Theoretical Prediction of the Spin–Spin Coupling Constants between

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Theoretical Prediction of the Spin Spin Coupling Constants between an Axis and Macrocycle of a Rotaxane Anna Pikulska and Mazgorzata Kauch Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warszawa, Poland

Magdalena Pecul* Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warszawa, Poland, and Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Tromsø, N-9037 Tromsø, Norway ABSTRACT: The indirect nuclear spin spin coupling constants between nuclei belonging to the axis and to the macrocycle of three structurally related rotaxanes have been calculated by means of density functional theory. It has been shown that the through-space axis-macrocycle proton proton coupling constants can be as large as 0.4 0.5 Hz and therefore of measurable values. The largest through-space axis-macrocycle carbon proton and nitrogen-proton coupling constants are 0.2 0.3 Hz. Visualization of coupling pathways by means of the coupling energy density method indicates that the larger proton proton couplings are indeed transmitted through the space between the coupled nuclei. Thus, it seems that measurement of such couplings should be possible and that indirect spin spin couplings can be actually transmitted through-space, with no covalent or hydrogen bonds between the coupled nuclei.

I. INTRODUCTION Spin spin coupling constants measured by nuclear magnetic resonance (NMR) spectroscopic experiments in nonoriented media (liquids, gas phase), so-called indirect or scalar spin spin coupling constants, result, roughly speaking, from polarization of electron spin and orbital motion by magnetic moments of nuclei. They have been usually interpreted as transmitted though covalent bonds between the atoms, although there has been experimental evidence as early as 19611,2 of so-called “through-space” coupling constants, that is, of substantial spin spin couplings between nuclei separated by several bonds, but in spatial proximity. Still, the first measurements of coupling constants between nuclei actually belonging to two different molecules, transmitted only through hydrogen bonds (between complementary pairs of nucleic bases in RNA3 and DNA4 and between fluorine-containing hydrogen-bonded organic molecules5) can be considered a breakthrough in this area of NMR spectroscopy: it has been unambigously shown that transmission of nuclear spin spin coupling does not require a conventional covalent bond and may take place through intermolecular interactions. In this situation, a question arises: can indirect spin spin coupling constants be transmitted by intermolecular van der Waals interactions weaker than hydrogen bonds? Or, in other words, what is the magnitude of indirect spin spin coupling constants between nuclei in different molecules, which are not bound (not even by hydrogen bond), but which are in spatial proximity? It has been already shown, also by our group,6 8 that theoretical calculations predict non-negligible values of spin spin r 2011 American Chemical Society

coupling constants between nuclei in molecules interacting only by weak van der Waals forces. However, such finding cannot be verified experimentally, since thermal motion of the molecules precludes such measurements. Thus, we decided to use as model compounds molecules which are composed of fragments not bound by covalent bonds but held together because of a steric barrier: rotaxanes. Rotaxanes are objects composed of two molecular moieties: a macrocycle, a ring-shaped rigid molecule usually containing aromatic groups, and an axis (or thread), onto which the macrocycle is mechanically interlocked. Thus, although there is no covalent bond between the macrocycle and axis, the components are held together by a so-called mechanical bond. This should in principle create a possibility of measurement of indirect coupling constants between a nucleus in the axis and a nucleus in the macrocycle, provided the couplings are large enough to be measured. To predict the values of such coupling is the aim of the present study. Rotaxanes have attracted much attention as molecular machines, thus axis-macrocycle spin spin coupling constants would be not only of theoretical interest (as evidence of indisputable “through space” transmission of indirect nuclear coupling) but also of practical interest as structural parameters of systems important in nanotechnology. Since the rotaxanes under study contain amino acid moieties (see below) they can also provide some Received: June 10, 2011 Revised: August 23, 2011 Published: August 24, 2011 10795

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Figure 3. The structure of the rotaxane Gly-L-Ala.

Figure 1. The structure of the rotaxane Gly-L-Leu.

Figure 2. The structure of the rotaxane Gly-L-Met.

indications concerning the analogous couplings in interlocking peptides with rotaxane-like structures.9 11 The calculations have been carried out for three rotaxanes, synthetized and described by Brancato et al.12 We have selected rotaxanes in which the axis and macrocycle are additionally stabilized by hydrogen bonds in order to minimize dynamics (mainly rotation of the macrocycle around the axis),13,14 which would impede experimental verification of our results. The hydrogen bonds are formed in them because of the presence of amino acid units: glycine and leucine in rotaxane named by Brancato et al.12 Gly-L-Leu and shown in Figure 1, glycine and metionine in rotaxane named by Brancato et al.12 Gly-L-Met and shown in Figure 2, and glycine and alanine in rotaxane named by Brancato et al.12 Gly-L-Ala and shown in Figure 3. However, the nuclei between which the

coupling constants are discussed here are not connected by hydrogen bonds. Apart from the prediction of the values of the coupling constants, we have carried out a visualization of them by means of the coupling energy density method,15,16 where the difference between electronic energy density for parallel and antiparallel nuclear spins is calculated. In this manner coupling paths can be established. The paper is organized as follows. First, the computational details are given and then the calculated coupling constants and their interpretation in terms of coupling pathways are discussed. The paper concludes with a summary.

II. COMPUTATIONAL DETAILS The geometry of the rotaxanes under study (with phenyl rings from stoppers replaced by hydrogen atoms to reduce computational requirements) has been optimized at the Hartree Fock level, using the 6-311G** basis set.17 Some test calculations have been also carried out directly for the crystallographic data, with C H bonds fixed at an arbitrary length of 1.0 Å. We present only the first set of results, since they correspond to a more physical situation (especially since our main interest is in the proton proton couplings), and proton proton distances would be significantly affected by the arbitrary positions of protons, distorting in turn prediction of proton proton coupling constants. The basis set Huz-II-su218,19 has been selected on the basis of preliminary results obtained for a small model of rotaxane (with only the closest axis macrocycle contacts kept, and all remaining atoms of the system removed), as performing well in comparison with the larger basis sets (such as Huz-IV-su418,19 or augcc-pVDZ-su2: aug-cc-pVDZ20 23 with s functions decontracted and two tight s functions added) and still having a moderate size. The B3LYP exchange-correlation functional has been chosen for all calculations, since this standard functional performs well for the spin spin coupling constants (see ref 24 for a review). In the nonrelativistic theory of Ramsey,25 the indirect nuclear spin spin coupling constants are composed of four terms: Fermi contact (FC) term, spin-dipole (SD) term, and dia- and paramagnetic spin orbit terms (DSO and PSO term, respectively). The most time-consuming SD term has been omitted in the calculations for large systems, since our preliminary results obtained 10796

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Table 1. Through-Space Proton Proton Coupling Constants between Axis and Macrocycle (J(H 3 3 3 H), in Hz) and Corresponding Internuclear Distances (R, in Å) Gly-L-Leu

a

Gly-L-Met

Gly-L-Ala

J(H 3 3 3 H)

R

J(H 3 3 3 H)

R

J(H 3 3 3 H)

R

H1 3 3 H1 3 3

3 H2 3 H3

0.26

2.42

0.30

2.64

0.31

2.73

0.22

3.40

a

3.52

0.36

2.46

H1 3 3 H5 3 3 H5 3 3 H5 3 3

3 H4

0.12

3.77

0.16

3.59

0.12

3.43

3 H2

0.34

2.71

0.11

3.39

0.07

3.93

3 H3 3 H4

0.38 0.59

2.43 2.15

a

3.19 2.35

0.36 0.23

2.75 2.75

0.44

Not calculated on account of large internuclear distances.

for a small model of rotaxane have shown that its contribution to the coupling constants under investigation is negligible. Following the calculations of spin spin coupling constants, the coupling energy density (CED) method relying on the double finite perturbation theory approach of Malkina and Malkin,15,16 have been applied for visualization and interpretation of the coupling pathways. The CED is the difference of two energy densities with parallel and antiparallel nuclear spins on the two nuclei of interest, defined in such manner that the space integral of it provides the reduced coupling constant between the nuclei. CED has been calculated with Perdew exchange-correlation potential and Huz-II-su218,19 basis set. The value of the perturbation parameter equal to 0.01 has been used. The cube of the dimension 29.5 au  22.5 au  17.0 au has been used. The calculations of the spin spin coupling constants have been carried out by means of the Dalton program.26 Gaussian 0927 has been used for geometry optimization, and the coupling energy density calculations and visualization of the coupling pathways have been carried out with a code based on ReSpect28 and deMon.29 The Molekel30 program has been used for the graphical output of the coupling energy density visualization.

III. RESULTS A. Axis Macrocycle H H Coupling Constants. The proton proton coupling constants between the nuclei belonging to the axis and to the macrocycle, as calculated at the B3LYP/HuzII-su2 level, are demonstrated in Table 1 for the three rotaxanes under investigation. The labels of the nuclei used in Table 1 and in the remaining part of the paper are shown in Figure 4. The pattern of the through-space couplings is similar in all three rotaxanes, but the internuclear distances depend on the substituent, affecting in turn the values of the coupling constants. The largest value of the J(H 3 3 3 H) couplings constant, 0.59 Hz, is predicted for the coupling between methylene proton from the axis and amino proton of the macrocycle in Gly-L-Leu, where the internuclear distance is 2.154 Å. The analogous coupling is only 0.23 Hz in GlyL-Ala, where the internuclear distance is 2.752 Å. Relatively substantial values (up to 0.38 Hz in Gly-L-Leu) are anticipated for the couplings between methylene protons from the axis and aryl protons from the macrocycle in two of the rotaxanes under study but not in Gly-L-Met, where the internuclear distances between these protons exceed 3 Å. All axis macrocycle proton proton couplings are predicted to be positive, similarly as the through-space proton proton coupling constants predicted for protons forming adjacent hydrogen

Figure 4. The numbering of the nuclei in the region of interest in rotaxanes.

Table 2. The Individual Contributions to the Total J(H 3 3 3 H) Coupling Constant in the Gly-L-Leu Rotaxane (in Hz) total

FC

DSO

PSO

H1 3 3 3 H2 H1 3 3 3 H3 H1 3 3 3 H4 H5 3 3 3 H2

0.26

0.22

5.01

4.54

0.22

0.02

2.31

2.10

0.12 0.34

0.05 0.02

1.43 3.78

1.26 3.46

H5 3 3 3 H3 H5 3 3 3 H4

0.38

0.12

4.82

4.32

0.59

0.12

6.36

5.65

bonds.31,32 It is also worth noting at this point that, in contrast to them, the dihydrogen-bond-transmitted proton proton coupling constants are predicted to be negative,8 even for a weakly bound complex LiH 3 3 3 H2. A possible explanation for this variation of the sign of the through-space proton proton coupling will be offered later in the article. It is instructive to examine the individual contributions to the coupling constants under investigation, shown in Table 2 for the Gly-L-Leu rotaxane. (They are analogous for the other two rotaxanes.) The through-space proton proton couplings are dominated by the interplay of the paramagnetic spin orbit (PSO) term (negative) and the diamagnetic spin orbit (DSO) term (positive), both terms much larger individually than the total coupling, with the FC term (usually negative) lowering the value of the total coupling usually approximately by half. The most time-consuming spin-dipole (SD) term has been neglected in the presented calculations, but tests have shown that it is usually 0.01 0.03 Hz for the couplings under study, and thus negligible. B. Axis Macrocycle Heteronuclear Coupling Constants. In addition to the through-space proton proton coupling constants, we have also calculated 15N 1H and 13C 1H axis macrocycle spin spin coupling constants. They are tabulated in Table 3, together with the individual contributions to them. Similarly as for the proton proton couplings, the SD contribution turned out to be negligible in test calculations and was omitted in subsequent modeling. The 15N 1H and 13C 1H coupling constants are, as a rule, smaller than the 1H 1H coupling constants and therefore less likely to be used in structural characterization of rotaxanes. However, their values still seem measurable. The largest of them are the 10797

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C 3 3 3 H3 coupling in the Gly-L-Ala rotaxane (0.30 Hz) and the N1 3 3 3 H1 coupling in the Gly-L-Leu rotaxane (0.26 Hz). Unlike the 1H 1H couplings, they are dominated by the FC term: even through the PSO and DSO terms are of larger absolute values, they nearly cancel each other, and the FC term determines the sign (positive) and magnitude of the total coupling. Only the smallest coupling under study, J(C 3 3 3 H4), has an opposite sign Table 3. The Axis Macrocycle J(C 3 3 3 H) and J(N 3 3 3 H) Coupling Constants and Their Individual Contributions (in Hz) Gly-L-Leu

total

FC

DSO

N1 3 3 3 H1 N2 3 3 3 H5 C 3 3 3 H2

0.26 0.12

0.24 0.14

0.22 0.22

C3 3 C3 3

PSO 0.20 0.24

0.22

0.14

0.85

0.77

3 H3

0.23

0.18

0.62

0.58

3 H4

0.05

0.02

0.80

0.74

total

FC

Gly-L-Met

DSO

PSO

N1 3 3 3 H1 N2 3 3 3 H5 C 3 3 3 H2 C 3 3 3 H4

0.12

0.10

0.18

0.17

0.15

0.15

0.21

0.20

0.12 0.13

0.07 0.06

0.64 0.72

0.59 0.66

Gly-L-Ala

total

FC

DSO

N1 3 3 3 H1 N2 3 3 3 H5 C 3 3 3 H2

0.06

0.04

0.22

0.20

0.13

0.11

0.20

0.18

C3 3 C3 3

PSO

0.09

0.05

0.49

0.45

3 H3

0.30

0.25

0.77

0.71

3 H4

0.15

0.09

0.57

0.51

than its FC term, but this may be simply caused by numerical noise. C. Distance Dependence. The distance dependence of the calculated 1H 1H coupling constants is shown in Figure 5 for all three rotaxanes under investigation. The total coupling decreases in an irregular fashion with the distance, but the dependence of the leading DSO and PSO terms is much smoother and less steep in the range of distances under study. The strongest distance dependence is exhibited by the FC term. This is as expected from the theory:33 the DSO and PSO terms should decay as R 2, while for the FC term a steep exponential decrease with the internuclear distance is predicted.32,33 The graph for the FC term shows that this contribution to a through-space 1H 1H coupling constant, negative for short internuclear distances, changes sign when the distance increases. Since the FC term depends more strongly on the internuclear distance than the other terms, it prevails for short internuclear distances (such as for the complexes bound by dihydrogen bonds8), causing the negative sign of the total coupling, while for the longer distances the spin orbit terms dominate, and the total coupling is positive. D. Visualization. As the final step we have used the coupling energy density method to obtain a better insight into the mechanism of transmission of the spin spin coupling constants between axis and macrocycle of a rotaxane. The example isodensity contours obtained for the H1 3 3 3 H3 coupling in Gly-L-Ala rotaxane (actually, for a model with the phenyl rings removed) are shown in Figure 6. The isosurfaces where the CED is 0.000112 are marked in blue and those where CED is 0.000112 are marked in orange. One can observe that the coupling is indeed a true through-space interaction: the most significant perturbations of electronic density take place in the space between the coupled

Figure 5. The dependence of the axis macrocycle JHH coupling constants on the internuclear distance: (a) the DSO term, (b) the FC term, (c) the PSO term, (d) the total coupling. 10798

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’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT We thank Olga Malkina for giving us access to the coupling energy density visualization program. This work has received support from the Wroczaw Centre for Networking and Supercomputing through a grant of computer time. ’ REFERENCES Figure 6. The isodensity contours obtained by means of coupling energy density method for the H1 3 3 3 H3 coupling in Gly-L-Ala rotaxane. The coupled nuclei are marked by black circles.

nuclei. However, this is not the case with smaller coupling constants, where the density changes are small and not well localized.

IV. SUMMARY AND OUTLOOK The indirect nuclear spin spin coupling constants between nuclei belonging to the axis and to the macrocycle of three structurally related rotaxanes have been calculated by means of density functional theory. The axis macrocycle J(H 3 3 3 H) coupling constants are predicted to be between 0.2 and 0.6 Hz for proton proton distances shorter than 3 Å, depending on the distance. The largest of the axis macrocycle 15N 1H couplings is 0.26 Hz, and of the 13C 1H couplings 0.30 Hz. Measurement of the intermolecular spin spin coupling constants of this magnitude is possible: Liu et al.34 have measured an intermolecular nitrogen nitrogen coupling constant as small as 0.136 Hz with an error of only 0.02 Hz, and there are several examples of measurements of the intermolecular carbon nitrogen coupling constants of 0.2 Hz (see for example ref 35 for a review). Proton proton coupling constants are usually more problematic because of proton exchange, but this should not be an issue in the case of aliphatic and aromatic protons in a rotaxane (or possibly a catenane). Also rotation of macrocycle around the axis can be, as mentioned before, to a large extent impeded. Thus, we consider such measurements to be a challenge for experimental NMR spectroscopists. The through-space axis macrocycle proton proton coupling constants are predicted to be positive, in contrast to the dihydrogen-bond-transmitted proton proton coupling constants.8 This is caused by the negative sign of the Fermi contact term for short proton proton distances, and steep decay of the FC term with the internuclear distance (so that the through-space axis macrocycle proton proton coupling constants are dominated by the spin orbit terms). The axis macrocycle 15N 1H and 13 C 1H coupling constants are predicted to be positive and dominated by the FC term. The visualization of the coupling path by means of the coupling energy density method indicates that the larger proton proton couplings are indeed transmitted through the space between the coupled nuclei. In the case of smaller coupling constants, the perturbations are too small to form any consistent pattern.

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