Charles 1. Wilkins and Charles E. Klopfenstein University of Oregon Eugene
Simulation of NMR Spectra Computers as teaching devices
T h e college chemistry course of today is vastly different from that of the past. This is an obvious and, indeed, a necessary result of tlie discoveries which are the basis of an experimental science. A concomitant search for new ways of teaching is a desirable and necessary result of the curriculum changes. Fortunately, today's technology provides new tools which can be adapted to instruction and, p~operlyused, can give the student a great deal of ins~ghtinto the theories of modern chemistry. The increased availability of computers for use in colleges and universities has made computer-based teaching aids practical as instructional devices. There are two basic ways in which computer programs can be used as teaching aids. First, programs may be written which will simulate a problem or phenomenon; second, programs may be written which can he applied to the mathematical treatment of experimental data. Both types have been developed a t the University of Oregon. Examples of the first type are the nuclear magnetic resonance (NMR) simulation program (described here) and programs which do Hiickel molecular orbital calculations (1,s). Others of this type could be developed to do such things as crystal field calculations (3) or electronic spectra calculations (4). These are all of value in demonstrating the power of the variational method and its wide applicability to chemical problems. The second type of program is exemplified by one we have developed which calculates firsborder rate constants and activation parameters by the least squares method. This is particularly useful in the demonstration of the great sensitivity of the activation parameters to small errors in experimental rate constants. Others have developed similar programs (5). Naturally, any program for the treatment of experimental data may be used to simulate problems merely by the introduction of "manufactured data." This gives these prog r a m an added utility. Many uses suggest themselves; to give one example, students in a physical organic chemistry class were mked to do Hiickel molecular orbital calculations for one or two organic compounds. The necessary input parameters were submitted by them and the calculations done by the computer. The students were then required to make an assessment of the meaning of the results and to make qualitative predictions concerning the chemical behavior of the compounds. Described here is an example of a spectrum simulation program, which may be used to simulate an hypothetical system or to analyze an actual system. This Financial support of the University of Oregon Computing Cente~ is gratefully acknowledged.
10 / Journal of Chemicol Education
computer program calculates the nuclear magnetic resonance (NMR) spectrum for a three-proton spin system. It may be used to illustrate the applicability of computer techniques to chemical problems and also to show graphically the relations of the appearance of the NMR spectrum to chemical shifts and coupling constants. It is particularly useful in the demonstration of the effects of varying the coupling constants or chemical shifts, in that a large number of theoretical spectra can be calculated in a short time. More complex NMR data may be treated when larger computers such as the IBM 7090 are available for use. Programs which calculate the theoretical spectra for systems with up to seven spins have been written ( 6 , 7 ) . I t is also possible to write programs which will analyze observed spectra by means of an iterative least squares treatment (7). The three-spin NMR calculator herein described could easily he adapted to such analysis. I t should be pointed out that the matrix diagonalization subroutine used in the NAIR program is general and may be used to diagonalize any real, positive-definite Hermitian matrix. Nuclear Magnetic Resonance Theory
The nuclear magnetic resonance spectrum arises from transitions between different energy states of nuclei. In the absence of an external magnetic field, a proton will have only one energy state. However, when a magnetic field is applied, two energy states will existone in which the proton's angular momentum vector lies in the direction of the applied field and one in which the vector is opposed to the field. Thus, the spin wave function may be a (opposed to the field) or 0 (aligned with the field). The energy for either state mill be given by E
=
.f
*X*d,
(1)
When there is a spin system of more than one nucleus the Hamiltonian operator may be divided into two parts. X = Xi01
+ Xu1
(2)
X(n represents the interaction of the nuclei with the applied field in the z direction and is given by
where y,+yo is the resonance frequency of an uncoupled nucleus i in an applied field H,,yo is some reference frequency, and y, is the the chemical shift in cycles per second relative to yo. 3CI,) represents the indirect spin coupling, i.e., the perturbation caused by interaction between the nuclei.
This perturbation operator is 3q11
-
W a v e Functions and Total Spin Components for the Two Proton Care (4)
Jdii).I(A j>i
where J t j are the coupling constants given in cycles per second. The spin operators I(,,, I(,,, and I(,, are defined conveniently in terms of matrices few generalizations. For the diagonal elements, bearing in mind that the wave functions are orthogonal and normalized and that an operator I(1) operates only on the nucleus (1) in the wave function a(l)a(2), the Xcol may be evaluated. For example: I,(1) operating on a(l)a(2) gives +'/zaa and 1,(2) operating on a(l)a(2) gives +'/paw. Substituting this into equation (3), we find that the Xto,part of X,, is 'lz(yl 7,). The X(,, portions of the diagonal elements may be evaluated from equation (4) with the numerical result that
and, using vector notation
+
,%K
I1 can thus'be seen tabatthe effect of the spin operators on a and 10 will be
(18)
= *'/J
