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Variational Approach to the Davydov Soliton 6411 PHYSICAL REVIEW A VOLUME 38, NUMBER 12 DECEMBER 15, 1988 Variational approach to the Davytiov soliton Qing Zhang Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Victor Romero-Rochin Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Robert Silbey Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received 14 December 1987; revised manuscript received 9 June 1988) The soliton states suggested by Davydov as approximate solutions to the Schrodinger equation for one-dimensional systems are examined from the standpoint of the variational principle. We show that the correct equations of motion are, in general, difFerent from those proposed by Davydov. In addition, we comment on the time evolution of these states. I. INTRODUcmxON q =e iqnagq=(g q}n an excitation on In a classic study of energy transport in one- In these forms, a„t(a„) create (destroy) ele- dimensional exciton systems with exciton-phonon cou- molecule n, J is the resonant-energy-transfer matrix ment between nearest-neighbor molecules, pling, Davydov' has proposed a soliton as a particular- p„and u„are mole- ly efficient transport state. Experimentally, Scott and the momentum and displacement operators of the Careri et al. have exanuned crystalline acetanilide and cule at site n, bq (b }create (destroy) a quantum of vibra- have discussed the results using Davydov's model. Alex- tional (phonon} energy of wave vector q with frequency ander and Krumhansl have also examined this system coq, ni is the mass of the molecule, k the intermolecular theoretically. In this paper, we concentrate on the constant, and finally y(yq ) is the exciton-phonon cou- theoretical situation, in particular, the derivation of the pling constant. The local [Eq. (la)] and extended [Eq. dynamical equations. We derive a new set of equations, (lb)] forms of H are related by different in some respects from Davydov's equations, and ' 1/2 show that these repair some of the problems of the latter. e iqna( b t (2a) Q~= q q Davydov's method is to guess a functional form for the 2Nm coq trial wave function which has a number of parameters. ' 1/2 Am coq By identifying these parameters as "momenta" conjugate eiqna(bt — (2b) to other "coordinates" in the problem, he is able to write 2N q q the equations of motion for these parameters in the form = of the classical Hamilton equations. This procedure has q =~g sin&g~ totz 2&tr/ttt (2c) ' been criticized Brown et al. ' and shown to have 1/2 by 2A sinqa flaws. In this paper, we examine this problem anew, and xq=-tx (2d) mNcoq ACOq by using the variational principle of time-dependent mechanics ' we derive corrected equations of quantum Davydov suggested two trial wave functions which we motion for the parameters in Davydov's trial wave func- call, following Brown et al. , ~D, (t) ) and ~Dz(t) ) given tions. by The system consists of molecules on a one-dimensional — lattice with nearest-neighbor separation a. The Hamil- ~D, (t)}=g A„(t)a„exp g[Pq„(t)b, P;„(t)bq] ~0) tonian for this system is 2 (3a) H= ea„a„+ a„+a„a„+,)+ g J(a„+, 2m and — — = —— + —,'tt(u„+, u„)z+ya ta„{u„+, u„,) {la) ~Dz(t) } g A„(t)a„exp g[y (t)p n m or, more generally, H= +[ca„a„+J(a„+,a„+a„a„+,)] (3b) + +fico btb + gyqfttoqa a (bq+b q) (lb) Note that the phonon part of ~Dz(t) } is independent of q, n the site of the excitation n, while the phonon part of where ~D&(t)) depends on n The next .step in the Davydov's 38 6409 1988 The American Physical Society 6410 QING ZHANG, VICTOR ROMERO-ROCHIN, AND ROBERT SILBEY 38 procedure is to identify ifiA„' and i AP»„as the momenta variational method described by Langhoff et al. based conjugate to A„and P»„ for ~D, ) and iAA„' and II„as on the Frenkel variational method. the momenta conjugate to A„and y„ in ~D2). Finally, Since we are using the variational method, we must in- by identifying the Hamiltonian function as the expecta- sert the condition that the trial state vector be normal- tion value of H over either ~D, ) or ~D2), Davydov as- ized at all times, i.e., serted that the evolution of the parameters is described by Hamilton's equations. This leads for ~D, ) to — Thiscondition for either ~D&) or ~D2) reducesto in'„= (D)' IHID,' ) = (H ), , aA„* =aA„' g A„'(t) A„(t)=1 . iAA'=— n „&H), , We therefore introduce a constraint into the variation, n (4a) Eq. (6), which now becomes i AP»„= &H&, , iAP»„=— &H&, , qn where the role of the Lagrange multiplier A, is discussed below. and for to ~D2) We may now derive equations of motion for the pa- rameters A„, A„, P»„, and P»„ in ~D& ) and A„, A„, y„, if&A„= (H)z, aA„' and II„ in ~Dz) by substituting the form of these trial state vectors for ~f) in Eq. (9) and performing the varia- iAA'=— tions with respect to these assumed independent parame- n „&H&, , n ters. The equations which replace Eq. (4) are (4b) a&a), Vn „(a),, =i' n BA„' rr n &a), . Vn — A, A„, From these starting points and making some further ap- q proximations, Davydov obtained his soliton solutions. ' B&H&, dA„' ' Recently, Brown et al. have shown that the two as- 8Aq dt sumptions of this method [(i) the form of the state vector and (ii) the equations of motion] are not equivalent to the d — +i~A «y —P«„'"„d —A. A P n Schrodinger equation. In particular, these authors show 2 '" dt '"„ dt q that the time evolution of ~D, ) or ~D2) found from (10a) Schrodinger's equation is different from that implied by B(H), dP (4a) and (4b). That is, these authors found that „d'(A„A„) ~4;(t) ) =e ' ' ~D;(0) ) is different from ~D;(t) ). In this article, we use the variational principle of quan- a&a &, dP,'„ tum mechanics to derive new equations of motion for the i fi A„A„'—,'ifi13» „—( A„—'A—„), parameters of the trial state vectors. By doing this we avoid the second of Davydov's assumptions while keep- and ing the first. We will comment on the form of the trial states later in this article. B(H) dA„1 dy dII "+y—11 ™ dt 2 )— dt dt II. VARIATIONAL PROCEDURE a&a&, The Schrodinger equation, BA„dt a Hly) =i~—ly&, at dy dII 11 — A„—A, A„ 2 dt 1 dt can be derived from the principle of least action (lob) a&a&, dr'. (6) err„ di a&a), dH„ if we allow and to be varied independently with ~P) (P~ ay. dt the requirement that the variations at the end points van- ish (see Appendix A). This procedure is equivalent to the In these equations, we have defined the following: 38 VARIATIONAL APPROACH TO THE DAVYDOV SOLITON 6411 out that the solutions of these and &H &, =&D, IHID, & equation. They point = Davydov's equations differ only by a time-dependent g EOA„' A„+ g QA„A„fico p'„p „ phase factor, so that although the soliton solution would q n still be obtained, differences in some of their predictions + g A„*A„fico (yqp „+yq'p „) compared to those of Davydov may occur. q, n We can examine the general validity of the form of ID, (t) & and ID&(t) & by looking at limiting cases. There A„'( + g A„+,J„+A„,J„), are two limits in which the dynamics of the problem can n be solved exactly: (a) J=0, general yq, i.e., no exciton where J„,J„are defined as motion and (b) yq =0, general J, i.e., no exciton-phonon interaction. ' —I'—' IP— —,I I'), exp ~ J„=J g (Pq, „Pq,„+i , q, , Pq, „+ In the case J =0, Brown et al. showed that q Davydov's approach applied to either ID, & or ID&& failed to reproduce the exact dynamics. We now show J =Jexpg(Pq„Pq„—t iIPq I iIPq, „—&I q that our approach, embodied in Eqs. (10) applied to ID, & =Jn-i yields the exact dynamics in this case. By setting J =0 in Eqs. (10a},we get so A „+ g Acoq (Pq„Pq„+gq Pq„+yq 'Pq„) = g A„'[(so+8')A„+J(A„+(+A„ i) q —— =i' A„—+i AA„Q '(P'„P '„) A, A„, dj , —„P„P where d(A„'A„) 8'= ' 1 —, g —II„+s(y„—y„ i) ficoq A& A (Pq&+g~& ) i%A& A&Pq&+ &'ifiP n dt We emphasize that the above set of equations is con- (12) sistent with quantum mechanics. Whether their solution, By multiplying Eq. (11) by A„' and then subtracting the for the initial condition ID, (0) & or ID&(0) &, agrees with result from its conjugate, it is easy to show that the exactly propagated state depends absolutely on the proposed structure of the trial state vector. d Iraq (A„*A„)=0, i.e., A„'(t)A„(t)=const, (13) If, for example, the trial state vector were given as a dt sum of a complete set of states of the Hilbert space of the Hamiltonian 0, with arbitrary coeScients, the set of so that Eq. (12) becomes equations obtained with the present method (which iris') A„'A„(P „+yq)=A„'A„ilPq„, (14) would have the form of Hamilton's equations) and those obtained by using the Schrodinger equation and the whose solution can be written down immediately: orthogonality condition of the basis mentioned, would differ the inclusion of the Lagrange multiplier.
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