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A. S. DAVYDOV and N. I. KISLURHA: Excitons in Molecular Chains 465 phys. stat. sol. (b) 59, 465 (1973) Subject classification : 13.5.1 Academy of Sciences of the Ukrainian SSR, Institute for Theoretical Physics, Kiev Solitary Excitons in One-Dimensional Molecular Chains BY A. S. DAVYDOV and N. I. KISLUKHA The change in time of the state of intramolecular excitation in an one-dimensional molecular chain is investigated, where its deformability is taken into account. It is shown that the excitation which is accompanied by a local deformation of the chain, can move uniformly along the chain, the size of the region under excitation remaining constant. These excitations in one-dirnsnsionsl molecular chains can be called particle-like exc.citona or solitary excitons. Es werden die zeitlichen hderungen des intramolekularen Anregungszustandes in einer ein-dimensionalen molekularen Kette untersucht, wobei deren Deformierbarkeit beriick- sichtigt wird. Es wird gezeigt, daB die Anregung der Kette, die durch eine lokale Deforma- tion begleitet ist, sich gleichformig langs der Kette bewegen kann, wobei die GroBe des angeregten Bereichs konstant bleibt. Diese Anregungen in ein-dimensionalen Molekiilketten konnen partikel-ahnliche Exzitonen oder solitare Exzitonen genannt werden. 1. One-Dimensional Molecular Chain The stationary states of the electron excitation taking crystal lattice defor- mation into account were considered in [I, 21. The change of such a state in time is of certain interest. The present paper is devoted to this problem. Consider a one-dimensional infinite molecular chain. The Hamiltonian of the electron excitation in the Heitler-London approximation has the form (see [3]) : H = 2 [E + D(n)l BZ Bn + 2' M l n ( T I - r n ) B,f Bn , (1) n h n where E is the excitation energy of an isolated molecule; D(n) = 2 Dzn P I - r n ) 9 N+n) D L , (rl - rn) is the change in the interaction energy between molecules 1 and n, when the molecule n is excited; X I , (rz - r,) are the matrix elements of the resonance interaction; r,, is the coordinate along the chain of the molecule n ; B:, B are the creation and annihilation operators of the electronic excitation over the molecule 12. The kinetic energy T of the molecules of the chain is 2 T =-.($), m 2 , where m is the mass of the molecule. In the nearest-neighbour approximation the potential energy of the chain is u = z u(R,) 9 (3) n 466 A. S. DAVYDOV and N. I. KISLUKIIA where Rn = rn+l - rn > 0 are the intermolecular distances; C A R: Ri u(R,) = ---, C , A , s, q > 0 are the constants defining the interaction of unexcited molecules; s > 4 . In the ground state 10) (without intramolecular excitations) the chain pos- sesses periodicity and the intermolecular distances obtained from the condition of a minimum of potential energy (3) are equal to s c W-d R = ( a ) * (4) In the state Iy) when the intramolecular excitation is present, the intermolec- ular distances R, are different from R and the potential energy of the chain changes by the value AU, as compared to the energy of the chain in the ground state. In the harmonic approximation AU = 2 [4&) - u(R)I x w 2 9 ( 5 ) 3, = - @?I 9 lenl < f (6) A where Taking into account (6), the formula (2) for the kinetic energy of the molecules can be transformed to T = =if[+ n Flpj. (8) Then the total energy E of the deformation of the chain is written as 2. The Nearest-Neighbour Approximation In the nearest-neighbour approximation the operator (1) takes on the form H 1 Z { [ E + D(Rn)] BR Bn + M(,Rn) [B,' B n i I + B$+I Bn1) . (10) n In (10) we adopt Dn-l,n(Rn-1) + Dn+l,n(Rn) D(Rrt) 7 Mn,n+l(Rn) Jf%+I,n(Rn) = N(Rn) * Let where a, p > 0 are the parameters of theory. considered below) In the case of a dipole-dipole resonance excitation (and only this excitation is b M(R, ) =-. R: Solitary Excitons in One-Dimensional Molecular Chains 467 We assume that the dipole moment of transition, d is oriented along the chain. Then in ( 1 2 ) b = - 2 da. Suppose that the following inequality is valid for distances close to R : where D r - D(R) > 0; M 3 - M ( R ) > 0. Using (6), (ll), and ( 1 2 ) ) D(Rn) and M(Rn) in (10) can be expanded near the point R . In D(Rn) we keep only the term linear in en, and in M(R,) the zero term, i.e. we set D(R,) x x - (1 + p e n / R ) D, Y ( R n ) = - M . Then (10) takes the form D > J f , (13) H = 2 { [ E + ( 1 + P en/R) Dl B$ - [BZ &+I + B,ft1 Bn]} . (14) n The operatorz of the excitation energy of the chain with its deformation energy taken into account reads The solutions of the Schrodinger equation X = H + E . (15 ) a i h - \Y(t)) = ;7e I!P(t)> at are looked for in the form provided The operator (15) is the function of a set of the values of en. Moreover, these values are dependent on the state l!P(t)). In the region of the chain where the density of the probability of finding the excitation la,(t)I2 is great, one should expect greater deformation of the chain which is defined by the values of en. From the condition of a minimum of the total excitation energy (Y(t)lY(t)) at every fixed moment of time t , and the fact that aenli3t is independent of en, we find @ - - 2 e (an(t)(2 . " - 2 w R Substituting (19) into (la), we obtain H = ,Z { [E - DI B,+ B, - M [Bnf Bn+1 + B2+1 BnI - G Ian(t)12 B,+ Bn) 9 (20) n where G = p 2 D2/2 w R2 is the parameter of nonlinearity. Further, according to (9) and (19) where Now the equation (16) takes the form 468 A. S. DAVYDOV and N. I. KISLUKHA where e = E - D f E . Using ( 2 2 ) and the fact that the system BklO) is complete and orthogonal, we obtain the equation for an(t) ih= + M [ ~ , + i ( t ) + ~ , - i ( t ) ] - e a n ( t ) + G l a n ( t ) I 2 a , ( t ) = 0 . (23) at In (23) we go over to a long-wave approximation and replace the differences by the derivatives. Thus a,@) 3 q(5, t ) , q(n, t ) = 4) > (24) where E = r / R is the dimensionless coordinate. Then the differential-difference equation (23) transforms to 3. The Particle-Lilre Excitation The normalized solution of the nonlinear equation (26) has the form of a soli- tary wave plv( f , t ) = -- _ _ _ _ _ - ~ cash [p (5 - 5 0 - t ) ] 7 where mz = h2/2 M R2 is the effective mass of the excitation in the chain with rigidly fixed molecules; m$ = m G2/24 M w R2 is the addition to the effective mass which is caused by a local deformation. + + v t . Consequently, the parameter defines the velocity V = v R of the excita- tion motion along the chain. To every value of v there corresponds an excitation energy of the chain, A Q. The excitation localized near the point to corresponds to the value v = 0. Its energy E - D - 2 M - Ga/48 M is less by the value G2/48 M than the excitation energy of the chain with rigidly fixed molecules. According to (28), the excitation is localized in the neighbourhood = Solitary Excitons in One-Dimensional Molecular Chains 469 Using (19), we find that in the excitation region the equilibrium distances be- tween molecules decrease by the value Thus, an appreciable change in the distances between the molecules, which is proportional to the parameter p, spreads over the region around the point to + v t . This region travels with constant speed along the chain. The average distance between the molecules in the excitation region is Therefore, when the inertia and deformability of a chain is taken into account, the motion of the excitation is accompanied by a local distortion of the chain and the velocity V of an excitation decreases, as the effective mass (29a) in- creases. Unlike the states of motion of free particles described by wave packets, the excitation state corresponding to the function (28) does not "spread" in the course of time. These excitations in one-dimensional molecular chains can be called particle-like excitons, or solitary excitons. The long-wave approximation used by us requires that the inequality A t > 1, i.e. p = G/4 M < z is satisfied. On the other hand, for the particle-like excitons the following inequality should be valid : where N is the total number of molecules in the chain; N > 1. transformed to the form of a simple harmonic wave At the very small values of the parameter p (p Q 1) the expression (25) can be Consequently, in this limiting case the excitation is practically nonlocalized. nal perturbation @ ( E , t ) which satisfies the inequality Excitations such as (28) can arise in the chain under the influence of an exter- We have not taken into account the processes of relaxation of the arising excitation. Phenomenologically the relaxation processes can be taken into ac- count by replacing the excitation energy (26) by the complex value hA'2 - i h y. Then an exponentially decreasing factor exp (- y t ) appears in the function (28) determining the average life-time (lly) of the excitation. 470 A. S. DAVYDOV and N. I. KISLUKHA: Excitons in Molecular Chains References [l] A. S. DAVYDOV, phys. stat. sol. 36, 211 (1969). [2] A. S. DAVYDOV, J. theor. Biol. 38, 559 (1973). [3] A. S. DAVYDOV, Theory of Molecular Excitons, Izd. Nauka, Moscow 1968; Plenum Press, (Received July 17, 1973) New York 1971.