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Mechanism of external magnetic field dependent fluorescence decays of gaseous carbon disulfide
Chemical Physics North-Holland
162
( 1992 ) 205-2 12
Mechanism of external magnetic field dependent fluorescence decays of gaseous carbon disulfide Yuichi
Fujimura
Department ofCherrustry. Faculty ofScrence. Tohoku Umversrty, Sendal 980, Japan
Hisaharu
Hayashi
The Instrtute ofPhwcal
and Chemical Research, Wako, Saltama 351-01, Japan
and Saburo
Nagakura
The Graduate Umversrty for Advanced Studies. Nagatsuda. Yokohama 227, Japan Received
17 July 199 1; in final form I4 January
1992
Mechamsms of the external magnetic field dependent fluorescence decays of gaseous molecules are theoretlcally investigated with particular attention bemg paid to the Renner-Teller effect m hnear molecules. A nonradIatIve decay rate constant mcluding both the Zeeman and mtramolecular nonadiabatIc interactions has been expressed in the time-independent Green’s function formahsm. A reduced form for the electronic Zeeman Interaction matrix element between the Renner-Teller coupled rovlbronic states has been derived. Applying this formalism to CS2, we find that Its field dependent ultrafast fluorescence decay may be explamed qualitatively by an external magnetic field-induced nonracbatlve transition between the Renner-Teller components. The fluorescence Intensity depletion from the ‘A* state of CS2 m the presence of an external magnetic field may also be explamed by a field-Induced nonradiative transitIon.
1. Introduction Magnetic quenching of molecular fluorescence in the gas phase was first found for carbon disulfide (CS) by Matsuzaki and Nagakura in 1974 [ 11. It is noteworthy that the fluorescence emitted from a nonmagnetic singlet state was shown to be influenced remarkably by ordinary magnetic fields below 1.3 T. Since this work, similar magnetic field effects on excited singlet states have been observed for a number of gaseous molecules such as sulfur dioxide [ 2 1, formaldehyde [ 3,4 1, glyoxal [ 5-7 1, methylglyoxal [ 8 1, pyrazine [ 9- 111, pyrimidine [ 12.13 1, and s-triazine [ 141. The role of intramolecular interactions such as spin-orbit and spin-rotation interactions as well as intermolecular ones in the nonradiative decay behavior of excited singlet states has been investigated for gaseous molecules by analyzing magnetic field effects on fluorescence intensities and time profiles. Thus, the study of magnetic field effects has become as powerful method for clarifying such dynamic processes [ 15- 17 1. In 197 1, Brus [ 18 ] found that the decay (I( t ) ) of the total emission of CS2 could be described by a biexponential function on ns+s time scales, Z(t)=A,exp(-r/ri)+A,exp(-_/T~). Here, Tf and Thereafter, 030 1-O 104/92/$
7,
(1.1)
are the fast and slow lifetimes, respectively, and A,/A, is the ratio of their zero-time amplitudes. in 1974, Matsuzaki and Nagakura [ 1] showed that the fast and slow components exhibited struc05.00 0 1992 Elsevier Science Pubhshers
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Y. Fujlmura et al /External
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magnetlcfield dependent Jluoresence decays
tured and broad fluorescence spectra, respectively, and found that the structured fluorescence (mainly assigned to the t + 3 (II) level of the lowest excited singlet state (T ‘AZ) ) decreased in intensity with increasing magnetic field strength (H) from 0 to 1.3 T. The observed magnetic quenching was attributed to a magnetically induced decrease in the collision-free lifetime of the fast component. Later, Orita et al. [ 19 ] reinvestigated the magnetic quenching of CS2 fluorescence with an improved apparatus. They found that the magnetic quenching of CS2 fluorescence originated from a magnetically induced decrease in Af and that the A,( 0.6 1 T) /A,( 0 T) value was independent of CS, pressure between 1 and 72 mTorr. Thus, Orita et al. [ 19 ] proposed that an intramolecular decay process could induce this magnetic quenching. Silvers et al. [ 201 confirmed the experimental results obtained by Orita et al. [ 19 1, but proposed a different mechanism for the magnetic quenching. They proposed that a detuning of the exciting N2 laser lines and CS, absorption ones occurs via a Zeeman broadening of the absorption ones. In order to clarify the mechanisms of the magnetic quenching of CS2 fluorescence, our group [ 2 1 ] measured the absorption intensity of gaseous CS? for individual N2 laser lines and fluorescence excitation spectra in the absence and presence of a magnetic field. The results showed that the detuning model proposed by Silvers et al. [ 201 was not applicable to the magnetic quenching of CS2 fluorescence. We suggested that a very fast intramolecular process was responsible for the decay, and that it might be influenced by a magnetic field, as proposed by Orita et al. [ 191. Such a fast process has, however, not yet been detected because we have no ps laser suitable for excitingC& through its t+3(II) band (~29655 cm-‘). The fluorescence from the second excited singlet state of CS2 (V ‘Bz) shows similar magnetic quenching [ 19,22,23]. Since a ps dye laser suitable for exciting CS, through the 6V band (at 30526 cm-‘) was available [ 241, we measured the time profiles (F(t) ) of its banded fluoresence on ps-ns time scales. We found that the observed F(t) curves could be described by biexponential decay functions, F(t)=A,
exp( --t/r,)+A?
exp( -f/r2)
.
(1.2)
Here, 7l is the fast lifetime on the ps time scale and 72 is the slow one which corresponds to 7f of eq. ( 1.1). We have found that the 7, and AZ/AI values decrease with increasing H from 0 to 1.51 T. But that the 7? value is almost independent of H. According to the decoupling model [ 5,11,17 ] of so-called intermediate case molecules [ 25 1. it is expected that a magnetic field decreases AZ/AI and does not change r, [ 25 1. This expectation is fulfilled in the cases of pyrazine [9-l 1 ] and pyrimidine [ 13 1. For the case of CS, however, 7, is decreased by a magnetic field, contrary to expectation. To solve this problem, we proposed [ 231 a direct mechanism involving Zeeman interactions [ 261 between the ‘BZ and ground states. The ‘A? and ‘BZ electronic states are produced by the Renner-Teller coupling. That is, a coupling between the electronic orbital angular momentum and vibrational angular momentum originating from the degenerate bending vibration of linear CS2 in the ‘A electronic state [ 271 lifts the orbital degeneracy and produces two states, ‘A7 and ‘Bz. The purpose of the present paper is to study theoretically the mechanism of the magnetic field effect on the ps fluorescence decay from the Renner-Teller coupled states of a molecule such as CS2, paying special attention to the possibility that the electronic angular momentum in a Renner-Teller coupled system may make a significant contribution to the Zeeman coupling. In other words, we investigate the possibility that the Zeeman interaction between the upper and lower Renner-Teller components plays a role in the magnetic field effect.
2. Nonradiative decay rate under the external magnetic fields Consider a model for the external magnetic field effects on the nonradiative decay from the upper component of the Renner-Teller coupled state of a molecule such as is shown in fig. 1. Here, a denotes an optically active rovibronic level in the upper component, {6} denotes the optically forbidden levels of the lower component, and (I} denotes the rovibrational manifold in the ground state and/or the rovibronic manifold in the lower triplet
Y. Fujimura et al. /E.xternal magnetlcfield dependentjuoresence
207
decays
“ZN
“ZN
Fig. 1. Model for the external magnetic field-induced nonradiattve decay from a Renner-Teller coupled state. a and {b} denote the optically active rovrbronic state of the upper Renner-Teller component, and the optically forbidden lower component, respecttvely. {/) refers to the rovibrattonal manifold in the ground state and/or rovtbromc manifold in the lower triplet states. These states are coupled wtth each other through the perturbations VZN= Vz+ VN, wtth the electromc Zeeman interaction V,and intramolecular nonadiabatic one V,.
states. These manifolds adiabatic interactions. The total Hamiltonian
are coupled to each other by both the external magnetic
field and intramolecular
non-
His given by
H=Ho+Vz+VN.
(2.1)
Here Ho is the zeroth-order rovibronic Hamiltonian in the Born-Oppenheimer basis set, and Vz, the Zeeman interaction operator, and VN, the intramolecular nonadiabatic interaction operator are given by vz=
c Ia>(bI(~z),Ll+ h
cc h
16)(4(k;),,+c.c.
(2.2)
Ib)(4(vN/,)h,+c.c.
(2.3)
/
and
vN=
c Ia>
h
I
In eqs. (2.2) and (2.3), ( Vz),b, (( Vz),,) and ( VN)nb, ((I’,),,) are the Zeeman interaction interaction matrix elements, respectively. An expression for the nonradiative decay rate constant from the optically active rovibronic derived by using the time-independent Green’s function (resolvent ) method [ 28 1. Thus,
and nonadiabatic state a, k,, can be
(2.4) where VzN= Vz+ I’,, and r,, the decay width of the rovibronic rh’2n
1
1
vZNib
state b evaluated
at E= EE, is given
126(E:-C’),
by (2.5)
/
l?b
is the energy, including the level shift, of the rovibronic state b. The nonradiative decay rate constant averaged over the initial distribution
is
given by
(2.6) where pa denotes the initial population
distribution
in the a state.
Y. FuJimura et al. /External
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magnetic field dependent fluoresence decays
3. Zeeman interaction between the Renner-Teller coupled rovibronic states Next, we derive a reduced expression for the matrix elements of the Zeeman interaction between the RennerTeller coupled rovibronic states of CS2. The electronic configurations with electronic angular momenta 1A 1= 1 and 2 are taken into account, and approximately expressed as linear combinations of singly excited electronic configurations constructed from the 3o,, In, and TC,MOs shown in fig. 2 [ 291. The approximate vibronic wavefor the upper and lower Renner-Teller components, respectively. functions represented by @Bvbcn,K and @At,ben,K, are expanded in the basis of the linear molecule, 1.4, z&,, I) [ 301,
(3.la)
+
Ci’$,;;‘( I 1 . z&n, , K- 1 ) + I - 1 , z&t , KS 1) ) ] .
(3.lb)
In eqs. ( 3.1 b) and ( 3.1 b), the total angular momentum about the principal axis of the bent molecule, K is given by K= (A+ I1,where I denotes the vibrational angular momentum. Subscript LJ,,~,,~ (with and without primes) denotes the vibrational quantum numbers of the bending vibration in the linear and bent geometrical structures, respectively. The contributions of the electronic angular momenta IA I= 1 and 2 are included in these equations since these are coupled to each other via configuration interaction in the bent structure. The dominant electronic configurations of the V system in the region 290-350 nm are recognized to be 7~: - np ( ]A I = 2) configurations, and
The rovibronic wavefunctions for the upper and lower Renner-Teller components are expressed in terms of the vibronic wavefunctions, eqs. (3. la) and (3. lb), vibrational ones, x. and rotational ones, r] as (3.2a) c2v
‘&h
2% -
Ill -r:9 ~a~~.~--_ lil” .ha ~______ 9
=:--
_--
2bl
----
%
__- ___
--__
___ _
_-- ___ =I- ---__ _
Ibq 5q La7 Y
5
4
‘a2 Lbz 3b2
t y+z
Fig. 2. Correlation of the lowest vacant and several occupied molecular orbitals of CS2 between Dmh and CzV point groups, and the convention of the axes. The configurations of the lower excited ‘Bz and ‘AZ electronic states are assumed to be constructed from the smgly excited contiguratlons 2x.-lrr, and 2a,-30. to evaluate the couplmg matrix elements.
Y FuJm’Wa et al. /External magnetrc$eld dependent fluoresence
209
deca.vs
and YA~d&f.Ka
- @Aw,en,Ka ~Au,J&,K,
n
(3.2b)
’ XAoa 9
The prime on IX indicates that the wavefunctions of the bending modes are omitted. The rotational of the Az electronic state is, for simplicity, expressed in the rigid rotor approximation as
respectively.
wavefunction
(3.3) where the subscripts v, J, M, and K are omitted. D& is the D function of rank J and M is the projection of the total angular momentum Jon the laboratory fixed Z axis. The rotational wavefunction of the B2 electronic state is expressed in a similar way. We are now in a position to derive a reduced expression for the Zeeman interaction matrix element, ( b l V, 1a). The operator of the external magnetic field H directed along the laboratory Z axis, V, is given in terms of the tensor of the electronic angular momentum [ 3 1 ] as (3.4) where D&g (0) is the component of the rotation matrix of rank 1 expressed in terms of the Euler angle 8. In eq. ( 3.4) T 8 ) (I), the tensor of electronic angular momentum with the spherical tensor component Q is given by t321 T!“(l)=-
T&“(l)=/_,
+I+
TL’{(l)=
+I-.
(3.5)
Here, I+ =I,+ il, and I- =I,-il,., in which I,, I,, and I= are the angular momentum coordinate frame, x, y. and z, respectively. The Zeeman interaction matrix element is expressed as
- ( -L,
vi,,,,
~~+LlT~‘(~)I-L~~~,~~,~~+L)l~(~Bl.h~bnrb~~l~06~’(~)l~Al~a~~n~~~
operators
n’
in the Cartesian
(3.6) Noting that < *L, 4,nt K~-LIT~)(I)I-~L,v~~~,,K~-L)=+LAG~,~,
forQ=O,
1
=o,
for Q#O
.
(3.7)
Eq. (3.6) can be written as (
%bJbMbKb
= - PogffA
where
1 vZ
I
yAvo&M&,
BA (&em
4
> ( VB~~J~M~KD
I 06;’ (Q) I VA~,~J~M~K,
> n’
(xB”~ I ~~~~~> :
(3.8)
Y. FUJi??Wa
210
et al.
/External magnetlcfield dependentjluoresence
decays
Rewriting the rotation matrix element of eq. (3.8) in terms of the Wigner 3J-symbols ( : : : ), we obtain for the reduced form of the Zeeman interaction matrix element the expression ( ylgL%JMhKh I vz I ~AL~‘aL&>=-~,g~~.,(~,“,,K)(-1)“-K[(2J1,+1)(2J,+1)l”Z (3.9) with K= K, = Kh, and M= M, = Mb. Eq. ( 3.9 ) contains the product of electronic ( Renner-Teller coupling), vibrational, and rotational terms. The contribution of the vibrational motion of the bending mode is included in the electronic term because the vibronic wavefunctions of the Renner-Teller coupled states are expressed in terms of the electronic and bending vibrational coordinates.
4. Results and discussion 4. I. Zeeman interaction between the upper and lower Renner-Teller components In the case where the rovibronic (2.6) can be simply expressed as
states of the lower Renner-Teller
k=FXCP,I(al~z,,lb)126(EIj-E~). a
component
consist of a dense manifold,
eq.
(4.1)
b
Furthermore, when all the allowed A4 states in electronic state a are simultaneously excited by an ultrashort pulse laser, the nonradiative decay rate averaged over M states is measured in the time-resolved fluorescence. In this case, the interference terms, the product of the interaction matrix elements ( Vz)a,,( VN)ab vanishes, and the rate constant may be expressed as a sum of two direct processes (after summing over M, and Mb, neglecting the effects of the lifting of M degeneracy, and noting that 1, ( - 1 )M ( : : : ) = 0 in eq. (4.1): k=k,+k,.
(4.2)
where kz= $1’
Vzlb) I*d(G-G)
a
1’ Pa I (al h
*
C’paI(a(V,lb)126(EZ:-E~). b
(4.3a)
and kN=$Cf
(4.3b)
In eqs. (4.3a) and (4.3b), primes denote sums taken over all quantum numbers except for the magnetic quantum ones. If {b} consists of a sparse manifold, and then the M dependence on Eh and rb as well as the effects of the lifting of the M degeneracy can safely be omitted. Under these conditions, the expression for the nonradiative decay constant is
rbI(alVzib)12
1
(4.4)
where r, is a sum of the damping constant due to the purely external magnetic fields r,(H) and that due to the intramolecular nonadiabatic couplings r,( H= 0). In the absence of an external magnetic field, the first term in
Y. Fu~mura
et al. /External
magnetlcfield
dependentfluoresence
211
decays
eq. (4.4) vanishes, and the expression represents the sequential decay mechanism (a~{ 6) *{I} ) induced by the intramolecular nonradiative interactions [ 281. From eq. (4.4), we can see that in the presence of an external magnetic field, a new channel of transitions between the rovibronic states in the Renner-Teller components opens up in addition to the channel originating from the nonadiabatic interaction. We call this new channel the external magnetic field-induced nonradiative transition. From the restrictions on the Wigner 3jsymbols in eq. ( 3.9 ), we find the selection rule for the new channel between the rovibronic states to be IJh_J,
I < 1
K,=K,,
and
M,=Mh.
(4.5)
As mentioned in section 1, the external magnetic field effect on the fluorescence decay of ‘Bz CS2 observed by Imamura et al. [ 22,23 ] is different from those in aromatic molecules such as pyrazine [ 9- 111 and pyrimidine [ 13 1. In aromatic molecules, the lifetime of the fast decay component (T, of eq. ( 1.3 ) ) is independent of an external magnetic field, and only the ratio of the pre-exponential factors is affected by the external magnetic field. The external magnetic field effects on the fluorescence of such aromatic molecules can be explained by the decoupling model. The external magnetic field dependent decay component (T, of eq. ( 1.3) ) observed for CS, ( ‘B>), on the other hand, is different from those in pyrazine and pyrimidine and can be explained by an external magnetic field-induced nonradiative transition. That is, an external magnetic field opens a new channel for the transition between the ‘B? and ‘A* rovibronic states. 4.2. Zeeman interaction between the lower Renner-Teller component (‘A j and the groundstate (‘C,t) It is well known that the fluorescence with a us decay rate from the ‘AZ state decreases in intensity in the presence of an external magnetic field through the enhancement of intramolecular energy redistribution processes as in the case of fluorescence from the ‘Bz state [ 1,19,21]. This phenomenon can also be explained in terms of the external magnetic field-induced nonradiative transition. The coupling matrix element (AZ 1Vz (C.J ) in the expression for the external magnetic field-induced nonradiative transition rate constant is
JJ’ (XAC. IXZLV >3
X where J,, KS and A4, denote the quantum
numbers
of angular motion in the electronic
(4.6) ground state ( ’ C ,’ ), and
Here, r and v%,,, denote the radial variable and quantum number of the doubly degenerate tively. From eq. (4.6), the selection rule for the external magnetic field-induced nonradiative to be
IJa-JsI&l,
K,=K,kI
and
vibration, transition
respecis seen
M,=M,.
The mechanism of the nonradiative transition from the ‘A1 state due to an external magnetic field was discussed by Matsuzaki and Nagakura [26], who proposed a direct mechanism on the basis of the fact that the coupling matrix element with 1,. was nonvanishing. As is shown above, the electronic configuration with electronic angular momentum/i = 1 in the ‘A2 state is necessary for the nonvanishing matrix element to exist. In a similar way, the ‘BZ state is also coupled to the ‘Cz state in the external magnetic field because ( Bz ( Vz ( Cz ) has a nonvanishing value. That is, the ‘Bz state has two channels for the external magnetic fieldinduced nonradiative transition; ‘B2*‘A2 and ‘B,*‘X p’. From the relation ] C(.4=2’ ] > ) C(,‘=‘) ] in eq. (3.6), and the selection rules for the channels derived in the present paper, and also from the energy gap law, the former
212
Y
FUJlWWa
et al. /E.~~ternal wzagnetlcfield dependentjluoresence decays
channel makes the dominant contribution to the external magnetic field-induced nonradiative transition from the ‘B2 state. Thus, it is qualitatively shown that the origin of the ps fluoresence decay from the ‘B2 state in an external magnetic field is associated with opening of the new channel due to the Zeeman interaction between the Renner-Teller coupled rovibronic ‘BZ and ‘A2 states.
Acknowledgement This work was supported by the Joint Studies Program ( 1987-l 988 ) of IMS. One of the authors (YF) would like to thank Professor S.H. Lin of the Arizona State University, and Professor N. Shimakura of the Niigata University for valuable discussions.
References [ 1 ] A. Matsuzaki and S. Nagakura, Chem. Letters ( 1974) 675; Bull. Chem. Sot. Japan 49 ( 1976) 359. [2] V.I. Makarov, N.L. Lavnk. G.I. Skubnevskaya and N.M. Bazhm, React. Kmet. Catal. Letters 12 (1979) 359. [3] N.I. Sorokm, N.L. Lavrik, G.I. Skubnevskaya, N.M. Bazhm and Yu. N. Molin. Dokl. Akad. Nauk SSSR 245 (1979) 657; Nom. Chim. 4 (1980) 395. [ 41 H. Orita. H. Morita and S. Nagakura, Chem. Phys. Letters 8 1 ( 198 1) 409; 86 ( 1982 ) 123. [5] H.G. Kiittner, H.L. Selzle and E.W. Schlag. Isr. J. Chem. 16 (1977) 264. [ 61 M. Lombardi. R. Jost, C. Michel and A. Tramer, Chem. Phys. 46 ( 1980) 273. [ 71 J. Nakamura, K. Hashimoto and S. Nagakura, J. Luminescence 24/25 ( 198 1) 763. [ 81 K. Hashimoto. S. Nagakura, J. Nakamura and S. Iwata, Chem. Phys. Letters 74 ( 1980) 228. [ 91 P.M Felker, Wm R. Lambert and A.H. Zewail, Chem. Phys. Letters 89 ( 1982) 309. [lo] Y. Matsumoto, L.H. Spangler and D.W. Pratt. J. Chem. Phys. 80 (1984) 5539. [ 111 N. Ohta and T. Takemura, J. Chem. Phys. 91 (1989) 4477. [ 121 Y. Matsumoto and D.W. Pratt, J. Chem. Phys. 81 (1984) 573. [ 131 N. Ohta, T. Takemura. M. FuJita and H. Baba, J. Chem. Phys. 88 ( 1988) 4197. [ 141 N. Ohta and T. Takemura, J. Chem. Phys. 94 ( 1990) 3466. [ 151 S.H. Lin and Y. Fupmura, m: Excited state, Vol. 4, ed. E.C. Lim (Academic Press, New York, 1979) p. 237. [ 161 U.E. Stemer and T. Ulrtch. Chem. Rev. 89 ( 1989) 5 1. [ 171 H. Hayashi. m: Photochemistry and photophysica. Vol. 1, ed. J.F. Rabek (CRC Press. Boca Raton, 1990) p. 59. [ 18 ] L.E. Brus. Chem. Phys. Letters 12 ( 197 1) 116. [ 191 H. Orita. H. Morita and S. Nagakura, Chem. Phys. Letters 81 (198 I ) 29. 33. [20] S.J. Silvers, M.R. McKeeverand G.K. Chawla, Chem. Phys. 80 (1983) 177. [ 211 T. Imamura. S. Nagakura, H. Abe, Y. Fukuda and H. Hayashi, J. Phys. Chem. 93 ( 1989) 69. [22] T. Imamura, N. Tamai. Y. Fukuda, I. Yamazaki. S. Nagakura, H. Abe and H. Hayashi. Chem. Phys. Letters 135 ( 1987) 208. [23] H. Abe, H. Hayashi, T. Imamura and S. Nagakura, Chem. Phys. 137 (1989) 297. [24] 1. Yamazaki, N. Tamai, H. Kume, H. Tsuchiya and Y. Oba, Rev. Sci. Instr. 56 ( 1985) 1187. [25] F. Lahmam, A Tramer and C. Tnc, J. Chem. Phys. 60 (1974) 4431. [26] A. Matsuzaki and S. Nagakura. Helv. Chim. Acta 61 ( 1978) 675. [27] Ch. Jungen. D.N. Malm and A.J. Merer. Can. J. Phys. 51 (1973) 1471. [28] Y. Fujimura, N. Shimakura and T. Nakajtma, J. Chem. Phys. 66 ( 1977) 3530. [29] J.W. Rabalaas, J.M. Mcdonald, V. Scherr and S.P. McGlynn. Chem. Rev. 71 ( 1971) 73. [30] J.T. Hougen, J. Chem. Phys. 41 (1964) 363. [31]W.M.Huo,J.Chem.Phys.52(1970)3110. [ 321 A.R Edmonds. Angular momentum m quantum mechanics (Princeton Univ. Press. Princeton. 1957) p. 81-85.
J.
162
( 1992 ) 205-2 12
Mechanism of external magnetic field dependent fluorescence decays of gaseous carbon disulfide Yuichi
Fujimura
Department ofCherrustry. Faculty ofScrence. Tohoku Umversrty, Sendal 980, Japan
Hisaharu
Hayashi
The Instrtute ofPhwcal
and Chemical Research, Wako, Saltama 351-01, Japan
and Saburo
Nagakura
The Graduate Umversrty for Advanced Studies. Nagatsuda. Yokohama 227, Japan Received
17 July 199 1; in final form I4 January
1992
Mechamsms of the external magnetic field dependent fluorescence decays of gaseous molecules are theoretlcally investigated with particular attention bemg paid to the Renner-Teller effect m hnear molecules. A nonradIatIve decay rate constant mcluding both the Zeeman and mtramolecular nonadiabatIc interactions has been expressed in the time-independent Green’s function formahsm. A reduced form for the electronic Zeeman Interaction matrix element between the Renner-Teller coupled rovlbronic states has been derived. Applying this formalism to CS2, we find that Its field dependent ultrafast fluorescence decay may be explamed qualitatively by an external magnetic field-induced nonracbatlve transition between the Renner-Teller components. The fluorescence Intensity depletion from the ‘A* state of CS2 m the presence of an external magnetic field may also be explamed by a field-Induced nonradiative transitIon.
1. Introduction Magnetic quenching of molecular fluorescence in the gas phase was first found for carbon disulfide (CS) by Matsuzaki and Nagakura in 1974 [ 11. It is noteworthy that the fluorescence emitted from a nonmagnetic singlet state was shown to be influenced remarkably by ordinary magnetic fields below 1.3 T. Since this work, similar magnetic field effects on excited singlet states have been observed for a number of gaseous molecules such as sulfur dioxide [ 2 1, formaldehyde [ 3,4 1, glyoxal [ 5-7 1, methylglyoxal [ 8 1, pyrazine [ 9- 111, pyrimidine [ 12.13 1, and s-triazine [ 141. The role of intramolecular interactions such as spin-orbit and spin-rotation interactions as well as intermolecular ones in the nonradiative decay behavior of excited singlet states has been investigated for gaseous molecules by analyzing magnetic field effects on fluorescence intensities and time profiles. Thus, the study of magnetic field effects has become as powerful method for clarifying such dynamic processes [ 15- 17 1. In 197 1, Brus [ 18 ] found that the decay (I( t ) ) of the total emission of CS2 could be described by a biexponential function on ns+s time scales, Z(t)=A,exp(-r/ri)+A,exp(-_/T~). Here, Tf and Thereafter, 030 1-O 104/92/$
7,
(1.1)
are the fast and slow lifetimes, respectively, and A,/A, is the ratio of their zero-time amplitudes. in 1974, Matsuzaki and Nagakura [ 1] showed that the fast and slow components exhibited struc05.00 0 1992 Elsevier Science Pubhshers
B.V. All rights reserved
Y. Fujlmura et al /External
206
magnetlcfield dependent Jluoresence decays
tured and broad fluorescence spectra, respectively, and found that the structured fluorescence (mainly assigned to the t + 3 (II) level of the lowest excited singlet state (T ‘AZ) ) decreased in intensity with increasing magnetic field strength (H) from 0 to 1.3 T. The observed magnetic quenching was attributed to a magnetically induced decrease in the collision-free lifetime of the fast component. Later, Orita et al. [ 19 ] reinvestigated the magnetic quenching of CS2 fluorescence with an improved apparatus. They found that the magnetic quenching of CS2 fluorescence originated from a magnetically induced decrease in Af and that the A,( 0.6 1 T) /A,( 0 T) value was independent of CS, pressure between 1 and 72 mTorr. Thus, Orita et al. [ 19 ] proposed that an intramolecular decay process could induce this magnetic quenching. Silvers et al. [ 201 confirmed the experimental results obtained by Orita et al. [ 19 1, but proposed a different mechanism for the magnetic quenching. They proposed that a detuning of the exciting N2 laser lines and CS, absorption ones occurs via a Zeeman broadening of the absorption ones. In order to clarify the mechanisms of the magnetic quenching of CS2 fluorescence, our group [ 2 1 ] measured the absorption intensity of gaseous CS? for individual N2 laser lines and fluorescence excitation spectra in the absence and presence of a magnetic field. The results showed that the detuning model proposed by Silvers et al. [ 201 was not applicable to the magnetic quenching of CS2 fluorescence. We suggested that a very fast intramolecular process was responsible for the decay, and that it might be influenced by a magnetic field, as proposed by Orita et al. [ 191. Such a fast process has, however, not yet been detected because we have no ps laser suitable for excitingC& through its t+3(II) band (~29655 cm-‘). The fluorescence from the second excited singlet state of CS2 (V ‘Bz) shows similar magnetic quenching [ 19,22,23]. Since a ps dye laser suitable for exciting CS, through the 6V band (at 30526 cm-‘) was available [ 241, we measured the time profiles (F(t) ) of its banded fluoresence on ps-ns time scales. We found that the observed F(t) curves could be described by biexponential decay functions, F(t)=A,
exp( --t/r,)+A?
exp( -f/r2)
.
(1.2)
Here, 7l is the fast lifetime on the ps time scale and 72 is the slow one which corresponds to 7f of eq. ( 1.1). We have found that the 7, and AZ/AI values decrease with increasing H from 0 to 1.51 T. But that the 7? value is almost independent of H. According to the decoupling model [ 5,11,17 ] of so-called intermediate case molecules [ 25 1. it is expected that a magnetic field decreases AZ/AI and does not change r, [ 25 1. This expectation is fulfilled in the cases of pyrazine [9-l 1 ] and pyrimidine [ 13 1. For the case of CS, however, 7, is decreased by a magnetic field, contrary to expectation. To solve this problem, we proposed [ 231 a direct mechanism involving Zeeman interactions [ 261 between the ‘BZ and ground states. The ‘A? and ‘BZ electronic states are produced by the Renner-Teller coupling. That is, a coupling between the electronic orbital angular momentum and vibrational angular momentum originating from the degenerate bending vibration of linear CS2 in the ‘A electronic state [ 271 lifts the orbital degeneracy and produces two states, ‘A7 and ‘Bz. The purpose of the present paper is to study theoretically the mechanism of the magnetic field effect on the ps fluorescence decay from the Renner-Teller coupled states of a molecule such as CS2, paying special attention to the possibility that the electronic angular momentum in a Renner-Teller coupled system may make a significant contribution to the Zeeman coupling. In other words, we investigate the possibility that the Zeeman interaction between the upper and lower Renner-Teller components plays a role in the magnetic field effect.
2. Nonradiative decay rate under the external magnetic fields Consider a model for the external magnetic field effects on the nonradiative decay from the upper component of the Renner-Teller coupled state of a molecule such as is shown in fig. 1. Here, a denotes an optically active rovibronic level in the upper component, {6} denotes the optically forbidden levels of the lower component, and (I} denotes the rovibrational manifold in the ground state and/or the rovibronic manifold in the lower triplet
Y. Fujimura et al. /E.xternal magnetlcfield dependentjuoresence
207
decays
“ZN
“ZN
Fig. 1. Model for the external magnetic field-induced nonradiattve decay from a Renner-Teller coupled state. a and {b} denote the optically active rovrbronic state of the upper Renner-Teller component, and the optically forbidden lower component, respecttvely. {/) refers to the rovibrattonal manifold in the ground state and/or rovtbromc manifold in the lower triplet states. These states are coupled wtth each other through the perturbations VZN= Vz+ VN, wtth the electromc Zeeman interaction V,and intramolecular nonadiabatic one V,.
states. These manifolds adiabatic interactions. The total Hamiltonian
are coupled to each other by both the external magnetic
field and intramolecular
non-
His given by
H=Ho+Vz+VN.
(2.1)
Here Ho is the zeroth-order rovibronic Hamiltonian in the Born-Oppenheimer basis set, and Vz, the Zeeman interaction operator, and VN, the intramolecular nonadiabatic interaction operator are given by vz=
c Ia>(bI(~z),Ll+ h
cc h
16)(4(k;),,+c.c.
(2.2)
Ib)(4(vN/,)h,+c.c.
(2.3)
/
and
vN=
c Ia>
h
I
In eqs. (2.2) and (2.3), ( Vz),b, (( Vz),,) and ( VN)nb, ((I’,),,) are the Zeeman interaction interaction matrix elements, respectively. An expression for the nonradiative decay rate constant from the optically active rovibronic derived by using the time-independent Green’s function (resolvent ) method [ 28 1. Thus,
and nonadiabatic state a, k,, can be
(2.4) where VzN= Vz+ I’,, and r,, the decay width of the rovibronic rh’2n
1
1
vZNib
state b evaluated
at E= EE, is given
126(E:-C’),
by (2.5)
/
l?b
is the energy, including the level shift, of the rovibronic state b. The nonradiative decay rate constant averaged over the initial distribution
is
given by
(2.6) where pa denotes the initial population
distribution
in the a state.
Y. FuJimura et al. /External
208
magnetic field dependent fluoresence decays
3. Zeeman interaction between the Renner-Teller coupled rovibronic states Next, we derive a reduced expression for the matrix elements of the Zeeman interaction between the RennerTeller coupled rovibronic states of CS2. The electronic configurations with electronic angular momenta 1A 1= 1 and 2 are taken into account, and approximately expressed as linear combinations of singly excited electronic configurations constructed from the 3o,, In, and TC,MOs shown in fig. 2 [ 291. The approximate vibronic wavefor the upper and lower Renner-Teller components, respectively. functions represented by @Bvbcn,K and @At,ben,K, are expanded in the basis of the linear molecule, 1.4, z&,, I) [ 301,
(3.la)
+
Ci’$,;;‘( I 1 . z&n, , K- 1 ) + I - 1 , z&t , KS 1) ) ] .
(3.lb)
In eqs. ( 3.1 b) and ( 3.1 b), the total angular momentum about the principal axis of the bent molecule, K is given by K= (A+ I1,where I denotes the vibrational angular momentum. Subscript LJ,,~,,~ (with and without primes) denotes the vibrational quantum numbers of the bending vibration in the linear and bent geometrical structures, respectively. The contributions of the electronic angular momenta IA I= 1 and 2 are included in these equations since these are coupled to each other via configuration interaction in the bent structure. The dominant electronic configurations of the V system in the region 290-350 nm are recognized to be 7~: - np ( ]A I = 2) configurations, and
The rovibronic wavefunctions for the upper and lower Renner-Teller components are expressed in terms of the vibronic wavefunctions, eqs. (3. la) and (3. lb), vibrational ones, x. and rotational ones, r] as (3.2a) c2v
‘&h
2% -
Ill -r:9 ~a~~.~--_ lil” .ha ~______ 9
=:--
_--
2bl
----
%
__- ___
--__
___ _
_-- ___ =I- ---__ _
Ibq 5q La7 Y
5
4
‘a2 Lbz 3b2
t y+z
Fig. 2. Correlation of the lowest vacant and several occupied molecular orbitals of CS2 between Dmh and CzV point groups, and the convention of the axes. The configurations of the lower excited ‘Bz and ‘AZ electronic states are assumed to be constructed from the smgly excited contiguratlons 2x.-lrr, and 2a,-30. to evaluate the couplmg matrix elements.
Y FuJm’Wa et al. /External magnetrc$eld dependent fluoresence
209
deca.vs
and YA~d&f.Ka
- @Aw,en,Ka ~Au,J&,K,
n
(3.2b)
’ XAoa 9
The prime on IX indicates that the wavefunctions of the bending modes are omitted. The rotational of the Az electronic state is, for simplicity, expressed in the rigid rotor approximation as
respectively.
wavefunction
(3.3) where the subscripts v, J, M, and K are omitted. D& is the D function of rank J and M is the projection of the total angular momentum Jon the laboratory fixed Z axis. The rotational wavefunction of the B2 electronic state is expressed in a similar way. We are now in a position to derive a reduced expression for the Zeeman interaction matrix element, ( b l V, 1a). The operator of the external magnetic field H directed along the laboratory Z axis, V, is given in terms of the tensor of the electronic angular momentum [ 3 1 ] as (3.4) where D&g (0) is the component of the rotation matrix of rank 1 expressed in terms of the Euler angle 8. In eq. ( 3.4) T 8 ) (I), the tensor of electronic angular momentum with the spherical tensor component Q is given by t321 T!“(l)=-
T&“(l)=/_,
+I+
TL’{(l)=
+I-.
(3.5)
Here, I+ =I,+ il, and I- =I,-il,., in which I,, I,, and I= are the angular momentum coordinate frame, x, y. and z, respectively. The Zeeman interaction matrix element is expressed as
- ( -L,
vi,,,,
~~+LlT~‘(~)I-L~~~,~~,~~+L)l~(~Bl.h~bnrb~~l~06~’(~)l~Al~a~~n~~~
operators
n’
in the Cartesian
(3.6) Noting that < *L, 4,nt K~-LIT~)(I)I-~L,v~~~,,K~-L)=+LAG~,~,
forQ=O,
1
=o,
for Q#O
.
(3.7)
Eq. (3.6) can be written as (
%bJbMbKb
= - PogffA
where
1 vZ
I
yAvo&M&,
BA (&em
4
> ( VB~~J~M~KD
I 06;’ (Q) I VA~,~J~M~K,
> n’
(xB”~ I ~~~~~> :
(3.8)
Y. FUJi??Wa
210
et al.
/External magnetlcfield dependentjluoresence
decays
Rewriting the rotation matrix element of eq. (3.8) in terms of the Wigner 3J-symbols ( : : : ), we obtain for the reduced form of the Zeeman interaction matrix element the expression ( ylgL%JMhKh I vz I ~AL~‘aL&>=-~,g~~.,(~,“,,K)(-1)“-K[(2J1,+1)(2J,+1)l”Z (3.9) with K= K, = Kh, and M= M, = Mb. Eq. ( 3.9 ) contains the product of electronic ( Renner-Teller coupling), vibrational, and rotational terms. The contribution of the vibrational motion of the bending mode is included in the electronic term because the vibronic wavefunctions of the Renner-Teller coupled states are expressed in terms of the electronic and bending vibrational coordinates.
4. Results and discussion 4. I. Zeeman interaction between the upper and lower Renner-Teller components In the case where the rovibronic (2.6) can be simply expressed as
states of the lower Renner-Teller
k=FXCP,I(al~z,,lb)126(EIj-E~). a
component
consist of a dense manifold,
eq.
(4.1)
b
Furthermore, when all the allowed A4 states in electronic state a are simultaneously excited by an ultrashort pulse laser, the nonradiative decay rate averaged over M states is measured in the time-resolved fluorescence. In this case, the interference terms, the product of the interaction matrix elements ( Vz)a,,( VN)ab vanishes, and the rate constant may be expressed as a sum of two direct processes (after summing over M, and Mb, neglecting the effects of the lifting of M degeneracy, and noting that 1, ( - 1 )M ( : : : ) = 0 in eq. (4.1): k=k,+k,.
(4.2)
where kz= $1’
Vzlb) I*d(G-G)
a
1’ Pa I (al h
*
C’paI(a(V,lb)126(EZ:-E~). b
(4.3a)
and kN=$Cf
(4.3b)
In eqs. (4.3a) and (4.3b), primes denote sums taken over all quantum numbers except for the magnetic quantum ones. If {b} consists of a sparse manifold, and then the M dependence on Eh and rb as well as the effects of the lifting of the M degeneracy can safely be omitted. Under these conditions, the expression for the nonradiative decay constant is
rbI(alVzib)12
1
(4.4)
where r, is a sum of the damping constant due to the purely external magnetic fields r,(H) and that due to the intramolecular nonadiabatic couplings r,( H= 0). In the absence of an external magnetic field, the first term in
Y. Fu~mura
et al. /External
magnetlcfield
dependentfluoresence
211
decays
eq. (4.4) vanishes, and the expression represents the sequential decay mechanism (a~{ 6) *{I} ) induced by the intramolecular nonradiative interactions [ 281. From eq. (4.4), we can see that in the presence of an external magnetic field, a new channel of transitions between the rovibronic states in the Renner-Teller components opens up in addition to the channel originating from the nonadiabatic interaction. We call this new channel the external magnetic field-induced nonradiative transition. From the restrictions on the Wigner 3jsymbols in eq. ( 3.9 ), we find the selection rule for the new channel between the rovibronic states to be IJh_J,
I < 1
K,=K,,
and
M,=Mh.
(4.5)
As mentioned in section 1, the external magnetic field effect on the fluorescence decay of ‘Bz CS2 observed by Imamura et al. [ 22,23 ] is different from those in aromatic molecules such as pyrazine [ 9- 111 and pyrimidine [ 13 1. In aromatic molecules, the lifetime of the fast decay component (T, of eq. ( 1.3 ) ) is independent of an external magnetic field, and only the ratio of the pre-exponential factors is affected by the external magnetic field. The external magnetic field effects on the fluorescence of such aromatic molecules can be explained by the decoupling model. The external magnetic field dependent decay component (T, of eq. ( 1.3) ) observed for CS, ( ‘B>), on the other hand, is different from those in pyrazine and pyrimidine and can be explained by an external magnetic field-induced nonradiative transition. That is, an external magnetic field opens a new channel for the transition between the ‘B? and ‘A* rovibronic states. 4.2. Zeeman interaction between the lower Renner-Teller component (‘A j and the groundstate (‘C,t) It is well known that the fluorescence with a us decay rate from the ‘AZ state decreases in intensity in the presence of an external magnetic field through the enhancement of intramolecular energy redistribution processes as in the case of fluorescence from the ‘Bz state [ 1,19,21]. This phenomenon can also be explained in terms of the external magnetic field-induced nonradiative transition. The coupling matrix element (AZ 1Vz (C.J ) in the expression for the external magnetic field-induced nonradiative transition rate constant is
JJ’ (XAC. IXZLV >3
X where J,, KS and A4, denote the quantum
numbers
of angular motion in the electronic
(4.6) ground state ( ’ C ,’ ), and
Here, r and v%,,, denote the radial variable and quantum number of the doubly degenerate tively. From eq. (4.6), the selection rule for the external magnetic field-induced nonradiative to be
IJa-JsI&l
K,=K,kI
and
vibration, transition
respecis seen
M,=M,.
The mechanism of the nonradiative transition from the ‘A1 state due to an external magnetic field was discussed by Matsuzaki and Nagakura [26], who proposed a direct mechanism on the basis of the fact that the coupling matrix element with 1,. was nonvanishing. As is shown above, the electronic configuration with electronic angular momentum/i = 1 in the ‘A2 state is necessary for the nonvanishing matrix element to exist. In a similar way, the ‘BZ state is also coupled to the ‘Cz state in the external magnetic field because ( Bz ( Vz ( Cz ) has a nonvanishing value. That is, the ‘Bz state has two channels for the external magnetic fieldinduced nonradiative transition; ‘B2*‘A2 and ‘B,*‘X p’. From the relation ] C(.4=2’ ] > ) C(,‘=‘) ] in eq. (3.6), and the selection rules for the channels derived in the present paper, and also from the energy gap law, the former
212
Y
FUJlWWa
et al. /E.~~ternal wzagnetlcfield dependentjluoresence decays
channel makes the dominant contribution to the external magnetic field-induced nonradiative transition from the ‘B2 state. Thus, it is qualitatively shown that the origin of the ps fluoresence decay from the ‘B2 state in an external magnetic field is associated with opening of the new channel due to the Zeeman interaction between the Renner-Teller coupled rovibronic ‘BZ and ‘A2 states.
Acknowledgement This work was supported by the Joint Studies Program ( 1987-l 988 ) of IMS. One of the authors (YF) would like to thank Professor S.H. Lin of the Arizona State University, and Professor N. Shimakura of the Niigata University for valuable discussions.
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J.