14 May 1999

Ž .Chemical Physics Letters 305 1999 139–146

The geometry dependence of the spin–spin coupling constants inethane: a theoretical study

Magdalena Pecul a, Michał Jaszunski b, Joanna Sadlej a,)´a Department of Chemistry, UniÕersity of Warsaw, Pasteura 1, 02-093 Warsaw, Poland

b Institute of Organic Chemistry, Polish Academy of Sciences, Kasprzaka 44r52, 01-224 Warsaw, Poland

Received 29 January 1999; in final form 24 March 1999

Abstract

The intramolecular surfaces of the spin–spin coupling constants in ethane have been analysed in ab initio calculations atthe MCSCF level. The Karplus equation has been modified by including the influence of the C–C and C–H bond lengths,and the CCH bond angle. The C–C bond length affects the vicinal H–H coupling constant in a wide range of torsion angles,while the influence of the other two geometric parameters is largest for the torsion angles near 1808. The one-bond couplingconstants are sensitive to bond stretching, while the change of the CCH angle mostly influences the C–C and geminal H–Hcoupling constant. The C–H coupling constants exhibit differential sensitivity to the changes of C–H bond length. q 1999Elsevier Science B.V. All rights reserved.

1. Introduction

The relationship between the vicinal H–H nuclearspin–spin coupling constant and the HCCH torsionangle is a valuable tool of conformational analysis inchemistry and biochemistry. The original Karplus

w xequation 1 has been repeatedly revised, both experi-Ž w x .mentally see Refs. 2–4 for a review and theoreti-

w xcally 4–12 . The aim of most of the empiricalcorrections is to take into account the substituent

w xeffects 2,3 , although the influence of the C–C bondlength and CCH bond angle on 3J has also beenHH

w xconsidered 3 .The theoretical works were, for a long time,

w xmostly semi-empirical 4,6,9,10 , with the exception

) Corresponding author. E-mail: sadlej@chem.uw.edu.pl

w xof Ref. 11 , because sophisticated ab initio calcula-tions of the spin–spin coupling constants in ethanewere not possible. Even in recent years only a few ab

w xinitio calculations for ethane 7,8,12 have been re-ported. The studies of 3J as a function of theHH

w xtorsion angle involved either the UHF 7 or MP2w xapproach 8 , both not being best suited to spin–spin

coupling constant calculations. The coupled-clustermethod has also been applied to study the Karplusrelation, but a relatively small basis set was used andonly the Fermi contact contribution has been consid-

w xered 12 . As far as we know, there has been noattempt in the past decade to take into account by abinitio methods the influence of the geometric param-eters other than the torsion angle on the 3J cou-HH

pling constants.Papers reporting ab initio calculations of the de-

pendence of spin–spin coupling constants other than

0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0009-2614 99 00363-2

( )M. Pecul et al.rChemical Physics Letters 305 1999 139–146140

3J on the geometric parameters are also scarce.HH

The intramolecular surfaces of the nuclear spin–spincoupling constants have been calculated only for a

w xfew polyatomic systems: methane 13 and its ana-w x w x w xlogues 14 , oxonium ion 15 , and water 16 . The

effects of the geometry changes are of particularinterest in the case of 1J and 1J , because theirCC CH

large, relatively easy to measure experimentally val-ues make them good candidates for parameters for

w xstructure elucidation 17 .In the present work the intramolecular surfaces of

the nuclear spin–spin coupling constants in ethanehave been calculated. Our attention has focused onthe influence of the geometry variations on the vici-nal proton–proton coupling constant. The parametersof a Karplus-type equation have been determined,including the corrections resulting from the changesof the bond lengths and the CCH bond angle. Thedependence of the remaining nuclear spin–spin cou-pling constants: 1J , 1J , 2J , and 2J on theCC CH CH HH

geometric parameters has been calculated as well.

2. Computational aspects

In our calculations the MCSCF method has beenused. Both theory and computational aspects ofspin–spin coupling calculations were described in

w x ŽRef. 18 for more recent applications, see e.g. Ref.w x w x.19 and the review in Ref. 20 . We use hereRASSCF wavefunctions, and all our calculationswere performed using the Dalton program systemw x21 . The calculation of the intramolecular surfacesof the spin–spin coupling constants has been pre-ceded by a study of the influence of the basis set andthe active space dimension on the calculated cou-pling constants. In these preliminary calculations we

w xhave used the experimental geometry 22 . The RASspace finally chosen is composed as follows: core sorbitals of C atoms inactive, no orbitals in RAS1space, 7 orbitals in RAS2 space, 20 orbitals in RAS3space and maximum 2 electrons allowed in RAS3space. Further extension of the active space, whileincreasing considerably the computational effort,does not improve significantly the agreement with

Ž .experiment see below . The surfaces were calcu-w xlated using the HIII IGLO modification 23 of Huzi-

w xnaga’s 24 atomic basis set, a medium-size basis setoptimal for our purpose. Test calculations with theHIV basis and other basis sets for the equilibriumgeometry indicated that all the coupling constantsobtained with the HIII basis are sufficiently accurateto study their geometry dependence.

For the calculation of the intramolecular spin–spincoupling constants surfaces the geometrical parame-ters of ethane have been varied as follows: R inCC

˚ Ž .the range 1.2351–1.7351 A 8 points ; R dis-CH˚Ž .tances all six identical in the range 0.994–1.194 A

Ž . Ž .5 points ; all the CCH angles a in the rangeŽ .101.17–121.178 5 points ; the torsion angle u every

Ž .108 7 points . Additionally, the spin–spin couplingconstants in ethane have been calculated for geome-tries with only one or two of the six R distancesCH

varied in a range as above. In order to establish adependence of the Karplus equation on the geometri-cal parameters other than u , this angle and one of theother parameters have been varied simultaneously: u

every 308 and the other parameters in the range asabove, with the exception of R , which was variedCC

˚ Žin a narrower range 1.4351–1.6351 A with the˚ .increment 0.05 A . Additionally, to check the valid-

ity of the relationship obtained, we have calculated acouple of points with two of the three parameters:R , R , a changed, and us08 or 608. To sumCC CH

up, the coupling constants in ethane were calculatedfor 54 different combinations of geometrical parame-ters. Ethane may be considered as a model systemshowing how the differences in the bond lengths andthe bond angle may affect the coupling constants in awide range of organic compounds.

Ž . ŽThe FC Fermi contact , PSO paramagnetic spin. Ž .orbital and DSO diamagnetic spin orbital terms

have been calculated for all the points of the surface.Ž .The SD spin dipole term has been omitted on the

basis of our test calculations for ethane and thew xresults reported for methane 13 .

3. Results and discussion

3.1. Equilibrium geometry results

The calculated spin–spin coupling constants forethane at the equilibrium geometry are shown in

( )M. Pecul et al.rChemical Physics Letters 305 1999 139–146 141

Table 1The calculated nuclear spin–spin coupling constants for ethane with the equilibrium geometrya

Calc. in this work Exp.Ž . Ž .Hz Hz

1 bJ 38.66 34.498–34.558CC1 bJ 122.00 125.19–125.238CH2 bJ y5.90 y4.655–y4.661CH3 cJ y14.82 y13.12HH3 3 31 bJ s 2= J 608 q J 1808 7.36 7.992–8.005Ž . Ž .Ž .HH av HH HH3

a ˚ ˚ w xEquilibrium geometry R s1.5351 A, R s1.094 A, as111.178 taken from Ref. 22 .CC CHb w xRef. 25 .c w xRef. 26 .

Table 1. These values, obtained with HIII basis setand RAS space finally chosen, include also the SDterm. The wave function used here was previouslyemployed for spin–spin coupling constants calcula-

w xtion in Ref. 25 . The agreement between calculatedand experimental values is satisfactory, the largestdiscrepancies occur for 1J and 2J . The extensionCC CH

Žof the active space inclusion of more virtual orbitals.and allowing higher than double excitation leads to

the decrease of the calculated 1J by ;2 Hz;CC

bringing it to a better agreement with experiment. Inthe case of 1J the extension of the active spaceCH

lowers the calculated value, increasing the discrep-ancy with the experiment. This discrepancy may beattributed to basis set incompleteness; e.g. one of ourtests shows that the addition of tight s orbital tos-decontracted cc-pVTZ Dunning basis set increases1J calculated by nearly 8 Hz. 2J and 2JCH CH HH

calculated with the more extended active space are inbetter agreement with experiment, similarly to 3J ,HH

but here the correlation corrections are small.

3.2. Reparameterisation of the Karplus equation

The least-squares fitting of the vicinal 3J cou-HH

pling constants, obtained by varying the torsion an-gle u by 108 and keeping the other geometricalparameters of ethane fixed at their equilibrium val-ues, leads to the following relation:

3J u s7.13y1.16 cos u q6.44 cos 2 u .Ž . Ž . Ž .HH

1Ž .

The square root of the residual variance is 0.14 Hzand decreases to 0.05 Hz with the addition of the

Ž . Ž .term containing cos 3 u . Eq. 1 is in reasonableagreement with the empirical relations for cyclohex-

w xanes and substituted ethanes 4 , although a directcomparison is impossible.

Let us now discuss the dependence of 3J on theHH

geometric parameters other than u . The modifica-3 Ž .tions of J u resulting from the variations inHH

R , R , and a are shown in Fig. 1.CC CH

The dependence of 3J on the R bond lengthHH CCŽ .Fig. 1a is significant for very small or very large u .The variation of R affects the depth of the KarplusCC

curve, but it does not change its asymmetry. It meansthat the term in the Karplus equation proportional to

Ž .cos u is practically independent of R .CC

The dependence of 3J on the R bond length,HH CH

displayed in Fig. 1b, is negligible for u in the rangeŽ .60–908 , typical for most unstrained organic com-pounds. The omission of the R dependence in theCH

w xempirical correlations 3 is therefore justified. Forthe torsion angle approaching 1808, however, thedependence of 3J on R is significant. The sameHH CH

happens, to some extent, when u is nearly 08. Theincrease of R results in a less symmetric KarplusCH

Ž .curve, that is the term proportional to cos u in-creases with R .CH

The most interesting feature of Fig. 1c, showingthe influence of the a bond angle on 3J , is thatHH

the curves cross near 508 and again near 1008. Theincrease of a causes thus an increase of 3J for uHH

outside this range and a decrease for u inside thisrange, typical for most hydrocarbons. This latter

( )M. Pecul et al.rChemical Physics Letters 305 1999 139–146142

3 Ž .Fig. 1. The modifications of the dependence of J Hz on theHH˚Ž . Ž .torsion angle u by the changes of a C–C bond length R A ,CC

˚Ž . Ž . Ž . Ž .b C–H bond length R A , and c CCH bond angle a deg .CH

observation is in agreement with semi-empirical re-w xsults for hydrocarbons 6 .

The dependence of the parameters in theŽ .Karplus-type Eq. 1 on R , R , and a is foundCC CH

to be, in a good approximation, linear in the rangeŽ w x.investigated same approach as in Refs. 3,6 . This

3 Ž .observation leads to the extended form of the J uHH

relation:3J u s7.324y20.39 D R q15.46 D RŽ .HH CC CH

q0.1547Day 1.286q0.44 D RŽ CC

q11.36 D R q0.2085 Da cos u. Ž .CH

q 6.561y18.66 D R q13.28 D RŽ CC CH

q0.1681 Da cos 2 u , 2. Ž . Ž .with the square root of the residual variance equal

˚Ž0.27 Hz D R , D R are given in A, Da inCC CH.degrees .Ž .Eq. 2 has been derived by fitting the calculated

3 Ž . Ž .J u with one besides the torsion angle geomet-HH

rical parameter varied. In order to assess the impor-tance of the omitted terms containing the products ofthe two geometrical parameters we have carried out

Table 23 Ž . Ž3 .A comparison of J derived by means of Eq. 2 J andHH HHfit

Ž3 .calculated ab initio J with two of the R , R , and aHHcal CC CH

parameters varied3 3

D R D R Da u J JCC CH HHfit HHcal

˚ ˚Ž . Ž . Ž . Ž . Ž . Ž .A A deg deg Hz Hz

y0.05 y0.05 0.0 0.0 13.70 13.51y0.05 y0.05 0.0 60.0 3.81 4.00y0.05 y0.05 0.0 120.0 4.50 4.12y0.05 y0.05 0.0 180.0 15.10 14.98q0.05 q0.05 0.0 0.0 11.49 11.27q0.05 q0.05 0.0 60.0 2.99 3.10q0.05 q0.05 0.0 120.0 4.87 4.51q0.05 q0.05 0.0 180.0 15.24 14.97

y0.05 0.0 y5.0 0.0 14.00 13.56y0.05 0.0 y5.0 60.0 4.13 4.26y0.05 0.0 y5.0 120.0 4.35 3.87y0.05 0.0 y5.0 180.0 14.45 14.08q0.05 0.0 q5.0 0.0 11.20 10.67q0.05 0.0 q5.0 60.0 2.67 2.67q0.05 0.0 q5.0 120.0 5.02 4.75q0.05 0.0 q5.0 180.0 15.90 15.51

0.0 y0.05 y5.0 0.0 11.16 11.140.0 y0.05 y5.0 60.0 3.41 3.700.0 y0.05 y5.0 120.0 3.09 2.980.0 y0.05 y5.0 180.0 10.51 10.990.0 q0.05 q5.0 0.0 14.04 13.950.0 q0.05 q5.0 60.0 3.39 3.550.0 q0.05 q5.0 120.0 6.29 6.300.0 q0.05 q5.0 180.0 19.83 20.14

( )M. Pecul et al.rChemical Physics Letters 305 1999 139–146 143

calculations with two of the three parameters: R ,CC

R , and a changed simultaneously and us08 orCH

608. The results are compared in Table 2 with 3JHHŽ .obtained from Eq. 2 . On the whole, the terms

Žcontaining two geometrical parameters and the tor-.sion angle are not very important in the geometry

dependence of 3J . The data in Table 2 are in aHH

satisfactory agreement for all the values of the geom-etry parameters investigated. The largest discrepan-cies occur for the simultaneous change of R andCC

a , the standard deviation then being 0.13 Hz. Thenonadditivity of the effects of the change in R andCC

R is less significant, and finally the changes ofCH3J when R and a are varied appear to beHH CH

nearly additive.

3.3. The influence of the geometry changes on thespin–spin coupling constants through one and twobonds

We shall now briefly discuss the influence of thegeometric deformations on the 1J , 1J , 2J andCC CH CH2J nuclear spin–spin coupling constants in ethane.HH

3.3.1. The influence of R on the coupling con-CC

stantsThe influence of the changes in R on theCC

coupling constants is shown in Fig. 2. 1J is theCC

most sensitive one to the R variations in terms ofCC

Fig. 2. The dependence of the 1J , 1J , 2J and 2J nuclearCC CH CH HHŽ .spin–spin coupling constants in ethane Hz on the C–C bond

˚Ž .length R ACC

Fig. 3. The dependence of the 1J , 1J , 2J and 2J nuclearCC CH CH HHŽ .spin–spin coupling constants in ethane Hz on the C–H bond

˚Ž .length R ACH

the reduced coupling constant, it increases stronglywhen R is shortened, but the elongation of RCC CC

introduces very slight changes. 1J increases withCH

elongation of R in nearly linear fashion. 2JCC CH

behaves in the opposite way: it decreases in absoluteŽ .value i.e. becomes less negative when the CC bond

length is increased. The elongation of R intro-CCŽ .duces only a minor decrease of the absolute value

of 2J .HH

3.3.2. The influence of R on the coupling con-CH

stantsThe effects of the simultaneous change in all RCH

are shown in Fig. 3. The stretching of the C–Hbonds increases significantly both 1J and 1J , theCC CH

1 Žchanges in J being the largest in terms of theCC.reduced coupling constant . The absolute value of

Ž . 2negative J is also increased by the stretching ofCH

R , more than it is reduced by the comparativeCH

elongation of R .CC

Prompted by the unexpected variation of 1JCHw xwith R bond length in methane 13 we decided toCH

investigate the behaviour of the coupling constantsŽ .with changes of one or two instead of all six C–H

distances. The most interesting results concer-ning 1J and 2J are summarised in Table 3.CH CH

1J in ethane clearly exhibits the same as inCHw xmethane ‘‘unexpected differential sensitivity’’ 13 :

the changes of 1J when R is varied areC1H1 C1H2

( )M. Pecul et al.rChemical Physics Letters 305 1999 139–146144

Table 31J and 2J calculated for ethane with CH distances variedCH CH

a b b aDistances varied All C1H1 C1H2 C2H4 C2H5 C1H1 and C2H5 C1H2 and C2H4

1˚Ž . Ž .R A J HzCH C1H1

0.994 106.89 118.71 116.15 122.25 121.90 118.66 115.851.044 114.68 120.80 119.22 122.23 122.06 120.78 119.051.094 122.23 122.23 122.23 122.23 122.23 122.23 122.231.144 130.08 123.52 125.19 122.25 122.41 123.53 125.371.194 138.73 125.07 128.08 122.30 122.60 125.08 128.48

2˚Ž . Ž .R A J HzCH C2H1

0.994 y5.01 y5.73 y6.18 y6.28 y5.22 y6.04 y5.411.044 y5.47 y5.82 y6.07 y6.12 y5.60 y5.99 y5.701.094 y5.97 y5.97 y5.97 y5.97 y5.97 y5.97 y5.971.144 y6.55 y6.16 y5.86 y5.81 y6.33 y6.01 y6.221.194 y7.27 y6.44 y5.75 y5.67 y6.69 y6.15 y6.46a Ž .Dihedral angle 1808 trans position .b Ž .Dihedral angle 608 gauche position .

larger than when R is varied. Columns 5 and 6C1H1

of Table 3 show that 1J does not changeC1H1

much when C2Hn bonds are varied. These changesŽare larger when R H1 and H5 are a gaucheC2H5

. Žpair is varied than when R H1 and H4 are aC2H4.trans pair is modified.

The variation of 2J with R bond lengthCH CHŽ .Table 3 is even more surprising: the changes of2 ŽJ are largest when R H1 and H5 are aC2H1 C2H5

.gauche pair is varied. They are also substantial, butŽof opposite sign, when R H1 and H4 are aC2H4

.trans pair is changed. The effects of the changes ofR and R on 2J are similar in magnitude,C1H1 C1H2 C2H1

but again opposite in sign. The changes of 1JC1H1

and 2J induced by simultaneous variation ofC2H1

C1H1 and C2H4, as well as C1H2 and C2H5 bonds,are approximately additive.

The dependence of 2J on the R bond lengthHH CH

is not very large, but, in contrast to its dependenceŽon R , it is not monotonic, having a maximum i.e.CC

.a smallest absolute value near the R equilibriumCH

value. Neither 2J , nor 3J exhibit differentialHH HH

sensitivity.

3.3.3. The influence of the CCH bond angle on thecoupling constants

The changes of the spin–spin coupling constantsŽwith the CCH bond angle a all the angles modified

.simultaneously are shown in Fig. 4. The variation ofa affects most significantly 1J , the increase of aCC

by 58 causing an increase of 1J by 17.5 Hz. 1JCC CH

decreases monotonically, but relatively little, with2 Žthe increase of a , similarly J the increase of aCH

2 . 2results in less negative J . As expected, J isCH HH

extremely sensitive to the changes in a .

3.3.4. The influence of the torsion angle on thecoupling constants

The internal rotation causes only slight, mono-tonic changes of constants other than 3J . TheHH

Fig. 4. The dependence of the 1J , 1J , 2J and 2J nuclearCC CH CH HHŽ .spin–spin coupling constants in ethane Hz on the CCH bond

Ž .angle a deg

( )M. Pecul et al.rChemical Physics Letters 305 1999 139–146 145

change of conformation from staggered to eclipsedform leads to an increase of 1J by ;0.7 Hz andCC

an increase of 1J by ;0.4 Hz. The negative valueCH

of 2J increases then by ;0.1 Hz, while theCH

absolute value of 2J undergoes the largest change,HH

increasing by ;0.9 Hz. The averaging over all u

increases the absolute value of the calculated 1JCC

by ;0.3 Hz, 1J by ;0.2 Hz, 2J by ;0.05CH CH

Hz, and 2J by ;0.4 Hz.HH

3.4. Contributions to spin–spin coupling constants

The relative importance of individual contribu-tions in the calculation of the spin–spin couplingconstants surfaces is also of interest, as in some

w xcases 17 only the FC term has been computed. Itfollows from our work that the changes caused bythe geometry deformations are in fact dominated bythe Fermi contact term, with one exception. Thechanges in 1J induced by the internal rotationCC

consist in 80% of the changes in PSO term, thechanges in FC being less than 1% of the total effect.The dominant importance of the changes of the FCterm with the variation of molecular geometry was

w xpreviously observed in other hydrocarbons 13 , but,as our results show, the omission of the PSO termcan, in some cases, lead to qualitatively wrong re-sults.

4. Conclusions

We have presented MCSCF calculations of theintramolecular nuclear spin–spin coupling constantssurface in ethane. The RAS SCF functions we haveused yield only an approximate description of thedynamical correlation effects. However, the agree-ment of different RAS SCF results and experimentindicates that the neglected effects are not very large.Moreover, they should not vary significantly withgeometry. The dependence of 3J on the bondHH

lengths and the bond angle has been incorporatedinto the Karplus equation. The variation of the C–Cbond length affects mostly the free term and the term

Ž .containing cos 2 u in the Karplus-type equation,while the changes of the C–H bond length and CCHbond angle influence the asymmetric term propor-

Ž .tional to cos u as well. Consequently, the influence

of these geometric parameters on 3J is largest forHH

the torsion angle approaching 1808. The variation ofthe CCH bond angle introduces changes in 3JHH

Ždifferent in the sign for the torsion angle in the 508,.1008 range and outside this range.

The most interesting feature of the intramolecularsurfaces of the C–H spin–spin coupling constants isthe differential sensitivity of both 1J and 2J toCH CH

the changes of the C–H bond length. The H–Hcoupling constants do not exhibit this property. 1JCC

is very sensitive to the changes of the bond lengthsand the bond angle, confirming its usefulness as aprobe of the molecular structure. The changes of thenuclear spin–spin coupling constants induced byvariations in the geometry of ethane are dominatedby the FC term, with, however, an interesting excep-tion of 1J dependence on the torsion angle, whenCC

the PSO term prevails.

Acknowledgements

We thank W.T. Raynes for numerous helpfulcomments. This work was supported by the 3 TO9A121 16 KBN Grant.

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