Download Theory of electric field effect on electronic spectra and electronic

Theory of electric field effect on electronic spectra and
electronic relaxation with applications to F centers
S. H. Lin
Department of Chemistry, Arizona State University, Tempe, Arizona 85281
(Received 3 December 1974)
A main purpose of this paper has been to present a microscopic theory of electric-field induced absorption
spectra; the related phenomenon, the Kerr effect, is studied by using the Kronig-Kramers relation. Our attention
is focused on the temperature effect, band shapes, moment relations, and the differences in field induced absorption
spectra between allowed transitions and symmetry-forbidden transitions. To illustrate the application of theoretical
results, we have investigated the field induced spectra of F centers in alkali halides. It is shown that the F band
is composite, consisting of three bands. We have shown that the technique of the electric-field induced optical
absorption can be used to resolve the hidden bands. We have also investigated the electric field effect on
radiative and nonradiative processes. It has been shown that from the measurement of the electric field dependence
of lifetimes of excited electronic states, one can determine the variation of radiationless transitions with the electric
field which in turn can be used to study the energy gap law and temperature effect in radiationless transitions.
Theoretical results have been applied to F centers in alkali halides. The feasibility of observing the electric field
effect on lifetimes of organic molecules has been discussed and the field strength required for observing the
electric effect for polar and nonpolar molecules has been suggested.
I- INTRODUCTION
When an electric field is applied to a system, the
electronic charge distribution, energy levels, and population of molecules will be affected, which in turn will
affect the absorption coefficient. The measurement of
field induced spectral changes provides a useful way
for determining excited state properties like dipole moments and polarizabilities, parameters describing intermolecular interactions, and the orientation of transition moments. 1,2 The theoretical basis of field induced
spectral changes has been detailed by Liptay and
Czekalla3 and Liptay,4 and it is believed that this experimental technique is now on sound footing.
In view of the recent active experimental interest in
field induced spectral changes, 1,2,5,6 in this paper we attempt to present a molecular theory which will treat
both field induced spectral changes and the related Kerr
effect. Attention will be focused on the temperature effect, and the difference in field induced spectra between
allowed transitions and symmetry-forbidden transitions
and the conventional equations used for the determination
of excited state properties will be critically examined.
We shall also derive the moment relations and show that
they can also be used to determine the excited state
properties. To demonstrate the application of our results, we shall discuss the electric-field induced spectra of F centers in alkali halides; it will also be shown
that the conventional F band is composite, consisting of
three bands. We shall show that the electro-optical absorption technique can be used to detect the hidden
bands. The band shape functions associated with the
field induced spectra will be investigated; the diffe1'ences
in the spectra for polar and nonpolar molecules are
shown.
For the Kerr effect (or electric birefringence), we
shall show that in addition to the conventional classical
expreSSion, the additional terms are obtained resulting
from the contribution from the derivatives of optical
polarizabilities. In this aspect, we derive the expres4500
sions for electric dichrOism, which has begun to attract
experimental attention but upon which little theoretical
investigation has been carried out, and then obtain the
expressions for the Kerr effect by using the KronigKramers transform.
In this paper, we also study the effect of an electric
field on radiative and non radiative transitions; the expressions for the dependence of the radiative and nonradiative rate constants on the field strength are derived. We shall show that in general the radiative process is less sensitive to the applied field than the nonradiative process and that from the measurement of the
electric field dependence of lifetimes of excited electronic states, one can determine the variation of radiationless transitions with the electric field which in turn
can be used to study the energy gap law and temperature
effect in radiationless transitions. Our theoretical results will again be applied to F centers in alkali halides.
We shall also discuss the feasibility of measuring the
electric field effect on lifetimes of organic molecules
and estimate the field strength required for observing
this electric field effect for polar and nonpolar molecules.
II. GENERAL THEORY
If we let the direction of the applied electric field to be
the z axiS, then in the adiabatic apprOximation the absorption coefficient with the optical polarization along
(parallel) to the z direction for the electronic transition
a- b can be expressed as 7,8
k~b(W) = 41T~W
L Lv
as c v'
PavD,,(av, bv') 5(Wbv'av - w),
(2.1)
where D .. (av, bv') represents the dipole strength D,,(av,
bv') = 1(av 1Z 1bv') 12 , and P av is the normalized Boltzmann factor. The factor o!s has been introduced to take
into account the medium effect arising from the electromagnetic field. Similarly, the absorption coefficient
with the polarization perpendicular (say the x direction)
The Journal of Chemical Physics, Vol. 62, No. 11, 1 June 1975
Copyright © 1975 American Institute of Physics
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4501
S. H. Lin: Electric field effect on spectra
to the external electric field is given by
k!b(W)
= 47T~
as fl,{;
p!~) Zav,av
Zav,av = L
v
\'
7'
L,PavDx(av, bv')Ii(Wbv',av- w ),
(2.9)
,
(2.2)
(2.10)
v
v
In Eqs. (2.1) and (2.2), x and z refer to the space-fixed
coordinates.
In the presence of the applied electric field, both
wavefunctions and energies are affected; their changes
that arise because a molecule is placed in a uniform
electric field F can be calculated by using the perturbation method. It follows that
P av =p!~) + FP!~) + F Z p!~) + ... ,
Ii ( Wbv',av -
(Z)_.1:...[
- 2'1f a
+
(')
( ) ] '(
zz bv - au av Ii W
2~Z (ZbV',bV' -
-
(0)
Wbv',av
W~~~,av)
Zav,av)zli"(w -
(2. 12)
etc. DAav, bv')(O) represents the dipole strength in the
absence of the electric field and DAav, bV,)(1), D,,(av,
bv')(Z), '" represent the changes of the dipole strength
due to the electric field; the expressions for D,,(av, bv,)(n)
are given in Appendix A. It should be noted that for centrosymmetric molecules, for allowed g- u transitions the
transition moment varies with the square of the field
strength and the dipole strength varies with the fourth
power of the field strength. In Eqs. (2.8)-(2.12),
a",,(av) and a",,(bv') denote the static polarizability of
vibronic states av and bv', respectively.
(2.3)
D,,(av, bv')=D,,(av, bv')(O)
+ FD.. (av, bv,)(1) + FZDz(av, bv')(Z) + ..• ,
(2.4)
and
W)
where F represents the effective field strength,
(2.6)
v
(lI_p(O) (Z
Z--)·,
P av
av {3 av ,av av,av'
(3--l/kT ,
Using Eqs. (2.3)-(2.5), Eq. (2.1) can be conveniently
written as
(2.7)
lf (
lf (
kifab (W )=kab
)(O)+Fk"ab (u) )(l)+Fzk ab
)(2)+ •••
W
W
(2.8)
+Z!V,4v,-Zav,av Z av,av)] ,
Z
W " " p(O)
-n.
L...JL...J
as c v v'
if ( )(0) 47T
k ab
W
=
(
av D" av, bv
,)(0)
(
(0)
where
)
(2.14)
Ii Wbv',av - W ,
Z
if ()(1)
" "
(0)
kab
W
= 47T W
L...J
L...J [{pO)D
av II (av, b v ')(0) + P(O)D
av z (av, bv ')U)} Ii (Wbv',av
-
--na .. c
v
(2.13)
,
W
)
(
b v ')(0) Ii ( Wbv',av + P(O)D
av "av,
W
)(1)]
,
(2.15)
v'
2
kif (W)(2) - 47T W L L [ {p(Z) D (av bv')(O) +P(1)D (av bv,)(l) + p(O) D (av bV')(2)} Ii( (0)
ab
- Q 1ic v v'
av
z,
av
z,
av
z,
Wbv·. av
s
_
)
W
U)D z (av, bv ')(0) + P(O)D
+{pav
av z (av, b v ')U)} Ii (Wbv',av-w )(1) + P(O)D
av II (av, b v ')(0) Ii (Wbv',av-W )(2)] ,
(2.16)
etc. Substituting Eqs. (2.6)-(2.12) into Eqs. (2.15) and (2.16), we obtain
k~b(W)(1) = !7T~~ L
s
v
4= P!~{{D,,(av,
bv')(O) + (3D.(av, bv')(O) (Zav,av - Zav,av)}
v
+
i
Ii(w~~l.av -
w)
w~~l.av~
D.(av, bv')(O) AZ(bv', av) Ii' (w -
(2.17)
and
+
i3(~ Z!.av - ~ Zaz".av + Z!v.av -
X
(Zav.av - Zav.av)} Ii' (w -
Zav.av Zav.av)]}
AZ(bv', av) = Zbv'.bV' - Zav.av
w)+
~
AZ(bv', av){D,,(av, bv,)(1) + (3D,,(av, bv')(O)
w~~l.av) + 2 n DII(av, bv')(O) Aallll(bv', av)Ii' (w - w~~l,av)
1
+ 2~2 D.(av, bv')(O) AZ(bv', av)2 1i ,,(w -
where
Ii(w~~).av -
w!~).av~
(2.18)
and
Aa .. (bv', av) = a ... (bv') - a"lI(av) .
J. Chern. Phys., Vol. 62, No. 11, 1 June 1975
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4502
S. H. Lin: Electric field effect on spectra
A similar expression can be given for k!b(W), The delta
function 6(w bv'.av - w) appears in k~b(W) and k!b(W) because we ignore the damping-effect; if the damping effect is included, instead of 6(Wbv'.av - w), we will have
the Lorentzian. 9 Although in the above discussion we
have restricted ourselves to the cases in which the polarization is either parallel or perpendicular to the external electric field, we can easily generalize our treatment to study the case in which the polarization is
oriented at a particular angle with the applied field by
simply replacing D.(av, bv') or D,,(av, bv') by Ie . Rbv'.av 12.
In this investigation, we shall study only the systems of
randomly oriented molecules.
e
In the above derivation, we have implicitly assumed
that in the system there exist molecular motions with
energies large and small compared with kT so that the
expansion like that given by Eq. (2.3) may be assumed
to hold. At low temperatures and in solid media, the expansion Eq. (2.3) is not valid, as in this case there exist
no molecular motions with energies small compared with
kT. In other words, in this case there is no contribution
from the orientational motion of molecules to the fieldinduced spectral changes; they arise from the electric
field effect on 6(w - Wbv'.av) and D.(av, bv') [or D,,(av,
bv')], with the former IJeing more important than the
latter (See Appendix A). We have also assumed that the
field is uniform over each molecule, that the fluctuations
in field due to the nonuniformity of the electronic distribution around a given molecule can be neglected, and
that each polar molecule may be treated as a point dipole and each polarizable molecule treated as a pointinduced dipole. Then the average fields in the system
can be used to describe the effective field F.
III. A SYSTEM OF RANDOMLY ORIENTED
MOLECULES
In this case, it can easily be shown that q~b(UJ)(l) [and
also q~b(W)(l)] vanishes due to the fact that the space
average of ZabZbcZcd and XabXbcXcd vanishes. Similarly,
it can easily be shown that if in general the polarization
direction forms an angle X with the direction of the external field, the absorption coefficient q~b(W) is related
to k~b(W) andk~b(W) by
(3.1)
From Eq. (2.13), we obtain the change in the absorption coefficient induced by the applied electric field as
D.k~b(W)
=F2[ k~b (w)12) + k~b (W)~2) + q~b (W)~2)] + . ..
,
(3.2)
where
(3.3)
1 D.O!•• (
) Fav
(0)
+ 21f
bv ', av
D. ( av, b')
v (0
)J 6' (
W -
(0 )
Wbv'
.av ) ,
(3.4)
and
2
If (
)(2)
k ab
W 3
= 41T
W """"
O!slfc~?-,
P(O)D (av bv')(O) _1_ D. ( '
)2,,(
av . '
2lf2 Z bv, av 6 W
-
(3.5)
(0»
Wbv!.av .
It is to be understood that the summations in Eqs. (3.3)-(3.5) includes the space average over the molecular orienta-
tion. Substituting Eqs. (2.6)-(2.8) into Eqs. (3.3)-(3.5) yields
k~b(W)J2)oo 41T:W 1:1: P!~)[(D.(av,
0sftC
v
bv,)(2»av+(3(D.(av, bv,)(1)Zav.av)av
v'
+t(3(D,,(av, bv') (0) {O!zg(av) - O!.Aav) + (3(Za2v.av - Za~.av)} )av] 6(w~~!.av - w) ,
(3.6)
and
(3.8)
where (- - - -)av denotes the space average over the
molecular orientation. Notice that Zav.av vanishes for
a system of randomly oriented molecules. It can easily
be shown that the terms involved D.(av, bv,)(1) and
D.(av, bV,)(2) are in general smaller than the remaining
terms and can be ignored unless those terms without
D.(av, bv,)(1) and/or D.(av, bV')(2) vanish (cf. AppendixA).
ordinates and molecule-fixed coordinates and carrying out
the spatial average over the molecular orientation, we obtain
(A.B.C.D.)avoon [(A' B)(C ·D)+ (A, C)(B 'D)
+ (A· D)(B' C)]
(3. ')
and
(AxB" C.D. )av oof5 (A, B)(C' D)
USing the Euler angles to relate the space-fixed co-
-i[(A'C)(B'D)+(A'D)(B'C)],
(3.")
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4503
S. H. Lin: Electric field effect on spectra
which are required in carrying out the space average of
molecular orientations for k~b(W) and k~b(W). It follows
that Eq. (3.8), which is easiest to simplify, can be
written as
harmonic oscillator approximation for molecular vibration (see Appendix B)
(a
2
1 '""' ~a
R ) (VI+21.) - Ii +"',
R.v •• v=R.(0)
a +-2L.J
I
8({i
°
(3.10 )
WI
where R!~) represents the dipole moment at the equilibrium position. A similar expression can be given for
RbV~.bV"
For nonpolar molecules, R!~) =0 =(a 2R.a /
aQ 1)0'
where AR(bv', av) =Rbv'.bV' - R. v•• v ·
To see how the dipole moments R. v••v and Rbv'.bV'
change with normal vibration, for simplicity we use the
In most cases, (a2Raa/aQ~)0 is small; this can be
seen from the use of the perturbation theory (cf. Appendix B):
(3. 11)
where (cp!o) I (aiI/aQI)o I cp~O», (cp!O) I (a2iI/aQ~)0 I cp~O»
etc., represent the matrix elements of the vibronic
interaction. Thus, in (a2R •.laQ~)0, second order vibronic couplings are involved. The dependence of the
polarizabilities a .. (av), a .. (av), etc. on normal coordinates can be discussed similarly (see Appendix B).
To simplify Eqs. (3.6) and (3.7), we notice that
(DAav, bv')(O) {a ...(bv') - a .... (av)} ).v
=fs-[lR.v•bv , 12 Tr{AQ(bv', av)}
+2Rbv'.av·AQ(bv',av)·R.v.bv'] '
where Q(av) represents the polarizability tensor of the
av vibronic state
_ '""' ' 2R gy•(0)CV Rc&u.
0)
(D .. (av, bv')(O)a .... (av».v
a (av ) - L.J
ev"
=y\ [ 1R av• bv ,12 Tr {Q(av)} + 2Rbv' .av • Q(av) • R av • bv']
(3.12)
and
(3.13)
U
Ecv
ll -
all
(3.14)
Eav
and AQ(bv', av) = Q(bv') - Q(av). Substituting Eqs. (3.12)
and (3.13) into Eqs. (3.6) and (3.7), we obtain
k~b(W)42) = k~b(U')~) + :'/T;~ 'Ev ~ p!~) [f5 { 1R av•bv,12 R av••v • AR(bv', av) + 2 (R. v.bV' • R.v.av)[Rav.bv' • AR(bv', av)]}
8
V
and
where
(Q(av»=
L P!~)Q(av) ,
v
k"ab (W)(0)
10 -
( 1R av•av 12) =L p!~) 1R av•av 12 ,
v
2
4'/T ""_w
as
f£{;
(3.17)
'""' L.J
'""' P(O)[(D
)] ii (W - Wbv'.av
(0»
L.J
av
• (av, bv ')(2» av+ f3 (D• (av, bv ,)(1)Zav.avav
,
v
v"
and
k "ab (W)(2)
20 -
4'/T
2
w "
''
~
L...J' L.J
as c v v'
p(O)(AZ(b'
(0 1
au
V , av )D
. • (av, bV,)(1»,(
av ii W - Wbv;
,av) •
(3.18)
(AZ(bv', av)D.(av, bv')(l»av, (D.(av, bv')(2»av, and (D .. (av, bv')(1)Zav.av>av are given in Appendix A; their contributions
in general are negligible (cf. Appendix A).
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4504
S. H. Lin: Electric field effect on spectra
The expressions for k!b(W)~2>, k!b(W)~2), and k!b(W)~2) can be obtained similarly. But sometimes it is convenient to
use ~k~b(W),
~k~b(U)
F2[ ~~b(WW) + ~~b(W)~2) + k~b(w)J2)1 + . "
=
,
(3. 19)
where
-i
-i
1R av •bv' 12 Tr {(Q!(av»} + fs i3{(2 - cos 2X) 1R av •bv' 121 R av •av 12 + (3 cos 2X- 1)(Rav•bv' . Rav.aY}
131 Rav. bV' 12( 1R av •av 12)] o(w - w~~J .av) ,
L~
v
v
(3.20)
p!~) [2 i3{(2 - cos 2 X) 1 R av• bv' 12Rav.av . ~R(bv', av)
2
2
+ (3 cos X- 1)(Rav• bv' • Rav.av) R av •bv' ~R(bv', av)} + (2 - cos X) 1Rav. bV'
12 Tr {~Q!(bv', av)}
+ (3 cos X- 1) Rav. bV' • ~Q!(bv', av) • Rav. bV' ] 0' (w - w~el.av) ,
2
(3.21)
and
1T2
k~b(W)~2) = 1: ~3C Lv ~
p!~) [(2 v
Q!s
cos 2X) 1R av•bv'
121 ~R(bv', av) 12 + (3 cos 2 X-1) 1R av• bv' . ~R(bv', av}j2] o"(w -
W~~!.av)
.
(3.22)
Notice that
k!b(W)~~) = 47T:W
as ftC
Lv Lv'
p!~) [(D:r(av, bv,)(2»av + 13 (D:r(av, bv,)(1) Zav.av )av] o(w - w~~l.av)
(3.23)
p(O)<~Z(b'
av
v ,av )Dx (av, bv ,)(1»
(3.24)
and
2
41T W " ' ' ' '
k.Lab ( W)(2)_
20 - ~ L..JL..J
as
c
v
v'
(0» .av ,
av U"'(W - Wbv'
and that k~b(W)~~) and k~b(W)~~) can be obtained from k~b(W)~~)' k!b(w)~5), k~b(W)~~)' and k!b(W)~) through the relation given by Eq. (3.1). So far we have been concerned with the case in which there exist molecular motions with energies
larger than kT; this is true in the gas phase and may be true in the liquid phase when the temperature is not very
low. Next we consider the low temperature condition in which no molecular motions have energies smaller than kT.
In this case, the expansion like that given by Eq. (2.3) is not valid, and we have
k~b(WW) =k~b(W)~~) ,
x ( )(2)
k ab
W2
(3.25)
2
_ kXab ()(2)
41T w
' " '" (0) [
2 1
12 {
}
W 20 + 15Q!s n2C L: ~ Pav (2 - cos X) R av•bv' Tr ~Q!(bv', av)
-
+ (3 cos X- 1) Rav. bV' . ~Q! (bv' , av) • Rav. bV' ] 0' (W - W~~). av) ,
2
(3.26)
and
k X ( )(2)
ab W 3
_
-
2
21T W
15Q!sn3C
L L p!~) [(2 - cos2X) 1R av•bv' 121 ~R(bv', av) 12 + (3 cos2X- 1) 1R av•bv' . ~R(bv', av) 12] 0" (W - W~~).av) .
v
(3.27)
v'
Equations (2.26) and (3.27) can be rewritten as
k~b(W)?) =k~b(W)~) + ~ LL p!~) [(2 5"
v
v'
cos 2X)Tr
{~Q!(bv', av)} + (3 cos2X-
1) Pav •bv' .
~Q!(bv', av) • Pav •bv'] ; - { k av•by' (w)O}
W
W
(3.26')
and
k~b(W)~2)= 10Wn 2 ~~ p!~)[(2-cos2x)I~R(bv',av)12+(3cos2X-1)lpav.bv,·~R(bv',av)12]
::2
{kav.b:(W)O} ,
(3.27')
where kav.bv'(w)O represents the absorption coefficient of a particular vibrational band and Pav• bv' denotes the unit
vector of R av• bv" It is to be understood that the summations over (v, v') in Eqs. (3.26') and (3.27') are only over the
vibrational bands and may not be the same as the summations over (v, v') in Eqs. (3.26) and (3.27). The detailed discussion of this point has been given elsewhere. 18 In other words, Eqs. (3.25)-(3.27') provide a method to determine
the excited state properties and transition moments R av• bv' of single vibronic states, provided the vibrational bands
are well resolved. These types of equations (setting v =0) have recently been employed by Mathies and Albrecht to
determine the excited state properties of azulene, benzane, and naphthalene. 5.8 When T is not low, similar relations
can be obtained from Eqs. (3.20)-(3.22). We find
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4505
S. H. Lin: Electric field effect on spectra
k~b(W)iZ) =k!b(U)ig) + t$ L L p!~)[H(2 - coszx)Tr[a(av)] + (3 cos\: -l)Pav,bv' . a(av) . Pav,bV'} v
v'
t Tr {( a(av»}
(3.20')
and
k!b(W)~2) = 10W1f2 LLP!~)[(2 -
COS2x)I
v v'
~R(bv', av) 12+ (3 COS2X -1)lpav ,bV' . ~R(bv', av) 12]a~2{kav'~'(w)O},
(3.22')
where the summations over v and v' and p!~) are over the vibrational bands.
For allowed transitions, the electronic transition moment may approximately be regarded as independent of normal
coordinates of vibration, i. e., Rav. bv' =R!~) <9 av I 9 bV')' where 9 av and 9 bv' represent the wavefunctions of nuclear
motion. In this case, k!b(W)!2) (n= 1, 2, 3) can be simplified as
k!b(WWl ) =k~b(W)~~) +
1~:2sW1f~
t IRab 12 Tr {a (a)} + /3{(Rab • Raa)2 - t IRab 121 Raa 12} ]
=k~b(W)~g)+-dr$(3 cos2 X-1)[Pab ' a(a)' Pab - t Tr{a(a)} + /3{(Pab . Raa)2 - t IR aa1 2}] kab(w)O ,
(3.28)
4
(3 cos 2X- 1}Fab(W)[Rab • a(a) . Rab -
2
k!b(W)~Z) =k!b(W)~~) + 15: ~2C F:b(w)[ ,B{(2 - cos 2 X) IRabI2R ••. ~R(b, a) + (3 cos 2X-l)(R.b . R•• )R.b . ~R(b,
s
an
+H(2 - COS2X)j R.b 12 Tr[~a(b, a)] + (3 cos 2 X - l)R. b • ~a(b, a) . R ab } ]
=k!b(W)~~) +
5% [.a{(2 - cos 2X) Ra• . ~R(b, a) + (3 COS2X -1)(Pab . Raa) P ab . ~R(b, a)}
1
• •
a
+ z{(2
- cos 2x) Tr [Aa(b, a)] + (3 cos 2X- 1) P ab . Aa(b, a) . P ab } ] aw
{k~
~t} ,
(3.29)
and
k~b(W)~2) = 121T2;s
F::(w)[(2 5a
c
s
= 1;/f2 [(2 - cos2X) I
cos 2X)IRab 121 AR(b, a) 12+ (3 cos 2 X -1) IR ab ' AR(b, a)1 2 ]
~R(b, a) 12 + (3 cos2 X -1) IFab • ~R(b, a) 12] 6{ kAb;)O} ,
(3.30)
where Fab(w) represents the band shape function
Fab(w)= L L
v
v'
P!~)1<9avI9bv')126(w-w!~1,av)'
(3.31)
and kab(w)o denotes the molecular absorption coefficient in the absence of the applied field,
kab(w)o =
341T2~
as c
I
Fab(w) Rab 12 .
(3.32)
For symmetry-forbidden transitions, the electron transition moment Rab varies with normal coordinates of the inducing modes; this can usually be treated by employing the Herzberg-Teller theory. Recently, the validity and
limitations of the Herzberg-Teller theory have begun to be critically examined. 10-12 For our purpose, we shall
assume it to hold; this assumption can easily be modified, however. According to the Herzberg-Teller theory, if
the inducing modes are nontotally symmetric, the variation of the electronic transition moment can formally be expressed as
(0)
~(aRAb) Q
Rab=Rab +"r'\'aQf 0 f+····
(3. 33a)
It follows,
[I
~ 1Rav,bV' 12 = IR!~) 12+ ~ (~~~b)oI2+R!~)' (~2~,~J(Vf +t) ~f + ...
(3. 33b)
For symmetry-forbidden transitions, R!~) =0. For many aromatic mole.cules (e. g., substituted benzenes), both
R!~) and (aRab /aQf)O may be equally important; in that case, the term R!~) . (a2Rab /aQ~)o in Eq. (3.39) may be ignored.
In other words, for these molecules the contribution to the dipole strength originates from substitution effect and
vibronic coupling.
J. Chern. Phys., Vol. 62, No. II, 1 June 1975
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4506
S. H. Lin: Electric field effect on spectra
Thus, using Eqs. (3.32) and (3.33) for symmetry-forbidden transitions, we find
k~b(W)~2) =k~b(W)W + 1~::'t (3
COS2X -
1)
~ 1e8~~b)0
r[- t
Tr {a(a)}+
p!~) . a(a) . p!~) + .B{- t 1Raa 12+ (p!~) . Raa)2} JFab(w)si
,
(3.34)
kX ( )(2)
ab W 2
_
-
2
kX ()(2)
4lT W
ab W 20 + 15 .,.2
a.n e
~ 1(~~~b)J [.B{(2 - cos X)Raa' AR(b, a) + (3 cos
2
2
X-
1)(P!~) . Raa)P!~). AR(b, a)}
+ H(2 - cos x)Tr[Aa(b, a)] + (3 cos 2X -1)P!~) . Aa(b, a)P!~)} ] F :b(W)si ,
2
(3.35)
and
(3.36)
""
in these tables, k~b(W)~~) and k~b(W)~~) have been ignored;
and for polar molecules, the contribution to Ak~b(W) from
the polarizability, which is usually less important than
that from the dipole moment, has also been ignored.
For the low temperature condition in the solid phase,
the approximate expressions for Ak~b(W) in Tables III. a
and III. b can be used by deleting those terms that involve .B.
where Fab(w).l represents a band shape function defined
by
p!~) represents the unit vector
and in this case the absorption coefficient induced by the
ith inducing mode in the absence of the applied field is
given by
kab ( W)0sl-_~
3a.Fie
I(~)
12 Fab(w).l·
8Ql 0
(3.38)
Thus kX(W)~2) and kX(W)~2) can be related to (8/8w){k ab
x (W)~l / w} and (8 2/8W2){kab(W)~1 / W}, respectively. Equations (3.34)-(3.36) can be reduced to Eqs. (3.28)-(3.30)
only when there is only one inducing mode or when p!~)'s
are the same for all the inducing modes. In other
words, the measurements of Ak!b(W) in this case will
provide us not only the excited state properties but also
some information about the transition moment induced
by each inducing mode (8R ab /8QI)0'
IV. MOMENT RELATIONS
Useful information can often be obtained from moment
relations. Here we shall derive only two lowest moments. Other higher moments can be obtained similarly. First we consider
[:
k~b(W)J = S d:: k~b(W) ,
(4.1)
which represents the area under the curve of k~b(W)/ W
vs W and is physically related to the strength of the
electronic transition. Substituting Eqs. (3.13)-(3.16)
into Eq. (4.1) yields
It should be noted that the choice of cos 2 X =t can sim-
plify the equation for Ak~b(W) considerably. In Tables I
and II, for practical purposes we give approximate expressions of Ak~b(W) for nonpolar and polar molecules;
where
(4.3)
(4.4)
TABLE 1. M~b(r,) for nonpolar molecules.
Symmetry-forbidden transitions
J3F2 (3 cos2x
M!b(W) = 10
" kab(w)~f[P,lt). a
-1)7
A
(a). P,It) -t Tr {a (a)} 1+ wF2
Ion "
t- [(2 -cos~)Tr{Aa (b,a)}
+ (3 cos 2X _1)P(f). Aa (b a)'
ab
,
p(f)l~
{kab(r-")~f}
ab a,,,
W
J. Chern. Phys., Vol. 62, No. 11, 1 June 1975
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4507
S. H. Lin: Electric field effect on spectra
TABLE II. ~~bh») for polar molecules.
Symmetry-forbidden transitions
f32F2
Ak~b(W)= 10 (3cos2X -1)
+ (3
"kab~U)~I[(Pa~)'Raa)2
A
f3wF2"
7'
-! IRoo 12I +~
7' [(2 -
cos~ -l)<PJ~)o Roo)PJ~)' AR(b,a)1 a~'u {kab(W)~I}
+ wF~ l: [(2 W
10". I
.
2
cos XlRaa'AR(b,a)
a2
cos2x)1 AR(b,a)12 + (3cos2X -1) Ip,g). AR(b,a)j21'0:':2
aW
{kab(W)~I}'
W
'
(4.5)
etc.
For a system of randomly oriented molecules, [(1/W)k~b(W)](l) vanishes and we obtain the change in the area under
the curve of (1/W)k~b(W) vs W induced by the electric field as
A[..! k~b(W~~ = 41T2;-2
Ct. L L
ftC
W
V
v'
p!~)[(DAav, bv' )(2»av+ f3(Zav.avDz(av, bv , )(1»av
t I
1- IR av•bv' 12 (Tr [Q(av)])}
+ t f32{fs IR av•bv' 121 R av •av 12 +ts IRav. bV' . R av •av 12 - 1- IR av• bv' 12 (I R av•av 12 >}] .
+ f3{fs- Rav. bV' 12 Tr [Q(av)] + ts Rav. bv' . Q(av) . Rav. bV' -
(4.6)
A similar expression can be given for the case of k!b(W), but sometimes it is convenient to have the expression
A[(l/w)k!b(w)] for practical application, which is given as
A[..!w.k!b(w)lJ =A[..! k~b(W~~
W
Ct r
+2
0
1T2
s C
2
L L p!~) [is (2 - COS2X) IR av•bv' 12 Tr {Q(av)}
v
v'
(4.7)
where
+ f3(Zav.av {cos 2XDz(av, bv')(1) + sin2xD.(av, bV,)(1)} >av] .
(4.8)
Equations (4.6) and (4.7) can also be obtaJ,ned from the use of Eqs. (3.16) and (3.20).
Next we consider
[k~b(W)]=
Jdwk~b(W)'
which represent the area under the curve of k~b(W) vs w, and will yield the frequency shift induced by the electric
field. As before, we rewrite Eq. (4.9) as
[k:b(w)] =[k~b(W)](O) + F[k: b(w)](1) + F 2[k: b(w)](2) + ... ,
(4.10)
where
(Ol .av D_(av, bv ')(0) ,
[k "ab (W )](0) -- -41T2
-",- "
£...J "
£...J pro)
av Wbv
as ftC
v
(4.11)
V'
":?-'
(0)
1 (Zav.av - ZbV' .bv' ) + f3wbv
(0)1 (
--)}]
[k "ab (W )](1) -- 41T2
lic " " P(O)[
av Wbv
.av D_(av, bv ')(1) + D_ (av, bv I)(O){' Ii
av Zav.av - Zav.av
,
Ct.
[k: b(w)](2) = Ct~~:
~ ~ P!~) [w~~l.avD_(av, bv' )(2) + D_(ab, bv'){l) {~ (Zav.av -
Zbv' .bV') + (3
W~~J.av(Zav.av -
(4.12)
Zav.av)}
J. Chern. Phys., Vol. 62, No. 11, 1 June 1975
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4508
S. H. Lin: Electric field effect on spectra
(4.13)
etc.
If the molecules are randomly oriented, then again [k~b(W)](l) =0 and Eq. (4.10) becomes
A[l.II(
"ab W)]-[k"()]
- abu) - [k"(
abW )](0)_4
-
1T2
2
F
1:
as ftC
'""''""'
(0)[W~v.av
(0)
L.JL.JPav
v
v'
<Dzav,bv
(
')(2»
(
,)(1){l-.;: (Zav.av-Zbv'.bv' )
av+ ~Dgav,bv
ft
/3W~~].av Zav.av}\v +t .B wJ~).av {* IR av•bv' 12 Tr[ a(av)] +tr R av•bv• . a(av) • R av•bv'
- -! IRav.bv·12 (Tr [a(av)]) + /3h\-1 Rav.bv·121 R av •av 12 +is (Rav•bv' . Rav.aY - -! IRav.~v' 12 (I R av•av 12) J}
+
+ 1:1f
[I R av•bv' 12 R av•av . AR(av, bv') + 2(Rav•bv' . Rav.av)Rav.bv· • AR(av, bv') J
+ 3~1f
{I R av•bv' I
Z
(4.14)
Tr[Aa(av, bv')] + 2Rav• bv ' . Aa(av, bv'). R av •bv·} ] .
Similarly, for k!b(W), we have
4 2 F2 '""'
A[k~b(w)J =A[k~b(W)]O + ~
L.J '""'
L.J p!~) [t /3wi~J.aJ& (2 - COS 2 X) IRav.bv.IZ Tr[a(av)]
0s"'C
v
v'
+fG- (3 cos 2X- I)R av •bv' . a(av) • R av •bv' - i IR av •bv' 12 (Tr [a(av)]) + /3[* (2 - COS 2 X) IRav.btl·121 Rav.av 12
+ ft (3 cos 2X- 1)(Rav• bv' . Rav.aY - -! IR av•btl' 12 <IR av•av 12 >J}
I
+ 1:1f {(2 - COS 2 X) R av •bv ·12 R av•av . AR(av, bv') + (3 cos 2 X- I)(R. v•bv' . Rav.av)Ratl.bv' . AR(av, bv')}
+ 3~1f {(2 - COSZX) R av • bv·12 Tr [Aa(av, bv') J+ (3 cos 2 X- I)R av •bv• . Aa(av, bv') . Rav. bV'} 1 .
I
(4.15)
It should be noted that in Eqs. (4.15), (4.14), (4.7), and (4.6), the terms involving Dx(av, bv')U), Dz(av, bv,)(1),
Dx(av, bV')(2), and Dz(av, bv')(Z) are, in general, negligible compared with other terms.
For allowed transitions, Eqs. (4.7) and (4.15) reduce to
A[;
k~b(W)]=A[; kab(W~o + ~~2 (3 coszX-l)[- t
2
Tr{a(a)}+P. b • a(a)' P ab + /3(- tlRaai +
iPab 'R•• 12)][k4b~w)OJ
(4.16)
and
2
2
A[k!b(W)] =A[k~b(W)Jo+ /3:0 (3 coszX -1)[ - t Tr{a(a)} + ab ' a(a)' ab + /3(- tlRaal 2 + Ip. b · R.aI )] [kab(w)O]
P
P
F2
~
~
+ 51f [ /3{(2 - cos 2 X)Raa . AR(a, b) + (3 cos 2 X- 1)(P. b . R •• ) Pab • AR(a, b)}
+ H(2 - cos 2 X) Tr[ Aa(a, b)] + (3 cos 2 X- I)Pab . Aa(a, b) . Fab } ] [kab;)O] ,
(4.17)
respectively. Notice that in contrast with A[k~b(W)/W], which arises solely from k!b(W)1 2), A[k~b(W)] arises from the
contributions of both k!b(W)~2) and k~b(W)~2).
Similarly, for symmetry-forbidden transitions, Eqs. (4.7) and (4.15) become
A[:;- k!b(W)]
=AB
k!b(W~O + /3:0
2
-1)~) -
(3 cos 2X
t Tr {a(a)} +
p!~) . a(a)· p!t) + /3(- t IRaa 12+ i p!t) • Raa 12)] [kllb:)~~
(4.18)
and
A[k!b(W)] =A[k~b(W)]O +
X[kab(W)~d +
~~2
(3 cos 2X- 1) ~ [- t Tr {a(a)} + p!t) • a(a) . p!t) + {3(- t IRaa 12 + Ii>!t) • Raa 12)]
F: 2)
5"
i
.a{(2 - cos 2 X}Raa . AR(a, b}+ (3 cos 2 X- 1)(i>!t)· Raa) p!~) . AR(a, b)}
+H(2 - cos 2 X) Tr[Aa(a, b)]+ (3 cos 2 X-1) i>!t) . Aa(a, b)· i>!t)} ][k4b~)~j] .
(4.19)
J. Chem. Phys., Vol. 62, No. 11, 1 June 1975
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4509
S. H. Lin: Electric field effect on spectra
TABLE III. Mk!b(lu)/'U) and ll[k~(ru)l for nonpolar molecules.
Symmetry-forbidden transitions
ll[k~(lu)l= /3F2 (3cos2X-l)L[-1.T {a(a)}+p(l>.a(P).p(I»)rkob(r.,)~jl
'v
]
10
3
j
~
/3F2
ab
r
ab [
J
'v
F2
Mk!b('U») = 10 (3 cos 2X -1) £.oJ [-! Tr{a (a)}+P'::>' a (P). pJL>)[kab(IU)~j)+ 1011 L [(2 - cos 2X)Tr {lla (a, b)}
A
A
j
j
+ (3cos 2x -l)P'::>.lla(P,b)'PJL»
Physically, [kab(W)~j] represents the area under the
curve of the absorption coefficient (in the absence of the
applied field) vs frequency induced by the ith inducing
mode. The expression [kab(W)~d / w has a similar meaning.
[kob:)~jJ
V. ELECTRICAL DICHROISM AND THE KERR
EFFECT
In Tables III and N, we give the approximate expressions for ll[k~b(W)/W] and ll[k~b(W)] for nonpolar and polar
molecules. Here ll[k~b(W)/W]O and ll[k~b(W)]O have been
ignored; and for polar molecules, the polarizability contributions to .1[k!b(W)/w] and ll[k!b(W)] have also been
ignored. If we are concerned with the field-induced
spectra in the rigid medium under the low temperature
condition, the expressions given in Tables N. a and
N. b can be used by deleting those terms which involve
(3. We should notice that in this case ll[k!b(W)/ wHo;
and for both polar and nonpolar molecules, ll[k~b(W)]
arises solely from the polarizability contribution.
Notice that ll[k~b(W)]O and ll[l/wk~b(w)]o are also proportional to F2 and are negligible because they involve
D.(av, bV')(l>, D.. (av, bV')(2), etc.
It is well known that the Kerr effect and electrical dichroism are related by the Kronig-Kramers relation. 2
Thus, in this section, we shall derive the expression for
the electrical dichroism and then apply the KronigKramers relation to obtain that for the Kerr effect. The
electrical dichroism for a particular electronic trans i.tion a- b is expressed as
(5.1)
The expressions for k!b(W) have been obtained in Sec.
III.
For a system of randomly oriented molecules, substituting Eqs. (3.19)-(3.22) into Eq. (5.1) yields
(5.2)
where
...c0>[(llD (av, bv ')(2)) /lP + (3 «llD av, bv ')(1) Z/lp,av )/IV
llkab ()
W 1 =4rwF2,,~
""_ L.- L.t rap
a.,""
+
*
V
v
,B{ 3R..",bv' • a(av)' R/IV,bu' - 1Rav,bu' 12T r [a(av)]} + ta-{3 2{ 3(Rap,bv' • R..p,/lp)2 - 1Rap,bu' 121 Rap,av 12} ] O(W - wbv~~lv) ,
(5.3)
TABLE IV. Mk!b(rV))/'u and ll[k~(ru») for polar molecules.
Symmetry-forbidden transitions
[k~«(,J)J = 13::
2
II
(3 cos2X
-1)~ (- il R".,12 + Ip,::>, R ... 12) eob~:)~']
~2F2
~
ll[k!b(r,J»)= 10 (3cos 2x-1)L (-iI R... 12+ Ip'::>·R... 12)[kob(W)~')+ 511 L [(2-cos2x)R.u,-llR(a,b)
A
,
j
+ (3 cos 2X -1) <PJt>· R".,)P'::> ·llR(a, b»)
[k/lb~V)~'J
J. Chern. Phys., Vol. 62, No. 11, 1 June 1975
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s.
4510
H. Lin: Electric field effect on spectra
-I Rdv,bv,12Rdv,dv' t.R(bv', av)} +ib- {3R bv ., dV' t.a(bv', av) . Rbv',dV -
bv ', dV 12 T r lt.a(bv', av)]}] Ii '(W - W~~).,dV) ,
(5.4)
1R
and
(5.5)
where t.D(av, bv')(2) =D z(av, bV')(2) - Dx(av, bv ')(2), and t.D(av, bv,)(1) =D z(av, bv')(1} - Dx(av, bv ')(1). As usual, the terms
involving t.D(av, bV')(2), t.D(av, bv')(1} are negligible compared with other terms.
In particular, for the rigid medium under the low temperature condition, Eqs. (5.3)-(5.5) reduce to
2 ' " ' " (O)(
(
')(2»
(
(0»)
( ) _ 41T2WF
t.kabWl«
L..JL..JPdvt.Dav,bv
dvOW-Wbv',dV'
v
C1. s ftC
(5.6)
v'
' ' ' ' (0)[( t.Zbv,avt.Dav,bv
( ,
)
(
')(1)) dV
W 2= 41T2WF2
t. k ab ()
ji2 "
L..JL..JPdV
a
c v v'
(5.7)
and
(5.8)
Applying the Kronig-Kramers transform,
2 1~ dw'
,
t.n(w)=A-P
'2
2t.k(w),
1T
0 W
-w
where A
=~Nc,
(5. ')
N being the Avogadro number, we obtain the electric birefringence (the Kerr effect) as
t.n/i>(w) =t.nab(w)l + t.n ab (w)2 + t.n db(w)3 ,
(5.9)
where
2
41TNF ", ~ (0)[( t.Dav,bv
(
')(2»
=--/i-L..J4PdV
as
v v
dV+{3
«t.Dav,bv ,)(1) ZdV,dV ) ]
dV
P Wbv,qy
(0)
1Tf3NF 2 "'p(O)
-w2 + 5
L..J av
as
v
W(O)2
bv,av
(5.10)
t2dw' 2
W
+
- W
. ~
41T NF2 '"
'" P (0) [(t.Z( bv,
, av ) t. D (av, bv ')(1» av
t.kab (W') 2=
L..JL..J
av
as
v
v'
Irs (3{3(R av ,bv' . Rav,av) Rav,bv' . t.R(bv', av) -
1 Rav,bv' 12
Rav,av . t.R(bv', av)}
+~{ 3R bv' ,av t.a(bv', av) . R bv' ,av - 1R bv' ,av 12 Tr [t.a(bv', av)]} ] P
2
W
(0)2
(0~2WbP'!Q!~
(Wbv',av -
2
(5.11)
W )
and
2
t.nab(wh =A - P
1T
Jor
.,
(5.12)
with
(5.13)
The above equation for the electric birefringence is true only when the optical frequency W is within an allowed band,
so that the contribution to t.n can be attributed to one particular electronic transition. In general, to obtain t.n we
have to sum Eq. (5.9) over all the possible transitions.
J. Chern. Phys., Vol. 62, No. 11,1 June 1975
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s.
4511
H. Lin: Electric field effect on spectra
At extremely low temperatures, Eqs. (5.10)-(5.12) reduce to
2
(0)
41TNF
"" "" (0)(
(
, (2»
Anab ()
W 1=
-.. - L.J L.J Pa. AD av,bv)
av P
CiSfL
v'
V
W""',a. 2'
(0)2
(5.14)
W bv ' ,av - W
2
(0)2
2
I
W + Wbv',av
-IR""',avl Tr[AQ(bv ,av)J) ] P ( (0)2 _ 2)2 ,
Wbv',av
W
(5.15)
and
2
(0) ( 2
(0)2
W
27T NF "" ""
(0)[ 3 I R .",,'· AR(
Anab (W) 3 = 15
..-3 L.J L.J P av
bv' , av )12 - IRa•• "", 121 AR (
bv' , av )12] P 2 bv',a.3w
«(0)2 +W""',a.)
2)3
•
av
as"
v
v'
(5.16)
W bv ' ,au - W
Notice that the phase difference between the two principal components of the light beam is given by
D=21Tl An
=
27T1KF2
A
(5.17)
'
where A is the wavelength of the light used, 1 is the path length of light, and K is the Kerr constant.
For allowed transitions, we may ignore the vibrational dependence of the transition moment R a•• bv '.
Eqs. (5.3)-(5.5) and (5.10)-(5.12) become
Akab(W)1=Akab(W)~+fotlF2kab(W)0[3Pab' Q(a)· P ab - T r{Q(a)}+t3{3(Pab • Raa)2-IRaaI2}],
2
3wF
Akab (W)2 = Akab (w)2o + -li-
[-.LIT t3 {A
3(Pab
+
0
to {3Pab '
In this case,
(5.18)
Raa)PAab ' AR(ba) - Raa' AR(ba) }
AQ(ab)· Pab-Tr[AQ(bam]a:[ka%W)O],
(5.19)
and
2
a [kab(W)OJ
Akab (Wh= WF2
lOli2 [3 IP ab ' AR (12
ba) - I AR (ba) 12] aw2
W
'
(5.20)
A
and
(5.21)
(5.22)
and
2
1TNF2 ( 2
(0)2)[
( ) a Q(a)0
( ) 11
()12
Anab ()
W 3 = 5a.li 2 3w + Woo
AR ba· a(w2)2' AR ba -"3 AR ba
Tr
2
{aa(w)2
Q(f)O}] ,
(5.23)
respectively, where
2
""
"" P (0)[( AD (av, bv ')(2»
Akab (W)01 -_ 4rwF
1;
L.J L..J
av
lls'''C
v
au
v'
(0) ,av) ,
+ t3 ( Zav.avAD (av, bv ')(1»]
av 6 (W - W"",
(0)(
('
)
(
,)(1» a.6'( W - Wb.'.a.
(0»
Akab (W)0_47T2WF2""""
2Ii! L.J L.J P av AZ bv , av AD av, bv
,
as c v v'
(5.24)
(5.25)
2
(0)
4 NF " " " (0)[(
(
')(2» a.+t3Za.,a.ADav,bv
(
(
')(1»]
W""'.av 2,
( )01 -__7T_..-_ "
Anabw
L.JL.JPa• ADav,bv
avP (0)2
(5.26)
2
2
(0)2
47TNF "" "" (0)(
('
)
(
,)(1» av P ( W(0)2+W""',av
Anab ( W) 0_
2-~ L..JL..J P av AZ bv ,av AD av,bv
2)2
(5.27)
Qs"
v
W~1V' ,au - W
v'
and
as'"
v
v'
W&v' ,au - W
Notice that
o unless we 0are dealing with the spherically symmetric system, we may ignore Ak a b(W)~ ,
Anab(wh, and Anab(w)z •
If we choose the principal axes of Q(a)o as the coordinate axes, we have the relations
3Q( a):
Q( a)o - T r [a( a)] Tr [a( a
>oJ ={ax< a)o -
a y( a )oH a,,(a) - a y ( a)}+ {ay(a)o - a.( a )oH ely ( a) - a.(a)}
+ {a.( a )0- a,,(a )o}{a.( a) - a,,( a)}
(5.28)
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4512
S. H. Lin: Electric field effect on spectra
and
3R •• o Cl{a)o
0
Raa - / Raa /2 Tr [Cl{a)o J ={Cl x{ a)o - Cly{a )o} (Xaa - Y aa )+ {Cly{a)o -
a~{ a )o}( Y aa - Zaa)
+{a~(a)o- ax{a)o}{Zaa-Xaa) ,
(5.29)
where aAa)o, a y { a)o, and Cl z { a)o are optical polarizabilities along their principal axes; a x{a), Cly { a), and Cl"( a) are
polarizabilities along these axes; and X aa , Y aa , and Z aa are the components of the permanent dipole moment in the
same directions.
The second and third terms in .:lnab{wh in Eq. (5. 21) represent the classical expression of the Kerr effect 2 • 13 which
is thought to be related to the orientation of molecules in the electric field; the second term is caused by the anisotropy of the induced moments, while the permanent moment leads to the third term.
After establishing the general relations between the electrical birefringence and electrical dichroism, from now
on, we will primarily be concerned with the electrical dichroism. For symmetry-forbidden transitions, Eqs. (5.3)(5. 5) can be written as
2
0 {3F
~ ()o [ 3PA ab(i) o Cl{a)oPA(ab/) -Tr [{)J
A(i) o Raa) 2- /Raa 12} J ,
( )
(5.30)
.:lkabWl=.:lkab{wh+WL..-kabWsi
Cla +(3 {3{Pab
i
Law)Jkab {W)O}
W si [(3{3{P~)
2
wF
.:lkab {W)2= .:lkab {w)g+5""If
/)
i
0
Raa)P~!)' .:lR{ba)- Raa' .:lR{ba)}
'
+ H3P~~)' .:lCl(ba)· p~)
-
Tr[.:lCl{ba)]}] ,
(5.31)
and
.:lk (w)
ab
2
3
= wF
10 n!
' " /)22 Jk ab {w)2i ([3/P(Oo .:lR{ba)/2_/.:lR {ba)/2]
~ /)w )
w
j
ab
(5.32)
Next let us find the moment relations for the electrical dichroism. We first consider [.:lk ab (w)/ w],
(5.33)
For allowed transitions, Eq. (5. 33) reduces to
[
.:lkab{W)]
W
2
{3F 41T2F2 /
12[ A
A
[
{A
2
= [.:lkab{W)~]
w
+W3Cl nC Rab
3PaboCl{a)oPab-TrCl{a)]+{33{Pab'Raa)s
1
Raa
/2}]
(5.34)
For symmetry-forbidden tranSitions, Eq. (5.33) becomes
[.:lk:{W)]
=[.:lka:,{W)rj
+
(31~2 ~ [3P~)' Cl{a)' P~:) _ Tr [Cl{a)J+ {3{3(P:~)o Raa)2-
/ Raa /2} ]tkab~)~iJ
(5.35)
Now we consider [.:lkab(w)],
+ {3[3{R av •bv'· Rav.av)2- /Rav.bv./2IRav.av/2]}+
2:
{3(R av •bv" Rav.av) R av •bll
0
.:lR{av,bv')
-/Rav.bv'/2Rav.avo .:lR{aV,bv')}+i{3R av •bv '· .:lCl{av,bv')· Ral1.bv'-/Rav.bv./2Tr[.:lCl{av,bv')]}] , (5.36)
where
[.:lkab{w)]O = !~%c2
~ ~ P~~) [W~lav <.:lD{av, bv,)(2) )al1+ (.:lD{aV, bV,)(l){ i.:lZ(av, bv')+ (3w~.av Zav.a.. })av ]
• (5.37)
In particular, for allowed transitions, Eq. (5.36) reduces to
2
[.:lkab{W)] = [.:lkab{w)]O+fn {3F 2 [kab(w)O] [3P ab • Cl{a)· P ab - Tr{Cl(a)}+ {3{3(P ab o Raa)2- /Raa/ }]
+~: [kab~)OJ
[,s{3{Pab
0
Raa)P all' aR{a, b) - Raa .:lR{a, b )}+ H3P ab
0
0
~Q{a, b)' P ab -
Tr [.:lQ{a, b)]}], (5.38)
and for symmetry-forbidden transitions, Eq. (5.36) becomes
[.:lkab(w)] = [.:lk ab (w)]o+fn-{3F 2
L: [kab{W)~i ][3P~)
0
Q{a)
0
P~) - Tr{Cl{a)}+ {3{3(PW • Raa)2-
IR aa/ 2 }]
j
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4513
S. H. Lin: Electric field effect on spectra
+
~~ :z;= [kab~)Z, ] [,B{3(P~) • Raa)P~)'
H3P~!)'
AR(a, b)- Raa' AR(a, b )}+
AO!(a, b)·
p~) - Tr [AO!(a, b)]}]
.
•
(5. 39)
We have tabulated the approximate expressions of Akab(w), [Akab(w)/w], and [Akab(w)] for practical application in
Tables V and VI, in which we have ignored Akab(w)~, Akab(w)g, [Akab(w)/w]O, and [Akab(w)]O and the polarizability
contribution to Akab(W)' [Akab(w)/w], and [Ak~(W)] of polar molecules. For the rigid medium under the low temperature condition, the expressions given in Tables V and VI can still be used, provided the terms involved f3 are
deleted.
VI. BAND SHAPE FUNCTIONS
1
Fab(w ) =21T
As we can see from the previous sections, for allowed transitions we have to consider the band shape
functions of Fab(w), F :b(W), and F~:(w); and for symmetry-forbidden transitions, we have to consider Fab(W)sh
F:b(w)st, and F:':(w)st. Let us start withF ab(w):
Fab(W)=4=~p!~)I<ea"leb,,')120(W-Wb<.?'!a,,).
(6.1)
Using the integral representation of the delta function,
Eq. (6.1) becomes
Fab(w) =;1T
1.:
dteif(W~)-W)~GI(t)
-
where
00
_00
(0)
)
",itbiwi
nwl
dtexp zt(Wba -w - Lt-2- coth 2kT
[.
~f3~2dj {coih::i - csch::i cos(Wi- ~~w;j}]
(6.5)
The integral appearing in Eq. (6. 5) can in general be
carried out by using the saddle-point method. 15-17 Here,
for SimpliCity, we shall assume thatLJ(f3~d~/2)>> I (the
so-called strong coupling case). In this case, we can
expand cos[w/ - (inw/2kT)] in power series of t; retaining terms up to t 2, we find
Fab(w)= J"-rr~ exp[- (Wba;W)2]
(6.2)
,
f
(6.6)
where
GI(t)= LLP!~~I<xa,,/QI)IXb,,'/Q;» 12
"I "I
X
exp[it{ (v; + t)w; - (v 1+ t)w I}] .
(6.3)
and
W-(O)=W(O)+
ba
ba
In Eqs. (6.2) and (6.3), the harmonic oscillator model
has been assumed for molecular vibrations. The exact
result of G I(t) has been obtained. 14 For small normal
frequency changes between the two electronic states,
G I(t) is given by14
_
[_ ~
tiwi J3jd;
tiwJ
GI(t) - exp
2 coth 2kT - 2 coth 2kT
It should be noted that the exact expression of the Gauss-
ian form of Fab(w) without assuming the smallness of
i:/s has been obtained by us. 1S Notice thatF~(w) and
F:;'(w) are given by
,
Fab(w)=
f3idj
tiwi
(,
inwl)]
+ 2 cSCh 2kTcos\Wlt-2kT
'
(6.4)
where f3~ =w/ti, Q; - QI =dl , and w; =wJ(l - t l ). Substituting Eq. (6.4) into Eq. (6.2) yields
",Md'Wf_ ",tlwi cothtiW..L
L.J
2
L.J 2
2kT
I
I
:;;ns (w ba -w)exp ~
-(0)
,,:(0)
\Wba
[
-w
D
)2]
(6.7)
and
F;;(w) =J~~3[-1+ 2(Wt~-w)2]exp[_
(W~~);W)2l
(6.8)
Symmetry-forbidden transitiOns
Akab(w) =
[
~~2~ kOb(r.tJ)~t [3P!1)'a (a).Ptl1) - T,.{a (a)} J+ ~:: ~ a~ tab~.tJ)~t}[3Ptl1). Aa (b,a)'pJ1) -T,.{Aa (b,a)} J
Akab(W)] = ~F2
W
[Akob(w)J =
10
L:t [kab(r.tJ)~t]J'"
[3PU>'cr(a)'P(U-T {a (a)}J
ab,.
.
W
~~ [kob(W)~tJ[3Ptl1)·a(a)·Ptl1)- T,.{a(a)}J + ~ ~ [~] [3Ptl1)'Aa (a,b)' Ptl1)- T..{Aa (a, b)} J
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4514
S. H. Lin: Electric field effect on spectra
Symmetry-forbidden tra,nsitions
~:~2 ~ kabh))~1[3(P,!Ll. Raa)2 _ 1Raa 12) + B~~F2 ~ a~ {kab~~)~I} [3(PdLl. Raa)P!Ll 'AR(b,a) -
Akab('JJ) =
~
~ k {kab(("')~I}
[3/P(O'AR(b a)/2 _I AR(b a) /2]
lOn- I a,,., 2
I,.,
ab'
,
+
[
AkabV,.,)]= p2F2
'v
10
[Akab(,,.,)]
=
~ [kabvv)~,l
w
I
R )2 _
J [3W(I).
ab
aa
Raa' AR(b,a)]
/2)
1n
£"aa
~2F2
fjF2
[k ( )0 ]
10
~ [kab(IJJ)~ll [3 (ildt l • Raa)2 - IR aal 2) + 5n ~ ~
The above equations for band shape function can also be
applied to vibrational bands of an electronic spectrum
by changing w~) and D accordingly. 18
[3(P,!Ll·Raa
)P.t l ' A R(a,b) -Raa'AR(a, b)]
obtain
F IW) =-..!£.~f"dt[(coth I'i(Ui+1)eit(",~g)+"'!·"')IIG It)
ab\ sI 4w1 27T • .,
2kT
J
P
Similarly, for symmetry-forbidden transitions, we
have
+( coth::i- ~eit("'fi:"'I·"')IfGi(t~
(6. 15)
or
Fab(w)sI = ~/COth
or
Fab(w)sI=
2~r: dtelt<"'~)·"')KI(t)~/Gi(t)
,
(6. 10)
where
K,(t)=
~~p!~! 1(X av , (Q,) IQIIXbvj(Q;»
v, VI
12
(6. 11)
Using the same technique as that for G itt),
be Simplified as
14,19
KI(t) can
(6.12)
Kdt)=K:O)(t)G,(t) ,
where
KlO)(t) =
i3? (COth ~; I
tanh ~:)+ i3f (coth X~
.
~
_tanh X~)
,
2~:2 tanh~ + Mtanh i} (i3t~ coth ~I +i3~ coth ~)
(6.13)
with JJ.; = - itw; and XI = itWI + (liwt/kT). If the normal
frequency change of the ith inducing mode between the
two electronic states is ignored, then the dominating
term of KlO)(t) can be expressed as
K(O)(t) =-..!£.[{COth l'iWi + l)e""'1 + (coth l'iWI I
4Wi \
2kT
2kT
1\'J e·""'!] .
(6. 14)
Substituting Eqs. (6.14) and (6.12) into Eq. (6.10), we
+
::i
::i -
~i (coth
+ l)Fab(W - Wi)
l)F ab(W + WI)
(6. 16)
In other words, the spectral band shape of a symmetryforbidden transition is the superposition of two ordinary
allowed bands F ab(W - WI) and F ab (w + WI) weighting by
the temperature factors coth(l'iwt/2kT) + 1 and coth(l'iwd
2kT)- 1, respectively. In the low temperature range,
the second term in Eq. (6. 16) is unimportant. Notice
thatFab(w)sI is not normalized; however, the normalization constant can easily be found to be (Ii/2w I )[coth(liw l /
2kT)].
Next we shall demonstrate the band shape functions,
Fab(w), F:..(w), andF::(w) by examples; for this purpose, we put Fab(w) in the dimensionless form as
r.
{1
(180-w)2]
Fab(w) =.J6'O; exp L60
•
F:
(6. 17)
F:':
(w) can be found by differentiating Eq.
b (w) and
(6. 17) with respect to w. The plots of F ab (w), F:W, and
F:':(w) against ware shown in Fig. 1. The choice of
D = 60 abd w~) = 180 actually corresponds to the F center
in KCl. 20
Now, if the vibrational bands for an electronic transition are resolved, then the over-all band shape func-
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4515
S. H. Lin: Electric field effect on spectra
are quite well resolved for F:b(w) andF:;(w). In Fig. 3,
show the curves for the case in which the vibrational
spacing is 20; in this case, the vibrational bands are
well resolved for Fab(w), F :,,(w), and F ::(w).
(a)
10
N08
)C
'"
~6...0
tE
4
2
....-;:;;;t::....-.L..,,-~-~~-.....-~~-...,I;;;--~2~00
160
W
As shown in the previous sections, at low temperatures and in the solid phase, F:b(w) [or F:"lb,,'(w)] appears in Ilk!~(w) and Ilkab(w) for nonpolar molecules, and
F::(w) [or Fa~.",,!(w)] appears in Ilk!b(W) and Ilkab(w) for
polar molecuies, while Fa,,(w) [or Fa"I""'(W)] appears in
the ordinary absorption spectra. Thus: from Fig. 2,
we can see that the resolving power is better for fieldinduced spectra than for ordinary absorption spectra and
that field-induced spectrafor polar molecules are slightly better resolved than those for nonpolar molecules.
VI. RADIATIVE AND NONRADIATIVE PROCESSES
It has been shown that when the adiabatic approximations are used in the zero order basis set, the expression for the non radiative rate constant for the electronic
transition a - b can be expressed in the golden rule form
as 14 - i6
(a)
10
8
160
170
w
180
190
2
200
(c)
2
160
220
w
(b)
8-
.., 4
o
x
~O~--------~------------~----
~
160
200
180
,-S
u..
220
W
FIG. 1. (a) The plot of Fa"~.tl) vs W; (b) the plot of F,{b(W) vs W;
and (c) the plot of F:& (,.tl) vs w.
160
tion for an allowed tranSition is simply the summation
of the individual vibrational band shape functionFau/hj (w)
weighted by the Boltzmann factor and Franck-Condon
factor, 18
180
w
200
2
(c)
2
(6. 18)
To demonstrate Eq. (6. 18), we set T:; 0 for convenience;
for the relative Franck-Condon factor, we choose l/v;l,
and for Fa" "",(w) we use Eq. (6.6). In Fig. 2, we plot
Eq. (6. 18) 'fo~ the case in which the spacing between
the vibrational bands is 10. As we can see, the vibrational bands in this case are not resolved for F 4b (w), but
220
W
FIG. 2. (a) The plot of Fab~") vs '''; (b) the plot of F,{IJ~") VS ,.tl;
and (c) the plot of F:.t, (w) vs '" with the vibrational spacing 10.
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s.
4516
H. Lin: Electric field effect on spectra
where Tav,bv' represents the Born-Oppenheimer coupling
strength defined by
(7.1)
a:
1(~auIH;ol~bu,>12= l-li2(<I>a9aul~ :~: Q:,)_n22 (ct>a9avl ~ :~b 9 bV') 12.
Ta»,bU'=
(7.2)
In the presence of an electric field F along the space-fixed z direction, ~a{e) will depend on F through P ov ', T aw,l>v',
and 5(EbV' -Eav) and can be conveniently expanded in power series of F as
Wba(.B) =wi~)(m + Fwi!) (13) + F2Wi~)(I3) + ••• ,
(7.3)
W(O)('B)- 21T "
(7.4)
ba
-
II
"P(O)T(O)
L;;:
~
bu'
av, bu'
5(E(0) _E(O»
av'
bu'
where
W;!) (e) =
2; L L p~~[{T!!!bVr+$(ZbV',bV'
v
)T!e~bV,}o(E~e~ -Ede»+T!~!bV·~Z(av,bv')o'(Ei~~
-ZbV',bV'
v'
-2
-E!e»],
(7.5)
-----
+ Zbv',bv' - ZbV',bv,ZbV'.bv')]
X~Z(av ,
bv')o'(E(O)
_E(Q»+.!.T(O)
{~Z(av , bv,)2 o "(E(0)
bv'
all
2 av,bu'
bv'
_E(Ol)+~
aw
a .(av
.,
bv')o'(E(fJ)
-E(Q»}]
b,,'
av
(7.6)
•
Using the same argument as that presented in the previous sections for absorption spectra, we can easily show that
the terms involving T~!!bV' and Ta~~bv' are, in general, negligible.
For a system of randomly oriented mOlecules, wi!) (13) =0, and Eq. (7.3) reduces to
~ Wba (I3) =Wba «(3) - W~~) (13) = F2[Wb~) «(3)1 + W~~) (13)2 + W~~)(!3>a] + ••• ,
(7.7)
where
W:~)(!3h =wi~)(eho + ; :
1; f.: P~~~ T~~~h,[Tr{a(bv')} - (T,.{a(bv')}) + e{ 1Rbv,bV 12 - (I Rbu',bU' 12 »]5(E~~ - E!~»
,
(7.8)
with
W(2) (0)
ba
I>
10
=21Tn
"
"
~ L...J
»'
R(Q)
bv'
U) Z
(OT
T(2»
IJ 41.1,11,,' b,,' ,b,,' +
av,but
"(E(O)
b1l
av u
-
E(O»
av
,
(7.9)
(7.10)
with
W~)(I3)20= 21T
L L p~~? (T~!!bV,~Z(av, bv'»avo'(E:e! -E:~»
n»
(7.11)
u'
and
(7.12)
I
Notice that W;!)(,sho and W;~)«(3)20 can be neglected for
practical calculation.
Here W:!)(,s)20 has been ignored. If we let ~E:~) represent the energy gap between the two electrOnic surfaces, Eqs. (7.13) and (7.14) can formally be written as
It has been shown that the dipole moment and polarizability vary only slightly with normal vibrations. If
we ignore this small effect, then W~!)(i3h is negligible,
and Eqs. (7.10) and (7.12) become
Wb~)(,s)2 = t[,gR"b· ~(a, b) + iT,.{~a(a, b)}]
(7.15)
and
Wb~)(!3)3=tl~R(a,b)12 a(~~~»Z W:~)(m,
x ".L...J "L...J R(O)
T(O) " '(E ltv'
(0)
bu' I2v,bp'V
»
u'
_
E au
(0»
(7.13)
and
Wb~)(i3)3=
3; I ~R(a, b)12 ~ ~ P~~T~~~b,,·511(E~~ -E!~».
(7.14)
(7.16)
respectively. Notice that W~~)(I3) is the rate constant of
radiationless transitions in the absence of electric field.
Equations (7.15) and (7.16) show that the measurements of the electric field effect on radiationless transitions provide us one way to determine the excited
state properties, provided we have the knowledge
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4517
S. H. Lin: Electric field effect on spectra
follows. Introducing the integral representation for the
delta function, W~~)
has been shown to be expressed
as!4
(m
(a)
10
W~~)(!3) =;2 L IR/(ab) 12
8
xl
j
t' dte-(jt/~ )ll.E~)K/t) II'
-~
(7.18)
The quantities R,(ab), Kj(t), and Gj(t) are defined in a
previous paper. 14 In Eq. (7.18), the Condon approximation has been used and the second term in Tav,bv' in Eq.
(7.2), which is at least 1 order higher than the first
term, has been ignored. It follows that
6
4
2
160
190
W
&
(0) ( )
a(AE(O»Wba
13
ba
-_ -
(b)
8
x
I ( )12
11i 3 ""
i...J R j ab
j
i~ dtte-(lt/~)U~~)Kj(t)
n'
-~
,
Gj(t) .
j
GJ(t)
J
(7.19)
.oj
fab(w)
and
x10 3
a(A~~~»2 W~~) (13) =-~ ~ IRj (ab) 12
0
-4
x
-8
i~ dtt2e-(it/~ )U~~)Kj(t)n' GJ(t)
-~
160
190
(7.20)
Applying the saddle-point method to Eqs. (7.19) and
(7.20) yields
o
220
W
J
a
a(AE~~»
~O) (
) _
.*
tt
(0) (
ba f3 - --;; Wba f3
)
(7.21)
and
(7.22)
160
w
220
FIG. 3. (a) The plot of F ."v,,) vs '''; (b) the plot of F~bV") vs ''';
and (c) the plot of F~6 (r,,) vs '" with the vibrational spacing 20.
of the dependence of the nonradiative rate constant on
the energy gap (the so-called energy gap law). On the
other hand, if we know the excited state properties, we
may use Eqs. (7.15) and (7.16) to determine the energy
gap dependence of the rate constant of radiationless
transitions; for nonpolar molecules, we can only determine &[W~~)(j3)]/[&(AE~~»], and for polar molecules,
we may determine both &[W~~)($)]/[&(AE~~»] and
a2[W~)(t1)]/[ &(AE:~)f]·
In particular, for rigid systems, if the temperature
is so low that no molecular motions have energies
smaller than kT, then Eq. (7.7) reduces to
AWba (t1) =
&(A!~~)Z Wb~)($)]
"" 1
i...J
J
Q'2.>2
, it* '" .,
1 "" }- ,
"2 I-'J
uJ wJ e
J =AWba +"2 ~ bJ wJ
(0)
,
(7.23)
J
where 13; =(~/1l)1/2 and tJ and dJ denote the .normal frequency and coordinate displacements, respectively. If
we introduce an average frequency w' on the left hand
side of Eq. (7.23), t* can be solved to yield!?
(7.24)
-I
can usually be apprOximated by the maximum frequency like the C-H stretching. For the strong coupling
case, t* is given by!8
W
~2 [Tr{Aa(a, b)} &(A~b~)W~~)(13)
+ 1 AR(a, b)12
where t* represents the saddle-point value of t and has
been obtained in connection with the discussion of temperature effect and energy gap law in radiationless transitions and energy dependence of radiationless transitions in isolated molecules. In other words, through the
measurement of the electric field effect on radiationless
transitions, we can determine t*, which can be used to
study the temperature effect, energy dependence of radiationless transitions in isolated molecules, and energy
gap law. It has been shown that for the weak coupling
case, t* is to be determined by!7
•
(7.17)
Next we shall attempt to find other relations for ~!)
x (13)2 and W~!) (l3h by using the saddle-point method as
(7.25)
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4518
S. H. Lin: Electric field effect on spectra
If we introduce an average frequency
it*
Awt~)
-'2
Sw
w',
then
coupling case, while the color center belongs to the
strong coupling case.
+Sw'
(7.26)
liw"
coth 2kT
where S '" t Li f3? ~. In this case, the band shape function for absorption spectra is given by
Next we turn to radiative processes. The radiative
rate constant Aba (j3) for the electronic transition b - a
can be expressed as
Aba(Fl) =K
LL
v
F
ab
(w)=
[T
J;D
exp[-..!(w(O) ",f3JdJ
D
ba + L..J
2
Wi
w
)2J
P bv ' w~v' ,av D(bv', av) ,
(7.27)
where K=4f.$/3lic 3 and
J
'" ~
-.ty
2 coth !!.Jl1
2k T -
v'
D(bv',av)= l<bv'IRlav)12", 1 <8 bv ' l~aI8av)12
(6.6)
,
where D '" LJ f3; ~ w7 coth (liw/2kT). Thus we can see that
if we ignore the normal frequency displacements t i , we
may approximate the numerator of Eq. (7.25) by the frequency at the absorption maximum and the denominator
of Eq. (7.25) by t D. It should be noted that the aromatic
molecules or other organic molecules belong to the weak
Here the factor f.$ has been introduced to account for the
medium effect due to the electromagnetic field. As for
radiationless transitions and absorption spectra, we expand A ba (f3} in power series of F,
(7.28)
where
(1)(f3)-K""
p(O)
(0)2 [f3 (0)
(Z bv',bv'- Z--)D(b'
~ AZ(av, b v')D(b'
A ba
~~
bv,Wlv',ov.Wbv',av
bV',bv'
v,av )(0) +1f
v,av )(0)
(0)
+Wbv',av
D(b'
v,av )(1)J ,
(7.29)
(7.30)
with
(2)(f3) 0 K~
"~
'"
A ba
-
p(O)
(0)2
bv' Wbv' ,av
)(2)
f.iI
(0)
(Z bV' ,bv
w bv' ,av D(b'
V ,av
+ ""Wbv'
,au
[(0)
t
Zbv'-,bv'
- ) D(b'
)(1) + ~
D(b'
)(1)J
V ,av
Ii ~ Z( av, b')
V
V ,av
-
(7.31)
and
(2)(f3) 1-K
" " p(O)
(0) D(b'
1. (0)
(av,bv ')} + 3f3
(0)
(
--)
A ba
~~
bv,Wbv',av
v,av)(O)[~{.!
Ii li A Z( av, b v ')2 +ZWbv',.vAa,u
Ii Wbv',av
Zbv',bv' -Zbv',bv'
AZ ( aV,bv ')
(0)2
+ W bv' ,av
{1.2""J!l[
- ( b')]
a.. (b')
v - a.. v +
02
p
(1
2
1 -2--
2
--)}]
"2 Zbv' ,bv' - '2 ZbV' ,bv' +Zbv',bV' - Zbv', bv' ZbV' ,bv'
(7.32)
•
In general, unless A~!) {f3) 1 vanishes, A~!) (f3)0 is negligible.
For randomly oriented molecules A~!) (f3) '" 0, and Eq. (7.28) reduces to
(7.33)
where
A(2)(f3)",A(2)(f3)
+K"''''P(O)w(O)
n(bv ' av)(O)[~IAR'av
b.
b.
0
~ ~ bv' bv' •• .,......,
li2
~,
+
bV')12+~~(0)
bv')}
21i bv' ,.v T r{Aot'av
~,
~W~~? ,.v 1R bv' ,bv' • AR(av, bV' ) 12 +~ f3 w~~?:.v{Tr[ ot(bv ' )] -
~v' .bv' 12 - <1~v"bv.12»)}]
<Tr[ ot(bv')])+ f3( 1
•
(7.34)
Ignoring the vibrational effect on dipole moments and polarizabilities, Eq. (7.34) becomes
A (2) (Il) _ A (0) (Il)[' AR(a, b) 12 Tr{Aot(a, b)} f31 R bb • AR(a, b) 12]
ba I-' - b. I-'
li 2w(0)2
+
21[wIO)
+
liw(O)
,
~
~
(7.35)
~
Here A~~) (f3)0 has been ignored. Substituting Eq. (7.35) into Eq. (7.33), we obtain
(7.36)
In particular, for rigid systems under low temperature conditions, Eq. (7.36) reduces to
A~(f3) -1 F2['AR(a,b)12 Tr{Aot(a,b)}]
A (0) (f3) - +
1[2W(0)2
+
2liw(0)
•
b.
ba
(7.37)
~
Equations (7.36) and (7.37) show that for polar molecules, the contributions from both dipole moments and polarizabilities are equally important to the electric field effect on radiative processes.
Combining Eq. (7.7) with Eq. (7.36), we obtain the electric field effect on the lifetime
T ~ (f3)
as
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S. H. Lin: Electric field effect on spectra
1
-«(3)
Tba
1
= T(O)
«(3)
ba
+F
2[{I~R(a,b)12
1f2W(0)2
ba
+
TT{~a(a,b)}
2lfw(0)
ba
+
4519
(3IRbb.~R(a,b)12}
Ifw(O)
ba
xA~~)«(3) +{t (3R bb • ~R(a, b) +~TT[~a(a, b)]} &(A~!~» W~~)«(3) +~I ~R(a, b)j2 &(A:!~»2 W~~)«(3)]
(7.38)
In particular, if we are dealing with rigid systems under low temperature conditions, Eq. (7.38) becomes
T
1 _
«(3) ba
1
T(O)
ba
«(3)
+F2[{I~R(a,b)12+TT[~a(a,b)]}A(0)«(3)+!T{Aa(a
b)} ( &(0»W~~)«(3)+~I~R(a,bW&(~:(0»2W~~)«(3)1.
1f 2w(O)2
2lfw(0)
ba
6 T
,
8 ~Eba
ba
J
ba
(7.39)
ba
It can easily be shown that, in general, the electric field effect on radiative processes is negligible in comparison
with that on nonradiative processes.
VIII. APPLICATION TO F CENTERS
Although the F center has been investigated for a number of decades, it is only recently that an understanding
of the states responsible for the emission of light is
emerging. 21 Swank and Brown22 were the first to measure the decay time of the F center luminescence. They
found that the radm,tive lifetime of the excited center was
approximately 2 orders of magnitude longer than the value to be expected from the oscillator strength in absorption. Of various explanations which they proposed for
this discrepancy the diffuse p-state model gained wide
,
23
acceptance as a result of Fowler's work. He was able
to show that the 2p-like state of the excited center would
become more diffuse as the surrounding ions adjusted to
the change in charge distribution following the optical
excitation. This would reduce the matrix element for
emission to the ground state. However, the recent work
of Bogan and Fitchen24 and Kiihnert25 could not be explained by the diffuse p-state model. Both these authors
studied the Stark effect on the relaxed excited state of
the F center by analyzing the electric-field-induced linear polarization of the luminescence. Their results
implied that the luminescent state of the F center has a
considerable amount of 2s character. Based on this
fact, Bogan and Fitchen assumed that the relaxed excited
state consisted of strongly mixed but not degenerate 2s2p states.
Grassano et al. have reported the electric field effect
on the absorption spectra of F centers in various alkali
halides 26 - 28 ; the effect was detected with the measuring
light polarized both perpendicular and parallel to the applied field. In this section, we shall apply our theoretical results presented in the previous sections to interpret their experimental results. The field-induced absorptic spectra of F centers reported by Grassano et ale
were measured at 55 OK (see Figs. 4-8). We shall assume that at this temperature, the low temperature
equation for ~~b(W) can be used,
Ak~b(W) 10a$1f
41T2W~2C Lv'
(21R..o,bV.12 Tr{Aa(bv', aO)}
- R..o bv' • ~a (bv', aO) • R..O,bV'] 6' (w -
w~~! ,aO) .
,
(8.1)
From Figs. 4-8, we can see that the F band is composite, consisting of three bands with the type of band shape
functions like F;b(W) discussed in the previous section.
This is consistent with the model proposed by Bogan and
Fitchen, if the three bands are assumed to arise from ,
Is - 2s' (mixed with 2PII), Is - 2p,. (or 2Pl')' and Is - 2PIl
(mixed with 2s). It follows that Eq. (8.1) can be written
as
Ak~b(W) = 1~:~~
1R..b 12 (2TT{Aa(ba)}
- Pab • ~a(ba). Pab ] F:b(w)
(8.2)
for allowed transitions. For vibronic-induced transitions like Is - 2s' and Is - 2P:, the same equation applies, provided that there is only one inducing mode; in
that case lR..b 12 in Eq. (8.~) should be replaced by
1(8R..b/&Qi)of(If/2wi), and P ab by p~!) (cf. Table I). For
3
KBr
2
" 1
'" Kab(w
x 103.nI-_..L=----_---\--_ _-J.--_ _+--_~
-1
FIG. 4. The plot of ~k~bV")
• • " experimental.
VB
r"
for KCl: -, theoretical;
FIG. 5. The plot of ~~b(r")
••• , experimental •
VB r"
for KBr: -, theoretical,
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s.
4520
H. Lin: Electric field effect on spectra
KI
NoCI
2
II
2
1
x~fM0
-1
-2
1.7
1.9
W(ev)
2.1
2.3
FIG. 6. The plot of Ak~bhJ) vs 'v for KI: -, theoretical;
••• , experimental.
25
2.7WCev) 2.9
3.1
FIG. 8. The plot of tl.k~b(w) vs tv for NaCI: -, theoretical;
••• , experimental .
convenience, we shall rewrite Eq. (8.2) as
(8.3)
as the strong coupling approximation can be applied to
F centers.
From Figs. 4-8, we can see that the F band of fieldinduced spectra consists of three bands, each of which
can be represented by ~k~b(W) given by Eq. (8.3). The
parameters A, w~~), and D for each separate band have
been determined for KCI, KI, KBr, RbCI, and NaCI and
are given in Table VII. The theoretically calculated
curves are compared with experimental results in Figs.
4-8; the agreement is reasonably good.
According to the model proposed by Bogan and Fitchen,
the 2s and 2p states are split as 2s', (2px 2py), and 2p;,
according to the energy levels. But the exact energy
spacings among them are not unequivocally determined.
If the model of Bogan and Fitchen is true, then from
Table VIII. a we can determine the energy levels of 2s',
(2p" 2py), and 2p;. Ignoring the differences in Stokes
shifts for various levels (i. e., assuming that Stokes
shifts are the same for these transitions), the energy
spacings between (2px 2py ) and 2s' and between 2p; and
(2A 2py) for KCI, KI, KB r , ~CI, and NaCI are given in
Table VIII.
alkali halides has been measured by Fitchen e tal. 29
For convenience, their results are reproduced in Fig.
9. In this case, the electric field dependence of life-
.10
KCI
T =4.2'K
• = I ..
-A
1:
.OS
.00~--------S~------~1~2------~I~~~10~3~-----­
[F(kv/c m ) I
KF
.OS
The electric field effect on lifetimes of F centers in
.00~------~S--------~------~lS~X~10~-----2
[F(kv/cm) I
.10
RbCI
8~
T=.4.2"\(.
NaF
• = I.
-41:
1:
*=ll
5)S
.00~~----~S~------~1~0------~lS~'~10~----­
[FC kv/cm )]2
W(ev)
FIG. 7. The plot of Ak~h) vs
•• • experimental.
t,J
for RbCl: -, theoretical;
FIG. 9. Plots of the electric field dependence of lifetimes for
(a) KCl, (h) KF, and (c) NaF •
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4521
S. H. Lin: Electric field effect on spectra
times of F centers can be obtained from Eq. (7.39) as
_1__ 1
F2[T..{~a(a,b)}A(0)(f3}
T b. (tl) - T ~~) ({3) +
2nw~~)
b.
TABLE VIII.
Energy spacings in F centers.
KCl
AE2 = O. 203 eV
AEI = 0.125 eV
(8.4)
As mentioned before, the contribution from the first
term in the square bracket in Eq. (8.4) is negligible
compared with the second term. Thus, from Fig. 9,
we can determine the variation of radiationless transitions of F centers with respect to the electric field; we
obtain the slopes of the plot of - A'T/ 'To vs F2 for F centers in KCI, KF, and NaF as 0.58x10· 5 cm2 /ky 2 , 0.40
5
2
5
2
X 10- cm / ky 2, and 0.32xlO- cm /ky2, respectively.
For F centers, Wb.(tJ) consists of the ordinary radiation-
KBr
AE 2 =0.138 eV
AEj = 0.125 eV
NaCl
AE j =0.176 eV
AE 2 =0.234 eV
KI
AE2 =0.134 eV
AEj =0.123eV
RbCl
AE j = 0.113 eV
AE 2 =0.149 eV
TABLE VII. Results of F centers. a
KCl
II
_
-3
[ (2.251- ,-,,)2J
o.k Ib Vv)j-5.180 X I0 (2. 251-w) wexp - 0.819xI0- 2
"
-3
[ (2.376 - ,-,,)2]
o.k ab ('-")2=-4.550XI0 (2.376-,-,,) I-"exp - 0.983xI0- 2
" )
-3
)
[(2.579-W)2J
AkabVv 3=1.155 x l0 (2.579-w Ivexp -1.037xI0-2
KBr
"( ) _
-2 (
)
[(1.965-W)2J
Aka& 'v 1 -1.618 x 10 1. 965 - w wexp - 0.614 x 10-2
" ( ) _
Ak'b
If (
2--2.057xI0
,-"
)
_
Akab 'v s-I.059 x l0
-2
(2
-2
(2
)
[ (2.090 _1-,,)2 ]
.090-1-" wexp - 0.7695xI0-2
)
[(2.228-1-,,)2]
.228-1-" wexp - 0.441 x 10-2
NaCl
If (
)
-s (
)
[(2.617 _1-,,)2 ]
2.617-lv 'vexp -1.483XI0-2
Akab'v j=I.616X10
"()
[(2.793-1-,,)2]
2.793-'v I-"exp -1.444x10-2
-S()
Akab -" 2=-1.611xI0
'
" ( )
027x-10,-" j2 ]
Akab
,-" S = 0.6805 x 10-3 (3.027 - 1-" ) '-" exp [(3.
- 1.300
2
KI
"( )
)
[(1.811-W)2J
1.811-w wexp - 0.650x10- 2
-2 (
AkabW j=2.554 X 10
)
-2 (1.934-w ) wexp [(1.
934 - ,-" )2]
Ak Ifab ( l-"2=-1.169x10
-0.871xl02
Ak~(w)s = 0.6383 X10-2 (2.068 -Iv) ,-" exp [- i\~~8x~~X]
RbCl
II ( )
(
)
[(2.000 - W)2]
Aksb 'v 1=0.8725 2.000-w wexp - 0.583xI0-2
A L" (
w)2]
'"""'ab
'v ) 2 = - O. 5632 ( 2.113 - w ) 'vexp [(2.113
- 0.911 x- 10-2
Aklf (w )
(
)
[(2.262 _ Iv)2]
ab 3=0.2509 2.262-w wexp -0.744XI0-!
~he
w values are in eV, and Ak~(W)h Ak~(W)2. and Ak'~V-")3
represent the three bands resolved from the spectra.
less transitions and thermal ionization. At 4.2 OK, the
contribution from thermal ionization is probably negligible.
Now the question arises as to whether it is possible to
measure the electric field effect on lifetimes of excited
electronic states of organic molecules, so as it is well
known that in F centers the electronic charge distribution is so widely spread that the polarizabilities of F
centers are much larger than those of organic molecules
[cf. Eq. (8.4)]. To estimate the order of magnitude of
Tr[~a(a, b)] for F centers, we shall assume that - AT/
'TopZ +- 0.4 X 10-5 cm 2 /ky 2 obtained from the slope of the
plot of - A 'Tho vs pZ shown in Fig. 9 and the average
phonon frequency is 2 X lOIS sec-I. It follows that from
Eqs. (8.4) and (7.21), we find that 10.01-10- 19 cms . In
other words, the polarizability of F centers is about 4
orders of magnitude bigger than that of ordinary nonpolar
organic molecules; to compensate for this, one can use
the field strength of 104 kY /cm which can be attained by
the pulse technique. 2 However, there should be no difficulty in studying the electric field effect on lifetimes of
solvated electrons because of diffuse electronic charge
distribution. 30 For polar organic molecules, the electric
field effect contributed from dipole moments is about 2
orders of magnitude bigger than that for nonpolar organic
molecules contributed from polarizabilities due to the
ratio of the electronic energy to the vibrational energy
[of Eq. (7.17)]; in this case, the electric field effect on
lifetimes can be observed by using the field strength of
103 kY /cm or by studying the molecule with big changes
in dipole moments like charge-transfer complexes or
large polar molecules.
In concluding this paper, it should be noted that, in our
discussion, the electric field F represents the effective
field and not the applied field; we have not attempted to
discuss the relation between the applied field and effective field, as this problem has been widely studied and
well documented. 29
ACKNOWLEDGMENT
The author wishes to thank H. P. Lin for helping in
the numerical calculation and for preparing the figures.
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S. H. Lin: Electric field effect on spectra
4522
APPENDIX
A
In this appendix, we are concerned with the calculation of D.(av, bv')(n) and the spatial average of some quantities
appearing in Sec. III. Using the Rayleigh-Schrodinger perturbation method, the wavefunction Way perturbed by the external electric field F can be found as
Z~",avZftv,av)J
_,T,(O) ,T.O)F +"'av
,T,(2) 1'-+'"
..,2
+ ••• -"'av +"'av
E O) EO)
av -
(A. 1)
cv"
A similar expression can be obtained for Woo' :
2Zav ,bv' «W~;! I Z I w:e»+(w~11 Z I Wd~»)
D• (av , bV')(2) =[(W(O)
IZ I w(l)
+(W(1)
IZ I w(O»12
+ 2Zav.bv' [(,T,(2)
IZ I W(O»
+ (,T,
(1) I Z I ,T.(1» + (.T, (0) I Z I,T,(2»J
au
bv t
av
but .
~bV'
av
~ 00'
';l"av
';l"l1v'
~av
D.(av, bv')(l)
(A.2)
=
(A.3)
etc.
Substituting Eq. (A. 1) into Eqs. (A. 2) and (A.3) yields
,
,
')(1)-2Z
" ZbY'~6v"ZCII",av
'" ZIlv"8f"Zav''',av
n(
:6 av, b v
aV by' (L..J
)
(0)
+ L..J
(
10)
t
cv"
E GV " -Eev
dv 'l '
Edv '" -E bv'
2Z
+
av,bv'
[_~'"
2
bv' -
cv"
cv"
(A.4)
,
I Zey" 'gv 12 _~ ",' IZcv"'bv' 12 +,,' Zby"CII" ("" Zev",4y'" Zgv""a y
E(0»2
2 L..J (EtO) EcO»)2 L..J E(O)
(0)
L..J
.(0)
( )
L..J (E(O)
cv"
+ , , ' ZeY",a~ (",'
L..J E(O) E 0) L..J
cv"
)
dv'"
-
av
cv"
Zev"'~v'"
Z"lf''''bv'
E() E 1 )
bv' -
tlv'"
cv"
-
b1"
av -Eel'"
CV"
dv'"
Eav -Eav'"
'Ny]
Zev" ,bY' Z'8' ,bY') " , ' , , ' ZbY' ,ay" ,Zatl" tCV"ZCII"
(O) E)
+ L..J L..J (E(O)
E(O»(E(O)
E(»
E bv'
cv"
cv"
dv ' "
4v'''1111
cv" a1'
-
Zev",gyZ~V,gv)
(0)0)
Eav -Eev"
,
(A.5)
etc.
Dx(av, bv')(n) can be ,obtained similarly and will not be given here. (D. (av, bv ' ) (2»av, (D.(av, bv')(1) Zav av av, and
(.IlZ(bv'av) D.(av, bv )(1»av can easily be calculated by using Eq. (3.9). For example, for (D.(av, bv')/l) Zav,a'>av, we
have
(D.(av, bv')(1) Zav,av)av
2
=15 L
CUi'
'E(O) ~ E(O) ([Rav,av' Rav,bV' ][R"v' ~ev" • Rev" ,av] + [Rav,av' R bv, ,ev" ][Rav,bV' • Rev" ,avl
ev"
a1'
2
+ [Rav,av' Rev" ,av] [Rav,bV' • Rbt>' ,ev"]) + 15
2: • EW) 1 _ E<O)
dv" ,
dv'"
([Rav,av' Rav,bv' ][R"v' ,4v'" • R4v ' " ,avl
b1"
+ [Rav,av • R bV ',4V'''] [Rav,bV' • Rav''',av] + [Rav,av • Rav''',avl[Rav,bv'' R"P',4V''']) .
(A.6)
In Eq. (3.6), using Eqs. (A.4) and (A. 5) we can see that, in general, D.(av, bV')(2) is smaller than fW.(av, bv')(l)
xZav,av by the ratio of kT to the energy difference between the two electronic states. Similarly, we can see that
D.(av, bV')(1) is smaller than fjD.(av, bv')(O) Zap,av by the same ratio. Thus, in general, the terms involving
D.(av, bv')(1), Dx(av, bv')(1), D.(av, bV')(2), and D,,(av, bV')(2) can be ignored unless the remaining terms vanish.
APPENDIX B
In this appendix, we study the effect of molecular vibration on dipole moments and polarizabilities. The electronic
Hamiltonian can vary with normal coordinates as 12 ,30
ail)
2
)
;
JO
1 , , ( 8 iI
H=Ho+ L..J 8Q Q; +2" L..J aQ 8Q
A
A
'" (
jO
j
/J
(Bl)
QjQJ + ....
To the second order approximation with respect to the vibronic coupling, the electronic wavefunction can be expressed as
cf>4 = cf>!O) + cf>!1) + cf>!2) + ••• ,
(B2)
where
ct>!1) =L:' ---.....,~~*.---
(B3)
c
,I,~2)=_ct>2!O) ,,'
'I'~
",
cf>!0)
L..J -'-----,-I-~~k-"..____-'- + L..J E(O) _ EUi)
C
C
4
c
2
[(,I,(0)1!.",,(_8_
1,1,(0»
'l'c 2 L..J 8Q _H_)QQ
8Q
i J '1'4
IJ
j
J 0
J. Chern. Phys., Vol. 62, No 11, 1 June 1975
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S. H. Lin: Electric field effect on spectra
+ ~'
~
4523
(<p~2) IL ( ail) Qi I<P~O»(<p~O) I L (&il) Qi I<P!O»]
°
&Qj
j
j
E!O) _ E~O)
8Qj
°
(B4)
•
Using Eqs. (B2)-(B4), we find the variation of the dipole moment RM with respect to normal coordinates Q i as
(B5)
and
(B6)
where
(B7)
Substituting Eqs. (B2)-(B4) into Eq. (B7) yields
2
(
~,~,
8 RAg) _
(<p~O) 1(8il) I<p!O»(<p!O) I( ail) 1<p~O»
BQj 0
(E(Ol_E(O»(E(OI_E(OI)
8Qj
8~
0-2L.., L..,
'¥i
C
d
0
tJ
C
~
(Ol~'
1(<p~O) I
8fI) I<p!O»
2
&Q
(E )-E I)
R"" +2 -RM L..,
C
d
(J
(0)
4
C
This indicates that (BZR",,/ &<i;)o depends on the second order vibronic couplings. It follows that
(
_~..,/O)
~(&2RM).J!.-.
R"v,41I ) -L..,1:'~vR"",.. v-R"" +~
_
(0)
"V
&~
tti
•
0
4
WI
nWi+ •••
coth 2kT
(B9)
.
In other words, R"",,,v varies only very weakly with Vi for small Vj' but since R"v,,,v and Rbu',bv' will have to be weighted by the Boltzmann factor and the Franck-Condon factor, the replacement of R"v,,,,, and Rbu"bu' by R"" and Rbb is a
good approximation.
Similarly, we consider a •• (av), aet:(bv'), (a". (av», and (a.. (bv'».
For a ... (av), we have
-2~' IZ4v ,ev,,1 -2 ~, 1(94y 1Z4e 1g e",,) 12
a..., (av ) - 4. E(Q)" _ E(O) - ~
EtO)" _ Etm
2
cu
Ct>
CI to>
cu
CU
(BlO)
4 SJ
Expanding Zoe in power series of Qj using Eqs. (Bl)-(B4) and notiCing that
E~~~' - E~~) '" E~O) - E!O) +
L
[(v;' +~)nw:' - (Vi +~)nWi] ,
(Bll)
j
we obtain
a ....(av) =
2~' Iz!~) \2\ <9 Ie cv") 12 [E(O) ~ E<Q) 4"
cv
d;' =Q;' - Qj'
where
found as
C
II
(EtD )
C
~ E(O»Z Li
{(V;'
4
+~)1Iw;' -
(Vi
+~)nwj}]
(Bl3)1
Again the variation of a.,(av) with Vj is small. The thermal average of a..(av) can easily be
_ ...(a) _ L..,L..,
~~,
(a..(av» -a
IZ!~) 12 t"2"2
"2 2 11
(E<OI_..,{())2
Wj d l +(Wj -Wj)-2 coth
Icc
.c.~
WI
nWiJ" ",
2kT
Z!~)
+4L..,L.., E.0I_E.0)
j
Cell
(az
llc ) "
--II
BQj
°d
j
+"', (Bl4)
where
(Bl5)
J. Chern. Phys., Vol. 62, No. 11, 1 June 1975
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4524
s.
H. Lin: Electric field effect on spectra
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