J. Phys. Chem. 1983,87,2895-2900
2805
Theory of Multlphoton Magnetic Circular Dichroism: Two Identical Photon Case S. H. Lln," Department of Chemistry, Arizona State University, Tempe, Arizona 85287
Y. FuJlmura,M. Salto, and T. Nakajlma Department of Chemistry, Faculty of Science, Tohoku University, Sendai 980, Japan (Received: December 27, 1982)
A theory of multiphoton magnetic circular dichroism (MCD) is developed for the two identical photon case. It is shown that, in the nonresonant intermediate state case, the transition probability of the two-photon MCD consists of A, B, and C terms as is well-known in the one-photon MCD. In the resonant case, another term which vanishes if both the initial and resonant electronic states are nondegenerate makes a contribution to the transition probability in addition to the A , B , and C terms. Expressions for those terms for the two-photon MCD of molecules are derived in the adiabatic approximation. Band shapes appearing in the two-photon MCD are discussed. Analytical expressions for the band shapes including the temperature effect have been derived in the displaced harmonic oscillator model. The expressions can be applied for both the weak and strong coupling cases. Model calculations of the band shapes are performed to demonstrate the effects of the potential displacement on the band shapes of the two-photon MCD.
1. Introduction When a magnetic field is applied along the propagation direction of circularly polarized light, circular dichroism in a medium can be observed: this is called magnetic circular dichroism (MCD). While natural CD is a relatively rare phenomenon, MCD is a property of all matter. One-photon MCD has been widely used by chemists and physicists to study electronic structures of molecules by measuring MCD in regions of electronic ab~orption.'-~ Multiphoton absorption spectroscopy is finding increasing application in the studies of excited electronic states of molecules, in particular of states which cannot directly be investigated by one-photon p r o c e ~ s e s . ~ , ~ The purpose of this paper is to develop a theory of two-photon MCD applicable to both degenerate and nondegenerate transitions, the temperature effect and resonance effect. In the present investigation we choose the case of two identical photons. Recently Wagni6re6has studied two-photon MCD. He has theoretically analyzed the polarization effects on two-photon MCD and derived the transition probability of two-photon MCD for nondegenerate initial and final states. Furthermore, the temperature effect is not included in his treatment. In the next section, a general theory of two-photon MCD both for the nonresonant and for the resonant intermediate states is developed. The transition probability for twophoton MCD is derived to first order in a magnetic field H. It is shown that, in the nonresonant case, the transition probability consists of A , B, and C terms as can be seen in the case of one-photon MCD. In the resonant case, on the other hand, another term, called the R term in this paper, makes a contribution to the transition probability in addition to the A, B, and C terms. The R term vanishes if both the initial and resonant electronic states are nondegenerate. (1)A. D.Buckingham and P. J. Stephens, Annu. Reu. Phys. Chem., 17,390 (1966). (2)P.J. Stephens, Adu. Chem. Phys., 35, 197 (1976). (3) D. Caldwell and H. Eyring, "The Theory of Optical Activity", Wiley-Interscience, New York, 1971. (4) P. M. Johnson, Acc. Chem. Res., 13, 21 (1980). (5)D.M.Friedrich and W. M. McClain, Annu. Rev. Phys. Chem., 31, 559 (1980). (6)G.WagniBre, Chem. Phys., 40,119 (1979). 0022-3654/83/2087-2895$01.50/0
In section 3, expressions for the transition probability for two-photon MCD of molecules are derived in the adiabatic approximation. The vibronic and nonadiabatic effects have been neglected. In section 4, band shapes for two-photon MCD are discussed. In the displaced harmonic oscillator model, analytical expressions for the band shapes including the temperature effects are derived by assuming a Lorentzian line shape. The expressions can be applied to both the weak and the strong electron-vibration coupling cases. To demonstrate the effects of the potential displacement on the band shapes of two-photon MCD, we perform model calculations of the band shapes using the expressions derived. 2. General Theory In this paper we consider only two-photon MCD due to the absorption of two identical photons. For this purpose, we shall use the semiclassical theory of radiation. In this case, the interction Hamiltonian between the system and the radiation field in the dipole approximation can be expressed as' e H'= -- [A, exp(itwR) exp(-itwR)].ii (2.1) 2mc
+ A*,
A,
where_ denotes the vector potential of the radiation field and P represents the total linear momentum operator of the system. For convenience, eq 2.1 can be rewritten as H' = Q exp(itwR) + Q* exp(-itwR) (2.2) where e V = - -A0.P 2mc - V* = -- e Ao*.P (2.4) 2mc If the dephasing (or damping) effects are taken into account, the transition probability for two-photon absorption is given by
--
(7)S.H.Lin, J. Chem. Phys., 55, 3546 (1971).
0 1983 American Chemical Society
The Journal of Physical Chemistty, Vol. 87,
2896
where Pa represents the Boltzmann factor, Yka, the dephasing (or damping) constant between k and a states, and fab(WR), the line shape function for a b, i.e.
-
-
for the Lorentzian line shape. Notice that as Yba 0, reduces to the delta function. We first consider two-photon MCD in-the nonresgnant intermqdiate state. Using the relation P b a = i m o b a R b a / e , yhere Rbais the matrix element of transition moment, and A, = A,$ where e^ represents the unit polarization vector, we can rewrite eq 2.5 as
fab(WR)
= KWR~C CPaMab(WR)fab(WR)
W(OR)
Lin et al.
No. 15, 1983
a b
Wba
=
- [(fiZ)bb
&)
f$)(UR)
=
- ( f i Z ) a a ] H / h + *..
[(fiZ)bb - ( f i Z ) a a l f L b ( W R ) / h
(2.20) (2.21)
where pZ denotes the Z component of the total magnetic moment (i.e., including the contributions from both orbital and spin). If the line-shape function is of the 6 function, then
Substitution of eq 2.20 and 2.21 into eq 2.17 yields
(2.7)
where K = Ao4/8a3c4and
In the case of two identical photons, MCD can be induced if the magnetic field is applied along the direction of light propagation. In this case, the transition probability of two-photon MCD is given by
AW(WR)= K W R 4 C C P a m a b ( W R ) f a b ( W R ) a b
(2.9)
where
&]
=
(5-
$)/21/2
e^r =
(5 + i7)/2'/2
(2.11)
Here and denote the unit vectors along the space-fixed X and Y directions, respectively. If the applied magnetic field H is not strong, A W ( q ) can be expanded in power series of H
+
AW(WR)= AW(,)(wR)
+ ...
(2.12)
For a randomly oriented system, AW(O)(W,) = 0. Due to the presence of the applied magnetic field both wave functions and energy levels of the system are affected. Thus we have
Pa = P'p + HP(,')+ ... aab(WR) fab(aR)
=
hMiV(WR)
= f(Bg)(WR)
+ HmiV(wR) + ...
+ Hf'L)(WR) + ...
(2.13) (2.14) (2.15)
As in the case of one-photon MCD, using eq 2.13-2.15, we can separate AW1)(wR) as
(8) P. M. Monson and W. M. McClain, J. Chem. Phys., 53, 29 (1970).
Theory of Multiphoton Magnetic Circular Dichroism
states are nondegenerate, then A:T(wR) = 0 and C$,)(W,) = 0 because in this case &,b is dso zero. It should be noted that just as in the case of one-photon MCD, from eq 2.35 we can see that, if all three terms are present, the relative contributions from the A, B, and C terms are approximately in the ratio of ( h T ~ ) - l : ( ~ b a ) - ' : ( ~where ~ - ' , @ba denotes the order of magnitude of zero-field state separations between the initial and final states. In other words, unless AiT(uR) and C!&)(W,)vanish, their contributions are more important than that of B:T(uR). We shall consider two-photon MCD in the resonant intermediate state. In this case, qaN wR, the H dependence both in the energy denominator wka(H) - wR and in the damping constant yka(H) in eq 2.5 has to be properly taken into account. It has been showngthat, for a system of randomly oriented molecules, the magnetic field dependence of the damping constant is expressed as yka(H) = ria + H2rk2,' + .... The H dependence of y k a ( H ) can therefore be neglected in the present treatment in which we restrict ourselves to first order in H. To take into account the H dependence of w,(H), we expand wka(Ei)to first order in H Wka(H)
=
- [(bZ)kk - (bZ)aa]H/h
Utilizing the expansion [@ka(H)- wR - iYka1-l = - WR - iTka)-'[l - ((bZ)kk - wR - iyka)] (2.36) (bZ)aa!H/ h we can derive the expressions for the transition probability of two-photon MCD in the resonant case. In addition to A ~ ~ ) ( O RAUI(~)(WR)~, )~, and Awl)(wR), with the damping constant yka, A ~ ' ) ( W R )called R resonance term in this paper makes a contribution to the resonant two-photon MCD:
where
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983 2897
the adiabatic approximation which is commonly used: la) Ik)
@aeav,
@kekv,
Ib) I @bebvb 11) @leiv,
(3.1)
where @ and 0 represent the electronic and nuclear wave functions, respectively. A. Nonresonant Case. Let us first consider two-photon MCD in the nonresonant intermediate state. The expression for the transition probability of the two-photon MCD of molecules in the nonresonant case can be obtained by substituting eq 3.1 into eq 2.35. In this paper, we restrict ourselves to two-photon allowed electronic transitions. In this case, we can make use of the Condon approximation; for example, for A ~ ' ) ( w R ) Aeq , 2.23 then becomes A W(')(WR)A= -( 1/ 15)KwR4AiV((wR)f6b(w~) (3.2) where f6b(wR) = ~~P(t\l(ObvblOavl) 12f'av,,bv,(WR)
(3.3)
Va vb
A i g ( a ~ )= (l/ga)CEZZ(wk'
-
a @ k k '
OR)-'(~~?A
-
(@&'R)k()(RE@'R&, X APga) + (Re@*R)kf) (Rtk' Rfta)(
[email protected]$ktx &aa) (R&,y'R&,)(R!k'R)kt x AP,~,)] (3.4) where Rzk = (@:lq@i), and and are the electronic wave functions at the equilibrium nuclear configuration in the ground state. Here the summations over a and p cover the initial a and final b electronic manifolds, respectively, and those over k and k' cover the intermediate electronic states and their manifolds. ga represents the degeneracy factor of the initial electronic state (i.e., = ga). In obtaining eq 3.2, we have used the approximations, (WL?~,:~,=( a ! ' - wR!-l, and the closure relation over the vibronic states in the intermediate electronic states. f'av,,bvb(WR) in eq 3.3 is a derivative of the line-shape function. Similarly, for A V ) ( U ~ )eq ~ ,2.28 can be written as WR)-lIm[
A w ' ) ( w R ) C = -( 1 /15)pKwR4cab'2'(oR)~eg)(WR) (3.5)
where @CUR)
~ t ~ l ( e b v b 1 0 a v . ) 1 2 ~ t ~ , b v b ( W R (3.6) )
= va vb
cLV(wR)
= ( l / g a ) c c c c (&) - UR)-'(Uk91 a @ k k '
(R%'R$kr)(
[email protected] Pas) + (R!k*ROkra)(Rg@'R$k'x Pas) + (R&@'R&fa)(R%*R$k~ Pas)] (3.7)
wR)-'lm[
Pas)
+ (
[email protected]$k/)(R!k'&a
where Finally we have bw')(WR)B
-( 1/ 15)K~R4Bi~(WR)f(~R) (3.8)
where 4
B#(wR) = CB,c2)(wR)ab i=l
(9) S. H. Lin and Y. Fujimura, "Excited State",Academic Press, New York, 1979, Vol. 4, p 237.
- -
(3.9)
For B / ' ) ( u ~we ) ~can ~ use the one given in the Appendix by making the substitutions a a,b p, etc. B. Resonant Case. we consider the two-photon MCD of molecules in the resonant intermediate state. For the two-photon allowed transitions, using the Condon approximation, the transition probabilities of the A term can be expressed as 1 AW(')(WR), = - G K W R ~ A ; ~ ) F ' ~ ~ ( W R(3.10) )
2898
Lin et al.
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983
and the band shape function
where
Dab(wR)
=
takes the form
- WR + iYka)-'(wfP?LI,ava
cCe$iCC((&k,av,
- iYka)-'(eavelekvk)
(ekvklebvb)
(ebv,lekv,')
(ekvk'leav.)
(3.21)
fit;,bvb(WR)
.
.
(3.12) From eq 3.10, we can see that F;,b(wR) plays the role of a band shape function for A W ( ' ) ( W ~ ) ~for the case in which the resonance effect on two-photon MCD is important. The resonance effect can of course enhance the MCD signal. Similarly the C term can be written as A w ( ' ) ( W R ) c = -( 1 / 15)pKwR4cLtz?$at(WR)
(3.13)
where
cig = (i/g,)ccccI,[(RO,,.R;J x a
@
K
K
'
(R$p.R:i, X pa,) + (R:p*R;,/)(RO,,.R$,X PJ + (RO,,. R:~,)(R:~R;;x pa,) + (R:~.R:~,)(RO,,.R;,x pa,)] (3.14)
(3.15) The B term in the resonant case is expressed as A W(')(WR)B = -( 1 / 15 ) K w R 4 B # F #
(wR)
(3.16)
In this section, it is shown that in the Born-Oppenheimer approximation each term of the two-photon MCD of molecules can be expressed by a product of the electronic factor and the band shape function. 4. Discussion
We have shown that the expression for the transition probability of two-photon MCD consists of the A, B , C, and R terms depending on the nature of the electronic state manifolds relevant to the transition. In the nonresonant intermediate state case, compared with those of the A, B, and C terms, the contribution of the R term is considerably small except for the case in which the intermediate electronic state is degenerate and a t the same time both initial and final electronic states are nondegenerate. In this case, the A and C terms vanish and the B and R terms are in the same order, these being approximately in the ratio of (U&)-':(@k,)-'. In the near resonant or the resonant case, on the other hand, the R term may make a significant contribution to the two-photon MCD spectrum: the relative contributions from the A, B, C, and R terms are approximately in the ratio of (hYba)-l: kT)-':( hYka)-'. It is interesting to compare two-photon MCD with ordinary two-photon absorption. From eq 2.7, the transition probability of two-photon absorption for a randomly oriented system is given by
(e&-':(
where
w(WR)
(3.17)
-
Vk
va vb
wR
Dab(WR)
=
(4.1)
K w R 4 c c p a ' o ' ( M i \ ) ( w R ) )fl,Ob)(WR) a b
where
(M~~(uR = )(1/15)CC(~lp,) ) - WR)-'(O~~A - wR)-'[(I&* k k',
+
+ (&'Rgk')
@b)
&,)]
(&.
= ( 1 / 1 5 ) L L i ) ( ~ ~(4.2) )
Here the following relation has been used'O
Equation 4.1 can be rewritten as w(wR)
= (1 / i5)KWR4CCp',o'Lii)(WR)f"(WR) a b
(4.4)
where L:?(W,) plays the role of dipole strength for twophoton absorption. Comparing L L ~ ( w R ) of two-photon absorption with ALv(wR), eq 2.24, and Ci?(wR), eq 2.29, of two-photon MCD, we can see that they are closely related. Thus using two-photon MCD, one can determine not only the magnetic moments (if exist) of initial and/or final electronic states but also the dipole strength of two-photon absorption. Next we compare AW(WR)of two-photon MCD with W(wR) of two-photon absorption: from the A term the ratio is approximately l A G a b l H / h T a b , from the C term the ratio is approximately ljlaalH/kT,and from the B term the ratio is approximately IjlablH/UEa.In other words, from the experimental point of view, it is easier to measure the two-photon MCD of degenerate transitions than that of nondegenerate transitions. (10)S. H.Lin, J. Chem. Phys., 62, 4500 (1975).
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983
Theory of Multiphoton Magnetic Circular Dichroism
21
19
I
23
19
2 wR/103 cm-1 Flgure 1. The band shape calculated by using eq 4.6, #& ) with parameters w = 800 cm-', AM = 1.0, yba= 100 cm-', and &/h = 20000 cm-'. It corresponds to the band shapes of the B and C terms of a nonresonant two-photon MCD in the low-temperature limit.
In section 3, we have derived the transition probability for two-photon MCD in molecules in the adiabatic approximation. Its transition probability is expressed as a product of the electronic factor and the band shape function. The electronic factor involves the electric dipole and the magnetic moment matrix elements. I t has been shown that there are several types of the band shape functions. In the nonresonant and nonde enerate intermediate state, the band shape functions &(wR) and f 6b(wR) make a significant contribution. These band shape functions take the same structure as those in one-photon MCD of molecules.'l In order to analyze the vibronic structure of MCD spectra, one finds it useful to derive analytical expressions for the band shape functions which can be applied not only to the weak electron-vibration coupling but also to the strong coupling. In a displaced harmonic oscillator model in which the vibronic Hamiltonians of the initial, intermediate, and final electronic states are written as
Ha =
hW
1
I
21
23
2890
I
2 uR/103 cm-' Flgure 2. The band shape calculated by using eq 4.7, f'ab(wR)with the same parameter set as that in Figure 1. It cwesponds to the band shape of the A term of a nonresonant two-photon MCD in the lowtemperature limit.
* 0 -
-*
-n
1 .o-
U
+
ll \
0 -
I
+ (I2) + 4 (4.5)
L
where w is the frequency of the vibrational mode, and A's are the displacements of the equilibrium points between the relevant electronic states, the band shape function, fi?(wR), can be expressed as
1
I
19
21
I
I
23
25
I
2'7
I
29
r
Flgure 4. The band shape calculated by using eq 4.7, the same parameters as those in Figure 3.
f'ab(WR)
with
form. An analytical expression for the band-shape function
f 6b(WR) can easily be obtained by differentiating eq 4.6 with respect to Yba
(EL?P,,/h+ w(k - 1 ) - 2WR)'
+ Yb:
(4.6)
where average vibrational quantum number, li, is given by li = [exp(phw) - 11-1, = ELo) - E,(O) , and the line-shape function is assumed to be characterized by a Lorentzian (11)D.J. Shieh, S. H.Lin, and H.Eyring, J.Phys. Chem., 76, 1844 (1972).
f 6b(WR)
OR:
2900
The Journal of Physical Chemisft-y, Vol. 87,No. 15, 1983
In Figures 1-4, the band shapes of the MCD in the nonresonant case are calculated to demonstrate the displacement parameter Aba dependence. The band-shape functions, r’,”d(wR) and f’ab(WR), in the low-temperature limit, ii = 0, are shown in Figures 1 and 2 , respectively. The parameters used are w = 800 cm-’, A b a = 1.0, Yba = 100 cm-l, and toa = 20000 cm-l. In Figures 3 and 4, the band shapes, and f;b(wR), are calculated with the same parameter set as those in Figures 1 and 2 except Aba = 3.0 and = 500 cm-’. From these figures, we can see that the band-shape function f/ab(WR) as well as f i ? ( w ~ ) can serve as a tool for analyzing the vibronic bands appearing in the MCD. In the resonant case, the band shape of MCD may be complicated compared with that in the nonresonant case because of the participation of the R term, and further because of the existence of two displacements, Abk and Aka. If only the initial electronic state is degenerate, the electronic factors of the A, C, and R terms neglecting the B term satisfy Ai? = RLV = -C‘2’ a b , and R&f = 0. In this case, three types of the band shape functions, F ( ~ ( w RF:b(wR), ), and RDab(wR),contribute to the resonant two-photon MCD. When the vibronic Hamiltonians, eq 4.5, are used, an analytical expression for F($(W,) can be written as12 1
~ k ~ ~ ) x/ 2 1
F : : ) ( ( ; ~ )= rexp[-(Z?i t 1 ) ( a , k 2 1
Yba
s
z z k = o l=o
[ {c($/h
A W ( k -
1)
X -
2WR)’
+
Yba2]
X
r!s!
nbk - ( P + q ) A k a (--I (FJ
4
2 _______ (cP:/h + w ( p - 4 A r - S)
(P’q)
\I
- LR-
I*
(4.8)
IYhI 1
Analytical expressions for the band-shape functions, and Raab(wR),can be obtained by differentiating eq 4.8 with respect to w ~ the : former originates from terms involving the differentiation of the Lorentzian line shape function, and the latter from those of the energy denominator with the resonant effect. One can calculate the band shapes of the resonant two-photon MCD by using the expressions derived above. A detailed discussion on the band shapes in the resonant case will be given in a subsequent paper. In this paper we have restricted ourselves to a displaced harmonic oscillator model. The treatment developed in this section can be extended to another potential surface model such as a distorted and displaced harmonic potential one. We have developed a theory of multiphoton MCD. Both the nonresonant two-photon MCD and the resonant MCD are treated in the two identical photon case. If the initial and/or intermediate electronic states are degenerate, the
F;b(wR)
(12) Y . Fujimura and S.
H.Lin, J . Chem. Phys.,
74. 3726 (1981).
Lin et al.
R term may contribute to the MCD as well as the ordinary A and C terms. It is shown that, in the adiabatic approximation and in the Condon approximation, each term of the two-photon MCD of molecules consists of a product of an electronic factor and a band-shape function. The effect of the breakdown of the Condon approximation and that of the adiabatic approximation will be reported elsewhere.
Acknowledgment. This work was supported in part by NSF and by a Grant from the Ministry of Education, Science and Culture.
