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The effective molecular polarizability in molecular crystals
Volume 36, number 2
CHEMICAL PHYSICS LEmERS
THE EFFECTIVE
MOLECULAR
PG. CUMMINS*,
D.A. DUNMUR
POLARIZABILITY
1 November 1975
IN MOLECULAR
CRYSTALS
Department of Clrernistry, University of Siwffield, Sheffield S3 7HF, UK
and
R.W. MUNN Department of Chemisfr): University of hfurlchesfer histitule Munchester hi60 I QD, UK
of Science md
technology,
Received 11 July 1975
The general solution for the effective polarizability in naphthalene-type crystals depends on two arbitrary parameters. These are eliminated by fixing one principal axis of polarizability parallel to a molecuiu -,k; numcricd results are given naphthalene.
I. Introduction In an earlier paper [ 1 ] , hereafter referred to as I, presented a theory relating the electric susceptibility of a molecular crystal to the effective moleczllar polarizabilities within the point dipole approximation. Effective polarizability tensors obtained by an iterative method were quoted for some aromatic hydrocarbons. Recently Chen et al. [2] have pointed out that the equations for the polarizabilities can have a continuum of solutions even when all symmetry constraints are applied. They show that in a particular case the equations are solved by polarizabilities dependent on an arbitrary parameter. Their solution is not the most general one, as they state, and they do not describe fully how it is derived. We had independently arrived at the same conclusions as Cnen et al., but have carried them somewhat further. In this paper we derive the _eenenl solution for the effective polarizability in cryst.als containing two molewe
culei per unit cell related by a twofold sciewaxis (such as naphthalene and anthcacene). The polarizability de-
pends on two arbitrary parameters. We argue that these may be eliminated by constraining the orientation of one principal axis of the po!arizability tensor; numerical results are given for naphthalene.
2. Effective
polarizabilities
In I we showed that for crystals with two equivalent molecules in the primitive unit cell, the local fields on the two sublattices are related to dimensionless polarizabilities fik by
Fl = Ml, .8,-F, + M12-B2*F,,
(1)
F2 = M12-P1-F1 + Ml1 -B2 -Fz,
(2)
where the iVlkkr depend on the electric susceptibility and the crystal structure. Writing each local field as dk*E, where dk is the local field tensor, we obtain (since the macroscopic field E is arbitrary) dl = Ml1 -81 -dl + M 12$, d, = M,,.Qd,
* Present address: Department of Chemistry, University of Southern California, Los Aweles, Gliforti 90007, USA.
for
Multiplying
.d2,
+ M,, $.6,. these eq-tions
(3) (4
on the right byd i’ -&I,
199 .
.
Volume 36, number 2
CHEMICAL PHYSICS
we find after rearrangement .&’ =M:II + M12’P> 8;! = Ml1 f M12.fy
-1
(9 .
(66)
The matrix p is given by p =s,.d~.d;‘.Bi’
(7)
and so is p1 of Chen et al. [2] , while p” is thekP2. To obtain the effective polarizabilities, we have to obtain p. The form of p is cons trained by symmetry requirements. The pola_&abiliti,:s must be symmetric, so that from eq. (5) M12-p must be symmetric. Further constraints arise from detailed consideration of the crystal symmetry. For sublattices related by a twofold screw axis parallel t? they axis (as in naphthalene), tha polarizabilities are related by 82 = U.&U,
(8)
w-here ‘-1
0 0 0 $?& 01 (9) 0 0 -1 ( i Using eq. (8) to derive pjl from eq. (5), we find that the result is identical with eq. (6) provided that u=
p-l
= U’P’U,
00)
LETTERS
1 November 1975
larizations, leaving the three components of their difference to assume arbitrary values; in the case considered here, the crystal symmetry removes one of these degrees of freedom. It may be remarked that there is a further-general but weak constraint on the variable parameters, since the resulting polarizability must have positive principd components.
3. Calculations and results To obtain well-defined polatiabilities, it is necessary to apply further constraints to the general solution. It is attractive io try to m&e the polarizability diagonal in the molecular axes, but this constraint provides three additional parameters instead of the required two, so that in generaI no such solution is obtainable. Two additional parameters majr however be obtained by constraining one principal axis of the polarizability We have assumed that the N axis of the effective
tensor. polar-
izability of naphthalene in the crystal remains along the out-of-plane C2 axis of the free molecule. Calculations have been performed by two methods. The first method starts from the assumed form of the polarizability /3 referred to the molecular axes, given in terms of the pk referred to the crystal axes by (k=1,2),
s = a&-.Zk
where the ak are direction
(13) cosine matrices
of the mole-
where we have used the fact that Ml1 and Ml2 cornmute with U . Since
cule with respect to the CiyStal axes. The effective polarizability satisfies the equation
d2=U-6*-u,
s;’ = Ml1 + M12-cs;_’ -“1l) -1.M 12 (14) derived in I [obtainable by eliminating p from eqs. (5) md (6)] . Substitution of fl and rearrangement gives the equation
(11)
eq. (7) is only satisfied if Ipl= 1, and using this result eq. (10) may be solved to give the elements of p in terms of four arbitrary parameters: P=
1 + bg/(l + d) g
b (i
be/( 1 + d) e
&l(l
it
1 + ek/(l
+ s,
8-X 1 *p-x, *p - B-X, + x, = 0, ’ f d)
(12)
.i :i Here d = (1 + bi + ek)‘i’ , and comparison with the case of identical sublattil:es (when p = 1) show that the,posit&e root should be taken. Two of the four parameters may be removed using the condition that M12’p must be symmetric, thus leaving the poiarizabilitics dependent on two 1:arameters. In terms of the arguments of Chen et al. [i] , the susceptibtity defmes t&three components o!_the sum of the sslblattice po-
(1%
where X1 =al*(MII X2
=al-M;i-Mll-d2,
X3 = al -Ml1
x4.
.M;i*MII
= al -M;:
-M;: Z,.
- wil,)-z,,
Cl6) 07)
.T2,
(18) (19)
With the assumed iorm of p, eqs. (1.5) give a quadratic
Volume
36, number 2
CHEhliCAL
PHYSICS
1 November
LETiERS
1975
Table 1
Effective polarizability tensor fl and p tensor for naphthalene crystal (the effective pohrizability u is the unit cell volume) 8
0 -0.625 0.627 0
equation
itself is obtained as o = saua, where
for the element
pN,v, for which one root is
positive and hence acceptable. Manipulation of eqs. (5) and (12) then provides unique values for the elements of p and hence for the remaining elements ofp. The second method derives a general algebraic expression for the elements of 8, including all symmetry constraints, and then sets to zero the elements PLN and PbfN. This yields two simultaneous equations in two
parameters, which can be written as simultaneous quadratic equations in one parameter, with coefficients depending on the ratio of the parameters. Solution of these equations yields the elements of p and hence of 8. The results given in table 1 show the effective polarizability of the naphthalene molecule referred to the free-molecule symmetry axes. Also given are the elements of p in the crystal axes (obc’). The molecular poIarizabiLity has substantial off-diagonal elements: the in-plane principal axes make an angle of about 35” with the free-molecule symmetry axes.
4. Discussion The apparent indeterminacy of the effective molecular polarizability (and hence of the local field) in crystals such as naphthalene is surprising. In crystals of higher symmetry and in crystals with one molecule per primitive unit cell both the polarizability and the local field are uniquely determined by the susceptibility [3]. For crystals in which the polarizability is not
unique, the question arises whether physical processes can justifiably be described in terms of single-molecule properties. However, since the microscopic treatments of many optical and electrical phenomena in crystals rely on such a description, it is worthwhile considering what constraints may be applied to the polarizability to generate unique values.
1.592
1.586
-0.417’
1.810
3.851
-1.276
-3.207
-8.594
3.259i
For molecules interacting weakly in the crystal, the molecular symmetry is likely to remain close to that of the isolated molecule, so that assumptions about the symmetry of the polarizability tensor are appropriate. In isolated aromatic hydrocarbon molecules, the polarizability component perpendicular to the mo. lecular plane is always less than the in-plane components. It is therefore reasonable to assume that this component is iess affected by the crystal environment. This provides the justification for our assumption that this axis remains perpendicular to the molecular plane 51 crystalline naphthalene. It must however be pointed out that f=ing one principal axis of the polarizability to-be parallel to some direction chosen on physical grounds, such as a molecular symmetry axis, does not necessarily lead to a physically acceptable solution for 8. For example in anthracene constraining a principal polarizability axis to be along the L or N molecular axes results in complex values for B and the local field tensor. Equivalently, the component of the molecular polarizability fiLiVcannot be made zero by any real values of the empirical parameters that determine 8. Real solutions for B of,anthracene can be generated
by allowing
the constrained
axis
of the polarizability to make. an angle of about 10” with the N axis. Clearly, different assumptions will have different effects on the interpretation of phenomena such as the Stark effect [3_,4]. Recently Chen et al. [S] have independently derived an alternative form of the general solution for & which like our solution necessarily depends on two arbitrary parameters. They have explored the physically acceptable solutions for fl and d for naphthalene and durene, and have discus.sed the uncertainty in the corresponding dipole moment changes deduced from Stark splittings.
201
Voiume 36, number
2
CHEMICAL PHYSICS LEI’TERS
1 November 1975
&nowledgernent
Referencgs
We thank the Science Research Council ior a studentship @IX). We have been-helped by discussions with Dr. N.J. Bridge, who independently derived the alge+k result in eq_ (12). We are aIso indebted to Miss W. Cheeseman for assistance with some of the calculations, and to Dr. DA!. Hanson for helpful c&espondence.
[l] P.G. Cummins, D.A. Dunmur and R.W. bfUM, Chem. Fnys. Letters 22 (1973) 519. [2] F.P. Chen, D.M. Hansonand D. Fox, Chem. Phys. Letters 30 (1975) 337. [3] P.G. Cuminins, Ph.D. Thesis, University o:ShefGld (1974). [4] D.A. Dunmur and R.W. Munn, Chem. Phys., to be pubfished. [5] F.P. Chen, D.M. Hanson and D. Fox, The Origin of Stzrk Shifts and Splittin@ in h!olecula~ Cry&l Spectra. I. The Effective Molecular Polzuizability and Local Electric Field: Durene and Naphthalene, unpublished.
CHEMICAL PHYSICS LEmERS
THE EFFECTIVE
MOLECULAR
PG. CUMMINS*,
D.A. DUNMUR
POLARIZABILITY
1 November 1975
IN MOLECULAR
CRYSTALS
Department of Clrernistry, University of Siwffield, Sheffield S3 7HF, UK
and
R.W. MUNN Department of Chemisfr): University of hfurlchesfer histitule Munchester hi60 I QD, UK
of Science md
technology,
Received 11 July 1975
The general solution for the effective polarizability in naphthalene-type crystals depends on two arbitrary parameters. These are eliminated by fixing one principal axis of polarizability parallel to a molecuiu -,k; numcricd results are given naphthalene.
I. Introduction In an earlier paper [ 1 ] , hereafter referred to as I, presented a theory relating the electric susceptibility of a molecular crystal to the effective moleczllar polarizabilities within the point dipole approximation. Effective polarizability tensors obtained by an iterative method were quoted for some aromatic hydrocarbons. Recently Chen et al. [2] have pointed out that the equations for the polarizabilities can have a continuum of solutions even when all symmetry constraints are applied. They show that in a particular case the equations are solved by polarizabilities dependent on an arbitrary parameter. Their solution is not the most general one, as they state, and they do not describe fully how it is derived. We had independently arrived at the same conclusions as Cnen et al., but have carried them somewhat further. In this paper we derive the _eenenl solution for the effective polarizability in cryst.als containing two molewe
culei per unit cell related by a twofold sciewaxis (such as naphthalene and anthcacene). The polarizability de-
pends on two arbitrary parameters. We argue that these may be eliminated by constraining the orientation of one principal axis of the po!arizability tensor; numerical results are given for naphthalene.
2. Effective
polarizabilities
In I we showed that for crystals with two equivalent molecules in the primitive unit cell, the local fields on the two sublattices are related to dimensionless polarizabilities fik by
Fl = Ml, .8,-F, + M12-B2*F,,
(1)
F2 = M12-P1-F1 + Ml1 -B2 -Fz,
(2)
where the iVlkkr depend on the electric susceptibility and the crystal structure. Writing each local field as dk*E, where dk is the local field tensor, we obtain (since the macroscopic field E is arbitrary) dl = Ml1 -81 -dl + M 12$, d, = M,,.Qd,
* Present address: Department of Chemistry, University of Southern California, Los Aweles, Gliforti 90007, USA.
for
Multiplying
.d2,
+ M,, $.6,. these eq-tions
(3) (4
on the right byd i’ -&I,
199 .
.
Volume 36, number 2
CHEMICAL PHYSICS
we find after rearrangement .&’ =M:II + M12’P> 8;! = Ml1 f M12.fy
-1
(9 .
(66)
The matrix p is given by p =s,.d~.d;‘.Bi’
(7)
and so is p1 of Chen et al. [2] , while p” is thekP2. To obtain the effective polarizabilities, we have to obtain p. The form of p is cons trained by symmetry requirements. The pola_&abiliti,:s must be symmetric, so that from eq. (5) M12-p must be symmetric. Further constraints arise from detailed consideration of the crystal symmetry. For sublattices related by a twofold screw axis parallel t? they axis (as in naphthalene), tha polarizabilities are related by 82 = U.&U,
(8)
w-here ‘-1
0 0 0 $?& 01 (9) 0 0 -1 ( i Using eq. (8) to derive pjl from eq. (5), we find that the result is identical with eq. (6) provided that u=
p-l
= U’P’U,
00)
LETTERS
1 November 1975
larizations, leaving the three components of their difference to assume arbitrary values; in the case considered here, the crystal symmetry removes one of these degrees of freedom. It may be remarked that there is a further-general but weak constraint on the variable parameters, since the resulting polarizability must have positive principd components.
3. Calculations and results To obtain well-defined polatiabilities, it is necessary to apply further constraints to the general solution. It is attractive io try to m&e the polarizability diagonal in the molecular axes, but this constraint provides three additional parameters instead of the required two, so that in generaI no such solution is obtainable. Two additional parameters majr however be obtained by constraining one principal axis of the polarizability We have assumed that the N axis of the effective
tensor. polar-
izability of naphthalene in the crystal remains along the out-of-plane C2 axis of the free molecule. Calculations have been performed by two methods. The first method starts from the assumed form of the polarizability /3 referred to the molecular axes, given in terms of the pk referred to the crystal axes by (k=1,2),
s = a&-.Zk
where the ak are direction
(13) cosine matrices
of the mole-
where we have used the fact that Ml1 and Ml2 cornmute with U . Since
cule with respect to the CiyStal axes. The effective polarizability satisfies the equation
d2=U-6*-u,
s;’ = Ml1 + M12-cs;_’ -“1l) -1.M 12 (14) derived in I [obtainable by eliminating p from eqs. (5) md (6)] . Substitution of fl and rearrangement gives the equation
(11)
eq. (7) is only satisfied if Ipl= 1, and using this result eq. (10) may be solved to give the elements of p in terms of four arbitrary parameters: P=
1 + bg/(l + d) g
b (i
be/( 1 + d) e
&l(l
it
1 + ek/(l
+ s,
8-X 1 *p-x, *p - B-X, + x, = 0, ’ f d)
(12)
.i :i Here d = (1 + bi + ek)‘i’ , and comparison with the case of identical sublattil:es (when p = 1) show that the,posit&e root should be taken. Two of the four parameters may be removed using the condition that M12’p must be symmetric, thus leaving the poiarizabilitics dependent on two 1:arameters. In terms of the arguments of Chen et al. [i] , the susceptibtity defmes t&three components o!_the sum of the sslblattice po-
(1%
where X1 =al*(MII X2
=al-M;i-Mll-d2,
X3 = al -Ml1
x4.
.M;i*MII
= al -M;:
-M;: Z,.
- wil,)-z,,
Cl6) 07)
.T2,
(18) (19)
With the assumed iorm of p, eqs. (1.5) give a quadratic
Volume
36, number 2
CHEhliCAL
PHYSICS
1 November
LETiERS
1975
Table 1
Effective polarizability tensor fl and p tensor for naphthalene crystal (the effective pohrizability u is the unit cell volume) 8
0 -0.625 0.627 0
equation
itself is obtained as o = saua, where
for the element
pN,v, for which one root is
positive and hence acceptable. Manipulation of eqs. (5) and (12) then provides unique values for the elements of p and hence for the remaining elements ofp. The second method derives a general algebraic expression for the elements of 8, including all symmetry constraints, and then sets to zero the elements PLN and PbfN. This yields two simultaneous equations in two
parameters, which can be written as simultaneous quadratic equations in one parameter, with coefficients depending on the ratio of the parameters. Solution of these equations yields the elements of p and hence of 8. The results given in table 1 show the effective polarizability of the naphthalene molecule referred to the free-molecule symmetry axes. Also given are the elements of p in the crystal axes (obc’). The molecular poIarizabiLity has substantial off-diagonal elements: the in-plane principal axes make an angle of about 35” with the free-molecule symmetry axes.
4. Discussion The apparent indeterminacy of the effective molecular polarizability (and hence of the local field) in crystals such as naphthalene is surprising. In crystals of higher symmetry and in crystals with one molecule per primitive unit cell both the polarizability and the local field are uniquely determined by the susceptibility [3]. For crystals in which the polarizability is not
unique, the question arises whether physical processes can justifiably be described in terms of single-molecule properties. However, since the microscopic treatments of many optical and electrical phenomena in crystals rely on such a description, it is worthwhile considering what constraints may be applied to the polarizability to generate unique values.
1.592
1.586
-0.417’
1.810
3.851
-1.276
-3.207
-8.594
3.259i
For molecules interacting weakly in the crystal, the molecular symmetry is likely to remain close to that of the isolated molecule, so that assumptions about the symmetry of the polarizability tensor are appropriate. In isolated aromatic hydrocarbon molecules, the polarizability component perpendicular to the mo. lecular plane is always less than the in-plane components. It is therefore reasonable to assume that this component is iess affected by the crystal environment. This provides the justification for our assumption that this axis remains perpendicular to the molecular plane 51 crystalline naphthalene. It must however be pointed out that f=ing one principal axis of the polarizability to-be parallel to some direction chosen on physical grounds, such as a molecular symmetry axis, does not necessarily lead to a physically acceptable solution for 8. For example in anthracene constraining a principal polarizability axis to be along the L or N molecular axes results in complex values for B and the local field tensor. Equivalently, the component of the molecular polarizability fiLiVcannot be made zero by any real values of the empirical parameters that determine 8. Real solutions for B of,anthracene can be generated
by allowing
the constrained
axis
of the polarizability to make. an angle of about 10” with the N axis. Clearly, different assumptions will have different effects on the interpretation of phenomena such as the Stark effect [3_,4]. Recently Chen et al. [S] have independently derived an alternative form of the general solution for & which like our solution necessarily depends on two arbitrary parameters. They have explored the physically acceptable solutions for fl and d for naphthalene and durene, and have discus.sed the uncertainty in the corresponding dipole moment changes deduced from Stark splittings.
201
Voiume 36, number
2
CHEMICAL PHYSICS LEI’TERS
1 November 1975
&nowledgernent
Referencgs
We thank the Science Research Council ior a studentship @IX). We have been-helped by discussions with Dr. N.J. Bridge, who independently derived the alge+k result in eq_ (12). We are aIso indebted to Miss W. Cheeseman for assistance with some of the calculations, and to Dr. DA!. Hanson for helpful c&espondence.
[l] P.G. Cummins, D.A. Dunmur and R.W. bfUM, Chem. Fnys. Letters 22 (1973) 519. [2] F.P. Chen, D.M. Hansonand D. Fox, Chem. Phys. Letters 30 (1975) 337. [3] P.G. Cuminins, Ph.D. Thesis, University o:ShefGld (1974). [4] D.A. Dunmur and R.W. Munn, Chem. Phys., to be pubfished. [5] F.P. Chen, D.M. Hanson and D. Fox, The Origin of Stzrk Shifts and Splittin@ in h!olecula~ Cry&l Spectra. I. The Effective Molecular Polzuizability and Local Electric Field: Durene and Naphthalene, unpublished.