- Email: [email protected]
benzoquinone interactions
Volume 26, number 1
THE
CHEMICAL PHYSICS LETfERS
ABSORPTION
CHLOROPHYLLlDIOXANE
SPECTRUM
OF CHLOROPHYLL
1 my
1974
A AND THE
AND CHLOROPHYLL/BENZOQUINONE
INTERACTIONS”
Joseph LE BRECH, Roger M. LEBLANC and Adel F. ANTIPPA Groupe de Recherche en Biophysique. Dhparrements UniversirEdu QuPbec, Trois-Rivieh,
de Physique et de Chimie-Biologie, Quhec, Canada
Received 11 June 1973 _ Revised manuscript received 2 January 1974 A molecular orbital calculation, using the “free-electron-network” method, is reported for the absorption spcctrum of chlorophyll a and the complexes chlorophyU/p-dioxane and chlorophyll/p-benzoquinone. For chlorophyil a, our results are in good agreement with experiment and are substantially the same as those obtained by Weiss using the self-consistent molecular-orbital method. Our results seem to indicate: (i) that chlorophyll a has 26 rather than 28 z electrons, (ii) that, in a chlorophyll complex, the central magnesium atom has a coordiiation number of 6, and (ii) that the cNorophyll/benzoquinone complex has an intense absorption band centered at 713 nm. If present in vivo this complex can provide a model for the P-700 energy trap.
1. Introduction The n-electron system of porphyrins has been the object of numerous theoretical calculations; Weiss et al. [I] have done extensive calculations using the selfconsistent field theory of molecular orbitals in the Pariser-Parr-Pople (SCM0 PPP) approximation. Recently, Weiss [2] has extended these calculations to chlorophyll and has given a thorough review of work done on their n-electron structure and their absorption spectra. On the other hand, Simpson 131, Kuhn [4,5], Nakajima and Kon [6] have approached the porphyrins by using the free-electron model. Their results agree with those of the preceding method. in the present work we study the absorption spectrum of chlorophyll a by using the free-electron-network molecular-orbital method (FEMO) developed by Ruedenberg and Scherr [7,8]. This FEMO method is much easier to use than the SCM0 PPP method, specially in the case df macromolecules. At the same time it gives r&ults in good agreement with Weiss’ talculations [2] and with expetiental results. In dealing with chlorophyll compIexes we will have occasion to apply the FEMO method to three-d&en-*Work suppoked in & by.tl&National Research C~u&il .. of Canada and the Quebec Ministry of Education. -. :
sional molecular structures. This is not difficult to justify_ It is sufficient to note that the free-electron network is a one-dimensional multiconnected configuration space which depends on the lengths and topology of the different branches but is independent of the angles between them. Thus a number of branches meeting at a point lead to the same subnetwork whether they are coplanar or not. This is related to the proof of Ruedenberg and Scherr [7] that corners in a bond path have no implications if the H electronic potential is continuous. Czikkely et al. [9] have already applied the FEMO method to three-dimensional molecular structures in the course of their study of the light absorption of aggregates of dye molecules. In this paper we will be concerned with the chlorophyll a molecule and the complexes chlorophyIl/dioxane and chl&ophyll/benzoqulnone. In each case there are three main factors that enter into the C&Xlation; the molecular structure, the FEMO,method of calculations and the absorption spectrum. Our generaI approach is as fo&ws: (i) using the known moleciIlar [email protected] ch@?phyll a [ IO] and the .ex+tiectqy d+&+ned absoiption specthm [1 I]; we s&idy’ the validity. of the FpMo .rn&hod and ,its st+it$it$ .-’ to clianges in the-mole++ st+ere; (iij.utigthe ix: .perimentally determined absorption [email protected]_;_ of &the .:-: ,, .-_ r.:‘. :, .:. ... :, : ., ...f ‘37 1
1 May 1974
CHEMICAL PHYSICS LETTERS
Volume 26, number 1
chlorophyll/dioxane complex [ 121 and the already ratified FEMO method of calculation, we determine the details of the proposed chlorophyllldioxane “interaction model” by requiring that the predicted spectrum accord with the experimentaI one; (iii) finally assuming a chIorophylI/benzoquinone interaction model analogous to the one determined above we use the FEMO method to predict the as yet experimentally undetemlined absorptio? spectrum of the chlorophyll/benzoquinone c0mp:ex-f.
(iv)
(v)
where F, = 2 cos K,, and D is the internuclear distance. This gives the energy spectrum of the B electrons of the conjugated network of the molecule. Using eq. (2) and the eigenvalues Fn, the normalized eigenvectors I& can be obtained and consequently the wavefunctions describing the movement of the H electrons of the network. The oscillator strengths of the diverse transitions are estimated from the expression [ 131
f = 1.085
x 1o-svQ2,
(4)
2. The FEMO method of calculation where v is the transition On supposing that the potential due to the u-bond skeleton remains constant throughout the free-electron network of a given molecule, the FEMO calculation [7] for obtaining the network’s electronic energy spectrum can be summarized by the following steps: (i) The determination of the symmetrical topological matrix f which represents the network. By a method similar to that used by Ruedenberg md Scherr [7], a general formula can be derived for the elements of the matrix F. Let theN atoms of the network be numbered from 1 to ZV,and letMk denote the number of network lines meeting at atom number k_ Then the elements of the matrix Fare given by Fii = 0,
W
if atoms i andi are not neighbors;
Fii=O, 2
=
if atoms i andi are neighbors.
(lb)
3. The absorption
(2)
-$ being the normalized (iii)
The evaluation
Neigenvalues aregiven by fE.
free-electron
eigenvedtor.
of the energy levels En using the eq. (2). These
F, obtained-from
E n = (&2&) K2 (3) n’ : _.___ :.- __‘- :-. xpenments cbtiducted to date for the detection of the
-.-..chl~ror;hylvbenzo.~~~onecorn&x are &t cqndusive since .’ thi b&z?quinorie va& p&s&e t&d h&been-abut two ordersof mq$tu@e smaller than qat.use{in+ chl@obhyll/ :
:di&~ne~.$xper&nts.
:,: ”
--’
::
spectrum of chlorophyll
by
a
The free-electron network used in calculating the absorption spectrum of chlorophyll a, as shown in fig. 1b, is determined by the conjugated system of bonds of the chIorophylI a molecule as shown in fig. la. The elements of the topological matrix Fare obtained from fig. lb by using eq. (l)_ The internuclear distance is set to D = 1.4 A all through the paper. A major difference between the conjugated system
The solution of the eigenvalue problem, F@=F$,
in &,
in the FEMO method Q is given compactly eq. (3.9) of ref. [ 14J_
(Mpj) ‘I2 (ii)
moment
:
Volume 26, number 1
CHEMICAL
1
PHYSICS LETI.ERS
Table 1 Absorption spectrum of chlorophyll a and of the complexes Cbi/dioxane and ChI/benzoquinone as calculated by tire The Xs are the wavelengths of the various transitions and thefls are the corresponding oscibator strengths _-_-. ____--__ Chi. a exptl. a)
Chl. a talc. b,
Chl/dioxane
‘)
Cbl/bentoquinone
h(nm)
f
0.44
743
0.34 0.20 0.13 0.06 0.32
688 676 630
0.14 0.28 0.14 0.15
832 713
0.14 0.50
594 565
442
0.14 0.12 0.10 0.15 0.15
416
0.26
WW 754’
661
86.2
659
615 575 530
12.6 6.8 3.4
612* 607 548 512
548 511
0.13
a.21
522 488
0.35
442
0.12
409 a) b, c) f)
71
434 415 411
d,
f
E x 1C3
113
FEMO method.
Mnd
f
Unm)
428
May 1974.
0.12 0.33
-
Taken from ref. [ 111 (E: extinction coefficient!. Calculated according to the model of fig. lb. Calculated according to the model of fig. 3b. Calculated according to the model of tig. 5b. This band disappears if the chlorophyll network has 26 instead of 28 n electrons.
trons. Thus we would expect a total of 28 7~electrons of bonds of porphyrin and that of chlorophyll is due to in the network. On the other hand, chlorophyll shows the central magnesium atom [ 151. Porphyrins have aromatic characteristics and thus we would expect the been the subject of intensive study [I, 3-61 and are Htickel4m + 2 rule [ 161 to apply giving 26 ‘IIelectrons. thought to be understood experimentally as well as theoretically_ ‘Thus we will be primarily preoccupied This would imply that each of the two nitrogen atoms is contributing only one n electron. Our results seem with the role played by the central magnesium atom. to support the Hiickel rule, since the absorption bands in determining the absorption spectrum of chlorothat are due to the two extra ‘ITelectrons have not as phyll a. Experimentally it is found that the central magneyet been observed experimentally_ Nevertheless,.&e sium atom is connected to only two of the four nitrowill proceed with our calculations on the basis of 28 gen atoms of,the pyrrole cycles, that these two nitron electrons and, in presenting our results, indicate by gen atoms are diagonally situated and are determined a star each absorption band that will be absent if there by the asymmetry of the chlorophyll molerAe, and were only 26 ‘Relectrons in the network. The wavelength A’and oscillator strengthfof the finally that the magnesium-nitrogen bonds are single abi&pfion spectrum of chlorophyll ai ca@ated acbonds [ IO]_ The situation-regarding the other two ni: trogen atoms is not very clear experimentally and it cording to the FEMO method and using the notwork, is suggested that a very weak interaction may exist be- .. of fig. lb; is presented in the 2nd column of table. 1 tween them and the magnesium atom. This interaction. and-agrees reasonably Well with the experimentally-de-is thought to be insuff%ent to produce even a weak termmed spectrum as presented &I the 1st &lumn:of bond. ThuS the free-electron network of fig. 1 b depicts the same table; The agreement is specially good for- .,-. the intense, *d exp&in%htaIiy~tieIl deternrined, bands. the experimental situation. : To deters&e the number of free electrons rr~the at‘tjfjl and 428 r& On the .other band;.the e&&ted .-network we note that each of the’ twelve double bonds ‘-. band at-512 nni @hauosdia~oistr&&h of 0.32 ha-1 contributes t& free.eIectr&. E&t+more each of. .. ‘:.po t$&ir$n~ti c.ounter_part,z$d & a &ortcomiugof _:;. thk two niti&ti atoms, .which’are~dnnected.to t&e, ._- .- ..- .rtfre&etfu$j The 6~_~ula~e,d,b~~.at_754 r+v$fh an .:. ‘. ‘. _,centraI ma~gncsium ator+& ; :. eontribute:. tti~o’~fr&elec; ,o~~~_?t~~~l~~~~~~.o~0:44 .is:dye,. ti, an e&+cG-r t&&.-::. -.- _:. .... ‘. T
Volume 26, number 1
CHEMICAL PHYSICS LETTERS
1 May 1974
! Table 2 Comparison of LCAO and FEMO methods as applied to chlorophyll a. Excitation for the principal bands Qv, Qy, aad B, of chlorophyll a Method
EQpm-‘)
fQ.Y
SCM0 PPP a)
15980 1.5170 15100
0.22 0.34 0.23
FEMO b, Exptl. a)
energies E and corresponding osciuator strengths,
EQ,mA
fQ_x
BB(cm-‘)
fB
17760 17020 17300
0.052 0.10 -
27330
1.92 0.81 1.1
23740 23260
a) Taken from ref. [2]. b, Calculated centers of gravity.
tion from the 14th to the 15th energy level. Thus this band will not be present in the spectrum if the 14th level is empty, which is the case if there are only 26 ?r electrons in the network.-The absence of this band from the experimental spectrum may be due to the insensitivity
of conventional
range of wavelengths.
in this
photomultipliers
In any circumstance
sults obtained are compruable to those of Weiss [2] with the model of four prbitals (table 2j. In order to verify the experimentally determined type. of bonding between the central magnesium atom and the four nitrogen atoms we repeat the calculation of the absorption spectrum of chlorophyll a assuming different
our calcu-
models
for the nitrogen-magnesium
bonding.
The results for
the different models tested are shown in table 3. In three models we have represented a weak nitrogen-magnesium bond by a free end attached to the nitrogen atom. Within the context of the free-elec-
lations suggest that either this band exists and will be found, or that otherwise the free-electron network of chlorophyll has 26 a electrons only. For the most important absorption bands, the re-
Table 3 Absorption spectrum of chlorophyll a. Calculated spectra by the FEN0 method for the five models of magnesium-nitrogen interactions shown below. The remaining parts of the free-electron networks are identical with the network of fig. lb. The X’s are the wavelengths of the various transitions and the f’s are the corresponding oscillator strengths Chl-a Expel
A (nm)
661 -
exlo-3
86.2 -
h(nm)
.f
h(nm)
f
Mm)
f
A(nm)
f
h(nnh
f
754+
0.44
824* 691
0.35 0.36
715*
0.52
713
0.50
721* 680
0.27 0.36
659 -
0.34 -
633*
645 -
0.05
._
-
-
-
-.
-
-
0.17 0.12 -
-
-,
-
587
0.23
._
_
-
553
0.07
566* 549
0.14 0.06
579 527
513
0.32
-
-
4.53. 443
-0.28
426
0.36 ._. _: -. _ :
,631 615
12.6
575
6.8
530
3.4
612* 607
0.20 0.13
548 -
0.06 0.32
.512 1 428
. . 113..
-
.409,
.71
-.
-
434 a15 4L1’
0.35 0.12 0133
:
___ ...
,-_ 4.
520 483 457.
-.
0.15
430. __
-. - : .:
-_._- .’ ‘.. a) Taken frorir ref. ill] (E: extindtioncoemcient); ‘_.:.. ’ ‘- .:I.’ ._“This bid disappears if the &e&o& has 26 Gratead 6f 28 i&&oonj:.--.
;
_-,
.._-, _-- ..
__.
.‘- :. ...._ _-- -:
+
.
._
0.12 .. 0.06 .O.lO ,520 0.11 0.19 -. 4.17 ‘0.11 _ _ 0.24. :- -, 441 0.16 0.03 -. :-
7 _
m_‘1 ..
-
:
... .-
._
550* .0.12 531 0.17 526 0.21 .477 :. 0.1% 452 ‘-- 0~13 ---
: -:432. :
.420 -
-0.13 .. 0.0s .. _
._ :., :
Volume 26, number 1
I May1974
CHEMICAL PHYSICS LETTERS
Table 4 Absorption spectrum of chlorophyll a. Spectra due to the majorsectionsof the free-electron network ofehkm@ylt a, as calculated by the FEW0 method. The h’s are tile wavelengths of the various transitions and thefs ate the conesponding o%illMor strength Chl-a b,
Chl-a exptl..a)
h (nm)
661
615 57.5 530
EXlO”
86.2
12.6 6.8
A(_)
f
754*
0.44
659
0.34
607
0.13
612* 548
i-4
0.32
428
113
409
71.
434 415 411
0.35 0.12 0.33
_
Porphine e,
A(nm)
A(nm)
f
A(um)
f
1149
i
--
539 531
I-
0.13
744
0.48
576
0.50
596
0.51
419 443 402
: 0.11
0.13
0.09 0.13
Taken from ref. Ill] (e: extinction coefficient). Calculated according to the model of fig. 2a. Calculated according to the model of fig. 2b. Calculated according to the model of fig. 2c.
e, *
This band disappears if the Chl-a network has 26 instead of28 P electrons.
according
950
0.08
0.17
a) k) c) d)
Calculated
0.08
782 0.60
-
0.06
512
Porphyrine cycle ef Chl-a d,
674
0.20
-
Chl-a without Mg ‘?I
-
481 416 -
0.49
465 450
0.13 0.83
391
0.28
0.10
to the model of fig. 2d.
tron model, this implies a partial mobility of the free electrons towards the magnesium atom. The model of column 2 is derived from the experimentally determined model of fig. I b. The results for the other models of bonding are presented in columns 3 to 6 and show that the FEMO method can distinguish between the different models of bonding and that only the absorption spectrum corresponding to the physical model.(column 2) agrees with experiment. To estimate the relative importance of the various sections of the chlorophyll molecule on the absorp tion spectrum, we present in table 4 the absorption spectra, calculated according to the FEMO method, of chlorophyll a (fig. 2a), chlorophyll a without the central magnesium atom (fig. 2b), the porphyrin-like cycle of chlorophyll (fig. 2c), and finally porphine (fig. 2d). The difference between columns 2 and 3 is a measure of the importance of the central magnesium atom; that between column 3 and.4 is due to the.two conjugated tails of the chlorophyti molecule, aud.fin- ‘. .’ Fig.12. Fredlectron r&works of: (a) chlorophyll a, (b):C& ‘ally .the difference between columns 4 and.5 measuies : a with&t its c&d mignesium, (c) porph$iin-$l$$yct& qf- -’ effect of the imperfection inthe porphyrin cyc!e.: -.:. ‘. ._ chl_:a, (4) porphir+~ .. :. ,;, ‘. -: .: .. .. .’
Volume 26. number I
CHEMICAL PHYSICS LEXTERS
1 May 1974
plying a chIorophylI/dioxane interaction_ Thus dioxane interacts with aggregates rather than individual molecules of chlorophyll. Further evidence to support this hypothesis comes from experiments with vapors of piperazine, morpholine, and piperidine. According to the interaction model presented here each dioxane molecuIe is bonded, through its two end oxygen atoms, to two chlorophyll moIecuIes. If this model is correct,
Fig. 3. Chl. a/pdioxane interaGtions:(a) model for the com(The Chl. a part not drawn is as shown in fig. i .)
pIex, (b) free-electron netwok
4. Absorption spectra of chlorophyll complexes 4-I_ Chloropltyllldioxatte
complex
The model proposed for the chlorophyll/dioxane interaction is shown in fig. 3a. The important features
of the model are that dioxane is bonded to the central magnesium atom, that it interacts with aggregates rather than single molecules of chlorophyll, and that the central magnesium atom has a coordination number of SIX in this complexed state. In the following we will justify these three features one by one. Experimentally no porphyrin-dioxane complexes exist [ 171. This strongly suggests that the dioxane molecule is not bonded to .the porphyrin cycle of chlorophyll. Hence it must be bonded to .the central magnesium atom. Furthermore the absorption spectrum ofchlorophyll is prackally ukhanged.when chlorophyll is dissolved in liquid dioxa& [ 181. This .&nplIt% the absence of an inter@ion between individ-. &xl &lorophyll &lecules &d’diox&e. On the other haqd, it @.ekperimentally observed-&t when dio&& yapoi-co_mes in co&t with chlorophyll iri the solid state [li] ,or in the monoinolecul~, itate [ 191 it displaces-the abso$ion band _?f the pigmeni, &Us im-. .i
the substitution of the two end oxygen atoms by two electron donors should lead to a similar interaction.1 Experimentally this is true in the case of piperazine and morpholine [ 191 where the two end oxygens are replaced by two nitrogens, and by one nitrogen and one oxygen, respectively. On the other hand, if the hypothesis that there is no interaction except in the aggregate state is true, then removing one of the end oxygens of dioxane, destroys the chIorophyll/dIoxane interaction_ Experimentally, vapors of piperidine (dioxane with end oxygens replaced by one nitrogen and one carbon) do not interact with chlorophyll [ 191. The third feature of the model concerns the coordination number of the central magnesium atom in the chlorophylI/dioxane complex. This is not yet settled experimentally, although various authors [ 12,201 have suggested that magnesiutm probably has a coordination number of SIX when chlorophyll is in the solid state or the monomolecular state. This would imply some sort of bonding of the magnesium atom to all of the four nitrogen atoms of the pyrrole cycle, in tiddition to its bonding to two dioxane molecules. To clarify this situation we test the three models of bonding shown in table 5. Model “a” assumes a coordination number of 4 for magnesium. Models “b” and “c” assume a coordination nurqber of 6 with two strong and two weak magnesium-nitrogen bonds for model “b” and four strong bonds for model “c”. The absorption spectra corresponding to the three models are presented in table 5. The calculations are based on the free-electron network of fig. 3b with the appropriate modifications in the case of each model. The FEMO m&hod is used aIi through-with an internuclear distance D = 1.4 &Comparison with &exierimental iesults clearly indicates that, in the &lo& phylI/dioxane’cotiplex, the cqorclm&i& number of .: the &&al magnesiti& atom-is SIX, qvith two strong “d two v&k tiagnesiGm%trogen bonds. This com@f-the chIor~ph$II/di~x&e-. :. ... .. pleies$he deter&&on interaction.&odel_.-As s&n f&m t&lfl;t$is x&&d&‘- ._-.. . . . . . .:- -;
CHEMICAL PHYSICS LETTERS
Volume 26, number 1
Table 5 Absorption spectrum of the Chl/dioxane complex. Calculated spectra by the FEMO method for the three models of Chl/dioxane interaction shown below. The remaining parts of the networks are identical with the network of fig. 3b. The X’s are the wavelengths of the various transitions and the fs are the corresponding oscillators strengths
Mg
coordination no. 4
Mg--N bonds-
Mg coordination no. 6.4 strong Mg-ti bondsh(nm)
Mgcoordination no. 6.2 strong
X(nm)
f
k(nm)
f
718 -
O-i3
743 688 676 -
0.14 0.28 0.14
659 656 -
0.14 0.27
606 544 509 -
0.15 0.07 0.26
630 548 511
442
-
f
-
-
645 -
0.05 -
585 545
0.09
481 480 453 422
0.17 0.09 0.23 0.10
0.20
-
leads to a displacement, caused by the interaction, of the whole spectrum of chlorophyll a towards the red; specifically the band Q-, calculated at 659 run is displaced to 688 nm as compared with an experimental displacement to 688-690 nm. An identical displacement of the absorption spectrum is obtained when a monolayer of chlorophyll is exposed to the vapors of piperazine or morpholine [20]. This agrees with the predictions of the free-electron model, since this model does not distinguish between dioxane, piperazine and morpholtie. 4.2.
ChlorophylZ/benzoqrtinone
1 May 1974
dioxane replaced by benzoquinone. The FEMO method, applied to the resulting network, then predicts a large bathochromic displacement of the absorption spectrum of chlorophyll a, with the absorption peak of the Q, band being calculated at 7 13 nm. The presence of a complex chlorophyll/quinone in vivo is not improbable, and could serve as the P-700 energy trap.
5. Discussion The use of the free-electron model is only justified in the case of those conjugated molecules for which the approximation of equivalence of all atoms is reasonably realistic. Among the chlorophylls, it is chlorophyll a, that has the least amount of heterogeneity. Thus even though the free-electron networks for the various chlorophylls do not differ appreciably, we would expect the general features of the resulting absorption spectrum to correspond most closely to that of chIorophyl1 a, and to resemble less and Iess the spectra of the other chlorophylls in order of increasing heterogeneity of their conjugated molecular structure. The treatment of the other chlorophylls would require a more sophisticated version of the FEMO method; one which accounts explicitly for the presence of heteroatoms. In considering the complexes of chlorophyll with dioxane or benzoquinone as a single free-electron network, the symmetry of the molecular structure permits us in as far as the FEMO calculation is concerned, to extend this network over only half a molecule of dioxme, or of benzoquinone, on either side of the chlorophyll a molecule. The resulting network is the same as that which would be obtained, in the unphysical situation, of one dioxane and one chlorophyll molecule interacting together, with the dioxane bent around the chlorophyll and attached to both sides of the central magnesium, thus forming a closed network.
complex
There are hardly any experimental results available on the chlorophyli/benzoquinone complex. But due to the similarities between dioxane and benzoquinone, it is reasonable to assume that the interaction model for both is the same. This leads to a molecular strutture, and result& free-electron network, similar to _ that shown in figs. ?a and 3b, respectively, but with
Acknowledgement
The computational phase of the carried out at the computer center du Quebec B Trois-R&i&es. One of like to thank Dr; Ky Toan Nguyen - puter prograniiniug. .. :
present work was of the UniversitC us cJ.L.B.) would fat help with com-
43
Volume
26, number 1
CHEMICAL PHYSICS LE’ITERS
References [ 1] C. Weiss. hf. Kobayashi and M. Goutennan, J. Mol. Spectxy. 16 (1965) 45 l_ [2] C. Weiss, J_ Mol. Spectry. 44 (1972) 37. [3] W.T. Simpson, J. Chem. Phys. 17 (1949) 1218. [4I H. Kuhn, J. Chem. Phys. 17 (1949) 1198. [S] 8. Kuhn and IV. Huber, Helv. Chim. Acta 42 (1959) 363. [6] T. Nakajima and H. Kon, J. Chem. Phys. 20 (1952) 750. [71 K. Ruedenberg and C. Scherr, J. Chem. Phys. 21 (1953) 156.5. (81 J-R. Platt, ed., Free-electron theory of conjugated molecules: a source book (Wiley, New York, 1964). [91 V. Czikkeiy, H.D. F&sterling and H. Kuhn, Chem. Phys. Letters 6 (1970) 11. [IO] E. Rabinowitch and Govindjee, Photosynthesis (Wiley, New York, 1969).
1 May 1974
[ li] C. HousSier a&l K. Sauer. J. Am. Chem. Sot. 92 (1970) 779. [I21 G. Sherman and E. Fujimori;Arch. Biochem. Biophys:130 (1969) 624. [I31 R.S. Mulliken and CA. Rieke, Rept. Pro&. Phys 8 (1941) 231. [14J N-S. Ham and K. Ruedenberg, J, Chem. Phys. 25 (1956) 7. [ISI J.L. Hoard,:M.J. Hamor and T.A. Hamor, J. Am. Chem. Sot. 8.5 (1963) 2334. [I61 E. Hiickel, Z. Physik 76 (1932) 628. 1171 B.B. Love and T.T. Bannister, Biophys. J. 3 (1963) 99. [IS] G-R. Seely and R.G. Jensen, Spectrochim. Acta 21 (1965) 1835. [I91 R.M. LebIanc, to be published. 1201 J.J. Katz, T.R. Janson, A.G. Kostka, R.A. Uphaus andG.L. Gloss, J. Am. Chem. Sot. 94 (1972) 2883.
THE
CHEMICAL PHYSICS LETfERS
ABSORPTION
CHLOROPHYLLlDIOXANE
SPECTRUM
OF CHLOROPHYLL
1 my
1974
A AND THE
AND CHLOROPHYLL/BENZOQUINONE
INTERACTIONS”
Joseph LE BRECH, Roger M. LEBLANC and Adel F. ANTIPPA Groupe de Recherche en Biophysique. Dhparrements UniversirEdu QuPbec, Trois-Rivieh,
de Physique et de Chimie-Biologie, Quhec, Canada
Received 11 June 1973 _ Revised manuscript received 2 January 1974 A molecular orbital calculation, using the “free-electron-network” method, is reported for the absorption spcctrum of chlorophyll a and the complexes chlorophyU/p-dioxane and chlorophyll/p-benzoquinone. For chlorophyil a, our results are in good agreement with experiment and are substantially the same as those obtained by Weiss using the self-consistent molecular-orbital method. Our results seem to indicate: (i) that chlorophyll a has 26 rather than 28 z electrons, (ii) that, in a chlorophyll complex, the central magnesium atom has a coordiiation number of 6, and (ii) that the cNorophyll/benzoquinone complex has an intense absorption band centered at 713 nm. If present in vivo this complex can provide a model for the P-700 energy trap.
1. Introduction The n-electron system of porphyrins has been the object of numerous theoretical calculations; Weiss et al. [I] have done extensive calculations using the selfconsistent field theory of molecular orbitals in the Pariser-Parr-Pople (SCM0 PPP) approximation. Recently, Weiss [2] has extended these calculations to chlorophyll and has given a thorough review of work done on their n-electron structure and their absorption spectra. On the other hand, Simpson 131, Kuhn [4,5], Nakajima and Kon [6] have approached the porphyrins by using the free-electron model. Their results agree with those of the preceding method. in the present work we study the absorption spectrum of chlorophyll a by using the free-electron-network molecular-orbital method (FEMO) developed by Ruedenberg and Scherr [7,8]. This FEMO method is much easier to use than the SCM0 PPP method, specially in the case df macromolecules. At the same time it gives r&ults in good agreement with Weiss’ talculations [2] and with expetiental results. In dealing with chlorophyll compIexes we will have occasion to apply the FEMO method to three-d&en-*Work suppoked in & by.tl&National Research C~u&il .. of Canada and the Quebec Ministry of Education. -. :
sional molecular structures. This is not difficult to justify_ It is sufficient to note that the free-electron network is a one-dimensional multiconnected configuration space which depends on the lengths and topology of the different branches but is independent of the angles between them. Thus a number of branches meeting at a point lead to the same subnetwork whether they are coplanar or not. This is related to the proof of Ruedenberg and Scherr [7] that corners in a bond path have no implications if the H electronic potential is continuous. Czikkely et al. [9] have already applied the FEMO method to three-dimensional molecular structures in the course of their study of the light absorption of aggregates of dye molecules. In this paper we will be concerned with the chlorophyll a molecule and the complexes chlorophyIl/dioxane and chl&ophyll/benzoqulnone. In each case there are three main factors that enter into the C&Xlation; the molecular structure, the FEMO,method of calculations and the absorption spectrum. Our generaI approach is as fo&ws: (i) using the known moleciIlar [email protected] ch@?phyll a [ IO] and the .ex+tiectqy d+&+ned absoiption specthm [1 I]; we s&idy’ the validity. of the FpMo .rn&hod and ,its st+it$it$ .-’ to clianges in the-mole++ st+ere; (iij.utigthe ix: .perimentally determined absorption [email protected]_;_ of &the .:-: ,, .-_ r.:‘. :, .:. ... :, : ., ...f ‘37 1
1 May 1974
CHEMICAL PHYSICS LETTERS
Volume 26, number 1
chlorophyll/dioxane complex [ 121 and the already ratified FEMO method of calculation, we determine the details of the proposed chlorophyllldioxane “interaction model” by requiring that the predicted spectrum accord with the experimentaI one; (iii) finally assuming a chIorophylI/benzoquinone interaction model analogous to the one determined above we use the FEMO method to predict the as yet experimentally undetemlined absorptio? spectrum of the chlorophyll/benzoquinone c0mp:ex-f.
(iv)
(v)
where F, = 2 cos K,, and D is the internuclear distance. This gives the energy spectrum of the B electrons of the conjugated network of the molecule. Using eq. (2) and the eigenvalues Fn, the normalized eigenvectors I& can be obtained and consequently the wavefunctions describing the movement of the H electrons of the network. The oscillator strengths of the diverse transitions are estimated from the expression [ 131
f = 1.085
x 1o-svQ2,
(4)
2. The FEMO method of calculation where v is the transition On supposing that the potential due to the u-bond skeleton remains constant throughout the free-electron network of a given molecule, the FEMO calculation [7] for obtaining the network’s electronic energy spectrum can be summarized by the following steps: (i) The determination of the symmetrical topological matrix f which represents the network. By a method similar to that used by Ruedenberg md Scherr [7], a general formula can be derived for the elements of the matrix F. Let theN atoms of the network be numbered from 1 to ZV,and letMk denote the number of network lines meeting at atom number k_ Then the elements of the matrix Fare given by Fii = 0,
W
if atoms i andi are not neighbors;
Fii=O, 2
=
if atoms i andi are neighbors.
(lb)
3. The absorption
(2)
-$ being the normalized (iii)
The evaluation
Neigenvalues aregiven by fE.
free-electron
eigenvedtor.
of the energy levels En using the eq. (2). These
F, obtained-from
E n = (&2&) K2 (3) n’ : _.___ :.- __‘- :-. xpenments cbtiducted to date for the detection of the
-.-..chl~ror;hylvbenzo.~~~onecorn&x are &t cqndusive since .’ thi b&z?quinorie va& p&s&e t&d h&been-abut two ordersof mq$tu@e smaller than qat.use{in+ chl@obhyll/ :
:di&~ne~.$xper&nts.
:,: ”
--’
::
spectrum of chlorophyll
by
a
The free-electron network used in calculating the absorption spectrum of chlorophyll a, as shown in fig. 1b, is determined by the conjugated system of bonds of the chIorophylI a molecule as shown in fig. la. The elements of the topological matrix Fare obtained from fig. lb by using eq. (l)_ The internuclear distance is set to D = 1.4 A all through the paper. A major difference between the conjugated system
The solution of the eigenvalue problem, F@=F$,
in &,
in the FEMO method Q is given compactly eq. (3.9) of ref. [ 14J_
(Mpj) ‘I2 (ii)
moment
:
Volume 26, number 1
CHEMICAL
1
PHYSICS LETI.ERS
Table 1 Absorption spectrum of chlorophyll a and of the complexes Cbi/dioxane and ChI/benzoquinone as calculated by tire The Xs are the wavelengths of the various transitions and thefls are the corresponding oscibator strengths _-_-. ____--__ Chi. a exptl. a)
Chl. a talc. b,
Chl/dioxane
‘)
Cbl/bentoquinone
h(nm)
f
0.44
743
0.34 0.20 0.13 0.06 0.32
688 676 630
0.14 0.28 0.14 0.15
832 713
0.14 0.50
594 565
442
0.14 0.12 0.10 0.15 0.15
416
0.26
WW 754’
661
86.2
659
615 575 530
12.6 6.8 3.4
612* 607 548 512
548 511
0.13
a.21
522 488
0.35
442
0.12
409 a) b, c) f)
71
434 415 411
d,
f
E x 1C3
113
FEMO method.
Mnd
f
Unm)
428
May 1974.
0.12 0.33
-
Taken from ref. [ 111 (E: extinction coefficient!. Calculated according to the model of fig. lb. Calculated according to the model of fig. 3b. Calculated according to the model of tig. 5b. This band disappears if the chlorophyll network has 26 instead of 28 n electrons.
trons. Thus we would expect a total of 28 7~electrons of bonds of porphyrin and that of chlorophyll is due to in the network. On the other hand, chlorophyll shows the central magnesium atom [ 151. Porphyrins have aromatic characteristics and thus we would expect the been the subject of intensive study [I, 3-61 and are Htickel4m + 2 rule [ 161 to apply giving 26 ‘IIelectrons. thought to be understood experimentally as well as theoretically_ ‘Thus we will be primarily preoccupied This would imply that each of the two nitrogen atoms is contributing only one n electron. Our results seem with the role played by the central magnesium atom. to support the Hiickel rule, since the absorption bands in determining the absorption spectrum of chlorothat are due to the two extra ‘ITelectrons have not as phyll a. Experimentally it is found that the central magneyet been observed experimentally_ Nevertheless,.&e sium atom is connected to only two of the four nitrowill proceed with our calculations on the basis of 28 gen atoms of,the pyrrole cycles, that these two nitron electrons and, in presenting our results, indicate by gen atoms are diagonally situated and are determined a star each absorption band that will be absent if there by the asymmetry of the chlorophyll molerAe, and were only 26 ‘Relectrons in the network. The wavelength A’and oscillator strengthfof the finally that the magnesium-nitrogen bonds are single abi&pfion spectrum of chlorophyll ai ca@ated acbonds [ IO]_ The situation-regarding the other two ni: trogen atoms is not very clear experimentally and it cording to the FEMO method and using the notwork, is suggested that a very weak interaction may exist be- .. of fig. lb; is presented in the 2nd column of table. 1 tween them and the magnesium atom. This interaction. and-agrees reasonably Well with the experimentally-de-is thought to be insuff%ent to produce even a weak termmed spectrum as presented &I the 1st &lumn:of bond. ThuS the free-electron network of fig. 1 b depicts the same table; The agreement is specially good for- .,-. the intense, *d exp&in%htaIiy~tieIl deternrined, bands. the experimental situation. : To deters&e the number of free electrons rr~the at‘tjfjl and 428 r& On the .other band;.the e&&ted .-network we note that each of the’ twelve double bonds ‘-. band at-512 nni @hauosdia~oistr&&h of 0.32 ha-1 contributes t& free.eIectr&. E&t+more each of. .. ‘:.po t$&ir$n~ti c.ounter_part,z$d & a &ortcomiugof _:;. thk two niti&ti atoms, .which’are~dnnected.to t&e, ._- .- ..- .rtfre&etfu$j The 6~_~ula~e,d,b~~.at_754 r+v$fh an .:. ‘. ‘. _,centraI ma~gncsium ator+& ; :. eontribute:. tti~o’~fr&elec; ,o~~~_?t~~~l~~~~~~.o~0:44 .is:dye,. ti, an e&+cG-r t&&.-::. -.- _:. .... ‘. T
Volume 26, number 1
CHEMICAL PHYSICS LETTERS
1 May 1974
! Table 2 Comparison of LCAO and FEMO methods as applied to chlorophyll a. Excitation for the principal bands Qv, Qy, aad B, of chlorophyll a Method
EQpm-‘)
fQ.Y
SCM0 PPP a)
15980 1.5170 15100
0.22 0.34 0.23
FEMO b, Exptl. a)
energies E and corresponding osciuator strengths,
EQ,mA
fQ_x
BB(cm-‘)
fB
17760 17020 17300
0.052 0.10 -
27330
1.92 0.81 1.1
23740 23260
a) Taken from ref. [2]. b, Calculated centers of gravity.
tion from the 14th to the 15th energy level. Thus this band will not be present in the spectrum if the 14th level is empty, which is the case if there are only 26 ?r electrons in the network.-The absence of this band from the experimental spectrum may be due to the insensitivity
of conventional
range of wavelengths.
in this
photomultipliers
In any circumstance
sults obtained are compruable to those of Weiss [2] with the model of four prbitals (table 2j. In order to verify the experimentally determined type. of bonding between the central magnesium atom and the four nitrogen atoms we repeat the calculation of the absorption spectrum of chlorophyll a assuming different
our calcu-
models
for the nitrogen-magnesium
bonding.
The results for
the different models tested are shown in table 3. In three models we have represented a weak nitrogen-magnesium bond by a free end attached to the nitrogen atom. Within the context of the free-elec-
lations suggest that either this band exists and will be found, or that otherwise the free-electron network of chlorophyll has 26 a electrons only. For the most important absorption bands, the re-
Table 3 Absorption spectrum of chlorophyll a. Calculated spectra by the FEN0 method for the five models of magnesium-nitrogen interactions shown below. The remaining parts of the free-electron networks are identical with the network of fig. lb. The X’s are the wavelengths of the various transitions and the f’s are the corresponding oscillator strengths Chl-a Expel
A (nm)
661 -
exlo-3
86.2 -
h(nm)
.f
h(nm)
f
Mm)
f
A(nm)
f
h(nnh
f
754+
0.44
824* 691
0.35 0.36
715*
0.52
713
0.50
721* 680
0.27 0.36
659 -
0.34 -
633*
645 -
0.05
._
-
-
-
-.
-
-
0.17 0.12 -
-
-,
-
587
0.23
._
_
-
553
0.07
566* 549
0.14 0.06
579 527
513
0.32
-
-
4.53. 443
-0.28
426
0.36 ._. _: -. _ :
,631 615
12.6
575
6.8
530
3.4
612* 607
0.20 0.13
548 -
0.06 0.32
.512 1 428
. . 113..
-
.409,
.71
-.
-
434 a15 4L1’
0.35 0.12 0133
:
___ ...
,-_ 4.
520 483 457.
-.
0.15
430. __
-. - : .:
-_._- .’ ‘.. a) Taken frorir ref. ill] (E: extindtioncoemcient); ‘_.:.. ’ ‘- .:I.’ ._“This bid disappears if the &e&o& has 26 Gratead 6f 28 i&&oonj:.--.
;
_-,
.._-, _-- ..
__.
.‘- :. ...._ _-- -:
+
.
._
0.12 .. 0.06 .O.lO ,520 0.11 0.19 -. 4.17 ‘0.11 _ _ 0.24. :- -, 441 0.16 0.03 -. :-
7 _
m_‘1 ..
-
:
... .-
._
550* .0.12 531 0.17 526 0.21 .477 :. 0.1% 452 ‘-- 0~13 ---
: -:432. :
.420 -
-0.13 .. 0.0s .. _
._ :., :
Volume 26, number 1
I May1974
CHEMICAL PHYSICS LETTERS
Table 4 Absorption spectrum of chlorophyll a. Spectra due to the majorsectionsof the free-electron network ofehkm@ylt a, as calculated by the FEW0 method. The h’s are tile wavelengths of the various transitions and thefs ate the conesponding o%illMor strength Chl-a b,
Chl-a exptl..a)
h (nm)
661
615 57.5 530
EXlO”
86.2
12.6 6.8
A(_)
f
754*
0.44
659
0.34
607
0.13
612* 548
i-4
0.32
428
113
409
71.
434 415 411
0.35 0.12 0.33
_
Porphine e,
A(nm)
A(nm)
f
A(um)
f
1149
i
--
539 531
I-
0.13
744
0.48
576
0.50
596
0.51
419 443 402
: 0.11
0.13
0.09 0.13
Taken from ref. Ill] (e: extinction coefficient). Calculated according to the model of fig. 2a. Calculated according to the model of fig. 2b. Calculated according to the model of fig. 2c.
e, *
This band disappears if the Chl-a network has 26 instead of28 P electrons.
according
950
0.08
0.17
a) k) c) d)
Calculated
0.08
782 0.60
-
0.06
512
Porphyrine cycle ef Chl-a d,
674
0.20
-
Chl-a without Mg ‘?I
-
481 416 -
0.49
465 450
0.13 0.83
391
0.28
0.10
to the model of fig. 2d.
tron model, this implies a partial mobility of the free electrons towards the magnesium atom. The model of column 2 is derived from the experimentally determined model of fig. I b. The results for the other models of bonding are presented in columns 3 to 6 and show that the FEMO method can distinguish between the different models of bonding and that only the absorption spectrum corresponding to the physical model.(column 2) agrees with experiment. To estimate the relative importance of the various sections of the chlorophyll molecule on the absorp tion spectrum, we present in table 4 the absorption spectra, calculated according to the FEMO method, of chlorophyll a (fig. 2a), chlorophyll a without the central magnesium atom (fig. 2b), the porphyrin-like cycle of chlorophyll (fig. 2c), and finally porphine (fig. 2d). The difference between columns 2 and 3 is a measure of the importance of the central magnesium atom; that between column 3 and.4 is due to the.two conjugated tails of the chlorophyti molecule, aud.fin- ‘. .’ Fig.12. Fredlectron r&works of: (a) chlorophyll a, (b):C& ‘ally .the difference between columns 4 and.5 measuies : a with&t its c&d mignesium, (c) porph$iin-$l$$yct& qf- -’ effect of the imperfection inthe porphyrin cyc!e.: -.:. ‘. ._ chl_:a, (4) porphir+~ .. :. ,;, ‘. -: .: .. .. .’
Volume 26. number I
CHEMICAL PHYSICS LEXTERS
1 May 1974
plying a chIorophylI/dioxane interaction_ Thus dioxane interacts with aggregates rather than individual molecules of chlorophyll. Further evidence to support this hypothesis comes from experiments with vapors of piperazine, morpholine, and piperidine. According to the interaction model presented here each dioxane molecuIe is bonded, through its two end oxygen atoms, to two chlorophyll moIecuIes. If this model is correct,
Fig. 3. Chl. a/pdioxane interaGtions:(a) model for the com(The Chl. a part not drawn is as shown in fig. i .)
pIex, (b) free-electron netwok
4. Absorption spectra of chlorophyll complexes 4-I_ Chloropltyllldioxatte
complex
The model proposed for the chlorophyll/dioxane interaction is shown in fig. 3a. The important features
of the model are that dioxane is bonded to the central magnesium atom, that it interacts with aggregates rather than single molecules of chlorophyll, and that the central magnesium atom has a coordination number of SIX in this complexed state. In the following we will justify these three features one by one. Experimentally no porphyrin-dioxane complexes exist [ 171. This strongly suggests that the dioxane molecule is not bonded to .the porphyrin cycle of chlorophyll. Hence it must be bonded to .the central magnesium atom. Furthermore the absorption spectrum ofchlorophyll is prackally ukhanged.when chlorophyll is dissolved in liquid dioxa& [ 181. This .&nplIt% the absence of an inter@ion between individ-. &xl &lorophyll &lecules &d’diox&e. On the other haqd, it @.ekperimentally observed-&t when dio&& yapoi-co_mes in co&t with chlorophyll iri the solid state [li] ,or in the monoinolecul~, itate [ 191 it displaces-the abso$ion band _?f the pigmeni, &Us im-. .i
the substitution of the two end oxygen atoms by two electron donors should lead to a similar interaction.1 Experimentally this is true in the case of piperazine and morpholine [ 191 where the two end oxygens are replaced by two nitrogens, and by one nitrogen and one oxygen, respectively. On the other hand, if the hypothesis that there is no interaction except in the aggregate state is true, then removing one of the end oxygens of dioxane, destroys the chIorophyll/dIoxane interaction_ Experimentally, vapors of piperidine (dioxane with end oxygens replaced by one nitrogen and one carbon) do not interact with chlorophyll [ 191. The third feature of the model concerns the coordination number of the central magnesium atom in the chlorophylI/dioxane complex. This is not yet settled experimentally, although various authors [ 12,201 have suggested that magnesiutm probably has a coordination number of SIX when chlorophyll is in the solid state or the monomolecular state. This would imply some sort of bonding of the magnesium atom to all of the four nitrogen atoms of the pyrrole cycle, in tiddition to its bonding to two dioxane molecules. To clarify this situation we test the three models of bonding shown in table 5. Model “a” assumes a coordination number of 4 for magnesium. Models “b” and “c” assume a coordination nurqber of 6 with two strong and two weak magnesium-nitrogen bonds for model “b” and four strong bonds for model “c”. The absorption spectra corresponding to the three models are presented in table 5. The calculations are based on the free-electron network of fig. 3b with the appropriate modifications in the case of each model. The FEMO m&hod is used aIi through-with an internuclear distance D = 1.4 &Comparison with &exierimental iesults clearly indicates that, in the &lo& phylI/dioxane’cotiplex, the cqorclm&i& number of .: the &&al magnesiti& atom-is SIX, qvith two strong “d two v&k tiagnesiGm%trogen bonds. This com@f-the chIor~ph$II/di~x&e-. :. ... .. pleies$he deter&&on interaction.&odel_.-As s&n f&m t&lfl;t$is x&&d&‘- ._-.. . . . . . .:- -;
CHEMICAL PHYSICS LETTERS
Volume 26, number 1
Table 5 Absorption spectrum of the Chl/dioxane complex. Calculated spectra by the FEMO method for the three models of Chl/dioxane interaction shown below. The remaining parts of the networks are identical with the network of fig. 3b. The X’s are the wavelengths of the various transitions and the fs are the corresponding oscillators strengths
Mg
coordination no. 4
Mg--N bonds-
Mg coordination no. 6.4 strong Mg-ti bondsh(nm)
Mgcoordination no. 6.2 strong
X(nm)
f
k(nm)
f
718 -
O-i3
743 688 676 -
0.14 0.28 0.14
659 656 -
0.14 0.27
606 544 509 -
0.15 0.07 0.26
630 548 511
442
-
f
-
-
645 -
0.05 -
585 545
0.09
481 480 453 422
0.17 0.09 0.23 0.10
0.20
-
leads to a displacement, caused by the interaction, of the whole spectrum of chlorophyll a towards the red; specifically the band Q-, calculated at 659 run is displaced to 688 nm as compared with an experimental displacement to 688-690 nm. An identical displacement of the absorption spectrum is obtained when a monolayer of chlorophyll is exposed to the vapors of piperazine or morpholine [20]. This agrees with the predictions of the free-electron model, since this model does not distinguish between dioxane, piperazine and morpholtie. 4.2.
ChlorophylZ/benzoqrtinone
1 May 1974
dioxane replaced by benzoquinone. The FEMO method, applied to the resulting network, then predicts a large bathochromic displacement of the absorption spectrum of chlorophyll a, with the absorption peak of the Q, band being calculated at 7 13 nm. The presence of a complex chlorophyll/quinone in vivo is not improbable, and could serve as the P-700 energy trap.
5. Discussion The use of the free-electron model is only justified in the case of those conjugated molecules for which the approximation of equivalence of all atoms is reasonably realistic. Among the chlorophylls, it is chlorophyll a, that has the least amount of heterogeneity. Thus even though the free-electron networks for the various chlorophylls do not differ appreciably, we would expect the general features of the resulting absorption spectrum to correspond most closely to that of chIorophyl1 a, and to resemble less and Iess the spectra of the other chlorophylls in order of increasing heterogeneity of their conjugated molecular structure. The treatment of the other chlorophylls would require a more sophisticated version of the FEMO method; one which accounts explicitly for the presence of heteroatoms. In considering the complexes of chlorophyll with dioxane or benzoquinone as a single free-electron network, the symmetry of the molecular structure permits us in as far as the FEMO calculation is concerned, to extend this network over only half a molecule of dioxme, or of benzoquinone, on either side of the chlorophyll a molecule. The resulting network is the same as that which would be obtained, in the unphysical situation, of one dioxane and one chlorophyll molecule interacting together, with the dioxane bent around the chlorophyll and attached to both sides of the central magnesium, thus forming a closed network.
complex
There are hardly any experimental results available on the chlorophyli/benzoquinone complex. But due to the similarities between dioxane and benzoquinone, it is reasonable to assume that the interaction model for both is the same. This leads to a molecular strutture, and result& free-electron network, similar to _ that shown in figs. ?a and 3b, respectively, but with
Acknowledgement
The computational phase of the carried out at the computer center du Quebec B Trois-R&i&es. One of like to thank Dr; Ky Toan Nguyen - puter prograniiniug. .. :
present work was of the UniversitC us cJ.L.B.) would fat help with com-
43
Volume
26, number 1
CHEMICAL PHYSICS LE’ITERS
References [ 1] C. Weiss. hf. Kobayashi and M. Goutennan, J. Mol. Spectxy. 16 (1965) 45 l_ [2] C. Weiss, J_ Mol. Spectry. 44 (1972) 37. [3] W.T. Simpson, J. Chem. Phys. 17 (1949) 1218. [4I H. Kuhn, J. Chem. Phys. 17 (1949) 1198. [S] 8. Kuhn and IV. Huber, Helv. Chim. Acta 42 (1959) 363. [6] T. Nakajima and H. Kon, J. Chem. Phys. 20 (1952) 750. [71 K. Ruedenberg and C. Scherr, J. Chem. Phys. 21 (1953) 156.5. (81 J-R. Platt, ed., Free-electron theory of conjugated molecules: a source book (Wiley, New York, 1964). [91 V. Czikkeiy, H.D. F&sterling and H. Kuhn, Chem. Phys. Letters 6 (1970) 11. [IO] E. Rabinowitch and Govindjee, Photosynthesis (Wiley, New York, 1969).
1 May 1974
[ li] C. HousSier a&l K. Sauer. J. Am. Chem. Sot. 92 (1970) 779. [I21 G. Sherman and E. Fujimori;Arch. Biochem. Biophys:130 (1969) 624. [I31 R.S. Mulliken and CA. Rieke, Rept. Pro&. Phys 8 (1941) 231. [14J N-S. Ham and K. Ruedenberg, J, Chem. Phys. 25 (1956) 7. [ISI J.L. Hoard,:M.J. Hamor and T.A. Hamor, J. Am. Chem. Sot. 8.5 (1963) 2334. [I61 E. Hiickel, Z. Physik 76 (1932) 628. 1171 B.B. Love and T.T. Bannister, Biophys. J. 3 (1963) 99. [IS] G-R. Seely and R.G. Jensen, Spectrochim. Acta 21 (1965) 1835. [I91 R.M. LebIanc, to be published. 1201 J.J. Katz, T.R. Janson, A.G. Kostka, R.A. Uphaus andG.L. Gloss, J. Am. Chem. Sot. 94 (1972) 2883.