J. Phys. Chem. 1985, 89, 4680-4687
4680
IL,-type band, if the cosine-square law (eq 5) is ~ a l i d . ~ ' . ~ ~ t(p)
=
t(O0)
cos* cp
(5)
Its validity seems, however, to be confined to an angular range of - 3 O O < cp < 90° for substituted anilines with strong donoracceptor c h a r a ~ t e r .For ~ ~ smaller twist angles, t is diminished with regard to the expected value. Thus, the combined effects i and ii lead to the anomalously low extinction coefficients for substituted indolines and julolidines. Broad angular distribution functions and the specific angle dependence of the energy of the 'L,-type state24 may also be (28) Braude, E. A,; Sondheimer, J. J . Chem. SOC.1955,3754.
responsible for the significant red shift of the absorption maximum of JULCN (33 700 cm-I) as compared to PYRBN (35 300 ~ 3 1 1 ~ ~ ) ~ ~ (compare also Figure 3 for the corresponding shift of the fluorescence spectrum).
Acknowledgment. We thank A. Flatow for recording the PE spectra. We acknowledge financial support from the Deutsche Forschungsgemeinschaft, the Fonds der Chemischen Industrie, and the BASF Aktiengesellschaft in Ludwigshafen. Part of this work was done under Project 05286 LI I of the Bundesministerium fur Forschung und Technologie (BMFT). Registry No. DMABN, 1197-19-9; DEABN, 2873-90-7; PYRBN, 10282-30-1; 2,5-PYRBN, 80887-45-2; PIPBN, 1204-85-9; EIN, 8993722-4; JULCN, 973 15-60-1
Enhancement of Molecular Fluorescence and Photochemistry by Small Metal Particles Purna Das and Horia Metiu* Department of Chemistry, University of California, Santa Barbara, California 931 06 (Received: November 8, 1984; In Final Form: June 4 , 1985)
We examine a quantum mechanical model for the enhancement of fluorescence yield and photochemical rates by the presence of a small particle capable to sustain electromagnetic resonances. The sphere lowers the absorption and emission efficiency by taking energy from the molecule and storing it into nonradiative modes, enhances the absorption by increasing the local field, and enhances the fluorescence by emitting efficiently the energy transferred from the emitting state to a radiative electromagnetic resonance of the particle. The present paper studies how the interplay of these effects modifies fluorescence intensity and the rate of photochemical reactions.
I. Introduction The spectroscopy and the relaxation dynamics of vibrational and electronic excitations of molecules and atoms adsorbed on metal surfaces have received a great deal of attention lately since they are involved in many surface processes such as surface phot~chemistry,'-~surface photoemi~sion,~ surface-enhanced Raman scattering (SERS),6,7 and photoluminescence.8-10 In the present article we examine molecular fluorescence and photochemistry in the presence of a surface. Qualitatively these phenomena are fairly well underst~cd.'~In all surface-enhanced processes the local field acting on the molecule is increased by reflection of the incident radiation. This increase is substantial if surface curvature is large," if the incident light excites surface electromagnetic resonances,l1-l6 or if solid (1) C. J. Chen and R. M. Osgood, Appl. Phys., A31, 171 (1983). (2)A. Nitzan and L. E. Brus, J . Chem. Phys., 75,2205 (1981). (3) G. M. Goncher and C. B. Harris, J . Chem. Phys., 77,3767 (1982). (4) G. M. Goncher, C. A. Parsons, and C. B. Harris, J . Phys. Chem.,
submitted for publication. (5) A. Schmidt-Ott, P. Schartenberger, and H. C. Siegmann, Phys. Reu. Lett., 45, 1284 (1980). (6) 'Surface Enhanced Raman Scattering", R. K. Chan and T. E. Furtak. Ed., Plenum, New York, 1982. (7) (a) H. Metiu, Prog. Surf. Sci., 17, 153 (1984); (b) H. Metiu and P. Das, Annu. Reu. Phys. Chem., 35, 507 (1984). (8) (a) G. Ritchie and E. Burstein, Phys. Rev. B, 24,4843 (1981); (b) J. F. Owen, P. W. Barber, P. B. Dorain, and R. K. Chang, Phys. Reu. Lerr., 47, 1075 (1981). (9) D. A. Weitz, S. Garoff, C. D. Hanson, T. J. Gramila, and J. I. Gersten, Opr. Left.,7,89 (1982); J . Lumin., 24/25,83 (1981). (10) A. M. Glass, P. F. Liao, J. G. Bergmann, and D. H. Olson, Opt. Lett., 5,368 (1980). (11) J. I . Gersten and A. Nitzan, J . Chem. Phys., 75, 1139 (1981); 73, 3023 (1980). (12) D. S. Wang, M. Kerker, and H. Chew, Appl. Opt., 19,2256, 4159 (1980). (13) D. S. Wang and M. Kerker, Phys. Reu. E , 24, 1777 (1981). (14) S. Efrima and H. Metiu, J . Chem. Phys., 70, 1602, 1939, 2297 (1979).
0022-3654185 12089-4680$01.50/0
particles are used to polarize each other.I7 This local field enhancement increases the rate of photochemical processes and the intensity of adsorbate fluorescence. The presence of the surface, however, causes energy transfer from the molecule to surface excitations, such as p l a ~ m o n s , ' ~electron-hole J~ pairs, or This results in a decrease of the excited population which causes a decrease of the photochemical rate. The total rate is established through the competition of these opposing effects. In the case of fluorescence an additional complication arises because the excited molecule polarizes the ~ ~ r f a ~ and e ~ ~ , ~ , ~ this emits at the molecular frequency, thus enhancing the emission intensity. This enhancement is particularly strong if the emission frequency matches that of a radiative surface resonance. Recently several theoretical models2~'e'6~22~23 predicting and/or explaining observed fluorescence8-I0 and p h o t o c h e m i ~ t r yof ~~ adsorbates have appeared in the literature. Our purpose in this article is twofold: to provide a more systematic derivation of the quantum mechanical behavior of the molecule-sphere system, and to carry out a more extensive numerical study of fluorescence and photochemical rates. Previous work included surface effects in an intuitive manner by replacing the oscillator strength and width appearing in the quantum theory for an isolated molecule with surface-dependent expressions (15) (16) ( 1982). (17) (18) (19) (20)
P. K. Aravind and H. Metiu, Chem. Phys. Lett., 74, 301 (1980). J. Arias, P. K. Aravind, and H. Metiu, Chem. Phys. Lett., 85,396
P. K. Aravind, A. Nitzan, and H. Metiu, Surf. Sci., 110, 189 (1981). R. R. Chance, A. Prock, and R. Silky, Adu. Chem. Phys., 37 (1978). H. Morawitz and M. R. Philpott, Phys. Reu. B., 10,4863 (1974). P. Avouris and B. N. J. Person, J . Phys. Chem.,88,837 (1984), and references therein. (21) (a) H. Metiu, Isr. J . Chem., 22,329 (1983); (b) K. M. Leung, G. Schon, P. Rudolf, and H. Metiu, J . Chem. Phys., 81,3307 (1984). (22) D. A. Weitz, S. Garoff, J. I. Gersten, and A. Nitzan, J . Chem. Phys., 78,5324 (1983). (23) X. Y. Huang, J . Lin, and T. F. George, J . Chem. Phys., 80,893 (1984). (24) S.Garoff, D. A. Weitz, and M . S . Alverez, Chem. Phys. Lett.. 93, 283 (1982).
0 1985 American Chemical Societv
Molecular Fluorescence and Photochemistry Enhancement
‘
I
I
I
I
I t
)IwI
Izrcry
I
I I
I I
The Journal of Physical Chemistry, Vol. 89, No. 22, 1985 4681 of absorption and the induced molecular dipole. The latter must be determined by a self-consistent procedure which includes self-polarization caused by the fact that the induced dipole polarizes the surface an! interacts with that polarization field. The incident field Eo(w,t) is reflected by the surface and this and called the produces a new field denoted here R(w,t).Eo(t,w),14 reflected field. A large number of formulae for the reflection tensor R(w) for various surface geometries are given in ref 7a. The molecular dipole induced by Eo(w,t) polarizes the surface and the polarization field at the point where the molecule is located is d e n ~ t e d ’ ~ , ’ ~ G(w,t).(ji(t,w)) C(w,t) is called here the image tensor and (,ii(t,w))
Figure 1. Schematic diagram of a three-level system. w and w’ are the incident and emitted light frequencies, respectively. E , , E,, and E , are the respective level energies. Solid vertical lines indicate stimulated transitions; dashed vertical lines indicate spontaneous (radiative) transitions; the wavy line indicates a radiationless transition.
suggested by the classical theory. The intuitive arguments are quite reasonable, but we prefer to derive the rate equations by using a detailed quantum model for the molecular processes while some of the effects of the sphere (specifically, the image field) is treated semiclassically. Our results confirm those of previous work. The theory uses a small (Rayleigh limit) sphere as a model of the surface. This can be experimentally realized in colloidal systems25or by preparing matrix-isolated metal sphere^.^^,^' To some extent a small sphere is a crude model of a rough surface; however, such extrapolations must be made with care since all the effects mentioned above could depend strongly on the shape of the surface.”-16,2s 11. The Model We develop a quantum mechanical model for the steady-state fluorescence or photochemistry of molecules adsorbed on a metal substrate. The system we have in mind consists of a.smal1 metallic particle or a rough metal surface covered with adsorbates and irradiated with an external light source. We are concerned with the influence of the substrate on the photochemistry of adsorbates. To this end we consider the incoherent light scattering from a molecule, idealized as a three level system, adsorbed on a dielectric substrate. II.1. The Three-Level System. The schematic diagram of the adsorbate-substrate system is shown in Figure 1. State 11) is the initial state (ground electronic level) and the states 111) and 1111) are excited electronic states. These are molecular states “dressed” by the presence of the substrate. State 111) is populated by absorption of energy from the light source. This is accompanied by a radiationless relaxation to state 1111) and radiative relaxations to state 11). If state 1111) is a dissociative continuum the molecule undergoes photochemical decomposition. If 1111) is a bound, discrete electronic level it can decay radiatively to the ground state 11). The presence of the surface modifies the transition 11) 111) and 1111) 11) but leaves 111) 1111) unchanged. The latter assumption excludes the chemical effects of the surface and it might be inaccurate for molecules chemisorbed at the surface. This is not a serious limitation here since we are concerned with long-range electromagnetic effects taking place many angstroms away from the surface. 11.2. Absorbing System: Spontaneous Emission. We consider the interaction of the external radiation with the molecule (only absorbing system 11) 111) is considered) t o determine the rate
-
-
-
-
(25) J. A. Creighton, C. G. Blatchford, and M. G. Albrecht, J. Chem. SOC., Faraday Trans. 2, 75, 790 (1979). (26) H. Abe, K. Manzel, W. Schulze, M. Moskovits, and D. P. DiLella, J . Chem. Phys., 74, 792 (1981). (27) H. Abe, W. Schulze, and B. Tesche, Chem. Phys., 47, 95, 2200 (1980). (28) P. C. Das and J. I. Gersten, Phys. Reu. E , 25, 6281 (1982).
y2[ji-(w)e-iut+ cc]
(11.1)
is the expectation value of the induced molecular dipole. To calculate the induced dipole self-consistently we include the field C(w,t).(,ii(t,w)) in the expression of the local field
Zl(w,r)
= y2[(I + R(U))-,!?~(~)+ C(w)-(,ii_(w))]e-’”’ + cc (11.2)
which is responsible for the polarization of the molecule. In this expression I is the unit tensor. The field-molecule Hamiltonian can be written as H = Ho+ V(t),where Hois the molecular Hamiltonian and V(t)is the perturbation due to the interaction with the radiation. In the interaction representation the radiation-molecule coupling V,(t)is given by
where Vis the volume of the cavity used for field quantization; the sum is carried over all the photon modes in the cavity; ai is the annihilation operator for a photon of frequency wk (to keep the notation simple we do not make explicit the fact that the photon polarization is included in ak and in the sum over modes; iik is the unit polarization vector; Gji is the transition dipole moment b); and wji (E, corresponding to a molecular transition l i ) - E i ) / h . In writing VI(t)we use the rotating wave approximation which neglects small, nonlinear terms. The image field has been treated semiclassically while the incident and the reflected fields are quantized. The field quantization is needed only to make sure that spontaneous emission is treated properly; other quantum electrodynamic effects are not expected to play a role in surface spectroscopy since we are working at intensities where the statistical properties of the photon are unimportant. The “image” term was treated semiclassically for convenience, since otherwise we would have to quantize the electrons and phonons in the solid to get the correct expression for the manner in which the polarized solid renormalizes the photon field.21 Note that in treating the semiclassical term we must disregard the photon states in the wave function. Thus in dealing with the quantized field we have 11) = Il)lnk + 1 ) and 111) = I2)lnk) where 11) and 12) are the molecular wave functions and Ink l ) and Ink) are field wave functions in the occupation number representation (a coherent state would be more adequate in describing laser radiation). However, in dealing with the semiclassical term we use 11) = 11 ) and 111) = 12). The equations of motion for the matrix elements of the density operatorz9 are
-
+
bii(t) =
i
- h ( i I [ V t ) , ~ ( t ) I I i+) C K ’ n ~ n n
(11.43)
nZ1
i
pij(t) = (-iwij - rij)pij(t)- h ( i l [ V ( t ) , p ( t ) ] p ) ,i # j
(II.4b)
4682
The Journal of Physical Chemistry, Vol. 89, No. 22, 1985
where (li),b)]= {lI),lII)}. V(t) is the interaction potential given by (11.3), but taken in Schrodinger representation; rijis a phenomenological parameter that characterizes the depopulation induced by interaction with the medium. If we consider a two-level system in the presence of a laser, but no surface, then the eqution of motion for the diagonal and off-diagonal elements of the density operator are
i PZZ(t) = -p[vo(t),P(t)l12) bll(t)
P Z I ( t ) = (-iwz1
- r21) P Z l ( t )
=
- rzpzz
-P22
i
- p[Vo(t),P(t)lll)
PIZ(t) =
PZl(t)*
(II.5a) (11.5b) (II.5c) (II.5d)
Here Vo(t) is obtained from eq 11.3 with R(w) = G(o) = 0 and written in Schrodinger picture. r2and I'21= r 2 / 2 are phenomenological parameters and rzhas both radiative and nonradiative components. The presence of the surface modifies both the radiative and the nonradiative decay rates. However, the present quantum mechanical model, as shown in the following derivations, does not produce the radiative correction due to the surface. Hence we include the surface-induced radiative correction in the width of the excited state (see eq 11.23). In the interaction representation, eq 11.4 can be written as p;](t)
= p~l(w)e-i(o-w21)'
PI1At) = P h ( t ) *
(II.6a) (II.6b)
Since (jL(t)) = Tr(iip(t)), the induced dipole ( f i ( t ) ) depends on itself through p ( t ) . However, eq 11.5, 11.6, and 11.3 can be solved to obtain self-consistent expressions for and (,&(w)):
(PI1 - Pzz)
I(w - w21 + irZl) +
ca+cw,, = (,ii-(w))*
(11.8b)
In these expressions the intensity of the light source appears through the factor n + 1, where n is the photon occupation number of the incident field, which is given by
n
+ 1 = VlEol2(8~h~)-'
(11.8~)
The term R(w) is particularly large if the light excites a radiative electromagnetic resonance of the surface. The polarization of the surface by the induced molecular dipole appears through G(w). This quantity acts as a molecular ~ e l f - e n e r g y since, ~ l ~ ~ by ~ ~polarizing the surface and interacting with the polarization charge, the molecule polarizes itself. As in any other areas of many-body theory the self-energy shifts and broadens the original state. Note that even though we treated the sphere by classical electrodynamics (29) K. Blum, 'Density Matrix Theory and Applications", Plenum, New York, 1981, Chapters 6 and 7. (30) (a) G . Korzeniwaski, T. Maniv, and H. Metiu, Chem. Phys. Left.,73, 212 (1980); J . Chem. Phys., 76, 1564 (1982); (b) G. Korzeniewski, E. Hood, and H. Metiu, J . Vac. Sei. Technol., 20, 594 (1982); J. Chem. Phys., 80, 6274 (1984).
Das and Metiu the structure of the theory is the same as the one obtained by a full quantum treatment of the electrons in the sphere and the two-level system. The numerical values of the self-energy are, however, different; a full quantum theory for the sphere introduces nonlocal electromagnetic effects, which are important at small (Le. about 5A30) distances. A detailed comparison between local and nonlocal theories can be found in ref 30b. Throughout this paper we use for C(w) a local theory. In this theory G(o) is enhanced whenever the dipole couples to the radiative or the nonradiative resonances of the surface. Note that if G = R = 0 the expressions derived above reduce to those for a two-level contains the system in the absence of the sphere. The width rZl dephasing and/or the depopulation caused by intramolecular and/or radiative transitions and the interaction with the medium. It is interesting to note that the term
is a 3 X 3 tensor. The tensorial character of this quantity does not mean that the lifetime is a tensor. It simply means that the lifetime depends on the orientation of the transition dipole with respect to the surface. In particular, parallel and perpendicular dipoles have different lifetimes. Since the lifetime depends on the rate of energy transfer from the dipole to the surface, the existence of the above effect in the theory is reasonable. The induced dipole (&(a))and the off-diagonal element p z l depend on the population difference (pI1 When the populations of the ground and excited state become equal, the induced dipole moment of the molecule vanishes and so does the surface-induced broadening and frequency shift. This is the wellknown saturation effect caused by intense radiation. If the external field is weak enough, we can approximate (pll - p z 2 ) N (poll p022) where poll = 1 and p022 0 are thermal equilibrium values of the populations. We shall not resort to this approximation since the strength of the surface-enhanced local field can be large for certain choices of parameters. The above calculations provide us with self-consistent expressions for ( E ) and pl2. To compute the rate os change of population we take the external field to be [I + R(w)].Eo(w). This expression does not contain the field G+i) since the latter is already included, by the use of the self-consistent procedure, in the self-energy of the two-level system. Using the expression of V,(t),modified as indicated above, and eq 11.4 and 11.7, we obtain
and p l l ( t ) = -pZ2(t), Equation 11.9 gives the rate of change of the population of state 111) by the emission_of a photon of frequency w , polarization ii, and wave-vector k . In addition to ra( twill ) be nonradiative decay diative contributions to ~ ~ ~there channels which are not included in eq 11.9. Although not necessary in principle, we shall make certain simplifications in the mathematics assuming that both the_induced dipole and the incident field have the form ,ii= pLi and Eo = E@, where i is the unit vector along the z axis. Then the image field due to the dipole induced in the surface, at the location of the dipole picks up only the z component of the induced dipole. So only C,,(w) = C ( w ) is nonzero. Thus we can rewrite the equations derived here as
(
Y ) l i z ( n
+ l)1/2(,ii21-(I+ R(w)).ii)jiI2/(w
-
w21
+i
~ ~ , )
Molecular Fluorescence and Photochemistry Enhancement
-
where
-
rSaand rseare the stimulated absorption and emission rates for the transition 11) III), and r$)is the corresponding spontaneous photon emission rate. The explicit expressions for various rates of absorption and emission are obtained by inspection of eq 11.9:
rsa= r$)(n+ 1) rse= rj;)n
(11.14) (11.15)
systems are coupled only through the radiationless transition 111) 1111). We denote by rzand r3the rates of decay of populations of levels 12) and 13), respectively, for an isolated molecule modeled as a three-level system. They include both radiative and nonra(i = 2, 3), diative parts, such that we can write ri = rl”)+ rYR) and treat the full ri as an empirical molecular parameter of the theory. The presence of the sphere simply renormalizes these decay rates, by adding to them the surface-induced terms discmsed in section 11.2, We thzrefore denote their values when the sphere is present by r2and r3.Written out explicitly these quantities are
f,
=
(11.22)
where Y21(W)
7’21(w) + I’”’z1(w)
(11.17)
and y’21(w) and y”zl(w) are real. For large n, rsaN rseas expected. They dep_end on the incident laser intensity through the relation n = l/lEo12/8aho. The total spontaneous emission rate is obtained by summing eq 11.16 over all wave vectors k and polarizations u. Using the fact that the number of modes per unit frequency interval and per unit solid angle is (for a given polarization) (11.18) we obtain the spontaneous emission rate per unit solid angle (about the propagation direction) and per unit frequency range: dZZ(2) 1zz1-(I +R(w))WZ~(W) - - sp - -2w3 (11.19) dw d o a h [(o ~ -~~ 2 -1 y”21(~))’ Y’~,(U)~]
+
In order to get the total spontaneous emission rate r$ we integrate the right-hand side of eq 11.19 with respect to w and Q. Approximating, as customary, y 2 ] ( w ) y21(w21),where wZ1is the transition frequency, we obtain the total spontaneous emission rate as
where Gzl 1 wZ1 + y”zl(W23]); y”21(w) is the shift of the upper level due to the presence of the surface. This shift is usually small compared to w21 and hence may be neglected. When y”21is neglected, the effect of the sphere on the spontaneous emission (within the approximations made here) appears through R(w,,) and we recover the usual formula when the sphere is absent: (11.21) which is identical with Einstein’s A coefficient. A similar inspection of eq 11.15 in the absence of the surface and in the limit 0 gives Einstein’s B coeffi~ient,~~ implying that the presence of a polarizable object renormalizes the Einstein coefficients via ijij and (I R(Bij)). 11.3. Steady-State Population. So far we have focused our attention on the absorbing system (11) 111)) alone and established expressions for the power absorption by the molecule in the presence of the sphere as well as the rate of spontaneous 11) transition. These formulae emission of photons via the 111) contain the level populations, which we now set out to determine. The emitting system consists of levels 1111) and 11) and fluorescence emission takes place via the spontaneous transition 1111) 11). Since the fluorescence intensity is low we do not include stimulated processes in the 1111) 11) transition. The absorbing and emitting
-
+
-
-
-
-
(31) R. Loudon, ‘Quantum Theory of Light”, Clarendon Press: Oxford,
1973.
The Journal of Physical Chemistry, Vol. 89, No. 22, 1985 4683
r3= r 3 ( l + q3{IR(w’)I2+ 2 Re R(w’))) +
,
where q I’$”)/r, and q3 = I’i3)/F3are the quantum efficiencies of oscillators 2 and 3, respectively; w is the frequency of the incident laser and w’ that of the fluorescent emission; and rjR) and rjNR) (i = 2,3) are the radiative and nonradiative components of ri,respectively. Note that the surface-induced decay rates depend on R(w) and Im G(w). Further, let us write rse= rsa. Then the equations of motion for the populations are (II.24a) P I I = rse(~22- P I I ) + f 2 ~ 2 2+ F 3 ~ 3 3 P22
= rse(P11 - ~ 2 2 )- f 2 ~ 2 2- K~22 P33
= KP22 - f3P33
(II.24b) (11.24~)
Steady-state solutions for pii (i = 1,2, 3) are obtained by requiring that pI1 = p 2 , = p33 = 0: pI1 =
(rse+ K + f , ) / [ ~+ f 2+ rse(2 + K / f , ) ] ~ 2 = 2
rsePll/(rse P33
+ K + 72)
= (K/f3)p22
(II.25a) (11.25b) (I I.2 5c)
The level populations are dependent on the molecule-surface separation through rseand r i (i = 2, 3). rsn is given by eq 11.15 and 11.16 where yZ1(w) is replaced by (r2+ K)/2. These 5teady-state values of the level populations can be used to compute ri(o)(eq 11.22 and 11.23). 11.4. Enhancement of Fluorescence. The spontaneous radiative decay of state 1111) causes a fluorescence signal of frequency a’. The dipole induced in the sphere by the emitting molecular dipole radiates at the frequency w’. Thus the radiation from the system consists of the coherent sum of the radiation for both molecular dipole and the dipole induced in the sphere. However, instead of going through the complications of determining the molecular (emitting) dipole moment at w’, and computing the fluorescence from the classical radiation formula, we write the power emitted by the molecule due to a spontaneous transition 1111) 11) as
-
PFI = Np33hw’r6i)T
(11.26)
where N is the total number of molecules in the system and rsp is the total rate of decay due to spontaneous emission from state IIII), and is given by 4 r(3)T = -G313p31.(I SP
h c3
+ R(G31)).;(*
(II.27a)
or
In writing eq II.27b we neglected the frequency shift caused by the sphere and averaged over the random orientations of the polarization direction F. If we denote the power emitted by isolated
4684
The Journal of Physical Chemistry, Vol. 89, No. 22, 1985
molecules by PF1 then the enhancement ratio for fluorescence is
Das and Metiu for the photochemical reaction to have any effect on the steady-state kinetics, so that p l l + p22 = 1. The steady-state solution of eq 11.29 yields n
(II.30a)
where p$) is the population of level 3 in the absence of the sphere. Since (P33/P33) = (P22/P22)(r3/r3) we can write PI1
= (1
- P22)
(II.30b)
where rsa= rse.Since K, is the number of times a molecule undergoes photochemical reaction per second, the number of molecules decaying per second by photochemical decomposition is given to be NK,pz2. In units of N , the photochemical rate is KpJ,, (II.28b) In a resonance situation (w w21 and W’ N ~ 3 1 we ) neglect the level shifts and assume rse
