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Volumn 104, Issue 8, 1996, Pages 2987-3002

Liquid theory for the instantaneous normal modes of a liquid. II. Solutions

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Indexed keywords


EID: 0000780932     PISSN: 00219606     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.471120     Document Type: Article
Times cited : (19)

References (88)
  • 26
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    • edited by J.-L. Martin, A. Migus, G. A. Mourou, and A. H. Zewail Springer-Verlag, Berlin
    • D. McMorrow, S. K. Kim, J. S. Melinger, and W. T. Lotshaw, in Ultrafast Phenomena VIII, edited by J.-L. Martin, A. Migus, G. A. Mourou, and A. H. Zewail (Springer-Verlag, Berlin, 1993).
    • (1993) Ultrafast Phenomena VIII
    • McMorrow, D.1    Kim, S.K.2    Melinger, J.S.3    Lotshaw, W.T.4
  • 33
    • 85033036119 scopus 로고    scopus 로고
    • note
    • Some hints that a spectrum of couplings can be calculated using species-projected densities of states are provided by Schvaneveldt and Loring, Ref. 6, in their recent work on the vibrational line shape of a solvated diatomic. In particular, their INM prescription for the generalized-Langevin-equation friction is highly suggestive of a spectrum of couplings.
  • 37
    • 5844256546 scopus 로고
    • edited by J. Szudy Ossolineum, Wroclaw
    • G. Seeley and T. Keyes, in Spectral Line Shapes, edited by J. Szudy (Ossolineum, Wroclaw, 1989), Vol. 5, p. 649;
    • (1989) Spectral Line Shapes , vol.5 , pp. 649
    • Seeley, G.1    Keyes, T.2
  • 45
    • 85033035288 scopus 로고    scopus 로고
    • The actual integral is over both real and imaginary frequencies, so it is not literally a cosine transform
    • The actual integral is over both real and imaginary frequencies, so it is not literally a cosine transform.
  • 47
    • 4043083965 scopus 로고
    • Our general approach to constructing sum rules for moments of densities of states is described in B.-C. Xu and R. M. Stratt, J. Chem. Phys. 91, 5613 (1989). The application there is to liquid-state electronic band structure.
    • (1989) J. Chem. Phys. , vol.91 , pp. 5613
    • Xu, B.-C.1    Stratt, R.M.2
  • 50
  • 51
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    • B.-C. Xu and R. M. Stratt, Ref. 21
    • B.-C. Xu and R. M. Stratt, Ref. 21;
  • 54
    • 0005248832 scopus 로고
    • It is shown in the Appendix of Z. Chen and R. M. Stratt, J. Chem. Phys. 97, 5687 (1992) that the replica trick leads to correctly normalized averages even though the replicated expressions have no denominators.
    • (1992) J. Chem. Phys. , vol.97 , pp. 5687
    • Chen, Z.1    Stratt, R.M.2
  • 55
    • 5544286423 scopus 로고
    • For a review of statistical-mechanical methods for handling liquids with internal degrees of freedom, see R. M. Stratt, Adv. Chem. Phys. 78, 1 (1990).
    • (1990) Adv. Chem. Phys. , vol.78 , pp. 1
    • Stratt, R.M.1
  • 58
    • 85033042673 scopus 로고    scopus 로고
    • note
    • There is an ambiguity in paper I that can lead to the idea that the w bonds only have a symmetry factor of 1/2 if the forked end impinges on a black circle. This idea is incorrect; the w bonds always carry the extra symmetry factor. In the text of Sec. III C of paper I, reference is made to Fig. 8 along with the statement that the bonds in that figure carry a symmetry factor of 1. The actual calculation in the paper correctly includes the factor of 1/2 on the w and ψ bonds.
  • 61
    • 85033056201 scopus 로고    scopus 로고
    • The term "vertex contraction" is defined and the process illustrated in paper I
    • The term "vertex contraction" is defined and the process illustrated in paper I.
  • 62
    • 0012677204 scopus 로고
    • and by Refs. 7 and 8
    • Because of the translational invariance of a liquid, instantaneous normal modes must conserve momentum exactly, even in approximate theories. This requirement is discussed by Y. Wan and R. M. Stratt, J. Chem. Phys. 98, 3224 (1993) and by Refs. 7 and 8.
    • (1993) J. Chem. Phys. , vol.98 , pp. 3224
    • Wan, Y.1    Stratt, R.M.2
  • 63
    • 85033060594 scopus 로고    scopus 로고
    • S. J. Schvaneveldt and R. F. Loring (preprint)
    • S. J. Schvaneveldt and R. F. Loring (preprint).
  • 64
    • 0000235403 scopus 로고
    • The parameters given here are the same ones used by Stassen and Gburski (Ref. 9)
    • M. F. Pas and B. J. Zwolinski, Mol. Phys. 73,471 (1991). The parameters given here are the same ones used by Stassen and Gburski (Ref. 9).
    • (1991) Mol. Phys. , vol.73 , pp. 471
    • Pas, M.F.1    Zwolinski, B.J.2
  • 65
    • 85033051762 scopus 로고    scopus 로고
    • We note, for completeness, that Stassen and Gburski truncated their forces and interaction tensors at a distance equal to half their system length
    • We note, for completeness, that Stassen and Gburski truncated their forces and interaction tensors at a distance equal to half their system length.
  • 67
    • 0003474751 scopus 로고
    • Cambridge University, London
    • For 108-particle diagonalizations, we used the Householder elimination routine, tred2, to generate a tridiagonal matrix, followed by the QL algorithm diagonalization routine tqli, from W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, London, 1994).
    • (1994) Numerical Recipes in C, 2nd Ed.
    • Press, W.H.1    Teukolsky, S.A.2    Vetterling, W.T.3    Flannery, B.P.4
  • 69
    • 85033050661 scopus 로고    scopus 로고
    • note
    • In test cases we found that the numerical results from the theory are sensitive to the position of the peak of the radial distribution function. The extremely rapidly varying character of even the renormalized potentials apparently makes the evaluation of the integrals appearing in the mean-field theory a rather delicate undertaking.
  • 70
    • 85033034884 scopus 로고    scopus 로고
    • note
    • st for points between the bin centers. It was found that a simple 3-point interpolation gave the same results as a 5-point interpolation.
  • 71
    • 85033055672 scopus 로고    scopus 로고
    • note
    • We use the routine newt, from Press et al (Ref. 39) for the root search. Because the root search is complex, we found it useful to assess the accuracy of the finite-difference calculation of the Jacobian matrix by checking that the Cauchy-Riemann equations were satisfied.
  • 72
    • 85033065751 scopus 로고    scopus 로고
    • note
    • The actual root search was performed on the equation obtained by subtracting both sides of Eq. (3.23) from X. Because the right-hand side is itself a function of λ-∑(λ), this modification saved a few complex additions and subtractions with each iteration. We should note that the root search occasionally converges to the unphysical branch of solutions, but this root can be made a physical root by changing the sign of ε, and multiplying Eq. (3.5) by a corresponding negative sign.
  • 73
    • 85033039666 scopus 로고    scopus 로고
    • note
    • Kr.
  • 74
    • 85033048793 scopus 로고    scopus 로고
    • note
    • It was found that the average eigenvalues from snapshots separated by more than five intermediate snapshots were statistically independent. This method of averaging blocks of data to estimate errors is discussed in Allen and Tildesley (Ref. 38), p. 192.
  • 75
    • 85033045514 scopus 로고    scopus 로고
    • note
    • s, would also produce such a collapse in our examples, but we are unaware of any real justification for this alternative.
  • 76
    • 85033058542 scopus 로고    scopus 로고
    • The resulting truncated distribution must then be scaled so that it is correctly normalized (Ref. 16)
    • The resulting truncated distribution must then be scaled so that it is correctly normalized (Ref. 16).
  • 78
    • 3342940332 scopus 로고
    • This kind of low frequency peak, one that is associated with a small number of heavy impurities, is reminiscent of the localized, low-frequency modes seen in simulations of soft-sphere glasses by B. B. Laird and H. R. Schober, Phys. Rev. Lett. 66, 636 (1991);
    • (1991) Phys. Rev. Lett. , vol.66 , pp. 636
    • Laird, B.B.1    Schober, H.R.2
  • 83
    • 0005295963 scopus 로고
    • Y. Wan and R. M. Stratt (Ref. 34)
    • Z. Chen and R. M. Stratt, J. Chem. Phys. ibid. 97, 5696 (1992); Y. Wan and R. M. Stratt (Ref. 34).
    • (1992) J. Chem. Phys. , vol.97 , pp. 5696
    • Chen, Z.1    Stratt, R.M.2


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