1789
J. Phys. Chem. 1993,97, 1789-1792
Ab Initio Quadratic Configuration Interaction Calculation of Indirect NMR Spin-Spin Coupling Constantst Ian Carmichael Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 Received: August 5, 1992
Finite-field perturbation theory is combined with the quadratic configuration interaction method of a b initio electronic structure theory to calculate the Fermi-contact component of the indirect spin-spin coupling constant between adjacent carbon nuclei in a range of bonding environments. Basis set requirements for accuracy are first investigated in ethane and a small modification of the outer-core, inner-valence region of the standard Dunning double-r contraction is shown to be satisfactory. Calculations with this basis set then allow the accurate determination of the contact component of 1 J ~ 3in~ ethane, ~ 3 ~ (34.1 Hz), cyclopropane, (13.9 Hz), cyclobutane, (27.4 Hz), and bicyclo[l.l.0]butane, (24.3 Hz), including a prediction (-13.6 Hz) of the coupling across the central bridge in this latter system.
Introduction
either a CONVEX C-1200r C-240computer. Exploratory studies employed the familiar 6-31G10 and 6-31G*11basis sets, and the Dunning double-{, [4s2p,2s], contraction1*of the Huzinaga (9s5p,4s) Gaussian basis set," abbreviated here as [4212]. The addition of polarization functions on both carbon and hydrogen wasinvestigated, givinga basis set whichmay bedenoted [421(21], as were various modifications of the outer-core inner-valence region. The effect of additional diffuse s- and p-functions and compact s-functions was also tested. In the correlatedcalculations the core electrons were held frozen and, with the full doublet basis set, the highest corresponding virtuals were deleted from the excitation space. In similar calculations, based on spin-unrestricted (UHF) reference determinants, of electron-nuclear isotropic hyperfine coupling in open-shell free radicals, the neglect of core-electron correlation has been shown1k19to be insignificant. The same is not true for the analogous free radical spin densitycalculationsbased on spinrestricted (ROHF) reference spaces, of course, where the neglect of core-electron polarization leads to large errors. A satisfactory descriptionof this core-polarization is apparently obtained from the UHF model, while valence-space effects require a more complete treatment of electron correlation effects. No further configuration selection is imposed in the present calculations, other than to limit excitations to the space of single and double replacementsin the UHFdeterminant togive theQCISD modela7 The effect of amplitudes due to triple excitations is recovered perturbatively and the resultant method denoted QCISD(T).7 To provide an estimate of the contributions from various levels of excitation, calculations were also performed using the simpler coupled-cluster doubles (CCD) model.20.21 The spin-spin coupling term, JN," is related22-24to a reduced coupling constant, KN,",
The determination of nuclear spin-spin coupling constants is a convenient tool in modem NMR spectroscopy,l often allowing molecular identification, structural analysis and even an assessment of conformational preferences. Inferences from experimentally observed splittings are customarily based on the results of semiempirical theoretical techniques which, while being the source of many useful correlations, can be of dubious reliability.2 Direct spin-spin interactions between magnetic nuclei are averaged out by molecular tumbling in solution leaving only indirect, electron-coupled, components. Ramsey' defined four contributions to these indirect nuclear spin-spin coupling mnstants; the orbitally based diamagnetic and paramagnetic spinorbit terms, and the electron-spin modulated dipolar and contact terms. Certain of the more important coupling constants, particularly those involvingcarbon and hydrogen nuclei, are known to be dominated by the Fermi contact mechanism. Although several ab initio studies of this term have appeared, these works have been limited to very small model systems. Additionally, treatments employing the more sophisticated computational approaches have often been further limited by a restriction to inadequate basis sets. Exceptions are found in the exhaustive polarization propagator study of HD by Oddershede et 81.: of methane by Geersten et al.5 and in the multiconfiguration SCF treatment of Vahtras et a1.,6 in which the effects of basis set completeness are probed. Here the quadratic configurationinteraction approach for the calculation of molecular energies7is combined with a finite-field (double-) perturbation theory treatment8 to calculate one-bond carbon-rbon spin-spin coupling constants, 1JCC,for saturated carbon-carbon linkages involving carbon atoms in a range of bonding enviroments. In particular, the ability of the current method to satisfactorily describe the influence on the coupling of the hybridization of thecoupledcarbons in a number of strained alicyclic systems is demonstrated. Basis set requirements for quantitative accuracy are elucidated and the applicability of various scaling methods for less computationally demanding schemes is investigated.
where YN is the magnetogyric ratio for the nucleus, N. In the finite perturbation treatment, this reduced coupling constant is obtained as an energy shift calculated in the presence of large hypothetical magnetic moments, C(N, located on the two coupled nuclei. Terms of the form
Computational Details
All calculationsreported here were performed with a modified version of the GAUSSIAN 90 series of programs9 running on
H&
'This is Document No. NDRL-3518 from the Notre Dame Radiation
H&t = c ( N ~ Q N ~ O )
are introduced into the Hartree-Fock Hamiltonian, where
Laboratory.
0022-365419312097-1789$04.00/0
c(NQN(O),
(B
1993 American Chemical Society
1790 The Journal of Physical Chemistry, Vol. 97, No. 9, 1993
TABLE I: F e d Contact Contribution to the C8rbon-c.rboa Nuclear Spi&spin Coupling Constrat, 'EJCIJC, in Ethnc. basis 6-31G 6-31G* 6-31G** [42121 [42112] [421121] ]52ll2] [631(41]
UHF 67.9 55.9 55.7 71.4 67.1 66.0 61.6 61.4
MP2 41.5 30.2 29.7 36.0 27.8 27.2 24.3 23.5
CCD 42.4 32.5 32.5 37.6 31.6 31.8 28.1 28.1
QCISD 47.9 37.3 37.7 44.4 39.1 39.1 35.2 35.4
QCISD(T) 47.4 36.3 36.6 44.0 38.0 37.9 34.1 34.1
a Values in Hz from calculations performed at the experimental geometry.30 The experimental value of I J I ~ CisI ~34.6 C Hz.26
QN(O)is the unpaired spin density at nucleus, N, and the reduced coupling is approximatedz5as the (second) derivativeof the energy
Fields of 0.002 atomic units proved sufficient to maintain numerical accuracy while eliminating higher order contributions to the derivatives,
Results a d Discussion Ethane. Basis Sets. Basis set requirements for accurate determination of the spin-spin coupling constants were first checked by a series of calculations of IJcc in ethane, where the experimentalvalue is well-known,z6and reliable estimates of the orbital and dipolar terms have been rep0rted,2~-~~ and are small. It should be pointed out that the non-contact contributions to indirect coupling constants are not always negligible. In particular, if at least one of the coupling nuclei is fluorine, the noncontact mechanisms appear to dominate the interaction. However for the saturated carbon-carbon linkages considered in the present study, noncontact components may be safely ignored. The results of these calculations,performed at the experimental geometry,30 are collected in Table I. It is immediately apparent that the SCF values, labeled UHF in Table I, do not provide a quantitativeestimateof the measured one-bond coupling constant, correlation corrections of more than 40% are found. Basis set effects are less pronounced but still important. At the SCF level, doubling the core representation,i.e. going from 6-31G to [4212], leads to an increase of 3.5 Hz in 'Jcc. This increment is offset by a decrease of about 4.4 Hz observed upon the introduction of shells of d-functions in the doubIe-{ basis on the carbon centers. The subsequent addition of pshells on the hydrogens provides a much smaller decrement of about 1 Hz. Similarresults for carboncarbon coupling in ethane, obtained at the SCF level with a [4212] basis set, have been reported by Laaksonen and Kowalewski.* Notice, however, that in the presence of shells of 6-component d-functions on carbon (the 6-31G*) basis, the calculated SCF coupling constants drop by a large amount, around 12 Hz. The magnitude of this decrease, attributable to the s-like "extra* component of the Cartesian d-functions, suggested that, for the full double-{ basis, a more flexible description of the outer-core, inner-valence region, than that provided by the standard Dunning contraction scheme, might be important. Hence the double-{ basis was modified by splitting off the outermost primitive of the outer core function on carbon and recontracting the remaining set to give a [5212] basis. Calculationswith this basis set produced a 5.5 Hz decrease in the estimate of IJcc, relative to the standard Dunning contraction scheme, independent of the presence of the polarizing d-shell. Little further changesensue upon the addition of p-shells on the hydrogens or indeed upon substantial extension of this basis set with diffuse s- and pfunctions on the carbons, and diffuse and compact (high-exponent) s-functions on hydrogen to give the j631111 basis set. These latter augmentations were
Carmichael
TABLE Ik Correlation Corrections to the Carbon-Carbon Oae-Boud Nuclear Sph-Sph Coapllno Conrtamt, 'JISCIJC, in Ethae basis 6-3lG* [421121] [52112]
SCF 55.9 66.0 61.6
doubles -23.4 -34.2 -33.5
singles 4.77 7.31 7.04
triples 4.98 -1.19 -1.08
QCISD(T) 36.3 37.9 34.1
Contributions (in Hz) from various levels of excitation within the quadratic configurationinteraction model, calculated at the experimental geometry; see Table I.
found to be importantin spin-restricted calculations of spin density distributions in open-shell s y s t e m ~ but ~ l ~are ~ ~apparently not required for the present problem. Ethane. Electron Correlation. The recovery of electron correlation by means of stcondsrder MaUer-Plesset perturbation theory, MP2,33 overestimates the correlation correction to VCC substantially, a trend which is often encountered. Comparison of the MP2 values with those obtained from calculationsin which a more complete treatment of the effects of electron correlation is made reveals the source of this overcorrection. The MP2 method exaggerates the effect of the inclusion of double replacementsin the UHF reference determinant; the MP2 doubles correction is about 4 Hz too negative. Additionallythe MP2 method gives no account of the effect of amplitudes due to single substitutions, so that a contribution of about +7 Hz, which is observed upon incorporating single excitations in the CCD model in a sizeconsistent way, to give the QCISD picture, is missed. The MP2 treatment also neglects triple and higher excitations, however, in the present case, these turn out to produce relatively minor changes; a triples correction of around -1 Hz is seen in the most complete calculations reported in Table I. In the correlated calculations doubling the number of basis functionsrepresentingthe carbon cores, leads to a decreasein the computed value of the indirect coupling constant, in contrast to the findings at the SCF level. Addition of shells of polarizing d-functions leads to similarly directed changes, as does the recontraction of the outer-core basis function. Other basis set improvements, such as the addition of gfunctions on hydrogen, result in negligible further changes in the computed values of I Jcc. An analysisof contributions to 'JCCfrom the various levels of excitation included in the correlated calculations is presented in Table 11. At the SCF level, the UHF referencedeterminant may be regarded as accounting for the so-called spin-polarization effects. Relative to the value of this term obtained using the [521)2] basis set, the popular 6-31G* basis is seen to afford a result about 6 Hz too low. This 10% shortfall is perhaps overcompensated by a 10Hz underestimationof the true electron correlation effect of incorporating certain amplitudes, particularly those due to configurations reached by double replacements in the UHF determinant, into the correlated wave function. In the CCD model, doubles amplitudes are treated to all orders by the inclusion a term of the form ijab
summing all the elementary double replacement operators $,in the defining equations of the method. Allowance is also made for a component of higher level, a simultaneous doubles term, through the presence of the operator pz2,required to ensure the size-consistency of the CCD prescription. In the QCISD model, the inclusion of single excitations is accomplished by the presence of a term
=
c.3: la
summing the elementary single replacement operators, $.' In
The Journal 6f Physical Chemistry, Vol. 97, No. 9, 1993 1791
Indirect NMR SpinSpin Coupling Constants
TABLE IIL Contact Coatribption~to tbe c.rboa-clrbon Nuclcu SpiaSpin Couplin# Co"t,' in Ethrne md Some
AliCytliCS molecule ethane cyclopropane" cyclobutane
SCF
MP2
61.6 24.3 33.9 3.78 60.0 12.2 bicyclobutaneg (edge) 49.7 11.1 bicyclobutane (bridge) -7.6 -23.9
QCISD(T)
expt
34.1 13.9 27.4 24.3 -1 3.6
34.6b 1 2.4d 28.4,29.1f (+)21* -17.5'
a IJIICIIC in Hz,calculated using the [52112] basis set. b See Table I for references. Experimental structure from ref 35. Reference 36. MP2/6-3 1G* optimized structure, present work. f Experimentalvalues for methylcyclobutane, ref 41. 8 Experimental structure from ref 45. * Reference 47. Experimental value for perdeutero-2,2,4,4-tetramethylbicyclobutane, ref 46.
addition the presence of terms of the form TIT2further modifies the final correlated contributions. In Table 11, the values in the 'singles" column accounts for this relaxation. The effect of amplitudes due to single excitations from the UHF reference space is similarly underestimated in calculations with the 6-31G* basis set, however the fact that this indirect term-its presence serves to modulate the doubles contribution-enters with the oppositesense, conspins to produce a QCISD/6-3G* estimate of IJcc in ethane only about 2 Hz higher than the value of 34.1 Hz obtained by calculations with the more complete [52112] basis. The inclusion of higher-order effects in the correlation treatment is limited to those recovered by the perturbative treatment of triples, and is seen to be similarly small for all the basis sets considered. Strained Cycloollunes. Given the level of agreement with experiment attained in QCISD(T) calculations with the [52112] basis set for ethane, it was decided to explore the one-bond I3CI3C couplingconstants in range of bonding environmentsutilizing this same level of theory. Saturated organic systemswere selected since it is well-known that in many unsaturated molecules, such as ethylene, the UHF solution to the formally closed shell problem suffers from a triplet instability. This problem may be obviated by a more suitable choice of reference wave f~nction,'~ but such systems were not considered here. In Table I11 results for calculations with the [52112] basis set for ~ J C Cin a some strained alicyclics are compared with the corresponding values for ethane. The structure of cyclopropane was taken from a Raman study3sin which rotational bands were resolved. As was the case in ethane, the SCF result greatly overestimates the experimental value of 12.4 Hz, for cyclopropane. Correlated calculations at the MP2 level again overcorrect the SCF value, due to a 4 Hz overestimation of the doubles correction and the neglect of a singles contribution of about 7 Hz, which is obtained in the QCISD approach. The QCISD(T) value, which contains a small negative contribution of about 1 Hz from theeffect of amplitudesdue to tripleexcitations from the UHF determinant, is 1.5 Hz above the experimental determination of Wardeiner et al.36 This small discrepancy may be due to the neglect, in the present calculations, of noncontact terms, which have been supposed to be larger in strained rings than in aliphatic systems. While this supposition was supported by a semiempiricalestimates of -3.87 Hz for the combined orbital termsj7 an ab initio equations of motion study,'* led to values more than an order of magnitude smaller for these contributions. This latter estimate is more in line with expectations from the present results. Calculations on cyclobutane were performed at the MP2/631G* optimized geometry obtained in the course of the present work. Similar structural parameters were reported39from a HF/ 6-31G* optimization. The inclusion of electron correlation leads to a slight shortening of the C-C bond length, r(CC) = 1.543 A, and a slight lengthening of the C-H bonds, r(CH) = 1.0935 and 1.0942 A at the equatorial and axial positions, respectively. Only
TABLE W. Correlation Corrections to the Carbon-clrbon OaC-Boad Nuclear spipspin Coupling Coastrst, ~JVIJC, in selected strained Rings. basis SCF doubles singlks triples QCISD(T) cyclopropane 33.9 -25.9 6.86 -0.92 13.9 cyclobutane 60.0 4 1 . 2 10.1 -1.43 27.4 bicyclobutane (edge) 49.7 -32.1 8.18 -1.47 24.3 bicyclobutane (bridge) -7.61 -12.1 6.46 4 - 3 4 -13.6 a Contributions (in Hz)from various levels of excitation within the quadratic configurationinteraction model using the [521121 basis set.See Table 111 for molecular geometries and experimental references. minor changes occur in the various structural angles, except that the dihedral folding angle is predicted to be 22O, rather that the value of less than 1 5 O found in the SCF calculation. The MP2 result is substatially closer to the value of -25O inferred from an electron diffraction experiment.@ The experimental value of 'JCCfor the unsubstituted compound is apparently not known, however, the results of the present investigationmay be compared with thevalues of 28.4 and 29.1 Hz obtained41 for the twodistinct carbon-rbon ring couplings in methylcyclobutane. That the effects of methyl substitution is indeed small in these systems is apparent from reports of I JCC = 13.3 Bz for methylcyclopropane42 and 14.4 Hz in l,l-dimethylcyclopropane.~7 By comparisonwith these values, an estimate of 28 Hz may be made for ~JCC in cyclobutane. The familiar pattern of substantial overestimation at the SCF level and a comparable underestimation with the MP2 correlation treatment is again observed for these species. The result of the QCISD(T)/[52112] calculation at 27.4 Hz is, however, in close agreement with the value inferred from experiment, indicating that the orbital and dipolar terms are still small. A more stringent test of the power of the QCISD method in the determination of indirect spin-spin coupling constants is provide by the case of bicyclo[ 1.l.O]butane. An early semiempirical study*' predicted a small negative value for the contact contribution to the carbon-carbon coupling across the central bridge. Similarly small and negative estimates were later also obtained44 for the orbital terms. The observed values of IJcc(bridge) in substituted bicyclobutaneshave indeed beenshown to be negative and the results of the present calculations,performed at the experimental on the unsubstituted molecule are compared in Table I11with the bridge coupling reported6 for the 2,2,4,4-tetramethyl derivative, in which the methyl groups were fully deuterated. The SCFvaluecomputed for IJCc(bridge) is already negative, but in this case underestimated, while the MP2 result again exaggerates the correlation correction. Bearing in mind that the orbital and dipolar components of the observed coupling constant have bcen shown to be small and negative, the QCISD(T)/[52112] estimate of -13.6 Hz is seen to be in accord with theexperimental finding. 'Jcc(edge) has alsobeencomputed and is reported in Table I11 together with an experimentally derived ~ a l u e . 4A~comparison with the QCISD(T) result suggests that the orbital and dipolar contributions to the edge couplings are also small and negative. The analysis of the contributions from the various levels of excitation incorporated into the correlation models is presented in Table IV. The MP2 model overestimates the large doubles term by between 4 and 7 Hz and misses the singles relaxation contribution of 6-10 Hz. The magnitude of the corresponding effects observed in MP2 calculations on unstrained systems, typified here by ethane, lie at the lower ends of these ranges. The inclusion in the correlated wave function of amplitudes due to triple replacements in the reference space leads to only a small decrement in the computed values of IJCC for all couplings considered here. To conclude this study a few comments on basis set effects in these species are necessary. Results from 6-31G calculations indicate that such a limited, valence double-f, description of the
1792 The Journal of Physical Chemistry, Vol. 97, No. 9, I993
oneparticle space is inadequate in that SCF values 'JCC of are overestimated by amounts ranging from 6 Hz in ethane to 15 Hz for the bridge coupling in bicyclobutane. This problem is often compounded by the fact that correlation corrections are underestimated in calculationswith the smaller basis set (by about 2-7 Hz),but is fortuitously compensated to some extent in the case of Vcc(bridge) in bicyclobutane, where thecomlation correction of almost 14 Hz,about 8 Hz larger than that obtained with the [52ll2] basis set at the same level of theory, brings the QCISD(T)/6-31Gvalue to -8 Hz. Correlationcorrectionsfor the [4212] basis set are essentially the same as those recovered with the more complete [52112] description of the oneparticle space, discrepancies of -2 to 10 Hz obtained at the SCF level are thus propagated to the QCISD(T) results. It is also worthwhile to point out that the basis set requirements for accurate estimates of the contact contribution to indirect carbon-carbon coupling in these saturated systems are much less stringent than those determined in the course of extensive calculations, with similar theoretical techniques, of isotropic splittings in free radicals.1c19 The magnetic perturbations are much larger in the open-shell systems, a fact which is reflected in the much smaller magnitudes required for the applied finite fields, and the polarization induced in the electron spin distribution is consequently much more pronounced. In summary the quadratic configuration interaction method, using a slight modification of a standard basis set, has been shown to be capable of yielding accurate values for 'JCCin a wide range of carbon-carbon bonding environments.
+
AcLao-t. The research described herein has been supported by the Office of Basic Energy Sciences of the United Stat- Department of Energy. Refere" ud Note ( I ) Marshall, J. L. Carbon-Carbonand carbon-proton NMR couplings: Applications toorganlcstereOChemlsfrya~conformationalaMlysis;Verlag Chemic International: Deerfield Beach, FL, 1983.
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