J. Phys. Chem. 1993,97, 2413-2416
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Can Molecules Have Permanent Electric Dipole Moments? W. Klemperer,'++K. K. Lehmann,'**J. K. C. Watson,# and S. C. Wofsyl Departments of Chemistry and Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts 021 38, Department of Chemistry, Princeton University, Princeton, New Jersey, and Herzberg Institute of Astrophysics, National Research Council, Ottawa, Canada Received: September 24, 1992; In Final Form: December 22, I992
The character of the first-order Stark effect in the energy levels of real symmetric tops is examined. Group theory shows that the only allowed levels are nondegenerate, and as a result time reversal symmetry requires that such eigenstates not have a nonzero orientation required for a permanent electric dipole moment and thus a first-order Stark effect. The terms responsible for the breaking of the nominal degeneracy of levels with fK units of angular momentum around the symmetry axis are shown to derive from off-diagonal higher-order distortion coupling (for ro-vibronic states of nondegenerate symmetry) and from spin-rotation and spin-spin interactions (for degenerate ro-vibronic states). A semiclassical model demonstrates that both effects can be viewed as caused by tunneling and as such decrease rapidly for increasing K. For a C3 symmetry symmetric top, the resulting splittings are a few kilohertz for K = 1 but only on the order of 10-4 H z for K = 2.
Introduction The Stark effect measures the interaction of a molecular dipole moment with an electric field, while the Zeeman effect measures the interaction of a molecular magnetic moment with a magnetic field. While both electric and magnetic dipoles are rank one tensor quantities, they have quite different symmetries. For example, though magnetic moments are common, no elementary particle can have a nonzero electric dipole moment by both parity and time reversal symmetries.'J While violation of parity is common in particle physics, violations of time reversal symmetry have been established in only one particular case.3 Attempts to measure a dipole moment for the free neutron, which provide a test of theories of time reversal symmetry breaking, have so far only found extremely small upper limitsS4 The same symmetry arguments which predict that elementary particles should have zero electric dipole moments apply to nondegenerate states of molecules as well. In fact, experimentaltests looking for violation of time reversal symmetry have included attempts to determine a small static dipole moment in a pure rotational state of TIF.S Some of the possible mechanisms that can produce such a static dipole require mixing states of different parity. One would expect higher sensitivities in molecules with nearly degenerate states of different parity and time reversal symmetries,since, then, a weak symmetryviolating interaction could induce a corresponding larger mixing of the levels and thus a greater nonzero dipole moment. As such, it may prove useful to these experiments to consider symmetrictop molecules which are traditionallyviewed as having true first-order Stark effects. In the discussion, we refer particularly to field free motion and the energy changes that Occur upon application of infinitesimal fields which negligibly perturb the field free motion but provide a convenient coordinate frame. Whileall moleculesposessdegeneratelevelsdueto the isotropy of space (labeled by the magnetic quantum number M), all these levels have the same parity, and thus this degeneracy does not in itself allow for an electric dipole moment. Some higher symmetry is required. For example, the first-order Stark effect of the hydrogen atom is an important and well-understood Occurrence that reflects the higher dynamical symmetry of the Authors for correspondence. Department of Chemistry, Harvard University. Princeton University. f Herzbcrg Institute of Astrophysics. 1 Department of Earth and Planetary Sciences, Harvard University. +
Coulombic potential. In particular, for the nonrelativistic spinfree hydrogen atom this symmetry is associated with the Lenz vector, A = (1/2e2me)[L X p - p X L] + r/r being a constant of thc motion.6 For the relativistic hydrogen atom there exist degeneracies between states of the same total angular momentum, j . The first-order Stark effect has been discussed in detail by Bethe and Salpeter.' However, the Lamb shift demonstrates that this extra symmetry is lifted when matter-radiation field interactions are considered. Hyperfine structure would also remove the exact degeneracies discussed here. In molecules, it is well-known that rigid spin-free symmetric tops are expected to have first-order Stark effects due to the degeneracy of the fK levels. This degeneracy is geometrical in nature, a consequence of the cylindrical symmetry of the inertial tensor being higher than that required for the general rigid rotor, namely that of the four element point group D2. The present definition of a dipole moment is in conflict with the traditional %hemicall definition of a dipole moment which is based upon the point group symmetry of a rigid molecule. The point group symmetry determines whether at the equilibrium geometry a molecule can have a nonzero dipole moment in the molecular axis system. However, the molecular axis system is not static but rotates with the molecule. Externalfieldsareapplied in the laboratory axis system, and it is the possible existence of a static projection of the dipole moment in this coordinate system with whichweareconcerncdin this paper. Ifonewantstoconsider the effects of rotation, even for a rigid molecule, one must go beyond the point group of a molecule and consider the molecular symmetry group, which is a subgroup of the complete permutation-inversion group which includes all 'feasible" operation^.^ If one considers the full permutation-inversion group for a molecule, it can be shown that the only energy levels allowed by quantum statistics belong to one of two nondegenerate representations (i.e., there are no exact degeneraciesdue to symmetry as discussed above for atomic hydrogen).*,9 Further, since these two representations are of opposite parity symmetryL0for all achiral molecules, the true molecular eigenstatesshould not have first-order Stark effects. For a chiral molecule the states allowed by the full permutation-inversion symmetry group are split into levelsof opposite parity due to tunneling that converts the molecule into its stereoisomer. The splitting due to this inversion can be estimated and is believed to be typically smaller than parityviolating terms in the molecular Hamiltonian." Thus, by considerationsof parity alone, one may not rule out the existence
0022-3654/93/2091-2413$04.00/00 1993 American Chemical Society
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2414 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
of a dipole moment in a chiral symmetric top (such as a molecule with symmetry C3 for example). When one considers the requirements of time reversal symmetry, however, even this possible exception can be ruled out. Even if we consider only a permutation subgroup of the full permutation-inversion group, the nuclear spin-ro-vibronic states allowed by quantum statistics will still be nondegenerate representations. Time reversal symmetry will still be applicable (though parity will not be for a chiral molecule), and by the same argument that applies to elementary particles,2we can conclude that such a nondegenerate state cannot have a dipole moment or thus a first-order Stark effect. The proof of this statement is as follows. Consider a molecule in a state (The index a labels degrees of freedom other than angular momentum.) To have a nonvanishing expectation value for the dipole moment operator requires that this dipole be along a conserved vector. The standard argument examines the angular momentum vector J , since rotation will average out any perpendicular components. Thus, an average dipole moment, given by the expectation value of p in state J,a, must be proportional to J.
= C(a)( # J a l J k J a ) Consider the effect of the time reversal operator. T-IpT = p (because p is defined in terms of particle positions and charges both of which are invariant under time reversal) while T-IJT = -J. If the state JIJa is nondegenerare, the time reversal symmetry of the Hamiltonian requires T$ja = eisJIJo. This equation will still hold if two levels of stereoisomers are degenerate since time reversal symmetry will not interconvert isomers (unlike parity). The above equation relating ( p ) and ( J ) is consistent with ( p ) invariant under time reversal and ( J ) changing sign only if C(a) is pure imaginary. Since both J and p are Hermitian, C(a)must be real; thus C(a) = 0. While the above arguments based upon symmetry are very powerful, because they transcend any specific molecular model, they are also very limited for the same reason. Thus, while they prove that no molecule can have a true first-order Stark effect, they do not give any indication as to what real physical effects remove all degeneracies. Another way of interpreting the lack of any dipole moment for an eigenstate is that, time averaged, the molecule must spend equal time pointing up, K M positive, or down, K M negative. But this gives no indication as to how long, if we prepare the molecule with an on average up orientation, it will stay that way. It is the magnitude of the splitting of this IKM degeneracy that must be examined. For this purpose, we need to explicitly look at a real molecular model. Consider the prototypical polar symmetrical top, noncoplanar XY3. First we will present a quantum mechanical treatment of the problem. This will be followed by a semiclassical treatment, which we believe gives a better physical model for why these effects occur. OkaIoexplicitly stated that the spin-rotation interactions lead to a lifting of all residual degeneracies, but to the best of our knowledge no one has given an analysis of the exact terms responsible for the nondegeneracy of the levels and how the splitting patten depends upon rotational quantum numbers. Below we show that, for a CH3Y molecules, the K = f l levels are split by terms coming from both the anisotropy of the IH spin-rotation interaction as well as from IH-Y spin-spin coupling. Both of there terms are on the order of a few kilohertz. For states with K = f 2 the coupling terms are second order in hyperfine interactions and totally negligible. Thus, in this case (and higher K = 3N f 1) the splittings predicted by symmetry are of no dynamical consequence and molecular orientation will be preserved, even without external fields, for time scales longer than any experiments currently feasible. ($Jobl$Ja)
Quantum Treatment If we consider the rotational wave functions for XY3, we have in the rigid rotor picture a degeneracy between levels with f K . I 2 Such states will have an average dipole moment proportional to KM. If we can find any interaction that couples, directly or indirectly, the states K,Mand -K,M, then the eigenstates will be Wang functions ( l / ~ / Z ) ~ I J , K , M ) f I J , - K , MSuch ) ~ . states do not havedipole moments. Thesplittingof the two Wang functions, in hertz, can be interpreted as the rate at which the molecule “flips” from pointing up to down and back up again, just as the splitting of the symmetric and antisymmetric tunneling levels in NH3 gives the rate of molecular inversion. The simplest case to consider is where atoms Y have zero nuclear spin, since in this case we do not have toconsider hyperfine effects. However, for Zy = 0, the only states that are allowed by BoseEinstein statistics are those with K = 3N, N = an integer.g Those rotational levels are nondegenerate, however, in the C3, point group. As first shown by Nielsen and DennisonI3 in their discussion of NH3, there exist off-diagonal sextic distortion interactions (1 /2)h3(J46 which cause a lifting of the degeneracy. In particular, 1 4 = 3 levels are directly connected by this interaction and thus have a splitting first order in h3. h3 is on the order of B S / w 4 my-3, where my is the mass of Y. For NH3, the value of h3 = 350 Hz.I4 1 4 = 6 levels are split by second4 = 9 levels are order coupling through the K = 0 level, while 1 split in third order by this interaction, etc. Thus, the splitting goes up as a high power of J but decreases exponentially with K , a sign that the physical origin of the coupling is some type of tunneling, much like the asymmetry splitting of an asymmetric top. Thus, it is a quantum effect which disappears rapidly as the mass or quantum numbers of the molecule increase. If one looks at symmetric tops with a higher-order axis (C4,CS,C6, ...), the same situation holds, namely, that the only states that are allowed by BoseEinstein statistics are split by centrifugal distortion terms. The higher the symmetry axis, the higher order the coupling, and thus the smaller the characteristicsize of the coupling constant. If we consider molecules for which ZY = I/*or higher, all values of K are allowed by quantum statistics. We will focus on the common case of Zy = ’ / 2 , but our conclusions will hold for higher spin as well. The levels for K = 3N f 1 will remain degenerate to all ,,den in the rotation-vibration Hamiltonian, since these levels form an E symmetry representation in the molecular symmetry group of the m ~ l e c u l e .These ~ levels are allowed by quantum statistics, however, because they can be combined with E symmetry nuclear spin functions to produce states with total nondegenerate symmetry. Since both the ro-vibrational and nuclear wave functions are E symmetry, it is natural to suspect that it is the interaction of these degrees of freedom that removes the degeneracy. In fact, the Hamiltonian for CH3D was worked out by Wofsy, Muenter, and KlempererlSunder the assumption of tetrahedral geometry. It was found that in a basis set where only the three H nuclear spins are coupled together that there is a coupling between levels with AK = f 2 ,
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In this equation c = (2/9)(c, - cg)[B(CH,D)/B(CH4)] = 3.0 kHz, and c, and cg are the two spin-rotation interaction constants for methane. c, is the spin-rotation interaction produced by rotation with Jparallel to a given C-H bond, and cg is for rotation perpendicular to that bond. MJ, M H ,and M D are the Z-axis projections of J and the two nuclear spins angular momenta; p~ and p~ are the magnetic moments of H and D nuclei. Wofsy, Muenter, and Klemperer used esu units and thus do not include the factor of 4 ~ ~ Thus, 0 . even without atom D, there is a splitting of the K = f l levels due to the spin-rotation interaction. The
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The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2415
second term reflects spin-spin interaction between the H and D nuclear spins (R is the distancebetween the nuclei in the molecule), with 3 p ” p ~ / 4 a p a 3 h= 9.28 kHz. Thus we see that the K = f l levels will be split by a few kilohertz, which implies that the orientation of such a molecule in zero field will only be preserved for a fraction of a millisecond. This is a time scale which can in principle be observed in the laboratory but requires that stray fields be reduced below -1 mV/cm. If we consider higher K values, they have no direct coupling by either the spin-spin or spin-rotation operators (which are second-rank tensors), but they will be coupled in higher order through order K levels. For example, K = 2 will be coupled through the K = 0 levels. This will produce a splitting that is smaller by the ratio of the K = 1 splitting to 4(A - B) (the K = 0 and 2 splitting). For CH3D, this ratio is on the order of and thus it will take about 1 h for K = 2 molecules to flip their orientation! Thus, this effect is totally negligible for all but K = 1 levels. Semiclassical Interpretation of K-Doubling Effects
While the above discussion provides a correct quantummechanical description as to how the apparent time-independent molecular orientation of a symmetric top is lost due to centrifugal and hyperfine interactions, it is appealing to have a semiclassical interpretation of this effect as well. In this section, we provide such a picture in the frameworkof motion on the rotational energy surface (RES), which has been used by Harter and Patterson16 to provide a physical visualization of rotational dynamics. As suggested by the quantum-mechanical treatment, the effects can be understood as due to tunneling. Consider a symmetric top which will be treated as prolate, although nothing below depends upon this choice. The classical rotational energy can be written as
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E(J,O,X) = J2[Acos2((?) B sin2(0)] where 8,x are the polar coordinates that determine the direction of the instantaneous angular momentum vector in the molecular axis system. For a given magnitude of angular momentum, in each direction 8,x, we draw a point at a distance E(J,O,X). This traces out the RES, which is a surface of revolution in the present case. Classically, the rotational motion of the symmetric top is represented by precession that scribesa circle on the RES. (More generally, the curve is the intersection of the constant angular momentum RES and a constant energy sphere.) Quantum states with a specific are a pair of quantizing circlesthat make angles given by cos(8*K) = fK/[J(J + 1)] as shown in Figure 1. All points on the RES surface have the same magnitude of angular momentum but differ in the direction of the angular momentum in the molecular axis system. In general, we can expect tunneling between these two degenerate states, as shown in Figure 1. For a given value of x , the minimum tunneling path will take us to the same x , Le., be a curve of constant longitude on Figure 1. For each x , we expect a tunneling amplitude t(X) proportional to [Harter and Paterson, eqs 4.2 and 4.41
In this formula, Jr is the angular momentum conjugate to the tunneling path, which will be on the equator for the minimal tunneling path y. Note that along the tunneling path y is real while Jr and the other angle used to describe the direction of J ( p in the notation of Harter and Patterson) are imaginary. JT is found by solving eq 4.6 of Harter and Patterson, which is
E =A(J~ - J,’) cos2 7 + B ( J ~- J,’) sin2 7 + CJ,’
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where E is the rotational energy of the state J,K and B = C for a symmetrictop. As y a/2(which is when the tunneling path crosses the equator), the value of . I , goes to i times infinity and
Figure 1. Rotational energy surface (RES)for a prolate symmetric top. The quantum states, J,K are shown for J = 8. 6 K = cos-’ [ K / ( J ( J 1))’/*] is shown for K = 7. Tunneling paths along arcs of constant x are shown for both K = f3 and f7. This figure was kindly provided by Prof. William G. Harter.
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we get a logarithmic divergence of the tunneling integral, and thus the tunneling rate is zero. The reason for this divergence is simply that coefficient of Jr is zero in the above equation for y = a/2. It can be shown that addition of terms diagonal in J and Kwill not change this situation; i.e., the fKlevels will remain exactly degenerate. If we consider a rigid asymmetric top, C < B, then the value of Jr stays finite for y = a/2 if we consider tunneling that takes the rotational angular momentum over the B rotational axis. We have a tunneling path of equal amplitude if we go over the -B rotation1 axis as well (see Harter and Patterson, Figure 4). We must add the amplitude of these two paths, but it can be shown that they will constructively interfere for any K. Now let us consider a distortable C3 rotor with the (l/2)h3(Jk)6 terms added to the rotational Hamiltonian. Now the classical“trajectories”of each K will dip down toward the equator six times on each precession around the symmetry axis. This will lead to six equivalent tunneling paths on each cycle of the motion. As each tunneling path goes across the equator, Jr has a finite value given approximately by i[(A - B)E/h3]1/6. As a result, the tunneling integral is convergent, and we have a nonzero tunneling amplitude for each of the six equivalentpaths. However, when one adds up the contributions to each path, it can be shown that for K = 3N the contributions constructively interfer and we get a nonzero tunneling rate, while for K = 3N f 1 there is exact destructive interference of the six contributions and the total tunneling rate is exactly zero. Let us now considerthe effect of includinghyperfine interactions in the rotational Hamiltonian. If all spins point in the same direction, then the sum of the hyperfine interactions give contributions that depend upon cos2 y but are independent of x . These terms are equivalent to a slight change in the A and B rotational constants but do not change the cylindrical symmetry of the RES and thus do not change the picture given earlier for a symmetric top. If, instead, one of the proton spins points up and the other two point down (or visa versa), then both the spin rotation and H-D spin-spin interactions lead to a contribution
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to the energy proportional to sin2 y cos2 x . These terms have the same functional form as making B - C nonzero. The direction of the B and C axes will be such that one passes through the proton with unique spin. As a result of this effective asymmetry, just as for the true asymmetric top treated earlier, one will get a finite tunneling integral, and constructive interference of the two equivalenttunneling amplitudes. This will allow for a splitting of the otherwise exactly degenerate K = 3N f 1 levels. It is exactly for theserotational levels that quantum statistics demands that the allowed nuclear spin functions have one spin pointing opposite to the other two. The conclusionof this section is that the spectroscopicsplittings that prevent a truly static orientation, and thus dipole moment, have a straightforward semiclassical interpretation. Because of spin-rotation and spin-spin interactions, a state with the spins of equivalent atoms pointing in different directions will behave like a very slightly asymmetric top.
Conclusions The present paper has presented an analysis of whether a true first-order Stark effect, or equivalently a static orientation, is expected in a symmetric top molecule. Group theory provides a rigorous answer of "no". But such an answer makes no reference to the time scales for the loss of orientation; if they exceed the time scales of an experiment that depends upon orientation, the molecular orientation will appear "static". Likewise, once the Stark coupling matrix element between nearby states exceeds their energy separation, the spontaneous reorientation will be quenched and a quasi-first-order Stark effect will be observed. The loss of orientation is due to a quantum tunneling phenomenon and as such is important only for the very lowest levels. It is interesting, however, that for CH3Xmolecules,even with extreme cooling of the rotational degrees of freedom in a supersonic jet, about one-half the molecules are in the J = K = 1 levels. These molecules can be readily oriented by an electrostatic hexapole focuser. Such molecules can then be extracted and used for orientational studies of scattering or reactions. It is clear that, for the fields that exist within the hexapole focusers,the hyperfine splittings discussed above will be negligibly small. The present analysis shows, however, that in a field-free environment the orientation of such states will be averaged out in a time scale of a fraction of 1 ms. If K = 2 levels are populated in the expansion, in a true field-free region, these molecules should retain their initial orientation for time scales much longer than any practical experiment. In recent experiments,Gandhi and Bernstein showed that they could maintain molecular orientation of CHJ only in the presence of a static field of at least 300 mV/cm.17 This is much higher than would be estimated by the above predicted
splittings. Further, they observed a loss of orientation for both the (J,K,M) = ( l , l , l ) and (2,2,2) states at about the same field. Based upon these facts, it is likely that the loss of orientation they observed reflects not a true spontaneous reorientation but a reorientation induced by precession about stray electric fields or by nonadiabatic transitions as molecules leave or enter regions of significant fields.
Note Added in hoof. The matrix elements of the hyperfine operator, off-diagonal in K, have been discussed by I. Ozier and W.L.Meerts(Can.J. Phys. 1981,59,15O)inthecontextoftheir Stark crossing experiments on symmetric tops. We thank Prof. Oka for bringing this reference toour attention. We have received a manuscript from R. J. Butcher, Ch. Chardonnet, and Ch. J. Borde who have reached a conclusion similar to the present work. Acknowledgment. We wish to thank Professors Ian Mills and Takeshi Oka for valuablesuggestionsand Prof. William G. Harter for helpful discussions and for providing the figure presented in this paper. This work was supported by the National Science Foundation and the donors of the Petroleum Research Fund, administered by the American Chemical Society. References and Notes (1) See for a recent review: Bernrreuther, W.; Suzuki, M. Rev. Mod. Phys. 1991, 63, 313. (2) Ramsey, N. F. In Aromic Physics, 7th 4.; Kleppner, D., Pipkin, F. M., Eds.; Plenum: New York, 1981. (3) Christenson, J. H.; Cronin, J. W.; Fitch, V. L.; Turlay, R. Phys. Rev. Lett. 1964, 13, 138.
(4) Altarev, I. S.; et al. Pis'ma Zh. Eksp. Teor. Fiz. 1986,44,360 [JETP Lett. 1986,44,460]; Pendlebury, M. J. in ProceedingsojtheNinthSymposium on Grand Unification,Aix les Bains, France, 1988; Barloutaud, R., Ed.; World Scientific: New York, 1988. (5) Cho, D.; Sangster, K.; Hinds, E. A. Phys. Rev. Left. 1989,63,2559. (6) Pauli, W. 2.Phys. 1926, 36, 336. English translation in: Sources of Quantum Mechanics; van der Waerden, B. L., Ed.; Dover: New York, 1968. Fock,V.A. 2.Phys. 1935,98,145. Seealso: Sung,S.M.;Hershbach, D.R. J . Chem. Phys. 1991, 95, 7437. (7) Bethe, H.; Salpeter, E. The Quantum Mechanics of One and Two Electron Atoms; Springer: Berlin, 1957; pp 238-241. (8) Watson, J. K. G. J. Mol. Specfrosc. 1974, 50, 281. (9) Bunker, P. R. Molecular Symmetry and Molecular Spectroscopy; Academic: New York, 1979. (10) Oka, T. J. Mol. Specrrosc. 1973, 48, 503. (1 1) Mason, S. F.;Tranter, G. E. Mol. Phys. 1984,53, 1091. Quack, M. Angew. Chem. Int. Ed. Engl. 1989, 28, 571. (12) For a review of the properties of symmetric top functions, see: Zare, R. N. Angular Momentum; Wiley: New York, 1988. (13) Nielsen, H. H.; Dennison, D. M. Phys. Rev. 1947, 72, 1101. (14) Townes, C. H.;Schawlow, A. L. Microwave Spectroscopy; .. Dover: New York, 1975. (15) Wofsv. . Klemwrer. .. S. C.: Muenter.. J. S.: . .W. J . Chem. Phvs. 1970. 53,'4oi)5. (16) Harter, W. G.; Patterson, C. W. J. Chem. Phys. 1984, 80, 4241. (17) Gandhi, S. R.; Bernstein, R. B. J. Chem. Phys. 1990, 93,4024.
