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On the theory of generalized rotational diffusion models
Volume 41. number 2
ON THE THECiRY OF GENERALIZED DKA.
15 July 1976
CHEMICAL PHYSICS LETTERS
ROTATIONAL
DIFFUSION
MODELS
McMAHON
Department of Physics, University of Tasmania, Hobart
7OOI, Australia
Received 9 February 1976 Revised manuscript received 29 I4farch 1976
A new Volterra type equa?ion is described from which correlation functions can be derived. A “memory kernel”, in contrast to the memory function is shown to arise more naturally out of the extended diffusion models suggested by Gordon.
1. Introduction Gordon [ 11 has proposed classical models for orientational relaxation in gases and liquids. He considered two main models, J-diffusion and M-diffusion. Their time dependent correlation functions have been cakulated either by performing multiple time integrals [l] or by a Fourier inversion technique [2] _ Fixman and Rider [3] have deveIoped coupled integrodifferential equations for orientational relaxation and have recently applied them to the rough spheres model [4] _ More recently, Eliot et al. [5] and Bliot and Constant [6] have described a method of calculating time correlation functions from a Volterra equation for J- and M-diffusion. Eagles and McCIung [7] have developed an efficient computing technique wherein the memory function is calculated numerically and employed to solve the Volterra equation_ Special advantages are to be had by empIoying a Volterra equation method, however the memory function technique is not easily gczrzlked to models that are perhaps more realistic than J-and M-diffusion for which the inadequacies are well docu,nented [8]. Below we derive a new Volterra equation which does treat more general models and is in practice simpler than the memory function apprcach.
2. Description of molecular orientations Let Jt’ and x’ be the spherical polar angles in the laboratory frame of reference of some vector fixed in a symmetric top molecule. Using the notation of Rose [9] for the Wigner matrices, our general tensorial vector can be written as
(1)
\‘Ib whici~ is 2 column vector of2 C+ 1 elements. Our rotational
and orientational relaxation theory is greatly simplified if changes in the lab-oratory frame orientation of the molecule is specified in body-fixed co-ordinates. In order to relate the angks 3/ and x specifying the body-fixed direction of the vector of interest to the angles I$’ and x’ we may introduce angles (L, p and 7 such that 378
CHEMKAL PHYSICS LETTERS
Volume 41, numb&r2
Dq$‘-$i,-
x’,o)=mGvY)Dz(ll
15 July 1976
-f n,-x,0),
(2)
where
Here a and 0 are the spherical polar angles of a reorientation axis in the body-fured frame and 7 is the angle of rotation between the body-fixed and laboratory framesThese angles are naturally changing continuously between collisions due to free molecular rotation. In p&tice we are interested in the time dependent evolution of the molecular symmetry axis. Thus, between collisions the motion of this-axis is determined by the precession rate Qt. Let &,, 6; and I&, f3* specify the initial directions of the angular momentum vector in the space-fixed and the body-fxed frames respectively. In some interval tl between some initial time and the time of the next collision the angles specifying the body orientation change from cro, PO and r. to ol, fll and yl according to the relation (3)
irr(~,,p,,~l)=a1c~b,eb,~20tl)bl(oro,~o,~o)=~z(aro,~o,~o)~I(~o,eO,~20tl). If there is a sequence of n collisions at times tl, t2, n-1
_.. fn
then eq. (3) can be generalized to
The product reads from left to right for increasing j_ After averaging over all initial orientations the sought after partia1 correlation function for n collisions can be written as
of the molecule,
f2
--.~~‘~~,,~,J-‘&@&)W,Jooh (4) 0 0 where the brackets ( ) indicate a remaining average over the probability distribution for the sequence of angular momentum variables Qz,, 80, Qo; -..; 9,) 8,) C2,. The correlation function is obtained by weighting eq. (4) over the Poisson distrrbution P,(r) = (1 /n!) (i/~#e-‘~e. Xj-
3. The generalized Volterra equation For brevity let us replace (of, Bi, Qi) by the variable I’i and let the equilibrium distribution function be F(l?) such that we have the normalisation condition J F@?) dI’= 1. The conditional probability of the sequence I’u, rl, _. . r, shall be denoted as JR (I’u, J?, , - _- r,) which has the markoffian property JJ&),
l-i, --a
r,~>=J,_rcro,r~,-..r*l>Kcr,lJrn),
(5)
where K is the collision kernel. In particular we have
J&y
~lcro~rl~=wo~rl)~
We shall define the average gn (i;,) ~~C&)=J-..@&,r,
9”.
= 1.
of any function X,, ro,
___ IJ,) as n-1
(6)
and the final average is defined as (X,,
=J&
(In)
dr-,. 379
VO~GZW
4 1,
where we
ncirnbr 2
CHGITCAL PHYSICS LETTERS
maydefine the matrix DJ
having replaced the dummy variables r, by I’. There is a convolution identity for G’(r, n) if n b2.
This is just
where the integration over 71 is between 0 and I$~~ r,, _f _ Further we have the special cases
For n > 1 we have the property t
G’(t,n)=SG’(~-r,n-l)B’(~,B,WT)dr, 0 where one has replaced rn by the dumuny variable T. Thus on employing the defmition equation (6) and eq. (5) we have fornB 1, t
~‘(r,n,~)=~~K(r:r)~~(t-~,n-1,r’>ir~(#,~,~~) dI-“dT, 0
and there is the special case for n = 0
G$,o,r)=ZP~w,szt)F(l?). Directly substituting for these into eq. (9) gives the sought after Voiterra equation,
t
(W
We may refer to K(I”, r)@@,8,sZr) as a kind of “memory kernel”. In general we have coupled integral equations for the (21-f- 1)2 matrix elements *A,(& r). The practical use of the memory function technique for stochastic models has been developed in detail only for the J-diffusion and M-diffusion models ES-71 .-For J-diffusion the memory function equation is 380
Voiume 41, number 2
CHEMICAL PHYSICS LEITERS
15 July 1976
where &f!(r) is the memory function. BIiot et al. [SJ have shown that M’(T) = .-+O
M:rr, (T)
(12)
and so in order to solve eq. (11) it is necessary to numerically calculate M/= (7) from dC&Jt)ldt
jM-&.&)C&Jt-~Jck,
= -
(13)
0
where the free rotation correlation function can be written in the notation of Rose [9]-as &&)
=S$l
( e’r%&(e)P>,
where ( ) denotes the average over the equilibrium distribution for 0 and Sz. Eq. (10) however offers a more direct procedure of calculating Cl(r). For J-diffusion the kernel is K(i+, r) = F(r), representing thermalisation of the angular momentum by collisions. Eq. (10) can be integrated over aI1 lT to give (14) where X’(f) =
j&t,
i-‘) di’,
X~~(t)=.SD’(~,B,~t)F(r)dr.
As F(r) is independent of Cp,eq. (14) is (21 f 1) Volterra equations for the Xi, (t)_ Writing C’(t) = e-“QX& we obtain
(t)
c
C’(t) = eeti7@ C&,Ct)+(l/ro)
jkqhoC&_(r)
C’(r-r)dT.
(15)
0
‘This equation has an advantage over eq. (11) since the latter requires fiI&_(i) to be ob;ained from eq. (13) whkreas eq. (15) can be solved directly. Also, on applying the Laplace transform to eqs. (11) and (13) and substituting for the Laplace transform of Cl(t) from eq. (15) one readiIy establishes eq. (12). A simiIar situation applies to M-diffusion. A memory function M*(T, 52) is defined where d&t,
a)ldt = -jM$-,
SZ)C’(?--7,Q)dr
0
and C’(t, S2) = e-r’To~~&,
(t, r) sin 8 d6 d@.
We also have M’(T, SL) = e- rh
M&e (7, St)
(17)
and M~‘,(T, S2) may be calculated analogously to eq. (13) with C&_(i) replaced by C&e (t, a) where tl
381
VoIumc 41, number 2
CHEMICAL PHYSICS LETTERS
1-S July 1976
ZW.model corresponds to coI&ions which thcrmalise the direction of the angular momentum fi.TJben co: (10) is ir&&ateci over ali 9 and Q we obtain on further manipulation,
but do not change
C’(t) = jC’(t, S-2) d_S2. This equation can be solved directly unlike eq. (16). As before;eq. (17) can be proved from eq. (18) by Lap&e transforms. It is conceivable that a general&&ion of eqs. (11) and (16) is possible for any stochastic model. This would require introduction of MI@, I”, I’) where f dC’(t, I’)/dt = -I M(L If, r ) C’ (t - 7, I“) dI+dr. J-Y !I However since properties of M(r, I”, I’) necessary to solve this equation can only be derived from eq. (IO), we con&de that the memory function technique has only limited application M-diffusion) and even there eq. (IO) is more advantageous.
[l] R.G. Gordon, 3. Chem. Phys. 44 (1966) 1830. [Zj A.G. St. Pierre and W.A. Steele, J. Chem. Phys 57 (1972) 4638.
[3j M. Fivman and K. Rider, J. Chem. Phys. 51(1969) [4] [S] (61 [7] [Sj 191
382
2425. I<. Rider and M. Fixman, J. Chem. Phys. 57 (1972) 2548. F. Bliot, C. Abbar and E. Constant, Mol. Phys. 24 (1972) 241. F. Bliot and E. Constant, Chem. Phys. Letters 18 (1973) 253. T-E. E&es and R-E-D. McClung, Chem. Phys. Letters 22 (1973) 414. B.J. O’Dell acd B.J. Berne, J. Chem. Phys. 63 (1975) 2376. &f-E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957).
to very special models (i.e. 5- and
ON THE THECiRY OF GENERALIZED DKA.
15 July 1976
CHEMICAL PHYSICS LETTERS
ROTATIONAL
DIFFUSION
MODELS
McMAHON
Department of Physics, University of Tasmania, Hobart
7OOI, Australia
Received 9 February 1976 Revised manuscript received 29 I4farch 1976
A new Volterra type equa?ion is described from which correlation functions can be derived. A “memory kernel”, in contrast to the memory function is shown to arise more naturally out of the extended diffusion models suggested by Gordon.
1. Introduction Gordon [ 11 has proposed classical models for orientational relaxation in gases and liquids. He considered two main models, J-diffusion and M-diffusion. Their time dependent correlation functions have been cakulated either by performing multiple time integrals [l] or by a Fourier inversion technique [2] _ Fixman and Rider [3] have deveIoped coupled integrodifferential equations for orientational relaxation and have recently applied them to the rough spheres model [4] _ More recently, Eliot et al. [5] and Bliot and Constant [6] have described a method of calculating time correlation functions from a Volterra equation for J- and M-diffusion. Eagles and McCIung [7] have developed an efficient computing technique wherein the memory function is calculated numerically and employed to solve the Volterra equation_ Special advantages are to be had by empIoying a Volterra equation method, however the memory function technique is not easily gczrzlked to models that are perhaps more realistic than J-and M-diffusion for which the inadequacies are well docu,nented [8]. Below we derive a new Volterra equation which does treat more general models and is in practice simpler than the memory function apprcach.
2. Description of molecular orientations Let Jt’ and x’ be the spherical polar angles in the laboratory frame of reference of some vector fixed in a symmetric top molecule. Using the notation of Rose [9] for the Wigner matrices, our general tensorial vector can be written as
(1)
\‘Ib whici~ is 2 column vector of2 C+ 1 elements. Our rotational
and orientational relaxation theory is greatly simplified if changes in the lab-oratory frame orientation of the molecule is specified in body-fixed co-ordinates. In order to relate the angks 3/ and x specifying the body-fixed direction of the vector of interest to the angles I$’ and x’ we may introduce angles (L, p and 7 such that 378
CHEMKAL PHYSICS LETTERS
Volume 41, numb&r2
Dq$‘-$i,-
x’,o)=mGvY)Dz(ll
15 July 1976
-f n,-x,0),
(2)
where
Here a and 0 are the spherical polar angles of a reorientation axis in the body-fured frame and 7 is the angle of rotation between the body-fixed and laboratory framesThese angles are naturally changing continuously between collisions due to free molecular rotation. In p&tice we are interested in the time dependent evolution of the molecular symmetry axis. Thus, between collisions the motion of this-axis is determined by the precession rate Qt. Let &,, 6; and I&, f3* specify the initial directions of the angular momentum vector in the space-fixed and the body-fxed frames respectively. In some interval tl between some initial time and the time of the next collision the angles specifying the body orientation change from cro, PO and r. to ol, fll and yl according to the relation (3)
irr(~,,p,,~l)=a1c~b,eb,~20tl)bl(oro,~o,~o)=~z(aro,~o,~o)~I(~o,eO,~20tl). If there is a sequence of n collisions at times tl, t2, n-1
_.. fn
then eq. (3) can be generalized to
The product reads from left to right for increasing j_ After averaging over all initial orientations the sought after partia1 correlation function for n collisions can be written as
of the molecule,
f2
--.~~‘~~,,~,J-‘&@&)W,Jooh (4) 0 0 where the brackets ( ) indicate a remaining average over the probability distribution for the sequence of angular momentum variables Qz,, 80, Qo; -..; 9,) 8,) C2,. The correlation function is obtained by weighting eq. (4) over the Poisson distrrbution P,(r) = (1 /n!) (i/~#e-‘~e. Xj-
3. The generalized Volterra equation For brevity let us replace (of, Bi, Qi) by the variable I’i and let the equilibrium distribution function be F(l?) such that we have the normalisation condition J F@?) dI’= 1. The conditional probability of the sequence I’u, rl, _. . r, shall be denoted as JR (I’u, J?, , - _- r,) which has the markoffian property JJ&),
l-i, --a
r,~>=J,_rcro,r~,-..r*l>Kcr,lJrn),
(5)
where K is the collision kernel. In particular we have
J&y
~lcro~rl~=wo~rl)~
We shall define the average gn (i;,) ~~C&)=J-..@&,r,
9”.
= 1.
of any function X,, ro,
___ IJ,) as n-1
(6)
and the final average is defined as (X,,
=J&
(In)
dr-,. 379
VO~GZW
4 1,
where we
ncirnbr 2
CHGITCAL PHYSICS LETTERS
maydefine the matrix DJ
having replaced the dummy variables r, by I’. There is a convolution identity for G’(r, n) if n b2.
This is just
where the integration over 71 is between 0 and I$~~ r,, _f _ Further we have the special cases
For n > 1 we have the property t
G’(t,n)=SG’(~-r,n-l)B’(~,B,WT)dr, 0 where one has replaced rn by the dumuny variable T. Thus on employing the defmition equation (6) and eq. (5) we have fornB 1, t
~‘(r,n,~)=~~K(r:r)~~(t-~,n-1,r’>ir~(#,~,~~) dI-“dT, 0
and there is the special case for n = 0
G$,o,r)=ZP~w,szt)F(l?). Directly substituting for these into eq. (9) gives the sought after Voiterra equation,
t
(W
We may refer to K(I”, r)@@,8,sZr) as a kind of “memory kernel”. In general we have coupled integral equations for the (21-f- 1)2 matrix elements *A,(& r). The practical use of the memory function technique for stochastic models has been developed in detail only for the J-diffusion and M-diffusion models ES-71 .-For J-diffusion the memory function equation is 380
Voiume 41, number 2
CHEMICAL PHYSICS LEITERS
15 July 1976
where &f!(r) is the memory function. BIiot et al. [SJ have shown that M’(T) = .-+O
M:rr, (T)
(12)
and so in order to solve eq. (11) it is necessary to numerically calculate M/= (7) from dC&Jt)ldt
jM-&.&)C&Jt-~Jck,
= -
(13)
0
where the free rotation correlation function can be written in the notation of Rose [9]-as &&)
=S$l
( e’r%&(e)P>,
where ( ) denotes the average over the equilibrium distribution for 0 and Sz. Eq. (10) however offers a more direct procedure of calculating Cl(r). For J-diffusion the kernel is K(i+, r) = F(r), representing thermalisation of the angular momentum by collisions. Eq. (10) can be integrated over aI1 lT to give (14) where X’(f) =
j&t,
i-‘) di’,
X~~(t)=.SD’(~,B,~t)F(r)dr.
As F(r) is independent of Cp,eq. (14) is (21 f 1) Volterra equations for the Xi, (t)_ Writing C’(t) = e-“QX& we obtain
(t)
c
C’(t) = eeti7@ C&,Ct)+(l/ro)
jkqhoC&_(r)
C’(r-r)dT.
(15)
0
‘This equation has an advantage over eq. (11) since the latter requires fiI&_(i) to be ob;ained from eq. (13) whkreas eq. (15) can be solved directly. Also, on applying the Laplace transform to eqs. (11) and (13) and substituting for the Laplace transform of Cl(t) from eq. (15) one readiIy establishes eq. (12). A simiIar situation applies to M-diffusion. A memory function M*(T, 52) is defined where d&t,
a)ldt = -jM$-,
SZ)C’(?--7,Q)dr
0
and C’(t, S2) = e-r’To~~&,
(t, r) sin 8 d6 d@.
We also have M’(T, SL) = e- rh
M&e (7, St)
(17)
and M~‘,(T, S2) may be calculated analogously to eq. (13) with C&_(i) replaced by C&e (t, a) where tl
381
VoIumc 41, number 2
CHEMICAL PHYSICS LETTERS
1-S July 1976
ZW.model corresponds to coI&ions which thcrmalise the direction of the angular momentum fi.TJben co: (10) is ir&&ateci over ali 9 and Q we obtain on further manipulation,
but do not change
C’(t) = jC’(t, S-2) d_S2. This equation can be solved directly unlike eq. (16). As before;eq. (17) can be proved from eq. (18) by Lap&e transforms. It is conceivable that a general&&ion of eqs. (11) and (16) is possible for any stochastic model. This would require introduction of MI@, I”, I’) where f dC’(t, I’)/dt = -I M(L If, r ) C’ (t - 7, I“) dI+dr. J-Y !I However since properties of M(r, I”, I’) necessary to solve this equation can only be derived from eq. (IO), we con&de that the memory function technique has only limited application M-diffusion) and even there eq. (IO) is more advantageous.
[l] R.G. Gordon, 3. Chem. Phys. 44 (1966) 1830. [Zj A.G. St. Pierre and W.A. Steele, J. Chem. Phys 57 (1972) 4638.
[3j M. Fivman and K. Rider, J. Chem. Phys. 51(1969) [4] [S] (61 [7] [Sj 191
382
2425. I<. Rider and M. Fixman, J. Chem. Phys. 57 (1972) 2548. F. Bliot, C. Abbar and E. Constant, Mol. Phys. 24 (1972) 241. F. Bliot and E. Constant, Chem. Phys. Letters 18 (1973) 253. T-E. E&es and R-E-D. McClung, Chem. Phys. Letters 22 (1973) 414. B.J. O’Dell acd B.J. Berne, J. Chem. Phys. 63 (1975) 2376. &f-E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957).
to very special models (i.e. 5- and