This article was downloaded by: [Moskow State Univ Bibliote]On: 05 November 2013, At: 11:09Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office:Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Molecular Physics: An International Journal atthe Interface Between Chemistry and PhysicsPublication details, including instructions for authors and subscriptioninformation:http://www.tandfonline.com/loi/tmph20

Coupled-cluster calculations of spin-rotationconstantsJURGEN GAUSS & DAGE SUNDHOLMPublished online: 03 Dec 2010.

To cite this article: JURGEN GAUSS & DAGE SUNDHOLM (1997) Coupled-cluster calculations of spin-rotationconstants, Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, 91:3,449-458

To link to this article: http://dx.doi.org/10.1080/002689797171346

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”)contained in the publications on our platform. However, Taylor & Francis, our agents, and ourlicensors make no representations or warranties whatsoever as to the accuracy, completeness, orsuitability for any purpose of the Content. Any opinions and views expressed in this publication arethe opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis.The accuracy of the Content should not be relied upon and should be independently verified withprimary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoevercaused arising directly or indirectly in connection with, in relation to or arising out of the use of theContent.

This article may be used for research, teaching, and private study purposes. Any substantialor systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, ordistribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use canbe found at http://www.tandfonline.com/page/terms-and-conditions

MOLECULAR PHYSICS, 1997, VOL. 91, NO. 3, 449± 458

Coupled-cluster calculations of spin± rotation constants

By JUÈ RGEN GAUSSInstitut fuÈ r Physikalische Chemie, UniversitaÈ t Mainz, D-55099 Mainz, Germany

and DAGE SUNDHOLMDepartment of Chemistry, PO Box 55 (AI Virtasen aukio 1), FIN-00014 University

of Helsinki, Finland

(Received 23 December 1996; accepted 11 February 1997)

Spin± rotation constants for H2, HF, N2, CO and F2 are calculated at the coupled-clustersingles and doubles level augmented by a perturbative correction for triple excitations togetherwith large uncontracted basis sets. Explicit values for the lowest rovibrational states of thesediatomics are obtained by solving the rovibrational SchroÈ dinger equation with the ® nite-element method. Good agreement between theory and experiment is obtained for H2 , HF andF2, while for N2 and CO a thorough comparison with experiment is hampered by inaccuraciesin the experimental numbers and the calculated values should be considered more reliable.

1. Introduction

The experimental determination of nuclear magneticshielding constants is based on measurements of spin±rotation constants [1], which are closely related to theparamagnetic part of the shielding with the nucleus ofinterest assigned as gauge origin. Combination withcalculated values for the diamagnetic shieldings togetherwith appropriate rovibrational corrections yields the so-called experimental’ shielding constants s . Hence, athorough comparison of theory and experiment shouldnot be based on single-point calculations of absoluteshieldings as done so far in most cases but rather basedon spin-rotation constants which alone can be deter-mined in the experiment.

Like the computation of nuclear magnetic shieldings,computation of spin± rotation constants is hampered byan origin dependence encountered in conventional ® nitebasis-set calculations. In addition, results from suchcalculations typically exhibit slow convergence to thebasis-set limit, thus rendering highly accurate calcula-tion of spin± rotation constants rather expensive. Fornuclear shieldings, the introduction of local gauge ori-gins [2± 7] provides an elegant solution to the above-mentioned problems. In particular, the use of explicitly® eld-dependent basis functions known as gauge-including atomic orbitals (GIAOs) or London orbitals[8]guarantees rigorous origin independence and ensuresrapid convergence to the basis-set limit [2]. However,these concepts have only recently been extended to thecalculation of spin± rotation constants [9]. Since noexternal magnetic ® eld is involved in the spin± rotationinteractions, ordinary GIAOs are not particularly use-

ful. The advantages of the GIAOs can be recovered byintroducing atomic orbitals which explicitly depend onthe rotation as perturbation parameter and thus resem-ble the GIAOs used in the calculation of magneticproperties such as nuclear magnetic shielding constants.The improved basis-set convergence due to the use ofthese perturbation-dependent basis functions have beendemonstrated in a previous paper [9] in Hartree± Fockself-consistent-® eld (SCF) calculations for HF, CO, N2

and H2CO.In this paper, we report accurate values for the spin±

rotation constants for H2, HF, CO, N2 and F2 calcu-lated at the coupled-cluster singles and doubles levelaugmented by a perturbative treatment of triple excita-tions (CCSD(T)) [10]; a method that provides in mostcases results of nearly quantitative accuracy (for exam-ple [11]). To facilitate comparison with experiment,theoretical values are given for speci® c rovibrationalstates with the required rovibrational corrections deter-mined from CCSD(T) potential curves and spin± rota-tion functions.

2. Theory

The spin± rotation tensor M characterizes the inter-action between the magnetic moment associated with agiven nuclear spin and the internal magnetic ® eldinduced by the rotation of the molecule. The corre-sponding energy corrections to the rotational levels areconveniently expressed as

n E = - IkM J, (1)

0026± 8976/97 $12.00 Ñ 1997 Taylor & Francis Ltd.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

11:

09 0

5 N

ovem

ber

2013

with Ik as the spin of the kth nucleus and J as therotational angular momentum.

Within the Born± Oppenheimer approximation, M

consists of a nuclear and an electronic part:

M = M(n) + M

(e) . (2)The nuclear contribution is given as

M(n) = a 2 g k å l

 Zl

´(Rl - Rk) ·(Rl - Rk)1- (Rl- Rk)(Rl - Rk)

|Rl - Rk|3I- 1,

(3)where the sum is over all nuclei except k. a denotes the(dimensionless) ® ne-structure constant, g k the gyromag-netic ratio for nucleus k, Zl the charge of nucleus l, I thenuclear inertia tensor, and Rl and Rk the positionvectors of nuclei l and k respectively. The electroniccontribution to M can be determined as the secondderivative of the electronic energy Eel with respect to Ik

and J:

M (e) = - d2Eel

dIk dJ. (4)

Its evaluation, however, requires expressions for ® rst-and second-order corrections to the electronicHamiltonian with respect to the perturbations Ik andJ. As both corrections involve single-particle operators,they are most conveniently expressed as the correspond-ing derivatives of the one-electron Hamiltonian h:

( ¶ h¶ J ) J= 0

= - I- 1lO, (5)

( ¶ h¶ Ik ) Ik= 0

= a 2 g klk

|r - Rk|3 (6)

and

( ¶ 2h¶ Ik ¶ J) J,Ik= 0

= - a 2 g k

´(r - Rk) ·(Rk - RO)1 - (r - Rk)(Rk - RO)

|r - Rk|3 I- 1 (7)

with the angular momentum operators lO and lk de® nedby

lO = (r - RO) ´ p (8)and

lk = (r - Rk) ´ p. (9)In equations (8) and (9), r and p denote the electronicposition and momentum operator respectively.Equations (5) ± (9) represent expressions valid for arbi-trary origins RO of the electronic angular momentum.

As shown in [9], origin independence in the calcula-tion of spin± rotation constants as well as improved

basis-set convergence can be ensured by employingbasis functions which explicitly depend on angularmomentum:

c (J) = exp [iI- 1J ´ (R¹ - RO) ·r]c (0), (10)

with c (0) denoting the usual perturbation-independentbasis functions centred at R¹ . In [9], these basis func-tions were called rotational London orbitals to stress theanalogy to the standard London orbitals (GIAOs) usedin nuclear shielding calculations.

The actual computation of M(e) proceeds in the same

way as for nuclear shieldings within the GIAOapproach. The expression for the second derivative ofthe electronic energy (cf. equation (4)) can be cast in thefollowing form [12]:

M(e) = - å

¹ t

D¹ t

¶ 2h¹ t

¶ J ¶ Ik- å

¹ t

¶ D¹ t

¶ J

¶ h¹ t

¶ Ik, (11)

where D¹ t represents the corresponding (e� ective) one-particle density matrix and ¶ D¹ t /¶ J its derivative withrespect to the angular momentum perturbation J. Greekindices label here atomic orbitals and h¹ t is de® ned asthe atomic orbital representation of the one-electronHamiltonian. Note that, as discussed in detail in [12],equation (11) holds for any quantum-chemical method;various approaches di� er only in the de® nition of thee� ective one-particle density matrix² D¹ t .

Since the derivative integrals ¶ h¹ t /¶ J and¶ 2h¹ t /( ¶ J ¶ Ik) are closely related to the GIAO integrals¶ h¹ t / ¶ B and ¶ 2h¹ t /( ¶ B ¶ Ik), one can calculate (asshown in [9]) spin± rotation constants with any existingGIAO code. The relevant expression to convert thecalculated shieldings to spin± rotation constants is

M(e) = 2g k[s (GIAO) - s dia(Rk)]·I- 1, (12)

where s dia(Rk) denotes the so-called diamagnetic con-tribution to s calculated with Rk as gauge origin andusing conventional ® eld-independent basis functions.

3. Computational details

The computational determination of spin± rotationconstants for speci® c rovibrational states involves thefollowing steps:

(1) determination of the spin± rotation constants M

as a function of the internuclear distance R;(2) solution of the nuclear SchroÈ dinger equation for

the rovibrational eigenstates;(3) averaging of the spin± rotation function M (R)

over the rovibrational wavefunctions.

450 J. Gauss and D. Sundholm

² Detailed expressions for the (e� ective) density D¹ t havebeen derived for most quantum chemical methods within thecontext of analytic gradient theory.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

11:

09 0

5 N

ovem

ber

2013

For the calculation of M (R), the CCSD(T) approachtogether with large contracted and uncontracted basissets have been employed. The basis used here for C, Nand O consists of a 13s8p3d2f primitive set contracted to8s5p3d2f (pz3d2f ), while for F the basis set which hasbeen used for the rovibrational averaging consists of theuncontracted 15s11p3d2f basis set. For H, the uncon-tracted 8s4p3d2f set has been employed. The sp parts (spart in the case of H) of these basis sets have beenoptimized at the Hartree± Fock SCF level as described in[13]; additional polarization functions have been takenfor C, N, O and F from [14] and for H from [15]. Toexplore remaining basis-set e� ects, additional single-point calculations at the equilibrium distance Re havebeen carried out for CO and N2 using the uncontracted15s11p3d2f basis sets. The basis-set dependence of thespin± rotation constants for HF and F2 was studied byperforming CCSD(T) calculations using large uncon-tracted basis sets. These basis sets have been constructedin the same way as those mentioned above [13, 16]. For Fin HF and F2, the following sets were used: 15s11p3d2f ,15s11p4d3f , 15s11p4d3f 2g, 15s11p4d3f 2g + spd,15s11p4d3f 2g + 2s2p2d, and 17s13p4d3f 2g. For H, thecorresponding sets are 8s3d2p, 8s4p3d, 8s4p3d2f ,8s4p3d2f + sp, 8s4p3d2f + 2s2p and 10s4p3d2f . Theexponents for the di� use functions were chosen asfollows: 0.07 (s), 0.04 (p), 0.15 (d) in the case of15s11p4p3d2f + spd, 0.07, 0.02 (s), 0.04, 0.013 (p) and0.15, 0.05 (d) in the case of 15s11p4d3f 2g + 2s2p2d,0.024 (s) and 0.1 (p) in the case of 8s4p3d2f + sp, and0.024, 0.08 (s) and 0.1, 0.03 (p) in the case of8s4p3d2f + 2s2p. To study the e� ect of g functions incalculations for F2, the pz3d2f set was augmented bytwo g functions. Spherical Gaussians have been used inall calculations.

To calculate the rovibrational corrections, the nuclearSchroÈ dinger equation

(- h2

2¹

d2

dR2 +h2J(J + 1)

2¹R2 + V (R)) W v,J(R) = Ev,J W v,J(R),(13)

with ¹ as the reduced mass and v and J as vibrationaland rotational quantum numbers respectively has to besolved. This has been accomplished using the ® nite-element technique [17]. The required potential curvesV (R) have been obtained for CO and N2 by spline ® ts toa set of energy points calculated with the CCSD(T)approach using the 13s8p3d2f primitives contracted to8s5p3d2f . For H2, HF and F2 the 8s4p3d2f and15s11p3d2f uncontracted H and F basis sets wereemployed. For F2, the Rice± Kassel± Ramsberger(RKR) potential curve deduced from experiment [18]has been used in addition. A more detailed description

of our approach to solving equation (13) can be found in[19].

Spin± rotation constants for speci® c rovibrationallevels M v,J are then determined as simple expectationvalue of the spin± rotation functions M (R) over therovibrational wavefunctions

M v,J = k W v,J(R)|M (R)|W v,J(R) l . (14)Corrections due to Thomas precession are important

for spin± rotation constants involving H. The Thomasprecession contributions are determined for H2 and HFusing the following equation [20, 21]:

M Thomas =h

4p M2c2RdEdR

. (15)

In equation (15), M is the mass of the nucleus, c thespeed of light and dE /dR the energy derivative withrespect to the internuclear distance. The rovibrationalaverage for M

Thomas is obtained by taking the appro-priate expectation value.

All quantum-chemical calculations in this work havebeen carried out using a local version of the ACES IIprogram package [22]; for the calculation of the spin±rotation constants the existing GIAO-CCSD andGIAO-CCSD(T) codes [12, 23] were employed togetherwith equation (12) for the conversion of shieldings intospin± rotation constants.

4. Results

4.1. H2

Tables 1 and 2 summarize the calculated values forthe spin± rotation constants of the three isotopomers H2,D2 and HD together with the available experimentaldata [24± 27]. Theoretical values are reported for the ® velowest rotational states with v = 0 and v = 1, whileexperimental numbers are only available for the ® rstfew rotational states with v = 0. Since basis-set conver-gence in the calculation of spin-rotation constants (aswell as the corresponding shielding constants) for H2

has been already investigated in [15], we refrain herefrom additional calculations. In [15], it was found thatthe 8s4p3d2f set provides results very close to the basis-set limit. For the remaining error, a conservative esti-mate of 200 Hz can be given for H2. This value corre-sponds to a paramagnetic shielding s p of about0.02 ppm. For HD at H, HD at D, and for D2, theerror estimates are 150, 25 and 15 Hz respectively. Itshould furthermore be noted that CCSD calculationsprovide for two-electron system such H2 the exactsolution to the electron correlation problem within thegiven one-particle basis, and thus they are equivalent tofull con® guration interaction (FCI) calculations.

Coupled-cluster calculations of spin± rotation constants 451D

ownl

oade

d by

[M

osko

w S

tate

Uni

v B

iblio

te]

at 1

1:09

05

Nov

embe

r 20

13

Thomas precession e� ects are for H2 of the order of afew hundred hertz² . This does not seem to be a largee� ect, but inclusion is essential when aiming for quanti-tative accuracy. For example, the discrepancy betweenthe calculated and experimental value for M v,J isreduced from 395 Hz down to 160 Hz when Thomasprecession e� ects are considered. As the remainingdi� erence is quite small, that is 160 Hz for M (1H),131 Hz for M ( 1H) in HD, 18 Hz for M (2H) in HD,and 9 Hz for M (2D), the current study provides anexcellent agreement between theory and experiment.As seen in the last columns of tables 1 and 2, thediscrepancies between calculated and measured spin±rotation constants are in all cases smaller than theestimated uncertainties.

4.2. HFResults of our basis-set convergence study for the

calculation of M (19F) and M (1H) are given in table 3.It is found that convergence is di� cult to achieve, asinclusion of higher-angular-momentum functions (g

functions on F and f functions on H) appears to beimportant. For M (19F), g functions on F and f func-tions on H contribute 2.32 kHz, while di� use functionsaccount for 0.96 kHz. By augmenting the sp space from15s11p to 17s13p, M (19F) increased by 0.44 kHz. Asseen in table 3, higher-order polarization functions arealso important for the spin± rotation constant at H. Theaddition of g functions on F and f functions on Hdecreases M (1H) by 0.54 kHz. Our best theoreticalestimate for M (19F) obtained by basis-set extrapolationstill deviates by 1.9 kHz from the corresponding experi-mental value [29].

Tables 4 and 5 report calculated and experimentalvalues [29± 31] for the lowest rovibrational states withv = 0 and v = 1 of the two isotopomers HF and DF.Thomas precession e� ects are negligible for M (19F),that is they amount to less than 1 Hz for all rovibra-tional levels given in table 4. For M (1H), they are of theorder of several hundreds of hertz, and for M ( 2D), thecorresponding correction is calculated to be of the orderof 30± 80 Hz.

The discrepancy between the measured spin± rotationconstants and the calculated values given in table 4 isabout 2± 4 kHz for M (19F). As seen in table 5, for

452 J. Gauss and D. Sundholm

Table 1. Calculated spin± rotation constants for H2 and D2 compared with experimental results.

Spin± rotation constant/kHz

v J Isotopomer Calculateda Calculatedb Measured Difference

0 1 H2 114.299 114.064 113.904 (30)c,d 0.1600 3 H2 111.021 110.933 111.10 (25)c - 0.170 5 H2 105.471 105.627 105.37 (32)c 0.260 7 H2 98.163 98.6280 9 H2 89.690 90.494

1 1 H2 112.362 111.6791 3 H2 109.046 108.5211 5 H2 103.440 103.1751 7 H2 96.078 96.1431 9 H2 87.569 87.995

0 1 D2 8.820 8.777 8.768 (3)e 0.0090 2 D2 8.769 8.733 8.723 (20)e 0.0100 3 D2 8.692 8.6560 4 D2 8.592 8.5670 5 D2 8.470 8.460

1 1 D2 8.725 8.5981 2 D2 8.673 8.5541 3 D2 8.596 8.4891 4 D2 8.494 8.4021 5 D2 8.371 8.298

a Without Thomas precession.b Including Thomas precession.c From [24].d From [25].e From [26].

² Similar results for the Thomas precession contributionshave been previously reported in the literature [28].

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

11:

09 0

5 N

ovem

ber

2013

M (1H) the di� erence is 600± 800 Hz. For the deuteratedspecies, the corresponding deviations are with 0.6 kHzfor M ( 19F) and 50 Hz for M (2D) somewhat smaller.However, by adding the basis-set corrections obtainedfrom the basis-set study, a signi® cant better agreementbetween calculated and measured spin± rotation con-stants is obtained. Our calculated values represent themost accurate theoretical values reported so far. Arecent computational investigation using polarizationpropagator methods [33], for example, reported valuesof - 325.61 and 74.25 kHz for M 0,1(

19F) and M 0,1(1H)

which deviate by 18.0 and 3.15 kHz respectively fromthe experimental values. The total correlation correctionto M (19F) amounts to about 30 kHz. The remainingerror amounts to about 6% of that contribution and itseems unlikely that the highly accurate CCSD(T)approach is so much in error for a molecule such asHF. Remaining basis-set errors should be also consid-ered rather small and of the order of 1 kHz. Thediscrepancies between the calculated rovibrational cor-rections to M (19F) for the v = 0 and J = 1 and v = 1and J = 1 states are 0.68 and 0.59 kHz respectively. Itmight therefore be of interest to investigate the impor-tance of relativistic e� ects.

Coupled-cluster calculations of spin± rotation constants 453

Table 2. Calculated spin± rotation constants for HD as compared with experimental results.

Spin± rotation constants/kHz

v J Isotopomer Calculateda Calculatedb Measured Difference

0 1 HD at H 85.938 85.731 85.600 (18)c 0.1310 2 HD at H 85.188 85.0260 3 HD at H 84.082 83.9860 4 HD at H 82.640 82.6290 5 HD at H 80.891 80.983

1 1 HD at H 84.732 84.1281 2 HD at H 83.974 83.4171 3 HD at H 82.856 82.3701 4 HD at H 81.400 81.0041 5 HD at H 79.633 79.346

0 1 HD at D 13.192 13.140 13.122 (11)c 0.0180 2 HD at D 13.077 13.0370 3 HD at D 12.907 12.8830 4 HD at D 12.686 12.6830 5 HD at D 12.417 12.440

1 1 HD at D 13.007 12.8561 2 HD at D 12.891 12.7521 3 HD at D 12.719 12.5971 4 HD at D 12.495 12.3961 5 HD at D 12.224 12.152

a Without Thomas precession.b Including Thomas precession.c From [27].

Table 3. Convergence to the basis-set limit in CCSD(T)calculations of the spin± rotation constants for HF. Allcalculations have been carried out at the equilibriumdistance (Re = 0´9165 AÊ ) with no Thomas precessioncorrection included.

Spin± rotation constant/kHz

Basis set for F/basis set for H Label M( 19F) M(1H)

15s11p3d2f/8s4p3d2f - 285.77 72.41pz3d2f/pz3p A - 287.73 74.0215s11p3d2f/8s3p2d B - 286.85 73.1215s11p4d3f/8s4p3d C - 286.14 72.4215s11p4d3f2g/8s4p3d2f D - 283.82 71.8815s11p4d3f 2g + spd/8s4p3d2f+ sp E - 282.88 72.0015s11p4d3f 2g + 2s2p2d/8s4p3d2f

+ 2s2p F - 282.86 72.0017s13p4d3f 2g/10s4p3d2f G - 283.38 71.90Estimated basis set limita - 282.42 72.02Experimentb - 280.49 72.02

a Calculated as D + (F - D) + (G - D).b From [29].

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

11:

09 0

5 N

ovem

ber

2013

Of interest is also a comparison of the calculatedspin± rotation functions with those derived from experi-ment. Table 6 contains the calculated derivatives of M

with respect to the internuclear distance R together withavailable data from experiment [29± 32, 34]. While agree-ment in case of ® rst derivatives is quite good, a compar-ison of the second derivatives is hampered by the largeuncertainty in values deduced from experiment.

4.3. F2

The basis-set convergence for the spin± rotation con-stant is given in table 7. As for HF, it is found that thebasis-set convergence is di� cult to achieve. Di� use s, pand d functions contribute - 0.29 kHz and g functions0.19 kHz. Thus for F2 the basis-set corrections aresomewhat smaller than for HF. By adding the estimatedbasis-set correction of table 7 to the M (F2) for the v = 0and J = 1 rovibrational state, a value of - 151.27 kHz isobtained. This value di� ers by 1.4 kHz from the experi-mental result. The remaining discrepancy between cal-culated and measured values may be due to relativistic

and correlation e� ects. Since in a recent coupled clusterstudy of the nuclear magnetic shielding constants of F2

[19] large contributions due to triple excitations werefound, the remaining correlation error might also benon-negligible. Calculated values for the spin± rotationconstant of F2 are given in table 8 together with avail-able experimental data [35]. The experimental RKRcurve as well as the calculated CCSD(T) potentialcurve have been used for the rovibrational averaging.As for the shielding constant [19], calculation of thespin± rotation constant for F2 is di� cult owing to anunusually large sensitivity of M (19F) on the internucleardistance. This is best illustrated by comparing in table 8computed values for v = 0, J = 1 based on the RKRand the CCSD(T) curves. The spin± rotation constantsdi� er by about 1.4 kHz, while the equilibrium distanceof both curves di� ers by less than 0.35 pm. The accuracyof the shape of CCSD(T) curve is demonstrated by theexcellent agreement obtained for the v = 0 and J = 0rovibrational energy in comparison to experiment. Itdi� ers by only 2.2 cm- 1 from the measured value. It

454 J. Gauss and D. Sundholm

Table 4. Calculated spin-rotation constants M ( 19F) for HF and DF compared with experimental results.

Spin± rotation constant/kHz

v J Isotopomer Calculateda Calculatedb Measured Difference

0 1 HF - 309.921 - 309.921c - 307.65 (2)d - 2.27- 307.637 (20)e - 2.280

0 2 HF - 310.528 - 310.528 - 308.263 (19) f - 2.2650 3 HF - 311.439 - 311.4390 4 HF - 312.655 - 312.6550 5 HF - 314.177 - 314.177

1 1 HF - 364.744 - 364.745 - 361.20 (2)d - 3.541 2 HF - 365.387 - 365.3881 3 HF - 366.351 - 366.3521 4 HF - 367.640 - 367.6411 5 HF - 369.253 - 369.254

0 1 DF - 158.940 - 158.940 - 158.356 (1)e 0.5840 2 DF - 159.106 - 159.1060 3 DF - 159.356 - 159.3560 4 DF - 159.689 - 159.6890 5 DF - 160.106 - 160.106

1 1 DF - 179.216 - 179.2171 2 DF - 179.391 - 179.3921 3 DF - 179.654 - 179.6551 4 DF - 180.004 - 180.0051 5 DF - 180.442 - 180.443

a Without Thomas precession.b Including Thomas precession.c The extrapolated value is - 306.31 kHz (see table 3).d From [29].e From [30].f From [31].

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

11:

09 0

5 N

ovem

ber

2013

Coupled-cluster calculations of spin± rotation constants 455

Table 5. Calculated spin± rotation constants M ( 1H) and M ( 2D) for HF and DF compared withexperimental results.

Spin± rotation constant/kHz

v J Isotopomer Calculateda Calculatedb Measured Difference

0 1 HF 72.082 71.932c 71.10 (2)d 0.830 2 HF 71.946 71.809 71.033 (27)e 0.776

71.128 (24)d 0.681

0 3 HF 71.742 71.6250 4 HF 71.471 71.3800 5 HF 71.134 71.075

1 1 HF 70.362 69.944 69.33 (3)d 6141 2 HF 70.231 69.8271 3 HF 70.036 69.6511 4 HF 69.777 69.4171 5 HF 69.454 69.126

0 1 DF 5.837 5.809 5.755 (19) f 540 2 DF 5.831 5.8050 3 DF 5.822 5.7980 4 DF 5.810 5.7900 5 DF 5.796 5.780

1 1 DF 5.746 5.6651 2 DF 5.740 5.6601 3 DF 5.732 5.6551 4 DF 5.720 5.6461 5 DF 5.706 5.637

a Without Thomas precession.b Including Thomas precession.c The extrapolated value is 71.88 kHz (see table 3).d From [29].e From [31].f From [30].

Table 6. Comparison of calculated spin± rotation constant derivatives² with experimental data.

Calculated Experiment

dM /d x /kHz d2M /dx 2 /kHz dM /d x /kHz d2M /d x 2 /kHz

M (F) in FH - 1437.930 - 155.277 - 1521 (90)a

- 1430 (50)b - 400 (200)b

- 1417c 48c

M (H) in FH - 338.439 996.683 - 230 (130)a

- 260 (20)b 500 (200)b

- 344c 1006c

M (F) in FD - 755.708 - 81.606 - 745c 25c

M (D) in FD - 27.303 80.407M (F) in F2 - 595.203 - 824.633 - 650d

² x = (R - Re) /Re.a From [31].b From [29].c From [32].d From [34].

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

11:

09 0

5 N

ovem

ber

2013

seems, as also indicated by recent large-scale CCSD(T)calculations [36], that the current CCSD(T) curve issimply shifted by about 0.35 pm to too long distances.This de® ciency of the CCSD(T) curve has a dramatice� ect in case of F2, while similar e� ects are rather smallfor the other investigated molecules.

4.4. CO and N2

Finally, calculated values for the spin± rotation con-stants of CO and N2 are reported in table 9.Comparison with experiment [37± 39] is here hamperedby the large error bars associated with M (17O) for CO[38] and the well known fact that the reported experi-mental value for M (15N) [39] is incorrect [40].Nevertheless, the calculations here prove useful, as

they provide the most accurate values to date for thesespin± rotation constants and that they might turn out tobe helpful in future experimental studies.

5. Conclusion

In this paper, accurate quantum-chemical calculationsof spin± rotation constants for a few diatomic moleculesusing the CCSD(T) approach together with large uncon-tracted basis sets have been presented. In addition, tofacilitate comparison with experiment, rovibrationalcorrections have been computed. In all cases goodagreement with experiment is obtained provided reliableexperimental values are available. The expansion co-

456 J. Gauss and D. Sundholm

Table 7. Convergence to the basis set limit in CCSD(T)calculations of the spin± rotation constants for F2. Allcalculations have been carried out at the equilibriumgeometry (Re = 1´412 AÊ ).

Basis set for F Label M(19F) /kHz

pz3d2f A - 150.2215s11p3d2f B - 150.8914s11p4d3f C - 150.9415s11p4d3f + spd E - 151.3515s11p4d3f + 2s2p2d F - 151.3417s13p4d3f G - 151.1715s11p4d3f H - 151.05pz3d2f 2g I - 150.03Estimated basis set limita - 151.27Experimentb - 152.7 6 0.3

a Calculated as B + (H- B) + (I- A) + (F- H) + (G - H).b From [34].

Table 8. Calculated spin± rotation constants for F2 comparedwith experimental results.

Spin± rotation constant/kHz

v J Calculateda Calculatedb Measuredc

0 1 - 156.301 - 154.908d - 156.85 (10)0 3 - 156.325 - 154.9320 5 - 156.368 - 154.9740 7 - 156.430 - 155.0350 9 - 156.512 - 155.116

1 1 - 164.652 - 163.3871 3 - 164.678 - 163.4141 5 - 164.726 - 163.4611 7 - 164.795 - 163.5291 9 - 164.886 - 163.618

a Calculated with CCSD(T) potential curve.b Calculated with the RKR potential curve.c From [35].d The extrapolated value is - 155.29 kHz (see table 7).

Table 9. Calculated spin± rotation constants for 13C16O (C), 12C17O (O) and 15N2 compared with experimental results.

M ( 13C) /kHz M ( 17O) /kHz M (15N) /kHz

v J Calculated Measureda Calculated Measuredb Calculated Measuredc

0 1 - 32.3732 - 32.70 (12) 31.3418 30.4 (12) 19.8573 22.0 (10)0 2 - 32.3731 31.34190 3 - 32.3730 31.3420 19.85730 4 - 32.3727 31.34220 5 - 32.3724 31.3424 19.8574

1 1 - 32.3508 31.4846 19.90811 2 - 32.3507 31.48471 3 - 32.3505 31.4848 19.90811 4 - 32.3503 31.48501 5 - 32.3500 31.4852 19.9082

a From [37].b From [38].c From [39].

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

11:

09 0

5 N

ovem

ber

2013

e� cients of the spin± rotation functions (M ( x ),x = (R - Re) /Re) for H2, HF, CO, N2 and F2 are listedin table 10.

For H2 and HF, accurate values (with small errorbars) for the spin± rotation constants have been deducedfrom the experiment. To achieve a similar accuracy inquantum-chemical calculations is a real challenge.Although satisfactory agreement is obtained by meansof the current high-level calculations employing largebasis sets, a high-level treatment of electron correlationand including Thomas precession e� ects, it should benoted that the remaining discrepancy between theoryand experiment is, in most cases, larger than the errorbars given for the experimental values.

The opposite is true for N2 and CO. Here, theaccuracy of the experimental values is not very high,and calculations provide the more accurate values.These might be helpful in future experimental investiga-tions.

This work was supported by the DeutscheForschungsgemeinschaf t (J.G.), the Fonds derChemischen Industrie (J.G.), the Academy of Finland(D.S.) and the Alexander von Humboldt-Stiftung(D.S.). In addition, the authors thank Dr S. P. A.Sauer (Odense) for providing his program for the con-version of paramagnetic shieldings into spin± rotationconstants, and Professor R. Ahlrichs (Karlsruhe) forgenerous support.

References[1] Flygare, W. H., 1978, Molecular Structure and

Dynamics (Englewood Cli� s, New Jersey: Prentice-Hall).[2] London, F., 1937, J. Phys. Radium, 8, 397.[3] Hameka , H., 1958, Molec. Phys., 1, 203.[4] Ditchfield , R., 1974, Molec. Phys., 27, 789.[5] Wolinski, K., Hinton, J. F., and Pulay, P., 1990, J.

Am. chem. Soc., 112, 8251.[6] Kutz elnigg, W., 1980, Isr. J. Chem., 19, 193;

Schindler, M., and Kutzelnigg, W., 1982, J. chem.Phys., 76, 1919.

[7] Hansen, A. E., and Bouman, T. D., 1985, J. chem. Phys.,82, 5035.

[8] Helgaker , T., and Jørgensen, P., 1991, J. chem. Phys.,95, 2595.

[9] Gauss, J., Ruud, K., and Helgaker, T., 1996, J. chem.Phys., 105, 2804.

[10] Raghavachari, K., Trucks, G. W., Pople, J. A., andHead-Gordon, M., 1989, Chem. Phys. L ett., 157, 479.

[11] Lee, T. J., and Scuseria, G. E., 1995, QuantumMechanical Electronic Structure Calculations withChemical Accuracy, edited by S. R. Langho�(Dordrecht: Kluwer), p. 47.

[12] Gauss, J., and Stanton, J. F., 1995, J. chem. Phys., 102,

251; 1995, ibid., 103, 3561.[13] Schaïfer, A., Horn, H., and Ahlrichs, R., 1992, J.

chem. Phys., 97, 2571.[14] Dunning, T. H., Jr , 1989, J. chem. Phys., 90, 1007.[15] Sundholm, D., Gauss, J., and Ahlrichs, R., 1995,

Chem. Phys. L ett., 243, 264.[16] Basis sets are available via FTP from host ftp.chemie.uni-

karlsruhe.de (login-id: ftp) from directory pub/basis.[17] Zienkiewicz , O., 1977, The Finite Element Method

(London: McGraw-Hill).[18] Colbourn, E. A., Dagenais, M., Douglas, A. E., and

Raymonda, J. W., 1976, Can. J. Phys., 54, 1343.[19] Sundholm, D., Gauss, J., and Schaïfer, A., 1996, J.

chem. Phys., 105, 11 051.[20] Reid, R. V., and Chu, A. H.-M., 1974, Phys. Rev. A, 9,

609.[21] Rebane, T. K., and Volodicheva, M. I., 1974, V est.

L eningr. Univ., 22, 55.[22] Stanton, J. F., Gauss, J., Watts, J. D., Lauderdale,

W. J., and Bartlett, R. J., 1992, Int. J. quant. Chem.Symp., 26, 879.

[23] Gauss, J., and Stanton, J. F., 1996, J. chem. Phys., 104,2574.

[24] Verberne, J., Oz ier, I., Zandee, L., and Reuss, J., 1978,Molec. Phys., 35, 1649.

[25] Harrick, N. J., Barnes, R. G., Bray, P. J., andRamsey, N. F., 1953, Phys. Rev., 90, 260.

[26] Code, R. F., and Ramsey, N. F., 1971, Phys. Rev. A, 4,1945.

[27] Quinn, W. E., Baker, J. M., LaTourrette, J. T., andRamsey, N. F., 1958, Phys. Rev., 112, 1929.

[28] Raynes, W. T., and Panteli, N., 1983, Molec. Phys. , 48,

439.[29] Bass, S. M., DeLeon, R. L., and Muenter, J. S., 1987, J.

chem. Phys., 86, 4305.[30] Muenter, J. S., and Klemperer, W., 1970, J. chem.

Phys., 52, 6033.[31] Muenter, J. S., 1972, J. chem. Phys., 56, 5409.

Coupled-cluster calculations of spin± rotation constants 457

Table 10. Calculated spin± rotation functions.

Molecule Nucleus Bond length/AÊ c0/kHz c1/kHz c2/kHz c3/kHz c4/kHz

H21H 1.401 24 115.628 - 438.162 1005.545 - 1823.603 2894.217

HF 19F 1.7291 - 283.439 - 1437.932 - 155.278 - 707.575 - 115.686HF 1H 1.7291 72.954 - 338.418 996.730 - 2214.061 4138.005F2

19F 2.6681 - 150.843 - 595.204 - 824.633 - 1251.801 - 2038.273CO 13C 2.1342 - 32.383 9.663 - 28.705 58.239 - 230.966CO 17O 2.1342 31.269 6.504 55.773 - 165.170 - 214.639N2

15N 2.0757 19.832 1.404 25.450 - 63.262 71.174

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

11:

09 0

5 N

ovem

ber

2013

[32] Hindermann, D. K., and Cornwell, C. D., 1968, J.chem. Phys., 48, 4148.

[33] Sauer, S. P. A., and Paidarova, I., 1995, Chem. Phys.,201, 405.

[34] Hindermann, D. K., and Williams, L. L., 1969,J. chem. Phys., 50, 2839.

[35] Oz ier, I., and Ramsey, N. F., 1966, Bull. Am. phys. Soc.,11, 23.

[36] Helgaker, T., Gauss, J., Jørgensen, P., and Olsen, J.,1997, J. chem. Phys. (to be published).

[37] Meerts, W. L., De Leeuw , F. H., and Dynamus, A.,1977, Chem. Phys. , 22, 319.

[38] Frerking , M. A., and Langer, W. D., 1981, J. chem.Phys., 74, 6990.

[39] White, R. L., 1955, Rev. mod. Phys., 27, 276; Baker,M. R., Anderson, C. H., and Ramsey, N. F., 1964,Phys. Rev., 113, 1533.

[40] Jameson, C. J., Jameson, A. K., Oppusunggu, D.,Wille, S., Burrell, P. M., and Mason, J., 1981,J. chem. Phys., 74, 81; Oddershede, J., and Geertsen,J., 1990, J. chem. Phys., 92, 6036.

458 Coupled-cluster calculations of spin± rotation constantsD

ownl

oade

d by

[M

osko

w S

tate

Uni

v B

iblio

te]

at 1

1:09

05

Nov

embe

r 20

13