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The optical activity of methane
CHEBIICAL
&lu~e 39, number 3.
1 May 1976
PHYSICS LE’JXERS
THE OPTICAL ACTIVITY~OF METHANE P.W. ATKINS Physical Chemistj
Labomtory,
Oxford OXI 3QZ. UK
and J.A.N.F. COMES Theoretical Chemistry Department, Oxford. OXI _?Q.Z, iJK
Received 9 February 1976
We examine whether a methane molecule is optically active by virtue of its rotation. We conclude that methane gas at room temperature can be regarded as an enantiomeric mixture of two components, each of which rotates the plane of C&t by about 1 rad m-r but in opposite directions.
The methane molecule has an electric dipole moment in its ground electronic and vibrational state [l] . It arises from the effects of molecular rotation reducing the tetrahedral molecule to axial symmetry; the effect is small and the magnitude of the dipole is only about 10m6 D. The question raised by this interesting observation is whether methane also shows other properties which simple ideas appear IO exclude. The property to which we address ourselves in this note is whether methane is optically active as a result of its rotation_ The classical argument underlying the optical activity can be stated quite concisely. A molecule is optically active if its polar&ability differs for right and left circularly polarizable light. A molecule rotating in the same sense as the electric vector of the light might have a polarizability different from that of counter-rotating molecule. Therefore a gas of methane molecules rotating in one sense will rotate plane polarized light. Of course an equilibrium ensemble of methane molecules rotates in both senses, and so the net activity is zero. Nevertheless it remains true that ordinary methane is properly regarded as optically inactive not because of the intrinsic inactivity of the molecules but because it is an enantiomeric mixture. Alternatively one can envisage an application of the birefringent properties of the molecules to a sorting process involving laser light scattering from a molecular beam. The magnitude of the intrinsic activity can be calculated without much difficulty_ We consider a molecule in the state IO, JKM) with the space-fmed z-axis along the direction of propagation of the light. For a path length fZ?through a sample of number density %, assuming a Lorentz local field correction, the angle of rotation is [2] S@= (.&~/2c) (jr,+--n,_)
= (i-!?Rw/4ceg)
(o+--
oXY),
where w is the frequency of the light and Q is the polarizability tensor. Note that this result relies on only electric dipole interactions, unlike conventional optical activity which requires one magnetic di;lole interaction. In normal circumstance (non-rotating molecules) epis symmetric and so S@= 0; in the presence of rotation it is not symmetric. The polarizability tensor in the presence of light of frequency w is [2] 8(W) = (l/E)
c r~t-J&&l(o,o -i-0) + &)&/(%O - o)] n and introduction of the sperical tensor components -er,l (I) = F 2- ‘I2 (d, C i$)
(and -e@
= d,) leads to 519
The matrkelements cz&.be ev&ated by standaid angular mbmentinn techniques, and k shbuld be interpreted as t$e stat& in,_J’K’M’) accessible from the statz.10,JKM). In tetrahedral symmetry (which is assumed to be mainin the sens:: that (Olr4(‘)lnX~~l$~10)
:J'
x
-K'
k
.J
it--K
K
after soee m*ip&tion
WE
-
k
J
Q
iii
(rii
T$~$z)
.E IZOn12-- 2J- J
2Acq
2LT+1
J'
-M’
we arrive at
e2w=L?(n & JCJ-tl) &
I(
t
n
,
1
Jo-o2
1
2Ji3 _:
1 &ln,JO-w
1
J+l 41
n,JO - a2
2
1.
-In this expression w~_~ n,JL = anO - 2BJ, and w~.+~& l), where I? iS the rotational constant and fiti,, are the electronic excitation frequencies_ We now obtain an approximate form of the last equation by retaining only terms first-order in the rotational energy [we take (2BJ)L Q (wzo - w2)]. Then one obtains the simp!e result #Jn,Jo
=
wno,
Jo
=
wllo
-+2&f
+
As expected, different signs of rotation are predicted for different senses of rotation of the molecules (different signs ofM). Henceforth we shall write the coefficient of M as 9i? so that
The magnitude of 66 can be estimated as follows. For a 1 m path length through a gas at 1 atm pressure at room temperature, takingB = 5.24 cm-‘, using 600 nm light, supposing that w,, x 2w, and with Ie.zon i =3 x IO- 3o C m, gives 6@ % 2 rad when M = 10.’ This calculation can be extended to a symmetric top and the angle of rotation then depends on the quantum
number K as well as M, but no significant differences appear. We thank the Institute de A&a Cultura, Lisbon for financial support to I.A.N.F.G. References (I] L. O&r, Phys. Rev.Lctteri 27 (1971),1329. [2] P.W. Atkins. Molecular quantum mechanics (Clarendon Press, Oxford, 1970).
520
&lu~e 39, number 3.
1 May 1976
PHYSICS LE’JXERS
THE OPTICAL ACTIVITY~OF METHANE P.W. ATKINS Physical Chemistj
Labomtory,
Oxford OXI 3QZ. UK
and J.A.N.F. COMES Theoretical Chemistry Department, Oxford. OXI _?Q.Z, iJK
Received 9 February 1976
We examine whether a methane molecule is optically active by virtue of its rotation. We conclude that methane gas at room temperature can be regarded as an enantiomeric mixture of two components, each of which rotates the plane of C&t by about 1 rad m-r but in opposite directions.
The methane molecule has an electric dipole moment in its ground electronic and vibrational state [l] . It arises from the effects of molecular rotation reducing the tetrahedral molecule to axial symmetry; the effect is small and the magnitude of the dipole is only about 10m6 D. The question raised by this interesting observation is whether methane also shows other properties which simple ideas appear IO exclude. The property to which we address ourselves in this note is whether methane is optically active as a result of its rotation_ The classical argument underlying the optical activity can be stated quite concisely. A molecule is optically active if its polar&ability differs for right and left circularly polarizable light. A molecule rotating in the same sense as the electric vector of the light might have a polarizability different from that of counter-rotating molecule. Therefore a gas of methane molecules rotating in one sense will rotate plane polarized light. Of course an equilibrium ensemble of methane molecules rotates in both senses, and so the net activity is zero. Nevertheless it remains true that ordinary methane is properly regarded as optically inactive not because of the intrinsic inactivity of the molecules but because it is an enantiomeric mixture. Alternatively one can envisage an application of the birefringent properties of the molecules to a sorting process involving laser light scattering from a molecular beam. The magnitude of the intrinsic activity can be calculated without much difficulty_ We consider a molecule in the state IO, JKM) with the space-fmed z-axis along the direction of propagation of the light. For a path length fZ?through a sample of number density %, assuming a Lorentz local field correction, the angle of rotation is [2] S@= (.&~/2c) (jr,+--n,_)
= (i-!?Rw/4ceg)
(o+--
oXY),
where w is the frequency of the light and Q is the polarizability tensor. Note that this result relies on only electric dipole interactions, unlike conventional optical activity which requires one magnetic di;lole interaction. In normal circumstance (non-rotating molecules) epis symmetric and so S@= 0; in the presence of rotation it is not symmetric. The polarizability tensor in the presence of light of frequency w is [2] 8(W) = (l/E)
c r~t-J&&l(o,o -i-0) + &)&/(%O - o)] n and introduction of the sperical tensor components -er,l (I) = F 2- ‘I2 (d, C i$)
(and -e@
= d,) leads to 519
The matrkelements cz&.be ev&ated by standaid angular mbmentinn techniques, and k shbuld be interpreted as t$e stat& in,_J’K’M’) accessible from the statz.10,JKM). In tetrahedral symmetry (which is assumed to be mainin the sens:: that (Olr4(‘)lnX~~l$~10)
:J'
x
-K'
k
.J
it--K
K
after soee m*ip&tion
WE
-
k
J
Q
iii
(rii
T$~$z)
.E IZOn12-- 2J- J
2Acq
2LT+1
J'
-M’
we arrive at
e2w=L?(n & JCJ-tl) &
I(
t
n
,
1
Jo-o2
1
2Ji3 _:
1 &ln,JO-w
1
J+l 41
n,JO - a2
2
1.
-In this expression w~_~ n,JL = anO - 2BJ, and w~.+~& l), where I? iS the rotational constant and fiti,, are the electronic excitation frequencies_ We now obtain an approximate form of the last equation by retaining only terms first-order in the rotational energy [we take (2BJ)L Q (wzo - w2)]. Then one obtains the simp!e result #Jn,Jo
=
wno,
Jo
=
wllo
-+2&f
+
As expected, different signs of rotation are predicted for different senses of rotation of the molecules (different signs ofM). Henceforth we shall write the coefficient of M as 9i? so that
The magnitude of 66 can be estimated as follows. For a 1 m path length through a gas at 1 atm pressure at room temperature, takingB = 5.24 cm-‘, using 600 nm light, supposing that w,, x 2w, and with Ie.zon i =3 x IO- 3o C m, gives 6@ % 2 rad when M = 10.’ This calculation can be extended to a symmetric top and the angle of rotation then depends on the quantum
number K as well as M, but no significant differences appear. We thank the Institute de A&a Cultura, Lisbon for financial support to I.A.N.F.G. References (I] L. O&r, Phys. Rev.Lctteri 27 (1971),1329. [2] P.W. Atkins. Molecular quantum mechanics (Clarendon Press, Oxford, 1970).
520