Importance of Triples Contributions to NMR SpinâSpin Coupling

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The Importance of Triples Contributions to NMR Spin-spin Coupling Constants Computed at the CC3 and CCSDT Levels Rasmus Faber, Stephan P. A. Sauer, and Jürgen Gauss J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b01003 • Publication Date (Web): 19 Dec 2016 Downloaded from http://pubs.acs.org on December 24, 2016

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Journal of Chemical Theory and Computation

The Importance of Triples Contributions to NMR Spin-spin Coupling Constants Computed at the CC3 and CCSDT Levels Rasmus Faber,∗,† Stephan P. A. Sauer,† and Jürgen Gauss‡ † Department of Chemistry, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark ‡ Institut für Physikalische Chemie, Universität Mainz, 55099 Mainz, Germany E-mail: [email protected] Abstract We present the first analytical implementation of CC3 second derivatives using the spin-unrestricted approach. This allows, for the first time, the calculation of nuclear spin-spin coupling constants (SSCC) relevant to NMR spectroscopy at the CC3 level of theory in a fully analytical manner. CC3 results for the SSCCs of a number of small molecules and their fluorine substituted derivatives are compared with the corresponding coupled cluster singles and doubles (CCSD) results obtained using specialized basis sets. For one-bond couplings the change when going from CCSD to CC3 is typically 1-3%, but much higher corrections were found for 1 JCN in FCN, 15.7%, and 1 JOF in OF2 , 6.4%. The changes vary significantly for multi-bond couplings, with differences of up to 10 %, and even 13.6% for 3 JFH in fluoroacetylene. Calculations at the coupled cluster singles, doubles, and triples (CCSDT) level indicate that the most important contributions arising from connected triple excitations in the coupled cluster expansion are accounted for at the CC3 level. Thus we believe that the CC3 method will become

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the standard approach for calculation of reference values of nuclear spin-spin coupling constants.

1

Introduction

Nuclear magnetic resonance (NMR) spectroscopy is the most common experimental technique for characterizing both organic and inorganic compounds. While standard applications for organic compounds mostly deal with hydrogen and carbon, the technique is nowadays applied to many more nuclei, particularly in inorganic chemistry. This ubiquitous use of NMR spectroscopy for the characterization of new compounds has led to an ever increasing demand for computational tools to calculate the two NMR parameters, chemical shift and indirect nuclear spin-spin coupling constant (SSCC). Today a number of theoretical methods can be employed for the calculation of SSCCs. Popular methods include the multi-configurational self-consistent field approach (MCSCF), 1,2 the second-order polarization propagator approximation (SOPPA), 3–6 coupled cluster (CC), 7–10 and density functional theory (DFT). 11–15 Among these, CC theory takes a special place since it provides a hierarchy of approximate methods which systematically approaches the full configuration interaction result, while each of the approximate methods retain the property of size-extensivity. 16 Although CC theory can only be applied to smaller molecules, CC results play an important role in providing reference values for less demanding methods. Benchmarking of DFT methods is of particular importance. A large number of functionals exist and, while performance typically improves on going from LDA over GGA to hybrid functionals, there is usually little reason to prefer one functional from a given class over an other a priori. Benchmarking against accurate CC results is often the best way of determining the computational accuracy of lower level methods, since theoretical values are rarely directly comparable to experimental ones due to the neglect of vibrational, 17,18 solvent/pressure 19 and relativistic

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effects. 20 While each of these effects can in principle be handled, at least if they are assumed to be additive, they add serious complications and often make it unclear in which step the primary error occurs. Since SSCCs are, in non-relativistic theory, 21–23 calculated as four separate terms, which are quite different, a functional good at predicting one of the terms might be less successful at predicting another. Thus benchmark studies ought to address each term individually. However, such benchmark studies can inherently only use theoretical values as reference values. The question thus arises which level in the coupled cluster hierarchy is sufficient for a good and balanced description of all contributions to the SSCC. For most molecular properties it is known that calculations at the CC singles and doubles (CCSD) 24 level give good results, but that some account of explicit treatment of triple excitations is needed in order to obtain very accurate results. However, the CC singles, doubles, and triples (CCSDT) 25 method is computationally too demanding for all but the smallest molecules. Instead the CCSD(T) approach, 26 using a non-iterative correction from perturbation theory, has become the method of choice for most applications. Two of the terms in Ramsey’s expressions for the SSCCs include the electron spin operator, leading to triplet terms in the perturbed wavefunction. Hartree-Fock (HF) calculations are susceptible to the triplet instability problem 27,28 often leading to useless results for SSCCs 3,22,29,30 and care must be taken not to import these problems into the ensuing CC calculation. 31 A consequence of this is that CCSD(T) cannot be used for the calculation of the triplet contributions to SSCCs. Instead plain CCSD has typically been used and only a few studies 10,31–33 have considered the effects of higher excitations on the quality of calculated SSCCs. Therefore little evidence exists regarding the reliability and possible shortcomings of CCSD theory for the calculation of SSCCs. While available previous results did not show large deviations between CCSD and CCSDT, it did appear that the CC3 method 34 will capture most of the deviation between the two. 10 The molecules employed in these early studies mostly featured hydrogen couplings.

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The main barrier for employing CC3 for the calculation of SSCCs has been the lack of software for calculating analytical second derivatives of the CC3 energy without enforcing singlet spin symmetry on the perturbed wavefunction. Calculation of SSCCs and other properties using analytical derivatives are vastly preferable to approaches based on numerical derivatives, not only because of lower computational costs, but also because they can be used in a black-box nature, unlike the numerical approaches where round-off errors in the differentiation step are always a concern. In this present work we present the first implementation of analytical second derivatives of the energy at the CC3 level in a spin-unrestricted framework, extending the previous spin-restricted implementation of Gauss and Stanton. 35 Previous experience has shown that SSCCs involving fluorine and double or triple bonds tend to be challenging to calculate with high accuracy. 36–41 For the fluorine-fluorine coupling in difluoroacetylene, it has been shown that the triples correction to the PSO contribution alone is on the same order of magnitude as the total SSCCs. 33 However, since there was no analytic implementation that could handle the triplet-type contributions, only the singlet contributions were investigated at the CC3 level. Thus it is of interest to investigate whether the addition of connected triple excitations can improve the accuracy with which SSCCs involving fluorine can be predicted. It is also interesting to see, how the addition of fluorine next to the coupled nuclei will affect both the SSCCs itself, as well as the importance of treating connected triples in the calculation of these. In the present work, we thus provide results on sets of related molecules which differ from each other by the replacement of a hydrogen atom with a fluorine atom. On the basis of these calculations we will discuss how important it is to include triples corrections at the CC3 level, and thus at which level of accuracy one really needs to consider these corrections. We will also assess how good CC3 is at approximating the CCSDT values, as well as establish best estimates for the various SSCCs included in this study.

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2

Theory

Nuclear spins of atoms in the same molecule are coupled. This coupling can be split into a direct, dipolar coupling and an indirect, electron-mediated coupling. However, in freely rotating molecules the dipolar coupling vanishes and only the trace of the indirect coupling remains. SSCCs observed in gas- or liquid-phase NMR spectroscopy can thus be obtained as J K,L =

∂2E 1 X , 3 α∈{x,y,z} ∂IαK ∂IαL

where E is the electronic energy and IαK is a component of the nuclear spin of nucleus K. In non-relativistic theory, 21–23 spin-spin interactions are mediated separately by both the electron spin and the electron angular momentum. If the nuclear magnetic field is assumed to arise from a point magnetic dipole, the nuclear spin-electron spin interaction is split into two contributions. One, the Fermi-Contact (FC) term, involves a δ function in the electron-nucleus distance with severe consequences for the basis-set dependence of SSCC calculations. 5,12,42,43 The finite-distance interaction is called the Spin Dipolar (SD) term. Since these interactions explicitly depend on the electron spin, they have triplet spinsymmetry and thus can only be handled by a program that allow the perturbed wavefunction to exhibit this symmetry, either by using explicit triplet symmetry adapted parameters or, as in the present case, via an implementation within a spin-unrestricted framework. The interaction with the electron orbital angular momentum leads to terms both linear and bi-linear in the nuclear magnetic moment, again giving rise to two separate contributions. These terms are referred to as the Paramagnetic (PSO) and Diamagnetic (DSO) Spin-Orbit interactions. The latter is typically evaluated as an expectation value, though a reformulation into a sum-over-states expression is possible. 44,45 While all four terms contribute to the same interaction, they can be calculated separately. There is a non-zero cross term between FC and SD interaction, which gives a traceless contribution to the total coupling tensor. Since in the present work we shall only consider 5



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isotropic coupling constants, this contribution can be ignored. In CC theory the wave-function is given via an exponentially parametrised excitation operator exp(Tˆ). Tˆ itself is defined as a linear combination of excitation operators. For the purpose of CC methods with triple excitations, such as CCSDT, CC3, 34 and CCSDT-n (n=1,1b,2,3), 46,47 Tˆ is truncated to contain single, double and triple excitation operators. Writing the HF state as |HFi the CC energy is defined as ˆ T |HFi. ECC = hHF|He ˆ

(1)

and by choosing a suitable projection space, {hµ|}, the CC amplitudes can be obtained by solving a system of non-linear equations ˆ ˆ Tˆ eµ = hµ|e−T He |HFi = 0.

(2)

For the purpose of calculating molecular properties as derivatives of the energy Eq. (1) is not ideal, since ECC is not stationary with respect to the CC amplitudes and thus the calculation of the n’th derivative of the energy would require knowledge of the n’th derivative of the amplitudes. Instead the CC Lagrangian is introduced 48,49

LCC = ECC +

X

eµ λµ ,

(3)

µ

which by construction is stationary with respect to the new parameters, λµ . These parameters can then be determined such that the Lagrangian is also stationary with respect to the amplitudes. Since the Hamiltonian contains only one and two-body interactions, the energy can be written in terms of the HF energy and the one- and two-particle CC density matrices. Using

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p, q, r, s as spin-orbital labels, the energy can thus be written

E = EHF +

X

Dpq fpq +

pq

X

Γpqrs hpq||rsi

(4)

pqrs

with the density matrices defined as ∂(LCC − EHF ) ∂fpq ∂(LCC − EHF ) = . ∂hpq||rsi

Dpq = Γpqrs

(5) (6)

and fpq as the elements of the Fock matrix and hpq||rsi as the antisymmetrized two-electron integrals in Dirac ordering. As D and Γ only depend on the CC amplitudes and Lagrange multipliers, the stationarity of the Lagrangian allows us to simultaneously ignore the first derivatives of D and Γ when differentiating Eq. (4), Thus a relatively simple expression for the first derivative w.r.t. some perturbation x is obtained dfpq X dhpq||rsi dEHF X dLCC Dpq Γpqrs = + + . dx dx dx dx pq pqrs

(7)

Full derivatives are used on the r.h.s. as fpq and hpq||rsi in general depend in three different ways on the perturbations, i.e., via the molecular orbital (MO) coefficients, via the basis functions, and via the operator. However, as mentioned above HF wavefunctions often suffer from triplet instabilities. Since dfpq /dx in Eq. (7) depends on the derivative of the MO coefficients, a singularity in the HF description will be imported directly into the calculation of the CC energy derivatives, which in turn renders the CC results unreliable. 31 The HF stability problems can be bypassed by taking derivatives in the unrelaxed way, which means considering the MOs to be independent of the perturbation. An expression for the unrelaxed first derivative is obtained by replacing the total derivatives on the right hand side of Eq. (7) with partial derivatives. As we consider only one-electron perturbations the last term in

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Eq. (7) vanishes. The second derivative thus becomes d2 LCC ∂ 2 EHF ∂Dpq ∂fpq ∂ 2 fpq = + + Dpq dydx ∂y∂x ∂y ∂x ∂y∂x

(8)

In order to calculate second-order properties in the unrelaxed scheme, one thus needs only the one-particle CC density matrix and its first derivative, requiring consequently the calculation of the T and Λ amplitudes and their first derivatives. This means that our formulation follows the asymmetric version for second derivatives. 50 Another option would be the symmetric version 49 which enforces both the (2n+1) and (2n+2) rules. The latter is advantageous, if, as in the present work, one calculates all SSCCs in a molecule, as the symmetric version requires the knowledge of only the derivative of T w.r.t all nuclei. However, if only a subset of SSCCs are calculated the asymmetric version maybe be equally or more efficient, as the derivatives of T and Λ w.r.t. one nucleus allows the calculation of all SSCCs involving that nucleus. A practical advantage of the asymmetric approach is that the derivatives of T and Λ can be discarded after use. The form of the CC Lagrangian given in Eq. (3) is valid even if the equations defining the CC amplitudes do not take the form in Eq. (2). Thus the framework described above can equally well be applied to methods such as CC3, 34,51,52 which are approximations to CCSDT. However, if explicit orbital relaxation is not included, one should ensure that the approximations in the amplitude equations are such that implicit orbital relaxation, i.e. via the T1 amplitudes, is fully included. This was the approach taken in the definition of the CC3 method. 34 The current implementation of spin-unrestricted second derivatives of the CC3 energy was added to the CFOUR 53 program, which is already capable of calculating second derivatives using the CCSD approach in a spin-unrestricted framework. 54 In the following we will therefore only discuss the additional terms that have to be evaluated in order to perform the calculations at the CC3 level.

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The full equations for the extra terms of the amplitude and Λ equations as well as the one and two-particle density matrices were given in a spin-orbital notation by Gauss and Stanton. 35 In the following we will make additional comments on how these can be evaluated in an efficient manner. The notation is the same as that of Gauss and Stanton, in that we use i, j, k, l, . . . as indices of occupied spin-orbitals and a, b, c, d, . . . as the indices of virtual spinorbitals. Two and three-body permutation operators generate the sum over antisymmetric and cyclic permutations of the indices, demonstrated here for an arbitrary tensor quantity a:

P (pq)apq = apq − aqp P (pqr)apqr = apqr + arpq + aqrp .

(9) (10)

Lower case t represent the amplitudes that parametrize the T operator Tˆ =

X

tai a† i +

ai

1 X abc † † † 1 X ab † † tij a ib j + t a ib jc k 4 abij 36 abcijk ijk

(11)

where a spin-orbital label p (p† ) represents an annihilation (creation) operator for the relevant spin-orbital. In spin-orbital form the CC3 Lagrangian, Eq. (3), can be written

LCC = ECC +

X ai

eia λia +

1 X ijk ijk 1 X ij ij eab λab + e λ , 4 abij 36 abcijk abc abc

(12)

where for example eia is obtained by inserting hµ| = hHF|i† a in Eq. (2). We assume that canonical orbitals are used and use the short hand notation for the orbital energy differences

ǫab··· ij··· = fii − faa + fjj − fbb + · · ·

(13)

The two-particle interaction intermediate, W , used in the following equations is the T1 -

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transformed fluctuation potential

Wpqrs =

X

¯ s′ ns tss ). (14) ¯ r′ nr trr )(δss′ + n ¯ q nq′ tqq′ )hp′ q ′ ||r′ s′ i(δrr′ + n (δpp′ − n ¯ p np′ tpp′ )(δqq′ − n ′

′

p′ q ′ r ′ s′

where np is one if p is an occupied spin-orbital and zero otherwise and n ¯ p = 1 − np . With the above definitions the CC3 T3 -equations can be written as tabc ijk

=

−1 (ǫabc ijk ) P (abc)P (ijk)

X

tae ij Wbcek

e

−

X

tab im Wmcjk

m

!

(15)

whereas the triples contribution to the singles and doubles equations are found as 

ab −1  1 ∆T tab P (ab) ij = (ǫij ) 2

∆T tai = (ǫai )−1 41

X aef

tijm Wbmef − 21 P (ij)

mef

X

tabe imn Wmnje +

mne

X me

tabe ijm [

X nf



hmn||ef itfn ]

(16)

X aef

(17)

timn hmn||ef i.

mnef

Eq. (15) implies that a triples amplitude only depends on the singles and doubles amplitudes. The implementation in CFOUR takes advantage of this by calculating only the triples amplitudes with a given set of occupied indices at a time. The triples contributions to the singles and doubles equation are calculated directly for each batch and storage of the full set of triples amplitudes is thus avoided. The Λ-equations can be handled in a similar manner. The triples equation is given by

λabc ijk

=

−1 (ǫabc ijk ) P (abc)P (ijk)

X

λij ae Wekbc

e

−

X

λim ab Wjkmc

m

+

!

λia hjk||bci

(18)

where the first two terms are analogs of similar terms in Eq. (15), and where the last term in Eq. (18) is a simple addition of N 6 cost. Similarly the triples contribution to the doubles equations simply contains analogs of the first two terms of Eq. (16). The triples contributions

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to the Λ1 -equations are, however, far more complicated than for the T1 -equations.1 The full term can be written as ∆T λia = (ǫai )−1

X

′ him||aeiDme

me

+ 12

X

′ 1 λim ef Wef am − 2

X

Wef ag Γ′ef ig

X

Wmena Γ′meni − 21

mef

+ 21

−

mne

′ λmn ae Wiemn

X

Wiemf Γ′aemf

X

Wmino Γ′mano ,

mne

mef

ef g

+

X

mno

(19)

where D′ , W ′ , and Γ′ are new intermediates that fall in two categories. Those that depend on the T3 amplitudes and those that depend on the Λ3 amplitudes. The intermediates of the first class are given as: ′ Dia =

1 4

X

aef λmn ef timn

(20)

mnef ′ 1 Wef am =− 2

X

g tef nmo hno||agi

(21)

′ Wiemn =

X

eg tfmno hio||f gi.

(22)

1 2

nog

of g

The second and third intermediate depends only on the T3 amplitudes and the two-electron integrals, neither of which change during the iteration of the lambda equations. Thus these need to be calculated only once. The D′ intermediate, however, depends on the Λ2 amplitudes, and thus the T3 amplitudes must be recalculated in each iteration of the Λ equations in order to compute this term. 1

The fact that additional complications only arise in the Λ1 equations is related to the fact the CC3 equations are non-linear only in the T1 -amplitudes. CCSDT-2 and CCSDT-3 include T22 in the triples equation and therefore also have slightly more complicated Λ2 equations.

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The four other terms in Eq. (19) depend on the intermediates Γ′abic =

1 2

Γ′iajk =− 12

X

(23)

ce λimn abe tmn

nme

X mjk ef

(24)

λef a tmi

mef

which must also be recalculated in every step of the Λ equations. After convergence of the Λ equations, the triples contributions to the one-particle density matrix are calculated:

X

ef g λmnj ef g tmni

(25)

X

bef λmno aef tmno

(26)

X

aef λmn ef timn +

1 ∆T Dij =− 12

mnef g

∆T Dab =

1 12

mnof g

∆T Dia =

1 4

mnef

+

X

∆T Dim tam −

1 2

X

Γ′iemn tae mn

mne

X

tei ∆T Dea .

(27)

e

m

The calculation of the occupied-occupied and virtual-virtual blocks of the density matrices requires that both the T3 and Λ3 amplitudes are known at the same time. In particular the occupied-occupied block causes some difficulties, since the current implementation computes only batches of triple amplitudes with fixed occupied indices. Calculation of Eq. (25) for g mnj each i thus requires that tef mni is recalculated for each λef g giving an additional computational

cost of No4 Nv4 . The equations defining the derivatives of the amplitudes can be found by differentiating the equations discussed above, thereby keeping in mind that the derivative Fock matrix is not diagonal. However, there are a few additional considerations. It is, for instance, convenient to split the triples amplitudes into a constant and an iterative part: ∂tabc ijk = ∂x

∂tabc ijk ∂x

!c

12

∂tabc ijk + ∂x

!i


(28)

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where the constant part can be taken as ∂tabc ijk ∂x

!c

∂fae X a ∂fme ebc − t tm ∂x ∂x ijk e m X ∂fme X ∂fmi abc e . t ti + − ∂x ∂x mjk e m

X

−1 = (ǫabc ijk ) P (abc)P (ijk)

(29)

c Thus (∂tabc ijk /∂x) needs to be calculated only once and its contribution to the singles and

doubles equations can be stored alongside the r.h.s. of these equations. Similarly the derivatives of the Λ3 amplitudes are also split into iterative and non-iterative parts. In addition a number of the contributions to the derivatives of the Λ1 equations are constant during the iteration of the perturbed Λ equations. Considering the derivative of Eq. ′ (20), we see that the part of ∂Dia /∂x that depends on the derivative of T3 is constant during

Λ iterations. While the T3 amplitudes thus must be recalculated in each Λ iteration, at least there is no need to also calculate their derivatives. If we consider the derivatives of the last four terms of Eq. (19), we see that each contain two kinds of contributions: Γ′ · (∂W/∂x) and (∂Γ′ /∂x) · W , the first of which is clearly constant. If we further consider the derivative of Γ′ , partitioning the derivative Λ3 amplitudes into constant and iterative parts, we find three kinds of contributions: Λ3 · (∂T2 /∂x), (∂Λ3 /∂x)c · T2 and (∂Λ3 /∂x)i · T2 . Again the first two can be considered constant during the solution of the derivative Lambda equations, and only the last need to be recalculated in each iteration. One final contribution to the Λ1 constant term arises, since the derivative Fock matrix is not block diagonal (ǫai )−1

X m

!

∂fie ∂fma X T . ∆ Dae − ∆ Dmi ∂x ∂x e T

(30)

As mentioned above, the steps with the highest scaling in the present first implementation scales as non-iterative No4 Nv4 , whereas formally only No3 Nv4 steps are needed. However, assuming that around 20 iterations are needed to converge the CC and Λ eequations, these contributions should not vastly increase the computational cost for small molecules. Increas-

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ing the basis set used to treat a given molecule changes only Nv , meaning that the relative cost of increasing the basis set size in our implementation is the same as in the ideal implementation and CCSD (No2 Nv4 ). The full CCSDT method has iterative terms that scale as No3 Nv5 , so our CC3 implementation should easily be two orders of magnitude faster than CCSDT, even if storage of the T3 amplitudes is not an issue. Our current first implementation uses a spin-unrestricted approach for the FC and SD terms. Using instead a triplet spin-restricted approach would speed up the calculation of these terms by a small constant factor, but will not greatly expand the set of systems that can feasibly be treated.

3

Computational Details

The geometries of the molecules were optimized using CCSD(T) 26,55 and the aug-cc-pCVQZ basis set, 56–58 except for difluroacetylene where the CCSD(T)/cc-pCVQZ geometry from a previous study 33 was used. Indirect nuclear spin-spin coupling constants were calculated using the basis sets of Benedikt et al., 59 which have been specifically optimized for the calculation of SSCCs. Calculations was carried out at the CCSD and CC3 levels using the aug-ccJ-pVTZ basis set. To study the triples effects beyond CC3, calculations were also carried out using the CC3 and CCSDT methods using the ccJ-pVDZ basis set. All calculations were performed with all electrons correlated and all parts of the SSCCs were calculated using unrelaxed derivatives. The triplet perturbations needed for FC and SD contributions, were handled by performing those calculations in a spin-unrestricted framework, all other calculations where performed in a spin-restricted fashion. The FC and SD terms at the CCSDT level were calculated using the MRCC program 60 of Kállay interfaced to CFOUR, whereas at the CC3 level they were calculated using a local version of CFOUR. Geometry optimizations, calculations of the PSO and DSO contributions, as well as the CCSD calculation of the FC and SD terms were performed using the public version of CFOUR. 53

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4

Results

In the following we present CC3 results for various nuclear spin-spin coupling constants in several small molecules. All reported results are for the 1 H,

13

C,

15

N,

17

O, and

19

F

isotopes. The molecules H2 O, CO, and HCN have previously been studied extensively, and they were part of the set for which Auer and Gauss presented numerical values of the SSCCs at the CC3 level. 10 Two other commonly studied molecules presented here are acetylene and formaldehyde. Acetylene has previously been studied using SOPPA, 5,61 SOPPA(CCSD), 5,62,63 RASSCF, 64 and EOM-CCSD. 65 Formaldehyde has been studied using SOPPA 61,66 and SOPPA(CCSD). 66 As mentioned above it has often proved to be difficult to accurately calculate the SSCCs of fluorine containing compounds. In order to study how CC3 changes this situation, the SSCCs of various fluorine substituted derivatives of the above molecules have also been calculated. Several of these have previously been included in studies using second order methods and CCSD by Del Bene et al. 65,67 and Kjær et al. 6

4.1

Importance of the individual contributions

In table 1 we present CC3/aug-ccJ-pVTZ values for each of the individual FC, SD, PSO, and DSO contributions for each SSCC. CCSD values are included for comparison. The conventional wisdom on SSCC calculations is that the FC contribution dominantes one-bond couplings, but decreases in importance relative to the other contributions as the distance between the coupled nuclei increase. This is clearly the case for most one-bond couplings presented here. For all 1 JCH couplings in the given set of molecules, the FC term appears to be around 250 Hz in magnitude, whereas the other contributions are mostly smaller than 1 Hz. Other kinds of one-bond couplings where the FC term dominates are 1

JCC , 1 JCF , and 1 JOH . However, two kinds of couplings show a different behaviour. The

couplings between C and an O or N nucleus bound to this carbon atom by a double or triple

15



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Page 16 of 39

bond are typically smaller in magnitude and the FC, SD, and PSO contributions are all of similar size. The other example is 1 JOF , for which the of FC, SD, and PSO contributions are all above 100 Hz in magnitude. Table 1: CCSD and CC3 values for all molecules in the study. All results calculated using unrelaxed derivatives and the aug-ccJ-pVTZ basis set. All results are given in Hz and for the 1 H, 13 C, 15 N, 17 O, and 19 F isotopes.

CO

H2 O

OHF

OF2

HCN

FCN

SSCC 1 JCO 1

JOH

2

JHH

1

JOF

1

JOH

2

JFH

1

JOF

2

JFF

1

JCN

1

JCH

2

JNH

1

JCN

1

JCF

2

JNF

CCSD CC3

FC 6.85 6.34

SD -5.09 -4.94

PSO 13.20 13.12

DSO 0.10 0.10

Total 15.07 14.62

CCSD CC3 CCSD CC3

-69.22 -68.88 -11.40 -10.93

-0.58 -0.60 0.99 0.99

-11.68 -11.67 9.25 9.29

-0.03 -0.03 -7.19 -7.18

-81.51 -81.19 -8.34 -7.84

CCSD CC3 CCSD CC3 CCSD CC3

114.02 -214.52 -456.11 -0.26 106.09 -211.71 -459.91 -0.26 -56.70 0.93 4.86 -0.11 -55.55 0.91 5.05 -0.12 24.22 -7.53 75.39 -1.98 23.01 -7.26 76.15 -1.97

-556.88 -565.79 -51.02 -49.70 90.10 89.93

CCSD CC3 CCSD CC3

128.43 -155.05 -242.10 -0.40 -269.13 120.58 -146.51 -225.66 -0.40 -251.99 77.99 411.65 723.15 -1.00 1211.78 97.68 431.61 799.05 -1.00 1327.34


-13.21 -13.13 253.53 249.63 -3.97 -3.98

-5.16 -5.02 0.52 0.52 -0.72 -0.71

CCSD 3.77 CC3 4.64 CCSD -355.31 CC3 -365.36 CCSD 42.18 CC3 44.98

-5.66 -5.50 -10.55 -9.67 -10.12 -10.20

-0.13 -0.07 -0.62 -0.60 -3.48 -3.40

0.04 0.03 0.39 0.41 0.62 0.62

-18.46 -18.19 253.82 249.95 -7.56 -7.47

-4.75 -0.03 -4.73 -0.04 -29.55 0.56 -28.55 0.57 17.04 0.41 17.92 0.41

-6.67 -5.63 -394.84 -403.00 49.52 53.12

Continued

16


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Table 1: CCSD and CC3 values for all molecules in the study. All results calculated using unrelaxed derivatives and the aug-ccJ-pVTZ basis set. All results are given in Hz and for the 1 H, 13 C, 15 N, 17 O, and 19 F isotopes. SSCC 1 JCC

HCCH

FCCH

FCCF

FNO

1

JCH

2

JCH

3

JHH

1

JCC

1

JCF

2

JCF

1

JCH

2

JCH

3

JFH

1

JCC

1

JCF

2

JCF

3

JFF

1

JNF

1

JNO

2

JOF

FC 172.09 165.83 243.85 240.39 46.27 47.86 8.62 8.30

SD 8.78 8.54 0.47 0.48 1.02 1.01 0.57 0.55

PSO 6.73 6.57 -0.79 -0.73 5.60 5.54 4.67 4.69

DSO 0.01 0.01 0.30 0.30 -1.35 -1.35 -3.59 -3.59

Total 187.61 180.96 243.83 240.44 51.55 53.07 10.27 9.95

CCSD 250.98 CC3 247.28 CCSD -248.37 CC3 -247.47 CCSD 14.07 CC3 13.02 CCSD 272.45 CC3 269.90 CCSD 61.29 CC3 63.04 CCSD -1.31 CC3 0.21

9.61 9.31 -10.43 -9.73 17.84 17.95 0.48 0.48 0.99 0.98 2.93 3.07

11.56 11.35 -22.11 -20.96 -4.29 -4.53 -0.81 -0.74 5.68 5.63 13.71 13.78

0.16 0.16 0.48 0.49 -0.88 -0.88 0.43 0.44 -1.12 -1.12 -2.62 -2.61

272.31 268.11 -280.42 -277.68 26.74 25.56 272.55 270.08 66.84 68.53 12.72 14.45

CCSD 381.63 CC3 376.08 CCSD -244.62 CC3 -241.57 CCSD 19.93 CC3 19.99 CCSD 7.10 CC3 8.17

10.53 10.18 -8.11 -7.52 18.92 19.10 33.30 34.45

15.36 15.09 -8.69 -8.08 6.98 7.14 -31.30 -38.22

0.31 0.31 0.59 0.59 -0.69 -0.69 -1.85 -1.84

407.83 401.65 -260.83 -256.58 45.14 45.54 7.26 2.56

-16.85 -12.16 0.74 0.69 -2.58 -2.45

26.43 40.17 -16.50 -15.93 155.55 166.55

-0.27 -0.27 -0.01 -0.01 0.26 0.26

146.39 152.73 -34.85 -34.14 119.20 130.22

CCSD CC3 CCSD CC3 CCSD CC3 CCSD CC3


137.09 124.98 -19.07 -18.90 -34.03 -34.13

Continued

17



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Page 18 of 39

Table 1: CCSD and CC3 values for all molecules in the study. All results calculated using unrelaxed derivatives and the aug-ccJ-pVTZ basis set. All results are given in Hz and for the 1 H, 13 C, 15 N, 17 O, and 19 F isotopes. SSCC 1 JCO

H2 CO

HFCO

F2 CO

1

JCH

2

JOH

2

JHH

1

JCO

1

JCF

1

JCH

2

JOF

2

JOH

2

JFH

1

JCO

1

JCF

2

JOF

2

JFF

FC 8.15 8.21 169.00 167.75 -5.63 -5.88 36.21 36.92

SD -1.69 -1.64 0.29 0.30 -0.76 -0.73 0.40 0.40

CCSD 6.44 CC3 6.49 CCSD -309.10 CC3 -314.60 CCSD 243.10 CC3 241.67 CCSD -19.25 CC3 -19.32 CCSD -9.92 CC3 -10.39 CCSD 182.57 CC3 190.06

-3.21 -3.13 1.19 1.18 0.46 0.46 -2.79 -2.87 -0.63 -0.61 -1.62 -1.50

-0.05 -0.05 0.72 0.72 1.17 1.18 0.33 0.33 0.52 0.51 -1.84 -1.83

14.93 14.67 -345.51 -352.41 243.80 242.40 47.11 48.74 -9.56 -10.01 175.74 182.86

CCSD 8.90 CC3 8.96 CCSD -254.84 CC3 -256.94 CCSD -4.14 CC3 -3.61 CCSD 128.97 CC3 144.15

-3.58 7.12 -0.13 -3.50 6.76 -0.13 -0.11 -38.64 1.16 0.00 -38.61 1.17 -4.73 47.21 0.32 -4.84 47.91 0.32 24.30 -257.23 -1.10 24.72 -267.98 -1.09

12.32 12.08 -292.43 -294.39 38.65 39.78 -105.05 -100.20

CCSD CC3 CCSD CC3 CCSD CC3 CCSD CC3

PSO DSO 20.72 0.00 20.41 0.00 -0.79 0.62 -0.78 0.63 3.07 0.50 3.11 0.50 3.49 -3.54 3.49 -3.53 11.76 11.36 -38.32 -39.70 -0.93 -0.91 68.81 70.60 0.48 0.47 -3.37 -3.87

Total 27.18 26.98 169.12 167.89 -2.82 -3.01 36.56 37.29

For the two-bond couplings the situation is more varied. 2 JOF couplings tend to have large PSO contributions, whereas 2 JCH couplings are primarily dominated by the FC contribution. The other couplings fall somewhere in between, with multiple contributions of similar size. Of particular interest is the 2 JFF of OF2 , which at 1327 Hz is more than twice as large as the second largest coupling in the set. This interaction can, however, not be measured by NMR spectroscopy, as the symmetry between the fluorine atoms would have to be broken 18


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but only one stable isotope of fluorine is available. The only three-bond couplings in our test set are those of the acetylenes. While the 3 JHH coupling in acetylene has only a small triples correction to the dominating FC term, the 3 JFH coupling in fluoroacetylene is dominated by the PSO term whereas the FC term is small and CC3 and CCSD predict different signs for it. Also the 3 JFF coupling in difluoroacetylene has a very large triples correction to PSO which combined with triples corrections of about 1 Hz for the FC and SD terms yield a total correction of almost 5 Hz. The values for CO, H2 O, and HCN given in table 1 closely agree with those of Auer and Gauss. 10 There are some minor differences in the FC term, but they are easily explained by the use of different basis sets. More importantly we reproduce their results for the size of the triples correction.

4.2

Size of triples corrections

The importance of the CC3 triples correction compared to the size of the CCSD values for the one-bond coupling constants are shown in figure 1. For the majority of the couplings, the triples corrections amount to about 2% of the total SSCC. An exception that stands out immediately, is the 1 JCN coupling in FCN, where the CC3 triples correction is 15.7%. However, this is largely due to the small size of the total coupling constant, which at the CCSD level is only -6.7 Hz. Compared to the same coupling in HCN, we see an increase in the CC3 correction, from just 0.3 Hz in HCN to 1.0 Hz in FCN, and since the total coupling constant is reduced by a factor of three by fluorine substitution, an even more pronounced effect is observed in the relative increase. Another exception is the 1 JOF coupling in OF2 , which has a CC3 triples correction of 6.3% or as much as 17.1 Hz. Again the presence of a nearby fluorine nucleus makes the triples corrections much more important, that is, the 1 JOF coupling in OHF has a CC3 correction of 1.6 % or 8.9 Hz, whereas the total SSCC roughly doubles on the fluorine substitution. Thus one may wonder if it is a general trend that the substitution of a hydrogen atom 19



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next to one of the coupled atoms by a fluorine atom increases the importance of the triples correction? Another example is the 1 JOH in water and OHF, where the CC3 correction increases from 0.3 Hz to 1.3 Hz. We observe, however, also a counter example, the 1 JCF coupling in mono- and difluoro-formaldehyde, where the magnitude of the CC3 correction decreases from 6.9 to 2.0 Hz. Also, it seems that only the first adjacent fluorine atom has a large influence on the triples contribution. If we look at the 1 JCO coupling of formaldehyde, the triples correction increases from 0.7% to 1.8% when going to the monofluoro derivative, but only further to 2.0% in difluoro formaldehyde. However, these differences are more due to a decrease in the size of the total SSCC and less due to changes in the absolute size of the triples correction. A similar trend is observed for the 1 JCC coupling of acetylene, an SSCC that increases significantly upon fluorine substitution, leading to a decrease in the relative size of the triples correction. The triples corrections to all the 1 JCH couplings are below 2%. All are negative and dominated by the FC contribution. Triples corrections to 1 JCF couplings are more difficult to generalize. They may be positive or negative and often all contributions are important, though the FC contribution is the largest in all cases except fluoroacetylene. Figure 2 shows the size of the CC3 triples corrections relative to the size of the total CCSD SSCC for the multi-bond SSCCs in the given set of molecules. We see that for multi-bond SSCCs, triples corrections are much more important than for one-bond SSCCs. Corrections between 3% and 6% are common. 3 JFF of difluoroacetylene has the highest relative triples correction, 64.7%. In this particular case the main difficulty is that the total SSCC is small compared to the individual contributions. The relative triples correction to the FC, 15.0%, and PSO, 22.1%, contributions are themselves large, but more in line with the observations for other two-bond SSCCs. Of the SSCCs actually shown in Figure 2, the largest triples correction is seen for the 3 JFH coupling in fluoroacetylene at 13.6%. The CC3 correction is in this case dominated by the FC contribution, which with 1.5 Hz is as large as the total CCSD value of -1.3 Hz. We have

21




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two further cases where the relative triples correction is almost 10%. These are 2 JOF in FNO and 2 JFF in OF2 , where in both cases the PSO term dominates the triples corrections. For the two-bond couplings we do not see a clear increase in the relative triples correction on the fluorine substitution. In some cases the total SSCC changes more upon fluorine substitution than does the triples correction. The 2 JOH coupling of fluoro formaldehyde, for instance, is a factor of three larger than the same coupling in formaldehyde. In most cases, however, the absolute triples correction also decreases, illustrated by the change in 2 JCF when going from mono- to difluoroacetylene which is associated with a reduction in the triples correction from 4.4% to 0.9%.

4.3

Accuracy compared to CCSDT

While it is clear that CC3 captures important additional effects that are not included in CCSD, CC3 is by its definition an approximation of CCSDT. Therefore we need to know what error is introduced by this approximation. In table 2 we compare the results of CC3 and CCSDT calculations using the aug-ccJ-pVTZ basis set for the three smallest molecules: CO, HCN, and H2 O. For the difference between the CC3 and CCSDT results, which we will refer to as the residual triples correction, ∆T , two values are given: The difference obtained from calculations using the aug-ccJ-pVTZ basis set and the difference obtained using the smaller ccJ-pVDZ basis set. In general the residual triples corrections are quite small. Comparing the residual triples corrections obtained with the double and triple zeta basis set, however, we see that there are some large relative differences, even though they are small compared to the total SSCCs. For instance, for the 1 JOH coupling in water the triples correction is four times larger when calculated using the larger basis set, but in absolute terms the difference amounts to only 0.03 Hz. The double zeta results can thus be used as qualitative estimates for the size of the residual triples correction. As it is not feasible to obtain CCSDT/aug-ccJ-pVTZ results for all molecules in the present study, we have used ccJ-pVDZ estimates for the residual triple 23



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Table 2: SSCCs calculated using the CC3 and CCSDT methods with the aug-ccJ-pVTZ basis set. The difference between the results obtained using the two methods are shown both as calculated using aug-ccJ-pVTZ (∆T (aTZ)) and the smaller ccJ-pVDZ (∆T (DZ)). All results are given in Hz for the 1 H, 13 C, 15 N, 17 O, and 19 F isotopes. Mol CO

SSCC 1 JCO

CC3 CCSDT ∆T (aTZ) ∆T (DZ)

FC 6.34 6.29 -0.05 0.01

SD -4.94 -5.01 -0.07 -0.06

PSO 13.12 13.12 0.00 -0.00

DSO 0.10 0.10 0.00 0.00

Total 14.62 14.50 -0.11 -0.05 -81.19 -81.16 0.03 0.01

H2 O

1

JOH


-68.88 -0.60 -11.67 -0.03 -68.85 -0.60 -11.67 -0.03 0.03 0.00 -0.00 -0.00 -0.01 0.00 0.01 0.00

H2 O

2

JHH


-10.93 0.99 -10.90 0.99 0.03 0.00 0.05 -0.00

HCN

1

JCN


-13.13 -13.07 0.06 0.10

-5.02 -5.06 -0.03 -0.03

HCN

1

JCH

CC3 249.63 CCSDT 249.97 ∆T (aTZ) 0.34 T ∆ (DZ) 0.28

0.52 0.52 0.00 0.00

HCN

2

JNH


-3.98 -0.71 -3.98 -0.71 -0.00 -0.00 -0.01 -0.00

24

9.29 9.30 0.01 0.00

-7.18 -7.18 -0.00 -0.00

-7.84 -7.79 0.05 0.06

-0.07 -0.07 -0.00 0.00

0.03 0.03 0.00 0.00

-18.19 -18.17 0.02 0.08

-0.60 0.41 249.95 -0.60 0.40 250.29 0.00 -0.00 0.34 0.00 -0.00 0.28 -3.40 -3.41 -0.01 -0.00


0.62 0.62 0.00 0.00

-7.47 -7.48 -0.01 -0.02

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corrections in the following discussion. In Fig. 3 the CC3 and the residual triples corrections to various one-bond nuclear spinspin coupling constants are compared. The general picture is that the CC3 triples correction is far larger than the residual triples correction and that CC3 tends to slightly overestimate the triples correction. However, in a number of cases CC3 strongly overestimates the triples contribution. Considering the 1 JCF coupling in fluoroacetylene for instance one notices that CC3 yields a triples correction of 2.7 Hz, with a residual of -1.9 Hz. This error originates in the FC contribution, where the residual correction is -1.7 Hz, whereas the remaining terms are well described by CC3. A large residual triples correction of -4.9 Hz, is also found for 1 JNF in FNO, compared to a CC3 correction of 6.3 Hz. This should, however, be seen in the light that the CC3 corrections for the FC and PSO contributions are -12.1 and 13.7 Hz, whereas their residual triples corrections are 1.4 and -4.8 Hz. So CC3 recovers most of the triples correction to each individual contribution. The 1 JNO coupling in the same molecule also shows a large deviation. In this case the PSO term is overestimated by CC3, leading to a triples correction that is almost twice the one predicted by CCSDT. In both mono- and difluoro formaldehyde, CC3 does accurately predict the triples correction to the PSO term of 1 JCO , but fails at predicting an equally large correction to the FC term with opposite sign. In the latter three examples, the residual triple corrections are numerically small, however, and the CC3 correction constitutes only around 2% of the total coupling constant. The CC3 and residual triples corrections for couplings between nuclei connected by multiple bonds are presented in Fig. 4. For these couplings CC3 performs very well. It captures by far the largest part of the triples correction in all but a single case. Once again, we see that in most cases CC3 slightly overestimates the size of the triples correction. The outlier is the 2 JFH coupling in OHF, where the CC3 correction is -0.17 Hz, but the residual triples correction is 0.63 Hz. While the residual triples corrections are less than 10% of the CC3

25




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value for the FC and SD terms, the CC3 triples correction to the PSO term is 0.76 Hz, with a residual triples correction of 0.56 Hz. A coupling that was left out in figure 4, is the 2

JFF of OF2 . While this coupling cannot be measured by usual gas- or liquid-phase NMR,

it is nonetheless remarkable because of its size. The CCSD value is 1212 Hz and the CC3 correction is 116 Hz, where the main contribution is a PSO correction of 76 Hz. The residual triples correction is only -7 Hz, which is little compared to the total size of the SSCC.

4.4

Comparison with Experiment

We have shown that triples corrections are significant for most SSCCs and that CC3 captures the majority of the effect. One may wonder how well the computed CC3 values reproduce experimental results. In Table 3 we compare some experimental results from the literature with our CCSD and CC3 results. We also give CC3+∆T estimates obtained by adding the residual triples corrections calculated using the ccJ-pVDZ basis-set to the CC3/aug-ccJpVTZ values. We believe that our CC3 calculations are the most accurate calculations so far reported for these SSCCs. The CC3+∆T values should offer a further improvement, but the ∆T estimate may be somewhat unreliable as discussed above. In many cases the agreement between our results and the available experimental values seems not to be too impressive at a first glance. Inclusion of triples effects appears to worsen agreement with experiment as often as it improves it. However, NMR experiments are typically performed in solution, and this can have a significant effect on the SSCC, as can be seen from the differences in the literature values given in some cases. In order to reproduce the value of a particular experiment one must thus account for the actual solvent used. Highly accurate experimental gas-phase values are preferable as they vastly simplify the comparison between theory and experiment. For such experimental gas-phase values the main difference with our computational results are then due to the lack of consideration of vibrational effects in the present study. Jackowski et al. 68 carried out gas-phase NMR experi27




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ments on acetylene and their observed SSCCs are given in Table 3. Vibrational corrections to the SSCCs of this molecule have been given in several previous publications including those of Wigglesworth et al., 62,63 Sneskov and Stanton, 69 and Faber and Sauer, 70 all of whom used correlated wavefunction methods. Wigglesworth et al. used SOPPA(CCSD) with SSCC surfaces expanded in symmetry coordinates, whereas the other authors used CCSD values for the SSCC together with second-order vibrational perturbation theory (VPT2). For the 1

JCC and 2 JCH couplings in acetylene their results agree reasonably well, and upon adding

the corrections of Faber and Sauer to our CC3+∆T estimates we obtain final values of 174.57 Hz and 49.18 Hz, which are in good agreement with the experimental results. The published vibrational corrections disagree, however, for the 1 JCH coupling. Adding the value of 4.87 Hz of Wigglesworth et al. to our CC3+∆T estimate yields 246.35 which is close to the experimental result, but the published CCSD vibrational corrections of Faber and Sauer, 2.56 Hz, and of Sneskov and Stanton, 0.78 Hz, leads to somewhat larger errors. For the final vicinal coupling, 3 JHH , the calculations of Wigglesworth et al. appear to overestimate the vibrational correction with a value of -1.27 Hz, whereas the vibrational correction of Faber and Sauer, -0.25, yields a final value of 9.54 Hz. While a further, careful study is probably needed for the vibrational corrections in acetylene, the available results clearly show that vibrational effects can explain the remaining differences between our CC3+∆T estimates and the experimental results. Table 3: CCSD, CC3, and CC3+∆T values for all SSCCs in the study (in Hz). Experimental results have been included for comparison, gas-phase results are marked with an *. The CC3+∆T estimates are calculated by adding the CC3 values calculated using the aug-ccJpVTZ basis, and the residual triples corrections, ∆T , calculated using ccJ-pVDZ. All results are given in Hz and for the 1 H, 13 C, 15 N, 17 O, and 19 F isotopes.

HCN CO

SSCC 1 JCN 1 JCH 2 JNH 1

JCO

CCSD -18.46 253.82 -7.56 15.07

CC3 CC3+∆T -18.19 -18.11 249.95 250.23 -7.47 -7.49 14.62

14.57

Experimental -19.1 71 18.5(1) 72 267.3(1) 261.7(4) -8.7 74 -7.4(1) 16.4(1)

Continued 29


75

72 73 73


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Page 30 of 39

Table 3: CCSD, CC3, and CC3+∆T values for all SSCCs in the study (in Hz). Experimental results have been included for comparison, gas-phase results are marked with an *. The CC3+∆T estimates are calculated by adding the CC3 values calculated using the aug-ccJpVTZ basis, and the residual triples corrections, ∆T , calculated using ccJ-pVDZ. All results are given in Hz and for the 1 H, 13 C, 15 N, 17 O, and 19 F isotopes.

H2 O

SSCC 1 JOH 2 JHH

OF2 FCN

-556.88 -51.02 90.10

-565.79 -49.70 89.93

-568.08 -49.68 90.56

1

JOF 2 JFF

-269.13 -251.99 1211.78 1327.34

-258.19 1320.35

1

-6.67 -394.84 49.52

-5.63 -403.00 53.12

-5.53 -404.02 53.11

187.61 243.83 51.55 10.27

182.16 241.45 52.67 9.95

JCC JCF 2 JCF 1 JCH 2 JCH 3 JFH

272.31 -280.42 26.74 272.55 66.84 12.72

268.11 -277.68 25.56 270.08 68.53 14.45

268.12 -279.53 25.37 270.11 68.58 14.37

1

407.83 -260.83 45.14 7.26

401.65 -256.58 45.54 2.56

401.82 -259.07 45.30 0.80

1

JCN JCF 2 JNF 1

1

HCCH

JCC 1 JCH 2 JCH 3 JHH 1

1

FCCH

FCCF

CC3 CC3+∆T -81.19 -81.18 -7.84 -7.79

JOF JOH 2 JFH

1

OHF

CCSD -81.51 -8.34

JCC JCF 2 JCF 3 JFF 1

300(30)

79

185.19 174.78(2)* 241.48 247.56(5)* 52.80 50.14(5)* 9.79 9.62(5)*

68

FNO

JNF JNO 2 JOF

146.39 -34.85 119.20

152.73 -34.14 130.22

147.86 -34.54 127.76

1

H2 CO

JCO JCH 2 JOH 2 JHH

27.18 169.12 -2.82 36.56

26.98 167.89 -3.01 37.29

27.07 168.01 -3.04 37.38

1

1

1

Experimental -80.6(1) 76 -78.22 * -7.11(3) 76

30

68 68 68

21

80

-287.3

81

2.1

81

172

82

40.2(1)

83

Continued


77,78

42.4(1)

83

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Table 3: CCSD, CC3, and CC3+∆T values for all SSCCs in the study (in Hz). Experimental results have been included for comparison, gas-phase results are marked with an *. The CC3+∆T estimates are calculated by adding the CC3 values calculated using the aug-ccJpVTZ basis, and the residual triples corrections, ∆T , calculated using ccJ-pVDZ. All results are given in Hz and for the 1 H, 13 C, 15 N, 17 O, and 19 F isotopes. SSCC JCO 1 JCF 1 JCH 2 JOF 2 JOH 2 JFH

CCSD 14.93 -345.51 243.80 47.11 -9.56 175.74

CC3 CC3+∆T 14.67 14.91 -352.41 -352.30 242.40 242.55 48.74 48.42 -10.01 -10.00 182.86 181.77

1

12.32 -292.43 38.65

12.08 -294.39 39.78

1

HFCO

F2 CO

JCO JCF 2 JOF 1

12.39 -296.04 39.77

Experimental 369 267

84 85

182

85

-308

86

In a recent study by Makulski et al., 78 the gas-phase value of the 1 JOH coupling of H2 O was determined to be -78.22 Hz. Combining our CC3 or CCSDT values of -81.19 Hz and -81.16 Hz with a SOPPA(CCSD) vibrational correction of 3.96 or 4.34 Hz 87 or a CASSCF vibrational correction of 4.58 Hz 88 leads to corrected values for this SSCC between -77.23 and -76.58 Hz. These values are reasonably close to the experimental result, but not closer than the values previously reported by Wigglesworth et al. 87 Noting the very small difference between CC3 and CCSDT in Table 2, further correlation effects are unlikely be significant. Using the aug-ccJ-pV5Z basis set instead of the aug-ccJ-pVTZ basis set, decrease the CCSD value for this SSCC by around 0.4 Hz, so up to half of the difference between our result and the experiment may be explained by the size of the basis set effects. The remaining differences are probably due to limits in the vibrational models that has been applied. As a simple system, water can be treated using high-accuracy vibrational methods in combination with models such as CC3, and we believe that such calculations would yield close agreement with the experimental value.

31



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5

Conclusion

We have presented an implementation of analytical second derivatives at the CC3 level within a spin-unrestricted framework. With this implementation it is now possible to calculate all four contributions to the indirect nuclear spin-spin coupling constants analytically at the CC3 level, while the previous spin-restricted implementation of CC3 only allowed the calculation of the paramagnetic and diamagnetic spin-orbit terms. The new implementation has been used to calculated new reference values for the SSCCs of a number of common small molecules and their fluorine-substituted derivatives using basis sets specially optimized for coupling constants. Our results show that CC3 provides important additional contributions to the SSCC, not covered by the previous standard reference method, CCSD. One-bond SSCCs typically have a triples correction of a few percent of the CCSD value. For multi-bond SSCCs the triples correction varies considerably, even the 2 JHH coupling in a molecule as common as H2 O has a correction of no less than six percent. While we found a few SSCCs where triples effects beyond CC3 appear to be important, the triples effects captured by CC3 were in most cases by far the most important. The residual effect of the triples contribution has been estimated using a smaller basis set and this can be added as a correction to the CC3 values to provide practically converged results for the SSCC in all but a few cases. Meaningful comparisons with experimental results are difficult, as most experiments in the literature have been carried out in solution. However, very good agreement with gasphase experimental results for acetylene and water were obtained by combining our results with previously published vibrational corrections.

Supporting Information Available The following files are available free of charge.

The ccJ-pVDZ values for all the SSCCs

considered in the present study calculated using the CCSD, CC3, and CCSDT methods. 32


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