Spin–spin coupling tensors by density-functional linear response theory

Spin–spin coupling tensors by density-functional linear response theory

Perttu Lantto^a)

NMR Research Group, Department of Physical Sciences, P.O. Box 3000, FIN-90014 University of Oulu, Finland

Juha Vaara^b)

Department of Chemistry, P.O. Box 55 (A.I. Virtasen aukio 1), FIN-00014 University of Helsinki, Finland Trygve Helgaker^c)

Department of Chemistry, University of Oslo, P.O. Box 1033, Blindern, N-0315 Oslo, Norway

共

Received 4 February 2002; accepted 1 July 2002

兲

Density-functional theory

共

DFT

兲

calculations of indirect nuclear magnetic resonance spin–spin coupling tensors J, with the anisotropic but symmetric parts being the particular concern, are carried out for a series of molecules with the linear response

共

LR

兲

method. For the first time, the anisotropic components of J are reported for a hybrid functional. Spin–spin tensors calculated using the local density approximation

共

LDA

兲

, the gradient-corrected Becke–Lee–Yang–Parr

共

BLYP

兲

functional, and the hybrid three-parameter BLYP

共

B3LYP

兲

functional are compared with previous ab initio multiconfiguration self-consistent-field

共

MCSCF

兲

LR results and experimental data. In general, the B3LYP functional provides reasonable accuracy not only for the isotropic coupling constants but also for the anisotropic components of J, with the results improving in the sequence LDA

→BLYP→B3LYP. Error cancellation often improves the total DFT spin–spin coupling when the magnitude of the paramagnetic spin–orbit contribution is overestimated, or when the spin–dipole

共

SD

兲

and Fermi-contact

共

FC

兲

contributions are far from the MCSCF values. For the ¹⁹F nucleus, known to be difficult for DFT, the anisotropic properties of heteronuclear, in particular ¹⁹F¹³C couplings are often more accurate than the poorly described isotropic coupling constants. This happens since the FC contribution is small at fluorine compared with carbon, leading to a small error in the total SD/FC term. With the recent implementation of the hybrid B3LYP functional, calculations of predictive quality for the J tensors are no longer restricted to small model molecules, opening up the possibility of studying the anisotropic components of J in large organic and biomolecules of experimental interest. © 2002 American Institute of Physics.

关

DOI: 10.1063/1.1502243

兴

I. INTRODUCTION

The indirect spin–spin coupling tensor between the mag- netic nuclei K and L,

J_KL⫽J_KL1⫹J_KL^S ⫹J_KL^A ,

共

1

兲

is one of the main parameters determining the nuclear mag- netic resonance

共

NMR

兲

spectrum.^{1– 4}In isotropic liquids and gases, the only spin–spin coupling observable is the coupling constant, J⫽¹3Tr J. In anisotropic environments such as liq- uid crystals and solids, the second-rank symmetric tensorial part J^Salso has an effect on the spectrum.^5–7The role of the antisymmetric, rank-1 part J^Ahas been discussed^8,9but so far not experimentally observed

共

see also Ref. 10

兲

.

The spin–spin coupling tensor J is a second-order mo- lecular property corresponding to the second derivative of the total electronic energy with respect to the nuclear mag- netic moments of the coupled nuclei.⁴From the point of view of theoretical calculations, it differs from many other second-

order properties in being more computationally demanding.¹¹ First, there are several contributing physical mechanisms that are highly dependent on the quality of the wave function near and at the nuclei; second, some of the mechanisms in- volve triplet perturbation operators; third, there are a large number of response equations to be solved for each magnetic nucleus. Hence, qualitatively correct results call for post- Hartree–Fock methods that include electron correlation and do not suffer from triplet instability.¹¹

Recent accurate ab initio calculations of J have been carried out using the multiconfigurational self-consistent- field linear response

共

MCSCF LR

兲

method,¹² the second- order polarization–propagator approach with coupled-cluster singles-and-doubles amplitudes, SOPPA

共

CCSD

兲

,¹³ and the equation-of-motion coupled-cluster singles-and-doubles

共

EOM-CCSD

兲

method.^{14 –16} Reference 11 provides a recent review of the field. The anisotropic spin–spin components, in particular, have been reported in Refs. 10, 17–26

共

see also Ref. 27

兲

. The first successful calculations of J using density- functional theory

共

DFT

兲

^{28 –30}have also given rise to an inter- est in the calculation of the anisotropic J^S with DFT.^31–33

The first general implementations of all the contributions to the full nonrelativistic J tensor, using the three main classes of exchange–correlation functionals—that is, the

a兲Author to whom correspondence should be addressed. Electronic mail:

[email protected]

b兲Electronic mail: [email protected]

c兲Electronic mail: [email protected]

5998

0021-9606/2002/117(13)/5998/12/$19.00 © 2002 American Institute of Physics

local-density approximation

共

LDA

兲

,³⁴the generalized gradi- ent approximation

共

GGA

兲

, and hybrid functionals—were in- troduced in Refs. 35 and 36. There, all the contributions to the spin–spin couplings were calculated using analytical de- rivative techniques. The calculated J have been found to im- prove from LDA to GGA and further to the hybrid function- als. The results obtained with the hybrid three-parameter Becke–Lee–Yang–Parr

共

B3LYP

兲

functional^37,38 are espe- cially promising: the overal accuracy is almost as high as with that of the much more elaborate ab initio calculations.³⁶ Nevertheless, the results for ¹J_FH in the HF molecule have been found poor with all these functionals. This has been related to deficiencies in their long-range behavior, occurring for elements having lone-pair electrons such as fluorine.^28,29 In this work, we examine the zeroth- and second-rank parts of J produced by DFT with the LDA,³⁴ BLYP

共

Becke–Lee–Yang–Parr

兲

,^39,40 and B3LYP functionals, re- stricting our attention to molecules and couplings for which recent MCSCF data exist. In addition, an accurate MCSCF calculation on FHF^⫺has been carried out explicitly for this paper—for details, see Ref. 41. However, our focus is on the performance of DFT for the anisotropic second-rank part of spin–spin coupling tensors of several types. Where possible, the anisotropic properties calculated using DFT and MCSCF theory are compared with experimental data.

II. THEORY AND COMPUTATION A. Spin–spin coupling tensor

The nuclear magnetic moment ␮K is related to the nuclear spin I_K through

␮K⫽␥K

ប

I_K,

共

2

兲

where␥Kis the nuclear gyromagnetic ratio. The J tensor can be calculated as the second derivative of the perturbed elec- tronic energy with respect to the nuclear spins as

共

in units of Hz

兲

J_KL⫽1 h

d²E dI_KdI_L

冏

_I

K⫽I_L⫽0

⫽

ប

2␲ ␥^K␥LK_KL.

共

3

兲

Here K is the reduced spin–spin coupling tensor usually re- ported in units of 10¹⁹ T²J^⫺1. Since it describes the magni- tude of the pure electronic coupling without the nuclear fac- tors, it is particularly useful for comparing couplings between different elements and isotopes.

In the nonrelativistic theory of Ramsey,^4,5 the nuclear spins are coupled by four distinct interactions, giving rise to five different contributions to the coupling tensor:

J⫽J^DSO⫹J^PSO⫹J^SD⫹J^FC⫹J^SD/FC.

共

4

兲

The diamagnetic nuclear-spin–electron-orbit

共

DSO

兲

contri- bution is a ground-state expectation value of the DSO opera- tor, bilinear in I_Kand I_L:

J_KL^DSO

⬀ 具

^DSO

共

KL

兲 典

^.

^共

⁵

^兲

The remaining four paramagnetic terms in Eq.

共

4

兲

can be calculated as linear response functions of the corresponding perturbation operators,¹² each of which is linear in the nuclear spins:

J_KL^PSO

⬀ 具具

^PSO

共

K

兲

; PSO

共

L

兲 典典

0,

共

6

兲

J_KL^SD

⬀ 具具

^SD

共

K

兲

; SD

共

L

兲 典典

0,

共

7

兲

J_KL^FC

⬀ 具具

^FC

共

K

兲

; FC

共

L

兲 典典

0,

共

8

兲

J_KL^SD/FC

⬀ 具具

^FC

共

K

兲

; SD

共

L

兲 典典

0

⫹

具具

^FC

共

L

兲

; SD

共

K

兲 典典

0.

共

9

兲

The purely zeroth-rank Fermi-contact

共

FC

兲

term is isotropic and usually gives the most important contribution to J. Like- wise, the purely second-rank spin–dipole/Fermi-contact

共

SD/

FC

兲

cross term usually dominates the anisotropic part J^S. By contrast, the paramagnetic nuclear-spin–electron-orbit

共

PSO

兲

and spin–dipole

共

SD

兲

mechanisms, as well as the DSO mechanism, contribute to all the rank-0, -1, and -2 parts of J.

The rank-0 and -2 contributions can be presented either in the principal axis system

共

PAS

兲

or in a molecule-fixed frame common to all tensors. The former way, defined by the principal values and the directions of the principal axes, is conventional in solid state NMR, whereas the molecule-fixed frame is usually used in liquid–crystal

共

LC

兲

NMR. There, the observable corresponding to J^S is denoted as J^aniso.^6,7 The anisotropy of the tensor

⌬

J⫽J_zz⫺ ¹2

共

J_xx⫹J_{y y}

兲 共

10

兲

and, in certain point-group symmetries, the asymmetry pa- rameter

␩⫽J_xx⫺J_{y y}

J_zz

共

11

兲

enter the expression of J^aniso. Here, the z axis is parallel with the principal molecular symmetry axis. Other combinations of tensor elements may also contribute when the molecular symmetry is low.⁶

The experimentally observable anisotropic coupling, D_KL^exp⫽D_KL⫹¹2J_KL^aniso, contains the direct bare-nucleus dipole–

dipole coupling D_KL

共

carrying information about the internu- clear KL distance

兲

as well as the indirect electronic coupling J_KL^aniso.⁶ To extract structural information from D_KL^exp, J_KL^aniso must be known. Accurate calculations of the full J_KL tensors—not just their isotropic parts—are therefore impor- tant.

B. DFT calculations

The DFT calculations were carried out by using a local version of the DALTON quantum chemistry program⁴²with a recent DFT implementation of NMR spin–spin couplings.³⁶ Three exchange–correlation functionals were used: LDA, GGA

共

BLYP

兲

, and hybrid

共

B3LYP

兲

. We have used the same geometries as in the previous10,17–24,27 and present⁴¹MCSCF studies. The basis sets are the same as in the best calculations of the cited works. Details of the geom- etries and basis sets are given in Table I.

In many cases, the segmented basis sets of Huzinaga,⁴³ as contracted and polarized by Kutzelnigg et al.,⁴⁴ were used. Because of their special contraction scheme,^45,46these compact basis sets

共

denoted HII, HIII, and HIV

兲

have been found to be particularly useful for spin–spin calculations.

While the small HII set provides qualitatively correct J ten- sors, quantitative accuracy is reached with the HIII and HIV sets augmented with tight primitives. By contrast, the correlation-consistent valence sets cc- pVXZ with

2

⭐

X

⭐

6^47,48 and their augmented counterparts aug-cc- pVXZ⁴⁸converge slowly for spin–spin couplings be- cause of insufficient flexibility in the core region.⁴⁶ For the correlation-consistent core–valence basis sets

TABLE I. Computational details for DFT calculations.^a

Molecule Basis set Geometry^b Reference^c

C2H2 HIV^d CC⫽1.207, CH⫽1.060, HCC⫽180.0 20

C2H4 HIV^d CC⫽1.338, CH⫽1.088, HCC⫽121.4 20

C2H6 HIII^e CC⫽1.535, CH⫽1.094, HCC⫽111.2 20

C6H6 HII^f CC⫽1.393, CH⫽1.082 18

HCN HIV^d CN⫽1.156, CH⫽1.064 17

HNC HIV^d CN⫽1.165, NH⫽0.994 17

CH3CN HIII^e CN⫽1.166, CC⫽1.461, CH⫽1.085, HCC⫽110.0 17 CH3NC HIII^e CN⫽1.424, CC⫽1.166, CH⫽1.094, HCN⫽109.1 17

HCONH2 HIII^e CN⫽1.352, CO⫽1.219, CH1⫽1.098, 19

NH2(trans to O)⫽1.001, NH3(cis to O)⫽1.002, OCN⫽124.7, HCN⫽112.7, CNH2⫽120.0, CNH3⫽118.5

CH3F HIII^e CF⫽1.383, CH⫽1.086, HCF⫽108.8 23

CH2F2 HIII^e CF⫽1.354, CH⫽1.092, HCF⫽108.8, HCH⫽112.9 23

CHF3 HIII^e CF⫽1.331, CH⫽1.086, HCF⫽110.3 23

p-C6H4F2g

HIIs3^h CF⫽1.368, CH⫽1.093, C7C8⫽1.399, C8C9⫽1.408, 21 C12C7C8⫽122.8, H1C8C9⫽121.4

FHF^⫺ HIVu(t4,d1)ⁱ HF⫽1.152, FHF⫽180.0 41

ClF3 cc- pVQZ^j,k ClFax⫽1.599, ClFeq⫽1.701, FaxClFeq⫽87.5 10

OF2 cc- pCVQZ^l OF⫽1.405, FOF⫽103.4 10

CH3SiH3 HIII^e CSi⫽1.864, CH⫽1.095, SiH⫽1.482, HCSi⫽110.9, CSiH⫽110.4

22

HF HIIIsu4^m HF⫽0.918 27

HCl HIIIsu4^m HCl⫽1.275 27

H2O HIIIsu4^m HO⫽0.968, HOH⫽75.5 27

H2S HIIIsu4^m HS⫽1.336, HSH⫽87.9 27

NH3 HIIIsu4^m HN⫽1.012, HNH⫽106.7 27

PH3 HIIIsu4^m HP⫽1.412, HPH⫽93.3 27

CH4 HIIIsu4^m CH⫽1.086, HCH⫽109.5 27

SiH4 HIIIsu4^m SiH⫽1.471, HSiH⫽109.5 27

LiF cc- pV5Zⁿ LiF⫽1.564 24

BF cc- pV5Zⁿ BF⫽1.263 24

NaF F: cc- pV5Z,ⁿNa:^o NaF⫽1.926 24

AlF aug-cc- pVQZ^p AlF⫽1.654 24

ClF aug-cc- pVQZ^p ClF⫽1.628 24

KF F: cc- pV5Z,ⁿK:^o KF⫽2.171 24

LiH cc- pV5Zⁿ LiH⫽1.596 24

Na2

o NaNa⫽3.079 24

KNa ^o KNa⫽3.499 24

aBasis sets and geometries common to present DFT as well as present共in the case of FHF^⫺兲and previous共see references兲MCSCF calculations. See the references listed in the last column for complete details as well as literature citations of the basis sets used.

bThe units for the bond lengths and angles are A˚ ngstro¨ms and degrees, respectively.

cReference to the earlier MCSCF calculation.

dC–F,关11s7 p3d1 f /8s7 p3d1 f兴; H,关6s3 p1d/5s3 p1d兴in the关primitive/contracted兴notation.

eSi,关12s8 p3d/8s7 p3d兴; C–F,关11s7 p2d/7s6 p2d兴; H,关6s2 p/4s2 p兴.

fC,关9s5 p1d/5s4 p1d兴; H,关5s1 p/3s1 p兴.

gThe numbering of nuclei refers to Fig. 1 in Ref. 21.

hHII augmented with three sets of tight s functions keeping the original contraction for C and F. C–F, 关12s5 p1d/8s5 p1d兴; H,关5s1 p/3s1 p兴.

iHIV fully decontracted and augmented with four sets of tight s共H兲and s p共F兲functions and one set of diffuse s pd共H兲and s pd f 共F兲functions. F,关16s12p4d2 f兴; H,关11s4 p2d兴.

jF,关12s6 p3d2 f 1g/5s4 p3d2 f 1g兴; Cl,关16s11p3d2 f 1g/6s5 p3d2 f 1g兴.

k共ax兲 denotes axial fluorine in the C2 symmetry axis and共eq兲the two equatorial fluorines. The 共ax兲/共eq兲 nomenclature has been interchanged in Ref. 10.

lO–F,关15s9 p5d3 f 1g/8s7 p5d3 f 1g兴.

mHIII basis set fully decontracted and augmented with four sets of tight s共H兲and s p共C–Cl兲functions. Si–Cl, 关16s12p3d兴; C–F,关15s11p2d兴; H,关10s2 p兴.

nLi–F,关14s7 p4d3 f 2g1h/6s5 p4d3 f 2g1h兴.

oUncontracted basis set: K,关26s17p1d兴; Na,关20s12p4d兴.

pF,关13s7 p4d3 f 2g/6s5 p4d3 f 2g兴; Al–Cl,关17s12p4d3 f 2g/7s6 p4d3 f 2g兴.

cc- pCVXZ,^47,49the situation is again different as the basis is well-converged already at the quadruple-zeta cc- pCVQZ level.

In the presentation of the results, the molecular axis frame is fixed and the anisotropic properties are defined as in the original papers. In most cases, this means that the z axis is along the principal molecular symmetry axis, although PAS is used for the molecules in Ref. 17. These choices are made to simplify the comparison of the DFT calculations with the previously reported MCSCF results.

C. The quality of the MCSCF reference data

For a complete specification of the MCSCF wave func- tions, we refer to the original papers and Table S2 in supple- mentary material obtainable from EPAPS.⁵⁰Here we restrict ourselves to some general comments.

The active spaces for HCN, HNC, CH₃CN, CH₃NC,¹⁷ and C₆H₄F₂

共

Ref. 21

兲

are of the multireference restricted- active-space

共

RAS

兲

⁵¹ type, with small RAS3 virtual- excitation orbital spaces. For these molecules, therefore, dy- namical correlation is only partly recovered.^17,21This is also true for C₆H₆, where the CC coupling tensors were calcu- lated with a small multireference RAS wave function and the CH tensors with a complete-active-space

共

CAS

兲

wave function.¹⁸Similarly, the small CAS expansions used for the diatomic molecules in Ref. 24 and the small single-reference RAS expansion used for OF₂ and ClF₃ in Ref. 10 can only be expected to provide qualitative results.

The single-reference RAS active space used for HCONH₂ was quite large but nevertheless allowed only single and double excitations into the virtual orbital space.¹⁹ The lack of higher excitations may compromise the accuracy, although the results are quite well converged in the sequence of calculations presented in Ref. 19. For the singly bonded systems C₂H₆,²⁰ CH₃SiH₃,²² and CH_4⫺nF_n (n⫽1,2,3),²³ static correlation is unimportant close to the equilibrium ge- ometry. Therefore, the large single-reference RAS wave functions used for these molecules should provide high- quality tensors. Likewise, the large multireference RAS treatments of the doubly and triply bonded C₂H₄, and C₂H₂ should be quite accurate.²⁰ Finally, the large multireference RAS functions used for FHF^⫺ in the present work and for the first-row hydrides²⁷ as well as the large single-reference RAS wave functions correlating also the semicore orbitals for second-row hydrides in Ref. 27 should ensure high qual- ity of the calculated J.

The present DFT results are compared also with experi- mental data, which contain relativistic, rovibrational, and solvent contributions. These contributions are not recovered by the DFT and MCSCF calculations discussed in this paper.

III. RESULTS AND DISCUSSION

In this section, we compare the DFT, MCSCF, and ex- perimental results of the CC, FF, FC, CH, HH, FX, and other spin–spin coupling tensors in turn, and comment finally on the DFT performance in general. For brevity, we have listed the contributions from the different mechanisms to the J ten- sors only for the anisotropic components. Tables including

all the individual contributions to the CC, FF, FC, CH, and HH isotropic coupling constants as well as the contributions to the anisotropic components of CH and HH coupling ten- sors, are obtainable as supplementary material from EPAPS⁵⁰

共

see Tables S3–S7

兲

.

A. CC spin–spin coupling tensors

The DFT J_CC tensors and their contributions are com- pared with MCSCF and experimental data in Table II. The isotropic couplings J_CCare usually well described by BLYP and B3LYP, the latter being the slightly better functional.

Except in C₂H₂, the total J_CCare either as good as or better than the MCSCF values—notably for benzene, where com- promises in the ab initio work had to be made.¹⁸ This is an excellent result in the sense that, for the simple hydrocar- bons, the MCSCF wave functions are in fact quite accurate.²⁰ The main differences in the total J between DFT and MCSCF—in particular, the overestimation by B3LYP for C₂H₂ and the usual underestimation by LDA—arise from J^FC, the other contributions being similar for the different methods

共

see Table S3

兲

. For C₂H₆, the slightly smaller J^FC obtained with B3LYP gives a total coupling that is closer to the experimental value than with MCSCF. For CH₃CN and C₆H₆, the B3LYP results are much closer to experiment than are the previous small-RAS MCSCF results, suggesting that B3LYP is also better for CH₃NC.^17,18MCSCF theory over- estimates the magnitude of J^FCin C₆H₆, probably because of the small number of active orbitals.¹⁸

The results for the anisotropic part of the spin–spin cou- plings improve in the sequence LDA→BLYP→B3LYP.

B3LYP is the only functional that gives the correct sign for

⌬

¹J_CC^SD/FC in C₂H₂. For

⌬

J_CC in CH₃NC and CH₃CN, the results obtained with the B3LYP functional and with a small MCSCF wave function are almost identical. However, DFT overestimates the PSO and SD contributions to

⌬

¹J_CC and

1J_CC,zz␩ ^{in C}2H₄ and, in particular, to

⌬

¹J_CCin C₂H₂. We recall that high-quality MCSCF wave functions are used for the simple hydrocarbons.

For the anisotropic couplings in C₆H₆, the situation is not as clear as for the isotropic couplings. For the meta and para couplings, the B3LYP values are a bit closer to experi- ment than are the MCSCF values, due to the smaller

⌬

J^SD/FC. For the ortho coupling, on the other hand, B3LYP is much further away from experiment than is MCSCF, be- cause of a negative

⌬

J^SD/FCwith B3LYP. However, since the experimental uncertainties are large for the anisotropic com- ponents, and since

⌬

J^SD/FC may be overestimated by MC- SCF

共

due to the insufficient treatment of electron correla- tion

兲

, we cannot rule out a near-zero or even negative true value of

⌬

¹J_CC^SD/FC.

In short, whereas LDA fails to produce reliable J_CC, BLYP works satisfactorily in nearly all cases. However, B3LYP is generally the most accurate functional for the PSO, FC, SD, and SD/FC contributions, resulting in the best total J_CC. For all functionals, the description of these paramag- netic contributions deteriorates for couplings over multiple bonds. Overall, the B3LYP carbon–carbon tensors are simi- lar to the MCSCF tensors whenever the latter approach has been pursued far enough

共

e.g., simple hydrocarbons

兲

. How-

ever, the B3LYP results are superior to the MCSCF results in large systems such as CH₃CN, CH₃NC, and C₆H₆, where it is difficult to correlate enough electrons in sufficiently large MCSCF spaces.

B. FF spin–spin coupling tensors

The DFT fluorine–fluorine tensors J_FFand their contri- butions are compared with MCSCF and experimental data in Table III. For J_FF, the LDA→BLYP→B3LYP improve- ment is evident. Even though DFT strongly overestimates the magnitudes of J_FF^PSO and J_FF^SD

共

see Table S4

兲

, the total J_FF coupling is—in accordance with previous experi- ence^{28 –30,36}—grossly underestimated due to the erroneous J^FC

关

e.g., for CH₂F₂, ¹J_FF^FC⫽140.0 Hz

共

MCSCF

兲

,⫺36.8 Hz

共

LDA

兲

,⫺3.3 Hz

共

BLYP

兲

, and 47.2 Hz

共

B3LYP

兲兴

.

The B3LYP functional gives the correct sign of J_FFin all molecules, but the results are still far from the MCSCF re- sults, which are accurate in the present singly bonded systems—the only exception is p-C₆H₄F₂, for which only a small MCSCF calculation was possible.²¹In this case, J^FCis only slightly underestimated by DFT

关

⁵J_FF^FC⫽11.2 Hz

共

MC- SCF

兲

, 6.4 Hz

共

LDA

兲

, 7.5 Hz

共

BLYP

兲

, and 8.2 Hz

共

B3LYP

兲兴

. However, this underestimation is compensated for by an overestimation of the dominant J^SDcontribution, resulting in an overall good⁵J_FFeven though the small J^PSOcontribution has a sign opposite that of the MCSCF result.

We now turn our attention to the anisotropic fluorine–

fluorine couplings. Occasional apparently good total LDA values of the anisotropic components occur by error cancel- lation. In general, however, the B3LYP functional performs best. The B3LYP overestimation of the magnitude of the PSO term is even more pronounced than for the isotropic couplings, producing, in combination with the overestimated

⌬

J^SD/FC, a too large total

⌬

²J_FFfor FHF^⫺

共

relative to accu- rate MCSCF

兲

. For CH₂F₂, the

⌬

²J_FFand²J_FF,zz␩ ^{are quite} good with both BLYP and B3LYP, but only because of a cancellation of the overestimated SD/FC and PSO terms of opposite signs. In the same manner, B3LYP reproduces the

共

somewhat unreliable

兲

MCSCF results for

⌬

⁵J_FF and

5J_FF,zz␩^{in p-C}6H₄F₂quite well due to error cancellation. In fact, the only molecule for which the anisotropy is correct without error cancellation is CHF₃. Here, the slightly smaller DFT

⌬

²J_FF^SD/FC value causes a small difference in

⌬

²J_FFrelative to the MCSCF result, which is of good qual- ity for this molecule.

Although the improvement in the sequence LDA

→BLYP→B3LYP is quite pronounced for J_FF, fluorine clearly poses a problem for the present functionals. Even for a fairly large system such as p-C₆H₄F₂, a simple MCSCF wave function is superior to B3LYP in comparison with ex- periment. In some cases, however, error cancellation be- tween the PSO contribution and the other paramagnetic con- tributions reduces the B3LYP error. The B3LYP anisotropic couplings are therefore potentially useful—in particular, for large systems, for which accurate MCSCF calculations are not feasible.

TABLE II. Comparison of the indirectⁿJCCcoupling tensors共Hz兲calculated by DFT LR and MCSCF methods.^a

Mol.^b Prop.^c Contr. LDA BLYP B3LYP MCSCF Expt.

C2H2 1J Total 169.9 195.1 198.5 181.2 185.04^d

Ref. 20 ⌬¹J DSO 6.8 6.8 6.8 6.8

PSO 63.5 62.8 67.1 54.4

SD ⫺11.1 ⫺14.0 ⫺14.4 ⫺10.2 SD/FC 14.6 5.0 ⫺1.0 ⫺3.4 Total 73.8 60.6 58.6 47.5

C2H4 1J Total 48.4 66.9 70.5 70.2 67.5

Ref. 20 ⌬¹J DSO 5.0 5.0 5.0 5.0

PSO 8.4 8.7 9.0 7.5

SD ⫺2.9 ⫺3.9 ⫺4.3 ⫺3.3 SD/FC 24.8 23.4 20.0 17.3 Total 35.4 33.2 29.7 26.5 11.0

1Jzz␩ ^DSO ⫺0.1 ⫺0.1 ⫺0.1 ⫺0.1

PSO ⫺26.2 ⫺25.6 ⫺26.3 ⫺22.7 SD ⫺3.0 ⫺4.5 ⫺5.5 ⫺4.1 SD/FC ⫺4.1 ⫺14.9 ⫺19.6 ⫺17.5 Total ⫺33.4 ⫺45.1 ⫺51.5 ⫺44.3 ⫺44

C2H6 1J Total 18.6 29.5 32.6 38.8 34.6

Ref. 20 ⌬¹J DSO 3.3 3.3 3.3 3.3

PSO ⫺2.6 ⫺2.4 ⫺2.3 ⫺2.3

SD 1.5 1.6 1.6 1.5

SD/FC 25.0 30.9 31.1 29.6 Total 27.2 33.5 33.7 32.1 56.0 CH3CN^{e 1}J Total 42.3 55.4 60.2 72.0 58.0

Ref. 17 ⌬¹J DSO 3.7 3.7 3.7

PSO 0.2 0.1 ⫺0.1

SD 0.7 0.8 0.8

SD/FC 30.4 34.9 34.3

Total 35.0 39.4 38.7 36.6 30

CH₃NC^{f 2}J Total ⫺10.9 ⫺11.6 ⫺9.1 ⫺5.2

Ref. 17 ⌬²J DSO 1.1 1.1 1.1

PSO 3.8 3.2 3.1

SD 0.0 0.1 0.2

SD/FC 5.2 6.8 7.3

Total 10.1 11.2 11.7 11.6

C6H6 1J Total 42.4 58.2 61.6 70.9 55.8

Ref. 18 ⌬¹J DSO ⫺2.1 ⫺2.1 ⫺2.1 ⫺2.3

PSO 10.5 10.3 10.6 10.3

SD 0.5 0.9 1.3 1.5

SD/FC ⫺10.3 ⫺8.0 ⫺5.3 2.3

Total ⫺1.4 1.2 4.6 11.0 17.5

2J Total 0.5 ⫺0.4 ⫺1.9 ⫺5.0 ⫺2.5

⌬²J DSO ⫺0.5 ⫺0.5 ⫺0.5 ⫺0.3

PSO 0.1 0.1 0.1 0.3

SD ⫺0.3 ⫺0.6 ⫺1.1 ⫺1.5 SD/FC ⫺2.9 ⫺4.8 ⫺6.9 ⫺11.3 Total ⫺3.5 ⫺5.8 ⫺8.3 ⫺12.7 ⫺3.9

3J Total 9.6 10.4 11.0 19.1 10.1

⌬³J DSO ⫺0.3 ⫺0.3 ⫺0.3 ⫺0.5

PSO ⫺0.9 ⫺0.8 ⫺0.8 ⫺0.7

SD 1.7 2.2 2.6 2.7

SD/FC 0.4 3.7 5.9 11.2

Total 0.8 4.8 7.5 12.8 9.5

aAll DFT calculations have been carried out using the same geometries and basis sets as in the MCSCF calculations. See Table I for details.

bReference is to the earlier MCSCF calculation. The original paper contains also the references to the experimental data.

cAnisotropic properties are defined as in the original papers: anisotropy as

⌬J⫽Jzz⫺(Jxx⫹Jy y)/2 and asymmetry as Jzz␩⫽Jxx⫺Jy y.

dEquilibrium value from Ref. 26.

eAnisotropy defined as⌬J⫽J33⫺(J11⫹J22)/2, where the principal values are ordered according to the convention兩J33兩⭓兩J22兩⭓兩J11兩.

fAs in footnote e but anisotropy defined as⌬J⫽J11⫺(J22⫹J33)/2.

C. FC spin–spin coupling tensors

The DFT J_FCtensor and its contributions are compared with MCSCF and experimental data in Table IV. As for J_FF, DFT fails in most cases to describe J_FC properly. Even though B3LYP outperforms BLYP and LDA in all cases, its overestimation of the magnitude of¹J_FC^FCis still significant for all molecules

共

see Table S5

兲

. The J_FC^PSO contribution is also poorly treated by DFT.

For the total²J_FC,³J_FC, and⁴J_FCin p-C₆H₄F₂, DFT is in closer agreement with experiment than MCSCF. It thus seems that, whereas DFT has difficulties with ¹J_FC, its per- formance for long-range couplings transmitted over the aro- matic ring structure is much better. There, the assignment of

the different couplings according to their magnitudes is al- ways the same as in the experimental data, contrary to MC- SCF for ³J_FCand⁴J_FC.

For the anisotropic components,

⌬

J_FCand J_FC,zz␩^{, DFT} somewhat surprisingly works very well compared with MC- SCF. Whereas DFT usually overestimates the magnitude of the PSO term, the DSO and SD values are satisfactory. As in the isotropic case, B3LYP works slightly better than BLYP, while the LDA results are usually worse.

To analyze the problem of DFT with the fluorine cou- plings further, we have, in Table IV, listed the individual contributions—that is, SD

共

F

兲

/FC

共

C

兲

and SD

共

C

兲

/FC

共

F

兲

of Eq.

共

9

兲

—to the dominant anisotropic SD/FC term of the ⁿJ_FC

TABLE III. Comparison of the indirect ⁿJFF coupling tensors 共Hz兲 calculated by DFT LR and MCSCF methods.^a

Mol.^b Prop.^c Contr. LDA BLYP B3LYP MCSCF Expt.

ClF3d 2J Total 519.1 558.7 503.6 404.0 403.0

Ref. 10 ⌬²J Total 437.1 616.1 637.0 720.0

2J33␩ ^Total ⫺491.2 ⫺435.5 ⫺391.3 ⫺192.0

FHF^{⫺e 2}J Total ⫺194.5 ⫺124.6 20.5 238.6 220.0^f

Ref. 41 ⌬²J DSO 26.8 26.9 26.9 25.9

PSO 688.8 678.1 646.3 501.0

SD 8.4 ⫺2.1 10.5 27.7

SD/FC 1290.6 1440.7 1421.4 1214.0

Total 2014.6 2143.6 2105.2 1768.6

CH2F2 2

J Total 243.8 247.2 292.2 346.2

Ref. 23 ⌬²J DSO ⫺17.0 ⫺17.1 ⫺17.1 ⫺17.1

PSO ⫺380.9 ⫺364.6 ⫺343.2 ⫺267.7

SD 2.8 4.0 1.7 ⫺3.3

SD/FC 180.8 157.1 108.6 25.8

Total ⫺214.3 ⫺220.5 ⫺250.1 ⫺262.2

2Jzz␩ ^DSO ⫺32.1 ⫺32.2 ⫺32.2 ⫺32.2

PSO 627.7 583.1 550.4 440.1

SD ⫺132.7 ⫺137.7 ⫺133.6 ⫺113.5

SD/FC ⫺430.5 ⫺485.1 ⫺474.0 ⫺434.0

Total 32.3 ⫺72.0 ⫺89.5 ⫺139.6

CHF3 2J Total ⫺27.0 ⫺5.6 53.2 152.4

Ref. 23 ⌬²J DSO ⫺16.8 ⫺16.8 ⫺16.8 ⫺16.8

PSO ⫺26.1 ⫺30.2 ⫺30.0 ⫺28.3

SD ⫺21.0 ⫺22.5 ⫺24.5 ⫺24.3

SD/FC ⫺98.7 ⫺129.1 ⫺144.5 ⫺162.8

Total ⫺162.6 ⫺198.7 ⫺215.9 ⫺232.1 ⫺200.0

p-C6H4F2 5J Total 18.6 20.1 22.6 21.6 17.4

Ref. 21 ⌬⁵J DSO 3.7 3.7 3.7 3.7

PSO ⫺42.2 ⫺38.9 ⫺33.0 ⫺16.4

SD ⫺3.8 ⫺4.7 ⫺4.8 1.9

SD/FC 7.4 ⫺2.4 ⫺10.4 ⫺25.4

Total ⫺34.8 ⫺42.3 ⫺44.6 ⫺36.2 ⫺36.5

5Jzz␩ ^DSO ⫺0.2 ⫺0.2 ⫺0.2 ⫺0.2

PSO 32.8 27.3 24.4 13.4

SD ⫺24.4 ⫺27.2 ⫺28.3 ⫺19.7

SD/FC ⫺2.6 ⫺18.4 ⫺28.1 ⫺31.6

Total 5.7 ⫺18.4 ⫺32.2 ⫺38.1 ⫺38.4

aSee footnote a in Table II.

bSee footnote b in Table II.

cSee footnote c in Table II.

dCoupling between the equatorial and axial fluorines. In the principal axis system, where the principal axes are ordered according to the convention兩J33兩⭓兩J22兩⭓兩J11兩. Anisotropy defined as ⌬J⫽J33⫺(J11⫹J22)/2 and asymmetry as J33␩⫽J11⫺J22.

eMCSCF calculations carried out presently. See Refs. 41 and 52.

fEstimated experimental value from Ref. 53.

TABLE IV. Comparison of the indirectⁿJFC coupling tensors 共Hz兲 calculated by DFT LR and MCSCF methods.^a

Mol.^b Prop.^c Contr. LDA BLYP B3LYP MCSCF Expt.

CH3F ¹J Total ⫺230.8 ⫺250.1 ⫺225.3 ⫺156.6 ⫺163.0

Ref. 23 ⌬¹J DSO 23.4 23.5 23.5 23.5

PSO ⫺89.3 ⫺78.0 ⫺79.0 ⫺75.4

SD 33.7 36.1 36.3 32.3

SD共F兲/FC共C兲 181.6 225.6 232.1

SD共C兲/FC共F兲 ⫺15.2 ⫺12.5 ⫺5.9

SD/FC^d 166.4 213.1 226.2 227.4

Total 134.3 194.8 207.0 207.8 350

CH2F2 1

J Total ⫺324.8 ⫺342.0 ⫺309.9 ⫺220.7 ⫺233.9

Ref. 23 ⌬¹J DSO 0.5 0.5 0.5 0.5

PSO 59.5 56.0 54.5 46.5

SD ⫺1.6 ⫺1.5 ⫺1.4 ⫺0.9

SD共F兲/FC共C兲 ⫺61.0 ⫺60.9 ⫺58.6 SD共C兲/FC共F兲 13.4 13.7 14.2

SD/FC^d ⫺47.6 ⫺47.2 ⫺44.5 ⫺35.8

Total 10.8 7.9 9.2 10.4 13.5

1Jzz␩ ^DSO ⫺16.0 ⫺16.1 ⫺16.1 ⫺16.0

PSO ⫺72.0 ⫺70.8 ⫺65.8 ⫺49.9

SD ⫺7.5 ⫺8.6 ⫺9.5 ⫺10.0

SD共F兲/FC共C兲 ⫺178.1 ⫺210.1 ⫺214.4 SD共C兲/FC共F兲 18.2 16.0 11.5

SD/FC^d ⫺159.9 ⫺194.1 ⫺202.9 ⫺204.4

Total ⫺255.4 ⫺289.5 ⫺294.2 ⫺280.3 ⫺360

CHF3 1J Total ⫺372.9 ⫺390.9 ⫺354.1 ⫺242.1 ⫺272.2

Ref. 23 ⌬¹J DSO ⫺7.8 ⫺7.9 ⫺7.9 ⫺7.9

PSO ⫺46.9 ⫺45.9 ⫺44.7 ⫺35.0

SD ⫺0.8 ⫺1.1 ⫺1.4 ⫺1.9

SD共F兲/FC共C兲 ⫺121.3 ⫺138.7 ⫺141.7 SD共C兲/FC共F兲 19.9 18.3 16.1

SD/FC^d ⫺101.5 ⫺120.4 ⫺125.6 ⫺128.5

Total ⫺157.0 ⫺175.3 ⫺179.7 ⫺173.3

p-C6H4F2 1J Total ⫺371.2 ⫺390.1 ⫺358.2 ⫺184.7 ⫺242.6

Ref. 21 ⌬¹J DSO 23.9 24.0 24.0 24.0

PSO ⫺44.0 ⫺39.8 ⫺42.9 ⫺59.6

SD 27.8 30.7 32.1 34.2

SD共F兲/FC共C兲 222.6 269.8 285.1 SD共C兲/FC共F兲 ⫺22.1 ⫺16.2 ⫺9.6

SD/FC^d 200.6 253.6 275.6 370.1

Total 208.2 268.5 288.7 368.8 400

1Jzz␩ ^DSO ⫺0.5 ⫺0.5 ⫺0.5 ⫺0.5

PSO ⫺72.6 ⫺69.1 ⫺65.7 ⫺31.0

SD 14.9 17.4 19.1 16.5

SD共F兲/FC共C兲 0.9 5.7 9.6

SD共C兲/FC共F兲 2.7 6.9 9.1

SD/FC^d 3.6 12.5 18.6 26.5

Total ⫺54.6 ⫺39.7 ⫺28.4 11.5 13

2J Total 7.5 16.3 21.7 42.5 24.3

⌬²J DSO 3.2 3.2 3.2 3.2

PSO ⫺11.7 ⫺10.8 ⫺10.2 ⫺6.9

SD ⫺1.7 ⫺2.3 ⫺2.7 ⫺1.6

SD共F兲/FC共C兲 ⫺11.1 ⫺15.6 ⫺16.5

SD共C兲/FC共F兲 5.0 3.4 1.5

SD/FC^d ⫺6.1 ⫺12.1 ⫺15.0 ⫺31.7

Total ⫺16.2 ⫺22.0 ⫺24.7 ⫺36.9 ⫺39

2Jzz␩ ^{DSO 1.2 1.2 1.2 1.2}

PSO ⫺17.4 ⫺15.5 ⫺16.0 ⫺11.9

SD ⫺4.4 ⫺5.9 ⫺7.1 ⫺7.3

SD共F兲/FC共C兲 34.6 29.7 23.4

SD共C兲/FC共F兲 ⫺1.2 ⫺3.7 ⫺5.9

SD/FC^d 33.4 25.9 17.6 ⫺1.4

Total 12.9 5.7 ⫺4.4 ⫺19.4 ⫺20.5

3J Total 6.3 6.8 5.2 3.5 8.2