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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 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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.") J. Chern. Phys., Vol. 62, No. 11, 1 June 1975 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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). J. Chern. Phys., Vol. 62, No. 11, 1 June 1975 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 J. Chern. Phys.• Vol. 62, No. 11, 1 June 1975 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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) J. Chern. Phys., Vol. 62, No. 11, 1 June 1975 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 J. Chern. Phys., Vol. 62, No. 11, 1 June 1975 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 J. Chern. Phys., Vol. 62, No. 11, 1 June 1975 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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- J. Chern. Phys., Vol. 62, No. 11, 1 June 1975 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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. J. Chern. Phys., Vol. 62, No. 11, 1 June 1975 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 J. Chern. Phys., Vol. 62, No. 11,1 June 1975 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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) J. Chern. Phys., Vol. 62, No. 11, 1 June 1975 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 J. Chem. Phys., Vol. 62, No. 11, 1 June 1975 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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, J. Chern. Phys., Vol. 62, No. 11, 1 June 1975 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 • J. Chern. Phys., Vol. 62, No 11, 1 June 1975 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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. J. Chern. Phys., Vol. 62, No 11, 1 June 1975 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 Downloaded 26 Aug 2011 to 140.113.224.113. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 4524 s. H. Lin: Electric field effect on spectra lR. M. 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