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Isotopic Chirality and Molecular Parity Violation.

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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
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