Supercomputer solutions

Quantum mechanics gives a mathematical description of the behavior of electrons that has never been found to be wrong. However, the quantum mechanical equations have never been solved exactly for any chemical system other than the hydrogen atom. Thus, the entire held of computational chemistry is built around approximate solutions. Some of these solutions are very crude and others are expected to be more accurate than any experiment that has yet been conducted. There are several implications of this situation. First, computational chemists require a knowledge of each approximation being used and how accurate the results are expected to be. Second, obtaining very accurate results requires extremely powerful computers. Third, if the equations can be solved analytically, much of the work now done on supercomputers could be performed faster and more accurately on a PC. 

Many institutions have hundreds, or even thousands, of powerful work stations that are idle for much of the day. There is often vastiy more power available in these machines than in any supercomputer center, the only problem being how to harness the power already available. There are network load-distribution tools that allocate individual jobs to unused computers on a network, but this is different from having many computers simultaneously cooperating on the solution of a single problem. 

One Important aspect of the supercomputer revolution that must be emphasized Is the hope that not only will It allow bigger calculations by existing methods, but also that It will actually stimulate the development of new approaches. A recent example of work along these lines Involves the solution of the Hartree-Fock equations by numerical Integration In momentum space rather than by expansion In a basis set In coordinate space (2.). Such calculations require too many fioatlng point operations and too much memory to be performed In a reasonable way on minicomputers, but once they are begun on supercomputers they open up several new lines of thinking. 

Levy (Chapter 6) has also explored the use of supercomputers to study detailed properties of biological macromolecule that are only Indirectly accessible to experiment, with particular emphasis on solvent effects and on the Interplay between computer simulations and experimental techniques such as NMR, X-ray structures, and vltratlonal spectra. The chapter by Jorgensen (Chapter 12) summarizes recent work on the kinetics of simple reactions In solutions. This kind of calculation provides examples of how simulations can address questions that are hard to address experimentally. For example Jorgensen s simulations predicted the existence of an Intermediate for the reaction of chloride Ion with methyl chloride In DMF which had not been anticipated experimentally, and they Indicate that the weaker solvation of the transition state as compared to reactants for this reaction In aqueous solution Is not due to a decrease In the number of hydrogen bonds, but rather due to a weakening of the hydrogen bonds. 

An example of this third solution is presented in this paper which shows how it is possible to achieve supercomputer speeds from a low cost laboratory computer. By placing the task on a local lab computer, it is now also possible to develop reasonable interactive molecular modelling tools which utilize energies and forces in real time. 

The combined fiuld fiow, heat transfer, mass transfer and reaction problem, described by Equations 2-7, lead to three-dimensional, nonlinear, time dependent partial differential equations. The general numerical solution of these goes beyond the memory and speed capabilities of the current generation of supercomputers. Therefore, we consider appropriate physical assumptions to reduce the computations. 

The vertical reactor simulations reported In this paper typically Involved 14,000 unknowns and took 25 CPU seconds per Newton Iteration on a Cray-2. The tracing of a complete family of solutions for one parameter (e.g. susceptor temperature) cost approximately 25 CPU minutes. The latter number underscores the advantage of using supercomputers to understand the structure of the solution space for physical problems which often Involve many parameters. 

Support for that portion of this work dealing with electrolyte solution theory has been provided to J.E. by the National Science Foundation (CBT-8811789) and by a grant of Cray X-MP time at the National Center for Supercomputing Applications. The authors wish to express appreciation for assistance provided by Nr. John Potter with the development of computer software used in the production of a movie of a nucleation event shown at this symposium. 

Although many routine programs for solving differential equations of all sorts are currently available at all levels of computing, from the desk-top to the supercomputer, it is desirable to have as much understanding as possible of the mathematics that lies behind them. Above all, it is essential to get as clear an idea as possible of what the solution looks like, before going to the computer. 

Three-dimensional models can handle complex cases with fewer empirical features. They require the solution of multidimensional, partial differential equations. These solutions are numerical tools and require the use of a supercomputer, or very long computation times on less sophisticated hardware. 

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