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On the theory of the alignment of a polar molecule in a non-polar solvent
Physica 83A (1976) 358-370 © North-Holland Publishing Co.
ON THE THEORY OF THE ALIGNMENT OF A POLAR M O L E C U L E IN A N O N - P O L A R SOLVENT J. BIEMOND and C. MACLEAN Seheikundig Laboratorium van de Vrije Universiteit, De Lairessestraat 174, Amsterdam, The Netherlands
and M. MANDEL Gorlaeus Laboratoria, Afdeling roar Fysische Chemie, Wassenaarseweg, Leiden, The Netherlands
Received 13 November 1975
The alignment average of a dilute polar substance dissolved in a non-polar solvent may be derived from the NMR spectrum of the polar component. Likewise, the Kerr effect is related to this quantity. In this paper a series expansion of the alignment of a polar molecule in a non-polar solvent is presented for a rigid-lattice model, first introduced by Van Vleck for the dielectric constant of a very dilute solution. Different results up to second order in the dipolar interaction are obtained for a simple, a body centred, or a face centred cubic lattice and a lattice whose lattice points are uniformly distributed in a continuum. The latter formula is compared with that calculated on the basis of the Lorentz local field model and with that of Buckingham in which the interaction between the dipoles themselves is neglected.
1. Introduction W h e n an external electric field is applied to a very dilute solution of polar molecules in a n o n - p o l a r solvent, the molecular dipoles o b t a i n a small average orientation along the field direction. Nuclear interactions like the q u a d r u p o l e coupling of a nucleus I >~ 1 in the polar molecule with its electronic e n v i r o n m e n t l ) , or the direct s p i n - s p i n interaction between a pair of spins 2'3) is then n o t averaged out as in the isotropic liquid. In suitable cases the a l i g n m e n t average, (~ cos 2 0~ - ½)E, where 0~ is the angle between the direction of the rigid dipole m o m e n t of a polar molecule i a n d the electric field E in the m e d i u m , can be derived from the N M R spectrum4). It has been p o i n t e d out by B u c k i n g h a m 5) that the a l i g n m e n t of a di358
A L I G N M E N T OF A P O L A R M O L E C U L E I N A N O N - P O L A R SOLVENT
359
lute polar component in a non-polar solvent is also closely related to the Kerr effect of the mixture (see also ref. 4). In this paper a series expansion is presented for the alignment of a polar molecule in a non-polar solvent. The non-polar molecules are regarded as forming a rigid lattice except for the substitution of a single polar molecule. In the theory of dielectric polarization such a model has been introduced by Van Vleck6), who represented the molecular polarizability by harmonic oscillators. Our method of calculation is essentially the same, but we use an expression for the potential energyT), which already contains scalar molecular polarizabilities. Contrary to the series expansion of the alignment of a one-component system of polarizable dipoles on a lattice 7) convergence of the present series is more easily fulfilled for ordinary liquids. It may therefore, in principle, be compared with experiment. However, its main significance is that it may serve as a standard to test assumptions in other theories of the alignment, for instance, those based on electrostatic arguments. Such arguments were originally developed in the theory of dielectric polarization, e.g. by Lorentz8). So, the series expansion for a continuum distribution of lattice points has been compared with the formulas derived by Buckingham and Lovering 2) for the Lorentz liquid model and by Buckingham 5) for a dilute solution in which the interaction between the polar molecules themselves has been neglected. Further, the series expansion of the alignment has been given for the simple cubic, body centred cubic and face centred cubic lattice.
2. The distribution function We consider a system of non-polar molecules with a diagonal polarizability tensor ~t2, which are placed on the fixed sites of a cubic lattice or uniformly distributed in a continuum. A single polar molecule with rigid dipole moment m and a diagonal polarizability tensor oq, replaces one of the solvent molecules at a lattice point. N2 is the number of the solvent molecules and N = N2 + 1 is the number of all molecules in the system. The position of the dipole i with respect to the laboratory frame of reference will be denoted by ri and its orientation by tat. The position and orientation of a non-polar molecule i' are fixed by r~, and tat,, respectively. It will be assumed, that the macroscopic volume F is of spherical shape and that the sample is subjected to an external electric field E e = Eeu. The total dipole moment p~ of the polar molecule i for a given configuration of the system (fixed position and orientation of all molecules) can be written as follows N 2
p~ = m~ + ~ , . ( E ~ -
~, T u , . t % , ) ,
J'=l
(1)
360
J. BIEMOND, C. MACLEAN AND M. MANDEL
where - T u, . p j, represents the field acting on dipole i, due to the total dipole m o m e n t of a non-polar molecule j ' . T u, is the dipole-dipole tensor of second rank, defined by Tu, - ru ,a (U - 3ru,ru,/r~/,);
(2)
U is the unit tensor and r u, = rj, - ri stands for the distance between the molecules i and j ' . In an analogous way the total dipole m o m e n t Pi, o f the non-polar molecule i' is given by N2
p , , = or,,. ( E ~ -
T~,~.p, -
~
T,,j,.pj,).
(3)
j'•i"
The statistical averages will be calculated at constant volume V, temperature T and total n u m b e r N with the help o f a classical, canonical ensemble described by a configurational distribution function ~ normalized to unity
f = f(r N, tou, E e)
~-
e-t~v/J,
(4)
where r N represents all position vectors r l , ..., rN and o~N all orientational unit vectors o91 . . . . . ton o f the N molecules. V = V (vu, o~N, E e) is the total potential energy o f the system in a given configuration, fl - 1/kT and J is the normalizing constant given by S = S "'" J e-aV drN doJN,
(5)
where dr N denotes dr1 ... drN and deJ N in an analogous way d ~ The average of a quantity X is then defined by
(X)E = ( X f ) .
-.. d ~ - .
(6)
The brackets ( ... ) o f the right-hand side o f (6) stand for an integration over the whole configuration space o f the system. F o r a system o f N polarizable dipoles, subjected to an external electric field, Mandel and M a z u r 9) derived an expression for the potential energy V. Generalization o f their 3N-dimensional notation to a mixture of N1 polar and N2 non-polar molecules leads to a formula for the potential energy V for a mixture, as has been shown in the appendix. We will give the result in the 3-dimensional form, since that is used in the following calculation. F o r the mixture o f (N2 + 1) molecules with diagonal polarizability tensors one obtains 7) N2
N2
V = V' +½ Z m i ' T u " P ° ' - p ° ' E ~ j'=l
ZP °'Ee i'=1
N2 - -
½E e" A , . E ~ - ½ ~ i'=1
E ~. A t , . E ~,
(7)
ALIGNMENT OF A POLAR MOLECULE IN A NON-POLAR SOLVENT
361
where/~o represents the dipole moment of the polar molecule i in the mixture in the absence of an electric field E e and m~ is the rigid dipole moment of the isolated molecule i in the absence of E ~. The additional dipole moment induced by the external field is represented by A~. E * (see appendix for the definitions o f p ° and At). Analogously, pO is the dipole moment of the non-polar molecule i' in the mixture, if E e is zero (my = 0) and Av • E ~ is the additional dipole moment o f / ' in the presence of E ~. The non-dipolar part V' of the intermolecular energy in (7) is assumed to be independent of the external electric field and the orientations of the molecules. By a combination of eqs. (4), (5) and (7) the distribution function f can be expanded as a power series in E ~. Up to quadratic terms in E ~ one obtains f=fo
t I + /~pO.E o + ~ N2 EP°'E
.
+ ½fl [E ~. A , . E ¢ -- ((E *. A , - E 0 f ° ) ] ]Nr2
+ ½fl ~ [E *. A , , . E ~ -- ~((Ee, A v ' E 0 f ° ) ]
1'=1
+ {,82 [(pO E 0 (pO. E o) _ ((pO. E .) (pO. E 0 f O ) ] N2
+ ~32 E [(pO E0 (po. E*) - ((pO. E0 (po EC)fo)] 1,'=1
N2
N2 E [(pO, EO (p~," EO - ((pO. EO (p~,. E , ) f o )
+ ½f12 E i'=l
J'=l
]) ,
(8)
where f o = exp(_flVO)/(exp(_flVO)) is the distribution function in the absence of an external electric field. The potential energy V ° is given by N2
V°=
V' + ½ ~ m~. T~j,.pj° = V' + V D.
j'=l
(9)
In deriving (8) use has been made of the assumption that no remanent polarization has to be taken into account; thus ((pO. EOfO) = 0 and
~ = ~ ((pO. EOfO) = O.
362
J. BIEMOND, C. MACLEAN AND M. MANDEL
3. Calculation of the alignment The alignment of the polar component can now be calculated with the dislribution function f i n the low field limit ((~ cos z 0 i -- ½ ) f ) = 3~/~ (Ee/r~l) 2 (3 ((m~. A~. mi)f°> - m 2 ((Tr A~)f°> N2
+ ~ [3 <(m,- Aj, o m/)f°> - m 2 ((Tr Aj,).f°>] ~/*= !
+ fl [3 <(m,. pO) (m,.p°)f°)
_ m2 < ( p o pO)fo)]
Nz
+ 213 E [3 ( ( m , . pO) ( m , . p o , ) f o ) _ m 2 ((pO .pj,)fo
o)]
j'=l N2
N2
~ [3 ((m,.p°) (m,.p°,)f°> _ m2 ((pO .pO)fo>]
+ fl ~
)
.
(10)
k'=t
j'=l
0~ is the angle between the direction of the rigid dipole moment m~ and the externally applied electric field E e = Eeu. In the derivation of (10) an average over all directions of u has been performed, assuming that there is no preferred direction for a spherical sample in the absence of an external electric field. It is noticed that eqs. (8)-(9) may be helpful to derive a series expansion for the alignment of a non-polar molecule ((-~:cos z 0v - ~)f> in a polar solvent, if the polarizability tensor of all molecules is diagonal. Experiments have been performed 1°) which show that this quantity can differ considerably from zero in some cases. Such a calculation, however, will not be given in this paper. The distribution function f o can further be developed by writing f o = f , exp ( - f i V D ) / ( e x p
(-flVO)f'),
(ll)
where f ' - e - a V ' / ( e - P V ' ) . The exponentials appearing in (1 l) may be expanded in a series of increasing powers in ft. The remaining averages with the distribution function f ' in the obtained expression for f o are performed in two steps: firstly, an average over all orientations ta N and secondly, an average over all positions r N. This last average is, of course, not meaningful for a rigid lattice: (Xf'>
--r --o
= X
,
(12)
where
2~' = S "'" S X f ' dtaN/S ... ~f' do N,
(13)
--r
X<" = j' ..-J' [X<" (f ".. J'.f' dta~)drN]lj '''" i (.~ "'" J'f' dtaN) d'N"
(14)
ALIGNMENT OF A POLAR MOLECULE IN A NON-POLAR SOLVENT
363
The average ( X f ' ) can often be simplified w h e n f ' does not depend on the molecular orientations as assumed. So far we have considered, quite generally, a system of molecules with a diagonal polarizability tensor. Now we will restrict ourselves to the case of isotropically polarizable molecules in the mixture. The orientation of the polar molecule i can then be characterized by to~ = ms/ms. The following relations can be used in order to simplify eq. (10) (~o[)"
--
OOa(Da
=
/(n + 1) for n even for n odd,
(15)
(16)
~t~o~0,,
o f X .13. "l. 6 to
aCt"aCUatOa
-
1½ (c~,0, 6>.o + O~,7 60,0 + 6~,6 60,~).
-
(17)
The superscript a, /3, 7 or 6 denotes a component of the unit vector tara, where a represents the polar or a non-polar molecule; ~0, stands for the Kronecker delta. Up to second order in the dipolar interaction the quantities pO po, A~ and Av for the considered system are, respectively, given by (7) 0 P i ~- m i
N2 + "1"2
E j'=l
(18)
Ti'J "° Tj,i'mi,
N2
2
po = -~x2Tvi" ra~ + ~x2 ~ Tvs," Ts,i" ms,
(19)
J' ¢ S" N2
As
=
~xlU -
J'=l
N2 +
N2
2 Tu' + "i"2 E Tu'" Ts'~
"1"2 E
0¢1"22 E k'¢j"
j'=l
N2 E j'=l
Ti,," Tj,k,,
(20)
N2
As, = -2U - oqo~2T,,s - 2
N2
E Ts,j, + "~"22 E T,,j,. Tj,s j'~i"
j':/:S"
N2
Nz
N2
k'=l
k'¢j"
j'¢i"
2 + ~a~X 2E Ts,s'T,k,+"23 Z
Z Tvj.'Tj,g,,
(21)
where al is the scalar polarizability of the polar molecule and -2 that of a nonpolar one. Up to quadratic terms in the dipole-dipole tensor the distribution function f o is then finally calculated to be, utilizing eqs. (15)-(19)
fo =f,
1 + ½/3.2m2 Z [~," Tij" TJ','°', - ½ Y , j , : T j . i r j'=l
.
(22)
364
J. B1EMOND, C. MACLEAN AND M. MANDEL
In an analogous way as f o the alignment of the polar c o m p o n e n t in a dilute solution with a non-polar solvent can be evaluated, resulting into N2
r
2 ( ( 3 c o s 2 0, -- ½ ) f ) = 51g(fimE~) 2 1 + 7c~a 2 ~ T u, :Tj,i j'=l 2
2
N2
+ ~2 E i'=1
N2
r
t
2
N2
E Trj, "T j,, + T6~x2~ j'~i"
i'=i
N2
r
2 Ti'i: T u, ) .
(23)
j'=l
In deriving eq. (23) the following properties of the dipole-dipole tensor have been used : Tr Tab = 0
and
Tab -----Tt, a,
(24)
where a and b denote the polar or a non-polar molecule on a lattice point. It is noticed that eq. (23) is also obtained when more than one non-polar molecule on tee lattice is substituted by a polar one, provided that dipolar interactions between the polar molecules are neglected. So far, in the derivation of (23) no use has yet been made of the lattice properties of the considered system. For a rigid lattice the bars - r have no significance and m a y be omitted. Further simplification m a y be achieved by application of the following relation, valid for a rigid lattice of cubic symmetry and a spherical sample. It can be based on Lorentz's calculation for the internal field in a spherical cavity, when the molecules are in a cubic a r r a n g e m e n t (8); it has been discussed in ref. 7 (see also ref. 6) and m a y be formulated as N
Tab: Tbo = o,
(25)
c:/:b
where a, b and c denote the polar or a non polar molecule on the fixed site of a cubic lattice. Utilizing the relation (25) the double sums in eq. (23) can be evaluated. One thus obtains up to second-order terms in the dipolar interaction ((2~ cos 2 O, - ½ ) f ) = ~gs ( f l m E e ) 2
(
-
)
1 + { (oqo~2 - ~z2) ~ T,j, : Tj,, .
(26)
j'=l
The obtained expression can be rewritten in a slightly different way by using the fact that Tu, : Tj,i = 6ru 6,
(27)
and by introducing a dimensionless quantity, defined by ( 3V ~2 N2 -6 P - \ 4r:N2 ] j,~l= ru' "
(28)
ALIGNMENT OF A POLAR MOLECULE IN A NON-POLAR SOLVENT
365
The values of P for the various types o'f cubic lattices can be calculated to be
P =
-0.479 0.414 0.412
simple cubic body centred face centred.
(29)
The continuum value of P may be obtained by replacing the sum on the r.h.s, o f eq. (28) by an integral. Thus o0
P = \4~N2
/
r -6
(4nNr2/V) dr
= 1.
(30)
a
Here a is the radius of a small spherical volume around molecule i to be exclucied from the integration. Assuming that polar and non-polar molecules have approximately the same dimensions, this volume has been taken equal to V/N = ~za 3, analogously to the definition of 0nsager's cavity. Furthermore N'-" N2. Comparison of this result for P with the values of eq. (29) shows, that this particular choice of the replacement of the lattice sum leads to a value P about twice as large. By utilizing eqs. (27) and (28) eq. (26) becomes finally ((~ COS20i -- ~')f) ----"-~5(flmEe)2( 1+4(°¢1°¢2
-°~2)(477N2"]2P) "k3V/
(31)
Comparison of this formula with the series expansion of the alignment of a pure polar liquid in ref. 7 shows that convergence of the present series is more easily fulfilled in ordinary liquids.
4. Comparison with other formulas for the alignment We will now compare the expression of the alignment of eq. (31) for the case of a continuum distribution of lattice points with formulas based on other liquid models. These models, originally developed in the theory of dielectric polarization, are often applied in theories of other dielectric phenomena.
4.1. Lorentz's model Firstly, an expression for the alignment based on the local field model of Lorentz will be compared with eq. (31). It has been derived by Buckingham and Lovering 2) for the case of a pure polar liquid and may be written as follows ((-~ cos 2 0, -- ½ ) f ) = ~
(flmEe)2.
(32)
366
J. BIEMOND, C. MACLEAN AND M. MANDEL
Comparison of eqs. (31) and (32) shows that they become identical in the limit of 4 (oqo¢2 - c~22)( 4 = N 2 / 3 V ) 2 P = 0. This is the case, if the dipolar factor P arising
from the dipolar interactions between polar and non-polar molecules Nz ~ j , = l Ti~-, : T~-,~vanishes. Such a result is not unexpected, since in Lorentz's model all interactions between a dipole and its neighbours are neglected: this model is only valid for gases at low pressure. However, eq. (32) is also obtained from (31) in the particular case where ~ = O~z, even if the dipolar interactions are taken into account. 4.2. B u c k i n g h a m ' s m o d e l Another description for the alignment of a dilute polar component in a nonpolar solvent has been given by Buckingham s) in a theoretical treatment of the Kerr effect of a dilute solution (see also ref. 4). Following Buckingham, a spherical sample of volume V containing N~ polar molecules with an axially symmetric polarizability and N2 non-polar, isotropically polarizable molecules is considered. It is assumed that the moment of the liquid sphere can be written as N2
M = Z P*,
(33)
i=l
where #* is the mean dipole moment of the spherical specimen when the i-th polar molecule is kept in a fixed position and orientation. Buckingham neglects interactions between the polar molecules and finds ((2a cos 2 0i - -})f) = ~ (/3 (0¢N - o%) + flz(/z*)2) (Ee) z ,
(34)
where (c~ii - ~a) is the difference between the static polarizabilities of an isolated polar molecule, parallel and perpendicular to the direction of the rigid dipole moment. For isotropically polarizable dipoles eq. (34) reduces to ((3 cos: 0, - ½)f} = ~ (fict*Ee)2.
(35)
Comparison of this equation and (31), if P = 1, shows that the dipole moment /z* may be related to the rigid dipole moment m by /z* =
(
2 ( 47~N2 "]e']½ 1 + 4(oqc%-~2)\---~/ / m.
It can be seen from this result that ,u* and m coincide for a~ = x2.
(36)
ALIGNMENT OF A POLAR MOLECULE IN A NON-POLAR SOLVENT
367
The quantity #* also appears in the theory of the dielectric constant of a dilute solution of a polar substance in a non-polar solvent, as, e . g . has been described by Buckingham s) [see, his eq. (11)]. Comparison of the latter equation with the series expansion of the dielectric constant with P = 1, derived by Van Vleck6), shows that in that case the same relation (36) is obtained.
5. Conclusion A series expansion up to second order in the dipolar interaction of the alignment of a polar molecule in a non-polar solvent is presented for a lattice model, first introduced by Van Vleck for the dielectric constant o f a dilute solution. Different results are obtained for a simple, a body centred, or a face centred cubic lattice and a lattice wEose lattice points are uniformly distributed in a continuum. The latter formula has been compared with that calculated on the basis of the Lorentz local field model and with that from Buckingham in which the interactions between the dipoles themselves have been neglected. These comparisons give some insight into how the models of Lorentz and Buckingham take account of the interactions between the polar and non-polar molecules.
Appendix The potential energy V of a mixture of N1 polar and N2 non-polar molecules, where N = N1 + N2, in the presence of an external electric field E ¢ will now be derived. All molecules are assumed to possess a diagonal polarizability tensor. A 3N-dimensional notation, as used by Mandel and Mazur 9) for a system of N isotropically polarizable dipoles, will be applied. In the 3-dimensional notation the dipole moment Pa, where a denotes a polar or a non-polar molecule, is given by N
p, = m, + ~a" (e e - ~ T,b "Pb).
(A.1)
b~ea
For a non-polar molecule the rigid dipole m o m e n t m, is, of course, zero. Further, the diagonal polarizability tensor ~i of a polar molecule i differs in general from the diagonal tensor ~i. of a non-polar molecule i'. In a 3N-dimensional space eq. (A.1) may be written as p
= m
+ a.
(E ~ -- T.p),
(A.2)
p, m and E ~ are 3N-dimensional vectors with projections p , , ma and E ~ in the 3-dimensional subspace, respectively. In this connection rn and E e denote 3N-
368
3. BIEMOND, C. MACLEAN AND M. MANDEL
dimensional vectors and no scalar quantities as elsewhere. Likewise, T is a symmetric tensor (with 3N × 3N elements) o f which the elements in the 3-dimensional space are given by (T)a~ = T,b; Tab = 0 for a = b. The polarizability tensor a in the 3N-dimensional space is a diagonal tensor of which the elements in the 3-dimensional space are defined by (a)ab = aaa C$ab = aa C)a,- Eq. (A.2) may alternatively be written as p = (U + a . T)-~ • (in + a . E~),
(A.3)
U is the 3N-dimensional unit tensor. The vector p m a y be separated into two parts: one, representing the dipole moment pO in the absence of an electric field E ~ and two, the additional dipole moment, A • E ¢, induced by the external field: p = p O + A . E e,
(A.4)
where pO = (U + a . T ) - I . m, A • E e
-- (U -1- a . T ) - 1
(A.5) •
a . E e.
(A.6)
The projections of pO and A • E e on the 3-dimensional subspace are, respectively, given by Nz
pao = ~ ( U + a . T ) a b
(A.7)
I"mb
b=l N ( A ° E e ) a ~" 2 ( U -{- a . b=l
T ) a b ' " (Xbb ° E ¢
(A.8)
The local electric field Fa acting on a molecule a is given [see, eq. (A.1)] by N
r~ = E ¢ - Z T a b ' P b
(A.9)
b:#a
or in 3N-dimensional notation • p.
F=E~-T
(A.10)
Using eqs. (A.1) and (A.9) the contribution to the potential energy V due to the deformation o f the molecules induced by the local electric field is
Pi V,. =
NE I f 1=1
F, . dp, +
Pit N2 ~ f i'=1
rai
Fv " dpv
0
F~ =
Nt f Z i=l 0
F i •oti. dF~ +
Fl, ' ~N2 I i'=1 0
F v " oti," dFi,.
(A.11)
ALIGNMENT OF A POLAR MOLECULE IN A NON-POLAR SOLVENT
369
When the tensor ~t and ~ , are diagonal, it is found that Vi, = ~-F. a . F .
(A.12)
By using the fact S . v = v • S when S is a symmetric tensor with 3N x 3N elements and v is a 3N-dimensional vector and by utilizing eqs. (A.4)-(A.6) this equation can be written as I/in = ½ m - T . p ° - ½ p ° . T - p ° - p ° . T . A . E +½E ~.A.E ~-½E e.A.T.A.E
~ e.
(A.13)
The dipole-dipole energy is given by
(A.14)
;qi. = ½p. 7 - . p .
By using again the property S . v = v • S and by introduction of eqs. (A.4)-(A.6) this equation becomes Vdlp = ½ p ° ' T ' p °
+p0.T.A.E
e+½E c.A.T.A-E%
(A,15)
It is noticed that the potential energies V~, and Vampboth contain a part independent of E c, a part linear in E e and a part quadratic in E e. The interaction energy of the molecules with the external electric field is given by VE~ = - p - E%
(A.16)
V' will denote possible other contributions to the potential energy V; it is assumed not to depend on the external electric field. Combination of eqs. (A.13), (A.15) and (A.16) yields the final result
v = V ' + ~ ° + V~ip+ VEo = V' + ½ m . T . p ° - p ° . E
e _ ½ E e . A . E e.
(A.17)
This potential energy may also be represented in the 3-dimensional form, as has been done in ref. 7 and, for N1 = 1 in eq. (7) of this paper.
310
J. BJEMOND,
C. MACLEAN
AND M. MANDEL
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
A.M.Vasil’ev, Zh. Eksp. Teor. Fiz. 43 (1962) 1526 [Soviet Physics - JETP 16 (1963) 10781. A.D.Buckingham and E.G.Lovering, Trans. Faraday Sot. 58 (1962) 2077. A. D. Buckingham and J. A. Pople, Trans. Faraday Sot. 59 (1963) 2421. J.Biemond and C.MacLean, Molec. Phys. 26 (1973) 409. A.D.Buckingham, Trans. Faraday Sot. 52 (1956) 611. .T.H.Van Vleck, Molec. Phys. 24 (1972) 341. J.Biemond, C. MacLean and M. Mandel, Physica 79A (1975) 52. H.A.Lorentz, The Theory of Electrons (Teubner, Leipzig, 1916). M. Mandel and P. Mazur, Physica 24 (1958) 116. J.Biemond and C.MacLean, Molec. Phys. 28 (1974) 571.
ON THE THEORY OF THE ALIGNMENT OF A POLAR M O L E C U L E IN A N O N - P O L A R SOLVENT J. BIEMOND and C. MACLEAN Seheikundig Laboratorium van de Vrije Universiteit, De Lairessestraat 174, Amsterdam, The Netherlands
and M. MANDEL Gorlaeus Laboratoria, Afdeling roar Fysische Chemie, Wassenaarseweg, Leiden, The Netherlands
Received 13 November 1975
The alignment average of a dilute polar substance dissolved in a non-polar solvent may be derived from the NMR spectrum of the polar component. Likewise, the Kerr effect is related to this quantity. In this paper a series expansion of the alignment of a polar molecule in a non-polar solvent is presented for a rigid-lattice model, first introduced by Van Vleck for the dielectric constant of a very dilute solution. Different results up to second order in the dipolar interaction are obtained for a simple, a body centred, or a face centred cubic lattice and a lattice whose lattice points are uniformly distributed in a continuum. The latter formula is compared with that calculated on the basis of the Lorentz local field model and with that of Buckingham in which the interaction between the dipoles themselves is neglected.
1. Introduction W h e n an external electric field is applied to a very dilute solution of polar molecules in a n o n - p o l a r solvent, the molecular dipoles o b t a i n a small average orientation along the field direction. Nuclear interactions like the q u a d r u p o l e coupling of a nucleus I >~ 1 in the polar molecule with its electronic e n v i r o n m e n t l ) , or the direct s p i n - s p i n interaction between a pair of spins 2'3) is then n o t averaged out as in the isotropic liquid. In suitable cases the a l i g n m e n t average, (~ cos 2 0~ - ½)E, where 0~ is the angle between the direction of the rigid dipole m o m e n t of a polar molecule i a n d the electric field E in the m e d i u m , can be derived from the N M R spectrum4). It has been p o i n t e d out by B u c k i n g h a m 5) that the a l i g n m e n t of a di358
A L I G N M E N T OF A P O L A R M O L E C U L E I N A N O N - P O L A R SOLVENT
359
lute polar component in a non-polar solvent is also closely related to the Kerr effect of the mixture (see also ref. 4). In this paper a series expansion is presented for the alignment of a polar molecule in a non-polar solvent. The non-polar molecules are regarded as forming a rigid lattice except for the substitution of a single polar molecule. In the theory of dielectric polarization such a model has been introduced by Van Vleck6), who represented the molecular polarizability by harmonic oscillators. Our method of calculation is essentially the same, but we use an expression for the potential energyT), which already contains scalar molecular polarizabilities. Contrary to the series expansion of the alignment of a one-component system of polarizable dipoles on a lattice 7) convergence of the present series is more easily fulfilled for ordinary liquids. It may therefore, in principle, be compared with experiment. However, its main significance is that it may serve as a standard to test assumptions in other theories of the alignment, for instance, those based on electrostatic arguments. Such arguments were originally developed in the theory of dielectric polarization, e.g. by Lorentz8). So, the series expansion for a continuum distribution of lattice points has been compared with the formulas derived by Buckingham and Lovering 2) for the Lorentz liquid model and by Buckingham 5) for a dilute solution in which the interaction between the polar molecules themselves has been neglected. Further, the series expansion of the alignment has been given for the simple cubic, body centred cubic and face centred cubic lattice.
2. The distribution function We consider a system of non-polar molecules with a diagonal polarizability tensor ~t2, which are placed on the fixed sites of a cubic lattice or uniformly distributed in a continuum. A single polar molecule with rigid dipole moment m and a diagonal polarizability tensor oq, replaces one of the solvent molecules at a lattice point. N2 is the number of the solvent molecules and N = N2 + 1 is the number of all molecules in the system. The position of the dipole i with respect to the laboratory frame of reference will be denoted by ri and its orientation by tat. The position and orientation of a non-polar molecule i' are fixed by r~, and tat,, respectively. It will be assumed, that the macroscopic volume F is of spherical shape and that the sample is subjected to an external electric field E e = Eeu. The total dipole moment p~ of the polar molecule i for a given configuration of the system (fixed position and orientation of all molecules) can be written as follows N 2
p~ = m~ + ~ , . ( E ~ -
~, T u , . t % , ) ,
J'=l
(1)
360
J. BIEMOND, C. MACLEAN AND M. MANDEL
where - T u, . p j, represents the field acting on dipole i, due to the total dipole m o m e n t of a non-polar molecule j ' . T u, is the dipole-dipole tensor of second rank, defined by Tu, - ru ,a (U - 3ru,ru,/r~/,);
(2)
U is the unit tensor and r u, = rj, - ri stands for the distance between the molecules i and j ' . In an analogous way the total dipole m o m e n t Pi, o f the non-polar molecule i' is given by N2
p , , = or,,. ( E ~ -
T~,~.p, -
~
T,,j,.pj,).
(3)
j'•i"
The statistical averages will be calculated at constant volume V, temperature T and total n u m b e r N with the help o f a classical, canonical ensemble described by a configurational distribution function ~ normalized to unity
f = f(r N, tou, E e)
~-
e-t~v/J,
(4)
where r N represents all position vectors r l , ..., rN and o~N all orientational unit vectors o91 . . . . . ton o f the N molecules. V = V (vu, o~N, E e) is the total potential energy o f the system in a given configuration, fl - 1/kT and J is the normalizing constant given by S = S "'" J e-aV drN doJN,
(5)
where dr N denotes dr1 ... drN and deJ N in an analogous way d ~ The average of a quantity X is then defined by
(X)E = ( X f ) .
-.. d ~ - .
(6)
The brackets ( ... ) o f the right-hand side o f (6) stand for an integration over the whole configuration space o f the system. F o r a system o f N polarizable dipoles, subjected to an external electric field, Mandel and M a z u r 9) derived an expression for the potential energy V. Generalization o f their 3N-dimensional notation to a mixture of N1 polar and N2 non-polar molecules leads to a formula for the potential energy V for a mixture, as has been shown in the appendix. We will give the result in the 3-dimensional form, since that is used in the following calculation. F o r the mixture o f (N2 + 1) molecules with diagonal polarizability tensors one obtains 7) N2
N2
V = V' +½ Z m i ' T u " P ° ' - p ° ' E ~ j'=l
ZP °'Ee i'=1
N2 - -
½E e" A , . E ~ - ½ ~ i'=1
E ~. A t , . E ~,
(7)
ALIGNMENT OF A POLAR MOLECULE IN A NON-POLAR SOLVENT
361
where/~o represents the dipole moment of the polar molecule i in the mixture in the absence of an electric field E e and m~ is the rigid dipole moment of the isolated molecule i in the absence of E ~. The additional dipole moment induced by the external field is represented by A~. E * (see appendix for the definitions o f p ° and At). Analogously, pO is the dipole moment of the non-polar molecule i' in the mixture, if E e is zero (my = 0) and Av • E ~ is the additional dipole moment o f / ' in the presence of E ~. The non-dipolar part V' of the intermolecular energy in (7) is assumed to be independent of the external electric field and the orientations of the molecules. By a combination of eqs. (4), (5) and (7) the distribution function f can be expanded as a power series in E ~. Up to quadratic terms in E ~ one obtains f=fo
t I + /~pO.E o + ~ N2 EP°'E
.
+ ½fl [E ~. A , . E ¢ -- ((E *. A , - E 0 f ° ) ] ]Nr2
+ ½fl ~ [E *. A , , . E ~ -- ~((Ee, A v ' E 0 f ° ) ]
1'=1
+ {,82 [(pO E 0 (pO. E o) _ ((pO. E .) (pO. E 0 f O ) ] N2
+ ~32 E [(pO E0 (po. E*) - ((pO. E0 (po EC)fo)] 1,'=1
N2
N2 E [(pO, EO (p~," EO - ((pO. EO (p~,. E , ) f o )
+ ½f12 E i'=l
J'=l
]) ,
(8)
where f o = exp(_flVO)/(exp(_flVO)) is the distribution function in the absence of an external electric field. The potential energy V ° is given by N2
V°=
V' + ½ ~ m~. T~j,.pj° = V' + V D.
j'=l
(9)
In deriving (8) use has been made of the assumption that no remanent polarization has to be taken into account; thus ((pO. EOfO) = 0 and
~ = ~ ((pO. EOfO) = O.
362
J. BIEMOND, C. MACLEAN AND M. MANDEL
3. Calculation of the alignment The alignment of the polar component can now be calculated with the dislribution function f i n the low field limit ((~ cos z 0 i -- ½ ) f ) = 3~/~ (Ee/r~l) 2 (3 ((m~. A~. mi)f°> - m 2 ((Tr A~)f°> N2
+ ~ [3 <(m,- Aj, o m/)f°> - m 2 ((Tr Aj,).f°>] ~/*= !
+ fl [3 <(m,. pO) (m,.p°)f°)
_ m2 < ( p o pO)fo)]
Nz
+ 213 E [3 ( ( m , . pO) ( m , . p o , ) f o ) _ m 2 ((pO .pj,)fo
o)]
j'=l N2
N2
~ [3 ((m,.p°) (m,.p°,)f°> _ m2 ((pO .pO)fo>]
+ fl ~
)
.
(10)
k'=t
j'=l
0~ is the angle between the direction of the rigid dipole moment m~ and the externally applied electric field E e = Eeu. In the derivation of (10) an average over all directions of u has been performed, assuming that there is no preferred direction for a spherical sample in the absence of an external electric field. It is noticed that eqs. (8)-(9) may be helpful to derive a series expansion for the alignment of a non-polar molecule ((-~:cos z 0v - ~)f> in a polar solvent, if the polarizability tensor of all molecules is diagonal. Experiments have been performed 1°) which show that this quantity can differ considerably from zero in some cases. Such a calculation, however, will not be given in this paper. The distribution function f o can further be developed by writing f o = f , exp ( - f i V D ) / ( e x p
(-flVO)f'),
(ll)
where f ' - e - a V ' / ( e - P V ' ) . The exponentials appearing in (1 l) may be expanded in a series of increasing powers in ft. The remaining averages with the distribution function f ' in the obtained expression for f o are performed in two steps: firstly, an average over all orientations ta N and secondly, an average over all positions r N. This last average is, of course, not meaningful for a rigid lattice: (Xf'>
--r --o
= X
,
(12)
where
2~' = S "'" S X f ' dtaN/S ... ~f' do N,
(13)
--r
X<" = j' ..-J' [X<" (f ".. J'.f' dta~)drN]lj '''" i (.~ "'" J'f' dtaN) d'N"
(14)
ALIGNMENT OF A POLAR MOLECULE IN A NON-POLAR SOLVENT
363
The average ( X f ' ) can often be simplified w h e n f ' does not depend on the molecular orientations as assumed. So far we have considered, quite generally, a system of molecules with a diagonal polarizability tensor. Now we will restrict ourselves to the case of isotropically polarizable molecules in the mixture. The orientation of the polar molecule i can then be characterized by to~ = ms/ms. The following relations can be used in order to simplify eq. (10) (~o[)"
--
OOa(Da
=
/(n + 1) for n even for n odd,
(15)
(16)
~t~o~0,,
o f X .13. "l. 6 to
aCt"aCUatOa
-
1½ (c~,0, 6>.o + O~,7 60,0 + 6~,6 60,~).
-
(17)
The superscript a, /3, 7 or 6 denotes a component of the unit vector tara, where a represents the polar or a non-polar molecule; ~0, stands for the Kronecker delta. Up to second order in the dipolar interaction the quantities pO po, A~ and Av for the considered system are, respectively, given by (7) 0 P i ~- m i
N2 + "1"2
E j'=l
(18)
Ti'J "° Tj,i'mi,
N2
2
po = -~x2Tvi" ra~ + ~x2 ~ Tvs," Ts,i" ms,
(19)
J' ¢ S" N2
As
=
~xlU -
J'=l
N2 +
N2
2 Tu' + "i"2 E Tu'" Ts'~
"1"2 E
0¢1"22 E k'¢j"
j'=l
N2 E j'=l
Ti,," Tj,k,,
(20)
N2
As, = -2U - oqo~2T,,s - 2
N2
E Ts,j, + "~"22 E T,,j,. Tj,s j'~i"
j':/:S"
N2
Nz
N2
k'=l
k'¢j"
j'¢i"
2 + ~a~X 2E Ts,s'T,k,+"23 Z
Z Tvj.'Tj,g,,
(21)
where al is the scalar polarizability of the polar molecule and -2 that of a nonpolar one. Up to quadratic terms in the dipole-dipole tensor the distribution function f o is then finally calculated to be, utilizing eqs. (15)-(19)
fo =f,
1 + ½/3.2m2 Z [~," Tij" TJ','°', - ½ Y , j , : T j . i r j'=l
.
(22)
364
J. B1EMOND, C. MACLEAN AND M. MANDEL
In an analogous way as f o the alignment of the polar c o m p o n e n t in a dilute solution with a non-polar solvent can be evaluated, resulting into N2
r
2 ( ( 3 c o s 2 0, -- ½ ) f ) = 51g(fimE~) 2 1 + 7c~a 2 ~ T u, :Tj,i j'=l 2
2
N2
+ ~2 E i'=1
N2
r
t
2
N2
E Trj, "T j,, + T6~x2~ j'~i"
i'=i
N2
r
2 Ti'i: T u, ) .
(23)
j'=l
In deriving eq. (23) the following properties of the dipole-dipole tensor have been used : Tr Tab = 0
and
Tab -----Tt, a,
(24)
where a and b denote the polar or a non-polar molecule on a lattice point. It is noticed that eq. (23) is also obtained when more than one non-polar molecule on tee lattice is substituted by a polar one, provided that dipolar interactions between the polar molecules are neglected. So far, in the derivation of (23) no use has yet been made of the lattice properties of the considered system. For a rigid lattice the bars - r have no significance and m a y be omitted. Further simplification m a y be achieved by application of the following relation, valid for a rigid lattice of cubic symmetry and a spherical sample. It can be based on Lorentz's calculation for the internal field in a spherical cavity, when the molecules are in a cubic a r r a n g e m e n t (8); it has been discussed in ref. 7 (see also ref. 6) and m a y be formulated as N
Tab: Tbo = o,
(25)
c:/:b
where a, b and c denote the polar or a non polar molecule on the fixed site of a cubic lattice. Utilizing the relation (25) the double sums in eq. (23) can be evaluated. One thus obtains up to second-order terms in the dipolar interaction ((2~ cos 2 O, - ½ ) f ) = ~gs ( f l m E e ) 2
(
-
)
1 + { (oqo~2 - ~z2) ~ T,j, : Tj,, .
(26)
j'=l
The obtained expression can be rewritten in a slightly different way by using the fact that Tu, : Tj,i = 6ru 6,
(27)
and by introducing a dimensionless quantity, defined by ( 3V ~2 N2 -6 P - \ 4r:N2 ] j,~l= ru' "
(28)
ALIGNMENT OF A POLAR MOLECULE IN A NON-POLAR SOLVENT
365
The values of P for the various types o'f cubic lattices can be calculated to be
P =
-0.479 0.414 0.412
simple cubic body centred face centred.
(29)
The continuum value of P may be obtained by replacing the sum on the r.h.s, o f eq. (28) by an integral. Thus o0
P = \4~N2
/
r -6
(4nNr2/V) dr
= 1.
(30)
a
Here a is the radius of a small spherical volume around molecule i to be exclucied from the integration. Assuming that polar and non-polar molecules have approximately the same dimensions, this volume has been taken equal to V/N = ~za 3, analogously to the definition of 0nsager's cavity. Furthermore N'-" N2. Comparison of this result for P with the values of eq. (29) shows, that this particular choice of the replacement of the lattice sum leads to a value P about twice as large. By utilizing eqs. (27) and (28) eq. (26) becomes finally ((~ COS20i -- ~')f) ----"-~5(flmEe)2( 1+4(°¢1°¢2
-°~2)(477N2"]2P) "k3V/
(31)
Comparison of this formula with the series expansion of the alignment of a pure polar liquid in ref. 7 shows that convergence of the present series is more easily fulfilled in ordinary liquids.
4. Comparison with other formulas for the alignment We will now compare the expression of the alignment of eq. (31) for the case of a continuum distribution of lattice points with formulas based on other liquid models. These models, originally developed in the theory of dielectric polarization, are often applied in theories of other dielectric phenomena.
4.1. Lorentz's model Firstly, an expression for the alignment based on the local field model of Lorentz will be compared with eq. (31). It has been derived by Buckingham and Lovering 2) for the case of a pure polar liquid and may be written as follows ((-~ cos 2 0, -- ½ ) f ) = ~
(flmEe)2.
(32)
366
J. BIEMOND, C. MACLEAN AND M. MANDEL
Comparison of eqs. (31) and (32) shows that they become identical in the limit of 4 (oqo¢2 - c~22)( 4 = N 2 / 3 V ) 2 P = 0. This is the case, if the dipolar factor P arising
from the dipolar interactions between polar and non-polar molecules Nz ~ j , = l Ti~-, : T~-,~vanishes. Such a result is not unexpected, since in Lorentz's model all interactions between a dipole and its neighbours are neglected: this model is only valid for gases at low pressure. However, eq. (32) is also obtained from (31) in the particular case where ~ = O~z, even if the dipolar interactions are taken into account. 4.2. B u c k i n g h a m ' s m o d e l Another description for the alignment of a dilute polar component in a nonpolar solvent has been given by Buckingham s) in a theoretical treatment of the Kerr effect of a dilute solution (see also ref. 4). Following Buckingham, a spherical sample of volume V containing N~ polar molecules with an axially symmetric polarizability and N2 non-polar, isotropically polarizable molecules is considered. It is assumed that the moment of the liquid sphere can be written as N2
M = Z P*,
(33)
i=l
where #* is the mean dipole moment of the spherical specimen when the i-th polar molecule is kept in a fixed position and orientation. Buckingham neglects interactions between the polar molecules and finds ((2a cos 2 0i - -})f) = ~ (/3 (0¢N - o%) + flz(/z*)2) (Ee) z ,
(34)
where (c~ii - ~a) is the difference between the static polarizabilities of an isolated polar molecule, parallel and perpendicular to the direction of the rigid dipole moment. For isotropically polarizable dipoles eq. (34) reduces to ((3 cos: 0, - ½)f} = ~ (fict*Ee)2.
(35)
Comparison of this equation and (31), if P = 1, shows that the dipole moment /z* may be related to the rigid dipole moment m by /z* =
(
2 ( 47~N2 "]e']½ 1 + 4(oqc%-~2)\---~/ / m.
It can be seen from this result that ,u* and m coincide for a~ = x2.
(36)
ALIGNMENT OF A POLAR MOLECULE IN A NON-POLAR SOLVENT
367
The quantity #* also appears in the theory of the dielectric constant of a dilute solution of a polar substance in a non-polar solvent, as, e . g . has been described by Buckingham s) [see, his eq. (11)]. Comparison of the latter equation with the series expansion of the dielectric constant with P = 1, derived by Van Vleck6), shows that in that case the same relation (36) is obtained.
5. Conclusion A series expansion up to second order in the dipolar interaction of the alignment of a polar molecule in a non-polar solvent is presented for a lattice model, first introduced by Van Vleck for the dielectric constant o f a dilute solution. Different results are obtained for a simple, a body centred, or a face centred cubic lattice and a lattice wEose lattice points are uniformly distributed in a continuum. The latter formula has been compared with that calculated on the basis of the Lorentz local field model and with that from Buckingham in which the interactions between the dipoles themselves have been neglected. These comparisons give some insight into how the models of Lorentz and Buckingham take account of the interactions between the polar and non-polar molecules.
Appendix The potential energy V of a mixture of N1 polar and N2 non-polar molecules, where N = N1 + N2, in the presence of an external electric field E ¢ will now be derived. All molecules are assumed to possess a diagonal polarizability tensor. A 3N-dimensional notation, as used by Mandel and Mazur 9) for a system of N isotropically polarizable dipoles, will be applied. In the 3-dimensional notation the dipole moment Pa, where a denotes a polar or a non-polar molecule, is given by N
p, = m, + ~a" (e e - ~ T,b "Pb).
(A.1)
b~ea
For a non-polar molecule the rigid dipole m o m e n t m, is, of course, zero. Further, the diagonal polarizability tensor ~i of a polar molecule i differs in general from the diagonal tensor ~i. of a non-polar molecule i'. In a 3N-dimensional space eq. (A.1) may be written as p
= m
+ a.
(E ~ -- T.p),
(A.2)
p, m and E ~ are 3N-dimensional vectors with projections p , , ma and E ~ in the 3-dimensional subspace, respectively. In this connection rn and E e denote 3N-
368
3. BIEMOND, C. MACLEAN AND M. MANDEL
dimensional vectors and no scalar quantities as elsewhere. Likewise, T is a symmetric tensor (with 3N × 3N elements) o f which the elements in the 3-dimensional space are given by (T)a~ = T,b; Tab = 0 for a = b. The polarizability tensor a in the 3N-dimensional space is a diagonal tensor of which the elements in the 3-dimensional space are defined by (a)ab = aaa C$ab = aa C)a,- Eq. (A.2) may alternatively be written as p = (U + a . T)-~ • (in + a . E~),
(A.3)
U is the 3N-dimensional unit tensor. The vector p m a y be separated into two parts: one, representing the dipole moment pO in the absence of an electric field E ~ and two, the additional dipole moment, A • E ¢, induced by the external field: p = p O + A . E e,
(A.4)
where pO = (U + a . T ) - I . m, A • E e
-- (U -1- a . T ) - 1
(A.5) •
a . E e.
(A.6)
The projections of pO and A • E e on the 3-dimensional subspace are, respectively, given by Nz
pao = ~ ( U + a . T ) a b
(A.7)
I"mb
b=l N ( A ° E e ) a ~" 2 ( U -{- a . b=l
T ) a b ' " (Xbb ° E ¢
(A.8)
The local electric field Fa acting on a molecule a is given [see, eq. (A.1)] by N
r~ = E ¢ - Z T a b ' P b
(A.9)
b:#a
or in 3N-dimensional notation • p.
F=E~-T
(A.10)
Using eqs. (A.1) and (A.9) the contribution to the potential energy V due to the deformation o f the molecules induced by the local electric field is
Pi V,. =
NE I f 1=1
F, . dp, +
Pit N2 ~ f i'=1
rai
Fv " dpv
0
F~ =
Nt f Z i=l 0
F i •oti. dF~ +
Fl, ' ~N2 I i'=1 0
F v " oti," dFi,.
(A.11)
ALIGNMENT OF A POLAR MOLECULE IN A NON-POLAR SOLVENT
369
When the tensor ~t and ~ , are diagonal, it is found that Vi, = ~-F. a . F .
(A.12)
By using the fact S . v = v • S when S is a symmetric tensor with 3N x 3N elements and v is a 3N-dimensional vector and by utilizing eqs. (A.4)-(A.6) this equation can be written as I/in = ½ m - T . p ° - ½ p ° . T - p ° - p ° . T . A . E +½E ~.A.E ~-½E e.A.T.A.E
~ e.
(A.13)
The dipole-dipole energy is given by
(A.14)
;qi. = ½p. 7 - . p .
By using again the property S . v = v • S and by introduction of eqs. (A.4)-(A.6) this equation becomes Vdlp = ½ p ° ' T ' p °
+p0.T.A.E
e+½E c.A.T.A-E%
(A,15)
It is noticed that the potential energies V~, and Vampboth contain a part independent of E c, a part linear in E e and a part quadratic in E e. The interaction energy of the molecules with the external electric field is given by VE~ = - p - E%
(A.16)
V' will denote possible other contributions to the potential energy V; it is assumed not to depend on the external electric field. Combination of eqs. (A.13), (A.15) and (A.16) yields the final result
v = V ' + ~ ° + V~ip+ VEo = V' + ½ m . T . p ° - p ° . E
e _ ½ E e . A . E e.
(A.17)
This potential energy may also be represented in the 3-dimensional form, as has been done in ref. 7 and, for N1 = 1 in eq. (7) of this paper.
310
J. BJEMOND,
C. MACLEAN
AND M. MANDEL
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
A.M.Vasil’ev, Zh. Eksp. Teor. Fiz. 43 (1962) 1526 [Soviet Physics - JETP 16 (1963) 10781. A.D.Buckingham and E.G.Lovering, Trans. Faraday Sot. 58 (1962) 2077. A. D. Buckingham and J. A. Pople, Trans. Faraday Sot. 59 (1963) 2421. J.Biemond and C.MacLean, Molec. Phys. 26 (1973) 409. A.D.Buckingham, Trans. Faraday Sot. 52 (1956) 611. .T.H.Van Vleck, Molec. Phys. 24 (1972) 341. J.Biemond, C. MacLean and M. Mandel, Physica 79A (1975) 52. H.A.Lorentz, The Theory of Electrons (Teubner, Leipzig, 1916). M. Mandel and P. Mazur, Physica 24 (1958) 116. J.Biemond and C.MacLean, Molec. Phys. 28 (1974) 571.