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Theory of electron spin alignment through nonradiative processes in naphthalene and anthracene
.Vc+um&7; number 5
..:
CHEMICAL
PHYSICS
1 December
LETTERS
1970
:i : :. :: :.
:
THEORY
.‘I
OF
THR0UG.H IN
ELECTRON
SPiN
NONRADIATIVE
NAPHTHALENE
Department of Chemistyv,
AND
ALIGNMENT PROCESSES
ANTHRACENE
B. R. HENRY Unirersil~ of Maniloba,
+
Winnipeg.
Canada
and W. SIEBRAND Dirisi&
o.f Chemistry.
National
Research
CounciL of Canada.
Ottaaya.
Canada
Recejved 6 October 1970
The three cornponcntsof the f!$plet-to-ground state nonrndintive rate constants of nnphthaleneand XIIthrncene are calculated thcorctir.!ly and compnred with experiment. ! 1
At very low temperatures (T Z 1%) the triplet state of an aromatic molecule can behave as three independent states, the rate of interconversion (spin relaxation) being slow compared to the rate of.radiative and nonradiative transitions to and from singlet states. In general rate constants for radiative and nonradiative processes involving triplets will then be different for the three components. so that these processes producf? spin alignment [ 1I. Van der Waals and coworkers [Z, 3) were the first to give an extensive discussion of this phenomenon with special reference to singlet-to-triplet intersystem crossing in heterocyclic systems. There the situation is relatively simple. the favored crossing being between an nz* and a zz* state bacause of the strong predominance of such coupling over that between two x-type states. After determining on this basis which triplet state is directly involved in the
have been made for naphthalene [5 1and anthracene [6] where the rates of radiationless deactivation of the lowest triplet state T1 have been determined for each of the 3 components of Tl. This note is principally concerned vith an analysis of the differences between these rites. Previously, we distinguished three mechanisms of intersystem crossing in aromatic hydrocarbons [4]: (1) a purely e!ectronic mechanism in which the spin-orbit operator couples two ii?i states in the nuclear equilibrium configuration; (2) a vibronic mechanism in which the coupling of type (1) is induced by in-plane vibrations; and (3) a vibronic mechanism involving spin-orbit coupling between xi and zcr states induced by outof-plane vibrations. The corresponding matrix
(1)
transition, the spin component(s) to be predominantly populated can be found by symmetry arguments. -The spin alignment is presumably con-
served during the vibrational and:or electronic relaxation to the lower vibrational levels of the lowest triplet state, on which the actual measurements are made. In aromatic hydrocarbons.
(2)
however, no nr* states are available. It turns out that there are competitive mechanisms for intersystem crossing 141which are, in principle, parti,ally distin@ishable on the basis of spin POlarization measurements. Such measurements -1 . * Issued RS NRCC No. 11635.
(3) where
533
Volume
7. number
I _1 December I970
CRRhIICAL PHYSICSLETTERS
5
all decay rate constant (k,.+ k,, + k,) in naphthalene. -To calculate k(S) = Kx + k,,: .ye note.tii+xt the matrix elements L&j and the energies LIE~,,~in eq. (3) ‘arb_esienti_ally~ the same as those we calculated.previously [7] for the radiative transitions T1 2 St)_ The vibronic integrals in (3) are readily calculatedIf we restrict ourselves to CH out-of-plane modes which can be shown [4] to yield the dominant contribution to mechanism (3). The amplitude of atom n in mode &_is written as Qkn = RC.fB&. where RcH,= 1.08A and f?k,r is the angle between the CH bond and the molecular plane. Expanding in atomic integrals we obtain
is a vibrational overlap integral. 81,, the spin-orbit coupling operator. pk. tihm’ and 81, are the reduced mass. angular frequency (in state GJ~?)and normal coordinate of mode k. 00 is the nuclear equilibrium configuration. the %'s are electronic wavefunctions and AE is an electronic energy difference. We have shown that mechanism (2) is mainly induced by deviations from Q. (Herzberg-Teller vibronic coupling) and vi; spin-orbit coupling, contrary to mechanism (3) which is mainly induced by the nonzero nuclear momenta (Born-Oppenheimer coupling) and ho spin-orbit coupling. Symmetry arguments then dictate that mechanism (2) (de)populates the outof-plane spin component (7z) of the triplet state. as does mechanism (l), and mechanism (3) the inplane components rx and r>,. Thus a measurement of the nonradiative rate constants in a spin-aligned system allows an immediate distinction to be made-between mechanisms (1) and (2) on the one hand and mechanism (3) on the other. In measurements on naphthalene [5] and anthracene [6] it was found that the observed triplet decay rate constants are in the order k.,- .. k, >> kz. It is known that radiative decay contributes mainly to k, 17.81 and is slow enough to be ignored for the present purpose [5.9]. so that the above rate constants may be assigned to TlSO crossing. Since mechanism (1) is forbidden by symmetry here. the observation that mechanism (3) is dok.\- + ky >> k, indicates minating mechanism (2). A similar conclusion has been reached [4. lo] on the basis of the observed effect of partial deuteration on the over-
s,rm
[41 Ta,, (k) = R&S*Jg where s’
elements
= (l*sH)-l~~.
SH = 3-1’2+i+21/~o]L)
h being an 1s hydrogen
orbital, and s and p. 2s and 2p Slater orbitals; po is directed along the CH bond. The coefficients Bnk in (4) depend on the K-MO coefficients Cn and are listed in tables 1 and 2. along with the corresponding coefficients in the expansion of &,b. All electronic integrals are evaluated with Z,ff = 3.25 for the carbon orbitals. The UP) are determined by taking averaged values of the frequencies of CH out-of-plane bending modes of the appropriate symmetry in the electronic ground state of naphthalene. The results are [ll]: fiwCH(blu) = 879 cm-l for the s component and fiwCH(au) = 905 cm-f for they component. Combining all these data with the results
1
for TI 4 So in n:phthalene
;
-_-Matris
-.---
element
---_--
a)
L(x’/fiKSf
-_
Tdim‘
.~
_---
------_-
_-__a) Throughout h)S’--
12-WC2
.
1
*
cl
2Cl
0
2
i 24-1/2c2
3-l/2c2
i
0
* 21/2q
<;
- J2c,
the table tho upper sign holds for Ii) = a: *, the lower sign for (1. LSH)-lj2: K .-
. C) -J:- .‘kjp& 171.
----_-
=.. TabQ,/d
L;;/iiKSf
(j-1/2C*
T 3-1/2c -fwiTim
h)
ah
.
and
Table
CH matrix
(4)
B,lk .
_ Ii) = x(J’. ato% n. with
i.
cg
-21/2c2. the numbering
as
in ref.
CHEMICAL PHYSICS LETTERS
Volume 7,. number 5
1 December 1970
Tnblc 2 CH matrix elements for T1 4 Sg in cmlhrxenc
,.:
>
Malrix element a)
L ‘-u)/ms*
L@‘/fmsr
ab
----_--
.ab
s-i/2cl
LniTim
--
------
- 2c1 i ,O_lkC2
r 24-l&2
=niLim
TuhRCH/.J.?
-----
1 2c2
-+ 12-1&s
0
i
3-mq
0
F 2112C.r
-
12-l&2
:c2
- 21/2c2
0
-
6_1/2Cs
I 2’kg
=s
:I) The symbols arc the snmc as in table 1.
listed in tables 1 and 2 and using Hiickel MO’s for the C,, we obtain from eq. (3):
(d3’ I2 n1r1napht
= 6 5 x low4 F(E) (cm-1)2
p(3) I2
= 6.7
nm anthr
e
X 10m4 F(E) (cm-l)2.
l,(2) I2 z lo-4 Nlll
. (5)
where F(E) is the Franck-Condon factor of the transition TlSO. Although such detailed calculations are not available for Ii:&, we can obtain an estimate if we accept the integrals calculated for benzene as representative for other aromatic hydrocarbons. The dominant term in eq. (2) for benzene probably involves the first excited singlet state (lB3u) as an intermediate state 1121. The corresponding spin-orbit integral Ljnl has been calculated [ 13 1 to be 0.33 cm-l. The corresponding vibronic integral Tab is expected to be of the same general form as eq. (4). For simplicity we take T . = R&, where RcC = 1.40A. i.e., we put the o&lap factor equal to unity. This estimate is based on the notion [4] that the inducing mode in benzene is the b3, carbon stretching mode (tiw = 1309 cm-l). The integrals are between zv states with and without nodes through the atoms and thus contain a one-center term for each carbon atom in a nodal plane (not counting the molecular plane). We have actually calculated these onecenter terms [41, but the total result turned out to be very close to the above rough estimate. Note, however, that the fraction of the carbon atoms located in a nodal plane is 1 for benzene, but only 1./5 and 1;‘7 for naphthalene and anthracene, respectively. Thus it seems likely that Tni is smaller for the latter molecules. The above estimates for Tni. Lirn, Jiw and RCC can be combined to give
F(E)
(cnl-1)2
‘(6)
for benzene and. as noted. possibly somewhat smaller values for naphthalene and anthracene. Comparing this result with the results for @i]i,. we find that theory predicts a ratio k(3) k(2) 4 7. in reasonable agreement with the experimental values of 19 and 14 for naphthalene [5 1and anthracene [6 1. respectively. In principle. one should be able to achieve more accurate results when comparing the I@) fit\‘) ratios. since this involves comparison of two components of the same mechanism. so that many errors will cancel. It turns out that the ratio ~~~~,(s). $:ij(_r) depends almost exclusively on the MO coefficients in our model. Using the numbering depicted in ref. (7 1. and denoting by G the factor involving the ratio of frequencies and reduced masses for the s and y modes. we obtain (considering only CA contributions)
$3) (_,.) Jp ) tr, 2 ?llll
II111
in naphthalene
ICH
= ;(2C;-C,2?C,4c2
= 5.7
, and
jJJ(3) (A-) d3)(v)i 2 ~1~1 CH IllI
zz $(2C; - C;)2 Ci4 G2 = 16
in anthracene. Equally high ratios are predicted for the higher polyacenes. The first of these numbers reduces to 5.1 when the CC contributions are taken into account [41. A similar reduction is indicated for anthracene. However. these ratios remain much larger than the experimental ratios [5.6] of 1.7 and 3.0. respectively, although they predict correctly that the s component dominates. The reason for the discrepancy may be the inaccuracy in the MO coefficients. Note that the ratio is extremely sensitive to C2_ Increasing C2
535
CHEMICAL PHYSICS LETTHF&
Voiume 7, number 5
in naphthalene by 10% at the expense of Cl,,would yield the correct result Note also that in our model the o orbital coefficients are all &ken to be equal in absolute magnitude. We have not yet extended the calculations to the $4 T1 crossing in naphthalene for which there are also data available [5 1. This transition shows the same order of rate constants as Tl-- SO. namely h,- _. k, .’ k,. Now the direct transition S1 d T is orbitally allowed, i.e.. HffLzl,lf 0. Since Hki:,), populates rz of Tl. :he experimental data can be taken as an mdic%tion that the transition involves a higher triplet state T3. A similar conclusion has been drawn from the temperature dependence of this infersystem crossing [ 141. If T3 belongs to B3u. & seems likely, the transition may be dominated by Hfin since Sl+T3 is then orbitally forbidden. However. since a B2u state involves excitation of an electron to and from the orbital above the highest empty and below the lowest filled 6 orbital. the calculation is more complicated than that for Tl+ SO.
‘.’
L
536
?
1 December 1270
REFERENCES
[I] M:Schwoere~nnd B. C. Wolf. Proc. XIVth Colloquy. Amcbre iNorth-Holland. Amsterdam. 1967) q . 87 [2] M. S. de droot.: 1; A. M.Hesselman.np$ J. H.‘& d, Waais. Mol.Phys. 12 (1967) 259. 13) M.&de Croat. I.A.M.He&elman. J.Schmidt and J. H. van der Wanls. Mot. Phys. 15 (1966) 17. 14)W.Siebrand. Chem. Phys. Letters 6 (1970) 192: B. R. Henry and W.Siebrand. J.Chcm. Phys.. submitted for oublication. 15) M.Schwoe&r and &%x1. Chem. Phys. Letters 2 (1966) 14: 6.(1970) 21: Z.Naturforsch. 24n (1969) .I
.
,
16) R. H. Clnrke. Chem.Phys. Letters 6 (1970) 413. [7] B. R. Henry and W. Siebrand. J. Chcm. Phvs. 51 (1960).2396.
[S) V.G.Krishna and L.Goodman. J. Chem. Phys. 37 (1962) 912. [S) J. Langelaar. R. P. H. Rettschnick nnd G.-J. Hoytink. J. Chem. Phys.. to be published. [lo) B..R. Henry and W.Siebrand: Chem. Phys. Letters 3 (1’969) 327. [ll) H.‘Luther and H.-J. Drcwitz. Z. Elektrochem. 66 (1962) 546. [12] D.M. Burlnnd and G. W. Robinson.. J. Chem. Phys. 51 (1969) 4546. [13) H. F. Hameka and L. J. Oqsterhoff. Mol.Phys. 1 (1958) 35s. [l-l] P. F. Jones and A.R. Cnlloway. J. Chim. Phys. XXc REunion Annuette (1970) 110.
..:
CHEMICAL
PHYSICS
1 December
LETTERS
1970
:i : :. :: :.
:
THEORY
.‘I
OF
THR0UG.H IN
ELECTRON
SPiN
NONRADIATIVE
NAPHTHALENE
Department of Chemistyv,
AND
ALIGNMENT PROCESSES
ANTHRACENE
B. R. HENRY Unirersil~ of Maniloba,
+
Winnipeg.
Canada
and W. SIEBRAND Dirisi&
o.f Chemistry.
National
Research
CounciL of Canada.
Ottaaya.
Canada
Recejved 6 October 1970
The three cornponcntsof the f!$plet-to-ground state nonrndintive rate constants of nnphthaleneand XIIthrncene are calculated thcorctir.!ly and compnred with experiment. ! 1
At very low temperatures (T Z 1%) the triplet state of an aromatic molecule can behave as three independent states, the rate of interconversion (spin relaxation) being slow compared to the rate of.radiative and nonradiative transitions to and from singlet states. In general rate constants for radiative and nonradiative processes involving triplets will then be different for the three components. so that these processes producf? spin alignment [ 1I. Van der Waals and coworkers [Z, 3) were the first to give an extensive discussion of this phenomenon with special reference to singlet-to-triplet intersystem crossing in heterocyclic systems. There the situation is relatively simple. the favored crossing being between an nz* and a zz* state bacause of the strong predominance of such coupling over that between two x-type states. After determining on this basis which triplet state is directly involved in the
have been made for naphthalene [5 1and anthracene [6] where the rates of radiationless deactivation of the lowest triplet state T1 have been determined for each of the 3 components of Tl. This note is principally concerned vith an analysis of the differences between these rites. Previously, we distinguished three mechanisms of intersystem crossing in aromatic hydrocarbons [4]: (1) a purely e!ectronic mechanism in which the spin-orbit operator couples two ii?i states in the nuclear equilibrium configuration; (2) a vibronic mechanism in which the coupling of type (1) is induced by in-plane vibrations; and (3) a vibronic mechanism involving spin-orbit coupling between xi and zcr states induced by outof-plane vibrations. The corresponding matrix
(1)
transition, the spin component(s) to be predominantly populated can be found by symmetry arguments. -The spin alignment is presumably con-
served during the vibrational and:or electronic relaxation to the lower vibrational levels of the lowest triplet state, on which the actual measurements are made. In aromatic hydrocarbons.
(2)
however, no nr* states are available. It turns out that there are competitive mechanisms for intersystem crossing 141which are, in principle, parti,ally distin@ishable on the basis of spin POlarization measurements. Such measurements -1 . * Issued RS NRCC No. 11635.
(3) where
533
Volume
7. number
I _1 December I970
CRRhIICAL PHYSICSLETTERS
5
all decay rate constant (k,.+ k,, + k,) in naphthalene. -To calculate k(S) = Kx + k,,: .ye note.tii+xt the matrix elements L&j and the energies LIE~,,~in eq. (3) ‘arb_esienti_ally~ the same as those we calculated.previously [7] for the radiative transitions T1 2 St)_ The vibronic integrals in (3) are readily calculatedIf we restrict ourselves to CH out-of-plane modes which can be shown [4] to yield the dominant contribution to mechanism (3). The amplitude of atom n in mode &_is written as Qkn = RC.fB&. where RcH,= 1.08A and f?k,r is the angle between the CH bond and the molecular plane. Expanding in atomic integrals we obtain
is a vibrational overlap integral. 81,, the spin-orbit coupling operator. pk. tihm’ and 81, are the reduced mass. angular frequency (in state GJ~?)and normal coordinate of mode k. 00 is the nuclear equilibrium configuration. the %'s are electronic wavefunctions and AE is an electronic energy difference. We have shown that mechanism (2) is mainly induced by deviations from Q. (Herzberg-Teller vibronic coupling) and vi; spin-orbit coupling, contrary to mechanism (3) which is mainly induced by the nonzero nuclear momenta (Born-Oppenheimer coupling) and ho spin-orbit coupling. Symmetry arguments then dictate that mechanism (2) (de)populates the outof-plane spin component (7z) of the triplet state. as does mechanism (l), and mechanism (3) the inplane components rx and r>,. Thus a measurement of the nonradiative rate constants in a spin-aligned system allows an immediate distinction to be made-between mechanisms (1) and (2) on the one hand and mechanism (3) on the other. In measurements on naphthalene [5] and anthracene [6] it was found that the observed triplet decay rate constants are in the order k.,- .. k, >> kz. It is known that radiative decay contributes mainly to k, 17.81 and is slow enough to be ignored for the present purpose [5.9]. so that the above rate constants may be assigned to TlSO crossing. Since mechanism (1) is forbidden by symmetry here. the observation that mechanism (3) is dok.\- + ky >> k, indicates minating mechanism (2). A similar conclusion has been reached [4. lo] on the basis of the observed effect of partial deuteration on the over-
s,rm
[41 Ta,, (k) = R&S*Jg where s’
elements
= (l*sH)-l~~.
SH = 3-1’2+i+21/~o]L)
h being an 1s hydrogen
orbital, and s and p. 2s and 2p Slater orbitals; po is directed along the CH bond. The coefficients Bnk in (4) depend on the K-MO coefficients Cn and are listed in tables 1 and 2. along with the corresponding coefficients in the expansion of &,b. All electronic integrals are evaluated with Z,ff = 3.25 for the carbon orbitals. The UP) are determined by taking averaged values of the frequencies of CH out-of-plane bending modes of the appropriate symmetry in the electronic ground state of naphthalene. The results are [ll]: fiwCH(blu) = 879 cm-l for the s component and fiwCH(au) = 905 cm-f for they component. Combining all these data with the results
1
for TI 4 So in n:phthalene
;
-_-Matris
-.---
element
---_--
a)
L(x’/fiKSf
-_
Tdim‘
.~
_---
------_-
_-__a) Throughout h)S’--
12-WC2
.
1
*
cl
2Cl
0
2
i 24-1/2c2
3-l/2c2
i
0
* 21/2q
<;
- J2c,
the table tho upper sign holds for Ii) = a: *, the lower sign for (1. LSH)-lj2: K .-
. C) -J:- .‘kjp& 171.
----_-
=.. TabQ,/d
L;;/iiKSf
(j-1/2C*
T 3-1/2c -fwiTim
h)
ah
.
and
Table
CH matrix
(4)
B,lk .
_ Ii) = x(J’. ato% n. with
i.
cg
-21/2c2. the numbering
as
in ref.
CHEMICAL PHYSICS LETTERS
Volume 7,. number 5
1 December 1970
Tnblc 2 CH matrix elements for T1 4 Sg in cmlhrxenc
,.:
>
Malrix element a)
L ‘-u)/ms*
L@‘/fmsr
ab
----_--
.ab
s-i/2cl
LniTim
--
------
- 2c1 i ,O_lkC2
r 24-l&2
=niLim
TuhRCH/.J.?
-----
1 2c2
-+ 12-1&s
0
i
3-mq
0
F 2112C.r
-
12-l&2
:c2
- 21/2c2
0
-
6_1/2Cs
I 2’kg
=s
:I) The symbols arc the snmc as in table 1.
listed in tables 1 and 2 and using Hiickel MO’s for the C,, we obtain from eq. (3):
(d3’ I2 n1r1napht
= 6 5 x low4 F(E) (cm-1)2
p(3) I2
= 6.7
nm anthr
e
X 10m4 F(E) (cm-l)2.
l,(2) I2 z lo-4 Nlll
. (5)
where F(E) is the Franck-Condon factor of the transition TlSO. Although such detailed calculations are not available for Ii:&, we can obtain an estimate if we accept the integrals calculated for benzene as representative for other aromatic hydrocarbons. The dominant term in eq. (2) for benzene probably involves the first excited singlet state (lB3u) as an intermediate state 1121. The corresponding spin-orbit integral Ljnl has been calculated [ 13 1 to be 0.33 cm-l. The corresponding vibronic integral Tab is expected to be of the same general form as eq. (4). For simplicity we take T . = R&, where RcC = 1.40A. i.e., we put the o&lap factor equal to unity. This estimate is based on the notion [4] that the inducing mode in benzene is the b3, carbon stretching mode (tiw = 1309 cm-l). The integrals are between zv states with and without nodes through the atoms and thus contain a one-center term for each carbon atom in a nodal plane (not counting the molecular plane). We have actually calculated these onecenter terms [41, but the total result turned out to be very close to the above rough estimate. Note, however, that the fraction of the carbon atoms located in a nodal plane is 1 for benzene, but only 1./5 and 1;‘7 for naphthalene and anthracene, respectively. Thus it seems likely that Tni is smaller for the latter molecules. The above estimates for Tni. Lirn, Jiw and RCC can be combined to give
F(E)
(cnl-1)2
‘(6)
for benzene and. as noted. possibly somewhat smaller values for naphthalene and anthracene. Comparing this result with the results for @i]i,. we find that theory predicts a ratio k(3) k(2) 4 7. in reasonable agreement with the experimental values of 19 and 14 for naphthalene [5 1and anthracene [6 1. respectively. In principle. one should be able to achieve more accurate results when comparing the I@) fit\‘) ratios. since this involves comparison of two components of the same mechanism. so that many errors will cancel. It turns out that the ratio ~~~~,(s). $:ij(_r) depends almost exclusively on the MO coefficients in our model. Using the numbering depicted in ref. (7 1. and denoting by G the factor involving the ratio of frequencies and reduced masses for the s and y modes. we obtain (considering only CA contributions)
$3) (_,.) Jp ) tr, 2 ?llll
II111
in naphthalene
ICH
= ;(2C;-C,2?C,4c2
= 5.7
, and
jJJ(3) (A-) d3)(v)i 2 ~1~1 CH IllI
zz $(2C; - C;)2 Ci4 G2 = 16
in anthracene. Equally high ratios are predicted for the higher polyacenes. The first of these numbers reduces to 5.1 when the CC contributions are taken into account [41. A similar reduction is indicated for anthracene. However. these ratios remain much larger than the experimental ratios [5.6] of 1.7 and 3.0. respectively, although they predict correctly that the s component dominates. The reason for the discrepancy may be the inaccuracy in the MO coefficients. Note that the ratio is extremely sensitive to C2_ Increasing C2
535
CHEMICAL PHYSICS LETTHF&
Voiume 7, number 5
in naphthalene by 10% at the expense of Cl,,would yield the correct result Note also that in our model the o orbital coefficients are all &ken to be equal in absolute magnitude. We have not yet extended the calculations to the $4 T1 crossing in naphthalene for which there are also data available [5 1. This transition shows the same order of rate constants as Tl-- SO. namely h,- _. k, .’ k,. Now the direct transition S1 d T is orbitally allowed, i.e.. HffLzl,lf 0. Since Hki:,), populates rz of Tl. :he experimental data can be taken as an mdic%tion that the transition involves a higher triplet state T3. A similar conclusion has been drawn from the temperature dependence of this infersystem crossing [ 141. If T3 belongs to B3u. & seems likely, the transition may be dominated by Hfin since Sl+T3 is then orbitally forbidden. However. since a B2u state involves excitation of an electron to and from the orbital above the highest empty and below the lowest filled 6 orbital. the calculation is more complicated than that for Tl+ SO.
‘.’
L
536
?
1 December 1270
REFERENCES
[I] M:Schwoere~nnd B. C. Wolf. Proc. XIVth Colloquy. Amcbre iNorth-Holland. Amsterdam. 1967) q . 87 [2] M. S. de droot.: 1; A. M.Hesselman.np$ J. H.‘& d, Waais. Mol.Phys. 12 (1967) 259. 13) M.&de Croat. I.A.M.He&elman. J.Schmidt and J. H. van der Wanls. Mot. Phys. 15 (1966) 17. 14)W.Siebrand. Chem. Phys. Letters 6 (1970) 192: B. R. Henry and W.Siebrand. J.Chcm. Phys.. submitted for oublication. 15) M.Schwoe&r and &%x1. Chem. Phys. Letters 2 (1966) 14: 6.(1970) 21: Z.Naturforsch. 24n (1969) .I
.
,
16) R. H. Clnrke. Chem.Phys. Letters 6 (1970) 413. [7] B. R. Henry and W. Siebrand. J. Chcm. Phvs. 51 (1960).2396.
[S) V.G.Krishna and L.Goodman. J. Chem. Phys. 37 (1962) 912. [S) J. Langelaar. R. P. H. Rettschnick nnd G.-J. Hoytink. J. Chem. Phys.. to be published. [lo) B..R. Henry and W.Siebrand: Chem. Phys. Letters 3 (1’969) 327. [ll) H.‘Luther and H.-J. Drcwitz. Z. Elektrochem. 66 (1962) 546. [12] D.M. Burlnnd and G. W. Robinson.. J. Chem. Phys. 51 (1969) 4546. [13) H. F. Hameka and L. J. Oqsterhoff. Mol.Phys. 1 (1958) 35s. [l-l] P. F. Jones and A.R. Cnlloway. J. Chim. Phys. XXc REunion Annuette (1970) 110.