J. Phys. Chem. 1984, 88, 5820-5826
5820
Opto-Vibronic Generation of Chirality via Jahn-Teller Distortion of Odd- and Even-Spin Moleculest Ying-Nan Chiu Center for Molecular Dynamics and Energy Transfer, Department of Chemistry, The Catholic University of America, Washington, D.C. 20064 (Received: March 13, 1984; In Final Form: June 4, 1984)
A mechanism is proposed for the production of chiral molecules through the cooperative action of polarized light and vibronic interaction. It requires a doubly degenerate electronic state of an achiral molecule to be split by a nondegenerate vibration of pseudoscalar symmetry in a Jahn-Teller effect. The splitting results in symmetrical double potential wells for the lower level, with one side of the well corresponding to one enantiomer and the other side to its mirror image, and with equal probability for each. It will be shown that the discrimination of one enantiomer against the other may be assisted either by the neutral-current coupling to a nuclear weak interaction or by a second-order pseudoscalar radiative interaction, both of which break the parity symmetry. The latter light-intensity-dependent interaction may give a lower energy for one enantiomer than the other by eV or more, and may be operative in geological time. It was found that only a limited class of molecules are amenable to this phenomenon, Le., those belonging to D2d, D4d, D6d, S4,and Sg. Spin singlet and doublet states are both considered. For odd-electron spin functions and for transformation of spin functions,the most general selection rule, for either spin-dependent or spin-independent operators, consistent with time-reversal permutational symmetry, is derived. New tabulation of Dzdf and Sql double groups are given to correct some misconceptions and minor errors in existing literature. The possible operation of this mechanism for chiral discrimination on the laboratory time scale is briefly discussed.
I. Introduction A large number of optical resolution methods’ make use of extraneous enantiomers to discriminate against one of the enantiomers in a racemic mixture. Some methods use extraneous circularly polarized light for this discrimination. Most of these methods are more destructive to one enantiomer than the other. In the case of photodiscrimination, the detailed molecular dynamics have seldom been elucidated. The theory of discriminating interaction2p3between chiral molecules addresses the intermolecular dynamics of the magnetic dispersion force.4 None of the above methods or theories address the philosophical question of enantiomeric excess in nature that is intrinsic to a molecule. In this regard, it was recognized5-’ that the mirror-image enantiomers are not exactly degenerate in this world (vs. the antiworld). The degeneracy is split by weak neutral-current coupling of electromagnetic and weak interaction, which has a parity-nonconserving pseudoscalar potential. C a l c ~ l a t i o n give s ~ ~an ~ ~energy ~ difference varying from to eV for different molecules. In this work we propose another pseudoscalar interaction coming from optical interaction that also splits the mirror-image degeneracy. In addition we shall show that when it is combined with JahnTeller vibronic coupling, it provides a nondestructive molecular mechanism for discriminating enantiomers. Although such splitting5 are small, they may well manifest themselves over a cosmic time scale. Based on this theory of molecular dynamics using high intensity laser light of the right wavelengths and with the right molecular vibrational frequency, some enantiomeric excess might be achieved on the laboratory time scale. Basically a left-handed or right-handed gaantiomer is a chiral molecule without parity (no rotation-reflection symmetry S,). For the conservation of parity of electromagnetic interaction, the enantiomer must be in dynamic equilibrium, albeit with a long lifetime, with its “degenerate” mirror image (in the sense of a double-well potential). In such a chiral molecule, not only an invariant scalar (e.g., R.R = x2 + yz + z2) but also a pseudoscalar (e.g., R.M = xl, + yl, + zl,) can belong to the totally symmetric irreducible representation of the molecular point because the rotation-reflection S,, which can change the sign of a pseudoscalar, is not a member of the group. Hence a pseudoscalar is invariant and is a physical observable of the molecule. Furthermore, because it changes sign under reflection, this observable will have exactly the opposite sign for the mirror-image enantiomer. When a parity-nonconserving pseudoscalar potential V$i +Presentedin part in the 39th Symposium on Spectroscopy in Columbus, OH, June 1984.
0022-3654/84/2088-5820$01.50/0
is present it gives unequal energies for the left and right enantiomers, giving an unsymmetrical double-well potential. This is accompanied by unequal mixing of left and right components, favoring one over the other, and thus the parity is broken. But, because the asymmetry of the wells is so small (AE eV or smaller) and the barrier is so high, though-resonance is broken, the resolution of one kind of enantiomer is slow. In order to detect these phenomena we must look for (1) a lower interconversion barrier (sections I1 and 111) (2) a larger pseudoscalar interaction (section IV). For (2), we find the pseudoscalar from the second-order radiative interaction12-16with randomly oriented precursors of an optically active molecule. For (1) we find a symmetry class of chiral molecules which has degenerate electronic states (e.g., Dzd has E containing x and y ) split by the Jahn-Teller coupling. The latter vibronic coupling comes from a twisting vibration that coincidentally has the same pseudoscalar symmetry (B, in DU). This pseudoscalar symmetry correlates coincidentally to the totally symmetric irreducible representation (A,) of a lower point group that is chiral (in this case D2). Therefore, in principle, the twisting vibration, which is a hindered internal rotation, may go to the left giving one optically active enantiomer, or may go
-
(1) Mason, S. F. “Molecular Optical Activity and the Chiral Discriminations”; Cambridge University Press: New York, 1982. (2) Craig, D. P.; Power, E. A,; Thirunamachandran, T. Chem. Phys. Lett. 1970, 6, 211;Proc. R. SOC.London, Ser. A 1971, 322, 165. (3) Craig, D. P.; Mellor, D. P. In “Topics in Current Chemistry”; Boschke, -. F. 0. L., Ed.; Springer-Verlag: New York, 1976;Vol. 63. (4) Chiu, Y. N.;,kgnney, A. V.; Brown, S . H. J. Chem. Phys. 1980, 73, 1422. Chiu, Y. N. j h y s . Rev. A . 1979, 20, 32. (5) Harris, R. A,; Stodolsky, L. Phys. Lett. B 1978, 78B, 313. Harris, R. A,; Stodolsky, L. J . Chem. Phys. 1981, 74, 2145. (6) Barron, L. D. Chem. Phys. Lett. 1981, 79, 392. (7) Barron, L. D. “Molecular Light Scattering and Optical Activity”; Cambridge University Press: Cambridge, England, 1982. (8) Zel’dovich, B.Ya.; Saakyan, D. B.; Sobel’man, I. I. JETP Lett. (Engl. Transl.) 1977, 25, 94. (9) Hegstrom, R. A,; Rein, D. W. Sandars, P. G. H. J . Chem. Phys. 1980, 73, 2329. Rein, D. W.; Hegstrom, R. A,; Sandars, P. G. H. Phys. Lett. A 1979, 71A, 499. (1 0) Cotton, F.A. “Chemical Application of Group Theory”; Interscience: New York, 1963. (1 1) Tinkham, M. “Group Theory and Quantum Mechanics”; McGrawHill: New York, 1964. (12) Hameka, H.F. “Advanced Quantum Chemistry”; Addison-Wesley: Reading, MA, 1965, Chapter 11. (13) Heitler, W. “The Quantum Theory of Radiation”; Oxford University Press: London, 1954. (14) Power, E.A. “Introductory Quantum Electrodynamics”;Longmans: New York, 1964. (15) Chiu, Y. N. J . Chem. Phys. 1969, 50, 5336. (16) Chiu, Y. N., submitted for publication in J. Chem. Phys.
0 1984 American Chemical Society
The Journal of Physical Chemistry, Vol. 88, No. 24, 1984 5821
Opto-Vibronic Generation of Chirality
TABLE I: Degenerate States in Parent Molecular Point Groups Split by Pseudoscalar Vibration in Jahn-Teller Effect’
I
k
achiral parent r state
direct product
E3 (D6d)
B l , BZ
rz
D,
‘Id
Figure 1. Figurative representation of the Jahn-Teller splitting of an E state in a Dzd molecule. The distorted molecule is chiral, has Dzsymmetry, and corresponds to either a left-handed ($q,) or right-handed (h) enantiomer. Dotted lines indicate that a pseudoscalar radiative interaction potential sets up the possibility for the degenerate enantiomers to split symmetrically, one up and one down. The thermodynamic equilibrium to lower energy is facilitated by internal BI twisting vibration in addition to the usual external collision due to translation. Upper case B1 is the symmetry of the parent D2dpoint-group molecule, lower case b2 and b3 are the symmetries of the distorted D2 molecular states.
to the right giving its mirror image. This is a precise simulation of the symmetrical double-well potential situation along the (pseudoscalar B,) vibration coordinate. These wells are perfectly amenable to modification by the abovementioned optical potential of exactly the same pseudoscalar symmetry (B,) (Figure 1). 11. Jahn-Teller Distortion by a “Pseudoscalar”Vibration For the static Jahn-Teller distortion to a lower point group of a molecule, the distorting vibrations must belong to the totally symmetric irreducible representation of this lower point group. For an achiral molecule to distort to a chiral molecule, the vibration must then be a pseudoscalar belonging to the totally symmetric irreducible representation of the chiral group. This pseudoscalar disallows any parity operator (S,) in this molecular subgroup that may change its sign. In this way parity of the molecule is destroyed. An example is the BI pseudoscalar vibration, in a Dzd molecule alluded to previously, which belongs to Al of the chiral subgroup D? This vibration will work in concert with the (R-M) radiative interaction (section IV) that also has pseudoscalar symmetry. The important points to look for here are that the direct product of the doubly degenerate state (not triply degenerate) contains the pseudoscalar representation (e.g., EZ = A I Bl [A,] Bz in &d) which is symmetric in a permutational sense. The state is split by the pseudoscalar vibration Bl which is a one-dimensional representation (being a “scalar”). See Figure 1 for the splitting of E of DZdinto x and y or b3 and bz of the Dz subgroup. Thus for the standard Herzberg-Teller expansion of the vibronic operator17 we get the first-order correction to energy, W, for Dzd. As an example
+
+
+
where from the irreducible representation of the Dzd point group we get xx
- YY C B1
~ H / ~ QCB B1 ,
xx
+ y y C Al
B1
= a (C2)
Bl (D6d)
= a1 (O6)
=a B I (D6d) = Bl ( D 6 d ) = a Bl (D6d)
E (4) A, [AI, B, B E2 (Sd A, [AI, B, B D,
pseudoscalar vibration
(2a)
(17) Herzberg, G. “Molecular Spectra and Molecular Structure”; Van Nostrand: Princeton, NJ 1966; Vol. 111.
(c6)
(4) (c3)
B (S4) = a (C2) B
6 )= a
chiral daughter states (subgroup)
(C4)
a, a (C2) bl, b2 (O6) b, b (c6) a,, a2 ( 0 3 ) a, a (CA b, b (Cz) b, b (C4)
‘ C , and C,, both have degenerate states (El, E2, E) that contain pseudoscalar A2 in their self-product. However, A2 is antisymmetric under permutation. Furthermore, when the pseudoscalar correlates to the totally symmetric “a” in the subgroup (c6 and C,), the degenerate state does not split. And when the degenerate state splits in a subgroup, Az does not correlate to the totally symmetric irreducible representation of this achiral subgroup.
where the symmetric direct products xx, y y guarantee satisfaction of time-reversal invariance’ 1~18required for a time-even vibronic operator over even-electron states (see below). An example of this type of molecule is a perpendicular allene19nz0in an excited state. The unique point here is that the resulting distorted molecule and negative vibrations is chiral and the positive vibrations yield respectively the right or left opposite enantiomers. I t is very different from the Jahn-Teller distortion of the squareplanar X4 (D4h) molecule into an X4 (DZh)molecule. In this later distortion by the nonsymmetric vz(QB, ) vibration, the E, electronic state composed of lx and 1, in D4his split into the b3&lX)and bzg(ly) states in D2,,. But the distorted rectangular molecule in this achiral group is the same molecule for positive and negative vibrations. It is also very different from the two-dimensional QEvibrational distortion of the generate E electronic statez1of an octahedral molecule (oh).The resulting elongated and compressed octahedrons belonging to the subgroup DPhare distinctly different structures and both are achiral. A comprehensive search among the point groupszz-z5shows that only a limited number of them have the above-mentioned properties we desire. These are given in Table I. In the examples in Table I, the doubly degenerate state of an achiral molecule may be split into an upper state with a single minimum and a lower state with symmetrical double minima, one on the side of positive vibration, and one on the negative side, each representing an optical enantiomer. The barrier heights separating the two potential wells vary from molecule to molecule. The case of very shallow wells is found in twisting ethylene (CzH4)in its ‘E state.z6 But the chance for going into each well is exactly equal. There is no Maxwell’s Demon sitting at the top of the barrier to pick one enantiomer over the other. However, the parity-nonconserving due to neutral-current coupling of pseudoscalar potential weak interacti~n,~***~ will lower one well with respect to the other
(eB,)
-eB,
vic,
(18) Lax, M. “SymmetryPrinciples in Solid State and Molecular Physics”; Wiley: New York, 1974; Chapter 10. (19) Chiu, Y. N. J. Am. Chem. SOC.1982, 104, 6937 and references therein. (20) Chiu, Y. N., accepted for publication in Isr. J . Chem. (21) Ballhausen, C. J. “Introduction to Ligand Field Theory”; McGrawHill: New York, 1962. (22) Atkins, P. W.; Child, M. S.; Philips, C. S. G. “Tables for Group Theory”; Oxford University Press: London, 1970. (23) Salthouse, J. A. Ware, M. J. “Point-Group Character Tables and Related Data”; Cambridge University Press: Cambridge, England, 1972. (24) Schutte, C. J. H. “The Theory of Molecular Spectroscopy”;NorthHolland Publishing Co.: Amsterdam, 1976. (25) Wilson Jr., E.B.;Decius, J. C.; Cross, P. C. “Molecular Vibrations”; McGraw-Hill: New York, 1955. (26) Merer, A. J.; Mulliken, R. S. Chem. Rev. 1969, 639.
5822 The Journal of Physical Chemistry, Vol. 88, No. 24, 1984
Chiu
TABLE 11: Basesa. Transformations.b and Characters in S,' and D,A' Double Grouus and under Timer ReversalC
S,' bases
E
S,
1 -W 1 -W* 2 -2 1 1 2 1 W* 1 W 2 2'12 1 i 1 -i 1 W* 1 0 1 --w 1 --w*
C : S,' -i
W*
i 0
W 2112
i
-W
-i 0 -1
--w*
-1
I
i
-W
-i
-W*
-i
W*
i
W
-2112
-i
E -1 -1 -2 -1 -1 -2 1 1 -1 -1 -1 -1
5,
-
C, S,' -
W
i
-W*
W* 2112
--w
-a*
-i 0 -i
-W
i
W*
-2112
c**
c,y
K
W 2112
-W W
i
--w*
W*
-i
--w
--w*
S[I
time reversal
-2112
0 -1 -1 -i i
i -i
Did'
-i i W
W*
means reflection By definition n2 = x r iy, w = exp(in/4). Compare tables in ref 17 and 29. Note S,' is a subgroup of D z d ' . udXmY with respect to the dihedral plane between the xz and the -yz planes. It is equal t o inversion multiplied b y a rotation with Euler angles 4 = -n/4, e = n, $ = n/4. Note that all of the extended double group operations (with a bar) will have signs for the transformations matrix as well as characters that are opposite t o the corresponding ones without a bar. Dr. G . Herzberg kindly called the writer's attention t o a paper by L. Couture and A. Le Pailler-Malecot (Mol. Phys. 1982,45, 663) which gave D,d characters (no matrices). Time reversal operation defined as K = u y K o where K, is the complex conjugation operation and uyol = ip, uyp = -io!.
and will favor one enantiomer over the other. The pseudoscalar optical interaction we shall propose (section IV) will also help the discrimination. These potentials work whether the vibrational coordinates are positive or negative. Thus we get, referring to eq 1 and 2, the unequal well energies W' (Figure 1)
where
is an effective, second-order, parity-nonconserving potential and is derived from the following second-order interaction
- Eb) z (x1xlu'pl$b3x) (xlPl$b)'(~blllx)/(Eo- Eb) (3e)
('rCl~lvpncl~$b)(~$blv~I',I'$x)/(EO
($b3Xl~.b'X)/(Eo- Eb)
where spin-orbit interaction V,, is needed to connect the ground singlet ('$) to the intermediate triplet (3$b). In eq 3e the Pauli spin operators u have been evaluated over the singlet ('x) and triplet (3x)spin wave functions giving (u#)(u-l) = Pel+ i0.P x 1. It is seen that P.1 is proportional to the optical rotatory strength operator R-M and is a pseudoscalar. The above discussions are applicable to even-electron ground singlet states. For the spin nonzero even-electron case, spin-orbit interactions will introduce new features in the splitting and in the barrier. Of special interest is the odd-electron situation. The 2E state of twisted ethylene is reported by Merer and MullikenZ6to have a shallow double well. Yet, overtly by Jahn's spin-selection no Jahn-Teller distortion exists for this twofold degenerate state. We shall explore the odd-electron case (section 111), not only because the spin selection rule and transformation need to be clarified and corrected in the literature, but also because the spin-orbit interaction provides an addition modulating factor for barrier height and tunneling probability. 111. Pseudo Jahn-Teller Distortion of Odd-Electron States Through Pseudoscalar Vibration In the case where the number of electrons is odd, Jahn'sZ7spin selection rule requires that only the antisymmetric direct product (rZ]will yield a nonvanishing first-order vibronic matrix element for Jahn-Teller distortion.28 Jahn also asserted that two-di(27) Jahn, H. A. Proc. R. SOC.London, Ser. A 1938, 164, 117.
mensional irreducible representations have antisymmetric direct products that always belong to the totally symmetric (A,) irreducible representation. Therefore, they will not sustain a nontotally symmetric vibronic interaction and will not distort. By this rule, a Du molecule which has Jahn-Teller distortionZEby B, vibration in an even-electron 'E state will suddenly become immune to distortion in an odd-electron 2E state. The latter has only twodimensional representations, ElI2and E3/2. Nature in this case should not be so capricious. In fact the spectroscopic results of Merer and Mulliken26indicated a double well potential also for the 2E state. We wish to explain the reason for this distortion and to clarify the spin-selection rules. Firstly, in the molecular groups of interest, DU and S4,we note that the characters of the S4 operator have been computedZ9the same as those of the C4 operation on a double-group basis function. This is not justified because, even though the spin functions in double-group representation are axial functions and are invariant under inversion, the reflection in S4 has more than an inversion (i) effect. Thus S4 = u C ~ = iC2C4
The extra C2 r ~ t a t i o nyields ~ ~ , ~extra ~ imaginary factors in C2a = -ia C2/3 = i/3 giving a character different from that of C4. Similarly, the udXs reflection plane between the x and y axis may be written as an inversion multiplied by a rotation with an axis between x and -y, Le., an axis perpendicular to the xy reflection plane. The latter rotation C2x*-Yhas Euler angles a = cp = - 3 ~ / 4 , / 3= 0 = T , and y = $ = 3 ~ 1 4 The . resulting rotation matrixg0exp((i/2)uZ(3?r/4)) exp((-i/2)uYn) &xp((-i/2)az(3a/4)) yields CZx,-Ya= -w/3 and C z x ~=~wp* a . Since these transformations are either not available or not correctly given in literature" we give them in Table 11. They will be used to determine the degenerate-state products and selection rules. Secondly, we note that the direct products r 8 r as given in all group character tables are good for spinless functions or functions with an even number of electrons. For spinless wave functions, linearly independent complex functions that span the same r may also give rise to the same direct product I'* @ r = I' 8 r (e.g., ( x iy); (x - iy) span the same irreducible representation E as (x,y) in D 2 J . Such direct products may be used for selection rules over spin-independent operators. However, for odd-electron, spin-dependent wave functions (and operators) and for time-reversal invariance the most general, correct direct product
+
(28) Jahn, H. A.; Teller, E. Proc. R. SOC.London Ser. A . 1937, 161, 220. (29) Flurry, Jr., R. L. "Symmetry Groups, Theory and Applications"; Prentice-Hall: Englewood Cliffs, NJ, 1980. (30) Brink, D. M.; Satchler, G. R. "Angular Momentum"; Oxford University Press: London, 1968; p 20 ff. (31) Chiu, Y . N. J . Chem. Phys. 1966,45, 2969.
The Journal of Physical Chemistry, Vol. 88, No. 24, 1984 5823
Opto-Vibronic Generation of Chirality TABLE 111: Direct Products of Degenerate Representationsain D2d’ (and S,’)b
with and without Time ReversalC
E = (KE)*, E* A , , [ A , ] , B l , Bz E3/2’EI/Z E,/,%El,, Ell, [ A , ] , A,, E (B,)(B,)(E) (K$L+,l,)*= ‘i$L+312 B , , B 2 , (E) (Al)(A2)(E) E [ A l l , A,, E B , , B , , (E) (K$L+3/2)* = Ti$*i/Z a See Table I for the definitions of the representation bases. Square brackets [I indicate antisymmetric products. Parentheses () indicate that both symmetric and antisymmetric products are possible when (K$J*ILU is combined with (K$u)*$u. See Table IV or eq 6 . For S4’,change A , and A, to A and B , and B, to B. The representation function to use for time-reversaloperation K is (K$,,J*. It is given underneath the usual double-grouprepresentation. It is mainly for its effect on spin functions and is applicable for spin-dependent, time-odd operators.
K(;) =
(@. ) --ICY
K Z =~(i)zp = K - z p
K Z= ~(-i)za = K + z a
A “transition” probability density constructed as (K+,)*+” that occurs in an integral will have the following behavior under time reversal:‘* KPO= KJ(K+,)*+,
d~ = KW,,$,)
(K$,&W,)
KZN(K$”,$,) = (-)”J(K+.)*+,
D,Al
Sdl
=
dT ( 5 )
where K~ = ( K + ) ~ = ( ~ i =) -1~ is the “eigenvalue” of the spin function (a or 0)under double time reversal and N is the number of electron spins. The most general “transition” probability that is time invariant (Le., symmetric under time reversal operator K ) comes from the following average, which amounts to a time-even projection operation 1/2( 1 + K ) : P =
Y20
+ K)Po = xu + KZNP,,)P0 =
+ (-)N(w”)*+rl (6)
X[(K$,)*$, TABLE IV Direct-Product Representations of Degenerate States in Dwr and S i
(4)
From the above it is clear that when N = even number of spins, this gives a symmetric (+ sign) permutation and when N = odd number of spins, this gives an antisymmetric (- sign) permutation of M , V . For spinless wave functions the time reversal operation gives the usual symmetric [r2]= p+ and antisymmetric (r2)= pcombination in conventional character tables for r 8 r Pi =
Y2(1 f
P,”)$,$U
=
X($,$, * $k,)
(7)
For an even number of spins, when the spin function is invariant under K, the above is also true (e.g., K(i/2’/’)(P@ + act) = (i/ 21/2)(&3 aa) belongs to E of DZd).If the operator is spinless (such as the vibronic operator) and the spin functions integrate to unity, only the spatial part (which responds to time reversal as if it were a complex conjugation) counts, and then the above simplified direct product pi may also be used to determine the selection rules. This is illustrated in Table IV for 8 r3/2. But, for a general operator H’occurring in a matrix-element integral Io = ](K$,)*HV,, dr, the effect of time reversal (compare eq 5) is
+
CHART I permutation symmetric (+) antisymmetric (-)
HI” time-even, a = +1 time-odd, a = -1 time-even, a = +1 time-odd, a = -1
Nb
case
even odd
1 2 3 4
odd
even
Perturbation operator. bNumber of spins. to consider is (KT)* 8 r which also covers the even-electron case. We have derived these for D21and S4’groups in Tables I1 and 111. The reason for this direct product lies in the antiunitary nature of the time-reversal operatorla K = uyyKowhere KOis the complex conjugation operator and uy is a Pauli spin matrix that gives Hence
(“>=(? p ) -1ff
KIo = KW,,H’$,) (K$,,KH‘K-’~$,)
= K(H*K$,,$J = (K$,,KH@K+,) = = ~K~~(K$,,H’$,) = 4-)N(K$u,H‘$,,) (8)
where KHtK-I = aH’with a = +1 or -1 for time-even or time-odd Hermitian operator, respectively. The most general time-invariant matrix element (integral) should be (compare eq 6 )
I = Y2(1
+ K)Io = f/z(1 + aK2NP,,)Zo = %[(W,?mJ+ ~(-)N(K+u,Hr$,)l (9)
The product a(-)N determines the symmetric or antisymmetric combination for use in the special kind of operators and the special number of spins (see Chart I). Case 1 applies to the Jahn-Teller effect in the ‘E state of a DU molecule. Case 3 applies to the effect in 2E. For spinless wave functions or an even number of electrons the general matrix-element integral reduces to
TABLE V Time Reversal Adapted Point-GroupSelection Rules@for Spin-Dependent and -Independent Operators over Odd-Electron Systems Dw‘ (1 - P,,)p, (for time-even operators) p o = (K+J*+. i (K$-,,)*$-, 3 (KT)* X r; (1 + P,,)p, (for time-odd operators)
*B”f $-,L= x r rxr=~,
(K+1/2)*+3/2+ (K+-I/Z)*$-~/Z C E-
3 BI X Ex i(r+rt- r-r-)(aa + PP) 3 B2 X Ey spinless operator B,;spin-dependent operator E spinless operator Bz; spin-dependent E
i(rtr++ r-r-)(aa - PO)
~
(K+I/z)*+~/z - (K$-I/Z)**-~/ZC E-
rxr=~,
(WI/2)*$.-3/2 + (K+-I/z)*+3/2 r x r = E- + E+
+ E+;
- Et;
c AI;
- (K+4/2)**3/2 c A,; r x r = E- + E~
(K$1/2)*$-3/2
“See eq 6 and the text following it.
+
+
i(rtrt - r-r-)(aa - PPI 3 BZ X E, i(r+r+ T X - ) ( ( Y O I pp) 3 Bl X Ey spinless operator B,; spin-dependent operator E spinless operator B,;spin-dependent operator E
i(rtr- + r-r+)(aP- Pa) 3 A I X A, spin-dependent operator AI
i(rtr- - r-?r+)(aP+ pa) 3 Az X A, spin-dependent operator A I
+ r-r+)(aP+ pa) 3 A I X A2 spin-dependent operator A2
z(r+r-- r-r+)(aP -) .0
i(rtr-
3 A2 X A] spin-dependent operator A2
5824 The Journal of Physical Chemistry, Vol. 88, No. 24, 1984 I =
+
f/2[(+p”v”) a(->N(+”,”+,)l
(10)
Thus for a time-even operator (a = +l), only a symmetric direct product is permitted for an even number of spins. For the same kind of operator, if it is spinless, even if there is an odd number of spins, as long as the spins integrate to unity effectively erasing their presence, the same equation with N = odd shows that only the antisymmetric direct product is permitted. These conclusions agree with Jahn and Teller’s “spin” selection rules. But, one has to carefully delineate the case in which spins integrate to unity. The above also shows that a time-odd ( a = -1) magnetic operator can in principle give a matrix’element for an odd number of electrons with symmetric permutation (case 2). These symm e t r y selection rules are necessary conditions for nonvanishing matrix elements. To have sufficient conditions for spin operators, spin-orbit interactions must be invoked. The above are time-reversal symmetry selection rules. They are necessary but not sufficient conditions. Superimposed on these must also be point group selection rules which must be compatible for the matrix element to be nonzero. Namely, it must not only be symmetric under time reversal but also totally symmetric under all symmetry operations, viz.
Even these symmetry rules may still not be sufficient for spindependent operators in doublepoint groups. Spin-orbit interactions may have to be invoked to have truly nonvanishing matrix eleit may be necessary ments. For example, for a state Ell2 =)::( to mix in another E l l , = ( t . )having the same transformation, viz. Iuoa)’
= laoa)
+ (*+PIAl+S-laoa) I*+@) AE
After all, without spin-orbit interactions that interconverts spin and orbital functions, making them equivalent, the double group, which treats spin and orbit on equal footing, is meaningless. Now, armed with these symmetry products and selection rules, we look at the spinless nonmagnetic, time-even Jahn-Teller vibronic operator that belongs to B1 symmetry and see that there is no B1 in the 8 I’ self-product of either EIl2= (2:)= ($
