR level oftheory (Table III). Due to systematic results obtained withDFT (Figure 1(b)), the estimate obtained for the relativis-tic contribution to the temperature derivative of shielding ca.+21 ppb/K (fitted in range T = 240–330 K) should be rea-sonable (see Table III). As it does not fully cancel with theNR result, the total temperature coefficient remains negative,although very small (dσ TOTAL

Xe /dT = −9 ppb/K). This is indrastic disagreement with the positive and large experimen-tal temperature derivative (dσ EXP

Xe /dT = +471.8 ppb/K) inRef. 50. The thermal effects on nuclear shieldings involvefavorable error cancellation: they are determined by the dif-ferences of the nuclear shielding constant in closely relatedsituations. This reduces the impact of the large BPPT errorin the atomic core region, which indirectly implies that thecalculated temperature dependence of Xe (and F, see below)shielding constant may nevertheless be of relatively high ac-curacy. This is corroborated by the reasonably good agree-ment of the shielding derivatives at 4-component and BPPTlevels (see Tables V and VI in the supplementary material51).The difference with experimental data could be due to largesolvent effect for XeF2 in the acetonitrile-d3 solution.

The interaction with the solvent usually increases theaverage bond length, inducing increasingly large relativisticeffects.9, 12 This eventually leads to a positive rovibrational ef-fect and its temperature derivative. This is in the present caseseen to arise mainly from the increase of the positive rela-tivistic SD-I contribution to the σ Xe

r and σ Xeθθ shielding deriva-

tives with the increase of the Xe-F bond length. In additionto this structure-induced effect, there are also direct solventeffects operating through the changes of the XeF2 electronicstructure, which probably are even more important. Large sol-vent effects are also seen in the temperature dependence of σ F

(vide infra).

D. Temperature dependence of 19F shielding

As expected, rovibrational effects are relatively moreimportant for σ F than for Xe. The ZPV correction of ca.−6.5 ppm is about 1.5% of the equilibrium value. Thermaleffects also increase this value much more rapidly than in the

FIG. 2. Temperature dependence of and rovibrational contributions to the19F shielding constant in 129XeF2. (a) Nonrelativistic (NR) and (b) relativis-tic (REL) rovibrational contributions at CCSD(T) and DFT/B3LYP levels,respectively. In panel (c), the total MP2, CCSD, and CCSD(T) results con-tain the relativistic contribution obtained at the DFT/B3LYP level.

case of σ Xe, reaching −9.2 ppm (2.2%) at 300 K. As seenin Table II, there is a negative relativistic contribution to σ F

at re. In addition, as shown in Fig. 2 (see also Table VIII inthe supplementary material51), the relativistic influence fur-ther enhances the negative rovibrational effect, unlike for σ Xe.

In contrast to the average 129Xe shielding, increasingthe proportion of EEX in the DFT functional improves boththe magnitude and temperature derivative of σ F at NR level.For both quantities, BHandHLYP provides practically simi-lar slight underestimation as MP2. This is also displayed bythe shielding derivatives in Table VI in the supplementarymaterial.51 Quantitative results for the NR rovibrational effectare only achieved at the CCSD(T) level, as CCSD gives sim-ilar results as MP2. This indicates that dynamical correlation

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214309-7 Lantto, Kangasvieri, and Vaara J. Chem. Phys. 137, 214309 (2012)

due to perturbative triples is important for the rovibrationaleffects on σ F at NR level.

The relativistic rovibrational contributions are quite in-sensitive to the choice of the DFT functional and this al-lows us to assume that the B3LYP estimate is likely to be afairly good one. As seen in Table VIII in the supplementarymaterial,51 unlike for σ Xe, the relativistic p-KE/OZ term haspractically no role in the rovibrational correction of σ F. Thesame finding was made also in the case of methyl halides,13

concerning σ C. Only the mutually heavily canceling scalarrelativistic p/mv and p/Dar terms, as well as spin-orbitFC-I and SD-I contributions, remain important. Particularlythe last of these is numerically large. As in the case of Xe,there is a good agreement between BPPT and 4-componentDirac results at both HF and DFT levels.

The 〈σ Fr 〉T term provides the main rovibrational contri-

bution both at NR and relativistic level of theory. It also playsa major role in the temperature dependence. Due to cancella-tion of the other fairly large terms, the bending 〈σ F

θθ 〉T term isalso significant for dσ F/dT at the NR and, hence, in the totalresults.

The temperature derivative at room temperaturedσ TOTAL

F /dT = −17 ppb/K is more than three times largerthan the experimental result dσ EXP

F /dT = −4.2 ppb/K inRef. 50. In the case of a seemingly similar discrepancyfor δXe, we argued (vide supra) that experimentally thesolution-induced, increased bond lengths cause an increase inthe positive relativistic terms. Here the same reasoning doesnot work, as the relativistic contributions further increase thenegative dσ F/dT. Hence, direct solvent effects, independentof geometry change, should be responsible for the differenceof the theoretical and experimental dσ F/dT. To quantitativelymodel the temperature dependence of shielding constantsmeasured in a condensed phase, a proper solvent treatment isnecessary. This is, however, beyond the scope of the presentwork.

TABLE IV. Calculated primary �Xe(131/129Xe) isotope shift in XeF2.a

T = 0 K T = 300 K

Method NR RELb TOTAL NR RELb TOTAL

BLYP − 13.4 6.1 − 7.3 − 6.3 2.8 − 3.5B3LYP − 16.9 5.8 − 11.1 − 8.2 2.6 − 5.6B3LYPc − 16.5 5.9 − 10.6 − 7.9 2.6 − 5.3D-B3LYPc,d,e − 16.4 5.1 − 11.3 − 7.8 2.2 − 5.6BHandHLYP − 22.3 5.1 − 17.2 − 11.2 2.1 − 9.1HF − 29.7 4.3 − 25.4 − 15.3 1.5 − 13.8HFc − 29.5 4.4 − 25.0 − 15.1 1.5 − 13.6DHFc,d − 29.5 2.6 − 26.9 − 15.1 0.4 − 14.7MP2 − 12.4 5.8f − 6.6 − 6.1 2.6f − 3.5CCSD − 16.7 5.8f − 10.9 − 8.4 2.6f − 5.9CCSD(T) − 12.3 5.8f − 6.5 − 6.0 2.6f − 3.5

aIn ppb. �Xe(131/129Xe) = σ Xe(129Xe) − σ Xe(131Xe).bBPPT calculations with FIVu6/HIVu2 basis set if not otherwise noted.cWith co-r basis set for F,41 27s25p21d1f for Xe.42

dRelativistic 4-component Dirac calculation.eCGO placed at Xe.fRelativistic BPPT contribution from DFT/B3LYP calculation.

FIG. 3. Electron correlation effects on the temperature dependence ofthe Xe primary isotope shift in XeF2. Rovibrational contributions atCCSD(T)/B3LYP level for nonrelativistic/relativistic contributions are alsodisplayed.

E. Primary Xe isotope shift

Although the primary isotope shift (Table IV) is tediousto measure experimentally, we present it here anyway as theeasier secondary Xe isotope shift is not present in XeF2 withonly one fluorine isotope 19F. The primary shift is shown inFig. 3 between the NMR-active isotopes of Xe, namely, 129Xeand 131Xe. In contrast to the rovibrational effect on absoluteshieldings, the isotope shifts are purely quantum-mechanicalin nature and, hence, they decrease with temperature. For theprimary Xe isotope shift we obtain ca. −6 ppb at 0 K and ca.−4 ppb at room temperature. The negative sign means thatthe lighter isotope is less shielded.

At first sight, the MP2 electron correlation treatment ap-pears to be enough at the NR level, as was the case for abso-

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214309-8 Lantto, Kangasvieri, and Vaara J. Chem. Phys. 137, 214309 (2012)

TABLE V. Experimental and theoretical secondary 1�19F(M/128Xe) isotopeshift in XeF2.a

Isotopomer T (K) NR REL TOTAL Exp.b

1�19F(129/128Xe) 257 −1.0 −0.3 − 1.4 . . .293 −0.9 −0.3 − 1.3 −1.2308 −0.9 −0.3 − 1.2 −1.1

1�19F(130/128Xe) 257 −2.1 −0.7 − 2.7 −2.6293 −1.9 −0.6 − 2.5 −2.5308 −1.8 −0.6 − 2.4 −2.3

1�19F(132/128Xe) 257 −4.0 −1.3 − 5.4 −5.1293 −3.7 −1.2 − 4.9 −4.7308 −3.6 −1.2 − 4.8 −4.6

1�19F(134/128Xe) 257 −6.0 −2.0 − 8.0 −7.5293 −5.5 −1.8 − 7.3 −7.0308 −5.3 −1.8 − 7.0 −6.7

1�19F(136/128Xe) 257 −7.9 −2.6 − 10.5 −9.9293 −7.2 −2.4 − 9.6 −9.1308 −6.9 −2.3 − 9.2 −8.8

aIn ppb. 1�19F(M/128Xe) = σ19F(128Xe) − σ19F(M Xe). Non-relativistic (NR) and rel-ativistic (REL) BPPT contributions from CCSD(T) and DFT/B3LYP calculations,respectively.bExperimental shifts measured in acetonitrile-d3 solution in Ref. 50.

lute shieldings, but the CCSD result reveals that this findingis due to error cancellation and CCSD(T) correlation is neces-sary for a quantitative treatment. Due to the error cancellationin general, for relative properties such as isotope and chemicalshifts, DFT performs satisfactorily at the NR level. IncreasingEEX induces, again, deterioration away from the benchmark-quality CCSD(T) data and, as the worst case, BHandHLYPgives almost twice too large a rovibrational correction. Fortu-itously, however, BLYP produces results almost right on thespot. The largest errors occur at low temperatures.

Results with both BPPT and 4-component Dirac meth-ods show that relativity is very important, canceling with theNR isotope shift by almost 50%. This is again due to BPPT-5contributions as seen in Fig. 3. There is a much smaller sensi-tivity to the choice of the DFT functional than in the NR part:BLYP and B3LYP produce practically similar results, whileBHandHLYP gives only a slightly smaller relativistic correc-tion and a similar T dependence. While the individual REL-pcontributions are quite large, they almost fully cancel eachother. Due to this and the smallness of the FC-I contribution,the SD-I term is mainly responsible for the relativistic isotopeshift and its temperature dependence. The role of the REL-pcontributions increases, however, with the admixture of EEX.

The 〈σ Xeθθ 〉T term is mostly responsible for the T depen-

dence of the primary Xe isotope shift. This is partly due to thealmost full cancellation of the NR and relativistic contribu-tions to 〈σ Xe

r 〉T , which results in almost constant contributionin the whole T range. In addition, heavy cancellation takesplace between the 〈σ Xe

rr 〉T and 〈σ Xerr ′ 〉T terms.

F. Secondary 19F isotope shifts

Similarly to the case of δF, relativity enhances the sec-ondary isotope shifts of the light 19F nucleus (Table V) incontrast to the primary isotope effect of the heavy Xe nu-

FIG. 4. Temperature dependence of the secondary 19F isotope shiftsin MXeF2. Computed at CCSD(T)/DFT(B3LYP) level for nonrelativis-tic/relativistic contributions. For the 1�19F(136/128Xe) isotope shift, also theCCSD(T)/DFT(BHandHLYP) result is shown. Experimental (EXP) isotopeshifts are taken from Ref. 50.

cleus. In the latter case, the relativistic contributions largelycancel with the NR part. Generally, relativistic effects, com-puted either with BPPT or 4-component Dirac methods, havea significant role in bringing the current theoretical values intoa quantitative agreement with experimental results50 for boththe magnitude and temperature dependence of the secondaryisotope shifts, as seen in Figure 4. A Devil’s advocate view-point on the good agreement that we have reached with BPPTcould be that the rovibrational effects (which is the presenttopic) are not able to discern between the fully relativistic andapproximate methods.

It is of paramount importance to treat electron correla-tion effects accurately (Table VI). As already noted, at theNR level, the good performance of the MP2 method, as ref-

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214309-9 Lantto, Kangasvieri, and Vaara J. Chem. Phys. 137, 214309 (2012)

TABLE VI. Calculated secondary 1�19F(136/128Xe) isotope shift in XeF2.a

T = 0 K T = 300 K

Method NR RELb TOTAL NR RELb TOTAL

BLYP − 21.3 − 4.2 − 25.5 − 11.8 − 2.4 − 14.2B3LYP − 17.3 − 4.1 − 21.4 − 9.4 − 2.4 − 11.7B3LYPc − 17.8 − 4.2 − 22.1 − 9.7 − 2.5 − 12.1D-B3LYPc,d,e − 17.9 − 3.8 − 21.7 − 9.7 − 2.2 − 11.9BHandHLYP − 11.6 − 3.6 − 15.2 − 5.8 − 2.2 − 8.0HF − 4.2 − 1.8 − 6.0 − 1.3 − 1.2 − 2.5HFc − 4.4 − 1.9 − 6.3 − 1.4 − 1.3 − 2.7DHFcd − 4.4 − 2.8 − 7.2 − 1.4 − 1.6 − 3.0MP2 − 12.0 − 4.1f − 16.1 − 6.2 − 2.4f − 8.5CCSD − 10.7 − 4.1f − 14.8 − 5.3 − 2.4f − 7.6CCSD(T) − 13.6 − 4.1f − 17.7 − 7.1 − 2.4f − 9.4

aIn ppb. 1�19F(136/128Xe) = σ F(128Xe) − σ F(136Xe).bBPPT calculations with FIVu6/HIVu2 basis set if not otherwise noted.c With co-r basis set for F,41 27s25p21d1f for Xe.42

dRelativistic 4-component Dirac calculation.eCGO placed at Xe.f Relativistic BPPT contribution from DFT/B3LYP calculation.

FIG. 5. Electron correlation effects on the temperature dependence of thesecondary 1�19F(136/128Xe) isotope shift in XeF2. Nonrelativistic/relativisticrovibrational contributions at CCSD(T)/B3LYP level are also displayed. Ex-perimental results taken from Ref. 50

erenced to the benchmark CCSD(T) data, is coincidental asthe higher-level CCSD again produces larger deviations. Lackof consistency is evident among the results by the currentDFT functionals: their differences at NR level exceed themagnitude of relativistic effects. As seen in Figure 5, forthe largest 1�19F(136/128Xe) isotope shift, DFT/B3LYP repro-duces experimental results already at the NR level, howeverfor the wrong reason. When comparing with the benchmarkCCSD(T) data, in contrast to Xe, BLYP is for the fluorineisotope shifts the worst functional and BHandHLYP performsbest.

As found for the other NMR parameters, the DFT perfor-mance is more systematic for the relativistic corrections of thesecondary isotope shifts of 19F. All DFT functionals remainwithin 1 ppb at T = 0 K, while the HF method clearly un-derestimates the relativistic correction. The systematic DFTperformance for the BPPT terms enables, when combinedwith benchmark NR isotope shifts obtained at CCSD(T) level,a quantitative analysis of the experimentally observed NMRisotope shifts as seen in Figure 4(c)).

Heavy cancellation of the 〈σ Xerr ′ 〉T rovibrational contri-

bution with the other terms takes place, as clearly dis-played in Fig. 5. The bending term 〈σ Xe

θθ 〉T affects heav-ily the low-temperature range and almost vanishes whenapproaching room temperature. Overall, the magnitude andtemperature trend of 1�19F(M

′/128Xe) mainly follow the〈σ F

r 〉T contribution.As shown in Figure 4 of the supplementary material,51

relativistic effects on the 19F isotope shift arise due to spin-orbit SO-I (FC-I and SD-I) contributions due to the cancella-tion of scalar relativistic p/mv and p/Dar contributions.

IV. CONCLUSIONS

We have shown that quantum-mechanical treatment ofthe nuclear motion on benchmark quality potential hypersur-face, combined with high-quality shielding constant surfaceswith ab initio nonrelativistic and DFT-estimated relativisticcontributions, reaches quantitative agreement with the experi-mentally observed 19F secondary isotope shifts in XeF2. Sim-ilarly, the estimated rovibrational effects on the absolute Xeand F shielding constants as well as the primary isotope shiftof Xe may be speculated to also be of predictive quality, apartfrom the significant solvent effects that we expect for theshielding constants. This calls for gas-phase measurements.

There is no systematic way to choose a DFT func-tional that would produce accurate nonrelativistic equilibriumshielding as well as the shielding surfaces for both Xe and F.As shown here, the currently existing purely DFT-based four-component (Dirac-DFT) or two-component (ZORA, X2C,etc) methods are not be capable of providing a quantitativetreatment of the chemical and isotope shifts of neither thelight or heavy nucleus in XeF2. This is due to the large DFTerrors in the correlation treatment, which mostly affect theNR contribution. Fortunately, correlation effects in relativis-tic corrections are described more consistently by DFT and,hence, the above-mentioned methods can be used for that asshown here by the close numerical agreement of the results

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214309-10 Lantto, Kangasvieri, and Vaara J. Chem. Phys. 137, 214309 (2012)

obtained by BPPT and 4-component Dirac methods for therovibrational effects on Xe and F shieldings.

The large error due to DFT is most probably one of themain reasons for the remaining discrepancies with experi-ment in recent studies of heavy-element rovibrational andisotope effects in other heavy-element systems.4–8 As the iso-tope shifts are seen here and elsewhere11, 12 to be quite insen-sitive to solvent effects, the inadequacy of DFT is a more plau-sible modeling deficiency than the neglect of solvent effects.8

As it is shown here that thermal effects are notable for boththe shieldings and their isotope shifts, errors also arise due tonot carrying out a quantum-mechanical treatment of the finite-temperature effects. Although probably not as large as for theabsolute shielding, errors in the shape of the potential energysurface induced by DFT may also occur.

The case of XeF2 shows again that the comparison ofin vacuo calculated and experimental temperature depen-dence of the absolute shielding or chemical shift is imprac-tical if the experiments are carried out in the solution phase,due to the large solvent effects. Therefore, only the rarelyavailable low-pressure gas experiments provide reasonabledata for such investigations.13 The isotope shifts are almostsolvent-independent and depend on the same potential energyand property surfaces as the rovibrational effects of absoluteshieldings. Therefore, the isotope shifts provide an excellentplatform for testing many different aspects of first principlestheoretical modeling involving the electron correlation treat-ment and the description of the motion of the nuclei.

ACKNOWLEDGMENTS

We thank the Academy of Finland (P.L. and J.V.), TaunoTönninki Fund, and U. Oulu (J.V.) for funding. Computationswere partially carried out at CSC-IT Center for Science Ltd.(Espoo, Finland).

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