Transport and capture of electronic excitation energy in the photosynthetic apparatus

J. theor. Biol. (1972) 36,223-235

Transport and Capture of Electronic Excitation Energy in the Photosynthetic Apparatus Gw

POW

Service de Biophysique, Centre d’Etudes NuclPaires de Saclay, B.P. no 2-9l-Gif-sur- Yvette, France (Received 30 November 1971) The different parameters which characterize the microscopic properties of energy transport in the photosynthetic unit are evaluated. It is shown that the rate determining step in the process of exciton capture is the charge separation which takes place at the reaction centre, and that the pigment heterogeneity promotes the capture of energy by the reaction centm. Finally, the model with restricted motion of excitation between photosynthetic units is discussed.

1.IEtroduction The energy transfers which take place in the photosynthetic apparatus, revealed by discovery of photosynthetic units (Emerson & Arnold, 1932) still give rise to many investigations. The properties of this energy transport have been studied from three points of view. (i) Microscopic properties of the energy transport: In this field the authors disagree on the nature of the transfers. According to the majority (Bay & Pearlstein, 1963; Pearlstein, 1967) the energy is carried by “localized excitons”, whereas Robinson (1967) claims that the carriers are “free excitons”, All the authors however (Bay & Pearlstein, 1963; Robinson, 1967; Knox, 1968) use a Markoffian random walk model to describe the exciton movement. (ii) Properties of the photosynthetic unit (PSU) as a whole: According to the hypothesis proposed for the microscopic processes, opinions also differ concerning the properties of the PSU. Some authors stress the time needed for the exciton to reach the reaction centre by diffusion whilst others as Robinson (1967), consider the rate determining stage to be the capture of the exciton. Thus the authors assign quite diflbrent values to the efficiency of the chemical reaction taking place at the reaction centre. (iii) Properties of a set of photosynthetic units: It has been admitted by Vredenberg & Duysens (1963) for photosynthetic bacteria, and by Joliot & 223

224

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PAILLOTIN

Joliot (1964) for Chlorella, that the PSU’s are not isolated. However, the properties of these two types of organism being fairly different, a separate model is used for each to describe the relations between PSU’s (Duysens, 1967). The model with free movement of excitation between units accounts for the properties of photosynthetic bacteria (Vredenberg & Duysens, 1963) whereas a restricted motion model is often used to describe the properties of algae (Joliot & Joliot, 1964). However, we showed earlier (Paillotin, 1968) that a fairly slow diffusion of the exciton in a set of PSu’s with unrestricted movement of excitation can also account for the experimental results obtained with algae. Finally, Clayton (1967) has proposed a more general model in which the two examples above are peculiar cases. However, the fact remains that the inter-PSU relations introduced in the restricted movement models still lack physical bases. The purpose of this paper is to re-evaluate all these properties of energy transport on the basis of results obtained recently (Paillotin, 1971) concerning the microscopic properties of energy migration in the PSU. We shall begin with a brief review of these results, presenting some mathematical derivations in the Appendix. From a numerical estimation of the different parameters characterizing this energy transport we shall then conclude, with Robinson (1967), that the chemical reaction which takes place at the reaction centre is the rate determining step in the process of exciton capture. Taking into account the heterogeneity of its component pigments we shall subsequently examine a number of properties of the PSU, and finally propose to describe the inter-PSU relations, a simple model which offers physical support to models including a restriction of excitation movement between units. 2. Microscopic (A)

Movement of Excitons

MASTER

EQUATION

We showed recently (Paillotin, 1971) that in the PSU the excitons are localized. This is due to the fact that, in the PSU, any excitation is accompanied by a local modification of the equilibrium positions of the nuclei; moreover the excitons spread out more slowly than the time of molecular displacement. Thus the motion of these localized excitons is a hopping process. However, the coupling between chlorophyll molecules in the PSU is strong enough so that the time between jumps is not much greater than the time needed for one jump (see section (c)). Such a situation has already been studied for electrons in semi-conductors with low mobility (Kudinov & Firsov, 1966). Their movement is not a Markoffian process. We previously derived (Paillotin, 1971) a similar master equation for excitons in the PSU.

ENERGY

TRANSPORT

IN

225

PHOTOSYNTHESIS

This equation includes a diffusion term with after effect. Let us introduce the probability P(n, t) of finding an exciton at the nth molecule at the instant of time t > 0, if it was created at t = 0. The time evolution of P(n, t) is described by the following equation:

dP(n, 0 ~

dt

= - t,-‘P(n,

t) + c j [Wm.(t - t’)P(m, t’)-

W&(t-

t’)P(n, t’)] dt’,

(1)

PRO

where t, is the lifetime of an excitation localized at site n; tn takes into account all desactivations (radiative and non-radiative) except the transfer of excitation to neighbouring sites. Wnm(t) is a function of excitation transfer from site n to site m and its explicit form is given in the Appendix [equation (A2)l.t Later we shall show that the properties of the PSU are such that equation (1) can be notably simplified as far as we are concerned by fluorescence or exciton capture (section 3). (B)

DESACTIVATION

AND

CAPTURE

OF EXCITONS

IN

THE

PSU

The right side of equation (1) includes two terms with quite different roles; the first is a desactivation and the second a diffusion term. The former expresses all the PSU desactivation processes including exciton capture at the reaction centre. It will be assumed hereafter that for any molecule outside a reaction centre the lifetime t,, does not depend on the site n, i.e. 2. = to if n does not correspond to a reaction centre. The properties of the reaction centres themselves seem not to differ fundamentally from those of other molecules. In particular the desactivation processes characteristic of these latter exist also for the reaction centres since they can be detected at low temperature (Cho, Spencer & Govindjee, 1966), under strong intensities (Krey & Govindjee, 1964) or in the presence of inhibitors (Krey & Govindjee, 1966). They can also retransmit their excitation energy to their surroundings as shown by the emission of luminescence (Lavorel, 1968). Their chief property is therefore of presenting an additional means of desactivation, illustrated by the reaction ZTxQ+Z+TQ-,

(2)

where T refers to the trapping molecule, TX its excited state, 2 and Q the primary electron donor and acceptor, respectively. This property is expressed by equation (1) on the condition that the excited state of the reaction centre is assigned a lifetime different from the time to allotted to the other molecules of the PSU. Thus if a chemical time tT t (A) refers to an equation in the Appendix.

226

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PAILLOTIN

is associated to the charge separation reaction (2), the condition necessary for the reaction centre to be effective is t* < 1,. (3) Several authors (Joliot, Barbieri & Chabaud, 1969; Kok, Forbush & McGloin, 1969) have shown recently that reaction (2) depends on the “state” of 2 and Q. This implies that the value of tT can depend on the “state” of Z and Q. Besides the desactivation terms analysed above, equation (1) also includes a diffusion term. Here again the reaction centres may perturb this diffusion, at least in their vicinity. However, we shall see that this is not a characteristic property of these reaction centres but a special case of the effect of pigment heterogeneity in the PSU. If the diffusion term were alone in equation (1) it would make the distribution P(n, t) tend towards a limit P(n, co). If the time defining the evolution of P(n, t) towards P(n, co) is much shorter than the time t,, and if only the general properties of the PSU are considered, it may be assumed that the distribution P(n, 03) is achieved at any moment. We shall demonstrate that this is the case for the PSU, neglecting at first the heterogeneity of the PSU pigments. (C)

PART

PLAYED

BY

DIFFUSION

IN

A

CHLOROPHYLL

U AGGREGATE

Let us consider an aggregate of chlorophyll a (chlu) molecules, all identical but possessing the properties of chla in vivo (for example, chla 668). As shown in the Appendix, two time units characterize the diffusion of excitons in such an aggregate: a “jump-over time” t,, i.e. the time needed for the exciton to make one jump [equation (A7)], and a “time between jumps” t,, [equation (Al2)]. In order to estimate these times we need to know three energies. The first one is the electronic interaction V between closest neighbours. Here we shall take the value obtained by Robinson (1967), V = 100 cm-‘. The second is the half-width 6R of the chla red absorption band in vivo. We shall adopt the value 60 = 450 cm- ’ obtained in the analysis into Gaussian components of the absorption band of Chlorella (Rabinowitch et al., 1967). The third is the Stokes shift AR between the maxima of the absorption and emission bands of chla in vivo. We shall choose the value An = 200 cm- ’ (Rabinowitch et al., 1967). One of the characteristics of chlorophyll molecules is precisely this low Stokes shift value, which means that the activation energy E, involved in the intermolecular energy transfers is very weak. According to equation (AS) E, N aAn = 50cm-‘. (4)

ENERGY

TRANSPORT

IN

PHOTOSYNTHESIS

227

Hence the time between jumps tn is relatively short. From equation (AS) tD = 0.07 psec. (5) This to value is very similar to that proposed by Robinson (1967). The jumpover time t, may be determined from expression (A7), which gives t, = 0.03 psec. (6) The first conclusion that we can draw from these numerical values is that the condition (AlO) t, Q tD necessary for the exciton motion to be Markoffian is not fulfilled and that consequently equation (1) is valid. However, the main fact to retain is that the times t, and to characterizing the trend of P(n, t) towards P(n, co) are much smaller than the desactivation time to. For to we take the value adopted by Brody (1957): t, 2? 15 nsec (7) and hence t,, r, 4 t,. It may thus be assumed that whatever the initial distribution, the limit distribution P(n, co) is reached in a time much shorter than t,. This can also be illustrated by calculation of the exciton diffusion length L. Indeed

L = aJto/tLl,

(8)

where a is the mean distance between molecules in the aggregate; therefore L = 460a. The radius R of an aggregate shaped as a flat disc containing 300 chla molecules is about lOa. Hence L = 46R B R. (9) We thus find that for a chla aggregate similar in size to the PSU the limit distribution P(n, co) is reached before any desactivation occurs. This is also true as shown below, for the real PSU where pigment heterogeneity is taken into account. 3. Properties of the Photosynthetic Unit According to the definition of Duysens (1967) a PSU consists of a reaction centre and the chlorophyll molecules most liable to transfer their excitation energy to this reaction centre when all centres are open. Finally, unless otherwise stated, the properties considered here will always refer to system II. In the case of PSU, the activation energy E. depends on the excitation energy of the molecule from which the jump originates and on the excitation energy of neighbouring molecules. As a result the time between jumps tD is dependent on the environment of each molecule in the PSU. As pointed

228

G.

PAILLOTIN

out earlier (Paillotin, 1971), the electron excitation energy En (0 + 0 transition) of each molecule n acts in reality as a potential for the exciton motion. Hence the conclusions of the previous section still apply to the PSU unless it contains several sub-units separated by potential barriers of height greater than kT. This condition is assumed to be fulfilled by an isolated PSU, otherwise part of the PSU would in practice be cut off from the reaction centre and would only contribute to the emission of “dead fluorescence” (Clayton, 1969). However, we shall see that this condition has no reason to be satisfied for a group of PSu’s. We may therefore presume that the limit distribution P(n, co) is obtained at each moment in the PSU. It does not correspond to a uniform distribution of excitation probability, as in the case of the aggregate of identical molecules, but as shown in the Appendix, equation (Al3), it complies with Boltzmann’s distribution law. (See also Davydov, 1964.) P(n, 00) = 2-l exp (- jL?$), WV where Z=Cexpt-B&J (11) m and /I = l/kT. From equation (1) we can obtain the following equation dP(t) 1 -=-dt to with p(t) = 1 P(n, 0 n and where P(T, t) is the probability that the exciton will be at the reaction centre at instant of time t. However, on the basis of our previous comment we can replace P(n, t) by the expression P(n, t) = P(n, co) P(t) (12) and hence from equation (12), to describe the time evolution of the PSU excitation probability P(t), we obtain the equation dP(t) 1 .-cm-(13) dt to where according to equation (10) P(T, a~) = Z-’ exp (-BET), ET being the excitation energy of the reaction centre. Indeed, equation (13) replaces the master equation (1) when t,, tn Q to and simplifies the theoretical discussion on the properties of the PSU. Although it can only be applied to the case of the isolated PSU, this equation is similar

ENERGY

TRANSPORT

IN

PHOTOSYNTHESIS

229

to the phenomenological equation used by Vredenberg & Duysens (1963) for a set of PSU’s. From equation (13) we derive the expression of the excitation lifetime tL in the PSU 1 t, = t, (14) lfP(T, c.o)(f&-1)’ Thus at room temperature, assuming that one molecule out of 300 is a reaction centre and that the difference between the excitation energy of this centre and the mean excitation energy of the PSU molecules is about 200 cm-’ (D&kg, Bailey, Kreutz 8z Witt, 1968), we have P(T, co) ‘y 1%. By then taking tL = 1.5 nsec (Murty, Cederstrand & Rabinowitch, 1965) we obtain from equation (14) t, = 20 psec. (15) This time corresponds to a non-radiative desactivation of the kind described by expression (2) (see also Pearlstein, 1967). Finally, if we define the capture efficiency by the ratio rD/fT we find that it is less than 1% as suggested by Robinson (1967). The microscopic properties of exciton motion in the PSU (especially the very low value of the activation energy E,, (4)) are such that the reaction governing the PSU desactivation kinetics is the charge separation occurring at the reaction centre. In more vivid terms the exciton loses no time between the moment of its creation and the moment when it is trapped at the reaction centre. In addition, as mentioned earlier, the effect of the reaction centre on the diffusion processes is only a special case of the pigment heterogeneity effect in the PSU, expressed by the Boltzmann factor, equation (12). Because of the trap depth the exciton capture probability is thus improved by a factor three. Only the properties of system II have been discussed in this section but a similar argument may be used for system I, although in this case the trap is deeper; thus according to equation (14) a trap depth of 500 cm-‘, all other things being equal, leads to a fluorescence yield five times lower than in the previous case where the depth was 200 cm- ‘. This may partly explain the low fluorescence yield of system I. Using a different approach, Seely (1971) has also discussed the pigment heterogeneity effect in system I. 4. Properties of a Group of Photosynthetic Units The model with free movement of excitation between PSU’s accounts for the experimental results obtained with photosynthetic bacteria (Duysens,

230

G.

PAILLOTIN

1967; Clayton, 1967) but seems to be unsuitable for algae. In this latter case, Joliot & Joliot (1964) proposed a model with loosely connected units. However, in the case of a free movement model the authors generally assume implicity that during its migration the energy is able to meet a large number of reaction centres, i.e. that the exciton diffusion length is very large compared with the inter-centre distance. Conversely, if the diffusion length were too short the PSU’s could be considered independent. On the other hand, if this length is of the same order of magnitude as the inter-centre distance the continuous model can suitably describe the results obtained with algae (Paillotin, 1968). A continuous model, i.e. a model with no boundary between units, can thus account for the properties of both photosynthetic bacteria and algae as long as in the former case the exciton diffusion length is much greater than the centre-to-centre distance, and in the latter case the two are of the same order of magnitude. As shown in section 2, however, the exciton diffusion length in a chlorophyll aggregate is very much greater than the size of the PSU. The diffusion length in an aggregate could only be of the same order of magnitude as the centre-to-centre distance in algae on condition that I--t0 21 10, tD i.e. t, iz O+Olt, = 0.15 nsec.

J

When allowance is made for expression (A9), this leads to an extremely low interaction energy V V = 2-5 cm-‘. Such a value is completely incompatible with the known properties of the chlorophyll molecule (Robinson, 1967), which means that a continuous model neglecting pigment heterogeneity cannot account for the properties of algae. It must therefore be assumed that for these organisms the exciton diffusion length is limited by a certain restriction of movement between units, the simplest cause of which appears to be the heterogeneity of the pigments in the PSU. Thus from chlorophyll b to chlorophyll a-682 identified by Diiring et al. (1968) there is an excitation energy difference of about 750 cm-‘, much larger than kT, even at room temperature. It was assumed in the previous section that this energy difference plays no part in an isolated PSU, but it can be very important in the case of different PSU’s. Its effects are illustrated in Fig. 1 which shows two identical and joined linear PSU’s, the pigments of highest excitation energy lying at

ENERGY TRANSPORT

IN PHOTOSYNTHESIS

231

the edge of each PSU. Again we suppose that these two units belong to photosystem II. In this very simple model (described with linear PSU’s for the sake of simplicity) each PSU is separated from its neighbours by a potential barrier of height H > kT. If at time t = 0 an exciton is created in PSU, (Fig. I), the Boltzmann equilibrium defined by expression (10) will be reached

G

6 Space

coordinate

Fro. 1. Two linear and joined P!W’s separated by an excitation energy barrier fL T. and Tb represent the PSU. and PSUbreactioncentres.Thesetwo unitsbelongto photosystem II.

immediately in this PSU only. A flux ~,,s will then be set up from PSU, to PSUs for example, and very gradually the excitation will migrate to this neighbouring unit. For a precise calculation of the flux p,,,, the excitation energy must be known at all points of the PSU, but by simple approximations (Chandrasekhar, 1943) an order of magnitude can be obtained in the case of 300 molecule units Pab

=

&

exp

(-bfi)

(16)

D

and if H = 750 cm- ’ pab = lo9 set- ‘. (17) From this value it may be shown that if the reaction centre of PSU,, where the exciton was originally created, is ineffective, the probability p that the exciton will be captured by the reaction centre of PSUs is equal to 80x, which seems too high compared with the value suggested by Joliot k Joliot (1964). A probability p of 50 % would imply a value of pai, equal to lo* set- ‘. This discrepancy, which moreover is not large enough to be significant, can be ascribed to the approximations made in expression (16), to disorder effects and to the presence of pigments other than chlorophyll in the PSU. Our model provides a physical basis for models with restricted movement

232

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PAILLOTIN

between PSU’s. It is obvious in fact that, according to the photosynthetic organism under consideration, the excitation movement restriction can vary widely and become negligible if the height of the potential barrier H is less than kT. Such a model, based on the simple hypothesis that the various pigments making up the PSU are not distributed at random, has interesting biological consequences. According to (12) the existence of an excitation energy gradient in the PSU promotes the capture of energy by the reaction centre (by a factor of about three for systems II and more than 10 for system I) but has the effect of creating real potential barriers between the different centres. However, these barriers are low enough to enable the exciton to cross them fairly quickly if the PSU from which it originated has a non-effective reaction centre. The energy gradient thus favours the localization of excitons at the reaction centre without at the same time over-restricting their movement from one PSU to another. REFERENCES BAY, Z. & PEARLSTEIN, R. M. (1963). Proc. nafn. Acud. Sci. U.S.A. So, 1071. BRODY, S. S. (1957). Rev. scient. Znstrum. 28, 1021. CHANDRASEKHAR, S. (1943). Rev. mod. Phys. 15, 1. CHO, F., SPENCER, J. & GOVINDJEE (1966). Biochim. biophys. Acta 126, 174. CLAYTON, R. (1967). J. theor. Biol. 14, 173. CLA‘YTON, R. (1969). Biophys. J. 9,60. DA~~D~v, A. S. (1964). Soviet Phys. Usp. 82, 145. DERRING, G., BAILEY, J. L., KREuTz, W. & WIT, H. T. (1968). Naturwissenschaften 220.

D-s,

55,

L. N. M. (1967). Brookhaven Symp. Biof. 19,71.

EMERSON, R. &ARNOLD, W. (1932). J. gen. Physiol. 16,191. JOLIOT, A. & JOLIOT, P. (1964). C. r. hebd. Sianc. Acad. Sci., Paris 258,4622. JOLIOT, P., BARBIERI, G. & CHABAUD, R. (1969), Photochem. Photobioi. 10, 309. KNOX, R. S. (1968). J. theor. Biol. 21,244. KOK, B., FORBUSH, B. & MCGLOIN, M. (1969). Photochem. Photobiol. 11,457. KREY, A. & GOVINDJEE (1964). Proc. natn. Acad. Sci. U.S.A. 52, 1568. KREY, A. & GO~INDIEE (1966). Biochim. biophys. Acta 120, 1. KUDINOV, E. K. & Fmsov, Yu. A. (1966). Soviet Phys. JETP 22, 603. LANG, I. G. & Fmsov, Yu. A. (1963). Soviet Phys. JETP 16, 1301. LAVOREL, J. (1968). Biochim. biophys. Actu 153,727. Mmn, N. R., CEDER~TRAND, C. N. & RAEIINOWIT~H, E. (1965). Photochem. Photobiol. 4, 917. PAILLOTIN, G. (1968). C. r. hebd. Siam. Acad. Sci., Paris 267, 529. P&mm, G. (1971). ZZZnt. Congr. Photosyn. Res. Stress (in press). PBARLSTEIN, R. M. (1967). Brookhaven Symp. Biol. 19,s. RABINOWITCH, E. I., SZALAY, L., DAS, M., MURT~, N. R., CEDERSTRANS, C. N. & GO~INDJEE (1967). Brookhaven Symp. Biol. 19, 1. ROBINSON, G. W. (1967). Brookhaven Symp. Biol. 19, 16. SEELY, G. R. (1971). ZZZnt. Congr. Photosyn. Res. Stresa (in press). VREDENEIERG, W. J. & Duwms, L. N. M. (1963). Nature, Land. 197,355.

ENERGY

TRANSPORT

IN

233

PHOTOSYNTHESIS

Appemlh (A)

MOVBMBNT

OF LOCALIZED

JXCITONS

IN

A MOLECULAR

AGGREGATE

The movement of localized excitons in a molecular aggregate when the intermolecular coupling is fairly weak, may be regarded as a nonMarkoBian random walk (Kudinov & Firsov, 1966; Paillotin, 1971). The equation expressing the time evolution of the exciton distribution P(n, t) over the sites n of the aggregate has the form

dP(n, 0 = ; i [I$$,&- t’)P(m, t’)- I&,,#- t’)P(n, t’)] dt’. dt

(The desactivation terms are omitted here.) The transfer function is given, in a unit system where h = 1, by the relation KAO = 21Lj”

ew (- GJ Rel&p {-C&N - 11

(Al) IV.,,,(t) 642)

with s nm= l/2 C kmx

qmJ2 coth WW

vu~(* +B,2) uv2)~ + if(Em_ E ) sinh cm(O= l/2 cx GLn-4m~2cos II x jI=l/kT,

i2=-1.

& is the frequency of the normal mode (x) introduced in the harmonic approximation to describe the displacement of nuclei and qxn the displacement of the equilibrium position of the oscillator (x) when the nth molecule is excited. V, is the electronic interaction between the molecules n and m. Em is the electronic excitation energy of the nth molecule (0 + 0 transition) including the van der Waals contribution. We also introduce the maximum rate of transfer F,, defined by the equation

(B)

JUMP-OVER

TIME

AND

TIME

BBTWBBN

JUMPS

Only the case of aggregates of identical molecules will be considered here in order to simplify the problem and hereafter the indices n and m will refer to neighbouring molecules. (As previously units are such that h = 1.) It follows from the form of equation (A2) that the IV,,,,,(r) values are only appreciable during a definite time r, which is the “jump-over time”, and

(A41

234

G.

The condition the aggregate is

PAILLOTIN

of weak intermolecular t,

coupling,

assumed to prevail in

4 t,

(A9

with t, = V- ’ exp (S), where V and S are the values of V,, and S,,,, for neighbouring molecules n and m, and t, is the coherent transfer time. In addition, when the condition (A5) is satisfied, the half-width 6R of the aggregate absorption band is given by the relation

1

6Q = 1.67 c &2,2 coth (fi,/3/2) “’ [x and hence from equations (A6) and (A4) we obtain 2: 12/6!J,

t,

where jumps”

t,

646)

@7)

is in psec and 6R is in cm -I. We can also define a “time between such that if F is the value of F",,,for neighbours n and m then

tD

= 1.

t,F

According to the definition that

(A3) (see also Lang & Firsov, 1963), it follows

t;‘=F=JGV’t,exp(-J?E,) with

4 = Z$ C (qxn--qxmYtanh 03W4). x Let us introduce the Stokes shift AR between the maxima of the absorption and emission bands of the aggregate. AC! can be written An=

C &-L x

whence

E, 21AR/4

(A@

and t,

= 1.2 ‘$ exp (BE,),

where t, is in psec and 6Q and V are in cm - ‘. The condition motion to be Markoffian (see Kudinov & Firsov, 1966) is tD %

b

i.e. 6R % I; exp( -pEJ2).

(A9 for the exciton (AlO)

ENERGY

TRANSPORT (C)

JZQUILIBRIUM

IN

235

PHOTOSYNTHESIS

DISTRIBUTION

According to equation (Al), the equilibrium distribution prove the equations c [FncnP(m, co)-FmP(n, oo)] = 0. But from equation (i2) W&-t-i/l) = Wm(f) exp [-/I@,-&)] and hence F = L, expC-PU%,-J&)1. Then the solution of equ:tion (Al 1) is P(n, co) = 2-l exp (-&) with 2 = z ev (-B&J.

P(n, a~) should (All)

6412)

16