The Role of Theory in Chemistry F. A. Matsen The University of Texas, Austin, TX 78712 While theory has played and continues to play an important role in the development of chemistry, it has received relatively little attention in the high school and undergraduate chemistry curricula. This is unfortunate since ignorince of the role of theory can seriously hinder both the acceptance and the utilization of modern theory. Before we proceed too far, we should agree on the meanings of important words that are used in discussing theory. We start a t the beginning and define science as a set of observations and theories about observations. We then defme theory as a device for making predictions and correlations of observations. A theory is composed of axioms, which are not necessarily self-evident, procedure, and the output of the procedure (Fig. 1).The axioms identify the system, select the procedure and its parameters, and interpret its output. Each theory is judged by the followingpragmatic criteria listed in the order of decreasing importance. How diverse is it? How accurate is it? How simple is it? Like all science, theories evolve; they do so because the basis of our scientific knowledge is constantly changing. The best theory a t a particular point in time is the theory that best satisfies the above criteria. I t is not to be judged on a political or a religious basis. For example, Darwinian evolution is a theory for making predictions and correlations of biological data. I t has been vigorously attacked on both political (Communist) and religious (fundamentalist) grounds. Despite these attacks and despite its frequent minor modifications i t remains the best theory for the study of biological change. Through the centuries i t has been found that absolutist concepts like "absolute truth," "objective reality," and "ultimate meaning" are not needed for science and are, in fact, counterproductive for its development. This view (as it relates to the quantum chemistry procedure) is humorously (but ac(see also curately) portrayed on the cover of THIS JOURNAL Figure 3). This is not to say that such concepts are not valid in other modes of thought or intellectual endeavor; they just have not been useful to the progress of science. A theory evolves as shown in Figure 2. The axioms are conceived in the mind of the theorist who also may double as an experimentalist. The outout of the theorv. are . oredictions and c~orrelationsthat may suggest new experiments to the experimentalist (dotted line in Fia. 21. The ~redictiomand co&elations are then compared with the observations. If the
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PROCEDURE
agreement between predictions-correlationsand observation isUeond" the theorv is a"eood" theorv, .. which isa o r a m a t i c valie judgment. If the agreement is poor-wh~ch occurs sometimes because new observations have been made-a better theory must be found by some theorist generating new axioms and a new cycle. This axiomatic-cum-oramatic (ACP) cycling is continued until the agreement between theoiy and experiment becomes "good." We call this orocess the ACP rp&trmolog) of science because (1) epif/umology is the acquisition and validation ot'knowledaeand (2) we wish todist&uish our simplistic view from themore erudite views of the professional philosophers. The ACP epistemology can be applied to areas outside of science. For example, all political beliefs can be considered to be theories for the common good that are hased on a certain set of axioms. Many political crises would not occur if the political antagonists would publicly admit the axiomatic character of their beliefs. Unfortunately "true believers," be thev on the extreme rieht or the extreme left. behave as thourh their beliefs are not b&ed on axioms at all but are self-evidtnt "truths" miraculously delivered to them on stone tablets. Theories of Matter There is a wide variety of chemical and physical theories from which we select as our example theories of matter (atoms, molecules, solids, nuclei, and elementary particles). We can illustrate the ACP epistemology with the familiar ball-andstick theory of molecules. This theory employs sticks and colored balls with holes drilled in them a t prescribed angles. The procedure consists of assembling the bklls and sticks;nro figures in all possiblc ways. The predictions of the theory inr h d e molecular geometry and the numtrr of isomers exwcted for the molecule inquestion. We all feel comfnnable with this ball-and-stick theory because i t operates in the three-dimensional, classical world of our senses and seems "real" to us. While i t is avery useful theory, i t has a number of significant failures. For example, it fails to predict both the geometry and the number of isomers of benzene. More seriouslv. .. it fails M, predict the electronic, vibrational, and rotational spectra of molecules. This failure to predict soectra is common toall classical theories and has made necessary the development of a new theory which includes predictive powers in this area. The nonclassical quantum theory is a theory that predicts
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Figure 1. T b structure of a theory
Figure 2. An ACP (axiomatic-cumgragmatic)cycle.
Volume 62
Number 5 May 1985
365
HAMlLTONlAN
SECULAR EQUATION DET([H(x)]- E[l]) = 0 I EIGENVALUES
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PREDICTIONS SPECTRA THERMODYNAMICS
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Flgve 3. The MMTM Procedwe, more diversely, more quantitatively, hut not more simply than the hall-and-stick theory. Quantum theory has two essentially equivalent versions: one concerned with wave mechanics (derived from Schroedhger's work) and one concerned with matrix mechanics (due to Dirac-Heisenberg) which is our choice. The procedure of the matrix mechanics theory of matter (MMTM) (which has taken 50 years t o develop) is shown in Figure 3. Its history is very interesting hut is not relevant to the application of the procedure. I t employs vector spaces and their bases, operators, matrices, secular equations, eigenvectors,eigenvalues, groups,' group algebras. etc.. concents that are well known tomathekatiEian'i but essentially &known to beginning physicists and chemists. In consequence MM'I'M may seem less "real" to these beginners than the classicill ball-and-stick theory. At the University of Texas we have developed a successful course for first-yeaistudents which requiresno prior knowledge of these concepts since they and MMTM are taught in ~arallel with the ~ " c k e ltheory-of organic chemistry.- he MMTM procedure can be applied uniformly to atoms, molecules, solids, nuclei, and elementary particles. I t is clear that the concept of structure is much simpler and more intuitive in the hall-and-stick theory than in the MMTM theory. The MMTM structure concept is that of a set of huildiug blocks (hasis vectors of a vector space) that are msembled under the supervision of ihe Hamiltonian into a physically significant set of structures (eieenvectors to the Hamiltonian): ~~~~~~~, The numerical'aGd algebraic calculations required in the MMTM orocedure can become ouite tedious hut fortunatelv many of ihem have been or can be programmed for person& computers. The calculations of MMTM [hen Imome triviid and operational familiarity is quickly acquired. Consequently, the challenging - - -Dart of MMTM hecomes the selection of the vector space and the Hamiltonian and then the interpretation of the output. ~
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' Symmetly is mathematically realized by the concept of a group. lb use factorssecular equations and assigns quantum n~mbers.For example the quantum numoer n. I. and mot the hyarogen atom have group theoretical origins.
366
Journal of Chemical Education
Molecular Theory In the previous section we dismssed the failure of the hall-and-stick molecular theory and suggested that MMTM is a better molecular theory. In the ab initio MMTM molecular theory the only parameters required by the procedure are Planck's constant, the charge and.mass of the electron, and the number and kind of nuclei. This theory can, for many molecules, predict with high accuracy their equilibrium geometries and their force constants. Unfortunately it predicts other properties, e.g., dissociation energy, electronic spectra, etc., with a lower accuracy. The accuracy can he increased by the use of larger vector spaces. a techniaue that can he verv difficult,veryexpensive, Adlor impoasibie with the current& available scalar computers. The ~ r o h l e mbecomes easier with supercomputers that employ vector andfor parallel processors and larger memories, hut there will always he some upper limit to the size of a molecule on which accurate ah initio calculations can he made. A theory which is less strongly computerdependent is the semiempirical molecular theory. Its procedure employs a smaller vector space and its parameters are determined by comparison of predictions with a small number of obsewations. An example of a semiemperical molecular theory is the s-electron theory of conjugated, unsaturated hydrocarhons. Here the size of the vector space is reduced by ignoring core and u-bonded electrons and employing a single a orbital for each carbon site. The size of the vector space can he further reduced by the use of either of two more approximate theories: the a Hiickel molecular orbital theorv. which resemhles the Bohr theory of the atom, and the i " a l e n c e bond theory, which resembles the hall-and-stick theorv. I t would he useful if the chemical education community had available a simple theory of matter which could he taught to beginning chemistry students in a useful way and which contained the elements of the more populsl theories. Matye Anne Fox and I have developed a s-electron theory based on the reduced Hiickel-Huhhard Hamiltonian, H(x) where x , called the fractional-valence-bond character, is a parameter such that a t x = 0 we recover Huckel theory and a t x = 1 we recover valence bond theory. Experimental data suggests that the "best" value of x is in the neighborhood of 0.5. By the pragmatic criteria outlined above this theory has some diversity, qualitative accuracy, and great simplicity. Further the simplicity of the Huckel-Huhhard theory makes i t a good theory with which to introduce the many-body aspect of MMTM and to relate it to other one-body problems.
The MMTM Course MMT.U isa freshman hu~luffi~hcmistryrourseat the i!nivereity uf'l'exasat Austin for srudcnts wh~,scoreGj0orabov~~,n rhc math SAT.It employs t h X~ I ' cpicttmology. vector spacea, operators. matrices, secular equations, eigenvalues, eigenvedors, groups, and voup algebras. MMTM treats theories of atoms, molecules, solids, microelectronics, nuclei, elementary particles, and relativity. The organic chemistry employs the HOckel-Hubhard theory and is described in the followine oaoer. MMTM satisfies th;freshman chernistrv ~~~~-~~.reauirementa in most I degree plans or may be offered for advanced credit. It has been taught to an average of 50 students per year for 16 years and has produced a large group of alumni who understand fully the role of I theory in both chemistry and physics. ~
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