Indirect SpinâSpin Coupling Constants in the Hydrogen Isotopologues

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Indirect Spin–spin Coupling Constants in the Hydrogen Isotopologues Piotr Garbacz, Maciej Chotkowski, Zbigniew Rogulski, and Michal Jaszunski J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b04855 • Publication Date (Web): 22 Jun 2016 Downloaded from http://pubs.acs.org on June 22, 2016

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The Journal of Physical Chemistry

Indirect Spin–Spin Coupling Constants in the Hydrogen Isotopologues Piotr Garbacz

∗1

, Maciej Chotkowski2 , Zbigniew Rogulski2 , and Michal Jaszu´ nski3

1

Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland Faculty of Chemistry, Biological and Chemical Research Centre, University of Warsaw, ˙ Zwirki i Wigury 101, 02-089 Warsaw, Poland 3 Institute of Organic Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland

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June 22, 2016

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Abstract

Studies of the spectroscopic properties of simple molecules are an excellent test bed for comparison of the results obtained in the experiment and in the theory. Moreover, the detailed understanding of these properties is of great importance for their applications as sensitive tools in physicochemical applications. In this paper we focus on accurate analysis of one of these properties, namely the indirect spin–spin coupling observed in NMR (nuclear magnetic resonance) spectroscopy. The spin–spin coupling constants can be observed in very small systems, such as molecular hydrogen, with its value of some interest for the study of basic physical interactions 1–3 and they are very useful in the studies of large systems, such as biomolecules, where they can be applied to determine the structure of the molecule. 4–6 The most accurate theoretical results are obtained for the smallest systems, in case of spin–spin coupling constant for the diatomic two-electron hydrogen molecule. This coupling is not directly observable by NMR for homonuclear diatomics, therefore it has to be studied for mixed hydrogen isotopologues, i.e., HD, HT, and DT. Our previous work 7 indicated that the discrepancy

The results of experimental and theoretical studies of indirect spin–spin coupling constants for hydrogen deuteride, hydrogen tritide, and deuterium tritide are described. The reduced coupling constants obtained from the gas-phase NMR (nuclear magnetic resonance) experiment conducted at 300 K are 2.338(1), 2.334(3), and 2.316(1) ×1020 T2 J−1 , while the ab initio values computed at the fullconfiguration-interaction level of the theory equal 2.349(3), 2.343(3), and 2.322(3) ×1020 T2 J−1 for HD, HT, and DT, respectively. The agreement of the experimental and theoretical results is improved when a proper treatment of the influence of nuclear relaxation on the NMR spectrum is applied. However, there is a minor discrepancy between experiment and the theory, exceeding the estimated error bars; potential sources of this discrepancy are discussed.

∗ Tel.

Introduction

+ 48 22 55 26 346, [email protected]

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by intermolecular interactions. J0 can be directly compared with the isolated molecule value derived from theory; however, the latter has to properly account for the zero-point-vibration (ZPV) and finite temperature at which the experiment is performed.

between the experimentally determined and computed indirect spin–spin coupling constant for hydrogen deuteride, 1 J(D, H) ≈ 43 Hz, does not exceed 0.1 Hz; further investigation suggested that this difference is larger, 0.17(5) Hz. 8 Therefore, it became of interest to compare also the results of state-of-art ab initio quantum mechanical computations with the results of the NMR measurements for the other two mixed hydrogen isotopologues: hydrogen and deuterium tritides. This comparison is not straightforward, since obtaining the experimental data of sufficient precision is a challenging task. We shall demonstrate that a careful interpretation of the measured spectrum is obligatory when high accuracy of the experimental results is required. The values of spin–spin coupling obtained from liquid-state NMR are perturbed by solvent-solute interactions. For example, dissolution of hydrogen deuteride in carbon tetrachloride decreases the multiplet splitting 0.5 Hz. 9 Therefore, in order to reduce the influence of intermolecular interactions one can take advantage of the gas-phase NMR experiment. However, in this case the influence of the nuclear relaxation on the NMR lineshape has to be analysed. This perturbation can be very large, for instance, the 1 H NMR doublet splitting observed for PH3 is approximately 25 Hz smaller than the spin–spin coupling. 10 An extended analysis is also mandatory for hydrogen isotopologues, since even for a well-separated multiplet this perturbation can reduce hydrogen deuteride triplet splitting observed in 1 H NMR spectrum by ≈ 1 Hz. 8 This effect varies greatly with the density of the sample and the gaseous solvent, e.g. for HD dissolved in CO2 it is much smaller then for pure HD. For sufficiently low pressure (a few bar) the HD multiplet observed in a pure gas collapses and only one broad line is observed. 11 The value of the coupling corrected for the influence of the nuclear relaxation on the NMR spectrum, J(ρ), is next extrapolated to the zero-density limit: J(ρ) = J0 + J1 ρ, (1)

2 2.1

Experimental methods Materials

Molecular tritium (activity 2.57 Ci/ml, pressure 0.48 bar, RC Tritec Ltd.), hydrogen deuteride (98 % D; Isotec), and carbon dioxide (99.8 %, Sigma Aldrich) were used as supplied. Tritium is a low energy beta emitter, therefore it must be handled according to relevant regulations for radioactive materials. In practice, the 1 mm glass walls of the ampoules which were used in the NMR experiments provide satisfactory protection. Prior to use platinum black (Sigma Aldrich) was washed by a 3:1 mixture of 98 % sulfuric acid and 30 % hydrogen peroxide and then three times by distilled water.

2.2

Preparation of the hydrogen isotopologues

The mixtures of hydrogen isotopologues were prepared by the exchange reaction between gaseous tritium and hydrogen deuteride: T2 + HD ⇀ ↽ HT + DT. First, the upper part below the glass joint of the breakseal ampoule with 2 Ci of the tritium gas was covered by approximately 10 mg of wet platinum black. The upper part of the ampoule was closed by a stopper with a valve. Then, the valve was connected to the vacuum pump and the platinum black was dried under vacuum for 1 hour. Next, hydrogen deuteride under pressure of 0.5 bar was transferred to the ampoule. Tritium and hydrogen deuteride were mixed after breaking by a small stainless steel cylinder a glass wall between two parts containing them. The gases were mixed by cooling the ends of the ampoule several times by turns in liquid nitrogen. The mixture was left for equilibration for 30 min. The equilibration time was estimated from the rate of exchange reaction between hydrogen and

where J0 is the rovibrationally averaged value of the spin–spin coupling for a single molecule, and J1 is the contribution to the spin–spin coupling induced 2


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deuterium, which is less than 5 min. Based on the 1 H, 2 H, and 3 H NMR spectra of the mixture one can estimate that after the exchange reaction the molar fractions of the isotopologues H2 :D2 T2 :HD:HT:DT are approximately 0.23:0.17:0.04:0.31:0.13:0.13. The fact that only some spin isomers of homonuclear hydrogen isotopologues contribute to the NMR signal was taken into account: 3/4 of H2 and T2 as well as 8/9 of D2 is visible for NMR measurement conducted at temperature 300 K.

2.3

and diffuse (1s1p1d) functions (see Table 1 of Ref. 7). Considering also the dependence on the potential, it was estimated that in the approach used the error of the final HD coupling constant does not exceed 0.05 Hz. Within the Born–Oppenheimer approximation, which is our starting point, the potential and the property curve for the reduced spin–spin coupling constant are (pointwise) exactly the same for all the isotopologues, thus there was no need to repeat now any of these calculations. Obviously, the differences appear when we incorporate the zero-point vibrational (ZPV) and temperature effects, and these have been for HT and DT computed in the same manner as previously done for HD. 7 Thus, in the numerical solution of the vibrational Schr¨ odinger equation for the dominant Fermi contact term the property curve determined applying the largest basis set has been used, an aug-pcJ-5+8s3p2d-1s property curve was used for the other components, whereas the potential-energy curve was calculated in the energy-optimized (uncontracted) aug-pV7Z basis set.

Preparation of the samples and measurements of NMR spectra

All the samples were prepared using a dedicated glass vacuum line. First, the mixture of hydrogen isotopologues was transferred to a glass ampoule 5 cm long, inner diameter of 2 mm and outer diameter of 3 mm. Then, a given amount of carbon dioxide of was transferred and condensed in the ampoule using liquid nitrogen. Finally, the ampoule was sealed by torch. 1 H NMR and 3 H NMR spectra were acquired using the Varian 500 MHz spectrometer. NMR measure4 Results ments of the hydrogen isotopologues mixtures with carbon dioxide were repeated three times at an inter- The spin–spin coupling constant for each of the isoval of one week. Data processing was performed using topologues was determined from the 1 H NMR and the Mathematica 10.0 18 and Origin 8.5 19 computer 3 H NMR spectra of the mixture of hydrogen isotopoprograms. logues and carbon dioxide using the line shapes analysis described in Ref. 8. The 1 H NMR spectrum of the mixture is a sum 3 Computational methods of signals from hydrogen, hydrogen deuteride, and hydrogen tritide: The applied computational methods, the calculations H H H and the results for the HD isotopologue have been deS H (ν) = SH (ν) + SHD (ν) + SHT (ν), (2) 2 scribed in Ref. 7. In particular, the coupled cluster 3 CCSD method is for a two-electron system equiva- while the H NMR spectrum of this mixture is a sum lent to full configuration interaction and the basis of signals from tritium, hydrogen tritide, and deuset dependence of the results has been analysed in terium tritide: detail. The accuracy of the computed results, reT T T S T (ν) = ST (ν) + SHT (ν) + SDT (ν), (3) 2 flecting their dependence on the potential curve and the property curve, has been examined for HD. In where ν is frequency. particular, the convergence of all the components of The signal of each hydrogen isotopologue is prothe coupling constant with the extension of the ba- portional to the real part of the sum of the elements N sis set has been studied. The finally used, largest of a matrix MN N ′ (ν) defined as: basis set, called aug-pcJ-5+8s3p2d-1s1p1d, was con−1 N N N structed adding to the aug-pcJ-5 basis tight (8s3p2d) MN , (4) N ′ (ν) = LN N ′ (ν) + RN N ′ (ν) 3



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where N, N ′ ∈ {H, D, T}, the matrix LN N N ′ (ν) de- is the inverse of the sum of two matrices:  H  scribes the structure of the multiplet, the matrix ∆ωHD − JHD 0 0 N (ν) describes the perturbation of the spectrum RN ′ N H H  , 0 ∆νHD 0 by nuclear relaxation, and N 6= N ′ . The elements LHD (ν) = ı H 0 0 ∆ν + J N HD HD of the matrix MN N ′ (ν) are parametrized taking into (7) account the constraints which follow from the Bloch  12 −c d f Redfield–Wangsness relaxation theory. Then, the H (8) RHD (ν) =  d −e d  , best fit of NMR lineshape to the experimental specf d −c trum allows to find the spin–spin coupling constants of the mixed isotopologues. with the notation in Eqs. (7) and (8) following that H T of Eqs. (5) and (6). The parameters c, d, e, f of the The signals SH (ν) and S (ν) are Lorentzian, T2 2 however, for mixed hydrogen isotopologues, i.e., HD, relaxation matrix are also non-negative and depend HT, and DT, the line shape is affected by nuclear spin on the proton transverse relaxation time, T2H , and relaxation and it is not a sum of Lorentz functions. the deuteron longitudinal relaxation time, T1D . The T (ν) for the signal 3 H NMR of the DT This effect was described by Pople 13 for a spin-1/2 matrix MDT H (ν). nucleus coupled to a spin-1 nucleus and details of the molecule is analogous to the matrix MHD 14 Spin-spin coupling constants of hydrogen deuteride appropriate formalism were given by Abragam. were In particular, for the 1 H NMR signal of hydrogen and tritide for a given density of carbon dioxide H determined from the best fit of the function S (ν) to H tritide, SHT (ν), the matrices are: 1 H NMR spectrum of the mixture. Fitting parameH H ters are ∆νHT , ∆νHD , JHT , JHD , a, b . . . , f , width H ∆ν − J /2 0 HT H and resonance frequency of the H2 signal. The same HT LHT (ν) = ı , H 0 ∆νHT + JHT /2 procedure was applied to 3 H NMR spectra using the (5) function S T (ν) which allowed to determine spin–spin coupling constants for tritides of hydrogen and deu −a b H , (6) terium. RHT (ν) = b −a Finally, the spin–spin coupling constants 1 J(D, H), 1 J(T, H), and 1 J(T, D) were extrapolated to the zero 1 below H H where ∆νHT = ν − νHT is the frequency offset from density-limit. The value of J(T, H) discussed 1 is an average of the results obtained from H NMR the frequency of the middle point of the HT doublet, 3 and H NMR spectra. H νHT . The middle point of the HT doublet is at the It is convenient to consider in the following the reH H frequency νHT = γH (1 − σHT )B0 /2π where γH is the duced spin–spin coupling constants, since the differH gyromagnetic ratio of the proton, σHT is the isotropic ences between values of spin–spin coupling constants nuclear magnetic shielding of the proton in the HT of hydrogen isotopologues are mainly determined by molecule, and B0 is the strength of the magnetic field. the nuclear gyromagnetic ratios. The reduced spin– JHT is the indirect spin–spin coupling between the spin coupling constant between nuclei N and N ′ is: proton and triton in the HT molecule. The parameters a and b of the relaxation matrix are non-negative and depend on the proton transverse relaxation time, T2H , and the triton longitudinal relaxation time, T1T . T T For the signal SHT , the form of the matrix MHT is H analogous to the matrix MHT , however, the a and b parameters do not have take the same values for these two cases.

1 1

K(N, N ′ ) = 4π 2

J(N, N ′ ) , hγN γN ′

(9)

where for mixed isotopologues of hydrogen the relevant constants (i.e. the gyromagnetic ratios for the proton, deuteron, and triton) are γH = 2.675222 × 108 s−1 T−1 , γD = 4.106629 × 107 s−1 T−1 , γT = 2.853498 × 108 s−1 T−1 . 15

H H Similarly, the matrix MHD (ν) for the signal SHD (ν)

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The 1 H/3 H NMR spectra of the mixture of hydrogen isotopologues as a function of the density of carbon dioxide are shown in Fig. 1. The HT doublets are well separated from the lines of the other hydrogen isotopologues, thus, their shapes are only slightly perturbed by the other signals. For HD and DT the influence of the nuclear relaxation on their multiplets splittings is pronounced. The reduced indirect spin–spin coupling constants for mixed hydrogen isotopologues for uncorrected data are 1 K(T, H) > 1 K(D, H) > 1 K(T, D), while application of the fitting procedure described above indicates that the relationship is 1 K(D, H) > 1 K(T, H) > 1 K(T, D). This order is in agreement with results of high-pressure gas-phase measurements reported by Aleksandrov and Neronov, 16 since the influence of the nuclear relaxation on the NMR spectrum decreases with pressure. The spin–spin coupling constants determined from the gas-phase NMR experiment and from quantum mechanical computations are shown in Table 1. The experimental uncertainties reported in parenthesis equal three standard deviations. The error bars of the ab initio results were estimated taking into account the basis set convergence and the accuracy of the potential energy curve; the role of the relativistic effects appears to be negligible (see Ref. 7). The ab initio computations approximately reproduce the experimental data, but there is a minor discrepancy which exceeds the estimated error bars. The discrepancy between theory and the experiment, (Kcalc − Kexp )/Kexp , is 0.44%, 0.39% and 0.24% for HD, HT, and DT, respectively. It is approximately proportional to the inverse of the reduced mass of the hydrogen isotopologue; adjusting for HD a proportionality constant (which becomes -318.82 u/Hz) these discrepancies are 1.50(43), 1.23(58), and 0.83(44) in agreement with the inverses of the reduced masses which are 1.50, 1.33, and 0.83 u−1 , respectively.

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Figure 1: The 1 H NMR (A) and 3 H NMR (B) spectra of the mixture of hydrogen isotopologues for different densities of the solvent gas, CO2 , black points, and the best fit of the functions S H and S T to these spectra, blue curves. Frequencies are referenced to ν0H = 500 609 356 Hz (A) and ν0T = 533 969 914 Hz (B). B0 = 11.75 T; T = 300 K.

A

B

Discussion and Conclusions

For each isotopologue the value of the indirect spin– spin coupling constant derived from the experiment 5



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Acknowledgments

Table 1: Indirect spin–spin coupling (J in Hz) and its reduced value (K in 1020 ×T2 J−1 ) in molecular hydrogen HD HT DT J exp. 43.12(1) 299.06(36) 45.56(2) calc. 43.31(5) 300.24(35) 45.67(5) K exp. 2.338(1) 2.334(3) 2.316(1) calc. 2.349(3) 2.343(3) 2.322(3)

The experimental part of this work was financed by the Polish Ministry of Science and Higher Education support for young researchers at University of Warsaw. (PG) This work was partly financed by the National Science Center (Poland) grant, according to the decision No. DEC-2011/01/B/ST4/06588. PG would like to thank Prof. A. Czerwi´ nski and Prof. M. Ka´ nska for providing the access to the equipment necessary for the preparation of the samples. Samples were prepared at the Biological and Chemical Research Centre, University of Warsaw, established within the project co-financed by European Union from the European Regional Development Fund under the Operational Programme Innovative Economy, 2007–2013.

may be perturbed by several factors. For instance, the coupling constant may be reduced by the interactions between molecules and the inner surface of the glass ampoule. Another possibility is the presence of a small residual coupling, originating from partial orientation of the molecules by the magnetic field. However, both these effects are orders of magnitude too small to affect the observed coupling constants. Moreover, let us note that the extrapolation to the zero-density limit removes the influence of the spin– spin coupling induced by intermolecular interactions between hydrogen molecules and carbon dioxide. Experimental data confirm that the spin–spin coupling density dependence for each isotopologue is linear, see N Eq. (1). The assumption that the matrices RN N ′ (ν) are symmetric is fulfilled, since the amplitude of the cross-relaxation terms is also negligible. We do not dare to speculate on the role of the spin-dependent interactions discussed in Refs. 1–3, thus, we do not envisage any simple explanation of the discrepancy between the theory and experiment arising from an experimental factor. In particular there is no obvious explanation reflecting the dependence on the reduced masses of the hydrogen isotopologues.

References [1] Ledbetter, M. P.; Romalis, M. V.; Jackson Kimball, D. F. Constraints on ShortRange Spin-Dependent Interactions from Scalar Spin-Spin Coupling in Deuterated Molecular Hydrogen. Phys. Rev. Lett. 2013, 110, 040402. [2] Salumbides, E. J.; Koelemeij, J. C. J.; Komasa, J.; Pachucki, K.; Eikema, K. S. E.; Ubachs, W. Bounds on Fifth Forces from Precision Measurements on Molecules. Phys. Rev. D 2013, 87, 112008. [3] Ubachs, W.; Koelemeij, J. C. J.; Eikema, K. S. E.; Salumbides, E. J. Physics Beyond the Standard Model from Hydrogen Spectroscopy. J. Mol. Spectrosc. 2016, 320, 1–12.

On the other hand, one may assume that this discrepancy is due to the neglect in the calculations of the nonadiabatic effects. Considering the proportionality of the discrepancies to the inverses of the reduced masses, this is a plausible explanation—the nonadiabatic effects include terms dependent on the reduced mass. 17 Unfortunately, a systematic and reliable analysis of all the nonadiabatic contributions to the spin–spin coupling constants is not presently possible.

[4] Karplus, M. Contact Electron-Spin Coupling of Nuclear Magnetic Moments. J. Chem. Phys. 1959, 30, 11–15. [5] Griesinger, C.; Soerensen, O. W.; Ernst, R. R. Two-Dimensional Correlation of Connected NMR Transitions. J. Am. Chem. Soc. 1985, 107, 6394–6396. 6


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[6] Barfield, M.; Smith, W. B. Internal H-C-C Angle Dependence of Vicinal 1 H–1 H Coupling Constants. J. Am. Chem. Soc. 1992, 114, 1574– 1581.

Estimation of the Magnetic Moment of the Triton. JETP Lett. 2011, 93, 305–307.

[8] Garbacz, P. Spin–Spin Coupling in the HD Molecule Determined from 1 H and 2 H NMR Experiments in the Gas-Phase. Chem. Phys. 2014, 443, 1–7.

[19] Origin 8.5; OriginLab, Northampton, MA, 2010.

[17] Pachucki, K. Finite Nuclear Mass Corrections to Electric and Magnetic Interactions in Diatomic Molecules. Phys. Rev. A 2010, 81, 032505. [7] Helgaker, T.; Jaszu´ nski, M.; Garbacz, P.; Jackowski, K. The NMR Indirect Nuclear Spin– Spin Coupling Constant of the HD Molecule. [18] Mathematica 10.1, Wolfram Research: Champaign, IL, 2015. Mol. Phys. 2012, 110, 2611–2617.

[9] Neronov, Yu. I.; Barzakh, A. E.; Mukhamadiev, Kh. NMR Study of Hydrogen Isotopes to Determine the Ratio of the Proton and Deuteron Magnetic Moments to the Eighth Decimal Place. Zh. Eksp. Teor. Fis. 1975, 69, 1872–1882, (JETP 42, 950–954 (1975)). [10] Garbacz, P.; Makulski, W.; Jaszu´ nski, M. The NMR Spin–Spin Coupling Constant 1 J(31 P,1 H) in an Isolated PH3 Molecule. Phys. Chem. Chem. Phys. 2014, 16, 21559– 21563. [11] Carr, H. Y. Nuclear Spin Multiplet Collapse in HD Gas. Physica A 1989, 156, 212–218. [12] Goldman, M. Formal Theory of Spin–Lattice Relaxation. J. Magn. Reson. 2001, 149, 160–187. [13] Pople, J. A. The Effect of Quadrupole Relaxation on Nuclear Magnetic Resonance Multiplets. Mol. Phys. 1958, 1, 168–174. [14] Abragam, A. The Principles of Nuclear Magnetism; Oxford: London, 1961; p 504. [15] Mohr, P. J.; Taylor, B. N.; Newell, D. B. CODATA Recommended Values of the Fundamental Physical Constants: 2010. Rev. Mod. Phys. 2012, 84, 1527–1605. [16] Aleksandrov, V. S.; Neronov, Yu. I. Study of the NMR Spectra of Hydrogen Isotopic Analogs and 7



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Indirect SpinâSpin Coupling Constants in the Hydrogen Isotopologues

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