Quadratic response calculations of the electronic spin-orbit contribution to nuclearshielding tensorsJuha Vaara, Kenneth Ruud, Olav Vahtras, Hans Ågren, and Jukka Jokisaari Citation: The Journal of Chemical Physics 109, 1212 (1998); doi: 10.1063/1.476672 View online: http://dx.doi.org/10.1063/1.476672 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/109/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A fully relativistic method for calculation of nuclear magnetic shielding tensors with a restricted magneticallybalanced basis in the framework of the matrix Dirac–Kohn–Sham equationa) J. Chem. Phys. 128, 104101 (2008); 10.1063/1.2837472 Relativistic calculation of nuclear magnetic shielding tensor including two-electron spin-orbit interactions J. Chem. Phys. 125, 164106 (2006); 10.1063/1.2361292 Calculation of binary magnetic properties and potential energy curve in xenon dimer: Second virial coefficient of129 Xe nuclear shielding J. Chem. Phys. 121, 5908 (2004); 10.1063/1.1785146 Perturbational ab initio calculations of relativistic contributions to nuclear magnetic resonance shielding tensors J. Chem. Phys. 119, 2623 (2003); 10.1063/1.1586912 Second- and third-order spin-orbit contributions to nuclear shielding tensors J. Chem. Phys. 111, 2900 (1999); 10.1063/1.479572

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Quadratic response calculations of the electronic spin-orbit contributionto nuclear shielding tensors

Juha Vaaraa)

Department of Physics and Measurement Technology, Linko¨ping University, S-58 183 Linko¨ping, Swedenand NMR Research Group, Department of Physical Sciences, University of Oulu, P.O. Box 333,FIN-90571 Oulu, Finland

Kenneth Ruud,b) Olav Vahtras, and Hans ÅgrenDepartment of Physics and Measurement Technology, Linko¨ping University, S-58 183 Linko¨ping, Sweden

Jukka JokisaariNMR Research Group, Department of Physical Sciences, University of Oulu, P.O. Box 333,FIN-90571 Oulu, Finland

~Received 20 February 1998; accepted 14 April 1998!

The electronic spin-orbit contribution to nuclear magnetic shielding tensors, which causes theheavy-atom chemical shift of the shielding of light nuclei in the vicinity of heavy elements, iscalculated as a sum of analytical quadratic response functions. We include both the one- andtwo-electron parts of the spin-orbit Hamiltonian and consider the interaction with both the Fermicontact and the spin-dipolar mechanisms.Ab initio calculations at the SCF and MCSCF levels arepresented for the1H and13C shielding tensors in the hydrogen and methyl halides. The applicabilityof different approximations to the full spin-orbit correction is discussed, and the calculated resultsare compared with experimental data, where available. ©1998 American Institute of Physics.@S0021-9606~98!30228-7#

I. INTRODUCTION

The electronic observables of nuclear magnetic reso-nance ~NMR! spectroscopy—nuclear shielding, spin–spincoupling and quadrupole coupling tensors—probe the elec-tron distribution close to nuclei, where the electrons have ahigher velocity than in the valence region. The effects ofspecial relativity on NMR parameters are therefore appre-ciable much earlier in the Periodic Table than for many othermolecular properties.1 Nuclear shielding constants of noblegas atoms show, for example, a relativistic effect of 6 ppmfor Ar, increasing to 352 ppm for Kr, and as much as 1398ppm for Xe, with the total shielding constants being 559,3596, and 7040 ppm, respectively.2,3

The most consistent approach for calculating the NMRparameters in molecules containing heavy elements is fullyrelativistic four-componentab initio methods.1,4 However,obtaining relativistic corrections perturbatively from nonrel-ativistic quantum chemical calculations is a computationallyless demanding alternative, in particular when electron cor-relation is important. The leading relativistic corrections tothe molecular electronic Hamiltonian are the mass-velocityand one- and two-electron Darwin terms—often denoted sca-lar relativistic corrections—and the one- and two-electronspin-orbit ~SO! coupling.5 As the mass-velocity and Darwincontributions may induce large perturbations in the elec-

tronic structure close to the nuclei, perturbation theory maynot be adequate. To circumvent this, techniques like thefrozen-core approximation,6 quasi-relativistic effective corepotentials,7 or a relativistic zeroth-order Hamiltonian8 havebeen used to calculate scalar relativistic corrections to NMRproperties.

Heavy-atom chemical shift, i.e., the large change in thenuclear shielding constants of light atoms in the vicinity ofheavy elements, is caused mainly by SO coupling.9–11 Therehas been a considerable interest in perturbation calculationsof the SO contributions to nuclear shieldings. Semiempiricalcalculations of the Fermi contact/one-electron spin-orbit cor-rection, FC~1!, were presented in Refs. 12–14. In Refs. 13and 14 the spin-dipole/one-electron spin-orbit correction,SD~1!, was also considered.Ab initio calculations of theFC~1! and SD~1! corrections at the unrestricted Hartree–Fock ~UHF! level have recently been presented by Nakatsujiet al.15 Malkin et al. calculated the FC~1! correction usingdensity-functional theory~DFT!.16 While the contribution ofthe SO Hamiltonian is formally a third-order property, Fukuiet al.used UHF wave functions to investigate a second-ordercontribution arising from the field dependence of the SOHamiltonian interacting with the FC operator.17 This contri-bution was in the hydrogen halides shown to be significantfor the heavy-element shieldings but negligible for the hy-drogen shieldings. Scalar relativistic effects have also beenfound to be unimportant for hydrogen shieldings.8,18

The present paper describes the first correlatedab initiocalculations of the SO contributions to the nuclear shieldingtensor,s. We also include for the first time the interaction of

a!Author to whom correspondence should be addressed. Permanent address:University of Oulu, FIN-90571, Oulu, Finland.

b!Permanent address: Department of Chemistry, University of Oslo, Blind-ern, P.O. Box 1033, N-0315 Oslo, Norway.

JOURNAL OF CHEMICAL PHYSICS VOLUME 109, NUMBER 4 22 JULY 1998

12120021-9606/98/109(4)/1212/11/$15.00 © 1998 American Institute of Physics

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the two-electron SO operator with the Fermi contact andspin-dipole operators, the FC~2! and SD~2! terms. Themethod is verified by calculations of the shielding tensors inthe hydrogen and methyl halides. In contrast to previous in-vestigations, which have employed the finite-perturbationmethod,15,16 we calculate the SO corrections analyticallyfrom quadratic response functions using self-consistent field~SCF! and multiconfiguration self-consistent field~MCSCF!reference wave functions.19 A similar approach was recentlyused to calculate some of the SO corrections to the spin–spincoupling constants in the XH4 series~X5C, Si, Ge, Sn! atthe SCF level.20

II. THEORY

In the effective spin Hamiltonian used to describe NMRexperiments, the nuclear shielding tensorsK couples the ap-plied external magnetic fieldB0 with the spinIK of nucleusK. The Cartesianet-component of the shielding tensor isdefined as the derivative of the electronic energyE:

sK,et5]2E~mK ,B0!

]mK,e]B0,tU

mK50,B050

, ~1!

wheremK5gK\IK is the nuclear magnetic moment associ-ated with IK , and the proportionality factor contains thenuclear gyromagnetic ratiogK . In nonrelativistic theory, theshielding tensor for a closed shell molecule is given by thesecond-order perturbation expression

sK,etnr 5sK,et

d 1sK,etp , ~2!

where the diamagnetic and paramagnetic contributions are

sK,etd 5

]2^0uHK,B0u0&

]mK,e]B0,tU

mK50,B050

~3!

and

sK,etp 5

]2

]mK,e]B0,t(nÞ0

^0uHK,PSOun&^nuHB0u0&1^0uHB0

un&^nuHK,PSOu0&

E02EnU

mK50,B050

, ~4!

respectively, with the operators in Eqs.~3! and ~4! being

HK,B05

e2\

2me

m0

4pgKIK•(

i

1~r iO•r iK !2r iOr iK

r iK3

•B0 , ~5!

HK,PSO5e\

me

m0

4pgKIK•(

i

l iK

r iK3

, ~6!

and

HB05

e

2me(

il iO•B0 . ~7!

In the above equations,r iK is the length of the vectorr iK

5r i2RK , giving the position of electroni relative tonucleusK, and r iO5r i2RO is the electron position vectorrelative to the gauge originO. l iK andl iO are the correspond-ing angular momenta.

As the operator linear in the external field,HB0, does not

involve the electron spinsi , the contributing excited statesun& in the summation are singlet states. Consequently, theparamagnetic spin-orbit operator,HK,PSO, appearing in Eq.~4! is a singlet operator, linear inIK . The sum-over-statesexpression forsK,et

p can be written as a linear response func-tion

sK,etp 5Np^^Ae ;Ct&&0 , ~8!

where the prefactor and second-quantized operators are

Np5e2

2me2

m0

4p, Ae5(

pq^pu

l K,e

r K3

uq&Epq ,

~9!

Ct5(rs

^r u l O,tus&Ers ,

with the singlet orbital excitation operatorEpq5apa† aqa

1apb† aqb .21

The field-free SO interaction Hamiltonian,5

HSO5e2\

4me2

m0

4pgeF(

LZL(

i

si• l iL

r iL3

2(i j

8~si12sj !• l i j

r i j3 G

[HSO~1!1HSO~2! , ~10!

gives a relativistic correction to the nuclear shielding at thethird order in a perturbation expansion. In Eq.~10!, ge

'2.0023 is the electrong-factor, ZLe is the charge ofnucleusL, and l i j 52 i\(r i2r j )3¹ i is a generalized two-electron angular momentum operator. We will in this workfocus on the shieldings of light elements and for this reasonneglect the contributions from the magnetic field dependenceof the angular momentum operators in Eq.~10!.17

HSO is an operator of rank one in spin space. Conse-quently, the operatorHK appearing in the products of matrixelements~with six permutations! in the numerator of a third-order sum-over-states expression for the SO-induced shield-ing

1213J. Chem. Phys., Vol. 109, No. 4, 22 July 1998 Vaara et al.

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sK,etSO 5

]2

]mK,e]B0,tF (

k,mÞ0

^0uHKuk&^kuHSOum&^muHB0u0&1permutations

~E02Ek!~E02Em! GUmK50,B050

~11!

must be a triplet operator. Two triplet operators that are lin-ear in IK are the FC operator

HK,FC54p

3

e\2

me

m0

4pgegK(

id~r iK !IK•si , ~12!

and SD operator

HK,SD5e\2

2me

m0

4pgegKIK•(

i

3r iKr iK21r iK2

r iK5

•si . ~13!

The spin-orbit corrections to the nuclear shielding can bewritten as a sum of four contributions, all expressible asquadratic response functions

sK,etSO 5sK,et

FC~1!1sK,etFC~2!1sK,et

SD~1!1sK,etSD~2! , ~14!

where we have forHK5HK,FC:

sK,etFC~n!5NSO ^Ae ;Be

~n! ,Ct&&0,0, ~15!

with the Ct operator being identical to the nonrelativisticcase@Eq. ~9!#, while the prefactor now reads

NSO5e4\2

8me4 S m0

4p D 2

ge . ~16!

Using second quantization,

Ae54p

3ge(

pq^pud~rK!uq&spq,e ~17!

and

Be~1!5(

rs^r u(

LZL

l L,e

r L3

us&srs,e , ~18!

for FC~1!, and

Be~2!52(

rstu^rt u

l 12,e

r 123

usu&~srs,eEtu2sru,edst!, ~19!

for FC~2!. In these expressions, the Cartesian components ofthe spin excitation operators are

spq,x51

2~apa

† aqb1apb† aqa!,

spq,y52i

2~apa

† aqb2apb† aqa!, ~20!

spq,z51

2~apa

† aqa2apb† aqb!.

Finally, for the terms withHK5HK,SD,

sK,etSD~n!5NSO (

n5x,y,z^^Aen ;Bn

~n! ,Ct&&0,0, ~21!

where

Aen5ge

2 (pq

^pu3r K,er K,n2denr K

2

r K5

uq&spq,n , ~22!

andBn(n) can be obtained from Eqs.~18! and ~19! for SD~1!

and SD~2!, respectively. The necessary triplet quadratic re-sponse functions and SO integrals have been implemented inthe DALTON quantum chemistry program22 by Vahtraset al.in Refs. 23 and 24, and this is the implementation used in thepresent work.

III. SAMPLE CALCULATIONS

We will use the above presented computational schemeto calculate the SO-induced shielding in two series of mol-ecules where this effect is substantial—the hydrogen andmethyl halides—using SCF and MCSCF wave functions. Inorder to be able to compare with experiment in anisotropicliquid crystal phases, not only the isotropic shielding con-stantssK5 1

3(sK,xx1sK,yy1sK,zz), but also the anisotropiesDsK5sK,zz2

12(sK,xx1sK,yy)—defined with respect to the

molecular symmetry axisz—are tabulated. Our main interestis in the1H shieldings in the HX~X5F, Cl, Br, I! series, andthe 13C shieldings for CH3X. For completeness, we alsopresent spin-orbit corrections to the heavy-atom shieldings inthe hydrogen halides and for the1H shielding in the methylhalides. However, the SO effect does not necessarily domi-nate the relativistic contribution to the shielding of the heavyatoms and the neglected scalar relativistic interactions shouldbe taken into account. This is most apparent in the examplementioned in the Introduction, as the SO-induced shieldingof the noble gas atoms is zero by symmetry.

A. Computational details

All the calculations presented here have been performedusing theDALTON program.22 The molecular geometries usedare collected in Table I. The nonrelativistic shielding tensors@Eq. ~2!# were calculated using gauge-including atomic or-bitals~GIAO! as described in Ref. 27. The calculations of theSO contributions to the shielding@Eq. ~14!# were carried outusing a common gauge origin placed on the halogen atom.

TABLE I. Molecular geometries used in the calculations.a

HXb CH3Xc

X r HX r CX r CH bHCX

F 0.9175 1.3907 1.0979 108.81Cl 1.2747 1.7859 1.0944 108.43Br 1.4141 1.9434 1.0930 107.69I 1.6046 2.1417 1.0939 107.66

aBond lengths in Å and angles in degrees.bGeometries taken from Ref. 13.cThe r a(300 K) geometries~Ref. 25! calculated as in Ref. 26 for the13C1H3X isotopomers with X519F, 35Cl, 79Br, and127I.

1214 J. Chem. Phys., Vol. 109, No. 4, 22 July 1998 Vaara et al.

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B. Hydrogen halides

The primary one-particle basis sets used are based on thecompilations of Ref. 28~Br and I! and Ref. 29~the otheratoms!, and are described in Table II. For the hydrogen ha-lides, we augmented the HIV basis by adding a total of twod and onef function on hydrogen, twof and oneg functionon F, and threef , two g and oneh function on Cl, Br, and I.These modifications gave negligible changes in the calcu-lated SCF shieldings, and all tabulated numbers have beencalculated without these additional polarization functions. Incontrast, the addition of tight functions to the decontractedHIV basis proved to be necessary to reach convergence ofthe SCF shieldings. The tight functions were added to allshells that are occupied in the free atoms, i.e.,s functions forH, s andp functions for F and Cl, ands, p, andd functionsfor Br and I. The exponents of the tight functions were ob-tained by multiplying the largest exponent in a given shell bya factor of 3. These sets withn additional sets of tight func-tions are denoted HIVun, and the different spin-orbit contri-butions to the isotropic shielding constants and anisotropiesare converged to within 1% in the SCF calculations withn53. These are the results we report together with our pri-mary sets collected in Table II.

For the hydrogen halides we used, apart from SCF, alsocomplete active space~CAS! SCF wave functions describedby 1000CAS6331 for HF, 3110CAS6331 for HCl, 7331CAS6331 forHBr, and 11;552CAS6331 for HI, in the notationinactiveCASactive. The four numbers in the different spaces de-note the number of orbitals in the A1, B1, B2, and A2 sym-metry species of the Abelian groupC2v used in the calcula-tions. The number of determinants in each of these wavefunctions is 128 283.

The calculated nonrelativistic1H shielding constants forthe hydrogen halides and the SO corrections are listed inTable III. Table IV contains similar data for the heavy atoms.The anisotropies of both the hydrogen and halogen shieldingtensors are collected in Table V, separated into nonrelativis-tic and SO contributions. The GIAO SCF nonrelativisticshielding tensors are found to be well-converged. All thecalculated parameters are within 1% of the HIVu3 resultsusing the HIV basis, and in most cases already with the HIII

basis. The tight functions added in the HIVun series (n50,1,2,3) have little effect on these parameters.

The spin-dipole contributions to the SO-induced shield-ing of hydrogen can be seen to converge rapidly and mo-notonously. The addition of tight functions hardly affects theSD contributions at all, whereas the convergence of the FCcontributions is only monotonous within the HIVun seriesof basis sets. For the FC contributions, results within 1% ofthe HIVu3 results are obtained forn52, with the signifi-cance of the tight functions most easily realized by the factthat the HIV set gives systematically 5% smaller contribu-tions ~in absolute value! for all parameters in both the SCFand MCSCF calculations.

The SO-induced contribution to the heavy atom shield-ings follows roughly the same pattern as that for hydrogen.The tight functions affect, however, also the SD contribu-tions and, in particular, the SD~1! term appears to be con-verged only to about 2% with the biggest basis set. The HIVset gives 1%–3% smaller results than those of the HIVu3basis, with the exception of thesX

SD(1) andDsXSD(1) contribu-

tions which converge significantly slower. The tight func-tions are less important for the total heavy atom shieldingcorrections than in the case of hydrogen.

Electron correlation affects the nonrelativistic shieldingparameters only slightly; the isotropic and anisotropic hydro-gen shieldings differ by less than 1% in the SCF and CAScalculations, while correlation changessX

nr by about 2%. Thechanges inDsX

nr due to correlation are larger, of the order of210%. Since restricted Hartree–Fock wave functions are notstable towards triplet perturbations, the inclusion of electroncorrelation can be expected to be more important for the SOcontributions to the shieldings. This is also observed; all pa-rameters for both hydrogens and halogens are systematicallyand significantly—mostly by 25%–30% and even more forhalogen anisotropies—reduced in the CAS calculations com-pared to the SCF results.

If we consider the importance of the different contribu-tions to the total SO-induced1H shielding constants andanisotropies, the positive FC~1! term is always found to bedominating. In absolute values, the ordering of the otherterms isuFC(2)u.uSD(1)u.uSD(2)u, where the two-electron

TABLE II. Primary series of one-particle basis sets used.a

HII HIII HIV

Hb @5s1p/3s1p# @6s2p/4s2p# @6s3p1d/5s3p1d#C,Fb @9s5p1d/5s4p1d# @11s7p2d/7s6p2d# @11s7p3d1f /8s7p3d1f #Clc @11s7p 2d/7s6p2d# @12s8p3d/8s7p3d# @12s8p4d2f /9s8p4d2f #Brd @16s13p10d/11s10p10d# @16s13p11d/12s11p11d# @16s13p12d2f /13s12p12d2f #Ie @20s16p13d/13s12p10d# @20s16p14d/14s13p12d# @20s16p15d2f /15s14p14d2 f #

aThe sets for Br and I are based on Ref. 28, while the rest are from Ref. 29. Only the innermost orbitals~1s,2p, 3d) were contracted.

bThe exponents of the polarization functions are given in Ref. 27.cThe exponents of the polarization functions are as follows. HII:d ~1.83, 0.46!, HIII: d ~3.2, 0.8, 0.2! ~Ref. 27!,HIV: d ~9.6, 2.4, 0.6, 0.15! ~Ref. 27! and f ~1.5, 0.5! ~Ref. 27!.

dThe exponents of the two~HII !, three~HIII ! and four~HIV ! d-type polarization functions were obtained bysuccessive divisions by a factor of 3, starting from the most diffuse existing primitive one. The exponents ofthe f -type primitives in HIV are~0.401~Ref. 30!, 0.133!.

ePolarization exponents for thed shell obtained as in footnote d. Thef exponents in HIV are~0.015~Ref. 31!,0.005!.

1215J. Chem. Phys., Vol. 109, No. 4, 22 July 1998 Vaara et al.

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terms always give a negative contribution. The dominance ofFC~1! increases with the atomic number of the halogen: inHF, uFC(2)u is roughly 30% of the FC~1! term, while it in HIis only 7%. Similar ratios apply for the absolute values of theSD~1! and SD~2! terms, the SD~1! term being 9% of the

FC~1! term in HF, and of the same positive sign, but only 3%for HI. These relative proportions apply to both the SCF andMCSCF results.

The present and previous15,16 calculations ofsH in thehydrogen halides are compared to experiment in Table VI,

TABLE III. Calculated1H nuclear shielding constants in the HX~X5F, Cl, Br, I! series.a

X Theory Basis snr sFC(1) sFC(2) sSD(1) sSD(2) sSO s tot

F SCF HII 29.02 0.192 20.061 0.019 20.006 0.14 29.16HIII 28.65 0.184 20.058 0.017 20.005 0.14 28.79HIV 28.39 0.179 20.056 0.017 20.005 0.13 28.52

HIVu3 28.39 0.190 20.059 0.017 20.005 0.14 28.54CAS HIV 28.83 0.134 20.042 0.012 20.004 0.10 28.93

HIVu3 28.84 0.142 20.045 0.012 20.004 0.11 28.94Cl SCF HII 31.10 1.042 20.193 0.052 20.010 0.89 32.00

HIII 30.76 1.050 20.194 0.055 20.010 0.90 31.66HIV 30.64 1.026 20.190 0.057 20.011 0.88 31.52

HIVu3 30.64 1.083 20.200 0.058 20.011 0.93 31.57CAS HIV 30.82 0.757 20.140 0.043 20.008 0.65 31.47

HIVu3 30.82 0.799 20.147 0.043 20.008 0.69 31.51Br SCF HII 31.19 5.630 20.552 0.224 20.022 5.28 36.47

HIII 31.00 5.580 20.548 0.227 20.022 5.24 36.23HIV 30.81 5.632 20.555 0.238 20.024 5.29 36.10

HIVu3 30.81 5.926 20.584 0.238 20.024 5.56 36.37CAS HIV 31.02 4.002 20.395 0.177 20.018 3.77 34.79

HIVu3 31.03 4.212 20.416 0.177 20.018 3.96 34.98I SCF HII 31.56 17.560 21.162 0.496 20.034 16.86 48.42

HIII 31.65 17.496 21.165 0.517 20.035 16.81 48.47HIV 31.50 17.550 21.173 0.538 20.039 16.88 48.38

HIVu3 31.51 18.447 21.232 0.538 20.038 17.71 49.22CAS HIV 31.58 11.901 20.797 0.396 20.028 11.47 43.05

HIVu3 31.58 12.510 20.837 0.396 20.028 12.04 43.62

aValues in ppm.

TABLE IV. Calculated X nuclear shielding constants in the HX~X5F, Cl, Br, I! series.a

X Theory Basis snr sFC(1) sFC(2) sSD(1) sSD(2) sSO s tot Expt.

F SCF HII 415.8 0.508 20.220 0.249 20.091 0.4 416.3HIII 413.9 0.427 20.179 0.178 20.069 0.4 414.2HIV 414.0 0.411 20.172 0.159 20.064 0.3 414.4

HIVu3 413.6 0.422 20.175 0.149 20.064 0.3 414.0CAS HIV 422.9 0.395 20.161 0.091 20.040 0.3 423.2

HIVu3 422.5 0.405 20.164 0.082 20.040 0.3 422.8 41066b

Cl SCF HII 959.4 3.287 20.912 1.651 20.369 3.7 963.1HIII 955.0 2.823 20.771 1.389 20.332 3.1 958.1HIV 947.5 2.622 20.700 1.032 20.273 2.7 950.2

HIVu3 946.8 2.678 20.712 0.930 20.273 2.6 949.5CAS HIV 969.1 2.229 20.604 0.520 20.169 2.0 971.1

HIVu3 968.6 2.277 20.615 0.427 20.169 1.9 970.5 952c

Br SCF HII 2625.3 29.952 24.117 13.200 21.709 37.3 2662.6HIII 2632.0 29.072 23.953 11.742 21.638 35.2 2667.2HIV 2634.2 28.428 23.862 10.693 21.598 33.7 2667.9

HIVu3 2633.9 28.747 23.903 9.581 21.616 32.8 2666.7CAS HIV 2682.4 22.593 23.221 4.946 20.997 23.3 2705.7

HIVu3 2682.0 22.835 23.254 3.928 21.011 22.5 2704.5 2617c

I SCF HII 4535.0 115.442 211.235 61.724 25.172 160.8 4695.7HIII 4551.3 109.522 210.552 55.000 24.968 149.0 4700.3HIV 4552.0 106.879 210.274 49.835 24.902 141.5 4693.5

HIVu3 4551.7 106.939 210.277 43.414 24.976 135.1 4686.8CAS HIV 4654.8 84.021 28.560 21.724 22.931 94.3 4749.0

HIVu3 4654.5 84.071 28.563 15.966 22.994 88.5 4743.0 4510c

aValues in ppm.bReference 32. Combination of theoretically calculated diamagnetic shielding and experimental spin-rotationconstant. The rovibrational corrections were estimated at210.4 ppm~Ref. 33! or 210.8 ppm~Ref. 34! atT5300 K.

cReference 35. Estimate based on experimental spin-rotation constants and a model calculation for diamagneticshieldings.

1216 J. Chem. Phys., Vol. 109, No. 4, 22 July 1998 Vaara et al.

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and Fig. 1 illustrates the trends. It is obvious that a state-of-the-art nonrelativistic calculation is uncapable of reproduc-ing the experimental trend ofsH , whereas all calculationsincluding the SO corrections reproduce at least qualitativelythe relativistic increase of the1H shielding. The overall mag-nitude of the SO contributions may be appreciated by con-sidering the ratiosH

SO/sHnr which is 0.004, 0.022, 0.13, and

0.38 for HF, HCl, HBr, and HI, respectively. The corre-sponding ratios for the anisotropies are roughly twice aslarge. The 12 ppm SO-shift of the hydrogen shielding in HIcorresponds in magnitude to the whole chemical shift rangeof 1H.

The magnitude of the dominating FC~1! contribution de-creases in the order UHF, DFT, CAS~Table VI!, with s tot

from the CAS calculation being in excellent agreement withexperiment. We note that the rovibrational contribution tosH in HF is estimated to be20.52 ppm,34 which accountsfor almost all the discrepancy of our CAS result compared toexperiment. An approximation of the full SO correction bythe FC~1! term only is seen to be a valid procedure due to therelative smallness of the other contributions and the partialcancellation of the FC~2! and SD~1! contributions. The error

introduced by only considering the FC~1! term ranges froman overestimation by 34% in HF to a modest 4% for HI. Dueto the negative sign of the two-electron SO corrections, theerror would be larger if only the one-electron terms werecalculated. The DFT results16 are slightly displaced from ex-periment towards higher shielding, partially due to the over-estimatedsH

nr , in particular for HF and HCl. The FC~1! con-tributions of the DFT calculations are quite close to our CASdata, the differences being of the order of the terms neglectedin Ref. 16. In contrast to the DFT and CAS calculations, thebasis set used in the UHF calculations15 may not be satu-rated, possibly causing a slight underestimation of the FC~1!contribution. However, we confirm the magnitude of theSD~1! contributions calculated in the UHF approach.

The shielding anisotropies@Fig. 1~c!# display an evenmore dramatic relativistic effect: TheDsH in HI is reducedto less than one-half ofDsH

nr by the SO correction, and therelativistic decrease of the anisotropy as a function of X isapparent. Our calculations verify the semiempirical relativis-tically parametrized extended Hu¨ckel ~REX! shieldinganisotropies calculated by Pyykko¨ et al.10

TABLE V. Calculated anisotropies of the1H and X shielding tensors in HX~X5F, Cl, Br, I!.a

X DsHnr DsH

SO DsHtot DsH

exp DsXnr DsX

SO DsXtot DsX

exp

F 23.10 20.17 22.93 2469b,c 88.4 0.3 88.7 10869b,d

Cl 22.05 21.08 20.98 2165b 270.3 1.5 271.8 300624b

Br 26.95 26.10 20.84 25e 666.3 10.6 676.9 760e

I 31.12 218.38 12.74 15e 1276.7 34.3 1311.0 1488e

aResults from the present CAS/HIVu3 calculation in ppm.bReference 36. Molecular beam electric resonance experiment.cThe rovibrational contribution atT5300 K was estimated at20.74 ppm~Ref. 34!.dThe rovibrational contribution atT5300 K was estimated at116 ppm~Ref. 34!.eFootnote c in Table IV.

TABLE VI. Comparison of theoretical and experimental1H shielding constants in the HX~X5F, Cl, Br, I!series.a

X Theory snr sFC(1) sFC(2) sSD(1) sSD(2) sSO s tot Expt.

F UHFb 27.33 0.17 0.02 0.18 27.52DFTc 30.04 0.17 0.17 30.21CASd 28.84 0.142 20.045 0.012 20.004 0.11 28.94 28.560.2e

Cl UHFb 29.76 0.88 0.04 0.92 30.69DFTc 31.72 0.89 0.89 32.61CASd 30.82 0.799 20.147 0.043 20.008 0.69 31.51 31.06f

Br UHFb 29.91 5.15 0.18 5.33 35.24DFTc 31.27 4.76 4.76 36.03CASd 31.03 4.212 20.416 0.177 20.018 3.96 34.98 34.96f

I UHFb 30.44 15.61 0.40 16.01 46.45DFTc 31.65 13.00 13.00 44.65CASd 31.58 12.510 20.837 0.396 20.028 12.04 43.62 43.86f

aValues in ppm.bReference 15.cReference 16. The nonrelativistic shielding constants~from Table 3 of the reference! are calculated with theBII basis corresponding to our HII set. The SO-corrections~from Table 2 of the reference! are calculated withthe large UP basis set, E128 integration grid and the finite perturbation parameterl50.0001.

dThis work. CAS/HIVu3 results from Table III.eReference 37. Gas-phase NMR measurement. The rovibrational contribution atT5300 K was estimated at20.52 ppm~Ref. 34!.

fReference 38. Gas-phase NMR chemical shifts with respect to CH4 converted into absolute shieldings usingsH(CH4)530.61 ppm~Ref. 37!.

1217J. Chem. Phys., Vol. 109, No. 4, 22 July 1998 Vaara et al.

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The magnitude of the different SO contributions is dif-ferent for the heavy atoms. For the shielding constantssX

SO,the SD~1! contribution is larger~in absolute value! thanFC~2! in HCl, HBr, and HI, while it is slightly smaller in HF.In HI, the SD~1! term is 20% of the dominating FC~1! term.For the anisotropiesDsX

SO, the SD~1! contribution is largerthan the negative FC~1! term and of opposite sign. Due to thecancellation of these two terms, the total SO contribution tothe shielding constants and anisotropies of the heavy atomsis small, ranging from 0.1% to 3% of the corresponding non-relativistic results. Considering that scalar relativistic effectsand rovibrational corrections have been neglected in thepresent work, our results compare favorably with experimentalso for the heavy atom shieldings.

C. Methyl halides

Because of the larger number of atoms and the fact thatwe can exploit onlyCs symmetry in calculations on the me-thyl halides, the computational task is significantly larger forthese molecules than for the hydrogen halides. For this rea-

son we tried to reach basis set convergence of the SO cor-rections by starting from the HII primitive set rather than byusing the HIVun sets. As our main interest focuses on the1Hand 13C shielding tensors, we improved the HII basis byadding tights ~H and C! and p ~C only! functions to thebasis obtained by decontracting the HII set for hydrogen andcarbon, while keeping the original contraction for the halo-gen. We denote these sets by HIIun, n indicating the numberof sets of tight functions added to hydrogen and carbon in amanner similar to that used for all atoms in the hydrogenhalides.

SCF calculations including up to six sets of tight func-tions where performed for CH3F. While changes of the orderof a few percent were observed fromn53 to n54, the totalSO corrections to the hydrogen and carbon shieldings werefound to be converged to within 1% of the HIIu6 resultsusing the HIIu4 basis, and the basis set convergence is betterfor the other methyl halides than for CH3F. For CH3F, theaddition of the polarization functions of the HIII set to allatoms~denoting this basis as HIIu4pIII) gave changes of theorder of a few percent in the dominating FC contribution tothe SO correction, and larger changes were observed for theSD contributions. A test calculation on CH3F using the po-larization functions of the HIV basis HIIu4pIV) showed thatthe results are converged to within 2% forsC

SO and within6% for sH

SO with the HIIu4pIII basis. As the HIIu4pIV basisbecomes prohibitively large for the heavier methyl halides,we tabulate only the results obtained with our HII and HIIIbasis sets~see Table II! together with the data calculatedusing the HIIu4pIII set.

For each member of the CH3X series, we used a re-stricted active space~RAS! SCF wave function, denoted by

RAS1inactiveRASRAS3

RAS2 with 4220RAS42

20 for CH3F, 4251RAS53

20 for CH3Cl,

4211;4RAS53

20 for CH3Br, and finally4217;7RASs53

20 for CH3I, withthe two digits in each entry denoting the number of orbitalsin the A8 and A9 irreps of theCs point group. A maximum oftwo particles ~holes! were allowed in the RAS3~RAS1!spaces. The active orbitals for all methyl halides account for60%–70% of the total MP2 occupation in the virtual orbitalspace, and the wave functions contain 6124 determinants forCH3F and 10 578 determinants for the other molecules.

Table VII reports the13C shielding constants from thepresent calculations, with the corresponding shieldinganisotropies listed in Table VIII. Table IX contains the1Hdata. The non-relativistic GIAO shielding tensors are con-verged with the HIII basis set, as a test calculation with thelarger HIV set changed the parameters by less than 1% in allcases exceptDsH in CH3Br, where a change of 1.8% wasobserved. Whereas the HII results deviate by as much as 4%from the HIII results, the addition of tight functions in theHIIun basis sets improves this. However, a more saturatedset of polarization functions than in the HII basis is necessaryin order to reach convergence.

Electron correlation increasessCnr by 2%–4%, while

RAS calculations using significantly larger active spacesgave changes of less than 1%, andsH

nr seems virtually unaf-fected by correlation. The correlation effects are larger in theshielding anisotropies. As for the hydrogen halides, correla-tion decreases the magnitude of the SO corrections to both

FIG. 1. Experimental~Ref. 38! and theoretical~a! 1H absolute shieldingconstants and~c! anisotropies in HX~X5F, Cl, Br, and I!. The deviation oftheory from experiment,sH

calc2sHexp, is shown in~b!. The present CAS/

HIVu3 results are shown for nonrelativistic calculations~nr!, with the fullspin-orbit correction~nr1SO!, and corrected only with the one-electronFermi contact term of the spin-orbit correction@nr1FC~1!#. The UHF andDFT results are from Refs. 15 and 16, respectively.

1218 J. Chem. Phys., Vol. 109, No. 4, 22 July 1998 Vaara et al.

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the hydrogen and carbon shieldings. For carbon, the decreasein the different contributions to the SO correction due toelectron correlation varies from 4% to 9% for CH3F, to 25%to 40% for CH3I. However, these changes are partially com-pensated for in larger RAS calculations for CH3F andCH3Cl. For the small SO effect on the hydrogen shieldings,the relative changes due to correlation are in some caseslarge.

The FC~1! term gives the largest SO contribution in allcases butDsH in CH3F, whereDsH

SO anyway is small. Asfor the hydrogen shielding of the hydrogen halides, thedominance of the largest contribution increases from CH3Fto CH3I, and the two-electron contributions are of the oppo-site sign of the corresponding one-electron terms. However,for the 13C shielding in all the methyl halides and the1Hshielding in CH3Br and CH3I, the SD~1! and FC~2! contri-

butions have the same sign. As these terms are of the sameorder of magnitude in methyl iodine, using the dominantFC~1! term alone will lead to an overestimation of the SOeffect by 15% to 20% for this molecule, and as much as100% in CH3F. For the carbon shieldings, the error for13Ccan be halved by neglecting only the two-electron contribu-tions.

The presentsC values are compared to other calcula-tions and experiment in Table X, and are illustrated in Fig. 2.Comparing first the nonrelativistic shielding constants calcu-lated using the UHF,15 DFT,16 and present MCSCF methods,we note that our values are systematically the highest. Elec-tron correlation is known to increase calculated shieldingconstants of main-group elements, as is also apparent fromTable VII. The DFT results forsC

nr are in all cases the lowest.This may be due to a systematic deshielding of the calculated

TABLE VII. Calculated13C nuclear shielding constants in the CH3X ~X5F, Cl, Br, I! series.a

X Theory Basis snr sFC(1) sFC(2) sSD(1) sSD(2) sSO s tot

F SCF HII 126.0 0.565 20.253 20.015 0.002 0.300 126.3HIII 122.7 0.547 20.250 20.021 0.003 0.280 123.0

HIIu4pIII 122.4 0.544 20.251 20.035 0.004 0.263 122.7RAS HIIu4pIII 124.4 0.504 20.235 20.033 0.004 0.240 124.6

Cl SCF HII 164.8 3.189 20.626 20.147 0.027 2.444 167.2HIII 162.6 3.127 20.622 20.134 0.026 2.397 165.0

HIIu4pIII 162.3 3.114 20.624 20.159 0.030 2.361 164.7RAS HIIu4pIII 168.5 2.745 20.552 20.123 0.023 2.093 170.6

Br SCF HII 173.2 16.616 21.595 21.127 0.111 14.005 187.2HIII 170.8 15.838 21.553 21.071 0.109 13.323 184.1

HIIu4pIII 170.5 15.895 21.559 21.106 0.113 13.343 183.8RAS HIIu4pIII 177.2 13.193 21.297 20.771 0.080 11.205 188.4

I SCF HII 194.3 51.306 23.253 24.665 0.305 43.694 238.0HIII 192.0 48.891 23.159 24.505 0.303 41.530 233.5

HIIu4pIII 191.7 49.171 23.173 24.572 0.307 41.733 233.5RAS HIIu4pIII 196.4 36.810 22.371 22.851 0.194 31.782 228.2

aValues in ppm.

TABLE VIII. Comparison of theoretical and experimental13C shielding anisotropies in the CH3X ~X5F, Cl,Br, I! series.a

X Dsnr DsFC(1) DsFC(2) DsSD(1) DsSD(2) DsSO Ds tot Expt.

F 97.6 20.87 0.33 0.05 20.02 20.51 97.1 113.560.7b

8764c

90620d

66615e

Cl 41.3 24.28 0.81 0.17 20.04 23.34 37.9 2265f

Br 32.9 220.18 1.95 1.02 20.11 217.32 15.5 29.060.7b

21065d

I 9.9 255.87 3.57 3.56 20.25 249.00 239.1 224.560.7b

2101615f

23066g

275620h

aWith respect to the molecular symmetry axis. The theoretical results are taken from the present RAS/HIIu4pIII

calculations. Values in ppm.bReference 39. NMR measurement in a nematic liquid crystal solution.cReference 40. NMR measurement in a nematic liquid crystal solution.dReference 41. NMR measurement in an Ar matrix.eReference 42. Early liquid crystal NMR result.fReference 43. Early liquid crystal NMR result.gReference 44. Early liquid crystal NMR result. An average of results from two supposedly identical samples istaken, and their difference is used as the error estimate.

hReference 45. Early liquid crystal NMR result.

1219J. Chem. Phys., Vol. 109, No. 4, 22 July 1998 Vaara et al.

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DFT shielding constants, similar to that previously observedfor 13C, 14/15N, and17O.52 All the nonrelativistic calculationsclearly underestimate the experimentalsC for X5Br and I,as was the case for the hydrogen halides. The total MCSCFsSO values are the smallest, mainly due to the fact that all thefour contributions have been included in these calculations.

The lack of gas-phase experimental data for all mol-ecules but CH3F makes a comparison of the theoretical andexperimental results difficult. Both the DFT and the MCSCFresults forsC are found to be larger than experiment. Theexperimental liquid-state data for CH3Cl, CH3Br, and CH3Iare likely to be 2–4 ppm deshielded due to medium effects.53

Furthermore, all the theoretical calculations~UHF, DFT andMCSCF! were done using rigid molecular geometries. Buck-ingham and Olega´rio54 recently reported nonrelativistic SCF

calculations of the13C and 1H shielding hypersurfaces forCH3F, CH3Cl and CH3Br. When the shielding constant isexpanded as a Taylor series in terms of the molecular sym-metry coordinates around the equilibrium geometry, thesedata can be used to estimate the vibrational contributions tothe shielding constant by combining with thermal averagesof the linear and quadratic symmetry coordinates.55 For themethyl halides, these averages are available from the forcefield calculations of Ref. 26, and combining the two sets ofdata leads to vibrational contributions at 300 K of24.2,23.3, and22.6 ppm for CH3F, CH3Cl, and CH3Br, respec-tively respectively~the isotopomers in question are listed inthe footnote to Table I!.

Another deshielding effect may be due to the neglect of

TABLE IX. Calculated and experimental1H shielding constants and anisotropies in CH3X ~X5F, Cl, Br, I!.a

X Theory snr sSO s tot s expb Dsnr DsSO Ds tot Dsexp

F RASc 27.66 0.04 27.69 26.61 4.07 20.01 4.06 5.260.2d

24.261.5e

27.060.5f

Cl RASc 28.92 0.04 28.97 27.90 4.60 0.01 4.61 1.160.9e

1.860.4f

Br RASc 29.17 0.05 29.22 28.30 5.75 0.15 5.89 1.360.6e

3.960.3f

I DFTg 29.5 20.1 29.4RASc 29.70 20.31 29.39 28.76 7.18 0.86 8.04 8.060.3h

6.560.3f

3.460.4e

1.8160.05i

aWith respect to the molecular symmetry axis. Values in ppm.bReference 46. Gas-phase NMR measurement.cThe result from the present RAS/HIIu4pIII calculation.dReference 40. NMR measurement in a nematic liquid crystal solution.eReference 47. Early liquid crystal NMR result.fReference 48. NMR measurement in a smectic liquid crystal solution.gReference 11.hReference 49. NMR measurement in a nematic liquid crystal solution.iReference 50. NMR measurement in a nematic liquid crystal solution.

TABLE X. Comparison of theoretical and experimental13C shielding constants in the CH3X ~X5F, Cl, Br, I!series.a

X Theory snr sFC(1) sFC(2) sSD(1) sSD(2) sSO s tot Expt.

F RASb 124.4 0.50 20.23 20.03 0.00 0.24 124.6 116.8c

Cl UHFd 163.0 2.64 -0.13 2.5 165.3DFTe 161.8 2.89 2.9 164.7RASb 168.5 2.74 20.55 20.12 0.02 2.09 170.6 168.1f

Br UHFd 172.1 14.08 20.98 13.1 185.2DFTe 171.0 13.84~16.69! 13.84~16.69! 184.8~187.7!RASb 177.2 13.19 21.30 20.77 0.08 11.21 188.4 183.6f

I UHFd 192.7 41.40 23.71 37.7 230.4DFTe 188.8 34.82 34.8 223.6RASb 196.4 36.81 22.37 22.85 0.19 31.78 228.2 215.3f

aValues in ppm.bThis work. The results from the RAS/HIIu4pIII calculation in Table VII taken for each X.cReference 51. Gas-phase NMR measurement.dReference 15.eReference 16. Calculations with the BII basis set and the finite perturbation parameterl50.001. For CH3Br,results for the larger UP basis are also shown~in parentheses!.

fReference 46. NMR measurement in the neat liquid. Chemical shifts with respect to methane converted intoabsolute shieldings using usingsC(CH4)5195.1 ppm~Ref. 51!.

1220 J. Chem. Phys., Vol. 109, No. 4, 22 July 1998 Vaara et al.

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scalar relativistic effects which may, in particular, be respon-sible for the relatively large overestimation ofsC in CH3I.56

Bearing all these limitations in mind, the three discussedtheoretical approaches perform equally well in comparisonwith the experimental13C shielding constants in the methylhalides.

The present calculations for the13C and 1H shieldinganisotropies can be compared with NMR data originatingmainly from experiments in nematic liquid crystal~LC! so-lutions ~Tables VIII and IX!. The theoretical results agreequite well with modern LC NMR data, where the anisotro-pies have been obtained using special solvent mixtures tominimize local solvent effects and vibrational correctionshave been included in the orientation parameter of the dis-solved molecule~see Refs. 49 and 57 and references therein!.

Gas-phase experiments have been reported forsH in themethyl halides.46 Our calculations overestimate these resultsby approximately 1 ppm. This difference may also be due torovibrational effects. Using the same procedure as outlinedfor the 13C shieldings, the vibrational corrections tosH at300 K are20.57,20.47, and20.40 ppm for X5F, Cl, andBr, respectively, and our remaining error is reduced to about0.5 ppm.

IV. CONCLUSIONS

We have presented SCF and MCSCF quadratic responsecalculations of the corrections to the nuclear shielding tensorinduced by electronic spin-orbit interaction. The one- andtwo-electron spin-orbit and both the Fermi contact and spin-dipolar interactions have been taken into account. Results forthe shieldings in hydrogen and methyl halides have beengiven. It is noted that, as in the calculation of NMR spin–spin coupling constants, basis sets that are converged for thenonrelativistic shieldings have to be augmented with tightbasis functions if accurate spin-orbit corrections to shieldingsare to be obtained. Electron correlation decreases the magni-tude of the spin-orbit corrections significantly, and the Fermicontact/one-electron spin-orbit@FC~1!# contribution is foundto dominate the total SO corrections to the1H and 13Cshieldings. The role of the other contributions is different inthe two series of molecules investigated: While it is safe to

neglect these contributions for the hydrogen halides, they arenon-negligible for the methyl halides as the FC~2! and SD~1!contributions do not cancel for these molecules. For the hy-drogen shielding constants in the hydrogen halides, our cal-culations are the most accurate so far presented. For the13Cshielding constants in the methyl halides, comparison withexperiment is complicated by the lack of gas-phase referencedata, and by larger rovibrational corrections than for the1Hshielding.sC in CH3I may in addition be affected by scalarrelativistic effects.

ACKNOWLEDGMENTS

This research was supported in part by Nordisk For-skerutdanningsakademi~NorFA, J.V.! and the Academy ofFinland ~J.J.!. The computational resources were partiallyprovided by the Center for Scientific Computing, Espoo, Fin-land. Support from the Norwegian Research Council~Pro-gram for Supercomputing! is gratefully acknowledged.

1P. Pyykko, Adv. Quantum Chem.11, 353 ~1978!; Chem. Rev.88, 563~1988!.

2G. Malli and C. Froese, Int. J. Quantum Chem.1S, 95 ~1967!.3D. Kolb, W. R. Johnson, and P. Shorer, Phys. Rev. A26, 19 ~1982!.4L. Visscher, T. Enevoldsen, T. Saue, and J. Oddershede, J. Chem. Phys.~submitted!.

5R. McWeeny,Methods of Molecular Quantum Mechanics~Academic,London, 1992!.

6G. Schreckenbach and T. Ziegler, Int. J. Quantum Chem.60, 753 ~1996!;61, 899 ~1997!.

7M. Kaupp, V. G. Malkin, O. L. Malkina, and D. R. Salahub, Chem. Phys.Lett. 235, 382 ~1995!.

8C. C. Ballard, M. Hada, H. Kaneko, and H. Nakatsuji, Chem. Phys. Lett.254, 170 ~1996!.

9N. Nakagawa, S. Shinada, and S. Obinata, The 6th NMR Symposium,Kyoto, 1967~unpublished!; Y. Nomura, Y. Takeuchi, and N. Nakagawa,Tetrahedron Lett.8, 639 ~1969!.

10P. Pyykko, A. Gorling, and N. Ro¨sch, Mol. Phys.61, 195 ~1987!.11M. Kaupp, O. L. Malkina, V. G. Malkin, and P. Pyykko¨, Chem. A: Eur. J.

4, 118 ~1998!.12I. Morishima, K. Endo, and T. Yonezawa, J. Chem. Phys.59, 3356

~1973!.13M. I. Volodicheva and T. K. Rebane, Teor. E´ ksp. Khim.14, 447 ~1978!

@Theor. Exp. Chem.~USSR! 14, 348 ~1978!#.14A. A. Cheremisin and P. V. Schastnev, J. Magn. Reson.40, 459 ~1980!.15H. Nakatsuji, H. Takashima, and M. Hada, Chem. Phys. Lett.233, 95

~1995!.16V. G. Malkin, O. L. Malkina, and D. R. Salahub, Chem. Phys. Lett.261,

335 ~1996!.17H. Fukui, T. Baba, and H. Inomata, J. Chem. Phys.105, 3175~1996!.18G. Schreckenbach, Ph.D. thesis, University of Calgary, Alberta, 1996.19J. Olsen and P. Jo”rgensen, J. Chem. Phys.82, 3235~1985!.20S. Kirpekar, H. J. Aa. Jensen, and J. Oddershede, Theor. Chim. Acta95,

35 ~1997!.21J. Olsen, inLecture Notes in Quantum Chemistry, edited by B. O. Roos

~Springer, Berlin, 1992!.22T. Helgaker, H. J. Aa. Jensen, P. Jo”rgensen, J. Olsen, K. Ruud, H. Ågren,

T. Andersen, K. L. Bak, V. Bakken, O. Christiansen, P. Dahle, E. K.Dalskov, T. Enevoldsen, H. Heiberg, H. Hettema, D. Jonsson, S. Kirpekar,R. Kobayashi, H. Koch, K. V. Mikkelsen, P. Norman, M. J. Packer, T.Saue, P. R. Taylor, and O. Vahtras,Dalton, an Ab Initio Electronic Struc-ture program, Release 1.0, 1997. See http://www.kjemi.uio.no/software/dalton/dalton.html.

23O. Vahtras, H. Ågren, P. Jo”rgensen, H. J. Aa. Jensen, T. Helgaker, and J.Olsen, J. Chem. Phys.97, 9178~1992!.

24O. Vahtras, H. Ågren, P. Jo”rgensen, H. J. Aa. Jensen, T. Helgaker, and J.Olsen, J. Chem. Phys.96, 2118~1992!.

25J. Lounila, R. Wasser, and P. Diehl, Mol. Phys.62, 19 ~1987!.26J. Vaara and Y. Hiltunen, J. Chem. Phys.107, 1744~1997!.

FIG. 2. Experimental~Refs. 51 and 46! and theoretical13C absolute shield-ing constants in CH3X ~X5F, Cl, Br, and I!. The present RAS/HIIu4pIII

results are shown for nonrelativistic calculations~nr! and with the full spin-orbit correction~nr1SO!. The UHF and DFT results are from Refs. 15 and16, respectively.

1221J. Chem. Phys., Vol. 109, No. 4, 22 July 1998 Vaara et al.

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130.237.81.26 On: Thu, 05 Feb 2015 14:32:47

27K. Ruud, T. Helgaker, R. Kobayashi, P. Jo”rgensen, K. L. Bak, and H. J.Aa. Jensen, J. Chem. Phys.100, 8178~1994!.

28K. Fægri~personal communication!.29S. Huzinaga,Approximate Atomic Functions~University of Alberta, Ed-

monton, 1971!.30B. C. Binning, Jr. and L. A. Curtiss, J. Comput. Chem.11, 1206~1990!.31S. P. Karna, M. Dupuis, E. Perrin, and P. N. Prasad, J. Chem. Phys.92,

7418 ~1990!.32D. K. Hindermann and C. D. Cornwell, J. Chem. Phys.48, 4148~1968!.33D. Sundholm, J. Gauss, and A. Scha¨fer, J. Chem. Phys.105, 11051

~1996!.34P.-O. Åstrand and K. V. Mikkelsen, J. Chem. Phys.104, 648 ~1996!.35T. D. Gierke and W. H. Flygare, J. Am. Chem. Soc.94, 7277~1972!.36F. H. de Leeuw and A. Dymanus, J. Mol. Spectrosc.48, 427 ~1973!.37W. T. Raynes inNuclear Magnetic Resonance, Specialist Periodical Re-

port, Vol. 7 ~Royal Society of Chemistry, Cambridge, 1978!.38W. G. Schneider, H. J. Bernstein, and J. A. Pople, J. Chem. Phys.28, 601

~1958!.39Y. Hiltunen, Mol. Phys.62, 1187~1987!.40J. Jokisaari, Y. Hiltunen, and J. Lounila, J. Chem. Phys.85, 3198~1986!.41K. W. Zilm and D. M. Grant, J. Am. Chem. Soc.103, 2913~1981!.42P. B. Bhattacharyya and B. P. Dailey, J. Magn. Reson.13, 317 ~1974!.43P. B. Bhattacharyya and B. P. Dailey, Mol. Phys.26, 1379~1973!.44C. S. Yannoni and E. B. Whipple, J. Chem. Phys.47, 2508~1967!.45I. Morishima, A. Mizuno, and T. Yonezawa, Chem. Phys. Lett.7, 633

~1970!.

46H. Spiesecke and W. G. Schneider, J. Chem. Phys.35, 722 ~1961!.47G. P. Ceasar, C. S. Yannoni, and B. P. Dailey, J. Chem. Phys.50, 373

~1969!.48A. J. Montana and B. P. Dailey, J. Magn. Reson.21, 25 ~1976!.49J. Jokisaari and Y. Hiltunen, Mol. Phys.50, 1013~1983!.50S. Raghothama, J. Magn. Reson.57, 294 ~1984!.51A. K. Jameson and C. J. Jameson, Chem. Phys. Lett.134, 461 ~1987!.52A. M. Lee, N. C. Handy, and S. M. Colwell, J. Chem. Phys.103, 10095

~1995!; J. R. Cheeseman, G. W. Trucks, T. A. Keith, and M. J. Frisch,ibid. 104, 5497~1996!; L. Olsson and D. Cremer,ibid. 105, 8995~1996!;G. Rauhut, S. Puyear, K. Wolinski, and P. Pulay, J. Phys. Chem.100,6310 ~1996!.

53K. Jackowski and W. T. Raynes, Mol. Phys.34, 465 ~1977!.54A. D. Buckingham and R. M. Olega´rio, Mol. Phys.92, 773 ~1997!.55W. T. Raynes, inNuclear Magnetic Shieldings and Molecular Structure,

edited by J. A. Tossel~Kluwer, Dordrecht, 1993!.56In the course of the work reported in Ref. 26, GIAO SCF calculations for

sC were carried out for CH3I both at nonrelativistic all-electron and quasi-

relativistic effective core potential~ECP! levels. The latter type of calcu-lation takes into account the scalar relativistic effects originating from theiodine atom. While the nonrelativistic result was 194.3 ppm, the use ofECP on the I atom led to the deshielding change of214 ppm.

57J. Lounila, M. Ala-Korpela, and J. Jokisaari, J. Chem. Phys.93, 8514~1990!; Y. Hiltunen and J. Jokisaari, J. Magn. Reson.75, 213 ~1987!.

1222 J. Chem. Phys., Vol. 109, No. 4, 22 July 1998 Vaara et al.

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