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CHINESE JOURNAL OF CHEMISTRY 2002, 20, 963-967 963
Internal Reorganization Energy Contributed by Torsional Motion in Electron Transfer Reaction between Biphenyl and Biphenyl Anion Radical
m, Wei*(FI$) SUN, Lin(#4+) Department of Applied Chemistry, College of Chemistry and Molecular Engineering, Peking University, Reijing 100871 , China
. . cmcerning the theoretical estimation of internal energy a m t r i i by the to16onal motion between biphenyl and biphenyl anion radical, direct calculation of self-exchange electroo trader d o n was investigated. With the introduc- tion of a proper average bond length and angle parameters cboodBp>, a multiple step reladion Nelson method wsp
developed to deal with the torsional reorganhtion energy. Basedon the above model, an estimation of plre t o m re-
achieved. The results of 0.140 and 0.125 eV of t o r s i d reor- ganization energy for a c m s d o n at the levels d4-31G and
of0.U eV obtained by Miller et d. piom the rate meaaue-
eStimatethe reoqgdzationenergyamtriibypue t o m motion of Bp.
organidon energy At,p with an appImimation d AtJ wsp
m m , rrspectively, are in good agN!mۦt W i t h the value
ments. This implies the efficiency and validity ofour method to
Keywordr multiple step relaxation Nelson method, internal re- organbation energy, torsional motion, biphenyl molecule, electron bander
Introduction
Electron transfer (ET) reactions are found to be an elemental step in many biological processes indeed. " It has been believed that the experimental measurement of ET reactions between biphenyl (Bp) and a series of or- ganic systems, designed by Miller et al. , is the first suc- cessful experimental observation of ET reactions in Mar- CUS' inverted ~ g i ~ n . ~ , * ~n the process of intramolecular
ET, the torsional movement of Bp has been expected to have a substantial effect on ET rates and affect the struc- ture of the spacer and acceptor more strongly than in an intermolecular ET process.
Neutral Bp is a very interesting molecule because of the existence of an approximately 45" torsional angle be- tween the two phenyl rings. Slightly different results have been found to vary in the range of 38"-47", depending upon the basis sets used in calculations .' Rubio et al . applied the geometry optimization using the complete ac- tive space self-consistent field method, and obtained a value of 44.3" for the torsional angle which is in agree-
ment with the gas-phase electron e t i o n ( 44.4 i 1.2") . 'O*" More recently, the torsional angle fi calculat- ed at the MP2/6-311+ G(2d,2p) level was 42.1", while fi calculated using various density functionals with Mer- ent basis sets was very close to 40". l2 On the other hand, geometry optimization indicates that all the carbon atoms in biphenyl anion radical ( Bp- ) lie on the same plane with an inter-ring torsional angle of o". 13*14
Ekperiments have been performed using both biphenyl and fluorine as the electron donors (BSN and 2FSN systems, Fig. 1) to examine the &ect of the inter- ring torsional motion of Bp, because the former wi l l un- dergo torsion of about 45", whereas the latter remains pla- nar in the ET p'ocess due to the existence of the tetrahe- dral carbon. Miller et al . found an eightfold change in the ET rate between biphenyl and fluorine systems. 'Ihis
* E-mail: [email protected] Received January 7 , 2002; revised May 22, 2002; accepted June 25, Uwn. Project supported by the Chinese Research Endowment of Hui-Chun Chin and Tsung-Dao Lee.
964 Reorganization energy MIN & SUN
difference can be attributed to the additional reorganiza- tion energy of 0.13 eV due to the torsional vibration of biphenyl. Is Such results lead to a conclusion that the tor- sion motion in these ET reactions which involve the biphenyl fraeplent wi l l make the internal mqpnimtion energy much higher than that of planar x-systems.
EEg. 1 FnuneakcthesofBSNand2FSN.
In theoretical researches, in older to compare with the compon+ experhntal results, models that are similar to BSN and 2FSN systems were chosen as the cal- culated molecules. However, results calculated by Li et
d. revealed that the difference between Mf- (9,9'- dimthyUuorene)/A and Bp-/A that the tomional motion in biphenyl contributes an extra 0.32 eV to the internal mrganhtion energy, when compared with the rigid co- planar case, which is largerbyO.19 eV than the value of 0.13 eV obtained by Miller et al , firm the rate measure- ments. l6
However, beside comparing the Merence of the in- ternal reorganidon energiea of two relating intramolecu-
calculating the internal reoqanhion eneqg contributed by tomiod movement of Bp h u g h one electmn self-ex- change reaction. Nelsen ever obtained the i n d mrga- d o n energy for self-exchange El' reactions using the AML semiempirical molec~lar orbital method. 17*18 ~n the present paper, with some considemtion of detailed dy- namic pmceas, a multiple step relaxation Nelson method was applied to deal with the torsional reorganhion ener- gy directly.
lar In pFocesses, few theore!tical papers exist that directly
Calculation method
Chemists have made attempts to gain insight into the reaction mechanism and to predict the rate constant by calculating the necessary dynamic parameters. In the de- velopment of electmn transfer theory, it was emphasized that the solvent contributions come h m modes of such low fkquencies that they can be treated with classically, but the internal mxpnht ion energies usually come fmm skeletal vibdons having hi& hquenciea which often xe- quire a quantum mechanical treatment. l 9 v m
Marcus ever presented a method where Ai is based on the reorganization of the intramolecular degrees of h- dom, assuming b n i c appmximation.21*p
According to this, the internal reorganization energy A i is the sum over all 3N-6 vibrational d e a . jjR and 6' are the force constants of the reactants and the pducts , respectively. Aqj is the change in bond lengths Aqj = I Q -qj I . The drawback of this method is the unam-
biguous assignment of the vibrational modes and the force constants to the bond length change^.^ Another reason might be that the time d e of electmn self-exchange does not allow complete normal mode vibrational motions of the neutral Bp and Bp- anion deal during the electmn t m m k ~ . ~ 'Iherefore the arcu us relation seems to be not valid for molecules like Bp, where tomional angle changes, as well as bond length and bond angle changes, are the main contribution to the geometrical m e - tion.
P R
Nelson methad
A different a p p m h , called Nelsen method, is to describe the intemal mqanh t ion energy by t h e d y - namic consideration. "?l8 For a self-exchange reaction, we can optimize the equilibrium geometries for B- and B, and thus, obtain the corresponding equilibrium energies E,(B-) and E,(B). On the other hand, it can be
Vol. 20No. 10 2002 Chinese Joumal of Chemistry 965
calculated the energy E,,(B) for neutral molecule B at the equilibrium configuration of B- , but E,, (B- ) for anion radical B - at the equilibrium configuration of B . l6 ?bus the internal reorganization energy A i can be obtained from Eq. ( 2 ) .
The internal reorganization energy of a cross-reaction D- t A+D + A - can be approximated as an average value,
Mul.tiple step m&n Nelson method
Nelsen method has been proved to be applicable and efficient in the internal reorganization energy evaluation . ?A
However, when the torsional reorganization energy needs to be separated hrn the whole internal reorganization en- ergy, such as in this paper, the concrete dynamic relation of the tomional motion, and bond length and bond angle changes should be taken into account, which can not be achieved in single step relaxation Nelson model. It is also difficult in treating with intemal reorganization energy of Bp-contained systems theoretically.
For instance, if we assume that the torsional motion takes place fmt and then the bond length and bond angles
change for Bp and Bp- in a self-exchange reaction, the torsional reorganization energy can be described as:
A , , I = A,(bond Bp, #, anion-bond Bp, co-plane, an- ion) + At(bond Bp- , co-plane, neutral+bond Bp- , 8 , neutral) (4)
where At(bond Bp, 8 , anion-bond Bp, *plane, an- ion) is obtained by varying the torsion angle of Bp to the co-plane structure and keeping its other bond length and angle unchanged, similarly I,(bond Bp-, co-plane, .neutral-bond Bp- , 8 , neutral) is the torsion barrier hrn the Bp- to the torsion confomtion without variation of other bond lengths and angles.
Here At(bond Bp, 8 , anion-tbond Bp, co-plane, anion) and A t ( bond Bp- , co-plane, neutral + bond BP- , # , neutral) can be expressed as following, respec- tively.
At(bond Bp, 8 , anion-bond Bp, co-plane, anion)
= E(bnd ~ p , C. anion) - E(tmwi ~ p , c o - p b , anion) ( 5 )
It is found that the potential value of equilibrium
B ~ . +, do,,) is very high. This is because that the negative charge of anion is quite repulsed not only by the # angle, but also by the eqdhrant bond lengths and an- gles of neutral Bp. So E ( m Bp, +, anion) can be evaluated appmximately .
.
where AE ( #+ -+ anion) stands for the potential rise
due to the interaction of 8 angle and negative charge of anion, while AE (bond Bp- +anion) is the potential rise due to the incompatibility of negative charge of anion and eqdbmnt bond lengths and angles of Bp. Analogous analysis can also be applied to E ( b o n d ~ p , -plane, mion) 9
E(bd Bp- , -place. neutral) and E(bond Bp- , I , neuhal) 9 re- spectively.
Here, we approximately regad AE (bond Bp- - neutral), AE( 8+ +neutral), AE(bond Bp- -$I, AE(bond Bp-+ +anion), AE(anion+ +co-plane) and AE( ceplanec +Bp- ) as 0. Obviously, A , in this case can be e x p r e s s e d as:
A t , l = AE ( $ + + anion) - AE (bond Bp+ -co- plane) + AE( ceplane- +neutral) - AE( bond Bp-+ -9) (8)
On the contmy, if we assume that the bond lengths and bond angles changes strictly take place earlier then the torsional movement for Bp and Bp- in a self-exchange reaction, the torsional reorganization energy would be de- scribed as:
A1,2= Al(bond Bp- , 8 , anion-bond Bp- , co-plane, anion) + A t ( bond Bp, -plane, neutral+ bond Bp, $, neutral) (9)
Following the same treatment above, A , can be obtained
966 Reorganization energy MIN & SUN
under this assumption:
&,2 = AE(#+ +anion) + AE(bond Bp-+ d#) t AE(co-plane+- -+neutral) t AE(bond B p t +
co-plane) ( 10')
However, in Eqs . (8) and ( 10) , A E (bond Bp+- +co- plane) and AE(bond Bp-+ +#) both come h the interntion of torsional movement and bond length and an- gle changes, and should not be included in the pure con- tribution of torsional motion to internal reorganization en- ergy. This implies that considering the low-fmpency tor- sional movement and high-frequency bond length and an- gle changes to be separated strictly is not convincing.
In principle, in order to reduce the interaction of torsional movement and bond length and angle changes to 0, proper average bond length and angle parameters < bond Bp > which meets the following equation can be induced :
A E ( # + + < b o n d B p > ) = AE(co-planec +<bond Bp> ) (11)
An imagined and detailed ET model in order to calculate torsional mrganimtion energy (Fig. 2) is shown consid- ering the following steps.
Reaction path
Fig. 2 Potumal energy curvea ofthe ETnection Bp-/Bp.
. To the relaxed neutral molecule (Nng) one electron is added. The energy difference in the neutral geometry (Ang) is equal to the vertical electmn affiity ( vEA) . Then, the anion changes its bond lengths and angles to < bond Bp > , yielding the relaxation energy A 1 i then its torsional angle changes from # to co-plane without any other variations, and this step yields h 2 ; at last, its bond
lengths and angles change fmm < bond Bp > to (bond Bp- ) and yield h3. Removing the electron without geom- etry relaxation results in the neutral molecule in the anion geometry ( Nag) correspondmg to the vertical ionization potential (VIP). Similarly the relaxation of the molecule to the neutral geometry (Nng) results in energy A4 + A s + A 6 . "berefore, once through the geometry optimization the <bond Bp> which suits for Eq. (11) can be ob- tained, A 2 t A s , the sum of both pure torsional reorgani- zation energies in the neutral molecule and in the anion radical, can be regarded as the calculation result of inter- nal reorganization energy contributed by torsional motion of Bp in a self-exchange reaction.
Estimation of torsional reorganization energy
Another simple appruach to estimate the contribution of torsional motion may exist. Comparing Eqs . ( 11 ) and ( 13) , the pure torsional reorganization energy A t, can be obtained by averaging the two hl , l , At,2 in Eqs. (8) and ( 10)
A t , ' + k . 2 L p = 2 (12)
Torsional potential barriers of 17.64 kJ-mol-' at the 4- 31Glevel and 14.86 kJ*mol-' at the DZP level (dun- ning' s 996p/Ss3p basis sets with polarization functions on all the atoms) have been obtained by varying the torsion angle of Bp to the co-plane structure and keeping the oth- er bond parameters unchanged. Similarly the torsion bar- rier fmm the Bp- to the torsion confomtion has been found tobe36.11 and32.16kJ*mol-'atthelevelsof4- 31G and HP/DZP, mpectively.p*25 As defied above, the value of A1,2 is the sum of these two torsional potential h e r s , and it is 53.75 and 47.02 kJ-mol-' at the lev- els of 4-31G and HP/DZE', respectively. In the case of A 1, 1 , in fact, the difference between AE( #+- + anion) and AE(bond Bp- +co-plane) is small, which is also
true between AE (co-plane+ +neutral) and AE (bond Bp-f + # I . As a result, Al,l can be simply regded aa 0, and thedore torsional reorganization energy of self- exchange reaction of h ,, is about 27 Id * mol - ' at the 4- 31G level and 24 kJ- mol-' at the HP/DZP level. Ac- cording to Eq. ( 3 ) , the i n t e d reorganization energy contributed by torsional motion of Bp is about 13.5 kJ * mol-' (or about0.140 eV) and 12 kJ*mol-' (or about
Vol. 20 No. 10 2002 Chinese Journal of Chemistry 967
0.125 eV) for a cross-reaction at the levels of 4-31G and HP/DZP, respectively.
Conclusions
'Ihe results of 0.140 and 0.125 eV for torsional re- organization energy at the levels of 4-31G and HP/DZP, respectively, are in good agreement with the value of 0.13 eV obtained by Miller et d . from the cross-reaction rate measurements , lS and better than indirectly calculation conclusions by comparing the internal reorganization ener- gy difference between two relating ET reactions. 16*25-28
'Ihis implies the efficiency and validity of OUT method to estimate the reorganization energy contributed by pure tor- sional motion of Ep. 'Ihe data suggest that the low-fre- quency torsional movement strictly takes place later than high-frequency bond length and angle changes and it may be ill-considered in evaluation of low-frequency torsional reorganization energy. Investigating long-distance in- tramolecular ET reactions that depend critically on these low-frequency torsional motions will need to address these issues to estimate the effect of torsional motions on solvent reorganization energy, internal reorganization energy and ET rates. 29-33
References
P e w , E. G. ; May, V . J , Phys. C h . A 2001, 105, 10176. Olofsson, J.; Larsson, S . J . Phys. C h . B 2001, 105, 10398. Ito, H . ; Nakatsuji, H. J. Camp. C h . 2001, 22, 265. Porath, D. ; Bezryadin, A . ; de Vries, S. ; Dekker, C. Na- ture 2000, 403, 635. Lower, S. K. ; Hochella, M. F. ; Beveridge, T. J . Science m1,292,1360. Win, I. A.; Omchic, J . N. Science#)oo, 290, 114. Miller, J. R . ; Calcatena, L. T . ; Qw, G . L. J. Am. C h . soc.1984,106,3047. Clw, G . L. ; calcaterra, L. T . ; Green, N. J . ; Penfield, K. W.; Miller, J . R. J . Phys. C h . 1986, 90, 3673. T s d , S.; Tanabe, K. J . Phys. Chem. 1991, 95. 139.
10
11
12
13
14
15
16
17
18
19
20
21 22
23
24
25
26 27
28
29
30
31
32
33
Rubio, M . ; Merchan, M . ; Orti, E. h r . Chim. A d a 1995, 91, 17. Ahenningen, A.; Bastiansen, 0.; Femholt, L. , Cyvin, B. N . ; Cyvin, S. J . J . Mol. strud. 1985, 128, 59. Arulmozhiraja, S.; Fujii, T. J . Chem. Phys. 2001, 11S, 10589. Takei, Y.; Yamaguchi, T . ; Osamura, Y. ; Fuke, K . ; Kaya, K. J . Phys. chem. 1988, 92, 577. B&, G.; P o d , 0. J . Phys. C h . 1989, 91, 2153. . Miller, J. R . ; Paulson, B. P . ; B a l , R . ; Qm, G . L. J . Phys. Chem. 1995, 99, 6923. Li, X. Y . ; Tong, J . ; He, F . C. C h . Phys. u)o, 260,283. Nelsen, S. F . ; Blackstock, S. C . ; Kim, Y. J. Am. Chem. Soc. 1981, 109,677. Nelsen, S. F . ; Yunta, M . J. R. J. Phys. Org. C h . m, 7, 55. S i h , P . ; Marcus, R. A . J. Am. C h . Soc. lB1, 103, 748. Sidem, P . ; Marcus, R. A . J. Am. C h . SOC. 1981, 103, 741. Marcus ,R.A. J. C h . Plrys.1956, 24,%6. Marcus, R. A.; Sutin, N. Biochim. Bwphys. A d a 1985, 811, 265. I i , X . Y . ; X i a o , S . Q . ; H e , F . C . S c i . C h i m , S r . B: Chem. 1999, 42, 441. Keltemr, A. M.; Landgraf, S.; Grampp, G . Spectrachim. A d a , P m A 2001, 57, 1959. L i , X . Y.; Tang, X. S.; Xiao, S. Q . ; He, F. C. J. Md. strud. 1995, 342, 181. Li, X. Y.; He, F. C . J . Camp. C h . 1999, 20,597. Z h a n g , X . D . ; W a n g , . Y . N . ; C u o , J . X . ; m , Q . Y . J. Photochm. Photobwl. A 1999, 121, 1. &, J. X.; Zhang, Q. Y. J . Photochan. Photobid. A 1997, 110, 247. Kumar, K.; Kumikov, I . V . ; Beratan, D. N . ; Waldeck, D . H . ; Z i m m t , M . B . J . Phys. b. Am, 102, 5529. Min, W.; Sun, L . h P h y s . - C h i m . Sin. 2001, 17, 924 (in Chinese). Davis, W. B . ; b, M. A . ; Wmielewski, M. R . J. Am. chon. soc.2001,123,87l. Skokov, S.; Wheeler, R. A. J . Phys. chsn. A 2000, 104, 6314. hng, Y. P.; McHale, J . L. J . Chem. Phys. 1997, lcb, 4%3.
(EOu)1075 LI, L. T . ; Z-ENG, G. C.)
