edited by JOHNW. MOORE
Bits and Pieces, 13 Most authors of Bits and Pieces will make available listings and/or machine-readable versions of their programs. Please read each description carefully to determine compatibility with your own computing environment before requesting mate& from any ofthe authors. Revised guidelines for authors of Bits and Pieces appeared in the Decemher 1982 issue of the JOURNAL.
Automatically Finding Eigenvalues in One Dimension and for a Simple Chemical Bond Robert Hunt Anderson
Western Michigan University Kaiarnazao, MI 49008 The one-dimensional Schrodinger equation is usually solved for a few simple potential energy wells. The average chemist^ student has some trouhle with the finite square well to say nothing of the twin-well problem. Since this is the simplest case of anvthine resembline a chemical hond, it is natural that " most chemists do not really have much conviction of the importance of Schrodinger's equation to chemistry. The reason appears to he the involved mathematics required to obtain the eigenvalues and eigenfunctions in all hut the simplest cases. In addition, for most well shapes, especially asymmetrical wells, there is no way except numerical methods to obtain the solutions. Melander ( I ) in 1962 proposed the twin square well as a model to increase the students' understanding of the chemical bond. Several programs have been described which calculate or plot psi versus X (2-5). With the exception of Johnson (2) these programs simply produce a solution of psi versus X. It is left to the student to trv different values of the enerev until "
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the energy until a well-behaved solution is found for the auantum number svecified. It handles asvrnmetrical potential
figures which they spent'30 minutes getting in an e&er assignment. The user need only write a simple FORTRAN function for the potential energy well. For a class assignment this may he furnished, and theprogram may then h&un with no knowledge of FORTRAN. It enables students to solve easily the Schrodinger equation for any V ( x ) .One option permits psi and X to be stored in a data file which can later be plotted usine a nlottine vroeram such as our GPPLOT (6). Figure 1 gives these results more easily than the mathematical procedure described by Melander (1). This program should he useful to acquaint students with the role of quantum chemistry in explaining the energy of the chemical hond. I t can also provide a better understanding of well-behaved functions and the relation between eigenfunctions and the corresponding eigenvalues. Thirdly, it can he used to obtain eigenvalues for various potential wells in a simple and uniform manner, thus separating the physical
Figure 1. The energy eigenvaluesof atwin square well potential of constant depth of 10 hamees as a function of R, the distance between the centers of the wells. Eo and E, show a bonding effect and correspond to symmetric wave functions.
internretation from themathematical process. The work of getting results is much less than by other methods. SCHROD is an interactive FORTRAN program with 490 statements and 240 comment lines. E x e c u ~ i o require ~s 12K 36-hit words or 24 pages in core on a DEC-10. The source program is 39K including the six subroutines. A HELP command gives instructions to the user. Documentation includes a listing and a sample execution. These are available for $4. The program is availahle on 9-track magnetic tape written in ASClI (or EBCDIC) at 1600 Bpi (or 800) 20 records per hlock, 80 characters per record for $15 to cover postage and tape charge. Make out check to Bob H. Anderson and send to the Chemistry Department a t the above address.
Real-Time Computer Simulation of Aqueous Equilibrium J. W. Schilling Trinity University San Antonio, TX 78284
The program described below illustrates that a model based on simple kinetic arguments explains why the reaction quotient approaches a constant value called the equilihrium constant. I t also illustrates the idea of the position of equilibrium; Le Chatelier's Principle; the mass action principle; the principles of electrical neutrality and mass balance; and the influence of the forward and reverse rate constants upon the value of the equilihrium constant. I t has both qualitative and quantitative fidelity to the actual chemical process being represented and is appropriate for physical chemistry and advanced general chemistry students. Volume 60 Number 1 January 1983
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