- Email: [email protected]
Reorientational correlation functions and memory functions in the J-diffusion limit of the extended rotational diffusion model
Volume
22, number
CHEMICAL
2
REORIENTATIONAL IN THE J-DIFFUSION
CORRELATION
PHYSICS
LETTERS
FUNCTIONS
LIMIT OF THE EXTENDED
1 October
AND MEMORY
ROTATIONAL
1973
FUNCTIONS
DIFFUSION
MODEL*
T.E. EAGLES** and R.E.D. McCLUNG University of Alberta, Edmonton, Alberta, Canada T6G 2G2
of‘Chemistry.
Department
Received
26 June 1973
New methods for calculating free rotor memory functions and reorientational correlation diffusion model are presented. The methods are much more efficient than previous methods, in other areas.
functions
with the J-
and have applications
1. Introduction In a previous paper [ 11, analytical expressions for the reorientational correlation functions for symmetric top molecules undergoing rotational diffusion in the J-diffusion limit were presented. These formulae have proven of limited value in practical computation of the correlation functions since these calculations involve first calculating the corresponding spectral densities and computing the correlation functions by numerical Fourier transformation. Since the calculation of the spectral densities involves a double quadrature to perform the necessary integrations, the computation of correlation functions in this way is rather time-consuming and the overall accuracy and reliability of the calculated functions is of the order of 1~~2%~when 32 X 32 Gauss-Legendre quadrature is employed. Bliot et al. [2] have recognized that the memory function approach to the description of the reorientational motions of molecules in liquids is particularly useful for the extended J-diffusion model. Berne and Harp [3] have shown that the reorientational correlation function G(f) and its associated memory function K(t) satisfy the Volterra equation
k(r) =
jduK(u)
G(t-u)
,
(1)
where 6(f) is the time-derivative of G(t)***. Bliot et al. [2] were able to show that the memory function K,(t), for molecules undergoing J-diffusion, was related to the memory function K,,(t), for molecules undergoing free rotation, by KJ(f) = KI:R(r) exp (-f/7J)
,
* (2)
where rJ is the angular momentum correlation time. In their work, Bliot et al. [2] have used a McLaurin expansion for the free rotor correlation function GFR(t) in order to generate the McLaurin expansion for K,:,(t). In their computations ofJ-diffusion correlation functions. they used analytical expressions involving multiple time
* Research supported in part by the National Research Council of Canada under Operating Grant A5887. ** NRCC Graduate Fellow 1970-1973. *** The units of time used throughout are the reduced units t(kT/l,)“*, where (kT/I,) “’ is the average rotation the freely rotating molecules. See ref. [ 11.
414
frequency
of
Volume 22, number 2
1 October 1973
CHEMICAL PHYSICS LETTERS
integrations analogous to those used for linear [4] and spherical [S, 61 mol&ules. This method is limited to T.I 2 0.5 [6] because the number of required multiple integrals becomes excessively large for smaller 7~. In this article, we describe procedures for calculating free rotor memory functions and J-diffusion reorientational correlation functions from the associated memory functions. The procedures are simpler than the expansion techniques employed by Bliot et al. [2] and are considerably more efficient than the Fourier transformation of spectral densities method [I]. Possible applications and extensions of the theory are also discussed.
2. Theory The computation of the J-diffusion reorientational cofrelation function describing the reorientation of the kth irreducible component of a spherical tensor of rank j, GJ(I,k)(t) , for a given angular momentum correlation time TJ, involves first the calculation of the free rotor memory function Kfkk)(t) from the free rotor correlation funcusing eq. (2). tion G$“‘(t), then the calculation of GJ(i3k)(t) from Kyk’(l), which is obtained from K$“(t) 2.1. Calculation of Kfik)(t) frorn,,G#E)(t) G&/‘(t) and c$$‘( 1) can be computed from analytical formulae for the symmetric top [7] , spherical top and linear [8] molecules, using Gauss-Legendre quadrature where necessary to evaluate the integrals. In the following, we shall deal specifically with the symmetric top since application of the formalism to other cases is straightforward. The numerical procedure for generating KFik’(f) for t = 0, A, 2A, . . . . nA, . . . is based on a trapezoidal approximation [9] to the integral in eq. (1) and uses the relations Kfik)(0)
= $$‘)(O)
K;ik’(A)
= -(2/A)
= j( jt 1) f k2 5 , 6fkk’(A)
(3)
- G$hk’(A) K$,k’(0)
,
(4)
and @ik)(nA)
n-1
= -(2/A)
i;$t’(nA)
- G~~$zA)K$~)(O)
- 2c
G$‘((n-Z)A)K;kk’(lA)
.
(5)
I=1
In eq. (3), the inertial asymmetry parameter [ = (l,-IZ)/l,, ular to and parallel to the symmetry axis of the symmetric multiplication by exp (--t/rJ).
where Z, and Z, are the moments of inertia perpendictop molecule. Ky.k)(r) is obtained from Kbjik’(f) by
2.2. Calculation of GF k)(t) from Kp k)(t) The procedure for computing G_$Ak)(t) at f = 0, A, 2A, . . . . nA, . . . . given K.pk)(t) on trapezoidal approximations for the integration in eq. (1) and in the equation . Gyk’(nA)
= Gyk)((n-
1)A) t
j:
duC$:i,k)(,)
x Gyk’((n-
at these values oft, is based
1)A) + QA[c$~~‘(YzA) + kyk’((n-
l)A)]
(6)
(n-l)A
The equations
required to effect the computation i;:i.k’(o)
Gyk)(0)
= 1,
Gyk)(A)
= [l -aA2~yk)(A)]/[l
+k’((n-
1)A) = -tiyk’((n-2)A)
are
=o,
(7) t ta2K:i,k)(0)] - (2/A)[G$j>k)((,-
,
(8) 1)A) - Gyk’((n-2)A)]
,
(9) 415
Volume 22, number 2
CHEMICAL PHYSICS LETTERS
1 October 1973
1.0
0.8 0.6
K(“ FR“( t)
0.4
K”FR “‘(O)
0.2 0 -0.2
1.0
0.8 06
0.4
0.2 0 -0.2
I
,
1
I
I
I
I
I
1
I
0
1
2
3
4
5
6
7
a
9
10
t
Fig. 1. Free rotor memory functions for symmetric rotors with inertial asymmetries ( as indicated: (a) j = 1, k = 1 spherical tensor component; (b) i = 1, k = 0 spherical tensor component. * and
G$kk)(,A) = [(;:i,k) ((n- 1)A) + &AGyk) ((H- 1)~) - +A~IC~~)(~A) n-1 -iA2
416
c G~k)(lA)$j*k)((n-l)A)]/[l I=1
t $A2fCyk’(0)]
_
(10)
Volume 22, number 2
CHEMICAL PHYSICS LETTERS
1 October 1973
b
d
u
2.0
4.0
6.0 t
80
100
0
20
40
6.0
8.0
100
t
Fig. 2. J-diffusion reorientational correlation functions with r~ values as indicated: (a) t = 10, j = 1, k = 1; (b) t = 10, j = 1, k = 0; (c) t = -0.5, j = 1, k = 1; (d) [ = -0.5, j = 1, k = 0.
3. Calculations Representative free rotor memory functions for symmetric tops with different inertia1 asymmetries are shown in fig. 1. In fig. 2, the J-diffusion reorientational correlation functions for prolate and ablate symmetric tops with several values of 7~ are given*. The J-diffusion correlation functions calculated using the memory function method agree with the corresponding functions calculated by Fourier transformation of the spectra1 densities computed using eq. (28) of ref. [ 1 ] . The accuracy of the memory function method is greater than the Fourier transform approach provided the time interval A used in the calculations is small enough **. Furthermore, the computation time involved in the present procedure is less than 2% of the time required in the previous method. . 4. Discussion We have employed the trapezoidal approximation in the numerical solution of eq. (l), rather than using a more sophisticated numerical integration formula. In test calculations using Simpson’s rule formulae, it was found that the calculated correlation functions and memory functions were much less stable than in calculations performed with eqs. (3)-( 10) above. * FORTRAN IV program listings, decks and sample data are available from the authors on request. ** In all calculations presented here, A = 0.02 was used. Comparison with calculations with A = 0.01 indicated accuracies greater than 0.5% were achieved in the calculations using A = 0.02. 417
Volume 22, number 2
CHEMICAL PHYSICS LETTERS
1 October 1973
The simple numerical techniques described above for computing free rotor memory functions and J-diffusion reorientational correlation functions for symmetric top molecules can be applied in other problems dealing with time correlations. A semi-classical J-diffusion model for molecules with quantized rotational levels can be derived from the memory function approach. It is expected that such a model will be more rigorous than the previous approach [lo] based on Gordon’s semi-classical treatment of transferring line amplitude among individual spectral lines [ 1 l] . It has been pointed out [l] that the generalization of the extended rotational diffusion models to asymmetric top molecules was an unlikely possibility. However, in the memory function approach, only the asymmetric top correlation function for the free rotor is required. We are presently involved in both the semi-classical model and the asymmetric top problems. Another interesting possibility is the determination of “experimental” memory functions from spectral band shapes. As shown above, it is necessary to know G(t), G(t) and G(0) in order to compute K(t). Provided the normalized band shape I(w) represents only reorientational effects, the correlation function and its derivatives can be computed by the Fourier transformatiohs:
G(r)
=.-l j_m
dwI((w) exp (iwt) ,
c(t) = in-’
s
--m
dwwI(w)
exp (iwt) ,
and
G(0) = +rl
1 dww2Z(w). _m (11)
By the procedure described above, the memory function K(t) associated with G(t) can be computed. Such “experimental” memory functions may be of use in the analysis of band-shape data in terms of molecular motion.
References [l] [2] [3] [4] [5] [6] [7] [8] [9]
[ 1O] [ 111
418
R.E.D. McClung, J. Chem. Phys. 57 (1972) 5478. F. Bliot, C. Abbar and E. Constant, Mol. Phys. 24 (1972) 241. B.J. Berne and G.D. Harp, Advan. Chem. Phys. 17 (1970) 63. R.G. Gordon, J. Chem. Phys. 44 (1966) 1830. R.E.D. McClung, J. Chem. Phys. 51 (1969) 3842; 54 (1971) 3248; R.E.D. McClung and H. Versmold, J. Chem. Phys. 57 (1972) 2596. R.E.D. McClung, J. Chem. Phys. 55 (1971) 3459. A.G.St. Pierre and W.A. Steele, Phys. Rev. 184 (1969) 172. W.A. Steele, J. Chem. Phys. 38 (1963) 2404. S.G. Mikhlin and K.L. Smolitskiy, Approximate methods for solution of differential and integral equations (American Elsevier, New York, 1967) p. 285. T.E. Eagles and R.E.D. McClung, J. Chem. Phys. (July 1, 1973), to be published. R.D. Gordon, J. Chem. Phys. 45 (1966) 1649; Advan. Magnetic Reson. 3 (1968) 1; W.B. Neilsen and R.G. Gordon, J. Chem. Phys. 58 (1973) 4131,4149.
22, number
CHEMICAL
2
REORIENTATIONAL IN THE J-DIFFUSION
CORRELATION
PHYSICS
LETTERS
FUNCTIONS
LIMIT OF THE EXTENDED
1 October
AND MEMORY
ROTATIONAL
1973
FUNCTIONS
DIFFUSION
MODEL*
T.E. EAGLES** and R.E.D. McCLUNG University of Alberta, Edmonton, Alberta, Canada T6G 2G2
of‘Chemistry.
Department
Received
26 June 1973
New methods for calculating free rotor memory functions and reorientational correlation diffusion model are presented. The methods are much more efficient than previous methods, in other areas.
functions
with the J-
and have applications
1. Introduction In a previous paper [ 11, analytical expressions for the reorientational correlation functions for symmetric top molecules undergoing rotational diffusion in the J-diffusion limit were presented. These formulae have proven of limited value in practical computation of the correlation functions since these calculations involve first calculating the corresponding spectral densities and computing the correlation functions by numerical Fourier transformation. Since the calculation of the spectral densities involves a double quadrature to perform the necessary integrations, the computation of correlation functions in this way is rather time-consuming and the overall accuracy and reliability of the calculated functions is of the order of 1~~2%~when 32 X 32 Gauss-Legendre quadrature is employed. Bliot et al. [2] have recognized that the memory function approach to the description of the reorientational motions of molecules in liquids is particularly useful for the extended J-diffusion model. Berne and Harp [3] have shown that the reorientational correlation function G(f) and its associated memory function K(t) satisfy the Volterra equation
k(r) =
jduK(u)
G(t-u)
,
(1)
where 6(f) is the time-derivative of G(t)***. Bliot et al. [2] were able to show that the memory function K,(t), for molecules undergoing J-diffusion, was related to the memory function K,,(t), for molecules undergoing free rotation, by KJ(f) = KI:R(r) exp (-f/7J)
,
* (2)
where rJ is the angular momentum correlation time. In their work, Bliot et al. [2] have used a McLaurin expansion for the free rotor correlation function GFR(t) in order to generate the McLaurin expansion for K,:,(t). In their computations ofJ-diffusion correlation functions. they used analytical expressions involving multiple time
* Research supported in part by the National Research Council of Canada under Operating Grant A5887. ** NRCC Graduate Fellow 1970-1973. *** The units of time used throughout are the reduced units t(kT/l,)“*, where (kT/I,) “’ is the average rotation the freely rotating molecules. See ref. [ 11.
414
frequency
of
Volume 22, number 2
1 October 1973
CHEMICAL PHYSICS LETTERS
integrations analogous to those used for linear [4] and spherical [S, 61 mol&ules. This method is limited to T.I 2 0.5 [6] because the number of required multiple integrals becomes excessively large for smaller 7~. In this article, we describe procedures for calculating free rotor memory functions and J-diffusion reorientational correlation functions from the associated memory functions. The procedures are simpler than the expansion techniques employed by Bliot et al. [2] and are considerably more efficient than the Fourier transformation of spectral densities method [I]. Possible applications and extensions of the theory are also discussed.
2. Theory The computation of the J-diffusion reorientational cofrelation function describing the reorientation of the kth irreducible component of a spherical tensor of rank j, GJ(I,k)(t) , for a given angular momentum correlation time TJ, involves first the calculation of the free rotor memory function Kfkk)(t) from the free rotor correlation funcusing eq. (2). tion G$“‘(t), then the calculation of GJ(i3k)(t) from Kyk’(l), which is obtained from K$“(t) 2.1. Calculation of Kfik)(t) frorn,,G#E)(t) G&/‘(t) and c$$‘( 1) can be computed from analytical formulae for the symmetric top [7] , spherical top and linear [8] molecules, using Gauss-Legendre quadrature where necessary to evaluate the integrals. In the following, we shall deal specifically with the symmetric top since application of the formalism to other cases is straightforward. The numerical procedure for generating KFik’(f) for t = 0, A, 2A, . . . . nA, . . . is based on a trapezoidal approximation [9] to the integral in eq. (1) and uses the relations Kfik)(0)
= $$‘)(O)
K;ik’(A)
= -(2/A)
= j( jt 1) f k2 5 , 6fkk’(A)
(3)
- G$hk’(A) K$,k’(0)
,
(4)
and @ik)(nA)
n-1
= -(2/A)
i;$t’(nA)
- G~~$zA)K$~)(O)
- 2c
G$‘((n-Z)A)K;kk’(lA)
.
(5)
I=1
In eq. (3), the inertial asymmetry parameter [ = (l,-IZ)/l,, ular to and parallel to the symmetry axis of the symmetric multiplication by exp (--t/rJ).
where Z, and Z, are the moments of inertia perpendictop molecule. Ky.k)(r) is obtained from Kbjik’(f) by
2.2. Calculation of GF k)(t) from Kp k)(t) The procedure for computing G_$Ak)(t) at f = 0, A, 2A, . . . . nA, . . . . given K.pk)(t) on trapezoidal approximations for the integration in eq. (1) and in the equation . Gyk’(nA)
= Gyk)((n-
1)A) t
j:
duC$:i,k)(,)
x Gyk’((n-
at these values oft, is based
1)A) + QA[c$~~‘(YzA) + kyk’((n-
l)A)]
(6)
(n-l)A
The equations
required to effect the computation i;:i.k’(o)
Gyk)(0)
= 1,
Gyk)(A)
= [l -aA2~yk)(A)]/[l
+k’((n-
1)A) = -tiyk’((n-2)A)
are
=o,
(7) t ta2K:i,k)(0)] - (2/A)[G$j>k)((,-
,
(8) 1)A) - Gyk’((n-2)A)]
,
(9) 415
Volume 22, number 2
CHEMICAL PHYSICS LETTERS
1 October 1973
1.0
0.8 0.6
K(“ FR“( t)
0.4
K”FR “‘(O)
0.2 0 -0.2
1.0
0.8 06
0.4
0.2 0 -0.2
I
,
1
I
I
I
I
I
1
I
0
1
2
3
4
5
6
7
a
9
10
t
Fig. 1. Free rotor memory functions for symmetric rotors with inertial asymmetries ( as indicated: (a) j = 1, k = 1 spherical tensor component; (b) i = 1, k = 0 spherical tensor component. * and
G$kk)(,A) = [(;:i,k) ((n- 1)A) + &AGyk) ((H- 1)~) - +A~IC~~)(~A) n-1 -iA2
416
c G~k)(lA)$j*k)((n-l)A)]/[l I=1
t $A2fCyk’(0)]
_
(10)
Volume 22, number 2
CHEMICAL PHYSICS LETTERS
1 October 1973
b
d
u
2.0
4.0
6.0 t
80
100
0
20
40
6.0
8.0
100
t
Fig. 2. J-diffusion reorientational correlation functions with r~ values as indicated: (a) t = 10, j = 1, k = 1; (b) t = 10, j = 1, k = 0; (c) t = -0.5, j = 1, k = 1; (d) [ = -0.5, j = 1, k = 0.
3. Calculations Representative free rotor memory functions for symmetric tops with different inertia1 asymmetries are shown in fig. 1. In fig. 2, the J-diffusion reorientational correlation functions for prolate and ablate symmetric tops with several values of 7~ are given*. The J-diffusion correlation functions calculated using the memory function method agree with the corresponding functions calculated by Fourier transformation of the spectra1 densities computed using eq. (28) of ref. [ 1 ] . The accuracy of the memory function method is greater than the Fourier transform approach provided the time interval A used in the calculations is small enough **. Furthermore, the computation time involved in the present procedure is less than 2% of the time required in the previous method. . 4. Discussion We have employed the trapezoidal approximation in the numerical solution of eq. (l), rather than using a more sophisticated numerical integration formula. In test calculations using Simpson’s rule formulae, it was found that the calculated correlation functions and memory functions were much less stable than in calculations performed with eqs. (3)-( 10) above. * FORTRAN IV program listings, decks and sample data are available from the authors on request. ** In all calculations presented here, A = 0.02 was used. Comparison with calculations with A = 0.01 indicated accuracies greater than 0.5% were achieved in the calculations using A = 0.02. 417
Volume 22, number 2
CHEMICAL PHYSICS LETTERS
1 October 1973
The simple numerical techniques described above for computing free rotor memory functions and J-diffusion reorientational correlation functions for symmetric top molecules can be applied in other problems dealing with time correlations. A semi-classical J-diffusion model for molecules with quantized rotational levels can be derived from the memory function approach. It is expected that such a model will be more rigorous than the previous approach [lo] based on Gordon’s semi-classical treatment of transferring line amplitude among individual spectral lines [ 1 l] . It has been pointed out [l] that the generalization of the extended rotational diffusion models to asymmetric top molecules was an unlikely possibility. However, in the memory function approach, only the asymmetric top correlation function for the free rotor is required. We are presently involved in both the semi-classical model and the asymmetric top problems. Another interesting possibility is the determination of “experimental” memory functions from spectral band shapes. As shown above, it is necessary to know G(t), G(t) and G(0) in order to compute K(t). Provided the normalized band shape I(w) represents only reorientational effects, the correlation function and its derivatives can be computed by the Fourier transformatiohs:
G(r)
=.-l j_m
dwI((w) exp (iwt) ,
c(t) = in-’
s
--m
dwwI(w)
exp (iwt) ,
and
G(0) = +rl
1 dww2Z(w). _m (11)
By the procedure described above, the memory function K(t) associated with G(t) can be computed. Such “experimental” memory functions may be of use in the analysis of band-shape data in terms of molecular motion.
References [l] [2] [3] [4] [5] [6] [7] [8] [9]
[ 1O] [ 111
418
R.E.D. McClung, J. Chem. Phys. 57 (1972) 5478. F. Bliot, C. Abbar and E. Constant, Mol. Phys. 24 (1972) 241. B.J. Berne and G.D. Harp, Advan. Chem. Phys. 17 (1970) 63. R.G. Gordon, J. Chem. Phys. 44 (1966) 1830. R.E.D. McClung, J. Chem. Phys. 51 (1969) 3842; 54 (1971) 3248; R.E.D. McClung and H. Versmold, J. Chem. Phys. 57 (1972) 2596. R.E.D. McClung, J. Chem. Phys. 55 (1971) 3459. A.G.St. Pierre and W.A. Steele, Phys. Rev. 184 (1969) 172. W.A. Steele, J. Chem. Phys. 38 (1963) 2404. S.G. Mikhlin and K.L. Smolitskiy, Approximate methods for solution of differential and integral equations (American Elsevier, New York, 1967) p. 285. T.E. Eagles and R.E.D. McClung, J. Chem. Phys. (July 1, 1973), to be published. R.D. Gordon, J. Chem. Phys. 45 (1966) 1649; Advan. Magnetic Reson. 3 (1968) 1; W.B. Neilsen and R.G. Gordon, J. Chem. Phys. 58 (1973) 4131,4149.