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Mathematical Challenges from Theoretical/Computational Chemistry
MATHEMATICAL CHALLENGES FROM THEORETICAL/COMPUTATIONAL CHEMISTRY
Committee on Mathematical Challenges from Computational Chemistry
Board on Mathematical Sciences and
Board on Chemical Sciences and Technology
Commission on Physical Sciences, Mathematics, and Applications
National Research Council
National Academy Press
Washington, D.C.
1995

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Mathematical Challenges from Theoretical/Computational Chemistry
NOTICE: The project that is the subject of this report was approved by the Governing Board of the National Research Council, whose members are drawn from the councils of the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine. The members of the committee responsible for the report were chosen for their special competences and with regard for appropriate balance.
This report has been reviewed by a group other than the authors according to procedures approved by a Report Review Committee consisting of members of the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine.
Support for this project was provided by the National Science Foundation, Air Force Office of Scientific Research, Department of Energy, Army Research Office, and Ciba-Geigy Corporation.
Library of Congress Catalog Card Number 95-68040
International Standard Book Number 0-309-05097-9
Copyright 1995 by the National Academy of Sciences. All rights reserved.
Available from:
National Academy Press
2101 Constitution Avenue, NW Box 285 Washington, DC 20055
B-515
Available on the Internet via the World Wide Web at the URL: http://www.nas.edu/.
Printed in the United States of America
COVER ILLUSTRATION: The catalytic binding site of the enzyme purine nucleoside phosphorylase, which plays a key rule in immune function, is shown in gray as a space-filling model. An inhibitor of the enzyme is shown in white. A tight fit between the enzyme and the inhibitor is required for binding and inhibitory activity, and a goal of structure-based drug discovery is the design of inhibitors that are geometrically (and chemically) complementary to an enzyme binding site.
The figure was computer generated and resulted from a study that involved calculating geometries of potential inhibitors "docked" in the enzyme binding site (Montgomery et al., 1993). The study involved energy minimization and Monte Carlo-like conformational searching using the MacroModel computational chemistry software (Mohamadi et al., 1990). Such a computationally intensive task could not have been carried out 10 years ago and was an integral part of a structure-based drug design effort (Montgomery, 1993; see also Bugg et al., 1993).
Figure courtesy of W. Guida, Pharmaceuticals Division, Ciba-Geigy Corporation.
References
Bugg, C.E., W.M. Carson, and J.A. Montgomery, 1993, Drugs by design, Scientific American 269:92–98.
Mohamadi, F., N.G.J. Richards, W.C. Guida, R. Liskamp, M. Lipton, C. Caufield, G. Chang, T. Hendrickson, and W.C. Still, 1990, MacroModel—An integrated software system for modeling organic and bioorganic molecules using molecular mechanics, J. Comput. Chem. 11:440–467.
Montgomery, J.A., 1993, Purine nucleoside phosphorylase: A target for drug design, Medicinal Research Reviews 13:209–228.
Montgomery, J.A., S. Niwas, J.D. Rose, J.A. Secrist, Y.S. Babu, C.E. Bugg, M.D. Erion, W.C. Guida, and S.E. Ealick, 1993, Structure-based design of inhibitors of purine nucleoside phosphorylase 1. 9-(arylmethyl) derivatives of 9-deazaguanine, J. Med. Chem. 36:55–69.

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Mathematical Challenges from Theoretical/Computational Chemistry
Committee on Mathematical Challenges from Computational Chemistry
FRANK H. STILLINGER,
AT&T Bell Laboratories,
Chair
HANS C. ANDERSEN,
Stanford University
LOUIS AUSLANDER,
City University of New York
DAVID L. BEVERIDGE,
Wesleyan University
ERNEST R. DAVIDSON,
Indiana University
WAYNE C. GUIDA,
Ciba-Geigy Corporation
PETER A. KOLLMAN,
University of California at San Francisco
WILLIAM A. LESTER, JR.,
University of California at Berkeley
YVONNE C. MARTIN,
Abbott Laboratories
GEORGE C. SCHATZ,
Northwestern University
TAMAR SCHLICK,
New York University and Howard Hughes Medical Institute
L. RIDGWAY SCOTT,
University of Houston
DEWITT L. SUMNERS,
Florida State University
PETER G. WOLYNES,
University of Illinois at Urbana-Champaign
Board on Chemical Sciences and Technology Liaison
KENDALL N. HOUK,
University of California at Los Angeles
SCOTT T. WEIDMAN, Study Director
TAÑA L. SPENCER, Project Assistant

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Mathematical Challenges from Theoretical/Computational Chemistry
Board on Mathematical Sciences
AVNER FRIEDMAN,
University of Minnesota,
Chair
JEROME SACKS,
National Institute of Statistical Sciences,
Vice Chair
LOUIS AUSLANDER,
City University of New York
HYMAN BASS,
Columbia University
PETER E. CASTRO,
Eastman Kodak Company
R. DUNCAN LUCE,
University of California at Irvine
PAUL S. MUHLY,
University of Iowa
GEORGE NEMHAUSER,
Georgia Institute of Technology
ANIL NERODE,
Cornell University
INGRAM OLKIN,
Stanford University
RONALD PEIERLS,
Brookhaven National Laboratory
DONALD ST. P. RICHARDS,
University of Virginia
MARY F. WHEELER,
Rice University
ROBERT J. ZIMMER,
University of Chicago
JON KETTENRING,
Bellcore,
Ex Officio Member
JOHN R. TUCKER, Director
JACK ALEXANDER, Program Officer
RUTH E. O'BRIEN, Staff Associate
BARBARA W. WRIGHT, Administrative Associate

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Mathematical Challenges from Theoretical/Computational Chemistry
Board on Chemical Sciences and Technology
ROYCE C. MURRAY,
University of North Carolina,
Co-Chair
EDWIN P. PRZYBYLOWICZ,
Rochester Institute of Technology,
Co-Chair
PAUL S. ANDERSON,
The Du Pont Merck Pharmaceutical Company
DAVID C. BONNER,
Premix, Inc.
PHILIP H. BRODSKY,
Monsanto Company
MARVIN H. CARUTHERS,
University of Colorado
GREGORY R. CHOPPIN,
Florida State University
FRED P. CORSON,
Dow Chemical Company
MOSTAFA EL-SAYED,
Georgia Institute of Technology
JOANNA S. FOWLER,
Brookhaven National Laboratory
BERTRAM O. FRASER-REID,
Duke University
JUDITH C. GIORDAN,
Henkel Corporation
JOSEPH G. GORDON II,
IBM Almaden Research Center
L. LOUIS HEGEDUS, W.R.
Grace & Co.
GEORGE J. HIRASAKI,
Rice University
DOUGLAS A. LAUFFENBERGER,
Massachusetts Institute of Technology
MARSHA I. LESTER,
University of Pennsylvania
W. HARMON RAY,
University of Wisconsin
GABOR A. SOMORJAI,
University of California at Berkeley
JOHN J. WISE,
Mobil Research and Development Corporation
DOUGLAS J. RABER, Director
MARIA P. JONES, Administrative Secretary
SYBIL A. PAIGE, Administrative Associate
TAÑA L. SPENCER, Senior Secretary
SCOTT T. WEIDMAN, Senior Program Officer
TAMAE M. WONG, Program Officer

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Mathematical Challenges from Theoretical/Computational Chemistry
Commission on Physical Sciences, Mathematics, and Applications
RICHARD N. ZARE,
Stanford University,
Chair
RICHARD S. NICHOLSON,
American Association for the Advancement of Science,
Vice Chair
STEPHEN L. ADLER,
Institute for Advanced Study
SYLVIA T. CEYER,
Massachusetts Institute of Technology
SUSAN L. GRAHAM,
University of California at Berkeley
ROBERT J. HERMANN,
United Technologies Corporation
RHONDA J. HUGHES,
Bryn Mawr College
SHIRLEY A. JACKSON,
Rutgers University
KENNETH I. KELLERMANN,
National Radio Astronomy Observatory
HANS MARK,
University of Texas at Austin
THOMAS A. PRINCE,
California Institute of Technology
JEROME SACKS,
National Institute of Statistical Sciences
L.E. SCRIVEN,
University of Minnesota
A. RICHARD SEEBASS III,
University of Colorado
LEON T. SILVER,
California Institute of Technology
CHARLES P. SLICHTER,
University of Illinois at Urbana-Champaign
ALVIN W. TRIVELPIECE,
Oak Ridge National Laboratory
SHMUEL WINOGRAD,
IBM T.J. Watson Research Center
CHARLES A. ZRAKET,
MITRE Corporation (retired)
NORMAN METZGER, Executive Director

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Mathematical Challenges from Theoretical/Computational Chemistry
CONTENTS
EXECUTIVE SUMMARY
1
Overview
1
Conclusions and Recommendations
2
Reference
5
1
INTRODUCTION
7
References
10
2
THE EMERGENCE OF COMPUTATIONAL CHEMISTRY
13
3
EXAMPLES OF CONSTRUCTIVE CROSS-FERTILIZATION BETWEEN THE MATHEMATICAL SCIENCES AND CHEMISTRY
19
Use of Statistics to Predict the Biological Potency of Molecules Later Marketed as New Drugs and Agricultural Chemicals
19
References
21
Numerical Analysis
21
References
22
Distance Geometry
23
References
25
Mathematics and Fullerenes
26
References
27
Quasicrystals
27
References
28
Chemical Topology
28
Combinatorics, Graph Theory, and Chemical Isomer Enumeration
28
Analysis of Molecular Spectra by Using Cayley Trees
29
Group Theory, Topology, Geometry, and Stereochemistry
29
Topology of Polymers
30
Knot, Links (Catenanes), and DNA
31
References
32
Graph Theory
34
Application of Graph Theory to Organizing Chemical Literature
34
Application of Graph Theory to Representation of Chemical Reactions
37
References
38
X-Ray Crystallography
40
Remark
42
References
42
4
MATHEMATICAL RESEARCH OPPORTUNITIES FROM THEORETICAL/COMPUTATIONAL CHEMISTRY
43
Introduction
43
References
43
Numerical Methods for Electronic Structure Theory
43
The N- and V-Representability Problems
48

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Mathematical Challenges from Theoretical/Computational Chemistry
References
50
Melding of Quantum Mechanics with Simpler Models
51
References
53
Molecular Dynamics Algorithms
54
Enhanced Sampling
54
Numerical Methods for Solving Ordinary Differential Equations
54
Symplectic Integrators
54
The Time Step Problem in Molecular Dynamics
55
Implicit Integration Schemes
56
Future Prospects
57
References
57
N- and V-Representability Problems in Classical Statistical Mechanics
59
References
63
Implications of Topological Phases
63
Theoretical and Computational Chemistry in Space of Noninteger Dimension
65
References
67
Multivariate Minimization in Computational Chemistry
68
Introduction
68
Problem Classification
69
The Complexity of Computational Chemistry Problems
69
Local Optimization Methods
71
Global Optimization Methods
72
Perspective
73
References
74
Locating Saddlepoints
77
References
78
Sampling of Minima and Saddlepoints
80
Efficient Generation of Points That Satisfy Physical Constraints in a Many-Particle System
85
Prototypical Problem
85
Variations on the Prototypical Problem
85
Simplest Strategy
86
Metropolis Monte Carlo Method
87
Relationship of These Problems to More General Optimization Problems
87
Molecular Diversity and Combinatorial Chemistry in Drug Discovery
87
Overview of the Drug Discovery Process
87
Sources of Molecular Diversity
88
Current Computational Approaches to Compound Selection
89
Opportunities for Improvements in Computational Approaches to Compound Selection
90
References
91
Statistical Analyses of Families of Structures
93
Quantum Monte Carlo Solution of the Schrödinger Equation
94
Variational Monte Carlo (VMC)
95
Diffusion Monte Carlo (DMC)
95
Green's Function Monte Carlo (GFMC)
96
Research Opportunities
96
References
96
Nonadiabatic Phenomena
96
References
99

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Mathematical Challenges from Theoretical/Computational Chemistry
Evaluation of Integrals with Highly Oscillatory Integrands: Quantum Dynamics with Path Integrals
99
Prototypical Problem
100
Discussion of the Problem
100
Stationary-Phase Monte Carlo Methods
101
Alternative Approaches to the Prototype Problem
102
Other Formulations and Solutions of the Basic Problem
102
References
104
Fast Algebraic Transformation Methods
105
References
108
5
CULTURAL ISSUES AND BARRIERS TO INTERDISCIPLINARY WORK
109
Motivation and Connections
109
Effects of Disciplinary Boundaries
110
Effects of the Curriculum
112
Language Differences
112
Toward a Fruitful Collaboration
114
References
115
6
CONCLUSIONS AND RECOMMENDATIONS
117
References
119
AFTERWORD
121
GLOSSARY
123

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Mathematical Challenges from Theoretical/Computational Chemistry
The National Academy of Sciences is a private, nonprofit, self-perpetuating society of distinguished scholars engaged in scientific and engineering research, dedicated to the furtherance of science and technology and to their use for the general welfare. Upon the authority of the charter granted to it by the Congress in 1863, the Academy has a mandate that requires it to advise the federal government on scientific and technical matters. Dr. Bruce M. Alberts is president of the National Academy of Sciences.
The National Academy of Engineering was established in 1964, under the charter of the National Academy of Sciences, as a parallel organization of outstanding engineers. It is autonomous in its administration and in the selection of its members, sharing with the National Academy of Sciences the responsibility for advising the federal government. The National Academy of Engineering also sponsors engineering programs aimed at meeting national needs, encourages education and research, and recognizes the superior achievements of engineers. Dr. Robert M. White is president of the National Academy of Engineering.
The Institute of Medicine was established in 1970 by the National Academy of Sciences to secure the services of eminent members of appropriate professions in the examination of policy matters pertaining to the health of the public. The Institute acts under the responsibility given to the National Academy of Sciences by its congressional charter to be an adviser to the federal government and, upon its own initiative, to identify issues of medical care, research, and education. Dr. Kenneth I. Shine is president of the Institute of Medicine.
The National Research Council was organized by the National Academy of Sciences in 1916 to associate the broad community of science and technology with the Academy's purposes of furthering knowledge and advising the federal government. Functioning in accordance with general policies determined by the Academy, the Council has become the principal operating agency of both the National Academy of Sciences and the National Academy of Engineering in providing services to the government, the public, and the scientific and engineering communities. The Council is administered jointly by both Academies and the Institute of Medicine. Dr. Bruce M. Alberts and Dr. Robert M. White are chairman and vice chairman, respectively, of the National Research Council.

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Mathematical Challenges from Theoretical/Computational Chemistry
LIST OF BOXES
BOX 2.1
The Schrödinger Wave Equation
14
BOX 2.2
Molecular Mechanics/Molecular Dynamics
15
BOX 2.3
Chemist, Mathematician, or Physicist?
16
BOX 3.1
Rational Drug Design
20
BOX 3.2
Research Opportunities in Parallel Computing
22
BOX 3.3
Protein Microtutorial
24
BOX 3.4
Clique Detection
36
BOX 4.1
Electronic Phase Transitions
52
BOX 4.2
Tutorial on Statistical Mechanics and the Importance of Minima and Saddlepoints in Condensed Matter Systems
60
BOX 4.3
Implications of Dynamic Chaos for Quantum Mechanical Systems
64
BOX 4.4
Nodal Properties of Wave functions
66
BOX 4.5
Automatic Differentiation
70
BOX 4.6
Comments on the Ambiguous Concept of ''Structure'' for Complex Molecules and Macromolecules
82
BOX 4.7
Implications of Dynamical Chaos at the Classical Level
84
BOX 4.8
Possibility of Intelligent Algorithms to Detect Novel Phenomena Automatically
90
BOX 5.1
American Chemical Society Curriculum Standards for Mathematical Course Work
111
BOX 5.2
Information Sources About Theoretical/Computational Chemistry
113
BOX 5.3
Information Sources About the Mathematical Sciences
114

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Mathematical Challenges from Theoretical/Computational Chemistry
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