Angewandte Chemie Chiral Molecules Isotopic Chirality and Molecular Parity Violation** Robert Berger, Guido Laubender, Martin Quack,* Achim Sieben, Jrgen Stohner, and Martin Willeke Dedicated to Richard N. Zare on the occasion of his 65th birthday Studies of isotope effects have a long tradition in providing fundamental insights into molecular spectroscopy and reaction dynamics,[1, 2] usually dealt with theoretically on the basis of the electromagnetic interaction that is parity conserving, i.e. remains unchanged under space inversion at the origin.[3–5] Isotope effects are frequently caused by mass differences of the isotopes. There are also isotope effects due to the different nuclear spins of the isotopes,[6] and, in principle, isotope effects can arise independent of mass and spin because of symmetry restrictions on the molecular wavefunction leading to different symmetry selection rules for different isotopomers.[7] Here we report the first quantitative investigations of a new isotope effect, which leads to a ground-state energy difference DpvE DpvH 00/NA for the enantiomers of molecules that are isotopically chiral, i.e. chiral only by isotopic substitution (Figure 1). This parity-violating isotope effect arises from the electroweak interaction between electrons [*] Prof. Dr. M. Quack, A. Sieben, Dr. M. Willeke Laboratorium fr Physikalische Chemie Eidgenssische Technische Hochschule Zrich 8093 Zrich (Switzerland) Fax: (+ 41) 1-632-1021 E-mail: martin@quack.ch Dr. R. Berger, G. Laubender Institut fr Chemie Technische Universitt Berlin Strasse des 17. Juni 135, 10623 Berlin (Germany) Dr. J. Stohner Zrcher Hochschule Winterthur (ICB-ZHW) 8401 Winterthur (Switzerland) [**] We thank Sieghard Albert and Michael Gottselig for help and discussions. Our work was supported financially by the ETH Zrich (including C4 and CSCS) and the Schweizerischer Nationalfonds. R.B. acknowledges financial support from the Volkswagen-Stiftung and computer time provided by the HLRN. G.L. thanks the Graduiertenkolleg 352 for a scholarship. Angew. Chem. Int. Ed. 2005, 44, 3623 –3626 Figure 1. Scheme showing all quantities of interest as explained in the text. For the example PF35Cl37Cl the relative magnitudes of the various DpvE values are to scale, with DpvE* for the case vi = 1 for all i, but the comparison with much larger zero-point and vibrational excitation energies is, of course, not to scale. The magnitude of the reaction enthalpy for the stereomutation reaction S = R is j DpvH00 j NA j DpvE j = 3.3 1013 J mol1 (at 0 K). and nucleons, mediated by the Z-boson, and thus depends upon nucleonic composition. Our calculations are of interest in relation to efforts of measuring DpvE in enantiomers,[5, 8] and they are also important for the fundamental understanding of isotope effects and molecular chirality. The present work opens a new avenue in this field by providing quantitative calculations on such chiral isotopomers in the framework of electroweak quantum chemistry[9] including the weak nuclear force. Since recent theoretical approaches predict absolute values of DpvE that can be orders of magnitude larger[9–12] than anticipated on the basis of earlier calculations,[14, 15] there is new hope that accurate measurements and calculations, particularly for molecules with light atoms, will provide additional insights into the standard model of high-energy physics.[5, 16] We refer here to recent articles with extensive further references.[4, 5, 10, 12] In this context we address and answer the following questions: 1. How large is DpvE in isotopically chiral systems compared to “ordinary” enantiomers? 2. Is DpvE dominated here by the parity-violating potential at the equilibrium geometry or by vibrationally averaging the parity-violating potential? 3. How does vibrational excitation change DpvE (i.e. DpvE*) in such systems compared to “ordinary” enantiomers where this question was addressed previously?[17] The answers to these questions will help in planning future experiments possibly including isotopic enantiomers. We study the phosphane derivatives PHDX (X = F,35Cl,37Cl,79Br,81Br) and P35Cl37ClY (Y = F,H,D) with these goals in mind. While isotopic chirality has been considered for some time,[3, 18–20] as an isotope effect through variation of DOI: 10.1002/anie.200462088 2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 3623 Communications ð2Þ Qw ðAÞ ¼ ZA ð14 sin2 qw ÞN A identity and mass of the isotopes, our work provides the first quantitative predictions of DpvE resulting from the parityviolating isotope effect. Here, qw is the Weinberg angle; we used sin2 qw = 0.2319 in e Figure 1 illustrates the quantities of interest. R-V pv and Sour calculations. The electroweak charge Qw differs for the V epv refer to the parity-violating potentials at the Born– various isotopes leading to the new parity-violating isotope Oppenheimer (BO) equilibrium geometries of the R and S effect discussed here. We first focus on the parity-violating enantiomers. Because of the antisymmetry of the paritypotential energy at the equilibrium geometries of the violating potential with respect to space inversion,[5] the electronic ground states V epv (and j DepvE j = j 2 V epv j ). absolute magnitude of the difference j DepvE j is just twice the The results reported in Table 1 for V epv correspond to the R absolute magnitude of each of the two potentials. Adding the configuration of the molecules 1–10. A negative value of V epv zero-point vibrational energy and the average parity-violating indicates a stabilization of the given structure, while the potentials in the vibrational ground states to the BO parity-conserving Table 1: Equilibrium contribution and vibrationally averaged parity-violating energy of various ordinary R potentials gives the ground state enantiomers and several isotopic R enantiomers.[a] energy levels including parity vioV epv E0pv Basis set for geometry Basis set for parity-violating lation for both enantiomers (R-E0pv, Compounds 1 in R configu[hc cm ] [hc cm1] optimization and for energy calculations 0 S-Epv). Their difference corre- ration 1D electronic potential sponds to the in principle measurcalculations able ground state energy difference 17 15 PHDF (1) 4.64 10 1.40 10 6-311G(d,p) 6-311G(d,p) (if tunneling is negligible) or to the aug-cc-pVDZ 5.73 1017 1.11 1015 6-311G(d,p) reaction enthalpy at 0 K PHD35Cl (2) 2.14 1017 1.70 1015 6-311++G(d,p) 6-311G(d,p) 0 0 0 j R-EpvS-Epv j = j DpvE j j DrH 0/NA j PHD37Cl (3) 2.14 1017 1.66 1015 6-311++G(d,p) 6-311G(d,p) (the sign depends on the convention PHD79Br (4) 4.09 1017 1.60 1014 6-311++G(d,p) 6-311G(d,p) 81 4.09 1017 1.63 1014 6-311++G(d,p) 6-311G(d,p) for the directed stereomutation PHD Br (5) 35 37 8.44 1015 7.50 1015 6-311++G(d,p) aug-cc-pVDZ reaction between R and S). PH35 Cl37 Cl (6) PD Cl Cl (7) 8.44 1015 8.97 1015 6-311++G(d,p) aug-cc-pVDZ Finally a similar definition applies aug-cc-pVDZ P35Cl37ClF (8) 1.53 1014 1.38 1014 6-311++G(d,p) to excited vibrational states PH35ClF (9) 2.83 1013 2.88 1013 6-311G(d,p) aug-cc-pVDZ (v1…v3N6) of the chiral molecule, P79Br35Cl1.09 1012 1.06 1012 6-311G(d,p) 6-311G(d,p) which occur as closely spaced F(10) doublets of levels for the two en- [a] V e /(hc) is the parity-violating potential calculated at the equilibrium geometry of the paritypv antiomers with the parity-vio- conserving Born–Oppenheimer potential. All geometry optimizations and one-dimensional (1D) cuts lating splitting j DpvE*(v1…v3N6) through the parity-conserving potential energy hypersurface were carried out on the level of second order j = j R-Evpv1 ...v 3N6 S-Evpv1 ...v 3N6 j pro- Møller–Plesset perturbation theory (MP2) within the frozen core approximation using Gaussian 98.[26] 0 vided that this splitting is much Epv/(hc) is the parity-violating potential averaged over the ground vibrational state as explained in the text. From the quantities given it is possible to calculate the parity-violating energy differences j DepvE j = larger than the hypothetical tunnelj 2 V epv j and j DpvE j = j 2 E0pv j (see text and refs. [11, 17] for methods and basis sets). Parity-violating ing splitting for the parity-conservpotentials were calculated with our modification[11] of the Dalton program.[27] ing case but much smaller than the separation of vibrational levels.[3, 8] The parity-violating potentials were calculated within our corresponding mirror image is destabilized by the same multiconfiguration linear response (MC-LR) approach to amount. The absolute values of V epv of systems that are chiral [11] electroweak quantum chemistry, here within the randomdue to substitution with different chlorine isotopes (6–8) are remarkably large, whereas the effect due to a deuteration (1– phase approximation (RPA). The approximate parity-violat5) is two to three orders of magnitude smaller. This can be ing Hamiltonian in SI units is given in Equation (1).[10, 11, 17] understood from the parity-violating operator Ĥpv given in Equation (1), which describes effectively a contact-like interN n X X ^ pv ¼ pGFpﬃﬃﬃ ð1Þ H Qw ðAÞ ½! p^ i ! s^ i ,d3 ð! r i ! r A Þþ action between each electron and each nucleus, depending on h me c 2 A¼1 i¼1 the weak nuclear charge Qw(A) [Eq. (2)]. The parity-violating potential arises as a sum of contributions from the various nuclei, where typically the heavier nuclei dominate.[10] ThereHere, GF (Fermi constant) is 2.222527 1014 Eh a30 = 62 3 fore, simple deuteration results in an absolute value of V epv 1.43586 10 J m (Hartree energy Eh and Bohr radius a0), that is small compared to systems that are chiral with different me is the electron mass, h is Plancks constant, and c is the chlorine isotopes. We report in Table 1 also V epv for the speed of light in vacuum. ! p^ i and ! s^ i are the linear momentum “ordinary” chiral systems P79Br35ClF (10) and PH35ClF (9). and spin operator of the electron i, and ! r i denotes its For the isotopically chiral molecules PH35Cl37Cl and position. ! r A is the position vector of nucleus A. d3 represents P35Cl37ClF the parity-violating potential is about one order the three-dimensional Dirac delta distribution and [.,.]+ the of magnitude smaller than for the “ordinary” chiral molecule anticommutator. The strength of the resulting effect is related PHClF. This can be understood by the electroweak charges of to the numbers of protons ZA and neutrons NA in the nucleus 35 Cl and 37Cl (Qw = 16.8 vs 18.8) differing by about 10 %. A, which enter the Hamiltonian by means of the electroweak For the isotopically chiral systems one would thus expect a charge [Eq. (2)]. 3624 2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org Angew. Chem. Int. Ed. 2005, 44, 3623 –3626 Angewandte Chemie decrease of V epv by an order of magnitude; one notes that for the symmetric equilibrium geometry of PF35Cl2 the atomic contributions to V epv for the two 35Cl nuclei are of exactly the same “normal” magnitude but cancel because of their different sign. In PF35Cl37Cl a difference of about 10 % remains with respect to exact cancellation. For a normal asymmetric molecule like PHFCl such a partial cancellation is not expected systematically and V epv is an order of magnitude larger. H–D substitution leads to a large relative change of the electroweak charge but to little change in the parity-violating potentials simply because neither H nor D contribute much to V epv. This explains the small parity-violating potentials with H/D isotopic chirality in Table 1. For isotopically chiral CHDTOH[20] we have discussed that those nuclei that lie in the Cs symmetry plane of the corresponding achiral isotopomers do not contribute to the parity-violating potential V epv. Thus, the values of V epv for PHD35Cl and PHD37Cl are identical (as also for the pairs PH35Cl37Cl(6)/PD35Cl37Cl(7) and PHD79Br(4)/PHD81Br(5)). Isotopic substitution, however, does not only influence the parity-violating potential at the equilibrium structure, it also modifies the molecular motion and changes the average ground state geometry of a molecule.[17] It is, for instance, well known that the average XD bond lengths are shorter than the average XH bond lengths (see refs. [3, 21] for a discussion of CH4 including its potentially chiral isotopomers). To include this effect in the calculation of DpvE we described the vibrational problem in the separable anharmonic adiabatic approximation (SAAA[17]). Our methods of calculation have been described in detail in refs. [11, 17], and we have checked by test calculations that the nonseparable coupling in Vpv(qi) as a function of the reduced normal coordinates qi is of minor importance.[17] We are allowed to talk of a measurable DpvE,[5, 8] as it is much larger than the tunneling splitting. The latter was roughly estimated for 2 as 1023 cm1 to 1020 cm1 using a simple WKB method. For the other compounds[22] the tunneling splitting should be even smaller because of the higher inversion barrier or larger tunneling mass (see ref. [23] and references therein). With the parity-violating potential Vpv(qi) and the vibrational wavefunction Yvi i we calculated the expectation value hEpvivi i of the parity-violating energy for the ith mode excited with vi quanta [Eq. (3)].[17] Figure 2 illustrates this for (R)-PHDF. hEpv ivi i ¼ hYvi i jV pv ðqi ÞjYvi i i ð3Þ Within the SAAA[17] we obtain the parity-violating energy E for a vibrational state with vibrational quantum numbers vi according to Equation (4). ðv 1 ,..., v 3N6 Þ pv Y 3N6 ðv 1 ,...,v 3N6 Þ Epv ¼h Yvi i jV epv þ i X 3N6 X i¼1 3N6 ¼ vi pv i hE i ð3N7Þ V Y 3N6 DV pv ðqi Þj i Yvi i i ð4Þ e pv i We use E0pv as the abbreviation for the ground state value with vi = 0 for all i. The ground state parity-violating energy difference can be calculated as j DpvE j = j 2 E0pv j or for excited Angew. Chem. Int. Ed. 2005, 44, 3623 –3626 Figure 2. The one-dimensional cut through the parity-conserving potential energy hypersurface (MP2/6-311G(d,p)) of (R)-PHDF along the reduced normal coordinate q6 (triangles, ordinate on the right). The displacement vectors of q6 are indicated for the different atoms. We show no scale for the probability density j Y06 j 2 (circles), but the scale may be defined by its maximum value (0.28) of j Y06 j 2 in the figure. The squares (ordinate on the left) represent the parity-violating potential (RPA/6-311G(d,p)) along q6. ðv 1 ,..., v 3N6 Þ states as j DpvE* j = j 2 Epv j . We applied the procedure illustrated for (R)-PHDF to all compounds 1–10. The results for E0pv are given in the third column of Table 1. For compounds 6–10 E0pv is similar to V epv, whereas for 1–5, which are chiral due to single deuteration, E0pv is up to three orders of magnitude larger than V epv. The strong vibrational dependence for 1–5 opens up the possibility of increasing or decreasing the parity-violating energy by selected excitation of fundamentals.[17] For (R)-PHDF excitation with one quantum in n6 for instance leads to Eð0,0,0,0,0,1Þ = 9.90 pv 1015 cm1 (positive), whereas an excitation of n4 with one quantum results in Eð0,0,0,1,0,0Þ = 1.22 1014 cm1 (negative). pv Simultaneous excitation of both modes leads to substantial compensation, with Eð0,0,0,1,0,1Þ = 8.57 1016 cm1. In (R)pv P35Cl37ClF, however, the parity-violating energy is predicted to depend less upon vibrational excitation. At some stage the nonseparable, anharmonic effects will certainly become important and can be accounted for.[17] We conclude that chirality caused by intermediate-mass isotopes can give rise to relatively large absolute values of the parity-violating energy difference DpvE. For 1–5, which are chiral due to deuteration, the vibrational averaging is important for DpvE. The absolute value of the vibrationally averaged parity-violating ground state energy difference for 4–5 is more than two orders of magnitude larger than DepvE at the equilibrium structure. For 6–10 DepvE and DpvE differ by less than 15 %, and their absolute values are only about an order of magnitude smaller than those obtained for “ordinary” chiral molecules with similar elemental composition. These results suggest that compounds that are chiral by isotopic substitution are well-suited candidates for theoretical and experimental work on molecular parity violation. Whereas detection of parity-violating effects in the frequency domain[24] may not be feasible for a compound such as P35Cl37ClF, the time t = h/(2 DpvE) for a transition between parity eigenstates induced by the parity-violating energy difference DpvE would correspond to about 600 s with an www.angewandte.org 2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 3625 Communications initial time-dependent signal for the change of parity being perhaps realistic on a millisecond timescale (see ref. [8] for such experiments and ref. [25] for a preliminary conference account of spectroscopic work in relation to parity violation in isotopically chiral P35Cl37ClF). Received: September 23, 2004 Revised: December 8, 2004 Published online: May 6, 2005 . Keywords: ab initio calculations · chirality · electroweak quantum chemistry · isotope effects · parity violation [1] F. Fernndez-Alonso, B. D. Bean, J. D. Ayers, A. E. Pomerantz, R. N. Zare, L. Baares, J. Aoiz, Angew. Chem. 2000, 112, 2860 – 2864; Angew. Chem. Int. Ed. 2000, 39, 2748 – 2752. [2] B. D. Bean, F. Fernndez-Alonso, R. N. Zare, J. Phys. Chem. A 2001, 105, 2228 – 2233; J. D. Ayers, A. E. Pommerantz, F. Fernndez-Alonso, F. Ausfelder, B. D. Bean, R. N. Zare, J. Chem. Phys. 2003, 119, 4662 – 4670. [3] M. Quack, Angew. Chem. 1989, 101, 588 – 604; Angew. Chem. Int. Ed. Engl. 1989, 28, 571 – 586. [4] M. Quack, Nova Acta Leopold. 1999, 81, 137 – 173. [5] M. Quack, Angew. Chem. 2002, 114, 4812 – 4825; Angew. Chem. Int. Ed. 2002, 41, 4618 – 4630. [6] R. R. Ernst, Angew. Chem. 1992, 104, 817 – 835; Angew. Chem. Int. Ed. Engl. 1992, 31, 805 – 823. [7] M. Quack, Mol. Phys. 1977, 34, 477 – 504. [8] M. Quack, Chem. Phys. Lett. 1986, 132, 147 – 153. [9] A. Bakasov, T. K. Ha, M. Quack in Proc. of the 4th Trieste Conference (1995), Chemical Evolution: Physics of the Origin and Evolution of Life (Eds.: J. Chela-Flores, F. Raulin), Kluwer, Dordrecht, 1996, pp. 287 – 296. [10] A. Bakasov, T. K. Ha, M. Quack, J. Chem. Phys. 1998, 109, 7263 – 7285; Erratum: A. Bakasov, T. K. Ha, M. Quack, J. Chem. Phys. 1999, 110, 6081. [11] R. Berger, M. Quack, J. Chem. Phys. 2000, 112, 3148 – 3158. [12] R. Berger in Relativistic Electronic Structure Theory, Part II— Applications (Ed.: P. Schwerdtfeger), Elsevier, Dordrecht, 2004, pp. 188 – 288. [13] G. Laubender, R. Berger, ChemPhysChem 2003, 4, 395 – 399. [14] R. A. Hegstrom, D. W. Rein, P. G. H. Sandars, J. Chem. Phys. 1980, 73, 2329 – 2341. 3626 2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim [15] S. F. Mason, G. E. Tranter, Mol. Phys. 1984, 53, 1091 – 1111. [16] L. Hoddeson, L. Brown, M. Riordan, M. Dresden, The Rise of the Standard Model, Cambridge University Press, Cambridge, 1997. [17] M. Quack, J. Stohner, Phys. Rev. Lett. 2000, 84, 3807 – 3810. M. Quack, J. Stohner, Z. Phys. Chem. (Oldenbourg) 2000, 214, 675 – 703; M. Quack, J. Stohner, J. Chem. Phys. 2003, 119, 11 228 – 11 240; M. Quack, J. Stohner, Chirality 2001, 13, 745 – 753. [18] J. Lthy, J. Retey, D. Arigoni, Nature 1969, 221, 1213 – 1215. [19] R. Harris, L. Stodolsky, Phys. Lett. B 1978, 78, 313 – 317. [20] R. Berger, M. Quack, A. Sieben, M. Willeke, Helv. Chim. Acta 2003, 86, 4048 – 4060. [21] R. Marquardt, M. Quack, J. Phys. Chem. A 2004, 108, 3166 – 3181. [22] S. Creve, M. T. Nguyen, J. Phys. Chem. A 1989, 102, 6549 – 6557. [23] R. Berger, M. Gottselig, M. Quack, M. Willeke, Angew. Chem. 2001, 113, 4342 – 4345; Angew. Chem. Int. Ed. 2001, 40, 4195 – 4198. [24] C. Daussy, T. Marrel, A. Amy-Klein, C. T. Nguyen, C. Bord, C. Chardonnet, Phys. Rev. Lett. 1999, 83, 1554 – 1557. [25] A. Sieben, R. Berger, M. Quack, M. Willeke in 18th Colloquium on High Resolution Spectroscopy (2003), Dijon, France, 2003, p. 161. [26] Gaussian 98 (Revision A.11.1), M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. Cheeseman, R. V. G. Zakrzewski, J. A. Montgomery, Jr., R. E. Stratmann, J. C. Burant, S. Dapprich, J. M. Millam, A. D. Daniels, K. N. Kudin, M. C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G. A. Petersson, P. Y. Ayala, Q. Cui, K. Morokuma, P. Salvador, J. J. Dannenberg, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. Cioslowski, J. V. Ortiz, A. G. Baboul, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, J. L. Andres, C. Gonzalez, M. Head-Gordon, E. S. Replogle, J. A. Pople, Gaussian, Inc., Pittsburgh, PA, 1998. [27] T. Helgaker, H. Jensen, J. P. Jørgensen, J. Olsen, K. Ruud, H. gren, T. Andersen, K. L. Bak, V. Bakken, O. Christiansen, P. Dahle, E. K. Dalskov, T. Enevoldsen, B. Fernandez, H. Heiberg, H. Hettema, D. Jonsson, S. Kirpekar, R. Kobayashi, H. Koch, K. V. Mikkelsen, P. Norman, M. J. Packer, T. Saue, P. R. Taylor, O. Vahtras, Dalton: An Electronic Structure Program, Release 1.2.1 ed., 1997. www.angewandte.org Angew. Chem. Int. Ed. 2005, 44, 3623 –3626

1/--страниц