J . Phys. Chem. 1988, 92, 7148-7156
7148
Electrlc Field Effects on the Prtmary Charge Separation in Bacterial Photosynthesis M. Bixon and Joshua Jortner* School of Chemistry, Sackler Faculty of Exact Sciences, Tel Aviv University, 69978 Tel Aviv, Israel (Received: March 29, 1988)
In this paper we present a theoretical study of the effects of external electric fields on the primary charge-separation process from the electronically excited singlet state (P*) of the bacteriochlorophyll dimer (P) to the bacteriophytin (H) in reaction centers of photosynthetic bacteria. The role of the accessory bacteriochlorophyll (B) was considered within the framework of three distinct processes, Le., the superexchange, the sequential, and the nonadiabatic/adiabatic mechanisms for the primary electron-transfer process. The electric field dependence of the rate-controlling electron-transfer rate, the quantum yield for the charge separation, and the fluorescence quantum yield were evaluated for both oriented and isotropic samples. The most interesting conclusions are traced to the manifestation of level crossing, whereas the crossing between the P'B- and P'Hlevels for the sequential mechanism predicts the unveiling of the P'B- intermediate at high external fields, while the level crossing between the P*B and P'B- levels for the nonadiabatic/adiabatic mechanism predicts an asymmetric field-dependent low-temperature retardation of the charge separation. These predictions have to be subjected to experimental scrutiny, which requires the establishment of the interrelationship between the external field and the internal field acting on the ion-pair states in the reaction center.
I. Introduction The primary process in bacterial photosynthesis is electron transfer across the membrane-spanning protein/pigment complex, Le., the reaction center (RC). All the mechanisms proposed for the primary charge separation from the singlet excited state (lP*) of the bacteriochlorophyll dimer (P) to the bacteriopheophytin (H) give a central role to the accessory bacteriochlorophyll (B), which according to X-ray structural analysis is located between P and H . B may either act as a short-lived genuine ionic intermediate or, indirectly, enhance the IP*-H electronic coupling. Two classes of mechanisms were advanced to account for the primary charge separation: One-step direct electron
-
'P*BH P+BH(1.1) which is mediated by superexchange interactions via the virtual state of P'B-H. Two-step sequential electron transfer via P+B-,bs presumably located at an energy between 'P*H and P+H-, is
- kl
k2
IP*BH P'B-H P'BHwith the formation of the intermediate constituting the rate-determining step, i.e., k2 >> k l . Another sequential model involves the energy-transfer mechanism that invokes an intermediate state PB+H-,9 presumably located at an energy below IP*, and that is effectively populated by the off-resonance interaction with the PB*H state. Recently, two observations have been made that argue against the applicability of the sequential mechanism: The temperature independence of the exchange integral, Le., the singlet-triplet splitting, for the P'BH- radical pair in qui(1) Woodbury, N . W.; Becker, M.; Middendorf, D.; Parson, W. W. Biochemistry 1985, 24, 7516. (2) Fischer, S. F.; Nussbaum, I.; Scherer, P. 0. J. In Antennas and Reaction Centers of Photosynthetic Bacteria; Michel-Beyerle, M. E., Ed.; Springer: Berlin, 1985; p 256. (3) Jortner, J.; Michel-Beyerle, M. E. In Antennas and Reaction Centers of Photosynthetic Bacferia; Michel-Beyerle, M. E., Ed.;Springer: Berlin, 1985; p 345. (4) Jortner, J.; Bixon, M. In Protein Structure Molecular and Electronic Reactiuity; Austin, R., Buhks, E., Change, B., De Vault, D., Dutton, P. L., Frauenfelder, H., Gol'danskii, V. I., Eds.; Springer-Verlag: New York, 1987; p 277. (5) Michel-Beyerle, M. E.; Plato, M.; Deisenhofer, J.; Michel, H.; Bixon, M.; Jortner, J. Biochim. Biophys. Acta 1988, 932, 52. (6) Haberkorn, R.; Michel-Beyerle, M. E.; Marcus, R. A. Proc. Natl. Acad. Sci. U.S.A. 1979, 70, 4185. (7) Marcus, R. Chem. Phys. Lett. 1987, 133, 471. (8) Chekalin, S.V.; Matveetz, Ya. A,; Shkuropatov, A. Ya.; Shuvalov, V. A.; Yartzev, A. P. FEBS Lett. 1987, 216, 245. (9) Fischer, S. F.; Scherer, P. 0. J. Chem. Phys. 1987, 115, 1 5 1 .
0022-3654/88/2092-7148$01.50/0
none-depleted R C sets a lower limit on the equilibrium energy of the intermediate P'B-H state,13 which implies that k l should be activated, in contrast to the activationless naturelbL2 of the primary charge separation.I3 Recent femtosecond experiments on the RCs of Rb. sphaeroides and Rps. viridis set a large lower limit on the ratio of the rate constants in scheme 1.2, with k 2 / k l 1 50 at 10 K.12 This result together with the experimental value kl = 8.3 X 10" s-l for Rb. sphaeroides a t 10 KL2results in k2 I 4.2 X loL3s-l at this low temperature. This extremely high rate constant for electron transfer from B- to H (or hole transfer from B' to P) provides strong evidence against the applicability of the sequential mechanism. On the basis of the Landau-Zener transition theoryL4we concludedL5that, providing a sequential mechanism prevails, k , is nonadiabatic, while k2 is adiabatic with k2 N w / 2 n , where w i= 100 cm-I (ref 16) is the characteristic frequency of the protein medium. Accordingly, we estimated k2 i= 3.3 X 10I2s-l, which is considerably lower than the experimental restriction k2 I 4.2 X loL3s-l for this rate. On the basis of the foregoing analysis one may conclude that the sequential mechanisms seem to be inapplicable and that the unistep, superexchange-mediated electron transfer should be favored. However, the superexchange mechanism is also fraught with some difficulties. In particular, the electronic matrix elements, which emerge from the analysis of the competition between superexchange and thermally activated electron transferL5-I7( VpB E 80 cm-' for the electronic coupling between 'P*BH and P'B-H and VBH= 480 cm-' for the electronic coupling between P'B-H and PBH-), are large,Ls"17implying a large singlet-triplet splitting of the P'H- radical pair, which is considerably higher than the experimental value.17 The consistency between the superexchange predicted and the experimental singlet-triplet splitting of P+Hcan be established by invoking the effects of structural relaxation of the prosthetic groups accompanying the primary charge-separation process.Isb (10) Rockley, M. G.; Windsor, M. W.; Cogdell, R. J.; Parson, W. W. Proc. Natl. Acad. Sci. U.S.A. 1975, 72, 2251. (11) Kaufmann, K. J.; Dutton, P. L.; Netzel, T. L.; Leigh, J. S.;Rentzepis, P. M. Science (Washington, D.C.)1975, 188, 1301. (12) Martin, J.-L.; Fleming, G.; Breton, J. Proc. NATO Adu. Res. Workshop React. Centres, Cadarache, 1987; in press. (1 3) Bixon, M.; Jortner, J.; Michel-Beyerle, M. E.; Ogrodnik, A,; Lersch, W. Chem. Phys. Lett. 1987, 140,626. (14) Levich, V. G.; Adu. Elecrrochem. Eng. 1965, 4, 249. (15) (a) Bixon, M.; Jortner, J.; Plato, M.; Michel-Beyerle, M. E. Proc. NATO Ad. Res. Workshop React. Centres, Cadarache, 1987; in press. (b) Michel-Beyerle, M. E.; Bixon, M.; Jortner, J., submitted for publication in Chem. Phys. Lett. (16) Bixon, M.; Jortner, J. J . Phys. Chem. 1986, 90, 3795. (17) Marcus, R. A. Chem. Phys. Lett. 1988, 146, 13.
0 1988 American Chemical Society
Charge Separation in Bacterial Photosynthesis
The Journal of Physical Chemistry, Vol. 92, No. 25, 1988 7149
An alternative nonadiabatic/adiabatic mechanism for the primary charge separation was very recently proposed." The first rate-controlling transition 'P*BH P+B-H is nonadiabatic with the rate k l , eq 1.2, while the intersection between the nuclear potential surfaces of P+B-H and P+BH- is adiabatic and activationless. To ensure the compatibility of this mechanism with the temperature independence of the singlet-triplet splitting of the P+BH- radical pair,13the free energy gap between the zero-order P*BH and P'B-H states should be low, Le., AG N -150 cm-l,17 while the activationless nature of k1'wi2implies a low to medium reorganization energy for this rate, which was taken to be" X = 800 cm-I. Additional experimental and theoretical studies are required to establish the nature of primary charge separation in the RC. Since the pioneering studies of Marcus,'* a central parameter in the electron-transfer theory has been the free energy gap between the reactant and product states. The modification of the energetics of the donor-acceptor system by the application of external electric fields is expected to modify the electron-transfer dynamics. The quantitative features of the electric field effects on the electron transfer in the RC, which constitute the subject of the present paper, may provide ways and means to distinguish between the superexchange, sequential, and nonadiabatic/adiabatic mechanisms for the primary process in bacterial photosynthesis. -+
11. Electric Field Effects on the Energetics and Kinetics of Charge Separation The nonadiabatic electron transfer rate is
k = (27r/h)VZF
(11.1)
where Vis the electronic coupling between the initial and final states, while F is the averaged Franck-Condon nuclear overlap factor. F is determined by the free energy gap AG between the equilibrium nuclear configurations of the reactants and the products, the nuclear reorganization energy E,, and the temperature. The reorganization energy is E , = E, X, where E, and A are the reorganization energies of the high-frequency intramolecular modes and of the medium modes, respectively. For the primary electron-transfer processes, which involve prophyrin prosthetic groups, the distortion of nuclear configurations accompanying electron transfer originates mainly from protein modes,I6 Le., A/E, >> 1. In contrast, for subsequent electrontransfer processes within the RC, which involve the quinone (Q), the contribution to E, from the distortion of the high-frequency (hw, N 1500 cm-l) modes of Q is s ~ b s t a n t i a l . ' ~As we shall mainly focus on the primary electron transfer, involving the P, B, and H prosthetic groups, we shall take E, A, with the medium reorganization energy originating from the protein modes. These nuclear protein modes can be characterized by an averaged fre100 cm-'. The factor F can then be expressed in quency hw the single-frequency approximationI6
+
F = (hw)-l((o
+ 1)/D)Pl2 exp[-s(28 + 1)]ZP[2s(D(o+ 1))'/2] (11.2)
where 17 = [exp(hw/kBT) - 11-I is the thermal population of the average protein mode, s = A / h w is the medium reorganization energy in frequency units, and p = -AG/hw is the free energy gap. Finally, I,,( ) is the modified Bessel function of the order p . The ET rate is given by eq 11.1 and 11.2. Equation 11.2 applies for exoenergic ET with AG < 0 (p > 0), while for endoenergic processes with AG > 0 one takes F(AG>O) = F(AG 2 mV/A. Figure 6 displays the field dependence of the fluorescence quantum yield. A marked electric field dependence of Yrjc)/Yf(O)is exhibited (Figure 6) with a fast increase in the range 5-10 mV/A. The temperature dependence of the quantum yield becomes more pronounced at low temperatures, exhibiting again the enhancement of the field dependent F with lowering the temperature. IV. Sequential Mechanism It will be interesting to confront the predictions of the superexchange model with those of the sequential model. A level scheme for the energetics of the sequential model is presented in Figure 7, where we have arbitrarily chosen AG(P+B-) = -1000 cm-l and AG(P+H-) = -2000 cm-I. We note in passing that this equilibrium energy of P+B- will imply a marked temperature
PI(?) = [AG(P+B-) - jI(P+B-).Z]/hu
(IV.4)
P,(7) = [AG(P+H-) - AG(P+B-) - [z(P+H-) - jI(P+B-)]-21/h~ (IV.5) sj = A j / h w ; j = 1, 2 (IV.6)
Assuming that both k , and k, are activationless, i.e., p j ( 0 ) N s, ( 8 = 1, Z), the high-temperature rates are
kl (z) / k I (0) = eXp [-(~(P'B-).?)2/ 4 A1 k~r ]
(IV .7)
k2(Z)/ k2(0) = exp [-( jI( P+H-)-Z - jI( P+B-)*7)2/ 4A2kBr] (IV.8) while the equilibrium constants are k-l(Z) /kl(z) = exp [-pl(z) h w / kB T] k-Z(Z)/k2(7) = exp[-p2(Z)hU/kBr] (IV.9) We consider the implications of the sequential mechanism for an oriented sample with the electric field oriented across the C2 axis of the RC.
The Journal of Physical Chemistry, Vol. 92, No. 25, 1988 7153
Charge Separation in Bacterial Photosynthesis l . ~ l ~ , , , , , ~ , , I ' ' ' , , ' ' I
I
,
,
,
I
I
/
I
J
I
I
SEQUENTIAL
F\
\
/'
1
E(rnV/%)
+ k2)for the sequential
Figure 8. Dependence of k , , k2,and k 2 / ( k 2
model at 300 K on the electric field along the C2 axis of the RC.
The rate-determining step kl is expected to exhibit a modest (25%) symmetric increase at 300 K over the entire c domain (Figure 8). k2(t) exhibits an appreciable symmetric decrease with increasing e (Figure 8). At the highest field t = *I5 mV/%, we find k2(e)/k2(0)= 0.19 at 300 K. The level crossing of EpB and Ep ,which for the level scheme of Figure 7 occurs at e = -13 mV/ff, results in an appreciable enhancement of k-2/k2with equilibration being achieved at large negative t values. As both kl and k2 are fast relative to k3,a kinetic bottleneck effect is created. Equilibration between P+B- and P+His expected to prevail, with the relative concentrations being given by
" X
t OOb
I
I
I
100
200
300
1 I
T(K)
Figure 9. Dependence of the orientational averaged initial rate kl on the
electric field and temperature far the sequential model.
J
(IV. 10) which together with eq IV.9 reveals that in large negative fields this ratio approaches unity (Figure 8). Accordingly, the sequential mechanism predicts the unveiling of the P+B- intermediate at large negative fields in oriented RC. This effect was independently advanced by Marcus.I7 The quantum yield for charge separation in oriented R C is (IV.11) y(t) = k l / ( k d + k l ) over the entire t domain, as the equilibrium constant k-l/kl for the level scheme of Figure 7 is negligible at all fields. In view will exhibit only of the weak e dependence of k,(c) (Figure 8) Y(E) a small deviation from Y(0). Of some interest are also quantitative predictions for the implications of the sequential model for isotropic randomly oriented RCs. The time-resolved decay of IP* is nonexponential. The initial rate is given according to (111.8)-(111.13) by ( k l ) ( k 1 ( c J , $ ) ) , where the angular averaging of k l , eq IV.2, is performed according to eq 111.10. In Figure 9 we display the field dependence and the temperature dependence of the initial rate. The initial rate ( kl(e))/kl(0) decreases monotonically with increasing e, the decrease being somewhat more pronounced at low temperatures (Figure 9). The sole effects of t on F imply that ( k l ( e ) ) / k l ( 0 ) 5 1 for all values of t. The temperature dependence of the initial rate reveals a modest increase with increasing temperature. The fluorescence quantum yield in isotropic samples is obtained in the form (IV. 12) Yf(€)/Yf(O) = ( 1 / ~ l ( w w ) ) From Figure 10 we infer that the yield for this mechanism exhibits an increase with increasing e, while at constant E it decreases fast with increasing temperature, the effect becoming more pronounced at higher values of t .
t
\
1
0
\
1
1
100
1
1
200
300
T(K)
Figure 10. Electric field and temperaturedependence of the orientational
averaged fluorescence quantum yield for the sequential model. The unveiling of the P+B- intermediate at large negative fields in isotropic samples is somewhat smaller than in oriented RC. In Figure 11 we present the field dependence of the steady-state concentration of P+B-. For feasible electric fields, e.g., c = 14 mV/%,, the steady-state concentration of [P'B-] is 0.087 at 300 K and 0.046 at 75 K. These values are considerably lower than the [P+B-]/[P+H-] ratios (Figure 8) obtained for oriented RCs. V. NonadiabatidAdiabatic Mechanism This new mechanism17 constitutes a variant of the sequential model, with the energetics of the potential surfaces being chosen in an attempt to ensure consistency with magnetic data,I3 weak temperature dependence of the rates12 and ultrafast relaxation of the P+B-H configuration.12 This m e c h a n i ~ m rests ' ~ on nonadiabatic crossing between the potential surfaces of P*BH and P+B-H and adiabatic crossing between the potential surfaces of P+B-H and P+BH-. For the rate-controlling nonadiabatic P*BH-P+B-H crossing, which is characterized by the rate k,, the
7154 The Journal of Physical Chemistry, Vol. 92, No. 25, 1988
Bixon and Jortner IO,
14
I
005
a Y
l2IO
E =8
1
1I
1
10-6
o,ooo
200
100
as a function of electric field (in mV/& and temperature for the sequential model. €(
0
5
lo4,10
mv/'
100
300
T(KI Figure 11. Relative steady-state concentration of P'B-
!
-5///
I
I
I
-]
200
300
T(K)
Figure 13. Temperature and field dependence of kl(c)/kl(0) for the nonadiabatic/adiabatic mechanism in oriented RCs with the field oriented across the C2 axis. The parameters are AG, = -150 cm-I, X = 800 cm-I, and hw = 100 cm-I; the field strengths are given in mV/A.
io3~
75K
-
150K
t 0'
I
I
2
I
1
4
I
I
6
I
I
8
€ (rnV/%) Figure 14. Electric field and temperature dependence of the initial
//i I
n -t
' u 3 2
1 0 - 1 - 2 - 3 P
Figure 12. Overall temperature dependence of the rate-controlling no-
nadiabatic rate kl for the nonadiabatic/adiabatic mechanism. The dependence of kl(300K)/kl(25K) on p = -AG/hw is presented for a few values of s = Xl/hw, with w = 100 cm-I. The electric field dependence of the ratio kl(300K)/kl(25K) is presented for p = 1.5. Note that noninteger values of p are obtained by interpolation. energetic parameters AGI = -150 cm-I and XI = 800 cm-' were proposed." We note in passing that for these parameters, Le., p = 1 or 2 and s = 8, the nonadiabatic ET theory, eq 11.1 and 11.2, implies a marked temperature dependence. The ratios of the rates (Figure 12) r = kl(300K)/kl(25K) are r = 4 for p = 2 and r = 10 for p = 1, which is in contrast with experiment.I2 Nevertheless, in view of the sensitivity of the temperature dependence of the ET rates on the energetic parameters (Figure 12) in conjunction with the possible role of zero-point energy effects,I7 we shall proceed to the analysis of electric field effects on the ET rates kl in oriented samples. Taking AGI(0) = -150 crn-I,l7 the P*BH and P'B-H states are expected to cross a t e = -5 mV/& while at e = -10 mV/A, AGl = 140 cm-'. For antiparallel (positive) electric fields the separation of the P*BH and P+B-H levels increases with increasing e, reaching AGI = -440 em-' at t = 10 mV/A. Figure 13 exhibits the temperature and field dependence of k l ( e ) / k l ( 0 )in oriented RCs for the parameters recommendedI7 for this mechanism.
charge-separationrate ( kl ) for the nonadiabatic/adiabatic mechanism in isotropic RCs. Energetic parameters and notation as in Figure 13. For oriented samples we find the following: kl (e) is expected to exhibit a marked temperature dependence at large negative electric fields (Figure 12 and 13), e.g., r = lo2 at t = -5 mV/A and r N lo3 for t = -10 mV/A. The low-temperature ET rate exhibits a dramatic retardation at large negative fields (Figure 13), e.g., for T = 20 K,kl(e)/kl(0) =6X at t = -5 mV/A and kl(t)/kl(0) N 3 X at e = -8 mV/A. The low-temperature field retardation effect is asymmetric in oriented RCs, as for large positive fields the ratios r and kl( t ) / k , ( O )exceed unity (Figures 12 and 13). At room temperature when kBT 2 AGI - ,ii(P+B-)G, even for high fields the negative field retardation of the ET rate is eroded, with k l ( e ) / k l ( 0 )being close to unity at 300 K (Figure 13). The low-temperature quantum yield for charge separation Y = kl/(kd + k , ) exhibits a decrease, which is accompanied by a marked increase of the fluorescence quantum yield Yf N kd/(kl + kd) = kd/kl at large negative fields. The low-temperature asymmetric field retardation effect on the ET rate in oriented RC manifests the effects of level crossing between P*B and P+B-. For isotropic nonoriented samples this low-temperature asymmetry of the ET rate is eroded due to angular averaging. In isotropic samples no low-temperature retardation of the averaged ET rate is observed, while the lowtemperature enhancement effect is retained for the fluorescence quantum yields (Figures 14 and 15).
The Journal of Physical Chemistry, Vol. 92, No. 25, 1988 7155
Charge Separation in Bacterial Photosynthesis
l
0'
l
2
l
l
l
4
l
6
l
l
8
E ( rn VI%)
Figure 15. Electric field and temperaturedependence of the fluorescence quantum yield for the nonadiabatic/adiabatic mechanism in isotropic samples. Energetics parameters and notation as in Figure 13.
For isotropic RCs we find the following: The temperature dependence of the averaged ET rate ( k , ( t ) ) becomes weaker with increasing e. The fluorescence quantum yield Ydt)/Yf(O)= ( kl(0)/kl(e)) exhibits a marked increase with increasing e at low temperatures (Figure 15). In view of the spherical averaging the relative fluorescence quantum yield reflects the contribution of the low values of k,(e), which originates from the negative field being oriented along the direction of the P+B- dipole moment. Accordingly, the considerable low-temperature high-field enhancereflects the effects of P*B-P+B- level crossing. ment of Yr(e)/Yr(O) The high-field enhancement of Yde)/YdO)is eroded at room temperature when kBT 5 AGl.
VI. Discussion We have explored some of the effects of electric fields on the kinetics of the primary charge separation in bacterial photosynthesis, focusing on the implications of the shifts of the energies of the ion-pair states on the electron-transfer dynamics. We shall consider first the qualitative and quantitative differences between electric field effects on the superexchange and sequential mechanisms, focusing on both oriented and isotropic samples. For the case of oriented samples we infer the following: The primary rate exhibits a marked asymmetric t dependence for the superexchange mechanism and a small symmetric t dependence for the sequential mechanism. The symmetric e dependence in the latter case may be, however, distorted if the primary sequential process is not strictly activationless, Le., when X exhibits a deviation (within 30%) from AG. The quantum yield for charge separation in the sequential mechanism is symmetric for positive and negative values of e, while for the superexchange mechanism the quantum yield is asymmetric. The superexchange mechanism predicts that Y(t)/Y(O) is lower for negative e than for positive t. This effect can be traced to the proximity of the energy levels for IP* and P+B- at large negative e. The sequential mechanism predicts the unveiling of the P+Bkinetic intermediate at high negative fields. This effect, which is specific for the sequential mechanism, is pronounced for oriented samples, being somewhat smeared out for isotropically oriented RCs. For isotropic, randomly oriented RCs we infer the following: Both the superexchange and the sequential mechanisms exhibit a nonexponential decay of the population of IP*. The primary rate, Le., ( k ( e ) )for the superexchange mechanism and ( k , ( c ) ) for the sequential mechanism, exhibits a similar qualitative pattern at low temperatures, decreasing with increasing e. At high temperatures ( T = 300 K) the superexchange mech-
anism for an activationless process implies that (k(t))/k(O)slightly decreases and then increases with increasing e, while for the sequential mechanism ( kl(c))/kl(0) decreases monotonically with increasing t. Only for the superexchange mechanism the ratio ( k ( e ) ) / k ( O )can exceed unity. The field dependence of the fluorescence quantum yield exhibits a qualitatively similar e dependence for the superexchange and sequential mechanisms, with the increase of Yf(c)/Yf(O)upon increasing e being somewhat larger for the superexchange mechanism. The currently available experimental information regarding electric field effects on the primary charge-separation process is rather meagre. The following information is of interest: Quantum Yield for Charge Separation in Oriented Samples. The theory predicts asymmetry of the quantum yield Y(e)vs t for the superexchange mechanism. Such an asymmetry is exhibited by the experimental quantum yield data for -15 mV/A < e < 15 mV/A.23 However, a semiquantitative fit of these experimental data can be accomplished only with an unphysical decay rate of IP*, kd N 9 X lo9 s-I, which considerably exceeds the radiative (kf N 3 X lo8 s-l) and intramolecular decay rates of IP*. Quantum Yield for Fluorescence in Isotropic Samples. The enhancement of the fluorescence quantum yield with increasing e is predicted for both the superexchange and for the sequential mechanisms. For the superexchange mechanism at 77 K Y f ( t ) / Y f ( 0assumes ) the value of 1.39 for e = 5 mV/A and 3.8 for t = 9 mV/A, while for the sequential mechanism at 77 K Y f ( t ) / Y f ( 0=) 1.25 for t = 5 mV/A and 2.8 for t = 9 mV/A. These predictions have to be confronted with the experimental ) 1.25 at t = observation of Lockhart and Boxer:22 Y f ( e ) / Y f ( 0= 9 mV/A at 77 K. There is a marked uncertainty in the calibration of the internal field, so that these significant experimental data cannot be currently utilized to assess the validity of the mechanisms for the primary charge separation. The quantitative discrepancies between the theoretical predictions for either the superexchange or the sequential model can be traced to two major factors that pertain to the RC and to the field effect. First, lack of detailed information regarding the nuclear potential surfaces in zero field precludes any detailed analysis. In fact, what we attempt to do is to infer from the field effects on the energetics of the ion-pair states. Second, the interrelationship between the externally applied field and the internal field acting on the ion-pair states requires a careful scrutiny. Local field corrections, surface polarization effects, and electrode contact effects among others21 have to be elucidated in this context. The nonadiabatic/adiabatic mechanism predicts a dramatic retardation of k , at high negative fields for oriented samples at low temperature. For high electric fields, t < -10 mV/A, we expect that k,(Z) < lo9 s-l at 25 K, which will become comparable to the radiative rate kd N 3 X lo8 s-I of P*, thus resulting in a marked decrease of the quantum yield for charge separation and an increase of the fluorescence quantum yield at low temperatures. A somewhat less pronounced but still substantial increase of the fluorescence quantum yield is expected to be exhibited in isotropic low-temperature samples at high t (Figure 15). For nonoriented RCs at 75 K the fluorescence quantum yield will be enhanced by Y,-(t)/Yf(0)= 7 at e = 5 mV/A and Yde)/Y,-(O)= 22 at t = 9 mV/A. This prediction results in Y f ( t ) / Y f ( 0values ) that are considerably higher than the experimental observation:2' Y & t ) / Y f ( 0= ) 1.25 at an external field e = 9 mV/A for isotropic samples. Furthermore, the asymmetry of the quantum yield for charge separation at high fields inferred from the semiquantitative analysis of this mechanism constituted a low-temperature effect, which will be eroded at room temperature. The experimental observation of a marked 30%decrease of Y at -15 mV/A at room t e m p e r a t ~ r ecan ~ ~ be accounted for by this mechanism only by assuming an unphysically high decay rate kd, as is the case for the superexchange mechanism. The most interesting conclusions emerging from our analysis of field effects on the mechanisms of primary charge separation in the RC can be traced to the manifestation of level crossing.
7156
J. Phys. Chem. 1988, 92, 7 156-7 160
The crossing between the P'B- and P'H- levels at large negative fields for the sequential mechanism predicts the unveiling of the B- intermediate in high external fields, which is, of course, unique for this sequential mechanism. The level crossing between the
P*B and P'B- levels at negative fields predicts the asymmetric field-dependent low-temperature retardation of the charge separation for the nonadiabatic/adiabatic mechanism. These qualitative predictions have to be subjected to experimental scrutiny.
Vapor-Liquid Equilibria in Flukls of Two-Center Lennard-Jones Molecules Sumnesh Gupta Department of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803- 7303 (Received: September 17, 1987; In Final Form: July 14, 1988)
Molecular dynamics simulationsare performed using 500 molecules for pure fluids of twecenter Lennard-Jones (126) molecules to obtain thermodynamic results, including the residual Helmholtz free energy, for several isotherms. Thermodynamic results obtained from these simulations are utilized to predict the vapor-liquid equilibria in these fluids. Simulation results have also been used to estimate the critical temperatures and critical densities of these fluids, and these are compared with the existing predictions from three theoretical methods. These comparisons show that the site-site Ornstein-Zernike equation, with the Percus-Yevick approximation, for these fluids overpredicts the critical temperatures, and a nonspherical reference potential based perturbation theory also shows small deviations whereas an approximate form of the zeroth-order Mayer function expansion cluster perturbation theory works well. These three methods predict well the critical densities of these fluids, within the combined uncertainties of the results.
Introduction Sitesite potential models can be very useful in modeling fluids of nonspherical molecules.' Interaction sites in these models are usually considered to be spherical: and these represent either the chemical groups such as CH3 or the individual atoms such as N, Br, etc. The simplest of these site-site models is the two-center potential model with Lennard-Jones (12:6) spheres as interaction sites. Several computer simulations of pure dense fluids using this potential have been reportedw for thermodynamic properties and microscopic structure. Kabadi and Steele'O have also recently reported studies of translational and rotational dynamics in fluids of these two-center Lennard-Jones (12:6) molecules. Simulations involving mixtures have also been reported."-15 Fluids modeled by the two-center Lennard-Jones potential have also been studied through the integro-differential distribution function theories'*2including the recent work of Monson,16 who has studied the effect of molecular elongation on critical properties. Perturbation theories have also been applied.',2J7-19 The nonspherical reference based perturbation expansions by Fischer17 and also by Kohler et a1.l8 in the center frameI9 have worked well. A similar expansion by Quirke and TildesleyIg in the site frame also works well. These methods have also been extended to mixtures.20+21In spite of all this, simulation data on fluids modeled ( I ) Gray, C. G.; Gubbins, K. E. Theory of Molecular Fluids; Clarendon: Oxford, 1984. (2) Streett, W. B.; Gubbins, K. E. Annu. Reu. Phys. Chem. 1977, 28, 373. (3) Streett, W. B.;Tildesley, D. J. Proc. R. Soc. London, A 1977,355,239. (4) Singer, K.; Taylor, A.; Singer, J. V. L. Mol. Phys. 1977, 33, 1757. (5) Cheung, P. S . Y.; Powles, J. G. Mol. Phys. 1975, 30, 921. (6) Romano, S.; Singer, K.Mol. Phys. 1979, 37, 1765. (7) Guillot, B.; Guissani, Y. Mol. Phys. 1985, 54, 455. (8) Barojas, J.; Levesque, D.; Quentrec, B. Phys. Reu. A. 1973, 7, 1092. (9) Wojcik, M.; Gubbins, K. E.; Powles, J. G. Mol. Phys. 1982, 45, 1209. (IO) Kabadi, V. N.; Steele, W. A. J. Phys. Chem. 1985, 89, 1467. (1 I ) Gupta, S.; Coon, J. E. Mol. Phys. 1986, 56, 1049. (12) Gupta, S . Fluid Phase Equilib. 1986, 31, 221. (13) Coon, J. E.; Gupta, S.; McLaughlin, E. Chem. Phys. 1987, 113.43. (14) Fincham, D.; Quirke, N.; Tildesley, D. J. J . Chem. Phys. 1986, 84, 4535. (15) 508. (16) (17) ( 1 8) (19)
Tildesley, D. J.; Enciso, E.; Sevilla, P. Chem. Phys. Lett. 1985, 100, Monson, P. A. Mol. Phys. 1984, 53, 1209. Fischer. J. J . Chem. Phys. 1980, 72, 5371. Kohler, F.; Quirke, N.; Perram, J. W. J. Chem. Phys. 1979, 71,4128. Quirke, N.; Tildesley, D. J. J . Phys. Chem. 1983, 87, 1972.
0022-3654/88/2092-7156$01.50/0
by using the two-center Lennard-Jones potential is considered to be limited even for pure l i q ~ i d s ,especially ~ ~ , ~ ~ for the residual Helmholtz free energy. Here we report systematic NVT molecular dynamics simulations to obtain thermodynamic properties of fluids modeled using the two-center Lennard-Jones (12:6) potential. These results have also been used to predict the vapor-liquid equilibria in these fluids. In this regard, it may be worthwhile to point out that P o w l e ~ ~ ~ has already shown that thermodynamic results from simulations can be used to reasonably predict vapor-liquid equilibria in these fluids. Simulation results have also been used to estimate the critical densities and temperatures in these fluids which in turn are compared with earlier theoretical predictions. Simulation Methodology For a pair of homonuclear molecules, 1 and 2, the two-center Lennard-Jones ( 1 2:6) potential model is given by'*2,25
Here w land w2 are the molecular orientations, i is an interaction site on molecule 1, j is an interaction site on molecule 2, r is the distance between the molecular centers, and t and u are the Lennard-Jones (12:6) interaction parameters.'v2 The site-site distances, rij, depend not only upon r, wl,and w2 but also upon the molecular elongation, I, which is a constant for rigid molecules. Thus, interaction between a pair of two-center Lennard-Jones (12:6) molecules usually is specified by t, u, and I* = l / u . Molecular dynamics simulations in the NVT ensemble have been performed for fluids of two elongation ratios, I* = 0.3292 (20) Fischer, J.; Lago, S . J. Chem. Phys. 1983, 78, 5750. (21) Enciso, E.; Lombardero, M. Mol. Phys. 1981, 44, 725. (22) Abascal, J. L. F.; Martin, C.; Lombardero, M.; Vazquez, J.; Baiion, A.; Santamaria, J. J. Chem. Phys. 1985. 82, 2445. (23) MacGowan, D.; Waisman, E. M.; Lebowitz, J. L.; Percus, J. K. J. Chem. Phys. 1984,80, 2719. (24) Powles, J. G. Mol. Phys. 1980, 41, 715. (25) Maitland, G. C.; Rigby, M.; Smith, E. B.; Wakeham, W. A. Inter-
molecular Forces: Their Origin and Determination; Clarendon: Oxford, 1981.
0 1988 American Chemical Society
