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85033064815
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note
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1 on the bond lengths.
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28
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85033063822
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note
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Others quantities describing the molecular geometry, such as in plane and out of plane bond angles, are simply not considered in the semiempirical model of Eq. (3).
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31
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0039147403
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The form of the electron-lattice coupling described by the semiempirical Hamiltonian used by G. König and G. Stollhoff, Phys. Rev. Lett. 65, 1239 (1990) is also such that polaron or soliton excitations cannot be supported. We disagree with the conclusions reached by these authors regarding the origin of the dimerization in conjugated system; indeed our results provide strong confirmation for the validity of the Peierls mechanism. We believe that the conclusions reached by König and Stollhoff regarding the relative unimportance of electron-phonon effects is due to their failure to independently fit the results of their ab initio calculation to the SSH Hamiltonian.
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edited by R. J. H. Clark and R. E. Hester Wiley, New York
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For recent studies regarding the vibrational structure of polyenes, see M. Gussoni, C. Castiglioni, and G. Zerbi, in Spectroscopy of Advanced Materials, edited by R. J. H. Clark and R. E. Hester (Wiley, New York, 1991), pp. 251-353;
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41
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85033044450
-
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note
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m for the ground state of neutral rings are proven in Ref. 13.
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43
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0004332272
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NAG, Oxford
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1) is represented by a P X P matrix. For N>6, if only the lowest few eigenvalues are sought, the Rutishauser algorithm implemented in the NAG Fortran Library - see NAG Fortran Library Manual Mark 16, (NAG, Oxford, 1993) - is considerably more efficient than Householder reduction to tridiagonal form followed by the QL algorithm;
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NAG Fortran Library Manual Mark 16
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44
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0003595562
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Springer, Berlin
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for this procedure, see B. T. Smith, J. M. Boyle, B. S. Garbow, Y. Ikebe, V. C. Klema, and C. B. Moler, Matrix Eigensystem Routines-EISPACK Guide (Springer, Berlin, 1976).
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Smith, B.T.1
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Klema, V.C.5
Moler, C.B.6
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46
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0000203351
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Similar behavior occurs within other independent electron schemes, see V. Bach, E. H. Lieb, M. Loss, and J. P. Solovej, Phys. Rev. Lett. 72, 2981 (1994).
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Bach, V.1
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47
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85033036656
-
-
note
-
m,s(=0,1) are the occupation numbers. Moving one electron from the Mth to the (M + 1)th state has the same effect on this sum as removing both the electrons present in the Mth state.
-
-
-
-
48
-
-
85033042616
-
-
note
-
u state.
-
-
-
-
49
-
-
85033050905
-
-
note
-
g state (for N = 4, 6, and 8) is the one obtained moving one of the two electrons from the Mth to the (M+2)th level.
-
-
-
-
50
-
-
85033042455
-
-
note
-
u state contains two charged solitons (Ref. 4) so that excesses of charge are concentrated at the solitons locations rather than being spread over the chain.
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51
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85033058279
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note
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0/5.
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59
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(a) L. Serrano-Andrés, R. Lindh, B. O. Roos, and M. Merchán, J. Phys. Chem. 97, 9360 (1993);
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(b) L. Serrano-Andrés, M. Merchán, I. Nebot-Gil, R. Lindh, and B. O. Roos, J. Chem. Phys. 98, 3151 (1993);
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ACES II, an ab initio program system, authored by J. F. Stanton, J. Gauss, J. D. Watts, W. J. Lauderdale, and R. J. Bartlett, Quantum Theory Project, University of Florida.
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81
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85033059853
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note
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0/5 so that the examples shown in these figures are representative of the predictions of the semi-empirical treatment for "realistic" values of the phenomenological parameters.
-
-
-
-
82
-
-
85033057723
-
-
note
-
A similar discrepancy is found in longer chains between the predictions of the SSH model and the results of the earlier ab initio calculations of Ref. 44. The discrepancy can be viewed as a result of the fact that while both calculations predict solitonic bond geometries, the two solitons are somewhat closer to the center of the chain in the SSH formulation. See Ref. 4 for details.
-
-
-
-
83
-
-
85033070704
-
-
note
-
m vanish [see Eq. (4)] and all bonds have the same length.
-
-
-
-
84
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-
0003877758
-
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Chemical Rubber, Boca Raton, Chap. 9
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See, CRC Handbook of Chemistry and Physics, 75th ed., edited by D. R. Lide (Chemical Rubber, Boca Raton, 1995), Chap. 9.
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R. M. Gavin, C. Weisman, J. K. McVey, and S. A. Rice, J. Chem. Phys. ibid. 68, 522 (1978).
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88
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-
85033067033
-
-
note
-
2.
-
-
-
-
89
-
-
85033046888
-
-
note
-
(i) as many times as this bond length occurs in the actual molecules. For example, the term containing the benzene bond length was assigned weight 6 [e.g., it appears six times in the sum of Eq. (6)], the term containing the butadiene outerbond was assigned weight 2 and the term containing the butadiene midbond was assigned weight 1.
-
-
-
-
90
-
-
0040326493
-
-
u state, it seems doubtful that the framework adopted by Jin and Silbey captures correctly the physics of the problem.
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Jin, B.Y.1
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91
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85033036130
-
-
note
-
0 falls within a couple of percents of 1.52 Ä in all the cases that we have explored.
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-
-
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