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Volumn 109, Issue 5, 1998, Pages 1670-1686

Variational calculation of macrostate transition rates

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EID: 0242454276     PISSN: 00219606     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.476882     Document Type: Article
Times cited : (14)

References (101)
  • 10
  • 17
    • 22244483490 scopus 로고    scopus 로고
    • note
    • Throughout the paper, subscripted Greek letters identify macrostates and are unrelated to regular Greek letters such as the inverse temperature β. Summations over these macrostate indices are over all macrostates.
  • 47
    • 0004086603 scopus 로고
    • The problem is similar to that of determining the relationship between the molecular wave functions and the localized basis sets in quantum chemistry. However, the quantum chemistry problem is simply solved (e.g., by MO-LCAO approximation) because the localized basis functions are roughly factorizable in terms of atomic orbitals [Hartree-Fock-Ruthaan approximation; W. A. Benjamin, New York This is not true for complicated multidimensional systems where the localized conformational macrostates are not factorizable in terms of the individual atomic coordinates
    • The problem is similar to that of determining the relationship between the molecular wave functions and the localized basis sets in quantum chemistry. However, the quantum chemistry problem is simply solved (e.g., by MO-LCAO approximation) because the localized basis functions are roughly factorizable in terms of atomic orbitals [Hartree-Fock-Ruthaan approximation; R. G. Parr, The Quantum Theory of Molecular Electronic Structure, (W. A. Benjamin, New York, 1964)]. This is not true for complicated multidimensional systems where the localized conformational macrostates are not factorizable in terms of the individual atomic coordinates.
    • (1964) The Quantum Theory of Molecular Electronic Structure
    • Parr, R.G.1
  • 48
    • 22244436321 scopus 로고    scopus 로고
    • note
    • α̇ J [P]dR, where P is the projection operator defined in Eq. (24). The current generated by the eigenfunction components that are outside the subspace corresponds to fast relaxation of the probability distribution within, not between, macrostates.
  • 50
    • 22244485761 scopus 로고    scopus 로고
    • note
    • β, resulting in an unbounded spectrum of relaxation rates which would lead to a singularity in the rate at t=0.
  • 51
    • 0042754937 scopus 로고    scopus 로고
    • in Global Minimization of Nonconvex Energy Functions: Molecular Conformation and Protein Folding: DIMACS Workshop, edited by P. Pardalos, D. Shalloway, and G. Xue of American Mathematical Society, Providence, RI
    • B. W. Church, M. Orešič, and D. Shalloway, in Global Minimization of Nonconvex Energy Functions: Molecular Conformation and Protein Folding: DIMACS Workshop, edited by P. Pardalos, D. Shalloway, and G. Xue, Vol. 23 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science (American Mathematical Society, Providence, RI, 1996), pp. 41-64.
    • (1996) DIMACS Series in Discrete Mathematics and Theoretical Computer Science , vol.23 , pp. 41-64
    • Church, B.W.1    Orešič, M.2    Shalloway, D.3
  • 52
    • 22244454459 scopus 로고    scopus 로고
    • note
    • obs, so the minimum decay condition can equivalently be imposed in either the occupation or observation representation.
  • 54
    • 22244449365 scopus 로고    scopus 로고
    • note
    • See Ref. 16 for further discussion. If H is invariant under a continuous symmetry group (e.g., rigid body translation and rotation), we will only be interested in group-invariant macrostates; therefore the sums in Eq. (24) can be restricted to group- invariant eigenfunctions, and degeneracies associated with irreducible representations of the group of rank >1 will not enter. If H is invariant under a discrete group of transformations, the natural requirement that all group-transformation-related macrostates be included in the set of macrostates (i.e., that the macrostates form a basis for a representation of the group) implies that the macrostate subspace will include either all or none of the degenerate eigenfunctions belonging to an irreducible representation of the group. Thus, there will be no symmetry-induced ambiguity in the subspace specification.
  • 55
    • 22244477145 scopus 로고    scopus 로고
    • note
    • This variational equation for σ is equivalent to the Rayleigh-Ritz variational equation for the first excited eigenfunction; see Ref. 16.
  • 56
    • 22244485100 scopus 로고    scopus 로고
    • note
    • The detW term ensures that the node is located between the two macrostates; the denominator will become small if σ is varied so much that the nodal surface does not separate the macrostates. However, this does not occur during normal numerical variation.
  • 57
    • 22244493026 scopus 로고    scopus 로고
    • note
    • 0 as R→∞ [A. Ulitsky and D. Shalloway (in preparation)]. Thus, by Eq. (15), the window and transition functions will go to constants.
  • 58
    • 22244455470 scopus 로고    scopus 로고
    • note
    • Note that erf(0.83x)-tanh(x)<0.02∀x. Thus the functions provide essentially equivalent parameterizations for the variational condition.
  • 59
    • 22244464281 scopus 로고    scopus 로고
    • note
    • For a system with N degrees-of-freedom, both the nodal surface and the transition "plane" are (N-1)-dimensional manifolds.
  • 60
    • 22244457109 scopus 로고    scopus 로고
    • note
    • Since η is a vector, Eqs. (40) represents N+1 conditions where N is the number of degrees-of-freedom.
  • 62
    • 22244440579 scopus 로고    scopus 로고
    • note
    • X and Z instead of the characteristic harmonic frequency.
  • 68
    • 22244449033 scopus 로고    scopus 로고
    • note
    • This method (see Ref. 66) was chosen because it uses only the objective function and does not require derivatives which are relatively expensive to accurately compute by Monte Carlo integration.
  • 69
    • 22244471421 scopus 로고    scopus 로고
    • note
    • The integrals in the right-hand-sides of Eqs. (47) were computed to 3% accuracy as determined by the variance (Ref. 65).
  • 70
    • 22244438599 scopus 로고    scopus 로고
    • note
    • XY<0.03.
  • 71
    • 22244462774 scopus 로고    scopus 로고
    • note
    • ξ=0.03.
  • 73
    • 22244436978 scopus 로고    scopus 로고
    • note
    • c.
  • 75
    • 22244473388 scopus 로고    scopus 로고
    • note
    • 1 to 1% accuracy.
  • 81
    • 22244477462 scopus 로고    scopus 로고
    • in preparation
    • D. Shalloway (in preparation).
    • Shalloway, D.1
  • 83
    • 22244454808 scopus 로고    scopus 로고
    • note
    • t) from the transition region conformations resulting from the replacement of atoms 3 and 4 with other pairs of adjacent atoms.
  • 84
    • 35248857016 scopus 로고
    • This procedure described in, was used. While multiple reaction paths are anticipated in multidimensional systems only one path connecting the two isoforms of the six-particle cluster was detected in 100 trials starting with different initial conditions.
    • This procedure described in A. Ulitsky and R. Elber, J. Chem. Phys. 92, 1510 (1990) was used. While multiple reaction paths are anticipated in multidimensional systems only one path connecting the two isoforms of the six-particle cluster was detected in 100 trials starting with different initial conditions.
    • (1990) J. Chem. Phys. , vol.92 , pp. 1510
    • Ulitsky, A.1    Elber, R.2
  • 85
    • 22244482839 scopus 로고    scopus 로고
    • note
    • As an additional check, transition rates were also calculated using the mean first-passage time evaluated from Langevin dynamics trajectories. Similar results were obtained.
  • 86
    • 22244460892 scopus 로고    scopus 로고
    • note
    • 2 using Eq. (49b) since only the coordinates of one atom enter Eq. (61).
  • 90
    • 22244442500 scopus 로고    scopus 로고
    • note
    • 0exp[-βw(q)]dq/∫exp[-βw(q)]dq, where w(q) is the potential of mean force along an ad hoc preselected direction and q=0 is the maximizer of w(q) in the barrier region.
  • 94
    • 22244457731 scopus 로고    scopus 로고
    • Variable-scale coarse-graining in macromolecular global optimization
    • in edited by L. Biegler, T. Coleman, A. R. Conn, and F. Santosa, Springer, New York
    • D. Shalloway, Variable-scale coarse-graining in macromolecular global optimization, in Proceedings of IMA Conference on Large Scale Optimization, edited by L. Biegler, T. Coleman, A. R. Conn, and F. Santosa, (Springer, New York, 1997).
    • (1997) Proceedings of IMA Conference on Large Scale Optimization
    • Shalloway, D.1
  • 96
    • 22244460560 scopus 로고    scopus 로고
    • note
    • β≈1, but this has not been proven.
  • 97
    • 22244485447 scopus 로고    scopus 로고
    • note
    • α to approach zero or one, but this has not been proven.
  • 101
    • 22244452391 scopus 로고    scopus 로고
    • note
    • 0).


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.