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In hitherto published papers [l to 41 the method used for the calculation of the energies of localized excitations in a linear polymer chain with a point defect has been described in inverse space. The transition into inverse space is made possible by the redefinition of the starting Hamiltonian. In the present note it will be shown that, if the redefinition of the Hamiltonian is performed and then only the operators of host molecule excitations are transferred into inverse space, we obtain also an equation which determines the energies of localized excitations. In that way, in the case of one point defect, a 2 x 2 determinant is obtained, instead of a 3 x 3 determinant obtained by the procedure used in earlier papers [l to 41.We shall consider a linear polymer chain containing a single substitutional impurity molecule, which is translationally and orientationally dislocated. Under the influence of light (transverse photon field), in the system considered collective and localized excitations arise. The collective excitations are Frenkel excitons [5]. The Hamiltonian of this polymer chain system, written in direct space, is given as [l to 51 H = C AfBSBn, + 1 (04 K m IfO> B$Bmf + AgB:Bsg n * s n*, m + s where A , is the excitation energy of the host molecules, B: and B, (n +s) are Bose operators which create and annihilate, respectively, the excitation on the host molecules, B: and Bs are Bose operators which create and annihilate, respectively, the excitation on the guest (substituted) molecule, whereas f denotes the excited state of the host molecule and g the excited state of the guest molecule.In order to obtain the Hamiltonian of an ideal polymer chain, of course, as a part of the Hamiltonian (l), we add to and subtract from the Hamiltonian (1) the following operator terms [l to 41: A,B,:B,f + c (04 v,, Ifo) [B,:B,f + ~n f B . 2 . n*s ') P. 0. Box 60,
In hitherto published papers [l to 41 the method used for the calculation of the energies of localized excitations in a linear polymer chain with a point defect has been described in inverse space. The transition into inverse space is made possible by the redefinition of the starting Hamiltonian. In the present note it will be shown that, if the redefinition of the Hamiltonian is performed and then only the operators of host molecule excitations are transferred into inverse space, we obtain also an equation which determines the energies of localized excitations. In that way, in the case of one point defect, a 2 x 2 determinant is obtained, instead of a 3 x 3 determinant obtained by the procedure used in earlier papers [l to 41.We shall consider a linear polymer chain containing a single substitutional impurity molecule, which is translationally and orientationally dislocated. Under the influence of light (transverse photon field), in the system considered collective and localized excitations arise. The collective excitations are Frenkel excitons [5]. The Hamiltonian of this polymer chain system, written in direct space, is given as [l to 51 H = C AfBSBn, + 1 (04 K m IfO> B$Bmf + AgB:Bsg n * s n*, m + s where A , is the excitation energy of the host molecules, B: and B, (n +s) are Bose operators which create and annihilate, respectively, the excitation on the host molecules, B: and Bs are Bose operators which create and annihilate, respectively, the excitation on the guest (substituted) molecule, whereas f denotes the excited state of the host molecule and g the excited state of the guest molecule.In order to obtain the Hamiltonian of an ideal polymer chain, of course, as a part of the Hamiltonian (l), we add to and subtract from the Hamiltonian (1) the following operator terms [l to 41: A,B,:B,f + c (04 v,, Ifo) [B,:B,f + ~n f B . 2 . n*s ') P. 0. Box 60,
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