However, unlike the molecular case where the Fock matrix elements are evaluated exactly, in the polymer case they are set to zero outside the domain running from cell -N to cell +N and approximated within this domain. In other words, where \j\ is greater than N, the number of neighbor cells used in (a)above,F1 is set to zero. Where \j\ is less than or equal to N, F ; can only be approximated. It cannot he determined accurately since, as seen from eq 2.14, (1) the evaluation of its electron repulsion terms involve D?; which is approximated as in (b) above. (2) N i n e a 2.14 has to he limited. although not necessarilv to the same value as in eq 2.13.~ong-&ge electron repulsion terms have to be neglected, otherwise the number of two-electron integrals becomes excessively large.
As indicated above both series converge, although eq 2.13 converges more slowly than eq 2.12. Both are truncated, sometimesas severely a s N = 1o r N = 2. These are referred to as the nearest neighbors and second nearest neighbors apnroximations. Recently. has been made usine- a ..much .nroeress " multipole expansion approximation to the most important neglected terms (15,16). (b) approximate, by numerical integration, the elements of the density matrix. Unlike the molecular case where the elements of the density matrix are obtained throueh the straiehtforward summation.
with polymers we are faced with the evaluation of a definite integral given by
Brillouin zone Although such integrals can be evaluated very accurately by muchmore sophisticated techniques of numerical integration (quadrature), in this case, since the approximation is not particularly sensitive, Simpson's rule may be used. Here, the function to he integrated is approximated by a curve and the Brillouin zone is divided up into an even number of evenly spaced k points (usually around 30 points). The integral is then approximated as the area under this curve. (c) approximate the lattice sums F ; occurring in the expression (eq 2.13). We have discussed the approximation of YPÈ(kin section (a) above, we are now looking a t an approximation within this annroximation. First we note that the matrix elements@ are similar to the Fock matrix elements met with in molecular calculations. The only difference being that in the polymer this element is evaluated over an atomic orbital \¡i the zeroth cell and an atomic orbital xi in the jth cell. ~~
~
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Taking into account approximations (a) though (c) we see t h a t a b initio calculations on polymers are considerably more approximate t h a n a b initio calculations on molecules. I n view of this one might well say of a b initio calculations on polymers, "There ain't no such animal." Recognition of this fact has lead t o t h e development of equally accurate alternative approaches (6-16). Literature Cited 1. Lowdin, P. 0. J. Chem. Phys. 1950,18,365. 2. Del Re. G.; Ladik,J.;Biczo, G.Phys. Rev. 1967,155,997. 3. Andr6, J. M. J. Chem. Phya. 1963.50.1536. 4. Andr6 J. J.; Ladik, J.; Delhalle, J. Electronic Structure of Polymers and Molecular Crystals: Plenum: New York, 1975. 5. Andr6, J. M.; Delhalle,J.; Ladik, J. Quantum Theory of Polymers, Reidel, 1978. 6. Andr6, J. M.; Bredaa, J. L.; Delhalle, J.; Ladik. J.; Leroy. G.; Moses, C, Recent Ads in the Quantum Theory of Polymers, Springer-Veriag: New York, 1980; Lecture Notes in Physics 113 113. e
7. Ladik, J.; Suhai, S. Theor. Chem. (London) 1981.4.49, 8. Kerteaz,M.Ado.QuantumChem. 1982,15,161. 9. Duke.6. J.; O'Leary,B. Chem. Phys. Letts. 1973,20,459. 10. Leroy, G.; Peeten,D.; Rosoux-Clarisse, P. Bull. Soc. Chim. Eek. 1976,85,629, 11. Dehalle, J,; Andr6, J. M.: Delhalle, S.; Pivont-Malherbe. C.; Clarisse, F.;Leroy, G.; Peetera,D. Theor. Chim. Ada 1977,43,215. 12. Duke,B.J.;O'Leary,B.Theor.Chim.Acta 1983.62.323. 13. Duke,& J.;O'Leary,B.J. Chem.Phys. 1983,79,3424. 14. Duke,& J.;O'Leary,B.Int.J. QuantumChem.Sym. 1984,18,407. 15. Harris, P. E.;Monkhorst. H. J. Phys. Rev. Letters 1969,23,1036. 16. Saunders, V. R. In "Ab initio Hartree-Pock Calculations for Periodic Systems";J. h e m Sc. Faraday Symposium 1984.13.79-84.
"Trivial Pursuit" for Chemists No one denies the importance of teaching descriptive chemistry, as many recent articles in this Journal1 attest. However, getting students to appreciate some fundamental aspects of chemical behavior can be a challenge. One method that has been successful in my Advanced Inorganic class is to use the hoard same "Trivial Pursuit" as the vehicle for presentation of descriptive facts. Students were assiened specific sections of their text dealine with descrintive chemistry for readine. Then a series of questions was prepared based on that reading assignment (60 questions2 were more than sufficient for a one-hour game with six participating teams). At the start of the game the class was randomly divided into six teams. To insure that the game consent was taken seriously, it was stated at the outset that each member of the winning team would receive five bonus points on the next hour exam. The rules of the actual game were used with the following modifications to allow the game to be completed within a one-hour period. First, all of the categories became "science" categories. If a team correctly answered a question while on a wedee space, they received that colored wedee and each of the six differently colored wedees had to be acquired as usual. ~econd,ifa &a& failed to correctly answera question, that question passed to the next team. heref fore, the instructor served as moderator. If no team could answer the question, the answer was read and a new question was drawn for the next team in succession. Upon answering a question correctly, a team was entitled to another roll of the die and a subsequent question as in the usualversion of the game. Once a team had the six wedges, they had to land in the center of the board and answer one final question to be declared the winners. This format made a nice break from the usual lecture class. Also, questions that were missed actually served to reinforce facts that were not "trivial".
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Zuckerman, J. J. J. Chem. Educ. 1986, 63,829; Lagowski, J. J. J. Chem. Educ. 1985, 62,915. 2Sample questions: (1) Name an element that occurs as a tetraatomic molecule. (2) What is the principal method of elemental chlorine gas production? Diana L. Sedney Williams College Wllllamsiown, MA 01167
Volume 85
Number 5
May 1988
383
