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Weak interactions and the stability of optical activity:

a two-level approach

M. CATTANI(1) and J. M. F. BASSALO(2) (1_{) Instituto de Física, Universidade de Sa˜o Paulo}

C.P. 66318, 05315-970, Sa˜o Paulo, SP, Brasil

(2_{) Departamento de Física, Universidade Federal do Pará - C.P. 66075, Belém, PA, Brasil} (ricevuto il 14 Aprile 1997; approvato il 27 Maggio 1997)

Summary. — Assuming the active molecule as a two-level system, we calculate the

racemization taking into account the effects of the weak interactions and of an external potential. We show that the weak interactions can play a fundamental role in the optical stabilization of enantiomers in compressed gases and liquids.

PACS 33.55.Ad – Optical activity, optical rotation; circular dichroism.

As is well known [1, 2], the optical activity of an optically active material changes with time. The sample, containing predominantly one stereoisomer, will become a mixture of equal amounts of each isomer. This relaxation process, which is called racemization, is due to the interaction of the active molecule with the environment. This interaction can be with the remaining molecules of the sample or with an external field. Many dynamic and static approaches have been proposed to describe the interaction of an active molecule with the environment, as is shown in details in recent papers [3-6]. However, these models are not completely satisfactory because they involve some phenomenological parameters whose identification and quantification is not immediate [3]. In preceding papers [7-11], we developed a two-level formalism to explain the environment racemization process when the active molecule is embedded in a gas, liquid or solid. In our approach all relevant physical parameters for the molecules of the active sample are clearly defined. Detailed calculations were performed for optically active dilute gases [8-11].

In a recent paper [9] we have investigated how the racemization depends on the effects of weak neutral currents. Taking into account the contributions of weak interactions and of an external potential we have roughly estimated the racemization in dilute gases. In the present work we analyse in details the racemization in dilute gases, compressed gases and liquids. We show that in compressed gases and liquids the weak interactions can play a fundamental role in the stabilization of the optical activity.

Optical activity [1, 2] occurs when the molecule has two distinct left and right configurations, NLb and NRb, which are degenerate for a parity operation, i.e.

P(x) NLb 4NRb and P(x)NRb 4NLb. Left-right isomerism can be viewed in terms of a

double-bottomed potential well and the states NLb and NRb may be pictured as molecular configurations that are concentrated in the left or right potential well. The double-bottomed potential well is assumed to have the shape of two overlapping harmonic potentials [8] with the two minima at the points x 42a and x4a. The coordinate x is involved in the parity operation P(x) and connects the potential minima. It may represent the position of an atom, the rotation of a group around a bond, some other coordinate, or a collective coordinate of the molecule. We indicate by v the fundamental frequency of each harmonic oscillator and by m the reduced mass of the particles vibrating between x 42a and x4a. We define H as the Hamiltonian of the double well which includes the parity-violating weak interaction. If parity is violated, the left and right sides of double-bottomed potential are no longer exactly symmetrical. In these conditions we have, aLNHNLb 4EL4 E02 e, aRNHNRb 4 ER4 E01 e and

aLNHNRb 4 aRNHNLb 4d, due to a small overlap of the NLb and NRb wave functions inside the potential barrier separating the two minima of the double well. E0 is the

energy of the fundamental left and right states in the absence of weak neutral currents and 2 e is the difference of energy between the left and right configurations due to the parity-violating interactions.

The parameter d, responsible for the natural tunneling, is given by d 4hL, where L, measured in Hertz, is written as

L 4 (2/p3 /2_{) v(mva}2_/ˇ)1 /2

exp [2mva2_{/ˇ] .}

(1)

The natural tunneling time is given by t 41/L.

Let us calculate the racemization of an optically-active sample assuming [11] that the relaxation process is produced essentially by transitions between the two fundamental vibrational states NLb and NRb. The interaction potential of the active molecule with the environment will be represented by U(t). So, the state function NC(t)b of the active molecule, represented by

NC(t)b 4 aL(t) NLb exp [2iELt/ˇ] 1aR(t) NRb exp [2iERt/ˇ] , (2)

will obey the equation (i/ˇ) ¯NC(t)b/¯t4 [H1U(t) ]NC(t)b. Thus, aL(t) and aR(t) are governed by the following differential equations:

.

/

´

a.L(t)

a.R(t)

4 2(i/ˇ)[aL(t)(E_{L1 ULL) 1aR(t)(d 1ULR}) ] , 4 2(i/ˇ)[aR(t)(ER1 URR) 1aL(t)(d 1URL) ] , (3)

where the matrix elements U_{nk, with n , k 4L and R, are given by Unk4 anNU(t) Nkb.} Defining an(t) 4

s

t

0Unn(t) dt and bn(t) 4an(t) exp

[

]

.

/

´

b.L(t) 42(i/ˇ) bR(t)(d 1ULR) exp

[

]

b.R(t) 42(i/ˇ) bL(t)(d 1URL) exp

[

]

Due to very small differences between NLb and NRb states we get ULR(t) 4URL(t). When enantiomers interact with achiral molecules we have ULL(t) 4URR(t). This occurs because achiral molecules cannot discriminate between enantiomers. These have identical chemical properties toward all achiral molecules. When enantiomers

interact with chiral molecules we have, in average, ULL(t) ` URR(t). This permit us to assume, in a first approximation, that ULL_{(t) 4U}RR(t) in Eqs. (4). In these conditions we obtain

.

/

´

b._{L(t) 42(i/ˇ) bR}(t)

(

)

b.R(t) 42(i/ˇ) bL(t)

(

)

We were not able to solve eqs. (5) for a general ULR(t). We will see, in what follows, how to solve these equations in two limiting cases: when ULR(t) is constant and when

ULR(t) changes rapidly with time. 1. – Compressed gases and liquids

In dense gases and liquids, multiple interactions dominate over binary interactions. In a first approximation for dense gases and liquids, as well for solids, the active molecule will be assumed to be inside a small cavity surrounded by the perturbing molecules that are regarded to be at rest. Under these conditions, the interaction potential created by the perturbing molecules inside the cavity may be taken to be static and equal to U1(x) in the region of the double-well potential. This potential U1(x)

can be estimated, for instance, when the active system is composed by dipolar molecules. This potential U1(x) can be estimated as follows [3, 12]: once a molecule is in

a left or right configuration, it has a non-zero average dipole moment d: then this dipole moment locally polarizes the surrounding which, in turn creates, at the position of d , a so-called reaction field Er4 2(e 2 1 ) d/( 2 e 1 1 ) R3, which is collinear with d, where e is the dieletric constant of the medium and R the radius of the cavity in which the molecule is embedded. Since the dipole matrix element of the active molecule between NLb and NRb configurations states is equal to zero, the reaction field Erwill be taken as interacting with the quadrupole moment of the active molecule. So, aLNUNRb 4 aLNU1(x) NRb 4W is given approximately by W`daLNQ(x)NRb/R4. Since aLNQ(x)NRb 4

u exp [2mva2_{/ˇ], where u is the quadrupole matrix element of the active molecule}

between left and right configurations [8] W can be written as

W 4 (ud/R4

) exp [2mva2_{/ˇ] .}

(6)

Substituting U_{LR(t) 4W, given by eq. (6) into eqs. (5) we verify that these equations can} be exactly solved [9], giving for the racemization r(t):

r(t) 4 (deff/D)2sin2(Dt/ˇ) ,

(7)

where deff4 d 1 W and D 4

(

)

.

In table I are shown the numerical values for L and W/h, measured in Hz, as a function of the frequency v, measured in rad/s. These values were calculated by using the following typical molecular parameters: a 41028_{cm , m 410}222_{g , m 410}223_{g , d 4}

10218 _{esu, u 410}226 _{esu, T 4300 K and N410}17_/cm3_{. In this table are also shown the}

values for the natural time tunneling t, defined by t 41/L, and the racemization time

Tr4 1 /l, measured in years.

According to recent calculations [13-19], the asymmetry in the spectra of mirror optically active systems produced by weak interactions can be estimated to be typically of the order of e/h ` 1023_{Hz for rotational and vibrational transitions and of the order}

Table I. – The parameters L , W/h , l, measured in Hz, t and Trin years, as functions of v, in rad/s. v Q 1013 _{L W/h l T} r t 5.0 2 .6 Q 1027 _{3 .1 Q 10}27 _{2 .6 Q 10}26 _{1 .2 Q 10}22 _{1 .2 Q 10}21 5.1 1 .0 Q 1027 _{1 .2 Q 10}27 _{1 .4 Q 10}26 _{2 .3 Q 10}22 _{3 .1 Q 10}21 5.2 4 .1 Q 1028 _{4 .7 Q 10}28 _{7 .2 Q 10}27 _{4 .4 Q 10}22 _{7 .8 Q 10}21 5.4 6 .4 Q 1029 _{7 .0 Q 10}29 _{2 .0 Q 10}27 _{1 .6 Q 10}21 _4.9 5.6 1 .0 Q 1029 _{1 .0 Q 10}29 _{5 .7 Q 10}28 _{6 .6 Q 10}21 _31.4 5.8 1 .6 Q 10210 _{1 .5 Q 10}210 _{1 .6 Q 10}28 _{2 .0 200} 6.0 2 .5 Q 10211 _{2 .3 Q 10}211 _{4 .5 Q 10}29 _{7 .1 1 .3 Q 10}3 6.5 2 .4 Q 10213 _{2 .0 Q 10}213 _{1 .9 Q 10}2 Q 10 _{1 .7 Q 10}2 _{1 .3 Q 10}5 7.0 2 .3 Q 10215 _{1 .7 Q 10}215 _{7 .8 Q 10}212 _{4 .1 Q 10}3 _{1 .4 Q 10}7 7.5 2 .2 Q 10217 _{1 .4 Q 10}217 _{3 .2 Q 10}213 _{9 .8 Q 10}6 _{1 .5 Q 10}9

From table I we see, for frequencies v F5.2 Q 1013/s, that L ` W/h and that these values are much smaller than e/h. This implies, according to eq. (7), that r (t) 4 (deff/e)2b1, showing that it would be possible to get optical stabilization for chiral

molecules with natural lifetimes t larger than one year [20]. The difference of energy e, due to weak interactions, in spite of being very small, would be able to block the transitions induced by the natural tunneling and by the static external field, between NLb and NRb. Finally, as the electric fields from perturbing particles very close to the active molecule undergo very rapid changes with time, the validity of the static approximation is doubtful. However, we must expect that a more realistic perturbing potential would be given by W 1DW(t), where DW(t) changes, not very abruptly, with time and that NWNcNDWN. In this way, a simple analysis of eqs. (5) would show us that the oscillating term would not modify our above conclusions about the optical stabilization.

2. – Dilute gases

When the optically-active system is a dilute gas, the dynamics of interactions of the active molecule is expressed in terms of binary collisions with the perturbing molecules of the sample. The durations of these collisions are very short (around 10211_{s for}

molecules at room temperature) but the frequency of binary collisions is very high since the molecular density is N ` 1017/cm3_{. This implies that the factor d 1U}LR(t), in eqs. (5), changes very rapidly with time, much more than the multiplicative term exp [6iet/ˇ]. In these conditions eqs. (5) can be integrated using the “impact approximation” [10, 21], giving

r (t) 4

[

]

where the transition probability l, measured in Hz, is shown elsewhere [10]. For a dipole-quadrupole interaction between active and perturbing molecules, l is given by [11]

l 413.0N(kT/m)1 /6(ud/ˇ)2 /3_{exp [22mva}2/3 ˇ] , (9)

eq. (9), measured in Hz, and the racemization lifetime Tr4 1 /l, measured in years, as a

function of the frequency v.

From eq. (8) and table I, we verify that the binary collisions are a very efficient agent for racemization and that no optical estabilization is possible in dilute gases. 3. – Conclusions

We have shown that it would be possible to have optical stabilization of enantiomers in dense gases and liquids and it would not be possible in dilute gases. It is very important to remark that our conclusions regarding the stabilization of enantiomers are limited to those molecules that principally racemize through simple inversion alone. As is well known, there are many other different racemization mechanisms [22]. In our work these processes are not considered.

* * *

The author, M. CATTANI, thanks Prof. A. DI. GIACOMOfor helpful discussions on the problem and for his kind hospitality during his visit at the Dipartimento di Fisica di Pisa, Italia. He also thanks the CNPq for financial support.

R E F E R E N C E S

[1] BARRON L. D., Molecular Light Scattering and Optical Activity (Cambridge University Press, Cambridge) 1982.

[2] MASONS. F., Molecular Optical Activity and Chiral Discrimination (Cambridge University Press, Cambridge) 1982.

[3] CLAVERIE, P. and JONA-LASINIOG., Phys. Rev. A, 33 (1986) 2245.

[4] LEGGETA. J., CHAKRAVARTYS., DORSEYA. T., FISHERM. P. A., GARGA. and ZWERGERW.,

Rev. Mod. Phys., 59 (1987) 1.

[5] SUAREZA., SILBEYR. and OPPENHEIMI., J. Chem. Phys., 97 (1992) 5101. [6] RAFFELTG., SIGEG. and STODOLSKYL., Phys. Rev. Lett., 70 (1993) 2363. [7] CATTANIM., J. Quant. Spectrosc. Radiat. Transfer, 46 (1991) 507. [8] CATTANIM., J. Quant. Spectrosc. Radiat. Transfer, 49 (1993) 325. [9] CATTANIM., J. Quant. Spectrosc. Radiat. Transfer, 52 (1994) 831.

[10] CATTANI M., Nuovo Cimento D, 17 (1995) 1083; J. Quant. Spectrosc. Radiat. Transfer, 54 (1995) 1059.

[11] CATTANIM., J. Quant. Spectrosc. Radiat. Transfer, 55 (1996) 191.

[12] BO¨TTCHERC. J. F., Theory of Electric Polarization (Elsevier, Amsterdam) 1952. [13] DIGIACOMOA., PAFFUTIG. and RISTORIC., Nuovo Cimento B, 55 (1980) 110. [14] HARRISR. A. and STODOLSKYL., Phys. Lett. B, 78 (1978) 313.

[15] MASONS. F. and TRANTERG. E., Mol. Phys., 53 (1984) 1091. [16] TRANTERG. E., Mol. Phys., 56 (1985) 825.

[17] MASONS. F. and TRANTERG. E., Proc. R. Soc. London, Ser. A, 397 (1985) 45. [18] TRANTERG. E., Chem. Phys. Lett., 115 (1985) 286.

[19] MACDERMOTTA. J., TRANTERG. E. and TRAINORS. J., Chem. Phys. Lett., 194 (1992) 152. [20] ARMSTRONGD. W., ZHOUE. Y., ZUKOWSKIJ. and KOSMOWSKA-CERANOWICZB., Chirality, 8

(1996) 39.

[21] CATTANIM., J. Quant. Spectrosc. Radiat. Transfer, 42 (1989) 83.