5286
J. Phys. Chem. 1996, 100, 5286-5290
NMR Spin-Spin Coupling Constants from Density Functional Theory with Slater-Type Basis Functions Ross M. Dickson and Tom Ziegler* Department of Chemistry, The UniVersity of Calgary, 2500 UniVersity DriVe NW, Calgary, Alberta T2N 1N4, Canada ReceiVed: July 12, 1995; In Final Form: January 2, 1996X
A scheme for calculating nuclear magnetic resonance spin-spin coupling constants has been implemented in the Amsterdam density functional program system. The applicability of the frozen-core approximation is discussed, and an orbital analysis shows that the core orbitals of one of the nuclei can be significant. Results calculated with a modest Slater-type (exponential) basis set are presented. It is shown that calculations of couplings to transition-metal nuclei are feasible.
1. Introduction As the experimental field of multinuclear NMR increases in importance, the ability to provide theoretical calculations of NMR properties for a wide variety of systems is desirable. Semiempirical theories are only reliable for analogues of already-understood systems, while traditional ab initio methods are constrained by the expense involved in treating electron correlation adequately. Density functional theory (DFT) provides a third path between these two methods and is now often the method of choice for investigating inorganic systems, especially those containing transition metals.1 The ability to calculate NMR parameters from density functional theory for such systems would be valuable. Work has recently been published on DFT calculations of chemical shielding tensors in transition-metal systems.2-5 In the present paper we address the calculation of nuclear spinspin coupling constants. Spin-spin coupling constants are difficult to compute. In the nonrelativistic formulation of Ramsey6 there are four terms, each of which has different requirements with respect to correlation and basis set. Malkin et al. provided the first practical implementation of these calculations in a density functional program, using a basis of Gaussian-type functions.7,8 In this paper we illustrate the application of their method in a basis of Slater-type rather than Gaussian functions and examine the feasibility of calculating spin-spin coupling constants in transition-metal systems. 2. Theory and Methods The theory of nuclear spin-spin coupling is well established; we review it here for the reader’s convenience and to establish symbols and terminology. The reader may wish to consult one of several excellent reviews on the subject for further information.9 Nuclear spin-spin coupling is the interaction energy of two nuclear magnetic moments. It can be shown that the reduced spin-spin coupling constant K is the second derivative of the energy of the system with respect to the nuclear magnetic moments b µA ) γApIˆA:10
Kxy(A,B) )
∂ 2E ∂µAx ∂µBy
(1)
* To whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, March 1, 1996.
0022-3654/96/20100-5286$12.00/0
It is a second-rank tensor, but we shall concern ourselves here only with the isotropic part, K ) (1/3)(Kxx + Kyy + Kzz). The reduced coupling constant K is preferred in theoretical discussions over the ordinary spin-spin coupling constant J,
J(A,B) ) hγAγBK(A,B)/4π2
(2)
because J is proportional to the product of the nuclear gyromagnetic ratios γ while K is independent of the identities of the nuclei. Reduced coupling constants are expressed in SI units of 1019 kg m-2 s-2 A-2 throughout this paper. The general theory of second-derivative properties11 tells us that for a pair of perturbations µA and µB
ˆ (µB) ˆ (µA) + µBH H ˆ )H ˆ 0 + µAH
(3)
then
[
]
∂ 2E ∂ ) 〈Ψ(µB)|H ˆ (µA)|Ψ(µB)〉 ∂µA ∂µB ∂µB
(4)
µA)µB)0
i.e., in order to calculate second derivatives of the energy we need the first derivative of the wave function. In density functional theory we avoid explicit reference to the wave function and reformulate this in terms of the KohnsSham molecular orbitals (MOs) and the density constructed therefrom. This only imposes a limitation in certain special cases, which we will discuss presently. There are four terms in the nonrelativistic Hamiltonian described by Ramsey:6 the diamagnetic and paramagnetic spinorbit terms (DSO and PSO), the spin-dipolar term (SD), and the Fermi-contact term (FC).
ˆ pso + H ˆ sd + H ˆ fc H ˆ )H ˆ dso + H
(5)
Each of these terms gives a separate contribution to the isotropic spin-spin coupling, and we shall examine three of these below. The spin-dipolar term, however, is both complicated to implement and generally very small,9 so it is not dealt with in the present work. 2.1. Diamagnetic Spin-Orbit Term. The interaction of the nuclear spin magnetic moment with the electronic orbital angular momentum gives rise to two terms, DSO and PSO. The DSO term can be evaluated as an integral over the ground state density, for which we use the default numerical integration scheme incorporated into the Amsterdam density functional program system (ADF).12,13 © 1996 American Chemical Society
NMR Spin-Spin Coupling Constants from DFT
J. Phys. Chem., Vol. 100, No. 13, 1996 5287
H ˆ dso ) µ02epβ
(IˆA‚IˆB)(b r kA‚b r kB) - (IˆA‚b r kB)(IˆB‚b r kA)
(4π)
3
γAγB∑ ∑ 2 AB k
dso KuV (A,B) )
TABLE 1: Comparative Results for Diamagnetic Spin-Orbit Contributions to Reduced Coupling Constants
(6)
3
rkA rkB
[(b r kA‚b r kB)δuV - rkAVrkBu]
µ02e2
∫F(br k) 16π m 2
3
3
d3 b r k,
rkA rkB {u, V} ∈ {x, y, z} (7)
e
Because this integrand has a rather different form than the charge- and energy-density integrands for which the numerical integration scheme in ADF was devised, it is prudent to ask whether reliable results are obtained by applying it to Kdso. In Table 1, we compare DSO couplings calculated with the ADF integration scheme with the Hartree-Fock results of Scuseria.14,15 The ADF integration precision parameter is set at 10-4. Scuseria’s results were compiled with a mixed numerical-analytic integration scheme due to Matsuoka and Aoyama,16 which is applicable only to Gaussian-type basis sets. The good overall agreement indicates that no special integration scheme is needed for the DSO term and furthermore that the differences between Hartree-Fock and local density approximation (LDA) densities are not usually significant for this property. 2.2. Paramagnetic Spin-Orbit Term. The paramagnetic spin-orbit operator is imaginary, which raises an interesting problem for a pure density functional theory.
H ˆ pso )
µ0βp
IˆN‚(b r kN × ∇ B k)
2πi
rkN3
∑N γN∑k
(8)
Since the charge density F is the sum of the squares of the occupied Kohn-Sham orbitals, F ) ∑iφ i*φi, an imaginary perturbation of the orbitals has no first-order effect on Fsalthough it does affect the current density B. j However, the usual density functional exchange-correlation potential Vxc is a functional of F and not of B, j and neither the Coulomb nor the kinetic energy operators contain any current-density dependence. Therefore, one obtains an uncoupled perturbed equation for Kpso which can be solved directly, rather than a coupled perturbed equation which must be solved iteratively as in the more general case. Letting {u, V} ∈ {x, y, z} signify the components of the nuclear magnetic moment vectors b µA and b µB, respectively, and F(u) the derivative of the Fock matrix F with respect thereto, we can calculate Kpso from
Fij(u) )
[
]
b r kA × ∇ Bk µ0β r φj(b) r d3b rk ∫φ (b) 3 2πi i rkA u Pij(u)
Fji(u)
) ni i - j
pso (V) ) ∑〈i|H ˆ pso |j〉Pij(u) KuV
(9)
(10) (11)
a
molecule
coupling
ADF
Scuseriaa
H2 H2O NH3 CH4 HF H2O NH3 CH4 C2H2 C2H2 C2H4 C2H4 C2H4 C2H2 CO CO2 N2
HH HH HH HH FH OH NH CH 1K(CH) 2K(CH) 1K(CH) 2K(CH) CC CC CO CO NN
-0.14 -0.59 -0.43 -0.29 0.05 0.05 0.07 0.09 0.11 -0.45 0.15 -0.22 0.09 0.00 -0.23 0.16 -0.29
-0.13 -0.59 -0.43 -0.28 0.11 0.10 0.09 0.09 0.13 -0.45 0.15 -0.22 0.08 0.01 -0.17 0.22 -0.34b
Reference 14. b Reference 15.
TABLE 2: Comparative Results for Paramagnetic Spin-Orbit Contributions to Reduced Coupling Constants molecule
coupling
ADF
H2 CH4 NH3 H 2O SiH4 PH3 H 2S CH4 NH3 H 2O HF CH3F SiH4 PH3 H 2S HCl CO N2 CH3F
HH HH HH HH HH HH HH CH NH OH FH FH SiH PH SH ClH CO NN CF
0.32 0.31 0.50 0.72 0.11 0.11 0.18 0.8 2.9 8.0 18.1 1.2 -0.2 1.4 5.2 13.2 -37.2 -39.0 13.7
deMona 0.30
0.6 17.5
-30.3 -31
CHFb
SOPPAc
0.4 0.22 0.39 0.59 0.08 0.07 0.13 0.5 2.4 7.5 17.4 1.1 -0.2 0.9 4.5 12.0
0.45 0.30
0.5 17.3
-36.2 -38
11.4
a
Reference 7, “pseudocoupled” SOS formulation. b Reference 9. c References 15 and 20-22.
and the theoretical underpinnings of the pseudocoupled method proposed in ref 18. In Table 2 we compare our uncoupled results with the pseudocoupled results (labeled “deMon”) and other theoretical calculations. While the pseudocoupled method provides an improvement for the C-H coupling in methane, it appears that the uncoupled approach performs better in N2 and CO (if one accepts the reliability of the SOPPA calculations). Other discrepancies are probably due to basis set and geometry considerations rather than to the theoretical treatment; we conclude that on the balance there is no clear advantage to the pseudocoupled correction, and we have therefore chosen not to implement it. 2.3. Fermi-Contact Term.
ij
The use of this uncoupled equation is an approximation since the exact (and unknown) exchange-correlation functional must have a B-dependence. j An identical problem arises in the calculation of NMR chemical shielding tensors, and Malkin et al. have shown that the error induced by the approximation Vxc[F] ≈ Vxc[F,jB] is occasionally important for chemical shielding.17,18 However, we lack confidence in both the efficacy19
H ˆ fc )
4µ0βp 3
∑N γN∑k δ(br kN)Sˆ k‚IˆN
(12)
The Fermi-contact interaction of a magnetic nucleus with the electron density at the nuclear position results in a small polarization of the total electron spin density which is “detected” at the other nucleus by the same contact mechanism. In many
5288 J. Phys. Chem., Vol. 100, No. 13, 1996
Dickson and Ziegler
cases this mechanism makes the largest contribution to the spinspin coupling constant. The Fermi-contact contribution can be calculated either by a coupled perturbed method analogous to the method sketched above for the PSO term (but now requiring an iterative solution, since Vxc does have spin dependence) or by a finite difference method where the first-order density matrix is determined by carrying out an SCF calculation with a perturbed Fock matrix. The two methods are formally equivalent, but the second is simpler to implement, so we have adopted it here. 2.4. Basis Sets. Basis set requirements for the calculation of spin-spin coupling constants vary not only from molecule to molecule but also from term to term. The spin-orbit terms usually converge with standard basis sets of double- or triple-ζ plus polarization quality, though it should be noted that at least two polarization functions are necessary for correct evaluation of the PSO term. The Fermi-contact term, on the other hand, has been shown to be very demanding of the basis set in some molecules. The most notorious example is hydrogen fluoride.23 This is because of the instability inherent in evaluating an approximate wave function at a single point. An error in any infinitesimal region may, in principle, have a vanishingly small effect on the variational energy and hence be poorly determined by the SCF procedure. The problem is compounded in the case of spinspin coupling constants by the occurrence of the δ function twicesonce for each nucleus. In order to examine a broad range of compounds in this study, we chose to employ a standard basis set of moderately high quality even though it is not converged for some difficult cases such as HF. We have made an all-electron modification of the standard ADF basis set “V” in which the cores are no longer frozen but have double-ζ flexibility, and the fit sets have also been adapted to allow for this additional flexibility. Valence orbitals are triple-ζ, and two sets of polarization functions are available on main group elements, one for each of the next two higher l-values. Hence, the basis set for hydrogen has 3 × 1s, 1 × 2p, 1 × 3d functions, carbon through fluorine have 2 × 1s, 3 × 2s, 3 × 2p, 1 × 3d, 1 × 4f, and silicon through chlorine have 2 × 1s, 2 × 2s, 2 × 2p, 3 × 3s, 3 × 3p, 1 × 3d, 1 × 4f. Basis sets for the calculations on transition-metal complexes have only one set of polarization functions: 2 × 1s, 2 × 2s, 2 × 2p, 2 × 3s, 2 × 3p, 3 × 3d, 3 × 4s, 1 × 4p. 3. Importance of Core Orbitals What is the role of the core electrons in the Fermi-contact term? This is an important question because all-electron calculations on transition-metal systems are extremely costly. On the other hand, the Fermi-contact mechanism involves spin polarization induced and detected through the amplitude of the orbitals at the nucleiswhich is largest for the core orbitals. We examined the role of the core orbitals in all-electron calculations in the following way. occ virt
〈| |〉
Kfc(A,B) ) ∑ ∑ i i
j
∂H ˆ fc ∂µ bB
j
∂Pij
∂µ bA
(13)
We construct the density matrix P in the basis of the unperturbed molecular orbitals, which can be classified as core, valence, or virtual (unoccupied). The first-order density matrix ∂P/∂µ bA will have nonvanishing entries that involve “excitations” between occupied and unoccupied MOs. These core-virtual and valence-virtual elements of P are the contributors to the FC coupling. Nonzero values are also found for core-core, corevalence, virtual-virtual elements, and so on, attributable to the
TABLE 3: Orbital Analysis of the Fermi-Contact Term of the Reduced Carbon-Oxygen Coupling Constanta molecule
CO
CO2
C 1s-virtual O 1s-virtual
-0.5 -15.7
-0.6 -40.1
total core-virtual valence-virtual
-16.2 -12.4
-40.7 -0.5
total
-28.6
-41.4
a
The carbon nucleus is the perturbing center in each case. Totals include small terms which are nonzero due to numerical error.
finite difference scheme and other numerical approximations, but these are invariably smallsless than 1% in total. At this point it becomes useful to distinguish between the two coupling nuclei. We define “perturbing” and “responding” nuclei thus: We take the derivative of P with respect to the nuclear magnetic moment of the perturbing nucleus and then multiply with the matrix elements of the Hamiltonian derivative with respect to the responding nucleus. In other words, the density matrix is perturbed by one nucleus, and the perturbation is detected at the responding nucleus. The coupling constants are formally symmetric with respect to the choice of perturbing and responding nuclei, but the individual orbital components are not. As a result of numerical errors in the individual terms, the total calculated Fermi-contact coupling is also slightly asymmetric, such that Kfc(A,B) is not precisely equal to Kfc(B,A). In Table 3 we show an analysis of the carbon-oxygen couplings in carbon monoxide and carbon dioxide. We first notice that the core-virtual contribution is not merely significant but actually dominant in both of these molecules. A further breakdown of the core-virtual contribution into carbon and oxygen core components reveals that the responding core is the important one, while the core about the perturbing nucleus makes only a small contributionsabout the same magnitude as the numerical error due to the finite difference scheme. We have also analyzed couplings in HF, SiH4, PH3, and H2S and observed that when H is the responding nucleus, corevirtual contributions are negligible. When the heavy atom is the responding nucleus, core-virtual contributions become meaningful. This suggests that the freedom of the core of the perturbing atom is irrelevant to the Fermi-contact coupling but that the core of the responding atom is frequently significant. Thus, couplings between transition metals and lighter atoms might be calculated efficiently by freezing the core electrons on the metal center and making it the perturbing nucleus while using all-electron bases on the responding light atom or atoms. 4. Results We have performed all-electron calculations of spin-spin coupling constants on a range of test systems, using the local spin-density approximation24 and the basis set described above. As mentioned above and described by Malkin et al.,7 the choice of the perturbing center makes a small difference in the calculated value of the Fermi-contact term. For instance, in this basis for hydrogen fluoride we find that choosing hydrogen as the perturbing nucleus leads to a Fermi-contact term of 16.38, while choosing fluorine leads to 16.06, a typical asymmetry of around 2%. Although we have not observed the improved results reported by Malkin et al., we follow their convention and take the lighter atom as the perturbing center in this section. Coupling constants for the 10-electron hydrides are shown in Table 4. The calculated geminal hydrogen-hydrogen couplings do not show the experimental trend across the row.
NMR Spin-Spin Coupling Constants from DFT
J. Phys. Chem., Vol. 100, No. 13, 1996 5289
TABLE 4: Reduced Coupling Constants in the 10-Electron Hydrides molecule
coupling
FC
DSO
CH4 NH3 H2O CH4 NH3 H2O HF
2K(H,H)
-0.55 -0.62 -1.03 39.6 50.6 41.4 16.4
-0.29 -0.43 -0.59 0.27 0.07 0.05 0.05
a
2K(H,H) 2K(H,H) 1K(C,H) 1K(N,H) 1K(O,H) 1
K(F,H)
PSO
total
exptla
0.31 0.50 0.72 0.8 2.9 8.0 18.1
-0.53 -0.69 -0.90 40.5 53.6 49.4 34.5
-1.03 -0.87 -0.60 41.3 50 48 46.9
Reference 9.
TABLE 5: Reduced Coupling Constants in Some Second-Row Hydrides molecule
coupling
FC
DSO
PSO
total
exptla
SiH4 PH3 H2S SiH4 PH3 H2S HCl
2K(H,H)
0.08 -0.77 -0.85 69.9 25.3 16.8 3.7
-0.20 -0.12 -0.15 0.0 0.0 0.0 0.0
0.11 0.11 0.18 -0.1 1.4 5.2 13.2
-0.01 -0.78 -0.82 69.8 26.7 22.0 16.9
0.23 -1.12
a
2K(H,H) 2
K(H,H)
1K(Si,H) 1K(P,H) 1K(S,H) 1
K(Cl,H)
84.9 37.8
TABLE 6: Reduced Coupling Constants in Some Light Hydrocarbons molecule CH4 C2H6 C2H4 C2H6 C2H6 C2H4 C2H4 C2H2 CH4 C2H6 C2H4 C2H2 C2H6 C2H4 C2H2 C2H6 C2H4 C2H2 a
(32
Reference 9.
The correct trend is reproduced in coupled Hartree-Fock (CHF) calculations, although the absolute magnitudes are very poor unless correlation is added.9 The approximate cancellation of the diamagnetic and paramagnetic spin-orbit terms also appears in the CHF calculations, leading to the conclusion that the Fermicontact term in these LDA calculations has the wrong behavior. The one-bond couplings between heavy atoms and hydrogen show (proportionally) much better agreement with experiment, with the exception of hydrogen fluoride. Results for hydrogen fluoride can be substantially improved by improving both the core and polarization flexibility of the basis set (H: 3 × 1s, 3 × 2p, 1 × 3d, F: 3 × 1s, 3 × 2p, 3 × 3d, 1 × 4f): The Fermicontact term then becomes 25.9, leading to a total coupling of 43.7, compared with an experimental value of 46.9. In the second-row hydrides shown in Table 5 the geminal proton-proton couplings seem to have the correct experimental trend, but the lack of experimental couplings for H2S makes this uncertain. The one-bond couplings to hydrogen are affected by a shortfall in the Fermi contact term of some 10-20 units, which increases monotonically across the row. This is probably due to basis set inadequacies for the Fermi-contact term, similar to those just described for hydrogen fluoride. In Table 6 we observe that the hydrogen-hydrogen couplings are also poor in the simple hydrocarbons, but the carbon-proton one-bond couplings are in quite good agreement with experiment. Two-bond carbon-proton couplings are somewhat worse, as might be expected due to their small magnitude. The carbon-carbon couplings in the series ethane-etheneethyne are qualitatively reproduced. The spin-dipolar term may be relevant to these couplings, particularly for C2H2.25 In Table 7, N2 and to a lesser degree CO are poor because the cancellation of different contributions is exceedingly sensitive to geometry.28 A similar geometry sensitivity may occur in CO2. The results for H2 are somewhat disappointing, but the overestimation of the Fermi-contact term here is probably due to the local spin-density approximation. The couplings to fluorine in CH3F are unsatisfactory; this is probably again due to an inadequate basis for fluorine. Neglect of the spin-dipolar mechanism may also be significant for the C-F coupling.29 The other couplings follow the trends just noted for other simple hydrocarbons: The proton-proton coupling is poor, but the carbon-proton one-bond coupling is just a few percent smaller than experiment.
coupling
FC
DSO
-0.55 -0.66 2K(H,H) 0.31 3K(H,H) trans 1.23 3K(H,H) gauche 0.22 3K(H,H) cis 0.57 3K(H,H) trans 1.07 3K(H,H) 0.15 1K(C,H) 39.6 1K(C,H) 40.2 1K(C,H) 47.6 1K(C,H) 79.5 2K(C,H) -0.6 2K(C,H) 1.3 2K(C,H) 14.3 1K(C,C) 39.6 1K(C,C) 103.5 1K(C,C) 262.2 2K(H,H)
2K(H,H)
PSO
-0.29 0.31 -0.24 0.26 -0.32 0.32 -0.26 0.24 -0.08 0.08 -0.09 0.06 -0.29 0.23 -0.30 0.36 0.1 0.8 0.2 0.6 0.1 0.3 0.1 -0.0 -0.1 0.1 -0.2 -0.4 -0.4 1.8 0.2 0.1 0.1 -13.3 0.0 7.5
total
exptla
-0.53 -0.64 0.31 1.21 0.22 0.54 1.01 0.21 40.5 41.0 48.1 79.6 -0.6 0.6 15.7 39.8 90.3 269.7
-1.03 0.21 (0.67) (0.67) 0.97 1.59 0.80 41.3 41.3 51.8 82.4 -1.5 -0.8 16.3 45.6 89.1 226.0
Reference 9.
TABLE 7: Reduced Coupling Constants in Some Other First-Row Compounds molecule coupling H2 N2 CO CO2 CH3F CH3F CH3F CH3F a
1K(H,H) 1K(N,N) 1K(C,O) 1K(C,O) 2
K(H,H) K(C,H) 1 K(C,F) 2K(F,H) 1
FC
DSO
28.0 39.0 -29.1 -41.4 -0.23 46.8 -104.7 1.88
PSO
exptla
total
-0.1 0.3 28.2 -0.3 -39.0 -0.3 -0.2 -37.2 -66.6 0.2 -11.7 -53.0 -0.25 0.24 -0.23 0.2 0.1 47.1 0.2 13.8 -90.9 -0.12 1.22 2.94
23.3 -20.b -40.1c -39.4c -0.80 49.4 -57.0 4.11
Reference 9. b Reference 26. c Reference 27.
TABLE 8: Reduced Coupling Constants in Three Transition-Metal Carbonyl Complexes molecule [V(CO)6]Fe(CO)5a Fe(CO)5 [Co(CO)4]-
coupling
FC
DSO
PSO
total
exptl
K(V,C) 1K(Fe,C ) ax 1K(Fe,C ) eq 1K(Co,C)
133 206 246 354
0.3 0.4 0.3 0.3
-6 -28 -9 -1
127 178 237 353
146b 239c 239 400 ( 20d
1
a Fe(CO) has a fluxional trigonal-bipyramidal structure, and the 5 couplings to the axial and equatorial carbonyls are experimentally indistinguishable. b Reference 30. c Reference 31. d Reference 32.
In general we note that any small coupling involving a balance of competing terms is liable to be poor for the usual reason that small errors in large numbers leads to a large error in their difference. There are additional problems with proton-proton couplings that seem to be associated with their small magnitude as well, but one-bond couplings involving heavy atoms are generally much larger and therefore quite reliable. In these cases there is a common tendency for the calculation to underestimate the coupling. 4.1. Transition Metal Carbonyls. We have carried out calculations on a series of three transition-metal complexes for which experimental couplings are known.30-32 One can see instantly from Table 8 that the calculated couplings are dominated by a single contribution (the Fermi-contact term), and so there is reason to believe a priori that the total couplings will reflect reality. This is confirmed by comparison with experiment. The reduced coupling constant increases going across the periodic row, and the calculated couplings are approximately 15% smaller than experiment. This suggests that further calculations of transition-metal coupling constants should be feasible. The remaining error could be due to the ap-
5290 J. Phys. Chem., Vol. 100, No. 13, 1996 proximate density functional, geometry, vibrational averaging, neglect of the spin-dipolar mechanism, or some combination of these. 5. Conclusions We have demonstrated that density functional calculations of NMR spin-spin coupling constants are feasible on transitionmetal compounds using Slater-type basis functions. Basis set requirements for the Fermi-contact term are stringent in some cases, as they are with Gaussian-type basis functions. Furthermore, contributions from core orbitals of the responding nucleus in the calculation of the Fermi-contact term can be significant, although use of the frozen core approximation on the perturbing nucleus may still be practical. With a standard, energy-optimized triple-ζ doubly-polarized basis set, we find that couplings between heavy atoms can be calculated with a typical error of less than 15%. Larger errors are observed (i) when the coupling constant has contributions of differing sign from the various mechanisms, (ii) when basis set requirements are more stringent, as for fluorine and chlorine, (iii) when the Fermi-contact term is less than about 1 × 1019 kg m-2 s-2 A-2, as in most proton-proton couplings, and (iv) when the spin-dipolar mechanism (neglected here) is significant. Errors arising through the Fermi-contact term might be reduced by optimization of the core basis-function exponents. The local spin-density functional used here has been superseded in some applications by gradient-corrected density functionals,33,34 but preliminary calculations have revealed no meaningful improvement in results with the potentials Vxc. We speculate that this is connected with the unphysical behavior of the common gradient-corrected potentials near the nucleus.35 We anticipate that further studies of the exchange-correlation potential, as opposed to the exchange-correlation energy density, will yield significant improvements for properties such as the nuclear spin-spin coupling constant. Acknowledgment. The authors thank NSERC of Canada for financial support and The University of Calgary for a postdoctoral fellowship for R.M.D. References and Notes (1) Ziegler, T. Chem. ReV. 1991, 91, 651. (2) Kaupp, M.; Malkin, V. G.; Malkina, O. L.; Salahub, D. R. J. Am. Chem. Soc. 1995, 117, 1851.
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