Environment racemization: A two-level approach

J. Quant. Spectrosc. Radiat. Transfer Vol. 46, No. 6, pp. 507-511, 1991

0022-4073/91 $3.00+ 0.00 Copyright © 1991 Pergamon Press plc

Printed in Great Britain. All rights reserved

ENVIRONMENT

RACEMIZATION:

A TWO-LEVEL

APPROACH

M. CATTANI lnstituto de Fisica da Universidade de S~o Paulo, C.P. 20516, 01498 Silo Paulo, Brasil

(Received 5 June 1991)

A ~ t r a e t - - A two-level quantum-mechanical approach is developed in order to explain how

racemization depends on the interaction of the active molecule with the environment. Relevant physical parameters for the molecules and the optically-active sample are defined. Our predictions are applied to (a) dilute gases and (b) dense gases, liquids or solids.

1. I N T R O D U C T I O N As is well known, l 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. We investigate how this relaxation process, which is called racemization, depends on interaction of the active molecule with the environment. Many dynamic and static approaches have been proposed to describe the interaction of an active molecule with the environment, as is discussed in detail by Claverie and Jona-Lasinio. 2 As pointed out by these authors, 2 the models are not completely satisfactory because they involve phenomenological parameters that are not immediately identifiable. Claverie and Jona-Lasinio 2 have developed a static approach that explains how active molecules exhibiting symmetric configurations become a single stereoisomer. In this paper, assuming that an active molecule changes its handedness only by tunnelling, a two-level quantum-mechanical formalism will be developed in order to explain the effect of the environment on the racemization process. Our approach can be applied when the active molecule is embedded in a gas, liquid, solid, or is subjected to a generic external field. 2 Since asymmetries in the spectra and in the optical activity induced by weak interactions are small, ~-3 only electromagnetic forces will be taken into account in our model. 2. A P E R T U R B E D

ACTIVE MOLECULE

Optical activity occurs 1 when the molecule has two distinct left and right configurations, [L ) and IR ) , which are degenerate for a parity operation, i.e., P IL ) = IR ) and P IR ) = IL ) . Left-right isomerism can be viewed in terms of a double-bottomed potential well 4,5 and the states IL ) and [R ) may be pictured as molecular configurations that are concentrated in the left or right potential well. The coordinate x is involved in the parity operation P and connects the two 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 define H0 as the Hamiltonian of each side of the double well and V0(x) as the potential barrier separating the two minima of the double well. In this picture, the IL ) and IR ) are eigenstates of H0, i.e., ( L I H o I L ) = ( R IHoiR) = Eo and we have a small overlap inside the barrier Vo(x ) so that ( L I VolR ) = ( R I VolL ) =6. We now develop a theory to explain the racemization of an optically-active sample assuming that the relaxation process is produced essentially by transitions between the two vibrational states [L ) and IR). The interaction potential of the active molecule with the environment will be represented by U(t). First, we assume that Vo(x) and U(t) are, in principle, equally important for transitions between 507

508

M. CATTANI

IL ) and IR ) configurations. In this case, the state function 14(t)) o f the active molecule will be represented by 1 4 ( t ) ) -- e-ie°t/h[aL(t)l L ) + aR(t)lR )]

(1)

and will obey the equation ihaJ~k(t))/dt = [ H 0 + V(x, t)]tg,(t)), where V(x, t)= Vo(x)+ U(t). Thus, aL(t) and aR(t ) are governed by the following differential equations: alL(t) = (--i/h)[aL(t)VLL + aR(t)VLR], a R ( t ) = ( - - i / h )[aR(t)VRR q-

aL(t)VRL],

(2)

where the matrix elements V,k, with n, k = L and R, are real functions given by V,k = (nlV(x, t)lk). D u e to left and right symmetry, VLR = VRL and VLL = VRR. Solving Eqs. (2.2), we get aL(t) = e

iOLL(t)[ae -iOLR(t) --I- b eiOLR(t)]/2,

a R ( t ) = e iOLL(O[ae -i°eR(O --

b eiOLR(t)]/2,

(3)

where a and b are constants determined by the initial conditions and

O,k(t) =

f0

Vo~dt/h.

I f at t = 0, the active molecule is p r e p a r e d so that 1 4 ( 0 ) ) = [ I R ) + IL)l/x/~, we have, from Eqs. (1) and (3), a = x / ~ and b = 0. Therefore, the molecular state will be described by the state function I~k+( t ) ) written as I 0+ (t)) = e-i=+(°l~b + (t)) = e ~+(')e -rE+'/hI+), where ~+(t)=

[ULL(t)+gLR(t)ldt/h, l + ) = [ I R ) + l L ) ] / , f 2

and E+ = E0 + 6. On the other hand, if 14(0)) = [ I R ) - I L ) I / , ~ , we see that a = 0 and b = ~/'2, resulting in the molecular state 10- (t)): 14 (t)) = e -i~-(t~] q~ (t)) = e ~-(')e-~e-'/hf--), where ~_ (t) =

f0

[ULL(t) -- ULR(t)] dt/h, I-- ) = [ I R ) - - I L ) l / x / - 2

and E_ = E o - 6 . These states, with + and - parities, respectively, are eigenstates of H = H 0 + Vo(x), i.e., H I 4 + ( t ) ) = E+ 1 0 + ( t ) ) . Molecules p r e p a r e d in + and - parity states m a k e a net contribution equal to zero to the chirality o f the sample. Since the initial left-right s y m m e t r y of these + and - states is not modified by U(t), no alteration o f the chirality of the system is produced by the external potential. We will n o w study the case when the active molecule is prepared at t = 0 in the left or right configuration. We will show that only in this case can U(t) contribute to the racemization process. Thus, solving Eq. (2) and assuming, for instance, that 14(0)) = I L ) , we get from Eqs. (1) and (3), aL(0) = 1, a R ( 0 ) = 0 and a = b = 1. Consequently, aL(t ) and aR(t) are given by aL(t) = e-i°~')[cos 0eR(t)], aa (t) = -- i e i0LL(,)[sin0LR (t)].

(4)

Using Eqs. (1) and (4), the state function of the active molecule, which is initially in a left configuration, will be written as I~kL(t)) = e-it°~')+ eo,/hl{COS[0LR( t ) ] l L ) - i sin[0LR ( t ) ] l R ) } .

(5)

The racemization is described by r(t) = I ( R 14L(t))l 2 = sin2[0LR(t)], where 0La(t) =

(LIVo(X) + U(t)IR) dt/h.

(6)

Environment racemization: a two-levelapproach

509

We first show that the well known quantum mechanical description of [~kL(t)> for an isolated molecule is obtained by introducing U ( t ) = 0 in Eq. (5). In this case, Eq. (5) becomes I@L(t)> = e-ieot/*[cos( ~ t / h ) l L > -- i sin( ~ t /h ) l R >]

(7)

and, consequently, r(t) = sin2(ft/h). According to Eq. (7), the molecule prepared in the left-handed state at t = 0 will oscillate back and forth between left and right forms with frequency co = 6/h. The quantum oscillation time z = 2re/co can be estimated in terms of the tunnelling rate through the barrier V0(x). 4 Taking into account the range of molecular parameters, ~we see that it may exist for values of r that are seconds, minutes, hours, and years up to billions of years. For alanine, for instance, r ~ 1 0 9 yr, according to the Hund 6 estimates. Using now our general predictions given by Eq. (6), we will investigate the effect of the perturbation U(t) on the chirality of the system in two limiting cases: (a) dilute gases and (b) dense gases, liquids and solids. Dilute gases By a dilute gas, we mean 5 that 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 which are very short compared to all other time scales; in particular, they are short compared to the tunnelling time. In other words, molecular interactions can be treated in the impact approximation. 7 We next consider how to estimate 0LR(t) in the impact approximation. According to Eq. (6), we see that

0LR(t) = 8t/h +

f0

ULR(t) dt/h.

Therefore, if the collision times are much smaller than the tunnelling time, then, during the time interval t, there is a large number of collisions between the active and perturbing molecules, with the tunnelling effect negligible compared with the collision effects. We calculate now the collision effects by using the well known impact-approximation formalism: We see that r(t) = e -:~', where the transition probability per unit of time 2 is given by s 2 = 2~N (2)

fo° ;: b db

vP(v) dv (sin2(0LR)>,

(8)

and N (2) is the density of perturbing molecules, b the impact parameter for the colliding particles, v the relative velocity between them, P ( v ) the Maxwell-Boltzmann velocity distribution, the brackets <...> mean an average over internal states of the colliding particles, and, finally, OLR=OLR(t

= -']-0(3) =

fo

ULR(t)dt/h =

f:

dt/h,

u(t) being the interacting potential between active and perturbing molecules. The racemization factor 2 is defined by Eq. (8) and may be calculated in detail 8 by taking into account all relevant parameters of the optically-active sample. However, we intend to obtain here only the main features of the relaxation phenomenon. Thus, we express the interaction potential u(t) simply by u(t) = 7 / R ( t ) p, where y represents the force constant for the interacting particles, 9'1° R ( t ) the distance between colliding molecules and p is equal to 3, 4, 5 . . . . if the interaction is a dipole-dipole, dipole~luadrupole, quadrupole-quadrupole, etc. interaction. Since for the active molecule, the dipole matrix element ( L Id l R > = 0, we see that p > 3. Thus, using this simplified potential and Eq. (8), 2 will be given by 8,H 2 = (7/h)2/(P - tW(2)(kT/m)(p - 3)/(2p-2),

(9)

where k is the Boltzmann constant, T the absolute temperature of the system and rn the reduced mass of the colliding particles. We see from Eq. (9) that racemization increases as the temperature increases: We note, however, that Eq. (9) can be applied only when the impact approximation is valid, i.e., when r > 1/2. This constraint implies that when a dilute gas approximation is satisfied, collisions will always destroy rather than stabilize the handedness of the sample)

510

M. CATTANI

Dense gases, liquids and solids In dense gases and liquids, multiple interactions dominate over binary interactions. In a first approximation for dense gases and liquids, as well as 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 potentials 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. Thus, instead of Vo(x), the potential barrier is given by the effective potential Ve~(x) = Vo(x) + U~(x). Defining 6ee~= ( L I V ~ ( x ) I R ) = ( L [ Vo(x) + U~(x)IR > and using Eq. (6), we obtain 0Lg(t)= 6~t/h and, consequently, r(t) = sin2(fart/h).

(10)

Equation (10) shows that the racemization time is given by an effective time ~,er= h/6e~. Depending on the perturbative potential U~ (x), the effective racemization time zoercan be larger or smaller than the natural tunnelling time z. Claverie and Jona-Lasinio 2 considered in their paper a perturbative potential UI (x) which is concentrated in a small special region compared with the size of the double well. They have shown that this type of perturbation would, in principle, stabilize molecules in a given left or right configuration. Numerical estimates for this localization have been obtained for dipolar molecules (NH3, AsH 3 and PH3) in the gaseous state. The potential U)(x) was obtained as follows: once a molecule is in a given left and right configuration, its dipole moment d~ polarizes the surroundings locally which, in turn, creates at the position of fll a reaction field Er )2 that is collinear with d). This U~ is given by U~ = - ( E - 1)d12/(2E + 1)a 3, where E is the dielectric constant of the medium and a the radius of the cavity in which the molecule is embedded. Since U~ is negative, it tends to stabilize the configuration by lowering the potential energy of the localized [ L ) or IR ) state. Finally, we study the effect of U(t) on the active molecules prepared in + and - parity states or when U(t) can be considered as a perturbation of Vo(x). In these cases, the energy difference E + - E_ = 26 and the left-right oscillation frequency co = 6/h depends only on the internal structure of the molecule. For these conditions, it is well known 8'~3 that the state function [ ¢ ( t ) ) of the active molecule can be represented in terms of the two eigenstates 14)+(t)) and 14)- (t)) of the operator H = H o + Vo(x), i.e. [tp(t)) = a+ (t)14)+(t)) + a_(t)14)_(t)), where 14)±(t)) = [ 4- ) e-e-+'/h, [ 4- ) = (1 R ) 4- [L))/x/~, E± = E0 4- 6, and HI4)± (t)) = E± [q)± (t)). In this last relation, [~k(t)) obeys the equation ihO[$(t))/3t = [ H + U(t)][$(t)). For these conditions, as is shown in the Appendix, the net contribution of U(t) to the chirality of the sample is equal to zero, independently of the initial state 1~,(0)).

Acknowledgement--The author thanks the Conselho Nacional de Pesquisas (CNPq) for financial support.

REFERENCES 1. S. F. Mason, Molecular Optical Activity and the Chiral Discriminations, Cambridge Univ. Press, Cambridge, U.K. (1982). 2. P. Claverie and G. Jona-Lasinio, Phys. Rev. A33, 2245 (1986). 3. A. Di Giacomo and G. Paffuti, Nuovo Cimento !i55, 110 (1980). 4. R. A. Harris and L. Stodolsky, Phys. Left. B78, 313 (1978). 5. R. A. Harris and L. Stodolsky, J. Chem. Phys. 74, 2145 (1981). 6. F. Hund, Z. Phys. 43, 805 (1927). 7. N. Baranger, Phys. Rev. 111, 494 (1958). 8. M. Cattani, JQSRT 27, 563 (1982). 9. C. J. Tsao and B. Curnutte, JQSRT 2, 41 (1962). 10. D. Robert, M. Giraud, and L. Galatry, J. Chem. Phys. 51, 2192 (1969). 11. S. Chen and M. Takeo, Rev. Mod. Phys. 29, 20 (1957). 12. C. J. F. B6ttcher, Theory of Electric Polarization, Elsevier, Amsterdam (1952). 13. M. Cattani, JQRST 42, 83 (1989).

Environment raeemization: a two-levelapproach

511

APPENDIX

Racemization Produced by U(t) when the Active Molecules are Prepared in the -t- and States or when U(t) can be Considered as a Perturbation o f Vo(x) As was pointed out in See. 2, the active molecule state function is represented by 8,13 I~ (t)) = a+ (t) I~b+ (t)) + a_ (t) I~b_ (t)),

Parity

(A 1)

where I(a+_(t))=l++_)e -iE'/h with I + + _ ) = ( I R ) + I L ) ) / x / ~ and the E ± = E o + _ 6 are eigenstates of H = H o + V o ( x ) , i.e., H l d p ± ( t ) ) = E ± l c ~ ± ( t ) ) . Since Iq/(t)) obeys the equation ih~ Iq/(t))/dt = [H + U(t)]l ~O(t)), a+ (t) and a_ (t) obey the following differential equations: ft+ (t) = ( - i / h ) [ a + (t)U++ (t) + a (t)U+_ (t) e-Ziat/h], ft_ (t) = ( - i / h )[a_ (t)U_ _ (t) + a+ (t)U_+ (t)

e2i&/h],

(A2)

where the real matrix elements U,k, with n, k = + and - , are given by (n I U ( t ) l k ) . Due to left and right symmetry, U+ + = U__ and U+_ = U_ +. As can easily be verified, 8'~3 Eqs. (A2) can be solved, in good approximation, by taking 6 = 0. In this case, we obtain a+ (t) = e-i°++(°[a e-i°+-(° + b ei° +- (')]/2, a_ ( t ) = e-i°++(')[a e -i°+-(') - - b e~°+-(')]/2,

(A3)

where the constants a and b are determined by the initial conditions and O,k(t) =

f0

(nlU(t)lk)

dt/h.

Assuming, for instance, that [0(0)) = I + ) , we find, from Eqs. (A1) and (A3), a = b = 1, which leads to a+ (t) = e -/°++<') cos[P+_ (t)], a_ ( t ) = - i e -i° ÷+¢t)sin[P+ _ (t)].

(A4)

Using Eqs. (A1) and (A4), we write I¢ ( t ) ) --

e-i°++(t){cos[O+

_

(t)]l 4~+ (t)) - i sin[P+ _ (t)] IqS_ (t))}.

(A5)

We are now able to calculate the racemization produced by U(t) as the transition probabilities between [~O(t)) and the IL ) and IR ) states. Using Eq. (A5), we see that left and right configurations appear with the same probability, i.e. I ( Z I~' (t))l 2 = I(R I~b(0)12 = { 1 - cos[P+ _ (t)]sin[0+ _ (t)]sin(26t/h)}/2. This relation implies that the net racemization produced by U(t) is equal to zero. Assuming that [qJ(0))= I L ) , we find from Eqs. (A1) and (A3) that a = 0 and b = x/~ which leads to a + ( t ) = - a _ ( t ) = e-i°(')x/~/2, where O ( t ) = O + + ( t ) - O + _ ( t ) . Consequently, we obtain from Eq. (A1) the relation I ~k (t)) = e- i0t,)e- ie0t/h[COS(Ot/h) IL ) - i sin(6 t/h) I R )],

which shows that racemization We have thus shown, for the the perturbation U(t) does not verified, this result is the same

is independent of the perturbation U(t). two particular initial states Iq~(0)) = I + ) and t qJ(0)) = [L), that change the chirality of the sample. Furthermore, as can be easily for any I~O(0)).