lndtwctional Approach to Molecular Electronic Structure Theory
Clifford E. Dykstra' and Henry F. Schaefer III University of California Berkeley 94720
Theoretical investigations of molecular electronic structure are an increasingly important part of fundamental chemical research (see, for example ( I ) ) . Understanding the methods and concepts used in theoretical calculations can be quite valuable in assessing the literature of electronic stru&ure studies. Because of this, most students encounter some discussion of theory and methods in quantum mechanics courses before beginning research. Unfortunately, i t is difficult, in a formal coirse oresentation. to exnlain theoretical anoroaches in much more than an abstract fo~malism.studentimay then he left with a knowledge of equations and derivations hut little idea of how to apply them. This, of course, makes it difficult for students to evaluate the aualitv of a eiven theoretical result and its relevance or importance to their own research. A successful oroeram for overcomine the limitations of .. formal theory presentations was developed fur a graduate ouantum mechanics course at the Universitv of California at ~erkeley.A primary feature of this prograh was assigning students to write a comouter . oroeram using- electronic .. structure methods presented in the lecture. and then letting them actuallv ~ e r f o r man ah inilio calculation at the level of accuracy typick~of many calculations which may he found in current literature. The results of student calculations are given below. The course in whi& this program was developed was the last of three (10-wk) quarter courses in quantum mechanics regularly offered by the Department of Chemistry. The experiment described here was carried out during the last half (5 wk) of the quarter. The course enrollment is generally small to moderate and consists primarily of first-year graduate students in physical chemistry. Students have generally been a t least introduced to the ideas of self-consistent-field (SCF) theory before taking this course. However, the presentation of course material did not assume any prior knowledge of electronic structure theory and relied only minimally on advanced quantum mechanical formalism. Thus, this instructional approach might he equally workable at an earlier level, such as with advanced undergraduates. After completing the lecture discussion of Roothaan's method (2). students were assiened to write a Fortran c l o s e d - s h e l i ' ~routine ~ ~ with a set of non-orthogonal basis functions. References (3),(4),and (5)provide clear discussions of methods herein referred to. Students were provided with a simple matrix diaeonalization suhroutine and a deck of puncded cards with overlap, kinetic energy, and nuclear attraction one-electron integrals and two-electron repulsion integrals with corresponding (ijlkl) labels for a selecied system. The four electron diatomic ion BeH- was a convenient choice for student calculations. A double-zeta basis set for Be and H atoms (6, 7) was reduced to eight functions by removing the Be p, and p, functions, which are not involved in the closed shell BeH+ ground state. The eight functions were of the contracted Gaussian type. Only unique non-zero ( 5 integrals were included in the punched card deck, so that there were 36 one-electron integrals and the number of two-electron integrals was less than 666. AU 12 students in the course were able to comolete accurate working programs in a period of two to three weeks. This included three students who had no prior Fortran experience. .
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'University of California Regents Fellow, 1975-1976. 310 1 Journal of Chemical Education
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lnternuclear distance (Bohrs) Theoretical potential energy curves for me 'Z+ electronic states of BeHt. Note tb inconed dissociatiar behaviw of tb SCF potential cmve.The grwnd state dissociation limit Is BeC + H, while the two higher 'Ttstates dissociate to Be
+ H'
and Bet(ls22p)
+ H.
Proerams were run a t the camnus comouter center on the C D 6400, ~ a relatively slow malhine by 'contemporary standards. The students were given integrals corresponding to different internuclear separations so that the class results provided a ootential curve as shown in the fieure. With the exception of large separations, conveigence was achieved after 10-15 iteration;. he compilation and exerution time wa3 always less than 16s. In writing their programs, studentsdeveloped a first-hand understanding of orthogonalization of orhitals, density matrices and Fock operators, and the calculation of the total enerw. The results of their HeH' calculations showed the formof molecular orbitals, highlighted difficulties of Hartree-Fock theorv in describing dissociation. and promoted some discussion of thequality of hasis sets. In addition, several students alao ralculated Mulliken . o o.~ u l a tions and net atomic charges. A ranee of methods in theoretical electronic structure were presencd in lecture, including semi-empirical methods and multi-configuration self-consistent-field (MCSCF). The second assignment in the course was to write a configuration interaction (CI) program to do a full valence CI calculation on BeHt. While developing their programs, students were given a discussion of imoortant topics related to ohtainine eorrelated wavefunctiods in the-lecture. Included were
transformation of integrals over basis functions to the molecular orbital basis, use of natural orbitals, and finally, the independent electron pair approximation (IEPA). Ten of the 12 students were able to complete the program in about two weeks. The figure gives the potential curve for the ground state and several excited states. The typical run time for the CI programs was about twice that otthe SCF program. Our experience in providing instruction in electronic structure suggests t h a t t h e understanding of the theory can be greatly enhanced by providing the opportunity to do a theoretical calculation. By carefully selecting the most important parts of a given method for a student to develop, oerformine" an actual calculation can be done in a reasonable amount of time. The choice of an appropriate test case is also important; BeHt seemed to be quite instructive. I t was sufficiently small that program efficiency need not be a prime consideration; yet it showed the value of SCF theory near the equilibrium geometry and the importance of CI in describing dissociation. Overall, we feel that this instructional approach covered many concepts and ideas which could not be fully explored in a lecture. For example, it is easy to prove Brillouin's theorem, hut understanding is aided by actually adding up terms to find zero-valued matrix elements. In some cases
Brillouin's theorem matrix elements were not zero and students were able to trace this to the use of a not fully converged SCF wavefunction and then check their conclusion. Finally, the fact that nearly all students completed both programs successfully and in a short period of time indicates that the approach of actually doing a catculation is something which might be readily incorporated into any physical chemistry course which includes a discussion of electronic structure. We thank the UC Berkeley Computation Center for their support of this instructional project. We are grateful to B. Brooks, D. Castner, J. Goble, S. Hansen, J. Kleckner, H. Luftman, E. Poliakoff, S. Siebener, L. Sterna, W. Swope, D. Trevor, and M. White for serving as guinea pigs for this educational experiment. Literature Cited
111 Srhneler. H. F.. 1Rdiloil. "Mudern Theoretical Chemirtry." Plenum. New Ywk. 1976. 121 1lnLhaan.C.C. J..Rncm. Mod. Phy~..23.W l l 9 i l l . (:I, l*r(dt~..I. A,. nrld ikwrlyc. I). L.. "Aormiximal~ M ~ , l w u l r r Orhitsl Them,." Mdirow-Hill. Nev York. 1970.Seechi~plc~r 2. 1.1) S r h n e l e r . H . F . . " T h e R l o ~ ~ n m a S ~ r i ~ ~ u r e i l A ~ ~ ~ m r : ~ n ~ l M ~ ~ / n . ~ ~ I ~ : A S ~ Quantum Mechanical Bosulb." Addiwn-W~xl?y,I < ~ i ~ d i nMnmrhurettr. c. 1978. 151 McWeeney.R.,andSu~lille.B.T.."MrthndrurM~,leculorUuun~umMechsnie~."A~. sdernic Press. London. 1969. 161 Y n h n y . D . 1Lsnd Schaefpr.H. F..J C h w . PI?)s.. 81.4921 ll974l. 53.2829 l19illl. (71 Dunninl.?'. H., J. C h r m W?>,X..
Volume
54. Number 5, May 1977 1 311
