10420

J. Phys. Chem. B 1998, 102, 10420-10430

Mechanism and Unidirectionality of the Electron Transfer in the Photosynthetic Reaction
Center of Rhodopseudomonas Viridis: SAC-CI Theoretical Study
J. Hasegawa† and H. Nakatsuji*,†,‡
Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering,
Kyoto UniVersity, Sakyo-ku, Kyoto 606-8501, Japan, and Institute for Fundamental Chemistry,
34-4 Takano Nishi-hiraki-cho, Sakyo-ku, Kyoto 606-8103, Japan
ReceiVed: July 2, 1998; In Final Form: September 16, 1998

The electronic mechanism and the origin of the unidirectionality of the electron transfer from photoexcitedspecial pair to bacteriopheophytin in the photosynthetic reaction center (PSRC) of Rhodopseudomonas (Rps)
Viridis are studied theoretically by using the SAC(symmetry adapted cluster)-CI (configuration interaction)
method. The effects of the surrounding proteins are considered by using the point charge model. The L-branch
selectivity of the electron transfer is explained by the asymmetry of the transfer integral, an electronic factor,
which originates from a small structural asymmetry of the PSRC: the L-side chromophores are locally closer
than the M-side ones, though the average separations are almost the same. The smallness of the charge
recombination rate is attributed to the difference in the electron localization between the LUMO and HOMO
of special pair. Protein effects on the unidirectionality are quite small as far as the electrostatic model isvalid, though the proteins keep the three-dimensional arrangement of the chromophores in the PSRC. A
mutation experiment for realizing M-side selectivity is suggested.

I. Introduction
Photosynthetic reaction center (PSRC) converts photon energy
into chemical energy through charge separations and initiates
biochemical reaction cycles.1 The structure of the PSRC of
Rhodopseudomonas Viridis (Rps. Viridis) was determined by
X-ray crystallography2 and we show in Figure 1 the L- andM-branches which contain seven chromophores arranged in a
pseudo C2 symmetry. They are special pair (P, bacteriochlorophyll b dimer), two bacteriochlorophylls b (BL and BM), twobacteriopheophytins b (HL and HM), menaquinone (MQ), andubiquinone (UQ) which are supported by the proteins consistingof about 1200 residues. The PSRC is embedded in themembrane proteins. Interestingly, the electron transfer (ET) in
the PSRC occurs unidirectionally:3,4 the electron in the excitedstate of P (P*) is transferred asymmetrically only along theL-branch, despite of its almost symmetrical alignment. The ET
is known to be fast, long-range, and significantly efficient.1 The
ET from P* to H occurs in 3 × 10-12s, from H- to MQ in 200
× 10-12 s, and from MQ to UQ in 100 × 10-6 s.1 The firststep, which is the main subject of the present study, is a very
fast process.
Marcus gave a basic formula describing the rate constant of
the ET reaction in a condensed medium as5k){(∆G + λ)1
2π
|H |2exp p IF (4πλRT)1/2
4λRT

}

2

(1-1)

where HIF is the transfer integral which describes the electronic
coupling between the initial and final states. ∆G is the freeenergy difference between the initial and final states and λ thereorganization energy associated with the ET reaction. This
* To whom correspondence should be addressed.
† Department of Synthetic Chemistry and Biological Chemistry.
‡ Institute for Fundamental Chemistry.

formula may be viewed as consisting of two factors: electronic
and thermodynamic. The former consists of the transfer integral
part and represents the width of the electron-transfer pathway.
The latter consists of the rest part depending on the energeticsof the system and shows a thermodynamic control.
On the basis of this formula, several theoretical studies were
performed to clarify the underlying mechanism of the electrontransfer and its unidirectionality. The energetics of this system
was studied by semiempirical quantum-chemical calculations,6,7
electrostatic calculations,8,9 and molecular dynamics simulation
calculations.10 These studies concluded that the ET along theL-branch is energetically favorable mainly due to the protein
electrostatic and electric-field effects.6,8-10 The transfer integrals
were also calculated using the semiempirical wave functions.7,8,11
In particular, Jortner and co-workers8,11 attributed the unidirectionality as being due to the geometrical and electronic-structural
asymmetry between the L and M-branches. However, becauseof the limited reliability of the semiempirical calculations, it isnecessary to perform more reliable ab initio calculations to
elucidate the mechanism and the underlying origin of this ET
reactions.
In our laboratory, we have developed SAC(symmetryadapted-cluster)12/SAC-CI(configuration interaction)13 method
as a reliable tool to study molecules in the ground, excited,
ionized, and electron-attached states. The accuracy of themethod had well been examined14 through the applications of
it to many different fields of chemistry and chemical physics.15,16
The method is simple enough to be useful and simple enoughto be reliable and therefore applicable even to relatively large
biological molecules in a good accuracy.16-22 This method has
been successfully applied to the ground, excited, and ionizedstates of various porphyrins, and detailed information on theelectronic structures of these compounds was obtained.16-22
In the preceding paper23 and in the previous communication,24
we have described the excitation spectrum of the PSRC of Rps.

10.1021/jp982870x CCC: $15.00 © 1998 American Chemical Society
Published on Web 11/11/1998

PSRC of Rhodopseudomonas Viridis

J. Phys. Chem. B, Vol. 102, No. 50, 1998 10421

Figure 1. Structure of the chromophores in the photosynthetic reaction center of Rps. Viridis. The nuclear coordinates are taken from ref 2.

Viridis1 by the SAC/SAC-CI method to give a definitive
assignment of the spectrum. We have calculated the groundand excited states of all the chromophores in the PSRC of Rps.
Viridis, P, BL, BM, HL, HM, MQ, UQ, and four different hemes,c-552, c-554, c-556, and c-559 in the environment of the proteins
replaced by an electrostatic model. We could be able to givethe assignment of all the peaks within about 1 eV from 1.25 to
2.5 eV. We could reproduce not only the ordinary excitation
spectrum but also the linear dichroism data.25 This calculation
was by far the largest calculation so far made by the ab initio
theoretical method including large amounts of electron correlations. It gave a firm basis for future photochemical studies of
the PSRC of Rps. Viridis.
In this paper, we apply the SAC/SAC-CI method to study
the electronic mechanism and the origin of the unidirectionalityof the ET in the PSRC of Rps. Viridis. Since this ET involves
many different electronic states of the chromophores in the
PSRC, the underlying theoretical method should be reliable forall the states involved. The SAC/SAC-CI method is indeed
very suitable to this type of study. The analysis of the calculated
results would provide us a basic picture of the ET reactions
and its unidirectionality. Further, based on such a picture, wewould be able to propose an interesting mutant system for further
understanding and development of PSRC: we call such a designa theoretical mutation which is a basis for an effective futurecollaboration between theoretical and experimental biochemists.
A short communication of the present study has been publishedelesewhere.24
In the next section, the calculational details are described.
The analyses of the unidirectionality and the efficiency basedon the transfer integrals calculated by the SAC/SAC-CI wave
functions are then described and a summary and concluding
remarks are given in the last section.

adopted in the present study. We first take up the electrontransfer process from P*B to P+B-. The SAC and SAC-CI
wave functions of P*, P+, B, and B- are expressed as

P*: ΨPEX ) RPexp(SP) ΦP0(2-1)

P+: ΨPI ) IPexp(SP) ΦP0(2-2)

B: ΨBG ) exp(SB) ΦB0(2-3)

B-: ΨBEA ) EBexp(SB) ΦB0(2-4)

where ΦP0 and ΦB0 represent the Hartree-Fock wave functions
for the ground states of P and B, respectively, namely, ΦP0 ) |φP1 φP2 ‚‚‚‚φNP P| and ΦB0 ) |φB1 φB2 ‚‚‚‚φNBB|. The operators R, I,
and E represent the symmetry-adapted singlet excitation, ionization, and electron attachment operators, respectively. S is thesymmetry-adapted excitation operator for the ground state. They
are actually the sum of the single and double replacement
operators. The subscripts, EX, I, G, and EA denote excited,
ionized, ground, and electron-attached states, respectively.
The initial and final states of the ET, P*B and P+B-,
respectively, are approximated by the products of the SAC/
SAC-CI wave functions of the fragments

P*B: ΨP*B = ΨPEX ΨBG) RP exp(SP + SB)ΦP+B
0

(2-5)

+ -

P+B-: ΨP B = ΨPI ΨBEA

II. Calculation of Transfer Integrals
We first explain the method of calculations of the transfer
integrals using the SAC/SAC-CI wave functions of thechromophores, since this would clarify our basic approximations) IPEB exp(SP + SB)ΦP+B
0

(2-6)

where ΦP+B
) |φP1 φP2 ‚‚‚‚φNP P φB1 φB2 ‚‚‚‚φNBB|. The operator IPEB
0

10422 J. Phys. Chem. B, Vol. 102, No. 50, 1998

Hasegawa and Nakatsuji

Figure 2. Electronic factors, |HIF|2 (×10-9 in au) for the electron transfers in the PSRC of Rps. Viridis (the upper value is in protein modelcompared with the experimental value given in the parentheses35 and the lower one is in a gas phase). Energy levels colored green are experimentallyestimated values37 and the black ones calculated by the SAC-CI method.

TABLE 1: Overlap Integrals between the Lower
Unoccupied MOs of Special Pair (P) and Bacteriochlorophyllb (B)P(1) L-branch
LUMO
B
LUMO+1
LUMO+2(2) M-branch
LUMO
B
LUMO+1
LUMO+2

SAC-CI wave functions

P: ΨPG ) exp(SP)ΦP0(2-10)LUMO

LUMO+1

H: ΨHG ) exp(SH)ΦH0(2-11)

0.000 005 2
0.000 747 4
0.000 120 7-0.000 110 1
0.000 439 1
0.000 160 2

H-: ΨEAH ) EHexp(SH)ΦH0(2-12)-0.000 033 4-0.000 353 50.000 198 8

0.000 085 6
0.000 116 2-0.000 098 7

in eq 2-6 is not symmetry adapted, so that we introduce thefor spin symmetry adaptation.
operator RP+B
I
+ -P+B-: ΨP B ) RP+B exp(SP + SB)ΦP+B.
0(2-6′)The details are given in the appendix.
Finally, the transfer integral of this ET reaction is given by+ -P*B to P+B-: 〈ΨP*B|H|ΨP B 〉(2-7)

where H is the Hamiltonian of the united system P+B. Similarly
for the next ET reaction, P+B-H to P+BH-, and for the charge
recombination reaction, P+B- to PB, the transfer integrals arecalculated by--

P+B-H to P+BH-: 〈ΨB H|H|ΨBH 〉+ -P+B- to PB: 〈ΨP B |H|ΨPB〉

(2-8)
(2-9)

For these processes, we additionally need the following SAC/

The constructions of the product wave functions and thecalculations of the transfer integrals are essentially the same asin the above case.
To calculate all the necessary quantities, we use the SAC/
SAC-CI wave functions for PG, P*, P+, BG(L), BG(M), B-(L),
B-(M), HG(L), HG(M), H-(L) and H-(M), where L and M
denote those in the L- and M-branches, respectively. Some ofthese wave functions have already been calculated in theprevious study,23 but others have been newly calculate during
this study.
In the calculation of transfer integrals, we assume an
orthogonality between the MOs of different chromophores, since
the calculated overlap integrals were very small as shown in
Table 1. The overlap integrals between the lower unoccupied
MOs of P and B were at most on the order of 10-4. Further,
for the sake of simplicity, we considered only the linked
configurations having the coefficients larger than 0.1 in the SAC/
SAC-CI wave functions.
III. Computational Details
The calculations of the SAC/SAC-CI wave functions for the
PSRC of Rps. Viridis are explained in some detail in thepreceding paper.23 The geometries of the chromophores aretaken from the X-ray crystallographic data2 (1PRC in Brookhaven

PSRC of Rhodopseudomonas Viridis

J. Phys. Chem. B, Vol. 102, No. 50, 1998 10423

Figure 3. Atomic contributions of BL and BM for the transfer integrals between P and B (a) for the electron transfer and (b) for the charge
recombination.

Data Bank26) and were shown in Figure 3 of ref 23. The
labeling of the atoms and rings was also given there. Somebranched fragments of the chromophores were simplified except
for the ones which may have π-conjugation with the porphyrinrings. We use Huzinaga’s (63/5)/[2s2p] CGTO’s for carbon,
nitrogen, and oxygen atoms27 and (4)/[1s] set28 for hydrogen.
For Mg, Huzinaga’s (533/5)/[5s/3p] set27 plus two p-type
polarization functions (ζ ) 0.045 and 0.143) and d-type
polarization functions (ζ ) 1.01) are used.23 Protein effect isintroduced by a point-charge model. Namely, the charges
reported previously29,30 for proteins and waters and the HartreeFock populations for the other chromophores which are obtainedby the present and previous23 calculations are placed at thecenters of the respective atoms.2
The SAC/SAC-CI wave functions for the ground and excitedstates of all the chromophores are the same as those calculated
in the previous spectroscopic studies of the PSRC.23 For theionized and electron-attached states, additional SAC-CI calculations are performed similarly to the previous case.23 At
least 2p-electrons are correlated in the active space and theperturbation selection17,31 is carried out for the double excitation
operators. For the electron attached states of B and H, theenergy threshold 1 × 10-6 au is used and for the ionized statesof P, the threshold 3 × 10-6 au is used. All single excitations
and the selected double excitations are included in the linked
term. For the Hartree-Fock SCF calculation, the HONDO
version 8 program,32 and for the SAC/SAC-CI calculation, theupdated version33 of SAC 85 program,34 was used.
IV. Examination of the Electron Transfer and Charge
Recombination Processes
We present the squares of the calculated transfer integrals|HIF|2 (hereafter, we call this value just transfer integral andthe unit is 1.0 × 10-9 au) for the ET process from P*B to P+Bin Table 2 and for the ET process from P+B-H to P+BH- in
Table 3. Table 2 also gives the transfer integrals for the charge
recombination process P+B- to PB. These are the values
between many lower states of the system and include the effects
of proteins in an electrostatic point-charge model.
The most relevant values in Table 2 for the ET process understudy are those from the lowest excited state of P, P*(21A)B to
the lowest ion-pair states, P+(12A)B-(12A) in the L- andM-branches, which are 25.21 and 1.65, respectively. The
transfer integral for the L-branch is much larger than the one
for the M-branch. We also calculated the transfer integrals from10424 J. Phys. Chem. B, Vol. 102, No. 50, 1998

Hasegawa and Nakatsuji

TABLE 2: Square of the Transfer Integrals |HIF|2 for the Electron Transter and the Charge Recombination between Special
Pair (P) and Bacteriochlorophylls b (B) (×10-9 au)(1) L-branch
P+(12A)B-(12A)
P+(22A)B-(12A)
P+(32A)B-(12A)
P+(12A)B-(22A)
P+(22A)B-(22A)
P+(32A)B-(22A)(2) M-branch
P+(12A)B-(12A)
P+(22A)B-(12A)
P+(32A)B-(12A)
P+(12A)B-(22A)
P+(22A)B-(22A)
P+(32A)B-(22A)

P(X1A)B

P*(21A)B

P*(31A)B

P*(41A)B

P*(51A)B

0.96
0.06
51.16
6.73
12.64
101.45

25.21
0.14
0.20
15.16
0.00
0.94

0.13
28.04
0.17
0.04
16.22
0.02

0.18
0.01
22.94
0.17
0.00
11.66

5.74
2.49
0.10
3.88
1.04
0.01

2.54
16.98
2.29
19.26
61.09
0.78

1.65
0.06
0.14
10.40
0.53
0.11

0.16
0.67
0.15
0.24
8.42
0.20

0.28
0.00
1.58
0.68
0.00
12.48

0.74
0.01
0.11
0.61
0.41
0.13

TABLE 3: Square of the Transfer Integrals |HIF|2 for theElectron Transter between Bacteriochlorophylls b (B) and
Bacteriopheophytins (H) (×10-9 au)(1) L-branch
H-(12A)
H-(22A)
H-(32A)(2) M-branch
H-(12A)
H-(22A)
H-(32A)TABLE 4: Estimation of the Energy Levels of the ElectronTransfer Pathway

B-(12A)

B-(22A)

B-(32A)energycontributiona

104.430.95
134.93

1.99
0.71
2.46

36.50
28.68
227.62

15.93
6.72
4.32

54.41
5.18
47.72

65.38
4.21
61.73

∆E or IP-EAb
E(+-)
relax(c)relax(p)ctotal∆higher excited states of P* to many different electronic statesof P+B-. Again the L-branch values are generally larger thanthe M-branch values, though the reality of these ET routes islimited, as discussed in the next section.
The first column of Table 2 gives the transfer integrals forthe charge recombination processes from P+B- to PB. The most
relevant value is 0.96 from the lowest state of P+B- in the
L-branch to the lowest state of PB: it is much smaller than thevalue for the ET, 25.21 for P*(21A)B to P+B-. The corresponding value for the M-branch is 2.54, again very small.
The transfer integral for the succeeding ET process, P+B-H
to P+BH-, is 104.43 for the L-branch and 15.93 for theM-branch as seen from Table 3. Again, the L-branch ET hasmuch wider route than the M-branch one. Though the values
for the processes involving the excited states of B-or H- arealso given in Table 3, they are less important for the present
purpose.
Thus, the result of the calculated transfer integrals reproduces
the L-branch selectivity in the sequential ET process, P*BH f
P+B-H f P+BH-. It also shows a low probability of theoccurrence of the charge recombination, P+B-H to PBH. The
calculated transfer integrals for the L-branch, 25.21 for P*BH
to P+B-H and 104.43 for P+B-H to P+BH-, are both in good
agreement with the experimentally estimated values 21 and 72,
respectively.35 These results are quite encouraging and seemto suggest that the L-branch selectivity and a high charge
separation yield (low charge recombination process) of the ET
in the PSRC of Rps. Viridis are controlled by the transfer
integral, i.e., the electronic factor in the Marcus formula.
To obtain a better understanding on the implications of the
Marcus formula, we tried to estimate the energy levels of theelectron-transfer pathway in the PSRC of Rps. Viridis. This
information is related with the thermodynamic factor in the
Marcus formula.
The energy of the electron transferred state relative to theneutral ground state may be calculated as

EET ) IP - EA + E(+-) + relax(chromophore) +relax(protein) (4-1)∆E or IP-EAb
E(+-)
relax(c)relax(p)c
total∆∆(exptl)

P+BP*

M

P+HL

M

L

1.42
0.00

In Gas Phase
4.41
4.45
-1.34
-1.49

4.66
-0.83

4.60
-0.85

0.00
1.42
0.00

-0.5
2.57
1.15

-0.5
2.46
1.04

-0.9
2.93
1.51

-0.9
2.85
1.43

1.41
0.00

In Protein
4.81
5.35
-1.26
-1.50

3.02
-0.71

5.57
-0.95

-0.5
3.05
1.64

-0.9
1.41
0.00

-0.9
3.72
2.31
-0.25

0.00
1.41
0.00
0.00

-0.5
3.35
1.94
-0.09

a Detailed accounts are given in the text. b Estimated by the SACCI method c The value estimated by Thompson and Zerner.6

where IP is the ionization potential of P, EA the electron affinityof B or H, E(+-) the Coulombic interaction energy between
the donor P+ and the acceptor B- or H-, relax(chromophore)
the relaxation energy of the chromophore, and relax(protein)
the relaxation energy of proteins and the surrounding medium.
The first two terms, IP-EA are rather easily calculated, at least
the vertical values, by the SAC-CI method. The term E(+-)
is calculated as a sum of the Coulombic interaction energies
between the gross charges of the donor and acceptor. The
relaxation energy of the chromophore, relax(chromophore), in
the environment of proteins and waters is rather difficult to
calculate, though the relaxation energy in a gas phase iscalculated within the scope of the SAC-CI method with theuse of the gradient technique recently established.36 The
relaxation energy of the protein medium is difficult to estimate
by the present ab initio method. Actually, the terms in eq 4-1
couple to each other and this coupling may be essential.
Table 4 shows the estimated energy levels of the electrontransferred state. ∆E which is the excitation energy of P and
the values of IP-EA for P+B- and P+H- are calculated by the
SAC-CI method with and without the electrostatic field dueto the proteins and waters. The value of E(+-) is estimatedby the above approximation. The relaxation energy, relax(chromophore), was not estimated. For relax(protein), we usedthe values estimated by Thompson and Zerner6 based on thecontinuum reaction-field model. The upper half shows the result
for a gas phase and the lower half the results in proteins. The
value relative to the energy level of P*, ∆, is given in the bottom
and compared with the experimentally estimated values for theL-branch37 given in the extreme bottom row. Unfortunately,

PSRC of Rhodopseudomonas Viridis
the present estimated ∆ values are by far different from theexperimentally estimated values. This is perhaps due to thepoor or no estimation of the relaxation energies of chromophores
and protein medium. A proper estimation of these relaxationenergies is certainly very important, but at present, we have toabandon it for a luck of sufficient knowledge and methodology.
Then, we decided to use the experimentally estimated valuesof ∆ when necessary.
V. Unidirectionality of the ET and Its EffectivenessWe present here a view on the mechanism of the unidirectionality and the effectiveness of the ET in the PSRC of Rps.
Viridis on the basis of the SAC-CI data presented in thepreceding section. Figure 2 shows a summary of the present
results pertinent for the succeeding discussions. It shows thecalculated transfer integrals, |HIF|2, in proteins and in a gas phasefor each process. The values in parentheses are those estimatedfrom the experimental data.35 They are proportional to the ET
rate constant as seen from the Marcus formula given in eq 1-1.
Only the electronic factors larger than 1.00 are shown. Theenergy levels of the two lower states in the L-branch coloredgreen are the experimentally estimated values,37 but for others,
we use the SAC-CI values calculated from the excitationenergies, ionization energies, and electron affinities. The
theoretical transfer integrals (including protein effect) for theETs from P to B and from B to H are 25.2 and 104.4,
respectively, which are in good agreement with the experimentally estimated values, 21 and 72, respectively. It indicates theadequacy of the present theoretical method.
Thus, the ET reaction from P to H is described as follows
based on the data summarized in Figure 2. First, the excitationof P occurs by a direct photoexcitation or by an energy transferfrom antenna molecule,1 and it would be relaxed to the firstexcited state, P*(21A), by an internal conversion as describedby Kasha’s rule. The ET from P to B, i.e., from P(21A) toP+(12A)B-(12A), would occur along the L-branch, since theelectronic factor of the L-side is about 15 times larger than thatof the M-side as shown in Figure 2. This electron transfer
occurs mainly from the LUMO of P to the LUMO of BL. The
unidirectionality is originated from the asymmetry of the transfer
integral: the calculated branching ratio |HL|2/|HM|2 = 15 inproteins (|HL2/|HM|2 = 35 in a gas phase) is large enough to
explain the experimental branching ratio kL/kM > 5.8 Here, theabove consideration is based on the sequential mechanism, since
the energy of the intermediate radical pair, P+B-, was experimentally estimated to be lower than that of the P* state.37,38
Even if the ET occurs faster than the Kasha’s internal conversion
within P, the L-side is also preferable to the M-side, since thetransfer integrals between the higher nearby states of P* and B(actually from P**(32 A) to P+(22A)B-(12A) and from P***(42A) to P+(32A)B-(12A)) are larger on the L-side than on the
M-side (see Figure 2).
A competitive reaction to the ET from B to H is the CR
reaction from P+B-H to the ground state of PBH. This CR isa transfer of an electron from LUMO(BL) to HOMO(P). The
transfer integral for the ET from B to H is 104.4 which is muchlarger than that for the CR 1.0, so that the CR is much lesspreferable to the ET: the branching ratio here to the CR is 1.0/
104.4, about 1% of that of the ET, which agrees reasonably
well with the so-called efficiency (quantum yield) of thephotosynthesis.
The successive ET from P+(12A)B-(12A) to P+(12A)H-(12A)
proceeds along the L-branch. This transfer occurs mainly fromJ. Phys. Chem. B, Vol. 102, No. 50, 1998 10425
LUMO(BL) to LUMO(HL). Further, we note that even if theelectron transfer from B to H might occur through higher
electronic states, the transfer integrals themselves are large
enough to permit such processes. From Figure 2 we see thatthe transfer integrals between the second and third higher statesof P+B-H and P+BH- are again very large in the L-branch incontrast to the corresponding values in the M-branch, thoughsuch higher electronic processes should scarcely occur for the
Kasha’s rule.
After the ET to H, the electron is known to be transferredfurther to MQ.1 This process (200 ps) occurs slower by about
2 orders of magnitude than the ET from P* to H (3 ps) and will
be studied separately. The side reaction of the ET to MQ is
the CR to the ground state, PBH. The electronic factor of the
ET to MQ was reported to be 0.48 in the previous article.39 On
the other hand, the electronic factor of the CR reaction by thesuperexchange mechanism is evaluated to be 0.053,40 which issmall enough to explain the high efficiency of the ET from H
to MQ.
The ET process from MQ to UB is a very slow process (100ns), and therefore, many other factors such as relaxations andvibrations may couple with the process. The nature of theprocess may therefore be different from that studied here andwill be studied in the forthcoming paper.
VI. Effect of Proteins on the Transfer Integrals
The roles of proteins in the ET reactions in the PSRC may
be classified into three types: (1) The static and dynamic
structural factors (The proteins hold the chromophores in thethree-dimensional arrays and vibrate together. The resultant
most probable geometries are understood as those reported by
the X-ray crystallography.), (2) the electrostatic effect on theelectronic structures of the chromophores, and (3) the electronic
factors through protein wave functions. Though factor 2 canbe included into factor 3, we consider each separately.
The structural factor, the static one in particular, is definitively
the most important one. As clearly shown in the succeeding
sections, the efficient ET mechanism and the unidirectionality
owe their origin to the three-dimensional effective array of thechromophores and its asymmetry between the L- and Mbranches.
On the other hand, the electrostatic effect of proteins, which
is taken into account by the point charge model, is small. In
Figure 2, the transfer integrals shown in the numerator were
calculated including the protein point-charge effect, while thoseshown in the denominator do not include such an effect, i.e.,
were calculated in a gas phase. These two values are close:the effectiveness of the ET and the unidirectionality could be
explained without introducing the protein electrostatic effect.
Table 4 shows the protein electrostatic effect on the energy
levels of the chromophores. They are not small, but it is difficultat present to estimate its role in the ET process in the PSRC,
though some authors believe it to be important.
The electronic factors of proteins through their electronic
wave functions are difficult to estimate, at present, especiallyfrom ab initio theoretical point of view. Particularly, there may
be another ET pathways through protein electronic states in asuperexchange mechanism. However, we believe such possibility is small, at least for the ET from P* to H, since it occurs
within 3 × 10-12 s, very fast, and since the density of theproteins in this region of the PSRC is small. Further, the
LUMOs of the B’s and H’s involved are much lower than those10426 J. Phys. Chem. B, Vol. 102, No. 50, 1998

Hasegawa and Nakatsuji

TABLE 5: Electronic Factors |HIF|2 (×10-9 au) in au with
Some Approximationsa
approximation

L

M

fLPLB )

B- f HP* f B
L/M

L

M

L/M

Fb
Hc

Single-Configuration Approximation
37.36
1.83
20.42 116.96 17.96
37.13
1.85
20.07 116.96 17.96

6.51
6.51

SAC-CI

25.21

Multiconfiguration
1.65
15.27 104.43

6.56

15.93

to B asExperimentally Estimated Values
21,d 14e
72,d 54ea Calculated with the point-charge protein model. b Fock approximation for total Hamiltonian. c Total Hamiltonian. d Reference 35. e Reference 41.

of the amino acid residues of proteins, and their HOMOs areagain higher than those of the proteins. Adequate HOMOLUMO gaps of the chromophores hinder other electronic
processes and makes the overall quantum yield of charge
separation very high.
On the other hand, for the ETs from H to MQ and inparticular, from MQ to UQ, the protein wave functions andprobably its dynamic effects would be important. We expect
an existence of the ET route due to the superexchangemechanism involving the lower excited states of proteins.

VII. Analysis of the Transfer Integrals
We now analyze the transfer integrals in some detail toelucidate the origins of the efficient ET mechanism and theunidirectionality. For this purpose, we need a convenientmethod of analysis of the transfer integrals defined by eqs 2-72-9. Table 5 shows various approximations of the electronic
factors, |HIF|2, for the ET from P*B to P+B- and from P+B-H
to P+BH-. The first two rows show the values due to the singleconfiguration approximation: in the upper one, the total
Hamiltonian was replaced with the Fock operator and in thesecond one the total Hamiltonian was used. We see that the
Fock approximation is very good for this system. However,the single-configuration approximation is not necessarily a good
approximation as seen in Table 5: an inclusion of the configuration interaction effect by the SAC/SAC-CI method reduces
the transfer integrals and makes them closer to the experimentally estimated values.35 The ratios between the values for theL- and M-branches are also shown in Table 5. For the ET from
P*B to P+B-, the SAC-CI method reduces the L/M ratio by
3/ . Thus, while the Fock approximation appears somehow
4
qualitative, it is clear that it describes well the dominant termof the transfer integral, and therefore, we will use this approximation for the analysis of the transfer integrals.
A. Factor for the ET from P to B. The unidirectionalityof the ET in the PSRC of Rps. Viridis originates from theasymmetry of the transfer integrals between the L- and Mbranches as shown in Figure 2. Comparing the transfer integrals
in protein and in gas phase, we see that the unidirectionality
can be explained without the protein effect. The protein effectis far from the origin of the asymmetry of the electronic factor.
To analyze the asymmetry origin, we approximate the transfer
integral for the ET from P* to B in a gas phase by the Fockmatrix element, fLPLB, where LP and LB denotes the LUMOs
of P and B, respectively, and further this element is decomposed
into the sum of the atomic contributions of the atoms belonging∑ ∑∑CrLPCsLBfrs

X∈Bs∈X r∈P)X∑ fLPLB

(8-1)

X∈B

where fXLPLB is the contribution of the atom X of B, C is the
MO coefficient and frs (r ∈ P and s ∈ B) is an AO Fock matrix
element. Note that, in the preceding sections, the transfer
integral is used in a squared form, but in the following analysis
it is used in a form of eq 8-1.
It is clear from eq 8-1 that the following three conditions
must be satisfied for the transfer integral fLPLB to be large. The
first one is the proximity condition. In order that frs has a large
value, the atomic orbitals r and s should be closely located,since frs depends exponentially on the distance between r and s.
The second one is the MO coefficient condition: namely the
product of the MO coefficients between the chromophores CrLPCsLB should be large. The third one is the nodal effect which
originates from the nodal character of the MOs. The summation
over the AOs may lead to a cancellation if the MO coefficients
have opposite signs due to some symmetry. This nodal effectmay be large when the donor or acceptor orbitals have nodal
structure as in porphyrin compounds.
The result of the decomposition of fLPLB is shown in Figure
3a. We see that BL has much larger atomic contributions thanBM and that the largest difference arises from the contributionsof rings III and II of B’s and that the largest atomic contribution
is due to the 6-carbon of ring III of BL (10.8 × 10-5), while
the largest contribution in BM is due to β-carbon (-6.4 × 10-5).
Now, what is the origin of the asymmetry in the transfer
integrals between P and BL and between P and BM? Figure 4
shows the geometries and the LUMO populations of P, BL, and
BM. First we examine the geometrical proximity factor. Rings
III of BL and BM are located close to rings I of PM and PL,
respectively. The center-to-center distances are 6.89 and 7.41
Å, respectively, and therefore, the L side is closer than the Mside by 0.5 Å. Rings II of BL and BM are close to rings V of
PL and PM, respectively, and in particular, the 4a-carbon of BL
is located at 4.7 Å from the γ-carbon of PL. However, in thiscase, the PL-BL and PM-BM center-to-center distances are
similar: 7.13 and 7.14 Å, respectively.
We next examine the MO coefficient condition using the MO
populations of the LUMO’s of P and B shown in Figure 4,
where the sign of the population indicates the phase of theorbital. For the B’s, the populations are almost the same
between the L and M sides: some amplitudes are calculated at
5- and 6-carbons of ring III, but 4- and 4a-carbons of ring II
have almost no populations. For P, the population gathers in
the vicinity of ring I of PL than in that of PM, which favors theM-side ET.
From the above analyses of both the geometrical proximity
condition and the MO coefficient condition, we conclude thatthe asymmetry in the contributions of rings III of BL and BM
depicted in Figure 3 is attributed to be due to the geometricalproximity effect. The distance between rings III of BL and ring
I of PM is closer by 0.5 Å than the distance between ring III of
BM and ring I of PL. The proximity condition overwhelms the
MO coefficient condition, since the latter favors the M-side ET.
On the other hand, the asymmetry in the contributions of rings
II of BL and BM shown in Figure 3 is due to the MO coefficientcondition. As seen in Figure 4, the LUMO of P is slightly
localized on the γ-carbon of PL, which lies closely to the 4-

PSRC of Rhodopseudomonas Viridis

J. Phys. Chem. B, Vol. 102, No. 50, 1998 10427

Figure 4. Geometries and LUMO populations of P, BL, and BM. The sign of the population indicates the phase of the orbital.

Figure 5. Atomic contributions of H for the transfer integrals between B and H.

and 4a-carbons of BL. In this case, the distance between ring
II of BL and ring V of PL is similar to the corresponding distance
between BM and PM as described above. Accordingly, thecontributions of the 4a- and 4-carbons of ring II of BL is due tothe MO coefficient asymmetry.
Between the above two factors, the ring III and ring II
contributions of BL and BM, the ring III contribution is larger
than the ring II contribution as seen from Figure 3, and therefore,
the origin of the asymmetry in the transfer integrals between P
and Bs is attributed to the proximity condition, i.e., to the 0.5
Å difference in the rings I-III distances shown in Figure 4.
B. Factor for the ET from B to H. The calculated transfer
integrals for the ET from B to H is also larger in the L sidethan in the M side. Figure 5 shows the analysis of the Fockmatrix element fLBLH between the LUMO of B and the LUMO
of H into the atomic contributions. The largest contributions
are 42.9 × 10-5 and 24.9 × 10-5 au of the 1-carbons of rings
I of HL and HM, respectively.
In Figure 6, the MO populations of the LUMO’s of B and H
in the L- and M-branches are shown. The MO populations arequite symmetric between the L- and M-regions. In both L- andM-regions, the most populated rings I of B and H are adjacent
to each other, so that the transfer integrals can be large upon
the fulfillment of the MO coefficient condition. Actually, thetransfer integral from BL to HL is twice (in square four times)
as large as that from P to BL. In a previous semiempirical
study,7 the transfer integral was calculated to be equal between
P to B and B to H, which is different from the present resultand from the experimentally estimated values.
The center-to-center distances between the porphyrin ringsof B and H are 10.7 and 10.6 Å for the L- and M-branches,
respectively, which are almost the same. However, the orientation of HL and HM are somehow different. The distance between
the 1-carbons of B and H, both having the largest coefficientin their LUMO’s, are 5.03 and 5.46 Å in the L and M sides,
respectively: the L side is closer by 0.43 Å than the M side.
This proximity in the L side owing to the orientational difference
results in a large asymmetry in the transfer integral through theasymmetry of the atomic contribution of the 1-carbon, inparticular.

10428 J. Phys. Chem. B, Vol. 102, No. 50, 1998

Hasegawa and Nakatsuji

Figure 6. Geometries and LUMO populations for (a) BM and HM and (b) BL and HL. The sign of the population indicates the phase of the orbital.

Figure 7. MO populations of (a) LUMOs of P and BL and (b) HOMO of P and LUMO of BL. The sign of the population indicates the phase of
the orbital.

We note a role of the nodal condition for this case, though it
is of secondary importance as seen from Figure 5. The atomic
contributions between 1- and 2-carbons of ring I of H’s have
opposite signs and are canceling, though incomplete. Namely,
the LUMOs of H’s have nodes between the 1- and 2-carbons,
which make their contributions different in sign.
This orientational asymmetry between HL and HM may be
attributed to the difference in the specific protein environments
surrounding H’s. HL has a glutamate side chain (GLU L104)
near the carbonyl group at ring V. The GLU L104 is thought
to have a hydrogen bond with the oxygen of the carbonylgroup42,43 and may control the orientation of HL, while in the
M-branch, GLU L104 is replaced with valine (M131)2,8 which
has no polar group.
C. Factor for the CR from B to P. The transfer integral
for the CR from BL to P is much smaller than that of the ET asshown in Figure 2, though both transfers are between the same

chromophores: the interatomic distances are the same between
the two processes. The ET occurs from the LUMO of P to the
LUMO of B, while the CR occurs from the LUMO of B to the
HOMO of P, and therefore the key difference should arise fromthe HOMO and LUMO of P. In Figure 3b, the Fock matrix
element for the CR process, fHPLB (HP ) HOMO of P and LB) LUMO of B), is analyzed and compared with the analysis
for the ET reaction shown in Figure 3a. The ring III contribution is reduced and a cancellation of the atomic contributions
occurs, resulting in a reduction of the total transfer integral.
In Figure 7, the population of the HOMO of P is comparedwith that of the LUMO of P. In the HOMO, the electron is
localized in the upper part of PM and the sign is different
between PM and PL, while in the LUMO the electron is localized
in the lower part of both PM and PL. Therefore, the distance
between the electron clouds of P and B is larger in the HOMO(P)-LUMO(B) pair than in the LUMO(P)-LUMO(B) pair.

PSRC of Rhodopseudomonas Viridis
Further, the antisymmetry between PM and PL of the HOMO of
P causes a cancellation of the atomic contribution in the transfer
integral. Thus, due to the MO coefficient condition and thenodal effect, the transfer integral for the CR is much smaller
than that of the ET. Thus, the inefficiency of the CR process,which implies the efficiency of the ET process, is regulated by
the difference in the localization between the HOMO and the
LUMO of P.

VIII. Summary and Concluding Remarks
We have studied in this paper the mechanism of the ET andthe origin of its unidirectionality in the PSRC of Rps. Viridis
from ab initio point of view. We have examined the ET processfrom P* to H through B in the sequential mechanism. The CR
process from B- to P+ is also examined. The electronic factors
were calculated by the SAC/SAC-CI method which is able tocalculate various electronic states of the chromophores of the
PSRC in a reasonable accuracy. The effect of the surrounding
proteins and waters was calculated by using the point-charge
model.
The unidirectionality of the electron transfer is explained by
the asymmetry of the transfer integrals, an electronic factor, for
the ETs from P to B and from B to H as shown in Figure 2.
The ratios of the transfer integrals between the L- and M-regions
are 15.2 and 6.56 for P to B and B to H, respectively (Table 5).
The ET from BL to HL is very efficient since the correspondingtransfer integral is very large, 104. The efficient feature of the
ET is also explained by the smallness of the CR process from
P+B- to PB.
We failed, however, to explain the energy levels of theelectron transferred states due to an inability of calculating therelaxation energies of the chromophores and the protein medium,the last two terms of eq 4-1, by our present ab initio
methodology. Reliable ab initio calculations of such quantities
are certainly very interesting subject to be solved in future. Thus,in the present study, we analyze the mechanisms of the electrontransfer and its unidirectionality only through the electronic
factor of eq 1-1, which is valid, for example, when the matchingcondition is always satisfied.
The transfer integrals are analyzed by decomposing them into
the atomic contributions and the three conditions, the proximity
condition, the MO coefficient condition, and the nodal effectare pointed out to be important. For the ET from P to B, themost important factor is the geometrical asymmetry between P
and B’s in the L- and M-branches (the proximity condition).
The L-side distance between ring I of PM and ring III of BL,
which fulfill the MO coefficient condition, is closer by 0.5 Åthan that in the M-side (see Figure 4). Since the transfer integral
depends exponentially on the distance, this local geometricasymmetry causes a large difference. The MO coefficient
asymmetry is another factor. For the ET from B to H, theasymmetry is attributed again to the geometric factor (theproximity condition): though the average distance between B
and H is almost the same between the M- and L-branches, theorientation of H relative to B is asymmetric and therefore thedistance between ring I of B and ring I of H is closer by 0.5 Å
in the L-branch than in the M-branch. The most efficient ET
route is from 1-carbon of ring I of BL to 1-carbon of ring I of
HL (see Figures 5 and 6). The smallness of the CR processfrom P+B- to PB is attributed to the difference in the orbital
localization between the LUMO and the HOMO of P (see Figure
7).

J. Phys. Chem. B, Vol. 102, No. 50, 1998 10429
Thus, basically, the unidirectionality of the ET is understood
to be due to the asymmetry in the three-dimensional arrangementof the chromophores in the PSRC. The L-side chromophores
are locally closer than the M-side ones, though the averageseparations are almost the same between the L- and M-regions.
The asymmetry in the MO distribution between the L- and Mregions is also a factor, but less important, since the exponential
dependence of the transfer integral on the distance is muchsteeper than the variations in the MO coefficients between theL- and M-sides. On the other hand, the inefficiency of the CR
is definitely due to the difference in the electron localization
between the HOMO and the LUMO of P: the origin is verymuch quantum.
As far as the present analysis is correct, the protein environment plays a decisive role, since the three-dimensional arrangements of the chromophores in the L- and M-branches aresupported by the proteins. However, the protein effects
accounted for by the electrostatic point-charge model were foundto be very small for the electron distributions in the chromophores and for the transfer integrals between the chromophores.
Thus, now that the origin of the L-branch selectivity of the
ET in the PSRC of Rps. Viridis is attributed to the local
closeness of the chromophores in the L-region relative to theM-region, we propose to do a mutation experiment to makethe M-side chromophores locally closer to each other than the
L-side ones, which should realize the M-side selectivity of theET. Nature offered a duality in the electron path, but only one
is utilized now. To realize the another path is certainly an
exciting experiment, which will lead to clarify many other facts.
Acknowledgment. This study was supported in part by aGrant-in-Aid for Scientific Research from the Japanese Ministryof Education, Science, Culture, and Sports, and by the New
Energy and Industrial Technology Development Organization(NEDO). J. H. gratefully acknowledges the research fellowships
from the Japan Society for the Promotion of Science (JSPS).
Appendix
The CT state, P+B-, defined by eq 2-6, is not a pure singlet
state, since it is defined by a product of the two doublet wave
functions, P+ and B-. In eq 2-6, the operators IP and EB are
the sums of the single (S) and double (D) ionization and electron
attachment operators
P

IP )

∑I

P

CPI (S)IPI (S) +∑I ∑J CPIJ(D)IPI (S)RPJ (S)

P

)

P

∑I IPI (S) {CPI (S) + ∑j CPIJ(D)RPJ (S)}

B

E )
B

∑I

B

CBI (S)EBI (S) +

∑I

(A-1)

B

∑I ∑J CBIJ(D)EBI (S)RBJ (S)

B

)

P

B

EBI (S) {CBI (S) +∑j CBIJ(D)RBJ (S)}(A-2)

where IPI , EBI , and RPI are ionization, electron attachment, andsinglet excitation operators, respectively. Using eq A-1 and A-2,

10430 J. Phys. Chem. B, Vol. 102, No. 50, 1998

Hasegawa and Nakatsuji

eq 2-6 is rewritten as
P+B-

Ψ

P

)

B

∑I ∑K

P

IPI (S) EBK(S) {CPI (S) +∑J CPIJ(D)RPJ (S)} ×

B{CBK(S) +(A-3)∑L CBKL(D)RBL (S)}exp(SP + SB)ΦP+B
0Here, the products of the excitator, IPI (S)EBK(S), are written as

IPI (S)EBK(S) ) aiaa†aa(A-4)

which is not a singlet operator, so that a spin contamination
occurs in eq A-3. We therefore introduce the spin-symmetry
adapted operator RP+B
as
I

RP+B
(S) ≡
I

1
†
)
(aiaa†aa + aiβaaβ
x2

(A-5)

in which the β-counterpart is added to eq A-4. Then, the puresinglet state of P+B- is written as

Ψ

P+B-

P+B

)

∑I

P

RP+B
(S) {CPI (S) +I∑J CPIJ(D)RPJ (S)} ×

B{CBI (S) +(A-6)∑J CBIJ(D)RBJ (S)}exp(SP + SB)ΦP+B
0

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