Mathematical Challenges from Theoretical/Computational Chemistry (1995) / Chapter Skim
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Mathematical Research Opportunities from Theoretical/Computational Chemistry
Mathematical Research Opportunities from Theoretical/Computational Chemistry
Pages 43-108
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From page 43... ...
The remaining sections "Molecular Dynamics Algorithms," "Multivariate Minimization in Computational Chemistry," and "Fast Algebraic Transformation Methods" contain discussions of topics from the mathematical sciences with specific insight into their relevance to theoretical/computational chemistry. Expositions and references are meant to give the mathematical reader sufficient insight and direction to be able into subsequently investigate the topic via deeper reading and especially by discussions with colleagues from the chemical sciences.
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From page 45... ...
The equation and its solutions are parameterized by the nuclear positions and charges. The eigenvalues Uk, viewed as functions of Ri, yield the potential energy surfaces for the molecule in its ground and excited states after including the repulsive internuclear interactions.
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From page 46... ...
For a finite basis of orthonormal molecular orbitals, it is possible to replace this equation by a different one that gives the same matrix element Hi, = < A, He ~~ > This is accomplished by utilizing an operator algebra that facilitates manipulation of Slater determinants, in which al is an annihilation operator and aft is a creation operator. A correspondence can be established between the function space formed by all possible linear combinations of Slater determinants and the vector space formed by the creation operators so that (5)
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From page 47... ...
Progress for somewhat larger systems can be made by use of the Hohenberg-Kohn and Kohn-Sham theorems to give a useful density functional theory. These assume that it is possible to find an effective one-body potential or so that, by solving the one-electron Schrodinger equation (-~/2V2 + a)
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From page 48... ...
In this regard, people have considered methods based on density matrices, density functionals, Monte CarIo diffusion equations, effective core potentials, and so on. In particular, because the energy of an atom or molecule is a linear function of the density matrix and the one- and two-body distribution functions derived from it, density matrix methods raise the hope that one could dispense with computing the associated 3N-dimensional wavefunction and deal with simpler three-dimensional density functions.
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From page 49... ...
is also called the two-body reduced density matrix. Similarly, the density matrix may first be Fourier transformed and then used to derive the one-body momentum distribution 7r(pl)
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Unfortunately, only an incomplete set of necessary conditions are known, but these are already so complex that further work in this area has been abandoned by chemists. In density functional theory, only the density p(r)
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From page 51... ...
As noted above, progress on this problem has been made when employing much simplified representations of the electronic structure of the system, which enable the solution of equations such as Equation (1) for the few "quantum mechanical" atoms as rapidly or more so than the classical molecular dynamical equations of motion, using Equation (10)
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Bash, and M Karplus, 1990, A combined quantum mechanical and molecular mechanical potential for molecular dynamics simulations, J
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From page 54... ...
Similar issues arise in other chemical applications, such as quantum mechanics, and the development of improved integration schemes will advance the systems and types of processes that can be simulated on modern computers. Nu~nerical Methods for Solving Ordinary Differential Equations Many problems in chemistry can be reduced to the solution of systems of coupled ordinary differential equations (ODEs)
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In particular, serious resonance problems have been noted (Man~ziuk and Schlick, 19951. Third, implicit schemes for multiple time scale problems increase complexity, since they involve solution of a nonlinear system or minimization of a nonlinear function at each time step.
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Noid, and B.G. Sumpter, 1994, Symplectic integrators for large scale molecular dynamics simulations: A comparison of several explicit methods, J
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Saville, 1978, Multiple time step methods in molecular dynamics, Mol.
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From page 59... ...
Furthermore, it appears in expressions for the X-ray and neutron diffraction patterns for the substance; consequently, these diffraction measurements constitute an experimental means for measuring girl for real substances. It should be added that girl is also one of the traditional results reported from computer simulations of N-body systems (Ciccotti et al., 1987~.
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Idlenoints..in.Condensed Mat · ~ .:.,, i: .
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important. Chemist~ is usu~ly~ a~low-temperatllre phenomenon-most chemicdil reactions are studied:~around ~:: ~ .
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Perhaps as a result of these shortcomings, the recent trend in classical statistical mechanics has been to rely heavily on direct computer simulation of condensed-phase phenomena. Because these simulations often require massive computational resources, a case can be made that revival of analytic predictive theory for girt would be favorable from the point of view of the "productivity issue" in theoretical and computational chemistry.
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From page 63... ...
Only in recent years has its significance been truly appreciated, however. One of the leaders in bringing out the significance of topology in quantum molecular dynamics was M
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From page 64... ...
Dvnamic..C:haos for Quantum Mechanical Systems: : :.,, -.:,:, ~=y.'.pheno.m, era.: s ': at"the: e' f:' i i i y f ' assi ch' ics ~:Qu tumid : ,. ~chmi~ ~e"nO.mena, :s^.
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From page 65... ...
Either a trajectory encircles a conical intersection (of Born-Oppenheimer energy surfaces) or it does not, leading to two values of the phase.
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As in the other applications, the notion of atomic and molecular quantum mechanics is unambiguously defined for D a positive integer; in other words, the Schrodinger wave equation and ..
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From page 67... ...
These latter afford convenient fixed points for refining the series summation attempts. The presumption that spaces with noninteger dimension were available as analytic tools for atomic and molecular quantum mechanics rests largely on simple observations such as the fact that the Ddimensional (hyper~spherical volume element, dtV(D)
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From page 68... ...
Such applications arise frequently in molecular modeling, rational drug design, quantum mechanical calculations, mathematical biology models, neural networks, combinatorial problems, financial investment planning, engineering, electronics, meteorology, and computational geometry. In applications that arise in computational chemistry (Scheraga, 1992; Schlick, 1992)
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From page 69... ...
Derivative information can often be exploited significantly for the optimization algorithm, but the benefits must be balanced with the additional costs involved. The Complexity of Computational Chemistry Problems Optimization problems frequently arise in molecular and quantum mechanical calculations in chemistry.
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From page 70... ...
is clearly important for reducing the severity of this problem and allowing more accurate representation of the BE 4.5 Automatic Differentiation ~~ ~ Automatic:diff~rent~at~n' essentially new: rat o st Tall: :1981 View: :1988: ~~ :::: ~ .
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that is complementary to potential energy minimization. While in theory information on all thermally accessible states should be observable, the restriction of the integration time step to a very small value with respect to time scales of collective biomolecular motions limits the scope of molecular dynamics in practice.
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. ~ In their attempt to find a global rather than local minimum, global optimization methods tend to explore larger regions of function space (see, e.g., Dixon and Szego, 1975; Floudas and Pardalos, 1991~.
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and functional transformations (e.g., Piela et al., 1989; Wu, 1994) appear promising to global optimization problems in computational chemistry, and further developments might be fruitful.
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Schulten, 1992, Accelerated molecular dynamics simulations with the parallel fast multiple algorithm, Chem.
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Schulten, 1991, Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions, Mol. Simul.
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73-142. Schlick, T., 1992, Optimization methods in computational chemistry, in Reviews in Computational Chemistry, Vol.
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From page 77... ...
on potential energy surfaces represents an important and challenging problem in computational chemistry. In chemical applications, special conformational-space search methods have been devised for locating minima on molecular mechanics-based potential energy surfaces.
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From page 78... ...
Clearly, more robust algorithms are still needed, and this is an area that mathematical optimizers may find very interesting. It is conceivable that algorithms for locating transition states on potential energy surfaces derived from calculations based on quantum mechanics could be employed for the location of confirmational transition states on molecular mechanics-derived potential energy surfaces once the minima have been located.
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From page 79... ...
Guida, and W.C. Still, An internal coordinate Monte Carlo method for searching conformational space, J
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From page 80... ...
The global geometry of the potential energy surface also enters into the study of nonbiological chemical problems such as those involving the structure and mechanical properties of amorphous materials. While crystalline solids can be studied through the analysis of the ground state and the first few excited states, glasses and other amorphous materials have a huge number of local structural configurations.
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From page 81... ...
The problem of broken ergodicity is one that is central to understanding the global topology of potential energy surfaces for such "random" systems. This problem plays a role both in the issues discussed here of biological macromolecules and amorphous materials, and in other optimization problems as well.
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From page 83... ...
These ideas are, however, based on the quasi-equilibrium statistical mechanics of such systems, and there are many questions about the rigor of this approach. Similarly, a good deal of work has been carried out to characterize computationally pathways on complicated realistic potential energy surfaces.
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From page 84... ...
::;;; ::-:: .'' 2.''"' '' ' ' '' ''' :"'':' ''" ' ' '4.7"'I' ' li' a' '' ' 's"'o' " ' ' ' ' 'i' 1"'' h' s'':atthe'" 1' ssi 1' L' vel":~:~:.
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From page 85... ...
In the statistical mechanics modeling of condensed phases, one typically is interested in restricted sets of particle configurations. For instance, one's interest is often restricted to only those states for which ~ri-rj~ 2 ~ for aD ~ |
From page 86... ...
The various steps in the random walk of the previous problem might be correlated in the sense that the probability distribution for si might depend on the value of si, or perhaps both s and si 2. In these problems, there are some constraints that are "easy" to satisfy (e.g., in the first problem each particle must be inside the cubic box; in the second problem, each step must have unit length)
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From page 87... ...
Molecular Diversity and Combinatorial Chemistryin Drug Discovery Overview of the Drug Discovery Process The discovery of new drugs is a time-consuming, risky, and expensive process. These things are true even though in the past 15 years there has been a dramatic increase in the number of threedimensional structures of proteins that can be used as scaffolds for the conceptual and computational aspects of drug design.
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From page 88... ...
Many of the improvements in computational chemistry discussed elsewhere in this report would also impact the ability to forecast affinity based on the structure of the ligand and the macromolecular target. However, additional opportunities exist for cases in which the structure of the macromolecular target is not known, cases for which the forecast is based on three-dimensional QSAR investigations (Kubinyi, 1993~.
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From page 89... ...
A final strategy to enhance molecular diversity results from computer programs that design molecules to meet specified three-dimensional criteria, typically based on the experimental structure of a protein binding site (Rothstein and Murcko, 19931. The programs design molecules to meet geometric criteria and include electrostatic complementarily at the level of force fields such as those used for molecular dynamics.
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From page 90... ...
Opportunities for Improvements in Computational Approaches to Compound Selection Molecular fingerprints are not the best descriptor to use to select compounds for bioactivity since the biological properties of compounds depend on their three-dimensional complementarily of shape and electronic properties with those of the target biomolecule. Clearly, we would like to consider the 90
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From page 91... ...
Since we typically would expect to investigate no more than I/100 as many clusters as original compounds, divisive methods might have an advantage because in this approach, clustering starts with one huge cluster, divides clusters into tighter ones, and could stop once the target number of clusters was formed. At this time, no method other than Jarvis-Patrick is known to the computational chemistry community that will group 105 or 106 objects in a time scale of less than a week (Willett et al., 1986; Willett, 1987; Whaley and Hodes, 1991~.
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From page 92... ...
Kim, and C.T. Lin, in press, Comparative molecular field analysis: CoMFA, in Linear Free Energy Relationships in Biology, M
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From page 93... ...
Is it punctuated or is it gradual as many have claimed in the past? Artificial evolution approaches may obviate the need to completely understand and design biological macromolecules, but there will be a large number of interesting mathematical problems connected with the design of efficient artificial evolution experiments.
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From page 94... ...
The underlying structure or formal expression also describes some unrelated random process, and therefore the deterministic problem can be solved numerically by a Monte Carlo simulation of the corresponding probabilistic problem. The essential feature common to all Monte Carlo computations is that at some point one will need to substitute for a random variable a corresponding set of values with the statistical properties of the random variable.
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From page 95... ...
The three terms on the right-hand side then correspond to diffusion with diffusion constant Di, a drift term associated with the trial function, and a branching term that derives this designation from the DMC equation being, in the absence of the first two terms on the right-hand side,a first-order kinetic equation. Because f can, in general, assume both positive and negative values, which would preclude the interpretation off as a probability for fermion systems, one alternative is to impose the nodes of the ground state wavefunction IT on ~ so that f is always positive.
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From page 96... ...
Lester, Jr., and P.J. Reynolds, 1994, Monte Carlo Methods in Ab Initio Quantum Chemistry, World Scientific, Singapore.
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From page 97... ...
The coupling is important if, for example, · two potential energy surfaces of the same symmetry cross in the BO approximation correction terms prevent such crossings or are close and more-or-less parallel over a moderate range of configuration space; · the electronic state of a polyatomic molecule is degenerate for a symmetric arrangement of nuclei; coupling leads to Jahn-Teller and Renner-Teller effects; · the nuclear velocities are large as in high-energy molecular collisions; · the molecule is in a high rotational state and has nonzero electronic angular momentum. There are two kinds of correction terms for the coupling of electronic and nuclear motions.
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This leads to a set of equations for the functions sbk(R) that determine the nuclear motion of the system, r—VR /(2~)
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are effectively a correction to the potential energy due to the coupling between the electronic and nuclear motions. The adiabatic approximation gives the best potential energy function.
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Discussion of the Problem A standard way of attacking this problem would be a Monte CarIo integration scheme in which points x are generated with a probability distribution of p (x) end f is evaluated at these points.
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Do~x,t) as the probability distribution.
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From page 102... ...
Other Formulations and Solutions of the Basic Problem The nature and dimensionality of the x space and the specific form of the p~xJ function are determined by the nature of the quantum mechanical problem to be solved. In general, however, the coordinates of x are Cartesian (or other)
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From page 103... ...
luls clttlCUlty becomes acute in the limit that z becomes purely imaginary, because the "probability distribution function" approaches unity and is not Formalizable. A variety of analytic continuation methods have been used to solve problems of this type.
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87:1648. Doll, J.D., 1984, Monte Carlo Fourier path integral methods in chemical dynamics, J
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79:5029-33. Wolynes, P.G., 1987, Imaginary time path integral Monte Carlo route to rate coefficients for nonadiabatic barrier crossing, J
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In density functional methods for calculating ground electronic states in quantum electronic structures, it is desired to approximate multilinear forms including integrals of the form ~~- pfr)
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With a rapid way to evaluate K, there might be a faster way to evaluate the bilinear form. (Alternative ways to evaluate p, based on similar considerations, have already been discussed.)
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From page 108... ...
97:4309-43 15. Draghicescu, C.I., 1994, An efficient implementation of particle methods for the incompressible Euler equations, SIAM J
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