We discuss the effect of electron correlation on the unexpected differential sensitivity (UDS) in the 1J(C–H) coupling constant of CH4 using a decomposition into contributions from localized molecular orbitals and compare with the 1J(N–H) coupling constant in NH3. In particular, we discuss the well known fact that uncorrelated coupled Hartree-Fock (CHF) calculations are not able to reproduce the UDS in methane. For this purpose we have implemented for the first time a localized molecular orbital analysis for the second order polarization propagator approximation with coupled cluster singles and doubles amplitudes—SOPPA(CCSD) in the DALTON program. Comparing the changes in the localized orbital contributions at the correlated SOPPA and SOPPA(CCSD) levels and at the uncorrelated CHF level, we find that the latter overestimates the effect of stretching the bond between the coupled atoms on the contribution to the coupling from the localized bonding orbital between these atoms. This disturbs the subtle balance between the molecular orbital contributions, which lead to the UDS in methane.

Isotope effects play an important role in different fields of chemistry, for example, reaction kinetics of atmospheric reactions. In the case of isotope effects on NMR spin-spin coupling constants (SSCC) one distinguishes1 between a primary isotope effect,

$\Delta _{P} ^{n}J(A-B)$
ΔPnJ(AB)⁠, on a coupling nJ(A-B) between nuclei A and B, where atom B has two isotopes and a secondary isotope effect,
$\Delta _{S} ^{n}J(A-B)$
ΔSnJ(AB)
, where a neighbor nucleus C has two isotopes. They have been measured in several molecules (see, e.g., Refs. 1–13) and can be reproduced by including vibrational corrections to the calculated coupling constants (see, e.g., Refs. 1 and 14–21).

A particularly interesting case of an anomalous isotope effect is the one-bond carbon-hydrogen SSCC in methane, where experimentally the secondary isotope effect,

$\Delta _{S} ^{a}J({\rm C}{\rm -}{\rm H})$
ΔSaJ(CH) = −0.356 ± 0.01 Hz was found to be ∼5 times larger than the primary isotope effect,
$\Delta _{P} ^{1}J({\rm C} { \hbox{-}{\rm H}})$
ΔP1J(C-H)
= −0.067 ± 0.06 Hz6 which could quantitatively be reproduced with vibrational averaging calculations at the SOPPA(CCSD) level leading to
$\Delta _{S} ^{a}J({\rm C}{\rm -}{\rm H})$
ΔSaJ(CH)
= −0.397 Hz and
$\Delta _{P} ^{1}J({\rm C}{\rm -}{\rm H})$
ΔP1J(CH)
= −0.154 Hz.16 Furthermore, the variation of this carbon-hydrogen coupling on changing the length of the bond of carbon to one of the other hydrogens,
$\frac{\partial ^{1}J_{({\rm C}{\rm -}{\rm H})} }{\partial R_{other}}$
1J(CH)Rother
, was predicted to be ∼5 times as large as the variation with respect to the bond to the coupled hydrogen,
$\frac{\partial ^{1}J_{({\rm C}{\rm -}{\rm H})} }{\partial R_{own}}$
1J(CH)Rown
.14 The phenomenon was named unexpected differential sensitivity (UDS)22 and was later found also in other molecules.18,19,23,24 In all cases the affected coupling is a one-bond coupling over a single bond, which implies that it is dominated by the Fermi contact (FC) term. It is therefore not surprising that the UDS can be studied by only considering the FC term.22 

Based on the comparison of molecules with and without UDS it was proposed that the absence of lone pairs would be a prerequisite for the UDS.23,24 Later, through an analysis of the one-bond couplings in CH4, NH3, and H2O in terms of contributions from localized occupied molecular orbitals (LMO), carried out at the coupled perturbed density functional theory (DFT) level with the B3LYP functional, Provasi and Sauer25–27 could show that the lone pairs are at least not directly responsible for the lack of UDS in neither NH3 nor H2O, as removing the lone-pair contributions to the couplings in these two molecules did not provoke an UDS. On the other hand, it turned out to be a subtle balance in the sensitivity of the bond-core, bond-other bond, and other bond-other bond contributions to changes in the bond length between the coupled atoms, Rown, which is responsible for the appearance of the UDS in methane and its absence in ammonia or water.

A still unsettled problem, however, is the connection between the UDS and electron correlation. Already in the first computational study on the UDS in methane it was observed that inclusion of electron correlation at the level of the second order polarization propagation approximation (SOPPA) or higher is necessary for reproducing the phenomena22 and the following SOPPA(CCSD) calculations lead to quantitative agreement with the experimental values.16 Uncorrelated calculations at the CHF level, however, fail to show an UDS for methane, while DFT/B3LYP calculations reproduce the phenomena again.25–27 Hence the present work is devoted to an analysis of the effect of electron correlation on each individual localized occupied molecular contribution to the one-bond coupling constants in CH4 and NH3 at the CHF and SOPPA(CCSD) levels, which will shed light on the failure of the CHF calculations.

The different quantum chemical methods for calculating SSCCs at the non-relativistic level28,29 are often described in the literature30,31 and we will therefore only discuss here their analysis in LMOs. Common to all localized molecular orbital analysis is that the isotropic SSCC, J(XY), between two nuclei X and Y is expressed as a sum over contributions from orbitals. In the oldest approach CLOPPA,25–27,32,33 which is implemented at the semi-empirical as well as CHF and DFT levels, the coupling constant can be decomposed into contributions from two occupied (i and j) and two unoccupied (a and b) localized molecular orbitals, i.e.,

$J_{ij}^{ab}(X-Y)$
Jijab(XY)⁠. This, however, requires that one calculates all eigenvalues of the molecular Hessian, which becomes basically impossible for any correlated wavefunction method. Alternatively,34–37 one decomposes the SSCC in contributions from only one occupied and one unoccupied localized orbital,
$J_{i}^{a}(X-Y)$
Jia(XY)
, or even only one occupied localized orbital, Ji(XY). This requires only a small modification of the typical implementation of SSCCs,30,31,38,39 which involves the contraction of a first order density matrix or solution vector NX perturbed by the nuclear magnetic moment of nucleus X with the property gradient PY of the perturbation due to the nuclear magnetic moment of nucleus Y, i.e.,
$J(X-Y) = \sum _{ai} N^X_{ai} P^Y_{ai}$
J(XY)=aiNaiXPaiY
. The transformation to localized molecular orbitals can hereby be carried out either directly on the elements of the solution vector,
$N^X_{ai}$
NaiX
, and property gradient,
$P^Y_{ai}$
PaiY
, before they are contracted to
$J_{i}^{a}(X-Y)$
Jia(XY)
or alternatively already directly after the solution of the Hartree-Fock or Kohn-Sham equations, i.e., before the eventual calculation of any correlated wavefunction and the calculation of the solution vector and property gradient.

For the current project we have implemented in the DALTON program40,41 such a decomposition into contributions from one occupied and one virtual localized molecular orbital,

$J_{i}^{a}(X-Y)$
Jia(XY)⁠, of SSCC calculated at the level of the second order polarization propagator approximation with coupled cluster singles and doubles amplitudes—SOPPA(CCSD).39,42 Here and in the previous implementation36,37 for the second order polarization propagator approximation38,43,44 the transformation to localized molecular orbitals according to the Foster and Boys localization scheme45 is carried out before the calculation of MP2 correlation coefficients or CCSD amplitudes and the solution of the SOPPA equations.29 This deserves two comments, because the SOPPA equations are in general not invariant to a unitary transformation of the orbitals, i.e., SOPPA results in canonical and localized molecular orbitals will not be exactly identical. Second, the implementation of the correlation coefficients and the SOPPA equations in the DALTON program assumes a diagonal Fock matrix as in the case of canonical orbitals. The numerical consequences of these approximations are illustrated for CH4 and NH3 in Sec. III.

In order to determine how electron correlation provokes the UDS in methane we have calculated LMO contributions to the FC term for the one-bond coupling constants in CH4 and NH3 at the CHF, SOPPA, SOPPA(CCSD), and DFT/B3LYP46,47 levels of theory with the aug-cc-pVTZ-J basis set.39,48 Calculations were carried out at the equilibrium geometries, RCH = 1.08580 Å for CH4 and RNH = 1.01240 Å, ∠HNH = 106.67° for NH3, and at geometries where one of the bonds was changed by ±0.05 and ±0.1 Å. In the following we will in addition to the normal coupling constant, J(XY), also discuss the so-called reduced coupling constant

$K(X-Y) = \frac{4\pi ^{2} J(X-Y)}{h \gamma _{X} \gamma _{Y}}$
K(XY)=4π2J(XY)hγXγY⁠.

In Table I we present the results for the four non-relativistic contributions to the SSCCs of CH4 and NH3 calculated with canonical and localized orbitals in order to investigate the effect of using localized orbitals in the SOPPA and SOPPA(CCSD) calculations. First of all, we note the excellent agreement between the experimental values and our SOPPA(CCSD) results. Second, it can be seen that the largest difference between canonical and localized results is for the FC term of NH3 calculated at SOPPA level. But this deviation amounts only to 2%, which implies that the approximations involved in employing localized orbitals in SOPPA calculations are insignificant. Interestingly, the localized and canonical orbital results are closer to each other in the SOPPA(CCSD) calculations. It can also be seen that the FC term dominates indeed the 1J(X-H) coupling constants. We will therefore only analyze this term in terms of localized molecular orbitals in the following.

Table I.

Comparison of contributions to 1J(XH) for CH4 and NH3 in [Hz] calculated at the SOPPA and SOPPA(CCSD) levels with canonical or localized orbitals. The calculations were carried out at the equilibrium geometries using the aug-cc-pVTZ-J basis set.a

 SOPPASOPPA(CCSD)
 CH4NH3CH4NH3
 Can.Loc.Can.Loc.Can.Loc.Can.Loc.
DSO 0.25 0.25 −0.07 −0.07 0.25 0.25 −0.07 −0.07 
PSO 1.53 1.51 −2.87 −2.89 1.49 1.49 −2.83 −2.84 
SD −0.01 −0.02 −0.13 −0.12 0.01 0.00 −0.14 −0.13 
FC 125.43 125.50 −60.31 −59.16 120.88 121.02 −58.69 −58.19 
Total 127.20 127.25 −63.38 −62.23 122.63 122.76 −61.74 −61.22 
 SOPPASOPPA(CCSD)
 CH4NH3CH4NH3
 Can.Loc.Can.Loc.Can.Loc.Can.Loc.
DSO 0.25 0.25 −0.07 −0.07 0.25 0.25 −0.07 −0.07 
PSO 1.53 1.51 −2.87 −2.89 1.49 1.49 −2.83 −2.84 
SD −0.01 −0.02 −0.13 −0.12 0.01 0.00 −0.14 −0.13 
FC 125.43 125.50 −60.31 −59.16 120.88 121.02 −58.69 −58.19 
Total 127.20 127.25 −63.38 −62.23 122.63 122.76 −61.74 −61.22 
a

Experimental values for CH4 120.9 Hz22 and NH3−61.45±0.03 Hz.4 

The LMO decomposition of the FC term to the reduced coupling constants K(X-H) in CH4 and NH3 calculated at the CHF, SOPPA, SOPPA(CCSD), and DFT/B3LYP levels with the aug-cc-pVTZ-J basis set are presented in Tables II and III, respectively. In addition to the decomposition at the equilibrium geometry also the changes in each LMO contribution on extending or contracting the bond between the central atom X and the first of the hydrogen atoms, H1, are shown. The UDS effect in CH4 is clearly seen in Table II. The three methods including electron correlation predict that the coupling to the other, non-coupled hydrogen, K(C–H2), varies more than the coupling to the coupled hydrogen, K(C–H1), by about 1.1 × 1019  T2 J−1 for SOPPA, 1.3 × 1019  T2 J−1 for SOPPA(CCSD) and 1.2 × 1019  T2 J−1 for B3LYP on contracting the bond and by 1.2 × 1019  T2 J−1 for SOPPA, 1.6 × 1019  T2 J−1 for SOPPA(CCSD) and 1.4 × 1019  T2 J−1 for B3LYP on extending the bond length in good agreement with the earlier results.16,25–27 In the CHF calculations the change in K(C–H2) is about 40% larger than at the correlated levels, but only about 53% (on contracting) or 68% (on extending) of the change in K(C–H1). In the case of NH3 in Table III, on the other hand, all methods agree on the fact that K(N–H1) changes more than K(N–H2) but here CHF predicts a smaller change in K(N–H1) than all the correlated methods. Furthermore, from Table II one can see that the largest change on extending or contracting the C–H1 bond happens for the contribution from the bonding orbital between these atoms, σ(C−H1), and not for any of the other bond contributions, σ(C−H2/3/4), implying that at the level of the individual localized orbital contributions there is no unexpected sensitivity of the individual orbital contributions, i.e., orbital UDS, as previously pointed out.25–27 

Table II.

Localized CHF, SOPPA, SOPPA(CCSD), and B3LYP orbital contributions to the FC term of K(C–H) of methane at equilibrium geometry, in [1019  T2 J−1], and their differences (ΔK = KfinalKequilibrium) due to a change in the bond lengths by ±0.1 Å.

ΔRCH1 CHFSOPPASOPPA(CCSD)DFT/B3LYP
in ÅContrib.K(C–H1)K(C–H2)K(C–H1)K(C–H2)K(C–H1)K(C–H2)K(C–H1)K(C–H2)
−0.1 Δcore(C) 0.47 −0.53 0.86 −0.42 0.87 −0.42 0.94 −0.40 
  Δσ(C−H1−) −6.48 −0.34 −3.68 −0.15 −3.27 −0.14 −4.00 −0.13 
  Δσ(C−H2) 0.65 −2.33 0.62 −1.86 0.57 −1.79 0.75 −1.95 
  Δσ(C−H3) 0.65 0.21 0.62 0.19 0.57 0.17 0.75 0.22 
  Δσ(C−H4) 0.65 0.21 0.62 0.19 0.57 0.17 0.75 0.22 
  Δ Total FC −4.07 −2.78 −0.97 −2.07 −0.67 −2.00 −0.84 −2.07 
0.0 core(C) 7.59 7.59 6.17 6.17 6.08 6.08 6.32 6.32 
  σ(C−H1eq.) 48.30 −1.42 40.52 −1.72 38.94 −1.65 43.36 −2.05 
  σ(C−H2) −1.42 48.30 −1.72 40.52 −1.65 38.94 −2.05 43.36 
  σ(C−H3) −1.42 −1.42 −1.72 −1.72 −1.65 −1.65 −2.05 −2.05 
  σ(C−H4) −1.42 −1.42 −1.72 −1.72 −1.65 −1.65 −2.05 −2.05 
  Total FC 51.63 51.63 41.54 41.54 40.06 40.06 43.55 43.55 
+ 0.1 Δcore(C) −0.45 0.49 −0.93 0.38 −0.94 0.38 −1.01 0.36 
  Δσ(C−H1 + ) 7.71 0.39 3.48 0.16 2.88 0.13 3.83 0.13 
  Δσ(C−H2) −0.68 2.39 −0.59 1.85 −0.55 1.77 −0.73 1.95 
  Δσ(C−H3) −0.68 −0.24 −0.59 −0.20 −0.55 −0.20 −0.73 −0.23 
  Δσ(C−H4) −0.68 −0.24 −0.59 −0.20 −0.55 −0.20 −0.73 −0.23 
  Δ Total FC 5.23 2.79 0.76 1.98 0.29 1.90 0.61 1.96 
ΔRCH1 CHFSOPPASOPPA(CCSD)DFT/B3LYP
in ÅContrib.K(C–H1)K(C–H2)K(C–H1)K(C–H2)K(C–H1)K(C–H2)K(C–H1)K(C–H2)
−0.1 Δcore(C) 0.47 −0.53 0.86 −0.42 0.87 −0.42 0.94 −0.40 
  Δσ(C−H1−) −6.48 −0.34 −3.68 −0.15 −3.27 −0.14 −4.00 −0.13 
  Δσ(C−H2) 0.65 −2.33 0.62 −1.86 0.57 −1.79 0.75 −1.95 
  Δσ(C−H3) 0.65 0.21 0.62 0.19 0.57 0.17 0.75 0.22 
  Δσ(C−H4) 0.65 0.21 0.62 0.19 0.57 0.17 0.75 0.22 
  Δ Total FC −4.07 −2.78 −0.97 −2.07 −0.67 −2.00 −0.84 −2.07 
0.0 core(C) 7.59 7.59 6.17 6.17 6.08 6.08 6.32 6.32 
  σ(C−H1eq.) 48.30 −1.42 40.52 −1.72 38.94 −1.65 43.36 −2.05 
  σ(C−H2) −1.42 48.30 −1.72 40.52 −1.65 38.94 −2.05 43.36 
  σ(C−H3) −1.42 −1.42 −1.72 −1.72 −1.65 −1.65 −2.05 −2.05 
  σ(C−H4) −1.42 −1.42 −1.72 −1.72 −1.65 −1.65 −2.05 −2.05 
  Total FC 51.63 51.63 41.54 41.54 40.06 40.06 43.55 43.55 
+ 0.1 Δcore(C) −0.45 0.49 −0.93 0.38 −0.94 0.38 −1.01 0.36 
  Δσ(C−H1 + ) 7.71 0.39 3.48 0.16 2.88 0.13 3.83 0.13 
  Δσ(C−H2) −0.68 2.39 −0.59 1.85 −0.55 1.77 −0.73 1.95 
  Δσ(C−H3) −0.68 −0.24 −0.59 −0.20 −0.55 −0.20 −0.73 −0.23 
  Δσ(C−H4) −0.68 −0.24 −0.59 −0.20 −0.55 −0.20 −0.73 −0.23 
  Δ Total FC 5.23 2.79 0.76 1.98 0.29 1.90 0.61 1.96 
Table III.

Localized CHF, SOPPA, SOPPA(CCSD), and B3LYP orbital contributions to the FC term of K(N–H) of ammonia at equilibrium geometry, in [1019  T2 J−1], and their differences (ΔK = KfinalKequilibrium) due to a change in the bond lengths by ±0.1 Å.

ΔRNH1 CHFSOPPASOPPA(CCSD)DFT/B3LYP
in ÅContrib.K(N–H1)K(N–H2)K(N–H1)K(N–H2)K(N–H1)K(N–H2)K(N–H1)K(N–H2)
−0.1 Δcore(N) 2.94 −0.18 3.09 −0.18 3.02 −0.20 3.27 −0.16 
  Δσ(N−H1−) −5.11 −0.64 −0.89 −0.34 −0.56 −0.31 −1.04 −0.32 
  Δσ(N−H2) 0.87 −2.15 0.84 −1.71 0.81 −1.70 1.07 −1.82 
  Δσ(N−H3) 0.87 0.25 0.84 0.25 0.81 0.25 1.07 0.31 
  ΔLP(N) 2.87 1.02 2.36 0.80 2.21 0.76 2.89 0.96 
  Δ Total FC 2.45 −1.69 6.25 −1.17 6.29 −1.19 7.26 −1.01 
0.0 core(N) 5.85 5.85 4.66 4.66 4.74 4.74 4.75 4.75 
  σ(N−H1eq) 69.26 −3.00 57.07 −3.45 55.65 −3.37 61.06 −4.20 
  σ(N−H2) −3.00 69.26 −3.45 57.07 −3.37 55.65 −4.20 61.06 
  σ(N−H3) −3.00 −3.00 −3.45 −3.45 −3.37 −3.37 −4.20 −4.20 
  LP(N) −7.24 −7.24 −6.25 −6.25 −5.88 −5.88 −7.31 −7.31 
  Total FC 61.87 61.87 48.57 48.57 47.77 47.77 50.09 50.09 
+0.1 Δcore(N) −3.50 0.00 −3.29 0.05 −3.18 0.06 −3.49 0.02 
  Δσ(N−H1 + ) 7.17 0.80 0.51 −0.26 −0.01 0.34 0.64 −0.31 
  Δσ(N−H2) −0.66 1.90 −0.60 1.47 −0.55 1.44 −0.78 1.53 
  Δσ(N−H3) −0.66 −0.24 −0.60 0.36 −0.55 −0.24 −0.78 0.35 
  ΔLP(N) −4.30 −1.26 −3.01 −0.96 −2.73 −0.90 −3.77 −1.16 
  Δ Total FC −1.95 1.20 −6.99 0.68 −7.03 0.71 −8.17 0.43 
ΔRNH1 CHFSOPPASOPPA(CCSD)DFT/B3LYP
in ÅContrib.K(N–H1)K(N–H2)K(N–H1)K(N–H2)K(N–H1)K(N–H2)K(N–H1)K(N–H2)
−0.1 Δcore(N) 2.94 −0.18 3.09 −0.18 3.02 −0.20 3.27 −0.16 
  Δσ(N−H1−) −5.11 −0.64 −0.89 −0.34 −0.56 −0.31 −1.04 −0.32 
  Δσ(N−H2) 0.87 −2.15 0.84 −1.71 0.81 −1.70 1.07 −1.82 
  Δσ(N−H3) 0.87 0.25 0.84 0.25 0.81 0.25 1.07 0.31 
  ΔLP(N) 2.87 1.02 2.36 0.80 2.21 0.76 2.89 0.96 
  Δ Total FC 2.45 −1.69 6.25 −1.17 6.29 −1.19 7.26 −1.01 
0.0 core(N) 5.85 5.85 4.66 4.66 4.74 4.74 4.75 4.75 
  σ(N−H1eq) 69.26 −3.00 57.07 −3.45 55.65 −3.37 61.06 −4.20 
  σ(N−H2) −3.00 69.26 −3.45 57.07 −3.37 55.65 −4.20 61.06 
  σ(N−H3) −3.00 −3.00 −3.45 −3.45 −3.37 −3.37 −4.20 −4.20 
  LP(N) −7.24 −7.24 −6.25 −6.25 −5.88 −5.88 −7.31 −7.31 
  Total FC 61.87 61.87 48.57 48.57 47.77 47.77 50.09 50.09 
+0.1 Δcore(N) −3.50 0.00 −3.29 0.05 −3.18 0.06 −3.49 0.02 
  Δσ(N−H1 + ) 7.17 0.80 0.51 −0.26 −0.01 0.34 0.64 −0.31 
  Δσ(N−H2) −0.66 1.90 −0.60 1.47 −0.55 1.44 −0.78 1.53 
  Δσ(N−H3) −0.66 −0.24 −0.60 0.36 −0.55 −0.24 −0.78 0.35 
  ΔLP(N) −4.30 −1.26 −3.01 −0.96 −2.73 −0.90 −3.77 −1.16 
  Δ Total FC −1.95 1.20 −6.99 0.68 −7.03 0.71 −8.17 0.43 

Comparing the LMO contributions in the three correlated calculations for both CH4 and NH3, one can see that with a few exceptions DFT/B3LYP gives the largest (in absolute values) LMO contributions and changes in these contributions followed by SOPPA while SOPPA(CCSD) typically gives the smallest LMO contributions.

We now turn to the question how electron correlation affects the orbital contributions and in particular the changes in these contributions on changing RCH1, i.e., the differences between the changes in the LMO contributions obtained at the CHF level and at the DFT/B3LYP, SOPPA or SOPPA(CCSD) level. In Figures 1 and 2 the differences between the CHF changes and SOPPA(CCSD) changes in the LMO contributions to

$\smash{{}^{1}K^{FC}_{X-H}}$
KXHFC1 are shown.

FIG. 1.

CH4: Variation of the difference in the CHF and SOPPA(CCSD) orbital contributions to

$^{1}K^{FC}_{X-H}$
KXHFC1 with the length of the coupled bond Rown and in the inset (up): the other bond Rother.

FIG. 1.

CH4: Variation of the difference in the CHF and SOPPA(CCSD) orbital contributions to

$^{1}K^{FC}_{X-H}$
KXHFC1 with the length of the coupled bond Rown and in the inset (up): the other bond Rother.

Close modal
FIG. 2.

NH3: Variation of the difference in the CHF and SOPPA(CCSD) orbital contributions to

$^{1}K^{FC}_{X-H}$
KXHFC1 with the length of the coupled bond Rown and in the inset (up): the other bond Rother.

FIG. 2.

NH3: Variation of the difference in the CHF and SOPPA(CCSD) orbital contributions to

$^{1}K^{FC}_{X-H}$
KXHFC1 with the length of the coupled bond Rown and in the inset (up): the other bond Rother.

Close modal

Among the contributions to K(C–H1), σ(C−H1 ± ) is not only the largest but also the one which is most affected by electron correlation followed by core(C), for which the correlation effects are similar though an order of magnitude smaller. The uncorrelated CHF calculations overestimate the negative/positive change in σ(C−H1±) on contraction/extension of the RCH1 bond by 76%/121% compared to SOPPA(CCSD), while the much smaller changes in the core contribution core(C) have the opposite sign and are underestimate at the CHF level.

Thus, when passing from SOPPA(CCSD) to CHF the absolute value of σ(C−H1±) increases by 3.21 × 1019  T2 J−1 on contraction and by 4.83 × 1019  T2 J−1 on extension of the bond to H1. The opposite sign core(C) contribution decreases in absolute value by 0.4 × 1019  T2 J−1 on contraction and by 0.49 × 1019  T2J−1 on extension which further enhances the overestimation of the changes in K(C–H1) on changing its own bond. Also the changes in the bonding orbital contribution on changing the other bond, as seen from the σ(C−H2) contribution to K(C–H2), are overestimated but only by about 30% or 0.54 or 0.62 × 1019  T2 J−1. The disappearance of the UDS in CH4 at the CHF level is thus solely due to the overestimation of the effect of changing the bond length on the contribution from the associated bonding orbital.

For NH3, Table III and Figure 2, the changes in the σ(N−H±) contributions are also most strongly affected by electron correlation as seen for CH4, and CHF overestimates the changes due to changing the corresponding bond length even more than in CH4. In addition, the changes in the LP(N) contribution are overestimated at the CHF level, but only by 30% or 57%. Overall due to the opposite signs of the changes in σ(N−H±) on one side and in LP(N) and core(N) on the other, the changes in the total FC term of K(N–H1) are smaller than in the correlated calculations quite contrary to CH4. Looking finally at the changes in K(N–H) on changing the other bond, as seen from the σ(N−H2) contribution to K(N–H2), we observe again that the CHF calculations overestimate this LMO contribution and therefore the change in the total FC term. Consequently due the subtle balance between the different LMO contributions, the dependence on changes in the own bond is reduced in the uncorrelated calculations in NH3 but still larger than the slightly increase dependence on the other bond.

In conclusion, we summarize that in both molecules the dominating effect is that the uncorrelated CHF calculations overestimate the effect of changing the bond length between two coupled atoms on the contribution of their localized bonding orbital σ(XH±) to the coupling constant between these atoms. However, this is not the only LMO contribution and summing all leads to the disappearance of the UDS in CH4 in CHF calculations, while in NH3 one gets a bit closer to an UDS in the CHF calculations.

Previous studies of 1J(XH) in terms of LMO contributions in saturated compounds at their equilibrium geometries have also shown that the main contribution to the coupling is given by the bond orbital, which links the coupled nuclei, and that such a contribution is the most affected by electron correlation.36,37 Recalling the well-known fact that the restricted Hartree-Fock method poorly describes dissociation of bonds and gives generally too large vibrational frequencies, i.e., too large curvature of the potential energy surface, it is probably reasonable to conclude that CHF calculations also overestimate the changes in the contributions from the bonding orbital to any molecular property on extending or contracting the associated bond. What makes the situation more complicated for indirect nuclear SSCCs is the fact that the contributions from the different LMOs have often different signs and that the total value of the coupling constants are the result of a subtle balance between these LMO contributions—a situation whose details only a LMO analysis as presented in this study can unravel.

M.N.C.Z. and P.F.P. acknowledge financial support from CONICET and UNNE (PI:F002-11 Res. 582/11). S.P.A.S. acknowledges financial support from DCSC and the Lundbeck foundation.

1.
C. J.
Jameson
and
H.-J.
Osten
, “
Isotope effects on spin-spin coupling
,”
J. Am. Chem. Soc.
108
,
2497
2503
(
1986
).
2.
Y. N.
Luzikov
and
N. M.
Sergeyev
, “
Deuterium isotope effects on the 13C-H coupling constants in acetylene
,”
J. Magn. Reson.
60
,
177
183
(
1984
).
3.
R. E.
Wasylishen
and
J. O.
Friedrich
, “
Deuterium isotope effects on the nitrogen chemical shift and 1J(N,H) in the ammonium ion
,”
J. Chem. Phys.
80
,
585
587
(
1984
).
4.
R. E.
Wasylishen
and
J. O.
Friedrich
, “
Deuterium isotope effects on nuclear shielding constants and spin-spin coupling constants in the ammonium ion, ammonia, and water
,”
Can. J. Chem.
65
,
2238
2243
(
1987
).
5.
E.
Kupce
,
E.
Lukevics
,
Y. M.
Varezhkin
,
A. N.
Mikhailova
, and
V. D.
Sheludyakov
, “
29Si-15N spin-spin coupling constants in silazanes
,”
Organomet.
7
,
1649
1652
(
1988
).
6.
B.
Bennett
,
W. T.
Raynes
, and
C. W.
Anderson
, “
Temperature dependences of J(C,H) and J(C,D) in 13CH4 and some of its deuterated isotopomers
,”
Spectrochim. Acta A
45
,
821
827
(
1989
).
7.
E.
Liepinš
,
V.
Gevorgyan
, and
E.
Lukevics
, “
Deuterium isotope effects in the 1H, 13C, and 29Si NMR spectra of phenylsilanes
,”
J. Magn. Reson.
85
,
170
172
(
1989
).
8.
Y. A. A.
Strelenko
,
V. N.
Torocheshnikov
, and
N. M.
Sergeyev
, “
Isotope effects in the NMR spectra of nitromethane
,”
J. Magn. Reson.
89
,
123
128
(
1990
).
9.
Y. A.
Strelenko
and
N. M.
Sergeyev
, “
Isotope effect due to 15N/14N substitution on 14N-14N coupling constants in nitrous oxide
,”
J. Mol. Struct.
378
,
61
65
(
1996
).
10.
N. M.
Sergeyev
,
N. D.
Sergeyeva
,
Y. A.
Strelenko
, and
W. T.
Raynes
, “
The 1H-2H, 17O-1H coupling constants and the 16O/18O induced proton isotope shift in water
,”
Chem. Phys. Lett.
277
,
142
146
(
1997
).
11.
N. D.
Sergeyeva
,
N. M.
Sergeyev
, and
W. T.
Raynes
, “
Deuterium-induced primary and secondary isotope effects on 13C,H coupling constants in halomethanes
,”
Magn. Reson. Chem.
36
,
255
260
(
1998
).
12.
N. M.
Sergeyev
,
N. D.
Sergeyeva
, and
W. T.
Raynes
, “
Isotope effects on the 17O, 1H coupling constant and the 17O-1H nuclear overhauser effect in water
,”
J. Megn. Reson.
137
,
311
315
(
1999
).
13.
W.
Schilf
,
J. P.
Bloxsidge
,
J. R.
Jones
, and
S.-Y.
Lu
, “
Investigations of intramolecular hydrogen bonding in three types of Schiff bases by 2H and 3H NMR isotope effects
,”
Magn. Reson. Chem.
42
,
556
560
(
2004
).
14.
W. T.
Raynes
, “
Theory of vibrational effects on properties of methane and its isotopomers
,”
Mol. Phys.
63
,
719
729
(
1988
).
15.
J.
Geertsen
,
J.
Oddershede
,
W. T.
Raynes
, and
G. E.
Scuseria
, “
Nuclear spin-spin coupling in the methane isotopomers
,”
J. Magn. Reson.
93
,
458
471
(
1991
).
16.
R. D.
Wigglesworth
,
W. T.
Raynes
,
S. P. A.
Sauer
, and
J.
Oddershede
, “
The calculation and analysis of isotope effects on the nuclear spin-spin coupling constants of methane at various temperatures
,”
Mol. Phys.
92
,
77
88
(
1997
).
17.
R. D.
Wigglesworth
,
W. T.
Raynes
,
S. P. A.
Sauer
, and
J.
Oddershede
, “
Calculated spin-spin coupling surfaces in the water molecule; prediction and analysis of J(O,H), J(O,D) and J(H,D) in water isotopomeres
,”
Mol. Phys.
94
,
851
862
(
1998
).
18.
R. D.
Wigglesworth
,
W. T.
Raynes
,
S.
Kirpekar
,
J.
Oddershede
, and
S. P. A.
Sauer
, “
Nuclear spin-spin coupling in the acetylene isotopomers calculated from ab initio correlated surfaces for 1J(C,H), 1J(C,C), 2J(C,H), and 3J(H,H)
,”
J. Chem. Phys.
112
,
3735
3746
(
2000
).
19.
R. D.
Wigglesworth
,
W. T.
Raynes
,
S.
Kirpekar
,
J.
Oddershede
, and
S. P. A.
Sauer
, “
Erratum: Nuclear spin-spin coupling in the acetylene isotopomers calculated from ab initio correlated surfaces for 1J(C,H), 1J(C,C), 2J(C,H), and 3J(H,H)
,”
J. Chem. Phys.
114
,
9192
(
2001
).
20.
S. P. A.
Sauer
,
W. T.
Raynes
, and
R. A.
Nicholls
, “
Nuclear spin-spin coupling in silane and its isotopomers: Ab initio calculation and experimental investigation
,”
J. Chem. Phys.
115
,
5994
6006
(
2001
).
21.
A.
Yachmenev
,
S. N.
Yurchenko
,
I.
Paidarová
,
P.
Jensen
,
W.
Thiel
, and
S. P. A.
Sauer
, “
Thermal averaging of the indirect nuclear spin-spin coupling constants of ammonia: The importance of the large amplitude inversion mode
,”
J. Chem. Phys.
132
,
114305
(
2010
).
22.
W. T.
Raynes
,
J.
Geertsen
, and
J.
Oddershede
, “
Unexpected differential sensitivity of nuclear spin-spin coupling constants to bond stretching in methane
,”
Chem. Phys. Lett.
197
,
516
524
(
1992
).
23.
S. P. A.
Sauer
and
W. T.
Raynes
, “
Unexpected differential sensitivity of nuclear spin-spin coupling constants to bond stretching in
${\rm BH}_4^-$
BH 4
,
${\rm NH}_4^+$
NH 4+
and SiH4
,”
J. Chem. Phys.
113
,
3121
3129
(
2000
).
24.
S. P. A.
Sauer
and
W. T.
Raynes
, “
Erratum: Unexpected differential sensitivity of nuclear spin-spin coupling constants to bond stretching in
${\rm BH}_4^-$
BH 4
,
${\rm NH}_4^+$
NH 4+
and SiH4
,”
J. Chem. Phys.
114
,
9193
(
2001
).
25.
S. P. A.
Sauer
and
P. F.
Provasi
, “
The anomalous deuterium isotope effect in the NMR spectrum of methane: An analysis in localized molecular orbitals
,”
ChemPhysChem
9
,
1259
1261
(
2008
).
26.
P. F.
Provasi
and
S. P. A.
Sauer
, “
Analysis of isotope effects in NMR one-bond indirect nuclear spin-spin coupling constants in terms of localized molecular orbitals
,”
Phys. Chem. Chem. Phys.
11
,
3987
3995
(
2009
).
27.
P. F.
Provasi
and
S. P. A.
Sauer
, “
Amendment: Analysis of isotope effects in NMR one-bond indirect nuclear spin-spin coupling constants in terms of localized molecular orbitals
,”
Phys. Chem. Chem. Phys.
12
,
15132
(
2010
).
28.
N. F.
Ramsey
, “
Electron coupled interactions between nuclear spins in molecules
,”
Phys. Rev.
91
,
303
307
(
1953
).
29.
S. P. A.
Sauer
,
Molecular Electromagnetism: A Computational Chemistry Approach
(
Oxford University Press
,
Oxford
,
2011
).
30.
T.
Helgaker
,
M.
Jaszuński
, and
K.
Ruud
, “
Ab initio methods for the calculation of NMR shielding and indirect spin-spin coupling constants
,”
Chem. Rev.
99
,
293
352
(
1999
).
31.
T.
Helgaker
,
S.
Coriani
,
P.
Jørgensen
,
K.
Kristensen
,
J.
Olsen
, and
K.
Ruud
, “
Recent advances in wave function-based methods of molecular-property calculations
,”
Chem. Rev.
112
,
543
631
(
2012
).
32.
A. C.
Diz
,
C. G.
Giribet
,
M. C. C. C. Ruiz
de Azúa
, and
R. H.
Contreras
, “
The use of localized molecular orbitals and the polarization propagator to identify transmission mechanisms in nuclear spin-spin couplings
,”
Int. J. Quantum Chem.
37
,
663
677
(
1990
).
33.
M. C. Ruiz
de Azúa
,
C. G.
Giribet
,
C. V.
Vizioli
, and
R. H.
Contreras
, “
Ab initio IPPP-CLOPPA approach to perform bond contribution analysis of NMR coupling constants: 1J(NH) in NH3 as a function of pyramidality
,”
J. Mol. Struct.: THEOCHEM
433
,
141
150
(
1998
).
34.
J. E.
Peralta
,
R. H.
Contreras
, and
J. P.
Snyder
, “
Natural bond orbital dissection of fluorine-fluorine through-space NMR coupling (JF,F) in polycyclic organic molecules
,”
Chem. Commun.
2000
,
2025
2026
.
35.
J.
Autschbach
, “
Analyzing molecular properties calculated with two-component relativistic methods using spin-free natural bond orbitals: NMR spin-spin coupling constants
,”
J. Chem. Phys.
127
,
124106
(
2007
).
36.
M. N. C.
Zarycz
and
G. A.
Aucar
, “
Analysis of electron correlation effects and contributions of NMR J-couplings from occupied localized molecular orbitals
,”
J. Phys. Chem. A
116
,
1272
1282
(
2012
).
37.
M. N. C.
Zarycz
and
G. A.
Aucar
, “
The analysis of NMR J-couplings of saturated and unsaturated compounds by the localized second order polarization propagator approach method
,”
J. Chem. Phys.
136
,
174115
(
2012
).
38.
M. J.
Packer
,
E. K.
Dalskov
,
T.
Enevoldsen
,
H. J. A.
Jensen
, and
J.
Oddershede
, “
A new implementation of the second order polarization propagator approximation (SOPPA): The excitation spectra of benzene and naphthalene
,”
J. Chem. Phys.
105
,
5886
5900
(
1996
).
39.
T.
Enevoldsen
,
J.
Oddershede
, and
S. P. A.
Sauer
, “
Correlated calculations of indirect nuclear spin-spin coupling constants using second order polarization propagator approximations: SOPPA and SOPPA(CCSD)
,”
Theor. Chem. Acc.
100
,
275
284
(
1998
).
40.
K.
Aidas
,
C.
Angeli
,
K. L.
Bak
,
V.
Bakken
,
L.
Boman
,
O.
Christiansen
,
R.
Cimiraglia
,
S.
Coriani
,
P.
Dahle
,
E. K.
Dalskov
,
U.
Ekstrøm
,
T.
Enevoldsen
,
J. J.
Eriksen
,
P.
Ettenhuber
,
B.
Fernandez
,
L.
Ferrighi
,
H.
Fliegl
,
L.
Frediani
,
K.
Hald
,
A.
Halkier
,
C.
Hättig
,
H.
Heiberg
,
T.
Helgaker
,
A. C.
Hennum
,
H.
Hettema
,
S.
Høst
,
I.-M.
Høyvik
,
M. F.
Iozzi
,
B.
Jansik
,
H. J. A.
Jensen
,
D.
Jonsson
,
P.
Jørgensen
,
J.
Kauczor
,
S.
Kirpekar
,
T.
Kjærgaard
,
W.
Klopper
,
S.
Knecht
,
R.
Kobayashi
,
H.
Koch
,
J.
Kongsted
,
A.
Krapp
,
K.
Kristensen
,
A.
Ligabue
,
O. B.
Lutnæs
,
J. I.
Melo
,
K. V.
Mikkelsen
,
R. H.
Myhre
,
C.
Neiss
,
C. B.
Nielsen
,
P.
Norman
,
J.
Olsen
,
J. M. H.
Olsen
,
A.
Osted
,
M. J.
Packer
,
F.
Pawlowski
,
T. B.
Pedersen
,
P. F.
Provasi
,
S.
Reine
,
Z.
Rinkevicius
,
T. A.
Ruden
,
K.
Ruud
,
V.
Rybkin
,
P.
Salek
,
C. C. M.
Samson
,
A. Sánchez
de Merás
,
T.
Saue
,
S. P. A.
Sauer
,
B.
Schimmelpfennig
,
K.
Sneskov
,
A. H.
Steindal
,
K. O.
Sylvester-Hvid
,
P. R.
Taylor
,
A. M.
Teale
,
E. I.
Tellgren
,
D. P.
Tew
,
A. J.
Thorvaldsen
,
L.
Thøgersen
,
O.
Vahtras
,
M.
Watson
,
D. J.
Wilson
,
M.
Ziolkowski
, and
H.
Ågren
, “
The DALTON quantum chemistry program system
,”
WIREs Comput. Mol. Sci.
4
,
269
284
(
2014
).
41.
Dalton, a molecular electronic structure program, Release DALTON2013.0 (2013), see http://daltonprogram.org/,
2013
.
42.
S. P. A.
Sauer
, “
Second order polarization propagator approximation with coupled cluster singles and doubles amplitudes - SOPPA(CCSD): The polarizability and hyperpolarizability of Li
,”
J. Phys. B: At., Mol. Opt. Phys.
30
,
3773
3780
(
1997
).
43.
E. S.
Nielsen
,
P.
Jørgensen
, and
J.
Oddershede
, “
Transition moments and dynamic polarizabilities in a second order polarization propagator approach
,”
J. Chem. Phys.
73
,
6238
6246
(
1980
).
44.
K. L.
Bak
,
H.
Koch
,
J.
Oddershede
,
O.
Christiansen
, and
S. P. A.
Sauer
, “
Atomic integral driven second order polarization propagator calculations of the excitation spectra of naphthalene and anthracene
,”
J. Chem. Phys.
112
,
4173
4185
(
2000
).
45.
J. M.
Foster
and
S. F.
Boys
, “
Canonical configuration interaction procedure
,”
Rev. Mod. Phys.
32
,
300
302
(
1960
).
46.
A. D.
Becke
, “
Density-functional thermochemistry. III. The role of exact exchange
,”
J. Chem. Phys.
98
,
5648
5652
(
1993
).
47.
C.
Lee
,
W.
Yang
, and
R. G.
Parr
, “
Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density
,”
Phys. Rev. B
37
,
785
789
(
1988
).
48.
P. F.
Provasi
,
G. A.
Aucar
, and
S. P. A.
Sauer
, “
The effect of lone pairs and electronegativity on the indirect nuclear spin-spin coupling constants in CH2X (X = CH2, NH, O, S): Ab initio calculations using optimized contracted basis sets
,”
J. Chem. Phys.
115
,
1324
1334
(
2001
).