INTERACTION OF TWO POL\RIS.\BLE MOLECULES
4.5 THE INTERACTION BETWEEN TWO CHIRAL MOLECULES
The energy shift between two non-identical chiral molecules with each centre possessing an electric and a magnetic dipole moment, is given by AE Q n
Im
— ^ > G .(w ) Im — ^ ) Ggo(w) ^“ 0 (4.5.1)including all terms second order in the transition moments. The contribution to the interaction energy from the product of the first order fields is evaluated using the electric and magnetic dipole dependent terms of the Maxwell fields linear in the sources (2.2.34) and
(2.2.55), and is - V j j Im L16K“E“c“n
K
\
(4.5.2)For the contribution from the zeroth and second order fields, use is made of the free fields (2.2.15) and (2.2.52), and the diagonal matrix elements of the quadratic fields (3.1.2) and (3.1.8), giving
— T m J L^-k ik.R,r ê» . —
'k,>- " o'
r -4 -4
- Im-! Y _ i ^ G%flw)lb^e / ' , '^et.' + (b^X-^+e^ie^^)e^e‘‘‘‘‘*j|. 8^.-cV -
(4.5.3)
Evaluating the terms of (4.5.3) using the appropriate polarisation sums, angular integrals and definitions of the tensor fields and adding the contribution from the first order fields (4.5.2), the interaction energy between two chiral molecules is
AE = -1 8n-€-c- S ° E >E q m •rdTmT^f-o(k R)f;p(k R) - 4 -t /LA qm qm *(k R)g;p(k R )] +
s S v
pn E >E D n W 2 ^ T ™ r - j 2 i d S u X ^ V t . . ( i c u ) . 16n E^c' m " ALL E 0 u “ + k ' III qm r 1I f^^( iuR ) f^y ( iuR) - g^-^( iuR)g^-^( iuR) I . (4.5.4)
When both molecules are in the ground state only the u-integral term of (4.5.4) survives, which when written explicitly in terms of transition moments after multiplication of the geometric tensors, is
duu e9 -2uR 1 T ,On no om m o '
l u V
u ^ R ^
u ^ R ^
u ^ R ^
u ® R ® J
(4.5.5) with = " U ^ j C ^ U ^ r i æ ^ L % u =The expression (4.5.5) applies when the orientations of molecules A and B are fixed relative to each other. It may also be written in terras of the molecular Dolarisability tensor G--(u).
- 7 - Iduu^e ( icu )G%( icu )^ 0 0
^ ^ f i i M l
I u“R “ u“R^ u^R^ u “R^ u“R^J
(4.5.7)
g Q Os gO
or in terms of the rotatory strength R - : = Im/J- m j . To deal with ^ ^ “ ÿ
molecules in the fluid phase, a rotational average of (4.5.5) is needed. By following the standard procedure [41J, the dispersion interaction for two freely rotating chiral molecules valid for all separation distances beyond electron overlap, is
Ign^E^hc^R^ m^n "(uT+k; )(uT+k; ) ^ u“R‘
0 0 O m O n
After the usual approximations, the far-zone limit is [48]
3 ^
A E „ = — ^ ; — --- (4.5.9)
- 3%-c;R- »rn
while the near-zone shift is [48,58]
AE z — =1— y
N7 ? 9 ? A /
l^Qm H»niG I iT^on -^no j/J .m I III .m 127t’e^c’R “ mTn ^mo'^^no
(4.5.10)
This completes the evaluation of the dispersion interaction between two chiral molecules, originating from the third term of (4.5.4), The result for all R is given by (4.5.8) while the results at large and small intermolecular separations are respectively given by (4.5.9) and (4.5.10). This interaction potential is discriminatory, dependent upon the relative chirality of the molecules of the pair. The polarisability tensor (w ) changes sign with enantiomer since a polar vector, is antisymmetric to inversion, in contrast to m which is symmetric. For molecules with absolute configurations R and S, the A(R)-B(R) and A(R)-B(S) interactions differ in sign. Since the rotatory strength maybe either greater or less than zero, it is not possible to determine the absolute sign of the interaction when the molecules are chemically distinct. For chemically identical molecules however, the energy shift for opposite isomers is attractive while that for like isomers is repulsive. The complete ground state interaction, along with the R
- 6
far-zone and R short range dependences agree with work carried out in the Schrddinger picture [6,38,58] and the calculation performed by Mavroyannis and Stephen [48] in the Lorentz gauge, in which only the limiting results were given. It is worth noting that the near-zone
result may be obtained from second order perturbation theory when the interaction is represented by an electric and a magnetic dipolar coupling term and the mixed electric-magnetic cross term is extracted [58j. This is examined in more detail in the final Section to this Chapter.
Returning to the general result (4.5.4) and examining the case in which molecule B is in the ground state and A is excited, the additional contribution from downward transitions from jp> from the second term of
(4.5.4) is i V i ; n E >E p n k'’ r"* pn (4.5.11)
which for an isotropic source and test becomes
12n"E-c- n E >E p n L k“ R“ k"* R^* k^ R^ J pn pn pn -* (4.5.12)
The far-zone limit of (4.5.12) is
-
y
G'(k..)|2 P".mnP|k: Gn-E^c^R- nE >E p n
pn on (4.5.13)
exhibiting an R dependence, being associated with real photon emission. This is also the dominant contribution as the asymptotic
_ 9
earlier. For small R, the near-zone limit of (4.5.12) is
I
' ' " p n " ' " - - " ! ° E >ED n
while that from the u-inte2ral is
, , , ^ > s«n(E„„) (4.5.13)
12i t V c V n,rn "P (E +!e |)
0 mo ' n p '
- 5
both terms exhibiting R “ dependences. The sum of (4.5.14) and (4.5.15) gives the total small R limit
T - T - T T ; - (4.5.16)
12n^E%c-R- rarn (E „+ E ) ° ALL E "P
n
which is composed of both real and virtual photon terms.
If both molecules are in electronically excited states, all three terms of (4.5.4) contribute. The first term of (4.5.4), the additional contribution due to excited molecule B is
° g qm qm qm
a m
with the second term of (4.5.4) given by (4.5.12), remembering that B is now excited, while the u-integral term can be written analogous to
18n"E:hc^R^ ^ (k“ +u“ )(k“ +u“ ) uTRT -I
0 m , n 0 qm pn
(4.5.18)
For large R the interaction energy has an inverse square dependence on separation, arising from the addition of the real photon exchange terms (4.5.12) and (4.5.17). In the near-zone, the downward transition contributions to the interaction energy shift are
E >E ° E >E
q m p n
) sgn(E )sgn(E ) --- (4.5.19) 12%^E:c^R" raTn ’ (|E |+|E |)
o ' mq' ' np' which simplifies to , , , 6 ; • (4.5.20) 127r“£“c“R mTn ( E + E ) 0 E )E mq np P n E >E q m
4.6 THE INTERACTION BETWEEN AN ELECTRIC DIPOLE POLARISABLE MOLECULE