- Email: [email protected]
Odd-rank order parameters in non-uniaxial orientational distributions
Volume 180. number 4
CHEMICAL PHYSICS LETTERS
24May 1991
Odd-rank order parameters in non-uniaxial orientational distributions M. van Gurp DSM Research, P.O. Box 18, 6160 MD Geleen, The Netherlands
and Y .K. Levine Department ofMo/ecu/ar Biophysics, Buys Ballot Laboratory, P. 0. Box 80000, 3508 TA Utrecht. The Netherlands Received 13 February 1990; in final form 4 February 199 1
The consequences of double-reflection symmetry operations for the orientational distribution function are discussed. A doublereflection operation is equivalent to a rotational transformation, and conserves the handedness of the molecular and macroscopic frames that are transformed. It is found that, in general, non-vanishing order parameters of both even and odd order exist for a biaxial phase containing biaxial molecules. This finding is confirmed by numerical computations of the order parameters using a simple form for the orientational distribution function.
1. Introduction Organic and biological molecules can be aligned macroscopically in orientationally ordered systems such as liquid crystals, lipid membranes and deformed polymers. The distribution of the molecular axes relative to a macroscopic Cartesian frame (in which the axis of alignment is often chosen along the Z axis), is most conveniently described in terms of a normalized orientational distribution function f( (Y,j?, y). Here (Y,8, y are the Euler angles describing the rotational transformation from the macroscopic to the molecular frame. It has been found useful to express the distribution functionf( a, /?, y) as a series expansion of the Wigner rotation matrix elements [l-3]:
The moments of the distribution function (02” > , the so-called order parameters, are simply the ensemble averages of the Wigner functions themselves. The orientational distribution function, eq. ( 1 ), is fully characterized only if the values of a sufficient 0009-2614/91/$
number of order parameters are known to ensure the convergence of the series. In practice this is an almost impossible task as up to (2L+ I)* order parameters of rank L must be determined experimentally. Nevertheless, the number of order parameters needed to characterize the distribution can be reduced drastically by exploiting the symmetry properties of both the system and the molecules. This is achieved simply by utilizing the fact that the explicit expressions of the Wigner rotation matrix elements can be factorized into separate functions, each of which is dependent on one of the Euler angles [ 4,5]. In this way, it can be demonstrated that for a phase possessing uniaxial symmetry around the laboratory Z axis, (0:” ) = (Oh: ) So,, while for axially symmetric molecules, (D”,l,) = ( DE0 )&,. While it is easy to determine the non-vanishing order parameter for systems and molecules with rotational symmetries, this is an impossible task for systems possessing mirror symmetries. The reason for this is that a mirror-symmetry operation transforms a right-handed frame into a left-handed one and vice versa. This change in handedness cannot now be expressed in terms of an equivalent rota-
03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
349
Volume180,number4
CHEMICAL PHYSICSLETTERS
tional transformation. Essentially, the problem boils down to the way the third Euler angle, y, is transformed in the reflection operations, as the mirrorsymmetry properties of spherical harmonics [ 6 ] can be used to evaluate transformation involving the two Euler angles (Yand @.Consequently, only the effects of mirror-symmetry operations on vectors and axially symmetric molecules can be handled readily. Nevertheless, it is important to point out that mirror-symmetry operations in uniaxial systems containing non-axially symmetric molecules can also be treated in a simple way [ 71. An elegant, though cumbersome, approach to this problem is to consider an orientational distribution of both right- and lefthanded frames [ 8 1. We shall show here that the non-vanishing order parameters can be determined for systems characterized by symmetry operations in which two subsequent reflections, or “turns” [ 91, are applied. In this case, the handedness of the molecular frames is conserved and it becomes possible to express the double mirror-symmetry operations as an equivalent rotation.
2. Double-reflection symmetries The simplest way of determining the effect of a double-reflection-symmetry operation on the order parameters is to follow the transformation of the Cartesian coordinates of the three basis vectors of the molecular frame. Reflection operations are represented by diagonal matrices, and consequently all the operations are commutative, i.e. a(XY)a( YZ) = a( YZ)a(XY). Here, a( PQ ) represents a reflection in the PQ-plane. Sincea(XY)=diag{l, 1, -l}anda(YZ)=diag{-1, 1, l}, their combined operation is represented by diag{ - 1, I, - 1). It is easy to show that this operation is equivalent to a rotation of 180” about the laboratory Y axis on making use of the explicit expressions of the matrix elements of the rotation operator R in Cartesian coordinates [ 4,5 1. We note that R(cu, b, y) effectively describes rotational operations in the same frame: a rotation around the Z axis by an angle (Y,followed by a rotation about the Y axis by an angle j?, and finally a rotation around the Z axis by an angle y [ 4,s 1. 350
24May 1991
Similarly, it can be shown that the double-reflection operation a( effects a rotation of 180” about the laboratory X axis, a=n/2, j.I=rc, y= -x/2. It turns out in fact that any general double-reflection operator can be expressed in terms of an equivalent rotation. This arises as the double-reflection operation always reverses the sign of two of the components of the Cartesian vector x. This change in sign can now be represented by a 180” rotation about the axis corresponding to the component whose direction has been conserved. It is now important to note that as each reflection operation can be represented by a diagonal matrix with elements + 1, the only mathematical manipulations involved in the rotational transformation R connecting the molecular and macroscopic frames are multiplications of the matrix elements by a phase factor of - 1. Hence, we can establish the symmetry properties of R under double-reflection operations by considering the explicit change in sign of every matrix element of the operator expressed in a Cartesian representation. As an example, we consider the double-mirror symmetry operation a( changing the transformational coordinates 9= (a, 8, y) of R into Q’ = (a’, p’ , y’ ) R(P)
= a(XY)a(XZ)R(O)
=diag{ 1, - 1, -l}
R(P) .
(2)
By writing down eq. (2) in its Cartesian matrix representation, one can show in a straightforward way that eq. (2) is satisfied for cr’ = - LY,j?’ = II- /3 and y’ = ft t y. In a similar way, the equivalent rotations corresponding to all 15 possible combinations of double reflections in the laboratory and molecular reference frames can be found. Furthermore, it can be shown clearly in this way that the inversion operator I=diag{- 1, - 1, - lj cannot be considered to be equivalent to a rotation with an angle of {rc+a, A-/?, nfy} as is sometimes claimed in the literature [ 10 1.
3, Results and discussion The equivalence between a double-reflection-symmetry operation and a rotational one is used in a
24 May1991
CHEMICALPHYSICSLETTERS
Volume180,number4
straightforward way to determine the non-vanishing order parameters in the expansion of the orientational distribution function, eq. ( 1). The two equivalent distribution functionf( 9) and f( 5)’ ) are now expanded according to eq. ( I ) and the explicit properties of the Wigner rotation matrix elements are used to relate D&,(Q) and D:,(J)‘). Finally, the nonvanishing order parameters are obtained by equating the coefficients of the same matrix elements in the expansions. These are shown in table 1. It can be seen from the relations shown in table 1 that if the macroscopic phase possesses three pairs of mirror-symmetries in the XY, XZ and YZ planes (also called biaxial, orthorhombic or orthotropic symmetry), we have (D~n)=(-)L(DL,,),
withmeven
(3)
This surprising conclusion arises from the combination of eqs. ( 3) and (4), which, for an arbitrary value of L, with m and n even, yields the condition (D”,, ) = (Ok,,, _” ) . This relation implies that (13;” ) =O, so that the lowest non-vanishing order parameters of odd rank are (D&) =(D)_2_-2) = -(DL)=---(0:-z). This result can in fact be checked simply by a numerical evaluation of the order parameters on expressing the orientational distribution function in terms of an effective mean orienting potential U(Q) acting on the molecules [ 11, f(Q)=exp[
-U(O)/kT]
.
(5)
The simplest potential describing a system of biaxial molecules embedded in a biaxial phase, has the form
and for a molecule of the same symmetry we have (Dk,)=(-)L(Dk_,),
withneven.
(4)
In the case of uniaxial systems for which ( D$,, ) = (D& >do, or axially symmetric molecules where (Oh,, ) = ( Dho ) d,,,,, all the order parameters of odd rank L satisfying eqs. (3) and (4) vanish identically as expected [ 11. However, eqs. (3) and (4) imply that this need not be true in general. In particular, it can be easily shown that the rank L of the non-vanishing order parameters may take on both even and odd values in a biaxial system containing biaxial molecules.
+dD:o(W+D%W
I).
(6)
We note that the numerical computations of the order parameter using this potential requires a numerical integration only over the angle /3, as the integrals over the Euler angles (Yand y can be expressed analytically in terms of spherical Bessel functions [ 111. The order parameters obtained from this potential with arbitrary values for the parameters a, b and c reproduce al the symmetry relations given in table 1. Significantly, we find that the odd-rank order parameters with L> 1 do not vanish, provided
Table I f(wP, Y)
Operation u(XU)u( YZ)
I
lu(XZ)
a(xr)u(xz) = IU(YZ)
a( YZ)a(XZ) = la(XY) u(xy)u(yz) - la(xz) u(xy)a(xz)-lu(yz)
u(yz)a(xz)
= lu(xy)
~(Jw4xv) ~(XoJW)
u(XY)u(xz)
am @(YZMP) a( YZ)u(xz)
U(X-m(XY) a(xz)aJz) u(XZ)u(xz)
(Df”> (-)L’“(D’_,,) (-)L
c-Y(Dk.> (-)“‘“(G-“> (-)“(D”,-.> ( - )“ok > (- )“+“(Dfin) (-)L-m(Dk-.) (-)L-"+"(Df_.> (- )=+“(%m > CD%n> (-)“<04,-.> (-)=+“+“(Dk,,)
(-)“(D%,> (-)"-"(D4,_,>
351
Volume 180, number 4
CHEMICAL PHYSICS LETTERS
the coefficients b and c in eq. (6) are both non-zero. Finally, we note that the symmetry properties of the order parameters given in table 1 can be used to determine the form of the correlation functions (D/2,02 ) which appear in the description of fluorescence depolarization and Raman-scattering experiments [ 10,121. This is most conveniently done using the Clebsch-Gordan series expansion [ 4,5 1,
xC(22L;m-n)(D,L_k,_.),
(7)
where C denotes the Clebsch-Gordan coefficient [ 13 1. Furthermore, since C( 2 2 3; 1 1) = C( 2 2 3; - 1 - 1) ~0, we see from eq. (5) that the order parameter (D& ) = - (DLzz) does not enter the expansion of the correlation functions and thus cannot be determined experimentally.
Acknowledgement We would like to thank Professor P.J. Brussaard for many useful discussions.
352
24 May 1991
References [ 1 ] C. Zannoni, in: The molecular physics of liquid crystals, eds. G.R. Luckhurst and G.W. Gray (Academic Press, New York, 1979) p. 57. [2] M. van Gurp, H. van Langen, G. van Ginkel and Y.K. Levine, in: Polarized spectroscopy of ordered systems, eds. B. Samori and E.W. Thulstrup (Kluwer, Donlrecht, 1988) p. 455. [3] V.J. McBtierty, J. Chem. Phys 61 (1974) 872. [ 41 A.R. Edmonds, Angular momentum in quantum mechanics (Princeton Univ. Press., Princeton, 1960). [5] M.E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957). [6] E. Merzbacher, Quantum mechanics (Wiley, New York, 1970). [7] J.J. Fisz, Chem. Phys. 114 (1987) 165. [ 81 H.J. Bunge, C. Esling and J. Muller, Acta Cryst. A 37 ( 1981) 889. [9] L.C. Biedenham and J-D. Louck, Angular momentum in quantum physics (Addison-Wesley, Reading, 198 I ) . [ lo] P.S. Pershan, in: The molecular physics of liquid crystals, eds. G.R. Luckhurst and G.W. Gray (Academic Press, New York, 1979) p. 385. [ 111 M. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions (Dover, New York, 1964). [ 121 C. Zannoni, A. Arcioni and P. Cavatorta, Chem. Phys. Lipids 32 (1983) 179. [ 131 Particle Data Group, Rev. Mod. Phys. 48 ( 1967) S36.
CHEMICAL PHYSICS LETTERS
24May 1991
Odd-rank order parameters in non-uniaxial orientational distributions M. van Gurp DSM Research, P.O. Box 18, 6160 MD Geleen, The Netherlands
and Y .K. Levine Department ofMo/ecu/ar Biophysics, Buys Ballot Laboratory, P. 0. Box 80000, 3508 TA Utrecht. The Netherlands Received 13 February 1990; in final form 4 February 199 1
The consequences of double-reflection symmetry operations for the orientational distribution function are discussed. A doublereflection operation is equivalent to a rotational transformation, and conserves the handedness of the molecular and macroscopic frames that are transformed. It is found that, in general, non-vanishing order parameters of both even and odd order exist for a biaxial phase containing biaxial molecules. This finding is confirmed by numerical computations of the order parameters using a simple form for the orientational distribution function.
1. Introduction Organic and biological molecules can be aligned macroscopically in orientationally ordered systems such as liquid crystals, lipid membranes and deformed polymers. The distribution of the molecular axes relative to a macroscopic Cartesian frame (in which the axis of alignment is often chosen along the Z axis), is most conveniently described in terms of a normalized orientational distribution function f( (Y,j?, y). Here (Y,8, y are the Euler angles describing the rotational transformation from the macroscopic to the molecular frame. It has been found useful to express the distribution functionf( a, /?, y) as a series expansion of the Wigner rotation matrix elements [l-3]:
The moments of the distribution function (02” > , the so-called order parameters, are simply the ensemble averages of the Wigner functions themselves. The orientational distribution function, eq. ( 1 ), is fully characterized only if the values of a sufficient 0009-2614/91/$
number of order parameters are known to ensure the convergence of the series. In practice this is an almost impossible task as up to (2L+ I)* order parameters of rank L must be determined experimentally. Nevertheless, the number of order parameters needed to characterize the distribution can be reduced drastically by exploiting the symmetry properties of both the system and the molecules. This is achieved simply by utilizing the fact that the explicit expressions of the Wigner rotation matrix elements can be factorized into separate functions, each of which is dependent on one of the Euler angles [ 4,5]. In this way, it can be demonstrated that for a phase possessing uniaxial symmetry around the laboratory Z axis, (0:” ) = (Oh: ) So,, while for axially symmetric molecules, (D”,l,) = ( DE0 )&,. While it is easy to determine the non-vanishing order parameter for systems and molecules with rotational symmetries, this is an impossible task for systems possessing mirror symmetries. The reason for this is that a mirror-symmetry operation transforms a right-handed frame into a left-handed one and vice versa. This change in handedness cannot now be expressed in terms of an equivalent rota-
03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
349
Volume180,number4
CHEMICAL PHYSICSLETTERS
tional transformation. Essentially, the problem boils down to the way the third Euler angle, y, is transformed in the reflection operations, as the mirrorsymmetry properties of spherical harmonics [ 6 ] can be used to evaluate transformation involving the two Euler angles (Yand @.Consequently, only the effects of mirror-symmetry operations on vectors and axially symmetric molecules can be handled readily. Nevertheless, it is important to point out that mirror-symmetry operations in uniaxial systems containing non-axially symmetric molecules can also be treated in a simple way [ 71. An elegant, though cumbersome, approach to this problem is to consider an orientational distribution of both right- and lefthanded frames [ 8 1. We shall show here that the non-vanishing order parameters can be determined for systems characterized by symmetry operations in which two subsequent reflections, or “turns” [ 91, are applied. In this case, the handedness of the molecular frames is conserved and it becomes possible to express the double mirror-symmetry operations as an equivalent rotation.
2. Double-reflection symmetries The simplest way of determining the effect of a double-reflection-symmetry operation on the order parameters is to follow the transformation of the Cartesian coordinates of the three basis vectors of the molecular frame. Reflection operations are represented by diagonal matrices, and consequently all the operations are commutative, i.e. a(XY)a( YZ) = a( YZ)a(XY). Here, a( PQ ) represents a reflection in the PQ-plane. Sincea(XY)=diag{l, 1, -l}anda(YZ)=diag{-1, 1, l}, their combined operation is represented by diag{ - 1, I, - 1). It is easy to show that this operation is equivalent to a rotation of 180” about the laboratory Y axis on making use of the explicit expressions of the matrix elements of the rotation operator R in Cartesian coordinates [ 4,5 1. We note that R(cu, b, y) effectively describes rotational operations in the same frame: a rotation around the Z axis by an angle (Y,followed by a rotation about the Y axis by an angle j?, and finally a rotation around the Z axis by an angle y [ 4,s 1. 350
24May 1991
Similarly, it can be shown that the double-reflection operation a( effects a rotation of 180” about the laboratory X axis, a=n/2, j.I=rc, y= -x/2. It turns out in fact that any general double-reflection operator can be expressed in terms of an equivalent rotation. This arises as the double-reflection operation always reverses the sign of two of the components of the Cartesian vector x. This change in sign can now be represented by a 180” rotation about the axis corresponding to the component whose direction has been conserved. It is now important to note that as each reflection operation can be represented by a diagonal matrix with elements + 1, the only mathematical manipulations involved in the rotational transformation R connecting the molecular and macroscopic frames are multiplications of the matrix elements by a phase factor of - 1. Hence, we can establish the symmetry properties of R under double-reflection operations by considering the explicit change in sign of every matrix element of the operator expressed in a Cartesian representation. As an example, we consider the double-mirror symmetry operation a( changing the transformational coordinates 9= (a, 8, y) of R into Q’ = (a’, p’ , y’ ) R(P)
= a(XY)a(XZ)R(O)
=diag{ 1, - 1, -l}
R(P) .
(2)
By writing down eq. (2) in its Cartesian matrix representation, one can show in a straightforward way that eq. (2) is satisfied for cr’ = - LY,j?’ = II- /3 and y’ = ft t y. In a similar way, the equivalent rotations corresponding to all 15 possible combinations of double reflections in the laboratory and molecular reference frames can be found. Furthermore, it can be shown clearly in this way that the inversion operator I=diag{- 1, - 1, - lj cannot be considered to be equivalent to a rotation with an angle of {rc+a, A-/?, nfy} as is sometimes claimed in the literature [ 10 1.
3, Results and discussion The equivalence between a double-reflection-symmetry operation and a rotational one is used in a
24 May1991
CHEMICALPHYSICSLETTERS
Volume180,number4
straightforward way to determine the non-vanishing order parameters in the expansion of the orientational distribution function, eq. ( 1). The two equivalent distribution functionf( 9) and f( 5)’ ) are now expanded according to eq. ( I ) and the explicit properties of the Wigner rotation matrix elements are used to relate D&,(Q) and D:,(J)‘). Finally, the nonvanishing order parameters are obtained by equating the coefficients of the same matrix elements in the expansions. These are shown in table 1. It can be seen from the relations shown in table 1 that if the macroscopic phase possesses three pairs of mirror-symmetries in the XY, XZ and YZ planes (also called biaxial, orthorhombic or orthotropic symmetry), we have (D~n)=(-)L(DL,,),
withmeven
(3)
This surprising conclusion arises from the combination of eqs. ( 3) and (4), which, for an arbitrary value of L, with m and n even, yields the condition (D”,, ) = (Ok,,, _” ) . This relation implies that (13;” ) =O, so that the lowest non-vanishing order parameters of odd rank are (D&) =(D)_2_-2) = -(DL)=---(0:-z). This result can in fact be checked simply by a numerical evaluation of the order parameters on expressing the orientational distribution function in terms of an effective mean orienting potential U(Q) acting on the molecules [ 11, f(Q)=exp[
-U(O)/kT]
.
(5)
The simplest potential describing a system of biaxial molecules embedded in a biaxial phase, has the form
and for a molecule of the same symmetry we have (Dk,)=(-)L(Dk_,),
withneven.
(4)
In the case of uniaxial systems for which ( D$,, ) = (D& >do, or axially symmetric molecules where (Oh,, ) = ( Dho ) d,,,,, all the order parameters of odd rank L satisfying eqs. (3) and (4) vanish identically as expected [ 11. However, eqs. (3) and (4) imply that this need not be true in general. In particular, it can be easily shown that the rank L of the non-vanishing order parameters may take on both even and odd values in a biaxial system containing biaxial molecules.
+dD:o(W+D%W
I).
(6)
We note that the numerical computations of the order parameter using this potential requires a numerical integration only over the angle /3, as the integrals over the Euler angles (Yand y can be expressed analytically in terms of spherical Bessel functions [ 111. The order parameters obtained from this potential with arbitrary values for the parameters a, b and c reproduce al the symmetry relations given in table 1. Significantly, we find that the odd-rank order parameters with L> 1 do not vanish, provided
Table I f(wP, Y)
Operation u(XU)u( YZ)
I
lu(XZ)
a(xr)u(xz) = IU(YZ)
a( YZ)a(XZ) = la(XY) u(xy)u(yz) - la(xz) u(xy)a(xz)-lu(yz)
u(yz)a(xz)
= lu(xy)
~(Jw4xv) ~(XoJW)
u(XY)u(xz)
am @(YZMP) a( YZ)u(xz)
U(X-m(XY) a(xz)aJz) u(XZ)u(xz)
(Df”> (-)L’“(D’_,,) (-)L
c-Y(Dk.> (-)“‘“(G-“> (-)“(D”,-.> ( - )“ok > (- )“+“(Dfin) (-)L-m(Dk-.) (-)L-"+"(Df_.> (- )=+“(%m > CD%n> (-)“<04,-.> (-)=+“+“(Dk,,)
(-)“(D%,> (-)"-"(D4,_,>
351
Volume 180, number 4
CHEMICAL PHYSICS LETTERS
the coefficients b and c in eq. (6) are both non-zero. Finally, we note that the symmetry properties of the order parameters given in table 1 can be used to determine the form of the correlation functions (D/2,02 ) which appear in the description of fluorescence depolarization and Raman-scattering experiments [ 10,121. This is most conveniently done using the Clebsch-Gordan series expansion [ 4,5 1,
xC(22L;m-n)(D,L_k,_.),
(7)
where C denotes the Clebsch-Gordan coefficient [ 13 1. Furthermore, since C( 2 2 3; 1 1) = C( 2 2 3; - 1 - 1) ~0, we see from eq. (5) that the order parameter (D& ) = - (DLzz) does not enter the expansion of the correlation functions and thus cannot be determined experimentally.
Acknowledgement We would like to thank Professor P.J. Brussaard for many useful discussions.
352
24 May 1991
References [ 1 ] C. Zannoni, in: The molecular physics of liquid crystals, eds. G.R. Luckhurst and G.W. Gray (Academic Press, New York, 1979) p. 57. [2] M. van Gurp, H. van Langen, G. van Ginkel and Y.K. Levine, in: Polarized spectroscopy of ordered systems, eds. B. Samori and E.W. Thulstrup (Kluwer, Donlrecht, 1988) p. 455. [3] V.J. McBtierty, J. Chem. Phys 61 (1974) 872. [ 41 A.R. Edmonds, Angular momentum in quantum mechanics (Princeton Univ. Press., Princeton, 1960). [5] M.E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957). [6] E. Merzbacher, Quantum mechanics (Wiley, New York, 1970). [7] J.J. Fisz, Chem. Phys. 114 (1987) 165. [ 81 H.J. Bunge, C. Esling and J. Muller, Acta Cryst. A 37 ( 1981) 889. [9] L.C. Biedenham and J-D. Louck, Angular momentum in quantum physics (Addison-Wesley, Reading, 198 I ) . [ lo] P.S. Pershan, in: The molecular physics of liquid crystals, eds. G.R. Luckhurst and G.W. Gray (Academic Press, New York, 1979) p. 385. [ 111 M. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions (Dover, New York, 1964). [ 121 C. Zannoni, A. Arcioni and P. Cavatorta, Chem. Phys. Lipids 32 (1983) 179. [ 131 Particle Data Group, Rev. Mod. Phys. 48 ( 1967) S36.