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Coherent potential theory for interacting bands: Phonons and excitons in substitutionally disordered molecular crystals
dc.contributor.authorHong, Hwei‐kwanen_US
dc.contributor.authorKopelman, Raoulen_US
dc.date.accessioned2010-05-06T23:05:38Z
dc.date.available2010-05-06T23:05:38Z
dc.date.issued1973-03-15en_US
dc.identifier.citationHong, Hwei‐Kwan; Kopelman, Raoul (1973). "Coherent potential theory for interacting bands: Phonons and excitons in substitutionally disordered molecular crystals." The Journal of Chemical Physics 58(6): 2557-2568. <http://hdl.handle.net/2027.42/71029>en_US
dc.identifier.urihttps://hdl.handle.net/2027.42/71029
dc.description.abstractA coherent potential approximation (CPA) theory for disordered molecular solids with interacting bands is reported here. This theory has a wide range of applications. Various examples of interacting bands can be cited, such as electronic states coupled via vibronic or spin‐orbit couplings, vibrational states with degeneracies in the gas phase or coupled by Fermi resonance, triplet magnetic sublevels coupled via exciton interactions, and phonons in general. The theory is developed using the self‐consistent condition with a single‐site and single‐band approximation. In particular, two approaches are adopted. In the first approach, a self‐energy is assigned for each subband. In the second approach, a common self‐energy is assumed for all the subbands. The two different approaches require different inputs to the theory. In one case, the entire dispersion relations of the pure system are called for; in the other, only the partial density‐of‐states functions for each degree of freedom are needed. It is also shown that in the limit of infinite dilution, the formalism reduces to the proper single‐impurity levels within the single‐band approximation.en_US
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dc.publisherThe American Institute of Physicsen_US
dc.rights© The American Institute of Physicsen_US
dc.titleCoherent potential theory for interacting bands: Phonons and excitons in substitutionally disordered molecular crystalsen_US
dc.typeArticleen_US
dc.subject.hlbsecondlevelPhysicsen_US
dc.subject.hlbtoplevelScienceen_US
dc.description.peerreviewedPeer Revieweden_US
dc.contributor.affiliationumDepartment of Chemistry, University of Michigan, Ann Arbor, Michigan 48104en_US
dc.description.bitstreamurlhttp://deepblue.lib.umich.edu/bitstream/2027.42/71029/2/JCPSA6-58-6-2557-1.pdf
dc.identifier.doi10.1063/1.1679538en_US
dc.identifier.sourceThe Journal of Chemical Physicsen_US
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dc.identifier.citedreferenceNotice that we have included the diagonal elements in H0.H0. For an isolated band, it is usually possible to write the mixed crystal Hamiltonian as (see Ref. 16), H  =  D+W,H=D+W, where D contains only the diagonal elements and W contains only the off‐diagonal elements. Furthermore, the pure crystal Hamiltonians of both A and B (HA0HA0 and HB0HB0) contain W:HA0  =  ϵA1+W,W:HA0=ϵA1+W, and HB0  =  ϵB1+W.HB0=ϵB1+W. Thus, it is possible to diagonalize both HA0HA0 and HB0HB0 with the same set of Bα(k,j).Bα(k,j). This is not true for the case of interacting bands. Now, HA0  =  εA+W,HA0=εA+W, HB0  =  εB+W.HB0=εB+W. Although εAεA and εBεB still contain only diagonal elements, they are no longer simple constant multiples of the unit matrix. The Bαf(k,j)Bαf(k,j)’s are different for components A and B. In other words, the density‐of‐states functions for A and B are not congruent to each other. Under these circumstances, it is more convenient to decompose the mixed crystal Hamiltonian as we do in Eq. (3.3a). The Bαf(k,j)Bαf(k,j) used throughout later sections is understood to be associated with the host (A) and not the guest (B). The same thing is true for phonons discussed in Sec. IV.en_US
dc.identifier.citedreferenceS. D. Colson, R. Kopelman, and G. W. Robinson, J. Chem. Phys. 47, 27, 5462 (1967); G. W. Robinson, Ann. Rev. Phys. Chem. 31, 429 (1970).en_US
dc.identifier.citedreferenceHere, we have put ⟨n,α,fRn,α,f⟩  =  Rnαf.⟨n,α,fRn,α,f⟩=Rnαf. It should be pointed out that RnαfRnαf is independent of n, α. We can define Rf ≡ Rnαf.Rf≡Rnαf. Thus, when we take the trace of the Green’s function, we have TraceG  =  ∑n∑α∑f⟨n,α,fRn,α,f⟩  =  Nσ∑fRf,TraceG=∑n∑α∑f⟨n,α,fRn,α,f⟩=Nσ∑fRf, where σ is the total number of molecules per primitive cell. Similarly, the over‐all density‐of‐states function g(E) is given by g(E)  =  (1/NσF)Img(E)=(1∕NσF)Im TraceG  =  (1/F)∑fImRf,TraceG=(1∕F)∑fImRf, where F is the total number of degrees of freedom.en_US
dc.identifier.citedreferenceThere appears to be some confusion with regard to different definitions of self‐energies by different authors, particularly those defined in Refs. 6, 16, and the present paper. Direct comparison is impossible because we are dealing with multibands and, as explained in Ref. 23, we have to absorb the diagonal terms into H0H0 also. However, in the limit of one band only our definitions of self‐energies are related to those of other authors by Uf  =  ∑1−ϵA,Uf=∑1−ϵA, Uf  =  ∑2+CAϵA+CBϵB−ϵA  =  ∑2+CBΔf,Uf=∑2+CAϵA+CBϵB−ϵA=∑2+CBΔf, where ∑1∑1 is used in Ref. 16; ∑2∑2 is used in Ref. 6. In other words, from Eq. (3.11), we have also Uf′  =  ∑2.Uf′=∑2. Care must be taken in comparing our results with those in Ref. 16. For example, using the above relations, Eq. (3.13) is found to be equivalent to ∑1  =  CAϵA+CBϵB+CACBΔf2/(Rnαf−1−CAϵB−CBϵA+∑1),∑1=CAϵA+CBϵB+CACBΔf2∕(Rnαf−1−CAϵB−CBϵA+∑1), which is given in Ref. 16 as ∑1  =  ϵ+CACBΔf2/(Rnαf−1+ϵ+∑1),∑1=ϵ+CACBΔf2∕(Rnαf−1+ϵ+∑1), because by their definition, ϵA  =  −ϵBϵA=−ϵB and so −CAϵB−CBϵA  =  CAϵA+CBϵB  =  ϵ.−CAϵB−CBϵA=CAϵA+CBϵB=ϵ.en_US
dc.identifier.citedreferenceTheir formulation is more general. In principle, it can be applied to cases where no interchange symmetry exists. However, as we point out in the text, their assumptions are questionable.en_US
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dc.identifier.citedreferenceFor some theoretical calculations see Ref. 10. Experimental work on molecular crystals using inelastic neutron scattering were reported by P. A. Reynolds, J. K. Kjems, and J. W. White, J. Chem. Phys. 56, 2928 (1972), G. Dolling and B. M. Powell, Proc. Roy. Soc. (London) 319, 209 (1970), and others.en_US
dc.identifier.citedreferenceFor example, see A. A. Maradudin, E. W. Montroll, G. H. Weiss, and I. P. Ipatova, Solid State Phys. Suppl. 3, 65 (1971).en_US
dc.identifier.citedreferenceSee Ref. 34, p. 361.en_US
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dc.identifier.citedreferenceSee, for example, calculations by D. A. Oliver and S. H. Walmsley, Mol. Phys. 17, 617 (1969); also assignments by M. Ito and T. Shigeoka, Spectrochim. Acta 22, 1029 (1966).en_US
dc.owningcollnamePhysics, Department of


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