Computer Simulation of NMR Spectra A. Ellison School of Science, Hull College of Higher Education, Cottingham Road, Hull. HU6 7RT, England The computer programs of J. D. Swalen (NMRITI NMREN) ( I ) and of A. A. Bothner-By and S. M. Castellano (LAOCN3) (2) have successfullv served the research requirements of the NMR spectrkcopist, and continue to urovide an essential tool for the analysis of comulex NMR spectra. These programs, however useful, do suffer from certain disadvantages which arise during the teaching situation and during comparison of the computed spectrum with the experimental spectrum. The origin of second-order spectra and the nature of nuclear soin annroximations such as AB. ABX., AB?. . svstem " .. -. etc... can only be thoroughly understood through the necessary, hut laborious. method of hand-calculation of a t least some NMR intensities and frequencies. The use of spin Hamiltonians, operator techniques, and of equations-of-approximation in this context have been well documented ( 3 )and form an essential nart of manv" oost-graduate studies. Once the com. putational techniques have been mastered and the principles of such calculations understood, it should be expected that the use of computer programs would relieve the monotony and effort reauired for suhseauent calculations. Unfortunatelv, it is not possible to access and subsequently output details the spin Hamiltonian matrix elements generated by the existing programs, LAOCN3 and NMRITINMREN, to provide a numerical check on the accuracy of hand calculation, or to inform on the values of the specific matrix elements which determine the magnitude of second-order effects and the appearance of obs&ed spectra. Computer programs which merely serve to evaluate the equations-of-approximation (4, 5 ) lack generality and do not proceed through the energy matrix to provide line frequencies and intensities. Johnson ( 6 ) does diagnonalize the energy matrix, but the program is severely restricted to the 3-spin case-the calculation of matrix elements is not reneralized and is accomolished throueh specific algorithms. For complex spectra, the programs LAOCN3 and NMRITINMREN often yield a large number of frequencies and intensities which must be either plotted bv hand to effect comparison with the experimental spectrum, or plotted using graphics plotting routines. The first alternative is direct, but extremely laborious, while the second is most often not possible for schools or medium-sized institutions through the sheer size of the computer programs required and the availability of suitable hardware. In addition. because the comuuter often generates manv more NMR lines than are observed, it often proves extremely difficult to relate the observed spectrum to that calculated. This is, of course, the result of the fact that the computer uroduces the totality of theoretical lines with an effective "theoretical spectrometer resolution" approaching infinity. In contrast, the actual resolution of the spectrometer producing the experimental spectrum is finite and dependent on many experimental parameters, so that the observed lines may
of
he envelopes of many unresolved resonances. Finally, in the matching of computed and actual spectra, all 'batch-processed' programs have the inherent disadvantage of delay in the resuhmission of the 'improved' data-set. Wilkins and Klopfenstein (7) used the first-order differential of the eigenvalues to deduce the effect on line position produced by changes in the values of coupling constant or of chemical shift. Although working well for first-order approximations, the method fails when second-order effects become important and the procedure does not overcome the time-delay factor. For these reasons this new PASCAL oroeram has heen . written to provide for the interactive analysis and display of the hirh resolution NMR suectra from suin usinr a . %nuclei . hard-&py or visual display unit. General Program Description The program calculates the frequencies and the intensities of lines expected in the NMR spectrum for a system of two to six nuclei with spin I = l/2. A set of chemical shift and coupling constant data chosen for the molecule of interest is used to generate a table of freauencies and intensities for the lines expected in the NMR sbectrum. The calculated 'stick' spectrum is displayed on the hard-copy or VDU terminal used to input the data. The frequency scale and spectrum line-resolution can be altered interactively so that, where necessary and reasonable, the calculated spectrum can be altered to afford comuarison with the observed suectrum. A better-fit to the
-
justment of the frequency-scale aidlor line-resolution parameters. The PASCAL program has been written and tested using a Norsk Data NORD-100 computer with 2 kh cache, 1 Mb main memory, and 180 Mb disk storage. The full six-spin version of the program has the followingstorage requirements: source . nrorram. ~. , 20 kb: run-time. 90 kb. These reauirements may be altered very easily as described below. T h e computed results have been checked for accuracy for all spin systems, from two to six spins. Systems consisting of two and three spins have been checked by manual calculations. Results for all systems have been compared with experimental spectra and with results from LAOCN3 and from NMRITINMRENI. The example running times given here are affected by queuing problems within the computer system used in the interactive mode and therefore may well not be truly representative. For a single calculation the run-time values are: 3-spin, 2 sec; 4-spin, 43 sec; 5-spin, 110 sec. Theoretical description of the program For the analysis of an S-spin system, a set of 29 hasis spin functions are symmetrically created hy an algorithm. For example, for a 3-spin system, 2:' = 8 functions have the form (mmnl, (cup~ul,(aLuPI . . . (PNPI,(@PI. HereaandP,inDirac Bra-Ket notation, represent the values +'iih and -'hh re-
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spectively of I,,,the z-component of spin angular momentum associated with nucleus i. The Hamiltonian matrix is computed from the high resolution spin Hamiltonian
.% =
S
S-1,s
zuJ,, ,=,+,z &it . ij
,=,
i=,
which has been used extensively and shown to hold within experimental error. Here v, is the resonance frequency (Hz) of the nucleus i in the absence of other nuclei, I,; is the zcomponent of the spin angular momentum operator associated with nucleus i, J , is the nucjear spin-spin coupling constant between nuclei i and j and I, is the nuclear angular momentum operator for nucleus i. The summations are over all S nuclei and all uairs of nuclei i i > G I . The elementsof the spin Hamiltonian matrix are calcilated from the formulae S
Hmm=
7-1
S
Z ivi x Em,)+ ,=, ,Xa =Z j=, j+1
(Jjk X
Em, X Emh)
where H,,, is the diagonal matrix element for wavefunction $, ( m adopts values between 1and 2") and E, is the expectation value of the integral ($; I I, I ),
where H,,
is the off-diagonal matrix element connecting basis
where 4, and @, are the spin components of basis functions and corresponding to the nuclei p and r. Non-zero off-diagonal elements only occur between wavefunctions having the same total spin (i.e., sum of a and 19spin components) and when only one spin compopent changes a 19and a through the operation of I+and I-. one The Hamiltonian matrix, of order ZS X ZS, is not simplified by symmetry, and is directly diagonalized using SYMQRTO (CACM alaorithm 254F21, wroducing the eigenvalues and normalizedeigenfunctions of the real &met& matrix. The transition frequencies are simple differences between these eigenvalues, in value order, ul,, =El -Em. The corresponding transition intensities are proportional to the matrix element . expression If,, .: (Z,Z,c;lPijcj,)' where cil and c j , are the coefficients of the basis functions in the ejgenfunctions xl and x,,. The transition moment operator, I, or f,, operates on any spin part of the eigenfunction xi, itself in general a linear combination function. If is a component function of x1 and is a component function of ~ 2 transition , from XI to ~2 is only possible, in part, if the sum of the a and components of differs from that of $e by +1or -1. A further condition for an allowed transition is that the sums of spins
+,
+,,
-
-
+
Calculated NMR Line Frequencies and Intensities for an ABX System, All J Positive, Relatively Large Jnx Value Line
in the rb component of X I and ~1 should contain only one zero. P, is the expectation value of the transition moment matrix integral
Frequency /Hz NMR LAOCN3
Intensity NMR LAOCN3
and has the value 0.5 for an allowed transition and the value zero if forbidden. Transitions can be accumulated into a swectral envelowe . bv. means of a 'degeneracy' parameter so that the experimental line resolution of the spectrometer can be imitated. Transitions whose energies are degenerate, as defined by this parameter, are deleted and their corresponding intensities summed. Spectrum display may consistof all Gssihle lines or of a selected group of lines a t the chosen 'resolution', the selected set of lines being displayed with normalized intensities. Capabilities of the Program The drogram accepts a set of chemical shift and coupling constant data which is considered to he representative of the molecule and which is obtained from tables, from consideration of the experimental spectrum, or from a set of "guessed valu~s. As a teaching aid, the program can yield as output the set of wavefunctions for the S-swin case reccuested. the calculated spin Hamiltonian, and the stationary-state eigenvalues and eieenfunctions. in addition to the transition energies, transieffects which particular matrix elements have on the appearance of NMR spectra, for example, in the differences between various approximations ABX, ABC, AB, AX, etc. Inwut and output are performed interactively with the comtion parameters, to conform to the experimental spectrum. As data, the program requests:
(mz)of the NMR spectrum; (b) an identifying legend; (c) the number of nuclear spins; (dl the supposed NMR frequency (!Hz) of each spin (chemicalshift !Hz); (el a set of coupling constant data; (f) a number of control parameters specifying (i) that full, intermediate spin Hamiltonian output is required; (ii) that the spectrum display is to occur on VDU or on h a d copy terminal; (iii) the resolution and frequency scale for the spectrum; (iu) the part of the spectrum to be displayed; ( u ) that . . . . a new display is required (new parameters (iii) and (a) the radiofrequency
(LU));
(ui) that a recalculation is required (new parameters (dl to
if)). As output, the program yields: (g) all input data; (h) when requested: (i) a table of spin wavefunctions used in the calculation; (ii) the calculated spin Hamiltonian matrix; (iii) an ordered list af eigenvalues and table of corresponding eigenfunctions; (i) a complete table of transition frequencies, (Hz) and (ppm), and intensities; ij) a re-computed table of frequencies and intensities appropriate to the spectrum resolution requested; (k) a spectrum display of the lines in (j) using the requested f r e ~ quency scale.
Comparison of the computed 'stick' spectrum with the experimental spectrum may require a new display using the original parameter set, i.e., 6 ) and (k), or recalculation of the spectrum using new parameters yielding (g) to ik). 426
Journal of Chemical Education
Figure 1. Computed NMR speclrum display for an ABX system. all Jpositive, relatively large J~~value. (a) using the program NMR and a hard-copy terminal: (b) using LAOCN3 and a graphics printer.
Examples of the Use of the Program
Figure 2. Computed NMR spectrum display for ABX systems (a)large Jsx, JAx negative; (b) small JBX,all Jvalues positive.
However, once the 'method'has been assimilated, no further useful purpose is achieved by the repetitious evaluation of the energies and intensities for different data-this task is better performed by the computer while retaining the facility for 'checkine' the numerical value of matrix eliments and their ~ ~ consequences. The molecules 1-uhenvlethanol and 2-uhenvlethanol serve ~
The table shows the calculated line intensities and frequencies for a simulated ABX-spin system with all J values positive, J s x having a relatively large value, using the new program reported here (NMR) and, for comparison, using LAOCN3. As data, U A = 110 Hz, U B = 100 Hz, ux = 190 Hz, JAB= 8 HZ,JBX= 15 HZand JAX= 5 Hz. The values of both frequency and intensity agree well with those from LAOCN3. Fieure l a shows the disnlaved soectrum obtained on a suppressed in these figures for the sake of clarit; of reproduction. The inserted 'break' between the two grouus of lines saves printing time and space. Full details of its implementation can he found in the program documentation. A graphics-printer display of the LAOCN3 result is shown in Figure l h for comparison. The value of this program in exhibiting the manner hy which the NMR spectrum of a three-spin, ABX system depends upon the sign of the J values and upon their relative magnitudes is demonstrated hy the spectra in Figure 2. I t is imoortant that students aonreciate the eross differences in .. spectra which can he observed, an appreciation only meaningfully . . achieved hv uerformine the hand calculations of at least one three-spin&ohlem, so chat the importance of certain matrix elements in the approximation can he understood.
.
~~
~
~
~~
A
or
constants, the computer program produces more lines for the aliphatic urotons of the molecules, each a four-soin svstem. than are actually observed using a 60 MHz spectrbmetkr. example, the calculation for 2-nhenvlethanol uroduces 14 lines,khereas only 6 lines are observed. By suitable choice of the spectrum resolution parameter in the program, the suectra considerations. I t is a simple matter ti adjust suhsequentl; the input shift and coupling constant data to obtain coincidence between the exuected and comnuted soectra. The program storecan he easily reduced for use by institutions with restricted computing facilities while still retaining considerable value as a teaching aid. Hand-calculations are perhaps only realistically vossihle for two- and three-soin systems, forkhich the essential arrays in the program caihe greatly reduced, saving considerable computer space. The arrays are dimensioned for six-spin systems and should he re-dimensioned [..a] where appropriate for three spins maximum, i.e., arrays Ti, T2, T6, T7, RA1 to RA5 in lines 3 to 16 of the program listing. The program has been submitted to the Quantum CbemVolume GO
Number 5
May 1983
427
~
-
-
Figure 3. Computer
simulated 60 MHz spectra of the aliphatic protons of: (a) I-phenylethanol;(b) 2-phenylethanol
istry Program Exchange, Indiana University, USA, f r o m where copies with complete documentation can be obtained. and of details of hand.calculations can copies of the also b e obtained from t h e a u t h o r for applicants i n t h e U.K. Or Europe Or from if a contribution postage is forwarded.
Literature Cited (1) Swden, J. D., (NMRITlNMRENI, "Computer Programsfor Chemistry,"DeTar, D. F. (Editor) Benjamin,New Y o ~ k 1968,Vol , 1 , C h 4. (2) ~ ~ t h n ~ ~ - ~ y , ~ . ~ . , s n d c a s t eM., i ~(LAOCN3),'Computer~rogramsforChemsno,s. istry:DeTar, D. F. (Editoil Benjamin,New Ynrk, 1958,Vol 1,Ch.3. (3, Abraham, R. J.. "Analysis of High Resoiuthm NMR Spectra: Elsevier, New York,
1971.
141
Acknowledgment
Wihere. K. B.."Phvsical O~zanicChemistry: John Wiley, NewYork, 1964, pp. 19&209
andj~~-5~3. Johns"". K. J.. "Numerical Method8 in Chemistry: pp. 92-111. (5) Juhnson,K. J..Wumerical Methods in Chemigry: (5)
The assistance Of Mr. D' Joslin' Director of puter Services, a n d members of his staff, is greatly appreciated.
Marcel Dekker. New York, 1980, MarcelDekker. New York, 1980,
PP. 429.437.
(7)
wilkins,C. ~ . , m ~d h p f e n d e i n , E.. ~ . J. CHEM. EDUC.,43,10,
(1966).
Contest for Instructional Computer Programmers Project SERAPHIM, an NSF-DISE-sponsored project far disseminating instructional modules in chemistry, will hold its second instructional computer program contest in conjunction with three meetings this summer. Judging of contest entries will he carried out at Chem Ed '83 (Indianapolis, IN, August 2-8, at the Workshop on Computers in Chemical Education-EAST (Richmond, KY, August 7-10), and at the Workshop on Computers in Chemical Education-WEST (Lawrence, KS, August 7-10). A prize of $200 will be awarded to the author of the instructional program judged best hy participants in the workshops, and prizes of $100 will be awarded for the best program written by an author in each of three categories: (1) secondary school teacher, (2) two-year college teacher, (3) four-year college teacher or non-teacher. Programs may deal with any aspect of chemistry at any level, from high school to graduate courses. All programs entered must he able to run on a t least one of the following microcomputers: Apple 11, Atari 800, CommodorePET 4000 series, Commodore 64, Radio ShackTRS-80 Model1 or 111,Radio Shack Color Computer, and IBM Personal Computer. All authors must agree to release their programs for distribution by Project SERAPHIM, whether or not the program wins a prize, and programs may not contain any copyrighted code that would prohibit their distribution by SERAPHIM. Criteria used in judging the programs will he the same as those used in writing software reviews for THIS JOURNAL (see the December 1982 issue far more details). All attendees at any of the three meetings listed above will be eligible to review programs and will he asked to complete program evaluation forms for all programs they interact with. The forms will be tabulated by Project SERAPHIM staff, and the winners will be announced in THIS JOURNAL in the fall. To enter the contest you must complete an entry form and mail it hy July 8 to: Dr. John W. Moore, ProjedSERAPHIM, Department of Chemistry, Eastern Michigan University, Ypsilanti, MI 48197. Programs on floppy diskettes must be received at the above address by July 29 in order to he distributed to the judging sites on time. If documentation is needed in order for a program to he used appropriately, that too must arrive by July 29. Copies of the review criteria, entry hlanks, and other information can he obtained from the address above at any time.
428
Journal of Chemical Education
