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Tunneling racemization in dense gases and liquids
Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 107}110
Tunneling racemization in dense gases and liquids M. Cattani *, J.M.F. Bassalo Instituto de Fn& sica, Universidade de SaJ o Paulo, C.P. 66318, 05315-970, SaJ o Paulo, SP, Brazil Departamento de Fn& sica, Universidade Federal do Para& , 66075-900, Bele& m, PA, Brazil Received 12 May 1999; accepted 19 May 2000
Abstract We calculate the racemization in dense gases and liquids produced both by natural tunneling and by random binary interactions between active and perturbing molecules. 2001 Elsevier Science Ltd. All rights reserved.
In recent works [1}3] we have studied the tunneling racemization of active molecules produced by molecular interactions in gases and liquids. In these studies we have applied a two-level approach developed [4] to explain the racemization process in gases, liquids or solids. In our papers the left}right isomerism is pictured as molecular con"gurations that are concentrated in the left (L) or right (R) sides of a double potential well. The double-bottomed potential well was assumed to have the shape of two overlapping harmonic potentials with the minima at the points x"!a and x"a. The coordinate x 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 u the fundamental frequency of each harmonic oscillator and by k the reduced mass of the particles vibrating between x"!a and x"a. We de"ne H as the Hamiltonian of the double well which includes the parity-violating weak interactions. If the parity is violated, the left and right sides of the double-bottomed potential are no longer exactly symmetrical. In these conditions, we have 1¸"H"¸2"E "E !e, 1R"H"R2"E "E #e and 1¸"H"R2"1R"H"¸2"d, due to the * 0 small overlap of the "¸2 and "R2 wave functions inside the potential separating the two minima of the double well. E is the energy of the fundamental left and right states in absence of weak currents and 2e is the di!erence of energy between the left and right con"gurations due to the parity violating interactions [5,6]. The natural tunneling parameter d is shown explicitly elsewhere [3].
* Corresponding author. Tel.: #11-3818-7035; fax: #11-3818-6749. E-mail address: [email protected] (M. Cattani). 0022-4073/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 0 ) 0 0 0 6 5 - 0
108
M. Cattani, J.M.F. Bassalo / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 107}110
We have calculated [7] the racemization of an optical active sample assuming that the relaxation process is due only to transitions between the two fundamental vibrational states "¸2 and "R2. The interaction potential of the active molecule with the environment was indicated by ;(t). So, the state function "t(t)2 of the active molecule, represented by "t(t)2"a (t)"¸2 exp (!iE t/ )#a (t)"R2 exp (!iE t/ ), * * 0 0 obeys the equation (i/h)(*"t(t)2/*(t)"[H#;(t)]"t(t)2. The racemization in dilute gases was extensively investigated [1,2,4,7] but, only in our last paper [3], we have considered dense gases and liquids that have been assumed as composed by dipolar molecules. This is a special case because a cooperative interaction mechanism [8] appears between the molecules of the sample. The mechanism is the following: once a molecule is in a left or right con"guration, 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 "eld E , which is P collinear with d. This coherent collective interaction between the dipolar molecules is clearly a non-linear e!ect. We have shown [3] that, in these conditions, taking into account the weak interactions, it is possible to get the optical stabilization. Our predictions are in agreement with those obtained by Claverie and Jona}Lasinio that have pointed out [8] that the reaction "eld could be able to induce optical stability. In the present work we consider a completely di!erent case: our sample is not composed by dipolar molecules, with no kind of cooperative interaction between active and perturbing molecules. So, we will assume that the molecular interactions are binary, additive and independent. In what follows, since the weak interactions will play a negligible role in the racemization process, we will take e"0. Thus, in these conditions, we have, according to our general formalism [1,2,4], that the racemization r(s) is given by r(s)"1 sin (# (s))2 *0
(1)
where # (s)"ds/ #u(s), *0
u(s)"
Q
; (t) dt/ and ; (t)"1¸";(t)"R2. *0 *0
In Eq. (1), the brackets mean an average over the e!ect of the binary collisions that are taken as independent and at random. The interaction ;(t) will be written as a sum of binary interactions given simply by u(t)"c/R(t)N, where c represents the force constant for the interacting particles, R(t) the distance between them as a function of the time t, p is equal to 4,5,2 if the interaction is dipole-quadrupole, quadrupole}quadrupole, and so on. Eq. (1) can be also written as [1,2] r(s)"()+1!Re1 exp [2ids/ #2iu(s)]2,.
(2)
M. Cattani, J.M.F. Bassalo / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 107}110
109
Indicating by (s)"exp(2iu(s)) and following the line shape theory at general pressures, developed by Anderson and Talman [9], it can be shown that
1 (s)2" exp !i
Q
u (t) dt/
*0
L ,
(3)
where n is the total number of perturbers and u (t)"c /R(t)N, c "1¸"c"R2 is the matrix *0 *0 *0 element of the force constant between the states "¸2 and "R2. Assuming that the perturbers describe straight-line paths, that collisions are independent, isotropic and at random, and making n and
(4)
where t(s)"t(R y/*)"2pR
r dr
> [1!exp (!ih(y))] dz, \
y"*s/R , R "(c / *) N\, h(y)" *0
W
[(z#u)#r]\N du,
* the average relative velocity between active and perturbing molecules and c the matrix element, *0 between "¸2 and "R2, of the force constant averaged over all remaining internal states of the active and perturbing molecules. We must note that, from Eqs. (1)}(4), we can obtain the racemization r(s) for dilute cases, calculated in our preceding paper [1,2], using the `impact approximationa to estimate the binary collisions. In dense gases and liquids, to a "rst approximation, molecules can be regarded as at rest. So, making *P0, t(s), de"ned by Eq. (4), is written as [9] t(s)"4p
R[1!exp (!ic s/ RN)] dR, *0
(5)
where
Im(t)"4p
R sin (c s/ RN)] dR *0
and Re(t)"4p
R[1!cos (c
*0
s/ RN)] dR.
Finally, from Eqs. (2)}(5) we obtain, since Im(t) 0, r(s)"()[1!cos (2ds/ )exp(!N Re(t))],
(6)
110
M. Cattani, J.M.F. Bassalo / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 107}110
with
NRe(t)"4pN
R[1!cos (c s/ RN)] dR *0
R sin (c s/2 R.) dR"j* s N, *0 where the racemization factor jH is de"ned by "8pN
x\N> N sinx dx. So, from Eqs. (6) and (7) we verify that the racemization r(t) is given by jH"(8p/p)N(c /2 ) N *0
(7)
(8) r(t)"()[1!cos (2dt/ ) exp (!j* t N)] showing that, in gases and liquids, random binary interactions between active and perturbing molecules produce a complete racemization of the sample. In dilute gases [1,2], the racemization decays temporarily as exp (!jt) and in dense gases and liquids, according to Eq. (8), more slowly, as exp (!j*t N), since p"4,5,2 and so on. M. Cattani thanks the CNPq for the "nancial support.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Cattani M. JQSRT 1995;54:1059. Cattani M. Nuovo Cimento 1995;17D:1083. Cattani M, Bassalo JMF. JQSRT 1999;61:299. Cattani M. JQSRT 1991;46:507. Di Giacomo A, Pa!uti G, Ristori C. Nuovo Cimento 1980;B55:110. Mason SF, Tranter GE. Mol Phys 1984;53:1091. Cattani M. JQSRT 1996;55:191. Claverie P, Jona-Lasinio G. Phys Rev 1986;A33:2245. Ch'en S, Takeo M. Rev Mod Phys 1957;29:20.
Tunneling racemization in dense gases and liquids M. Cattani *, J.M.F. Bassalo Instituto de Fn& sica, Universidade de SaJ o Paulo, C.P. 66318, 05315-970, SaJ o Paulo, SP, Brazil Departamento de Fn& sica, Universidade Federal do Para& , 66075-900, Bele& m, PA, Brazil Received 12 May 1999; accepted 19 May 2000
Abstract We calculate the racemization in dense gases and liquids produced both by natural tunneling and by random binary interactions between active and perturbing molecules. 2001 Elsevier Science Ltd. All rights reserved.
In recent works [1}3] we have studied the tunneling racemization of active molecules produced by molecular interactions in gases and liquids. In these studies we have applied a two-level approach developed [4] to explain the racemization process in gases, liquids or solids. In our papers the left}right isomerism is pictured as molecular con"gurations that are concentrated in the left (L) or right (R) sides of a double potential well. The double-bottomed potential well was assumed to have the shape of two overlapping harmonic potentials with the minima at the points x"!a and x"a. The coordinate x 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 u the fundamental frequency of each harmonic oscillator and by k the reduced mass of the particles vibrating between x"!a and x"a. We de"ne H as the Hamiltonian of the double well which includes the parity-violating weak interactions. If the parity is violated, the left and right sides of the double-bottomed potential are no longer exactly symmetrical. In these conditions, we have 1¸"H"¸2"E "E !e, 1R"H"R2"E "E #e and 1¸"H"R2"1R"H"¸2"d, due to the * 0 small overlap of the "¸2 and "R2 wave functions inside the potential separating the two minima of the double well. E is the energy of the fundamental left and right states in absence of weak currents and 2e is the di!erence of energy between the left and right con"gurations due to the parity violating interactions [5,6]. The natural tunneling parameter d is shown explicitly elsewhere [3].
* Corresponding author. Tel.: #11-3818-7035; fax: #11-3818-6749. E-mail address: [email protected] (M. Cattani). 0022-4073/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 0 ) 0 0 0 6 5 - 0
108
M. Cattani, J.M.F. Bassalo / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 107}110
We have calculated [7] the racemization of an optical active sample assuming that the relaxation process is due only to transitions between the two fundamental vibrational states "¸2 and "R2. The interaction potential of the active molecule with the environment was indicated by ;(t). So, the state function "t(t)2 of the active molecule, represented by "t(t)2"a (t)"¸2 exp (!iE t/ )#a (t)"R2 exp (!iE t/ ), * * 0 0 obeys the equation (i/h)(*"t(t)2/*(t)"[H#;(t)]"t(t)2. The racemization in dilute gases was extensively investigated [1,2,4,7] but, only in our last paper [3], we have considered dense gases and liquids that have been assumed as composed by dipolar molecules. This is a special case because a cooperative interaction mechanism [8] appears between the molecules of the sample. The mechanism is the following: once a molecule is in a left or right con"guration, 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 "eld E , which is P collinear with d. This coherent collective interaction between the dipolar molecules is clearly a non-linear e!ect. We have shown [3] that, in these conditions, taking into account the weak interactions, it is possible to get the optical stabilization. Our predictions are in agreement with those obtained by Claverie and Jona}Lasinio that have pointed out [8] that the reaction "eld could be able to induce optical stability. In the present work we consider a completely di!erent case: our sample is not composed by dipolar molecules, with no kind of cooperative interaction between active and perturbing molecules. So, we will assume that the molecular interactions are binary, additive and independent. In what follows, since the weak interactions will play a negligible role in the racemization process, we will take e"0. Thus, in these conditions, we have, according to our general formalism [1,2,4], that the racemization r(s) is given by r(s)"1 sin (# (s))2 *0
(1)
where # (s)"ds/ #u(s), *0
u(s)"
Q
; (t) dt/ and ; (t)"1¸";(t)"R2. *0 *0
In Eq. (1), the brackets mean an average over the e!ect of the binary collisions that are taken as independent and at random. The interaction ;(t) will be written as a sum of binary interactions given simply by u(t)"c/R(t)N, where c represents the force constant for the interacting particles, R(t) the distance between them as a function of the time t, p is equal to 4,5,2 if the interaction is dipole-quadrupole, quadrupole}quadrupole, and so on. Eq. (1) can be also written as [1,2] r(s)"()+1!Re1 exp [2ids/ #2iu(s)]2,.
(2)
M. Cattani, J.M.F. Bassalo / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 107}110
109
Indicating by (s)"exp(2iu(s)) and following the line shape theory at general pressures, developed by Anderson and Talman [9], it can be shown that
1 (s)2" exp !i
Q
u (t) dt/
*0
L ,
(3)
where n is the total number of perturbers and u (t)"c /R(t)N, c "1¸"c"R2 is the matrix *0 *0 *0 element of the force constant between the states "¸2 and "R2. Assuming that the perturbers describe straight-line paths, that collisions are independent, isotropic and at random, and making n and
(4)
where t(s)"t(R y/*)"2pR
r dr
> [1!exp (!ih(y))] dz, \
y"*s/R , R "(c / *) N\, h(y)" *0
W
[(z#u)#r]\N du,
* the average relative velocity between active and perturbing molecules and c the matrix element, *0 between "¸2 and "R2, of the force constant averaged over all remaining internal states of the active and perturbing molecules. We must note that, from Eqs. (1)}(4), we can obtain the racemization r(s) for dilute cases, calculated in our preceding paper [1,2], using the `impact approximationa to estimate the binary collisions. In dense gases and liquids, to a "rst approximation, molecules can be regarded as at rest. So, making *P0, t(s), de"ned by Eq. (4), is written as [9] t(s)"4p
R[1!exp (!ic s/ RN)] dR, *0
(5)
where
Im(t)"4p
R sin (c s/ RN)] dR *0
and Re(t)"4p
R[1!cos (c
*0
s/ RN)] dR.
Finally, from Eqs. (2)}(5) we obtain, since Im(t) 0, r(s)"()[1!cos (2ds/ )exp(!N Re(t))],
(6)
110
M. Cattani, J.M.F. Bassalo / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 107}110
with
NRe(t)"4pN
R[1!cos (c s/ RN)] dR *0
R sin (c s/2 R.) dR"j* s N, *0 where the racemization factor jH is de"ned by "8pN
x\N> N sinx dx. So, from Eqs. (6) and (7) we verify that the racemization r(t) is given by jH"(8p/p)N(c /2 ) N *0
(7)
(8) r(t)"()[1!cos (2dt/ ) exp (!j* t N)] showing that, in gases and liquids, random binary interactions between active and perturbing molecules produce a complete racemization of the sample. In dilute gases [1,2], the racemization decays temporarily as exp (!jt) and in dense gases and liquids, according to Eq. (8), more slowly, as exp (!j*t N), since p"4,5,2 and so on. M. Cattani thanks the CNPq for the "nancial support.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Cattani M. JQSRT 1995;54:1059. Cattani M. Nuovo Cimento 1995;17D:1083. Cattani M, Bassalo JMF. JQSRT 1999;61:299. Cattani M. JQSRT 1991;46:507. Di Giacomo A, Pa!uti G, Ristori C. Nuovo Cimento 1980;B55:110. Mason SF, Tranter GE. Mol Phys 1984;53:1091. Cattani M. JQSRT 1996;55:191. Claverie P, Jona-Lasinio G. Phys Rev 1986;A33:2245. Ch'en S, Takeo M. Rev Mod Phys 1957;29:20.