Smoothing techniques of global optimization: Distance scaling method in searches for most stable Lennard-Jones atomic clusters

Journal of Computational Chemistry

Volumn 18, Issue 16, 1997, Pages 2040-2049

  Pillardy, Jarosław ^a   Piela, Lucjan ^a  

^a UNIVERSITY OF WARSAW   (Poland)

Author keywords

Atomic clusters; Global optimization; Potential energy averaging; Potential energy deformation; Potential energy smoothing; Stable structures

Indexed keywords

EID: 0009239593     PISSN: 01928651     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1096-987X(199712)18:16<2040::AID-JCC8>3.0.CO;2-L     Document Type: Article

References (22)

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2. See review articles: (a) J. E. Straub, In New Developments in Theoretical Studies of Proteins, R. Elber, Ed., World Scientific, Singapore, 1995; (b) L. Piela, K. A. Olszewski, and J. Pillardy, J. Mol. Struct., 308, 229 (1994) (c) M. Vásquez, G. Nemethy, and H. A. Scheraga, Chem. Rev., 94, 2183 (1994) (d) R. Elber, Curr. Opin. Struct. Biol., 3, 260 (1993).
3. See review articles: (a) J. E. Straub, In New Developments in Theoretical Studies of Proteins, R. Elber, Ed., World Scientific, Singapore, 1995; (b) L. Piela, K. A. Olszewski, and J. Pillardy, J. Mol. Struct., 308, 229 (1994) (c) M. Vásquez, G. Nemethy, and H. A. Scheraga, Chem. Rev., 94, 2183 (1994) (d) R. Elber, Curr. Opin. Struct. Biol., 3, 260 (1993).
4. See review articles: (a) J. E. Straub, In New Developments in Theoretical Studies of Proteins, R. Elber, Ed., World Scientific, Singapore, 1995; (b) L. Piela, K. A. Olszewski, and J. Pillardy, J. Mol. Struct., 308, 229 (1994) (c) M. Vásquez, G. Nemethy, and H. A. Scheraga, Chem. Rev., 94, 2183 (1994) (d) R. Elber, Curr. Opin. Struct. Biol., 3, 260 (1993).
5. See review articles: (a) J. E. Straub, In New Developments in Theoretical Studies of Proteins, R. Elber, Ed., World Scientific, Singapore, 1995; (b) L. Piela, K. A. Olszewski, and J. Pillardy, J. Mol. Struct., 308, 229 (1994) (c) M. Vásquez, G. Nemethy, and H. A. Scheraga, Chem. Rev., 94, 2183 (1994) (d) R. Elber, Curr. Opin. Struct. Biol., 3, 260 (1993).
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7. The first article that discusses a smoothing of the potential energy function by the convolution transformation was F. H. Stillinger, Phys. Rev. B, 32, 3134 (1985). The article deals with the glass transition for amorphous substances and gives general formulas concerning the exponential decrease of the number of minima when the Gaussian smoothing of the potential energy hypersurface is performed. In L. Piela, J. Kostrowicki, and H. A. Scheraga, J. Phys. Chem., 93, 3339 (1989) a similar idea has been introduced independently with the following differences: smoothing is introduced as going beyond any convolution; a global minimization strategy has been constructed that consists of two steps, convexation of the potential yielding the single minimum, and the so-called reversing procedure that hopefully links the convex potential to the global minimum; and the strategy has been applied numerically.
8. The first article that discusses a smoothing of the potential energy function by the convolution transformation was F. H. Stillinger, Phys. Rev. B, 32, 3134 (1985). The article deals with the glass transition for amorphous substances and gives general formulas concerning the exponential decrease of the number of minima when the Gaussian smoothing of the potential energy hypersurface is performed. In L. Piela, J. Kostrowicki, and H. A. Scheraga, J. Phys. Chem., 93, 3339 (1989) a similar idea has been introduced independently with the following differences: smoothing is introduced as going beyond any convolution; a global minimization strategy has been constructed that consists of two steps, convexation of the potential yielding the single minimum, and the so-called reversing procedure that hopefully links the convex potential to the global minimum; and the strategy has been applied numerically.
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18. One may think of even a simpler feature of an ideal smoothing method. Such a method should produce a coarse grained hypersurface with the single minimum exactly in the position of the global minimum of the original hypersurface. Going beyond the convolution idea one may imagine such a method as a "wheelbarrow smoothing," which in one dimension may be described as follows: one creates a trajectory (it will represent a coarse grained function) of the axis of a wheel rolling over the function from below. The method has a most precious feature that at any radius of the wheel the coarse grained function has the global minimum exactly at the position of the global minimum of the original function. When the radius is zero one has the original function; when the radius goes to infinity one ends up with a coarse grained function with only a single minimum (when the global minimum is nondegenerate). Work on this problem is in progress in our laboratory.
19. J. Pillardy and L. Piela, J. Phys. Chem., 99, 11805 (1995).
20. The number of minima was computed from a formula obtained in M. R. Hoare, In Advnces in Chemistry and Physics, I. Prigogine and S. A. Rice, Eds., Wiley, New York, 1979, p. 40. This was done by fitting to values of N < 13 and extrapolating to large N.
21. J. Pillardy and L. Piela, manuscript in preparation.
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