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Internal Reorganization Energy Contributed by Torsional Motion in

By Allen Kennedy,2014-04-04 21:50
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In the theoretical estimation of internal reorganization energy contributed by the tortional motion between biphenyl and biphenyl anion radical,

Series of Selected Papers from Chun-Tsung Scholars,Peking University (2002)

Internal Reorganization Energy Contributed by Torsional

    Motion in Electron Transfer Reaction between Biphenyl and

    Biphenyl Anion Radical

    ?Min, Wei (闵玮) Sun, Lin (孙琳)

    Department of Applied Chemistry, College of Chemistry and Molecular Engineering,

    Peking University, Beijing 100871, China

In the theoretical estimation of internal reorganization energy contributed by the

    tortional motion between biphenyl and biphenyl anion radical, direct calculation

    in self-exchange electron transfer reaction was investigated in our present paper.

    With the introduction of a proper average bond length and angle parameters

    <bond Bp>, a multiple step relaxation Nelson method was developed to deal with

    the torsional reorganization energy. Based on the above model, an estimation of

    pure torsional reorganization energyλ with an approximation of λ was t,pt,1

    achieved. The results of 0.140 eV and 0.125 eV for torsional reorganization

    energy for a cross-reaction 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 al. from the

    rate measurements. This implies the efficiency and validity of our method to

    estimate the reorganization energy contributed by pure torsional motion of Bp.

Keywords multiple step relaxation Nelson method, internal reorganization

    energy, torsional motion, biphenyl molecule, electron transfer

Introduction

    Electron transfer (ET) reactions are found to be an elemental step in many

    1-6biological processes indeed. It has been believed that the experimental measurement of ET reactions between biphenyl (Bp) and a series of organic systems,

    designed by Miller et al., is the first successful experimental observation of ET

    7,8reactions in Marcus’ inverted region. In the process of intramolecular ET, the torsional movement of Bp has been expected to have a substantial effect on ET rates

    and affect the structure of the spacer and acceptor more strongly than in an

     ? E-mail: minw@chem.pku.edu.cn

    Accepted by Chinese Journal of Chemistry

    Project supported by the Hui-Chun Chin and Tsung-Dao Lee Chinese Undergraduate Research

    Endowment.

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    Series of Selected Papers from Chun-Tsung Scholars,Peking University (2002) intermolecular ET process.

    Neutral Bp is a very interesting molecule because of the existence of an

    approximately 45?torsional angle between the two phenyl rings. Slightly different

    results have been found to vary in the range of 38?~47?, depending upon the basis

     9sets used in calculations. Rubio et al. applied the geometry optimization using the

    complete active space self-consistent field method, and obtained a value of 44.3?for

    the torsional angle which is in agreement with the gas-phase electron diffraction (44.4

    10,11?1.2?). More recently, the torsional angleφ calculated at the MP2/6-311+G (2d,2p) level was 42.1?, while φ calculated using various density functionals with

    12different basis sets was very close to 40?. On the other hand, geometry

    -optimization indicates that all the carbon atoms in biphenyl anion radical (Bp) lie on

    13,14the same plane with an inter-ring torsional angle of 0?.

    BSN

    (-) N

    2FSN (-)

    Fig. 1 Frame skethes of BSN and 2FSN

Experiments have been performed using both biphenyl and fluorine as the electron

    donors (BSN and 2FSN systems) to examine the effect of the inter-ring torsional

    motion of Bp, because the former will undergo torsion of about 45?, whereas the latter remains planar in the ET process due to the existence of the tetrahedral carbon.

    Miller et al. found an eightfold change in the ET rate between systems with biphenyl

    and fluorine. This difference can be attributed to the additional reorganization energy

    15of 0.13 eV due to the torsional vibration of biphenyl. Such results lead to a conclusion that the torsion motion in these ET reactions which involve the biphenyl

    fragment will make the internal reorganization energy much higher than those of

    planar π-systems.

    In theoretical researches, in order to compare with the corresponding experimental

    results, models that are similar to BSN and 2FSN systems were chosen as the

    calculated molecules. However, results calculated by Li et al. revealed that the

    -difference between Mf (9,9-dimethylfluorene) /A and Bp/A that the torsional motion in biphenyl contributes an extra 0.32 eV to the internal reorganization energy,

    when compared with the rigid co-planar case, which is larger by 0.19 eV than the

     16value of 0.13 eV obtained by Miller et al. from the rate measurements. However, beside comparing the difference of the internal reorganization energies of

    two relating intramolecular ET processes, few theoretical papers exist that directly

    calculate the internal reorganization energy contributed by torsional movement of Bp

    through one electron self-exchange reaction. Nelsen ever obtained the internal

    reorganization energy for self-exchange ET reactions using the AM1 semiempirical

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    Series of Selected Papers from Chun-Tsung Scholars,Peking University (2002)

    17,18molecular orbital method. In the present paper, with some consideration of

    detailed dynamic process, a multiple step relaxation Nelson method was applied to

    deal with the torsional reorganization energy directly.

    Calculation method

    Marcus relation

    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

    development of electron transfer theory, it was emphasized that the solvent

    contributions come from modes of such low frequencies that they can be treated with

    classically, but the internal reorganization energies usually come from skeletal

    vibrations having high frequencies which often require a quantum mechanical

    19,20treatment.

    Marcus ever presented a method whereλ is based on the reorganization of the i21, 22 intramolecular degrees of freedom, assuming harmonic approximation.

    RPffjj2 (1) ???q?ijRPf?fjjj

    According to this, the internal reorganization energyλ is the sum over all 3N-6 iRPvibrational modes. f and f are the force constants of the reactants and the products, jjPRrespectively. Δq is the change in bond lengths Δq=?q-q?.The drawback of jjjjthis method is the unambiguous assignment of the vibrational modes and the force

    23constants to the bond length changes. Another reason might be that the time scale of

    electron self-exchange does not allow complete normal mode vibrational motions of

    -24the neutral Bp and Bp anion radical during the electron transfer. Therefore the Marcus relation seems to be not valid for molecules like Bp, where torsional angle

    changes, as well as bond length and bond angle changes, are the main contribution to

    the geometrical reorganization.

    Nelson method

    A different approach, called Nelsen method, is to describe the internal

    17,18reorganization energy by thermodynamic consideration. For a self-exchange

    - reaction, we can optimize the equilibrium geometries for Band B, and thus, obtain

    -the corresponding equilibrium energies E(B)and E(B). On the other hand, it can be eqeqcalculated the energy E(B) for neutral molecule B at the equilibrium configuration n-eq---16of B, but E(B) for anion radical B at the equilibrium configuration of B. Thus n-eq

    the internal reorganization energyλ can be obtained from Eq.(2) i---λ(B/B) = E(B)+ E(B)E(B)E(B) (2) in-eqn-eqeqeq--The internal reorganization energy of a cross-reaction D + A?D + A

    can be approximated as an average value,

    --- λ(D/A)=[λ(D/D) +λ (A/A)]/2 (3) iii

Multiple step relaxation Nelson method

     419

    Series of Selected Papers from Chun-Tsung Scholars,Peking University (2002) Nelsen method has been proved to be applicable and efficient in the internal

    24reorganization energy evaluation. However, when the torsional reorganization

    energy needs to be separated from the whole internal reorganization energy, such as in

    this paper, the concrete dynamic relation of the torsional 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 internal

    reorganization energy of Bp-contained systems theoretically.

    For instance, if we assume that the torsional motion takes place first and then the

    -bond length and bond angles changes for Bp and Bp in a self-exchange reaction, the torsional reorganization energy can be described as:

    λ=λ(bond Bp,φ, anion?bond Bp, co-plane, anion) t,1t --+λ(bond Bp, co-plane, neutral?bond Bp,φ, neutral) (4) t

    whereλ(bond Bp,φ, anion?bond Bp, co-plane, anion) is obtained by varying the t

    torsion angle of Bp to the co-plane structure and keeping its other bond length and

    --angle unchanged, similarlyλ(bond Bp, co-plane, neutral?bond Bp,φ, neutral) is t - the torsion barrier from the Bpto the torsion conformation without variation of other

    bond lengths and angles.

    -Hereλ(bond Bp,φ , anion?bond Bp, co-plane, anion) and λ(bond Bp, tt-co-plane, neutral?bond Bp,φ, neutral) can be expressed as following, respectively,

    λ(bond Bp,φ , anion?bond Bp, co-plane, anion) t

    = EE (5) φ(bond Bp,, anion)(bond Bp, co-plane, anion) --λ(bond Bp, co-plane, neutral?bond Bp,φ, neutral) t

    = EE (6) φ(bond Bp-, co-plane, neutral)(bond Bp-,, neutral)

    It is found that the potential value of equilibrium E is very high. This φ(bond Bp,, anion)is because that the negative charge of anion is quite not preferred not only by the φ

    angle, but also by the equilibrant bond lengths and angles of neutral Bp. So Eφ(bond Bp,,

    anion) can be evaluated approximately

    E=ΔE(φ?? anion)+ΔE(bond Bp??anion) (7) φ(bond Bp,, anion)

    whereΔE(φ?? anion) stands for the potential rise due to the interaction of φ

    angle and negative charge of anion whileΔE(bond Bp??anion) is the potential rise due to the incompatibility of negative charge of anion and equilibrant bond lengths

    and angles of Bp. Analogous analysis can also be applied to E, E(bond Bp, co-plane, anion)(bond

    Bp-, co-plane, neutral) and E, respectively, φ(bond Bp-,, neutral)

    Here, we approximately regardΔE(bond Bp??neutral), ΔE(φ??neutral), Δ

    -E(bond Bp??φ), ΔE(bond Bp??anion), ΔE(anion??co-plane) andΔ

    -E(co-plane??Bp) as 0. Obviously, λ in this case can be expressed as t

    λ =ΔE(φ?? anion)ΔE(bond Bp??co-plane) t,1-+ΔE(co-plane??neutral)ΔE(bond Bp??φ) (8) On the contrary, if we assume that the bond lengths and bond angles changes take

    -place first and then the torsional movement for Bp and Bp in a self-exchange reaction, the torsional reorganization energy would be described as:

    --λ=λ(bond Bp,φ, anion?bond Bp, co-plane, anion) t,2t

    +λ(bond Bp, co-plane, neutral?bond Bp,φ, neutral) (9) t

    Following the same treatment above,λ can obtained under this assumption t

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    Series of Selected Papers from Chun-Tsung Scholars,Peking University (2002)

    - λ =ΔE(φ??anion) +ΔE(bond Bp??φ) t,2

    +ΔE(co-plane??neutral) +ΔE(bond Bp??co-plane) (10)

    -However, in Eq (8) and Eq (10),ΔE(bond Bp??co-plane) andΔE(bond Bp??φ)

    both come from the interaction of torsional movement and bond length and angle

    changes, and should not be included in the pure contribution of torsional motion to

    internal reorganization energy. This implies that considering the low-frequency

    torsional movement and high-frequency bond length and angle 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 introduced

    ΔE(φ??<bond Bp>) =ΔE(co-plane??<bond Bp>) (11) An imagined and detailed ET model in order to calculate torsional reorganization

    energy (see Fig. 2) is shown considering the following steps:

    Potential Value

    λt,p+=λ2λ5Neutral Bp