- Email: [email protected]
Correlation analysis of ESR spectra on a small computer
JOURNAL
OF MAGNETIC
67,42-54
RESONANCE
( 1986)
Correlation Analysis of ESR Spectra on a Small Computer ANN
MOTI-EN*
AND JORG SCHREIBER~
Laboratory of Molecular Biophysics, National Institute of Environmental Health Sciences, P.O. Box 12233, Research Triangle Park, iVorth Carolina 27709 Received September 19, 1985 A correlation approach to computer solutions of electron spin resonance spectra has been adapted for use on a small computer. The correlation analysis is capable of finding solutions for complex, well-resolved spectra, including those containing spin-l species such as nitrogen. Even for spectra that can be solved readily by hand, a tuning procedure using correlation as a criterion for goodness of fit saves much time and effort. A fast procedure for adjusting hyperfme constants to within approximately 0.05% of the scan range is presented. Fast methods of calculation for other parts of the correlation analysis are described and their application to the aminophenoxy and the tetramethylaminophenoxy radicals is shown. 0 1986 Academic Pm, Inc. INTRODUCTION
The most difficult part of interpreting an electron spin resonance spectrum is often extracting the hyperline constants, which contain information about electron spin distribution, and thus information about the chemical environment of the free electron. The traditional trial-and-error methods aided by several rules of thumb (I) can be time-consuming if not impossible, especially if spectra are noisy or poorly resolved. Several attempts have been made recently to assign ESR hypetfine coupling constants of radical spectra in solution with the help of computers. Dunham et al. (2) reported the use ofthe fast Fourier transform algorithm to solve the methyl viologen free radical spectrum. In another approach Liiders (3) used a roll-up transformation to extract the coupling constants of the hydroxymethyl free radical. Due to the complexity of the transformations involved, neither of these methods has been used on a small laboratory computer. Grampp and Schiller (4) used the autocorrelation function of an ESR spectrum to demonstrate the solutions of the naphthalene anion and the N,N,N’,N’-tetramethyl phenylenediamine cation radical spectra. However, the autocorrelation spectrum alone often does not provide all the information needed for a solution. Based on slightly different approaches by Allen (5) and Ziegler and Hoffman (6), Jackson (7) recently outlined a correlation approsch to solving ESR spectra. In this * Present address: Department of Chemistry, Duke University, Durham, N.C. 27706. t To whom correspondence should be addressed. 0022-2364186 $3.00 Copyright Q 1986 by Academic Press, Inc. All rigbts of reproduction in any fom reserved.
42
ANALYSIS
OF
ESR
SPECTRA
ON
A SMALL
COMPUTER
43
analysis, the criterion for a solution is that the correlation of the experimental spectrum (Y&i)) with the simulated solution ( Ytim(z)) should be a maximum: c Ye&) * ysi&) correlation (= product function P) = ’
LZ YZindOl”*’
[II
Jackson found that by using the increase in the correlation, or product function P, as a criterion for improving the analysis, solutions could be found even for spectra of weak intensity. The analysis begins with a single line as test spectrum; hyperfme coupling constants and multiplicities are found one at a time and incorporated into the increasingly complex test spectrum until it matches the experimental spectrum. We have adapted Jackson’s approach for use on a small computer and tested its usefulness in solving complex spectra with many overlapping lines. We show here that the correlation analysis is capable of finding solutions for complex, well-resolved spectra. Even for spectra that can be solved readily by hand, the tuning procedure using correlation as a criterion for goodness of fit saves much time and effort. EXPERIMENTAL
Spectra were accumulated and analyzed on a Hewlett-Packard 9835B desktop computer interfaced to a Varian E-109 electron spin resonance specfrometer. The computer contains hardwired BASIC and a total of 128K of memory space for data and program storage. Experimental spectra of the radicals analyzed here were obtained using methods presented elsewhere: paminophenol (AMN) was oxidized with horseradish peroxidasel H202 (8, 9), and duroaminophenol (DAP) was synthesized by Volker Fischer and oxidized with AgzO in ethanol (8). THE
ANALYSIS
Our procedure is written in BASIC and consists of six steps, each called from a master program: PREP, AUTOCORRELATION, MATCH, FIRSTMATCH, SEEK, and TUNE. This organization basically follows that of Jackson (7). The major constraint in adapting the Jackson analysis for use on a small computer was computation time. We have developed a number of shortcuts to reduce the run time to no more than several hours for each segment as discussed below. The program consists of BASIC subroutines (available on request) overlaid on the data collection and manipulation routines supplied by Varian (E-935 Data Acquisition System). Experimental, product function, and simulated (test) spectra were stored in three 4096-point arrays.
PREP The PREP program prepares the experimental spectrum already stored by the computer for further analysis. The baseline is zeroed and the experimental spectrum is centered in the data array. Centering will allow, in the TUNE program, a comparison of the product function values with a test spectrum without searching for maximum
44
MOTTEN
AND
SCHREIIBER
overlap between the test and experimental spectra. Starting with a. provisional center obtained by eye, the exact center is found by a “mirroring” process as follows. Since the ESR spectrum is symmetric to inversion through the center, the left side of the spectrum, up to the provisional center, is copied into the test array, then the mirror image (reversed left-to-right and inverted) of the left side up to the center is copied into the right side of the test array starting at the next higher point after the center. The product function P for the test array and the data array is then calculated. The provisional center is moved one point at a time until a maximum is obtained for P, at which point the center of the spectrum has been found. The experimental spectrum is then shifted to bring the new center to the center of the data array and any empty space in the array is filled in by zeroes. If the right side of the spectrum is substantially broadened contrary to the assumption of inversion symmetry, as is often the case for radicals with a large nitrogen splitting, the “mirrored” spectrum is used for further analysis. If the center cannot be found with certainty, neither the use of the “mirrored” spectrum nor the application of the TUNE routine is possible using our subroutines, but the AUTOCORRELATION and MATCH/SEEK parts of the analysis can still be carried out. AUTOCORRELATION
The AUTOCORRELATION spectrum (4) is calculated to serve as a guide to possible splittings. The AUTOCORRELATION (Eq. [2]) is a measure of the overlap of the experimental spectrum with a copy of the experimental spectrum which is displaced along the x axis. Maxima in the AUTOCORRELATION spectrum represent repetition of lines or groups of lines due to splittings or sums or differences of splittings. In general, all possible separations between lines can be AUTOCORRELATION peaks, but the several largest generally contain most of the real splittings. The formula we used is: AUTOCORRELATION
(n) = C Y(i)*Y(i i
+ n) -
2 Y(i)*C i
Y(i + n)
Nf-Tl
where Y are the points of the experimental spectrum, the sums run from i = 1 to i = N - n, n is the displacement in points and N is the total number of points in the spectrum. In Eq. [2], the AUTOCORRELATION varies between 0 and 1; we use an additional scaling factor of 400 for optimal graphic display, as reflected in the numbers in Table 1. Using the formula in Eq. [2], calculation of the A.UTOCORRELATION spectrum took approximately 20 h. The following two modifications shorten the run time to approximately 2 4 h: (a) The initial sums in Eq. [2] are calculated once for n = 0 and subsequent sums are formed by subtraction of the appropriate single terms. (b) The
ANALYSIS
OF ESR SPECTRA ON A SMALL
COMPUTER
45
AUTOCORRELATION spectrum is locally sufficiently smooth that calculation of every sixth point yields all the relevant maxima; for the 50 largest peaks a more exact local maximum is found by calculating autocorrelation values point-by-point around each peak. The run time can be shortened further by stopping the calculation when the displacement is greater than a reasonable estimate of the largest splitting. In our program, rather than displaying the AUTOCORRELATION spectrum we list the AUTOCORRELATION peaks in order of size, starting with the highest. Each peak is separated from the origin by a distance corresponding to a real splitting or a sum or difference of real splittings. The position of the highest peak usually represents a real splitting, although we have found a few cases where this is not true. If hyperfme splittings are available for analogous radicals, it is often possible at this point to select trial splittings from the AUTOCORRELATION list and to put these splittings directly into TUNE. If it is not possible to guess the splittings then in many cases MATCH/SEEK cycles can be used to find the splittings one by one. MATCH MATCH locates the points of best agreement between the experimental spectrum and a test spectrum. The initial test spectrum consists of one line and produces a maximum in the MATCH spectrum corresponding to each line in the experimental spectrum. As the test spectrum becomes more complex, the MATCH peaks near the center become taller. When the test spectrum consists of the final solution, the center line of the MATCH spectrum is always the tallest peak in the MATCH spectrum and cannot be increased by adding another splitting to the test spectrum. The MATCH spectrum consists of the product function P(n) of the test and experimental spectra where the test spectrum is displaced by n points from the center: N/2
MATCH(n)
C K,& = P(n) = i=-N’2 N
+ 4* yt,&) [31
[C Estm1’2 i=l
where N is the number of points of the test spectrum. The range of MATCH (n) is -2047 + N/2 G n < 2048 - N/2 in a 4096-point spectrum. In effect, the test spectrum is moved to the left side of the experimental spectrum and “stepped” to the right one point at a time, calculating P at each point. MATCH 1, the MATCH spectrum of the initial single line with the experimental spectrum, is then stored on tape. In the Jackson analysis the MATCH spectrum is calculated in each MATCH/SEEK cycle and used to find a provisional center for the search for the next hyperfine splitting constant. In our modification, the first MATCH spectrum rather than the experimental spectrum is used as a basis for obtaining subsequent MATCH spectra. This is possible because the test spectrum is a superposition of J single lines, and the first MATCH spectrum is the correlation product of one single line with the experimental spectrum. Thus instead of multiplying many points of the test and experimental spectra together to get one point of the subsequent MATCH spectrum, we only need to add up J points from the first MATCH spectrum. A more detailed explanation can be found in Appendix 1.
46
MO’ITEN
AND
SCHREIHER
FIRSTMATCH
The FIRSTMATCH program was created to deal with a problem described in greater detail by Jackson, namely, that in the first MATCH/SEEK cycle a sum or a difference of splittings rather than a true splitting could be found to have the highest correlation product P. In FIRSTMATCH the splitting is constrained to have a value given by AUTOCORRELATION, the correlation coefficient P for each set of spins and multiplicities is calculated from the first MATCH spectrum as described for SEEK in Appendix 2. The FIRSTMATCH subprogram can also provide a way to assign the most probable multiplicities to hyperfme constants suggested by AUTOCORRELATION, thus allowing the user to skip the MATCH/SEEK cycles. SEEK
In this subprogram the product function P is used to extract one hyperhne constant and its multiplicity from a MATCH spectrum. In Jackson’s approach, each point in his “a-spectrum” is the product function obtained using a test spectrum composed of the currently known hyperfine splittings plus one splitting of magnitude a. As a is varied, the “a-spectrum” is generated. A separate “a-spectrum” is calculated for each possible multiplicity. The maximum P(a) in the set of “a-spectra” gives the next hyperfine splitting and multiplicity. Our adaptation departs from Jackson’s method at two points. First, as it is prohibitive for our computer to generate a new simulation for each point as in Jackson’s “aspectrum”, we use the current MATCH spectrum as a basis for calculating the product function values of the next hyperthre coupling set. (For details see Appendix 2.) With this approach, run time is only a few hours even for very complex spectra. Second, we have noticed that difficulties are encountered when one splitting is exactly half of another one, therefore we can generate many different multiplicities including a number of multiplicities representing this special case. The maximum P(a) is found for each multiplicity. At this point, either the hfs/multiplicity giving the largest P is selected or we use our knowledge of the molecule to select the next combination. If no improvement can be made in the product function from the last round then the solution is complete. Otherwise the new splitting is incorporated into the test spectrum for generation of a new MATCH spectrum and a new SEEK cycle. TUNE
In the last step the splitting constants are adjusted, one at a time, to give the largest product function between the simulated and experimental spectrum. This is done in two stages. First, we run a fast TUNE routine in which one splitting is omitted from the simulation and the final trial spectrum is built up in a manner analogous to the creation of MATCH from MATCH 1. In this routine we make use of the fact that solution ESR spectra are composed of a superposition of smaller spectra with fewer hyperfine constants. For example, a spectrum consisting of two proton splittings a1 and a2 can be simulated by adding two simulations of a one-proton (a,) spectrum, separated by
ANALYSIS
OF ESR SPECTRA
ON A SMALL
COMPUTER
47
a distance u2, with an intensity ratio of 1: 1. For fast tuning of a spectrum of n sets of inequivalent nuclei, we simulate n - 1 of the sets once, then create the final test spectrum by superposition using the nth hyperfine constant and its corresponding multiplicity and calculate the product function. Now new test spectra can be created by superposition while varying the nth hyperIme constant, without the necessity of simulating the first n - 1 constants again. When the product function is a maximum, another hyperfme constant is chosen for tuning in the same manner. The second TUNE stage consists of a slow fit with a new simulation for each variation in the splitting constants. The fast fit will refine the splitting constants to an accuracy of 2-3 x-axis points in 2-3 h; the slow fit can then run overnight to find the best fit to the desired tolerance and to adjust the linewidth. The difference between the fast and the slow TUNE is that in the fast TUNE, roundoff errors are greater and splittings are constrained to correspond to an integral number of points. In cases where very accurate hyperhne values are not necessary, the fast TUNE process gives quite an adequate fit.
FIG. 1. CORRELATION analysis of the 4-aminophenoxy radical. (A) Experimental spectrum. (B) Simulation after CORRELATION and TUNE-hyperfine parameters in Table 3. (C) First MATCH spectrum between experimental spectrum and a single Lorentzian line with linewidth 0.04 G. (D) Second MATCH spectrum using one nitrogen nucleus (a N = 5.24 G). (E) Third MATCH spectrum using one nitrogen and two equivalent protons (an = 5.24 G, uH = 5.65 G). (F) Fourth MATCH spectrum using one nitrogen, two equivalent protons, and two more equivalent protons (ar+ = 5.24 G, au = 5.65 G, au = I .84 G).
48
MOTTEN
AND
SCHREIBER
For convenience in programming, our version of TUNE requires that the experimental spectrum be centered in the array. This is conveniently done using the PREP subroutine as discussed earlier. RESULTS
AMN-Aminophenol The aminophenoxy cation radical presents a case of a complex radical whose ESR spectrum can be easily analyzed. The noise is low and the resolution is good; the splittings are well separated and there is very little overlap between lines. In this case the CORRELATION analysis led directly to a solution. In Fig. 1A the experimental spectrum is shown; in Table 1 the correlations returned by each part of the program are listed. Of the ten largest AUTOCORRELATION peaks listed in Table 1, the first four are close to the four real splittings and the remainder are sum and difference combinations of the first four. The first MATCH spectrum (Fig. 1C) was obtaine:d using a trial spectrum consisting of one Lorentzian line 0.04 G wide. Correlation values for the largest AUTOCORRELATION peaks given by FIRSTMATCH (Table 2) were: 5.64 G, two protons, P = 2094; 5.24 G, 1 nitrogen, P = 2880; 1.84 G, two protons, P = 2101; 2.67 G, 2 protons, P = 1752. For the first trial coupling we chose one spin-l nitrogen nucleus with a splitting of 5.24 G. From the second MATCH spectrum (Fig. 1D) the maximum SEEK correlation values for several spin/multiplicity combinations are shown in Table 2. We chose two equivalent spin- 1 hydrogen nuclei with a splitting of 5.65 G for the second set. The third MATCH/SEEK cycle (Fig. lE, Table 2) gave a maximum P for two protons at 1.84 G. The fourth cycle (Fig. lF, Table 2) gave a maximum for two protons at 2.69 G. The fifth cycle did not give an improved correlation for any combination, thus the
TABLE
1
AutocorrelationValues Autocorrelation point number 578 537 188 273 41 767 1115 810 230 307
Hyperhe
splitting 63 5.643 5.243 1.835 2.665 0.400 7.488 10.886 7.908 2.246 2.997
for AMN Correlation (Normalized 262.1 252.6 230.6 209.0 187.3 183.3 174.3 162.0 157.2 150.4
value to 400)
ANALYSIS
OF ESR SPECTRA ON A SMALL
COMPUTER
TABLE 2 Correlation Values for AMN FIRSTMATCH combination
Hypertkte splitting 63
1H IN 2H 3H
5.643
1832 1964 2094 1919
1H 1N 2H 3H
5.243
2411 2880 2741 2632
1H IN 2H 3H
1.835
1827 1973 2101 1864
IH 1N 2H 3H
2.665
1555 1481 1752 1523
I st SEEK lH IN 2H 3H
5.244 5.244 5.645 5.244
3194 3835 2796 3479
2nd SEEK IH 1N 2H 3H
5.645 5.645 5.645 5.645
4177 4554 4826 4457
3rd SEEK 1H IN 2H 3H
1.826 1.836 1.836 1.836
5064 5474 5841 5205
4th SEEK 1H 1N 2H 3H
2.616 2.686 2.686 2.676
5602 6193 6690 5838
5th SEEK 1H 1N 2H 3H
2.637 1.816 1.807 1.826
5984 6378 6575 5840
Correlation value
50
MOT-I-EN AND SCHREIBER TABLE 3 Hypertine Splitting Constants for pAminophenoxy Radical Fit after TUNE, aqueous solution” UN ak, 46 45
5.24 5.65 2.69 1.84
’ The hyperline splitting constants for the paminophenoxy radical in ethanol are (10) uN = 5.10 G, aEH1 = 5.10 G, ar6 = 3.25 G, and a& = 1.35 G.
analysis was finished. The simulated spectrum using the splitting constants refined by TUNE is shown in Fig. 1B. The final splitting constants (Table 3) are comparable to the published values for aminophenoxy radical in a different solvent. Duroaminophenol
The duroaminophenoxy radical (Fig. 2A) offers an excellent example of a complex spectrum where correlation can be used as a criterion for assigning multiplicities. Since hyperline values were available for analogs such as aminophenol, we used only the AUTOCORRELATION and TUNE programs, with starting parameters for TUNE taken from AUTOCORRELATION values. Because aN and a& were nearly the same size, the two possible solutions were not visually distinct. Therefore, the TUNE program was run for two combinations (Fig. 2). While the best fit for starting parameters of aN = 4.04 G, a& = 3.75 G (Fit A) gave a correlation of 33,504, the best fit for initial values of aN = 3.75 G, a& = 4.04 G (Fit B) gave a correlation of 33,127. The assignment in Fit A was confirmed by V. Fischer using isotopic substitution. Each TUNE program ran overnight. The final fit parameters are presented in Table 4. Note that the relative magnitudes of aN and a& are noi. the same as those of the aminophenoxy radical in aqueous solution. DISCUSSION
We have shown that the CORRELATION approach can be used to solve complex spectra with many splitting constants including spin-l species such as nitrogen. The TUNE routine in particular is exceptionally useful for refining the coupling constants once an approximate solution has been found. The correlation approach also provides a good criterion for choosing between two very similar solutions, although isotopic substitution should also be used where feasible. Our best results in finding a solution based on AUTOCORRELATION or MATCH/ SEEK results have been obtained for well-resolved spectra without extensively over-
ANALYSIS
OF ESR SPECTRA ON A SMALL
COMPUTER
51
FIG. 2. Application of the TUNE program on the duroaminophenoxy radical. (A) Experimental spectrum. (B) Simulated spectrum using the hyperfme coupling constants of Table 4, Fit A. (C) Simulated spectrum using the hyperfme coupling constants of Table 4, Fit B.
lapping lines. The product function in our experience is much more sensitive to overlap than to noise, a point which might be considered when acquiring data. For spectra with very broad overlapping lines the TUNE program does not always find a maximum at the correct splittings, even for simulated spectra with no noise. In the special case where one splitting constant is nearly twice another one, we have found particular problems with false maxima in the SEEK program. There is, however, no problem with TUNE in this special case. Within the limitations of excellent resolution (even with noisy spectra) for MATCH/ SEEK and good resolution for TUNE, the correlation approach can be very useful for complex spectra. We have shown that it is practical to run all of the programs needed to solve and refine spectra on a small computer, and the correlation approach should be feasible to use in most laboratories, thus reducing substantially the actual working time spent to solve and/or optimize a spectrum.
52
MOTTEN
AND SCHREIBER TABLE 4
Hypertine Splitting Parameters for Duroaminophenoxy Radical (after TUNE) Fit A’
Fit B
4.03 3.70 3.22 0.34
3.70 4.03 3.22 0.34
aN ah &, ah,
’ Shown to be correct by isot’opic substitution. APPENDIX
1: CALCULATING
MATCH
SPECTRA
The test spectrum, if it consists of more than one line, can be symbolically represented as a superposition of J lines: Y,,(i) = ;; 4.i a)* G(i)
141
j=l
where Z(j, a) represents the stick spectrum and G(n) is a factor derived from the shape function. The first MATCH spectrum using one line can be represented as N/2
MATCHl(n)
C Y&i + n - N/2)*G(i) N,,2 = Pi(n) = i=-N’2
[51
1,=X,2G2(i)11’2
where Nis the number of points in the one-line spectrum. Thus any MATCH spectrum from a multiline test spectrum consisting of J lines can be calculated from MATCH 1 by superposition using J c Pl(4 4.d) MATCH(n) = ‘SW 161 test1 with i running over the width of the test spectrum. This is mathematically equivalent to using Eq. [3]. Calculation of the numerator is thus speeded up substantially and calculation of the denominator need only be performed once. For all practical purposes, MATCH 1 replaces the original experimental spectrum. This is equivalent to using Eq. [5] to transform the spectrum into a more convenient form. Subsequent MATCH spectra and SEEK operations do not require the experimental spectrum. An interesting aspect of the transformation in :Eq. [5] is that it can be used as a noise-filtering process (7). If the linewidth in G(i) in Eq. [5] is chosen slightly lower
ANALYSIS
OF
ESR
SPECTRA
ON
A SMALL
COMPUTER
53
than the intrinsic experimental linewidth and the transformation is done twice, S/N is improved without loss of resolution and without third derivative sidebands. In the case of the duraminophenoxy radical (Fig. 2, intrinsic Lorentzian linewidth 0.065 G), S/N increased by a factor of five (data not shown) when a test spectrum with a single line of 0.03 G was used for the double-MATCH transformation. APPENDIX
2. THE
SEEK
SUBROUTINE
With SEEK, we want to find the maximum correlation between the experimental spectrum and a trial spectrum in which one hyperfine splitting constant is varied, normalized according to the procedure of Jackson. We calculate the hyperfme splitting giving the best product function by adding single points from the current MATCH spectrum in much the same way as the current MATCH spectrum was built up from MATCH 1. For each MATCH/SEEK cycle, the MATCH 1 spectrum is loaded from the tape where it was saved; the trial spectrum is simulated, a MATCH spectrum is calculated from MATCH1 and the trial simulation parameters; and the SEEK subroutine finds a new splitting and multiplicity from the MATCH spectrum. The correlation for SEEK can be represented by
c Y,est(i, 4* Ye&) p(a)= ’ [C Y&(i, u)p2 .
[71
The trial spectrum can be represented as the sum of several trial spectra from the previous level, added using a new coupling constant a and intensity factors Z derived from the desired multiplicity. For example, for a single proton splitting, with two equally intense lines (m = 2, Ii = Z2 = 1) the resultant spectrum contains two components, a trial spectrum from the previous level and the trial spectrum displaced by a. The correlation is then 5 Z(s)*MATCH(center
P(a) = S=l [C YLG>l “2
+ s*a)
PI
where “center” is the highest point in the MATCH spectrum and Z(s) represents the “stick spectrum” for the trial multiplicity. Each term in this expression, correlated with the experimental spectrum, was calculated when the most recent MATCH spectrum was generated. That is, instead of generating a new trial spectrum, and multiplying point-by-point with the experimental spectrum, we can simply add single points from the MATCH spectrum to get the numerator in Eq. [3]. For a single proton splitting, two points are added; one point is the highest point of the MATCH spectrum and the other is either -a or +a from the highest point of the MATCH spectrum. (Both possibilities are calculated and the largest sum is chosen.) The normalization constant must still be calculated for each point P(u). To speed up this calculation, the trial spectrum is represented as a sum of individual lines:
54
MOT-TEN AND SCHREIBER
Y,,(i) = ;: Z(j, a)*G(i)
[91
j=l
as introduced in Eq. [4]. The denominator may then be found by taking the sum of the squares of the individual lines plus the overlap between the lines: Y&(i) = 5 Z*(m, a)*G(i)*
t?l=l
+ 2* ; m=2
mgl Z(m)*l(n)*G(i)*G(i
+ (m - n)*a)
[lo]
n=l
The overlap sums G(i) * G(i + (m - n) * a) = ci Y,(i) * Y, (i + (m - n) * a) are calculated as needed and stored for use with later multiplicity combinations. In practice we only calculate P(u) for every sixth point, then a more exact local maximum is calculated point-by-point around each peak. ACKNOWLEDGMENTS We gratefully acknowledgehelpful discussion with Professor Richard Jackson and thank him for his kindness in sending us copies of his FORTRAN correlation programs. We also thank Dr. Volker Fischer for providing the duroaminophenoxy spectrum and Dr. Paul West for providing the paminophenoxy spectrum. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
J. E. WERTZ AND J. R. BOLTON, “Electron Spin Resonance,” McGraw-Hill, New York, 1972. W. R. DUNHAM, J. A. FEE, L. J. HARDING, AND H. J. GRANDE, J. Magn. Reson. 40, 351 (1980). K. LODERS, Wiss. Z. Karl-Marx-Universit& Leipzig, Math.-Nat. R 33(4), 366 (1984). G. GRAMPP AND C. A. SCHILLER,Anal. Chem. 53,560 (1981). L. C. ALLEN, Nature (London) 196,663 (1962). E. ZIEGLER AND E. G. HOFFMANN, Z. Anal. Chem. 240, 145 (1968). R. A. JACKSON,J. Chem. Sot. Perkin Trans. 2, 523 (1983). P. R. WEST, V. FISCHER, L. S. HARMAN, AND R. P. MASON, presented at the Chemical Institute of Canada 68th Canadian Chemical Conference, Kingston, Ontario, 1985. 9. P. D. JOSEPHY,T. E. ELING, AND R. P. MASON, Mol. Pharmacol. 23,461 (1983). 10. H. B. STEGMANN AND K. SCHEFFLER,Z. Naturforsch. B 19, 537 (1964).
OF MAGNETIC
67,42-54
RESONANCE
( 1986)
Correlation Analysis of ESR Spectra on a Small Computer ANN
MOTI-EN*
AND JORG SCHREIBER~
Laboratory of Molecular Biophysics, National Institute of Environmental Health Sciences, P.O. Box 12233, Research Triangle Park, iVorth Carolina 27709 Received September 19, 1985 A correlation approach to computer solutions of electron spin resonance spectra has been adapted for use on a small computer. The correlation analysis is capable of finding solutions for complex, well-resolved spectra, including those containing spin-l species such as nitrogen. Even for spectra that can be solved readily by hand, a tuning procedure using correlation as a criterion for goodness of fit saves much time and effort. A fast procedure for adjusting hyperfme constants to within approximately 0.05% of the scan range is presented. Fast methods of calculation for other parts of the correlation analysis are described and their application to the aminophenoxy and the tetramethylaminophenoxy radicals is shown. 0 1986 Academic Pm, Inc. INTRODUCTION
The most difficult part of interpreting an electron spin resonance spectrum is often extracting the hyperline constants, which contain information about electron spin distribution, and thus information about the chemical environment of the free electron. The traditional trial-and-error methods aided by several rules of thumb (I) can be time-consuming if not impossible, especially if spectra are noisy or poorly resolved. Several attempts have been made recently to assign ESR hypetfine coupling constants of radical spectra in solution with the help of computers. Dunham et al. (2) reported the use ofthe fast Fourier transform algorithm to solve the methyl viologen free radical spectrum. In another approach Liiders (3) used a roll-up transformation to extract the coupling constants of the hydroxymethyl free radical. Due to the complexity of the transformations involved, neither of these methods has been used on a small laboratory computer. Grampp and Schiller (4) used the autocorrelation function of an ESR spectrum to demonstrate the solutions of the naphthalene anion and the N,N,N’,N’-tetramethyl phenylenediamine cation radical spectra. However, the autocorrelation spectrum alone often does not provide all the information needed for a solution. Based on slightly different approaches by Allen (5) and Ziegler and Hoffman (6), Jackson (7) recently outlined a correlation approsch to solving ESR spectra. In this * Present address: Department of Chemistry, Duke University, Durham, N.C. 27706. t To whom correspondence should be addressed. 0022-2364186 $3.00 Copyright Q 1986 by Academic Press, Inc. All rigbts of reproduction in any fom reserved.
42
ANALYSIS
OF
ESR
SPECTRA
ON
A SMALL
COMPUTER
43
analysis, the criterion for a solution is that the correlation of the experimental spectrum (Y&i)) with the simulated solution ( Ytim(z)) should be a maximum: c Ye&) * ysi&) correlation (= product function P) = ’
LZ YZindOl”*’
[II
Jackson found that by using the increase in the correlation, or product function P, as a criterion for improving the analysis, solutions could be found even for spectra of weak intensity. The analysis begins with a single line as test spectrum; hyperfme coupling constants and multiplicities are found one at a time and incorporated into the increasingly complex test spectrum until it matches the experimental spectrum. We have adapted Jackson’s approach for use on a small computer and tested its usefulness in solving complex spectra with many overlapping lines. We show here that the correlation analysis is capable of finding solutions for complex, well-resolved spectra. Even for spectra that can be solved readily by hand, the tuning procedure using correlation as a criterion for goodness of fit saves much time and effort. EXPERIMENTAL
Spectra were accumulated and analyzed on a Hewlett-Packard 9835B desktop computer interfaced to a Varian E-109 electron spin resonance specfrometer. The computer contains hardwired BASIC and a total of 128K of memory space for data and program storage. Experimental spectra of the radicals analyzed here were obtained using methods presented elsewhere: paminophenol (AMN) was oxidized with horseradish peroxidasel H202 (8, 9), and duroaminophenol (DAP) was synthesized by Volker Fischer and oxidized with AgzO in ethanol (8). THE
ANALYSIS
Our procedure is written in BASIC and consists of six steps, each called from a master program: PREP, AUTOCORRELATION, MATCH, FIRSTMATCH, SEEK, and TUNE. This organization basically follows that of Jackson (7). The major constraint in adapting the Jackson analysis for use on a small computer was computation time. We have developed a number of shortcuts to reduce the run time to no more than several hours for each segment as discussed below. The program consists of BASIC subroutines (available on request) overlaid on the data collection and manipulation routines supplied by Varian (E-935 Data Acquisition System). Experimental, product function, and simulated (test) spectra were stored in three 4096-point arrays.
PREP The PREP program prepares the experimental spectrum already stored by the computer for further analysis. The baseline is zeroed and the experimental spectrum is centered in the data array. Centering will allow, in the TUNE program, a comparison of the product function values with a test spectrum without searching for maximum
44
MOTTEN
AND
SCHREIIBER
overlap between the test and experimental spectra. Starting with a. provisional center obtained by eye, the exact center is found by a “mirroring” process as follows. Since the ESR spectrum is symmetric to inversion through the center, the left side of the spectrum, up to the provisional center, is copied into the test array, then the mirror image (reversed left-to-right and inverted) of the left side up to the center is copied into the right side of the test array starting at the next higher point after the center. The product function P for the test array and the data array is then calculated. The provisional center is moved one point at a time until a maximum is obtained for P, at which point the center of the spectrum has been found. The experimental spectrum is then shifted to bring the new center to the center of the data array and any empty space in the array is filled in by zeroes. If the right side of the spectrum is substantially broadened contrary to the assumption of inversion symmetry, as is often the case for radicals with a large nitrogen splitting, the “mirrored” spectrum is used for further analysis. If the center cannot be found with certainty, neither the use of the “mirrored” spectrum nor the application of the TUNE routine is possible using our subroutines, but the AUTOCORRELATION and MATCH/SEEK parts of the analysis can still be carried out. AUTOCORRELATION
The AUTOCORRELATION spectrum (4) is calculated to serve as a guide to possible splittings. The AUTOCORRELATION (Eq. [2]) is a measure of the overlap of the experimental spectrum with a copy of the experimental spectrum which is displaced along the x axis. Maxima in the AUTOCORRELATION spectrum represent repetition of lines or groups of lines due to splittings or sums or differences of splittings. In general, all possible separations between lines can be AUTOCORRELATION peaks, but the several largest generally contain most of the real splittings. The formula we used is: AUTOCORRELATION
(n) = C Y(i)*Y(i i
+ n) -
2 Y(i)*C i
Y(i + n)
Nf-Tl
where Y are the points of the experimental spectrum, the sums run from i = 1 to i = N - n, n is the displacement in points and N is the total number of points in the spectrum. In Eq. [2], the AUTOCORRELATION varies between 0 and 1; we use an additional scaling factor of 400 for optimal graphic display, as reflected in the numbers in Table 1. Using the formula in Eq. [2], calculation of the A.UTOCORRELATION spectrum took approximately 20 h. The following two modifications shorten the run time to approximately 2 4 h: (a) The initial sums in Eq. [2] are calculated once for n = 0 and subsequent sums are formed by subtraction of the appropriate single terms. (b) The
ANALYSIS
OF ESR SPECTRA ON A SMALL
COMPUTER
45
AUTOCORRELATION spectrum is locally sufficiently smooth that calculation of every sixth point yields all the relevant maxima; for the 50 largest peaks a more exact local maximum is found by calculating autocorrelation values point-by-point around each peak. The run time can be shortened further by stopping the calculation when the displacement is greater than a reasonable estimate of the largest splitting. In our program, rather than displaying the AUTOCORRELATION spectrum we list the AUTOCORRELATION peaks in order of size, starting with the highest. Each peak is separated from the origin by a distance corresponding to a real splitting or a sum or difference of real splittings. The position of the highest peak usually represents a real splitting, although we have found a few cases where this is not true. If hyperfme splittings are available for analogous radicals, it is often possible at this point to select trial splittings from the AUTOCORRELATION list and to put these splittings directly into TUNE. If it is not possible to guess the splittings then in many cases MATCH/SEEK cycles can be used to find the splittings one by one. MATCH MATCH locates the points of best agreement between the experimental spectrum and a test spectrum. The initial test spectrum consists of one line and produces a maximum in the MATCH spectrum corresponding to each line in the experimental spectrum. As the test spectrum becomes more complex, the MATCH peaks near the center become taller. When the test spectrum consists of the final solution, the center line of the MATCH spectrum is always the tallest peak in the MATCH spectrum and cannot be increased by adding another splitting to the test spectrum. The MATCH spectrum consists of the product function P(n) of the test and experimental spectra where the test spectrum is displaced by n points from the center: N/2
MATCH(n)
C K,& = P(n) = i=-N’2 N
+ 4* yt,&) [31
[C Estm1’2 i=l
where N is the number of points of the test spectrum. The range of MATCH (n) is -2047 + N/2 G n < 2048 - N/2 in a 4096-point spectrum. In effect, the test spectrum is moved to the left side of the experimental spectrum and “stepped” to the right one point at a time, calculating P at each point. MATCH 1, the MATCH spectrum of the initial single line with the experimental spectrum, is then stored on tape. In the Jackson analysis the MATCH spectrum is calculated in each MATCH/SEEK cycle and used to find a provisional center for the search for the next hyperfine splitting constant. In our modification, the first MATCH spectrum rather than the experimental spectrum is used as a basis for obtaining subsequent MATCH spectra. This is possible because the test spectrum is a superposition of J single lines, and the first MATCH spectrum is the correlation product of one single line with the experimental spectrum. Thus instead of multiplying many points of the test and experimental spectra together to get one point of the subsequent MATCH spectrum, we only need to add up J points from the first MATCH spectrum. A more detailed explanation can be found in Appendix 1.
46
MO’ITEN
AND
SCHREIHER
FIRSTMATCH
The FIRSTMATCH program was created to deal with a problem described in greater detail by Jackson, namely, that in the first MATCH/SEEK cycle a sum or a difference of splittings rather than a true splitting could be found to have the highest correlation product P. In FIRSTMATCH the splitting is constrained to have a value given by AUTOCORRELATION, the correlation coefficient P for each set of spins and multiplicities is calculated from the first MATCH spectrum as described for SEEK in Appendix 2. The FIRSTMATCH subprogram can also provide a way to assign the most probable multiplicities to hyperfme constants suggested by AUTOCORRELATION, thus allowing the user to skip the MATCH/SEEK cycles. SEEK
In this subprogram the product function P is used to extract one hyperhne constant and its multiplicity from a MATCH spectrum. In Jackson’s approach, each point in his “a-spectrum” is the product function obtained using a test spectrum composed of the currently known hyperfine splittings plus one splitting of magnitude a. As a is varied, the “a-spectrum” is generated. A separate “a-spectrum” is calculated for each possible multiplicity. The maximum P(a) in the set of “a-spectra” gives the next hyperfine splitting and multiplicity. Our adaptation departs from Jackson’s method at two points. First, as it is prohibitive for our computer to generate a new simulation for each point as in Jackson’s “aspectrum”, we use the current MATCH spectrum as a basis for calculating the product function values of the next hyperthre coupling set. (For details see Appendix 2.) With this approach, run time is only a few hours even for very complex spectra. Second, we have noticed that difficulties are encountered when one splitting is exactly half of another one, therefore we can generate many different multiplicities including a number of multiplicities representing this special case. The maximum P(a) is found for each multiplicity. At this point, either the hfs/multiplicity giving the largest P is selected or we use our knowledge of the molecule to select the next combination. If no improvement can be made in the product function from the last round then the solution is complete. Otherwise the new splitting is incorporated into the test spectrum for generation of a new MATCH spectrum and a new SEEK cycle. TUNE
In the last step the splitting constants are adjusted, one at a time, to give the largest product function between the simulated and experimental spectrum. This is done in two stages. First, we run a fast TUNE routine in which one splitting is omitted from the simulation and the final trial spectrum is built up in a manner analogous to the creation of MATCH from MATCH 1. In this routine we make use of the fact that solution ESR spectra are composed of a superposition of smaller spectra with fewer hyperfine constants. For example, a spectrum consisting of two proton splittings a1 and a2 can be simulated by adding two simulations of a one-proton (a,) spectrum, separated by
ANALYSIS
OF ESR SPECTRA
ON A SMALL
COMPUTER
47
a distance u2, with an intensity ratio of 1: 1. For fast tuning of a spectrum of n sets of inequivalent nuclei, we simulate n - 1 of the sets once, then create the final test spectrum by superposition using the nth hyperfine constant and its corresponding multiplicity and calculate the product function. Now new test spectra can be created by superposition while varying the nth hyperIme constant, without the necessity of simulating the first n - 1 constants again. When the product function is a maximum, another hyperfme constant is chosen for tuning in the same manner. The second TUNE stage consists of a slow fit with a new simulation for each variation in the splitting constants. The fast fit will refine the splitting constants to an accuracy of 2-3 x-axis points in 2-3 h; the slow fit can then run overnight to find the best fit to the desired tolerance and to adjust the linewidth. The difference between the fast and the slow TUNE is that in the fast TUNE, roundoff errors are greater and splittings are constrained to correspond to an integral number of points. In cases where very accurate hyperhne values are not necessary, the fast TUNE process gives quite an adequate fit.
FIG. 1. CORRELATION analysis of the 4-aminophenoxy radical. (A) Experimental spectrum. (B) Simulation after CORRELATION and TUNE-hyperfine parameters in Table 3. (C) First MATCH spectrum between experimental spectrum and a single Lorentzian line with linewidth 0.04 G. (D) Second MATCH spectrum using one nitrogen nucleus (a N = 5.24 G). (E) Third MATCH spectrum using one nitrogen and two equivalent protons (an = 5.24 G, uH = 5.65 G). (F) Fourth MATCH spectrum using one nitrogen, two equivalent protons, and two more equivalent protons (ar+ = 5.24 G, au = 5.65 G, au = I .84 G).
48
MOTTEN
AND
SCHREIBER
For convenience in programming, our version of TUNE requires that the experimental spectrum be centered in the array. This is conveniently done using the PREP subroutine as discussed earlier. RESULTS
AMN-Aminophenol The aminophenoxy cation radical presents a case of a complex radical whose ESR spectrum can be easily analyzed. The noise is low and the resolution is good; the splittings are well separated and there is very little overlap between lines. In this case the CORRELATION analysis led directly to a solution. In Fig. 1A the experimental spectrum is shown; in Table 1 the correlations returned by each part of the program are listed. Of the ten largest AUTOCORRELATION peaks listed in Table 1, the first four are close to the four real splittings and the remainder are sum and difference combinations of the first four. The first MATCH spectrum (Fig. 1C) was obtaine:d using a trial spectrum consisting of one Lorentzian line 0.04 G wide. Correlation values for the largest AUTOCORRELATION peaks given by FIRSTMATCH (Table 2) were: 5.64 G, two protons, P = 2094; 5.24 G, 1 nitrogen, P = 2880; 1.84 G, two protons, P = 2101; 2.67 G, 2 protons, P = 1752. For the first trial coupling we chose one spin-l nitrogen nucleus with a splitting of 5.24 G. From the second MATCH spectrum (Fig. 1D) the maximum SEEK correlation values for several spin/multiplicity combinations are shown in Table 2. We chose two equivalent spin- 1 hydrogen nuclei with a splitting of 5.65 G for the second set. The third MATCH/SEEK cycle (Fig. lE, Table 2) gave a maximum P for two protons at 1.84 G. The fourth cycle (Fig. lF, Table 2) gave a maximum for two protons at 2.69 G. The fifth cycle did not give an improved correlation for any combination, thus the
TABLE
1
AutocorrelationValues Autocorrelation point number 578 537 188 273 41 767 1115 810 230 307
Hyperhe
splitting 63 5.643 5.243 1.835 2.665 0.400 7.488 10.886 7.908 2.246 2.997
for AMN Correlation (Normalized 262.1 252.6 230.6 209.0 187.3 183.3 174.3 162.0 157.2 150.4
value to 400)
ANALYSIS
OF ESR SPECTRA ON A SMALL
COMPUTER
TABLE 2 Correlation Values for AMN FIRSTMATCH combination
Hypertkte splitting 63
1H IN 2H 3H
5.643
1832 1964 2094 1919
1H 1N 2H 3H
5.243
2411 2880 2741 2632
1H IN 2H 3H
1.835
1827 1973 2101 1864
IH 1N 2H 3H
2.665
1555 1481 1752 1523
I st SEEK lH IN 2H 3H
5.244 5.244 5.645 5.244
3194 3835 2796 3479
2nd SEEK IH 1N 2H 3H
5.645 5.645 5.645 5.645
4177 4554 4826 4457
3rd SEEK 1H IN 2H 3H
1.826 1.836 1.836 1.836
5064 5474 5841 5205
4th SEEK 1H 1N 2H 3H
2.616 2.686 2.686 2.676
5602 6193 6690 5838
5th SEEK 1H 1N 2H 3H
2.637 1.816 1.807 1.826
5984 6378 6575 5840
Correlation value
50
MOT-I-EN AND SCHREIBER TABLE 3 Hypertine Splitting Constants for pAminophenoxy Radical Fit after TUNE, aqueous solution” UN ak, 46 45
5.24 5.65 2.69 1.84
’ The hyperline splitting constants for the paminophenoxy radical in ethanol are (10) uN = 5.10 G, aEH1 = 5.10 G, ar6 = 3.25 G, and a& = 1.35 G.
analysis was finished. The simulated spectrum using the splitting constants refined by TUNE is shown in Fig. 1B. The final splitting constants (Table 3) are comparable to the published values for aminophenoxy radical in a different solvent. Duroaminophenol
The duroaminophenoxy radical (Fig. 2A) offers an excellent example of a complex spectrum where correlation can be used as a criterion for assigning multiplicities. Since hyperline values were available for analogs such as aminophenol, we used only the AUTOCORRELATION and TUNE programs, with starting parameters for TUNE taken from AUTOCORRELATION values. Because aN and a& were nearly the same size, the two possible solutions were not visually distinct. Therefore, the TUNE program was run for two combinations (Fig. 2). While the best fit for starting parameters of aN = 4.04 G, a& = 3.75 G (Fit A) gave a correlation of 33,504, the best fit for initial values of aN = 3.75 G, a& = 4.04 G (Fit B) gave a correlation of 33,127. The assignment in Fit A was confirmed by V. Fischer using isotopic substitution. Each TUNE program ran overnight. The final fit parameters are presented in Table 4. Note that the relative magnitudes of aN and a& are noi. the same as those of the aminophenoxy radical in aqueous solution. DISCUSSION
We have shown that the CORRELATION approach can be used to solve complex spectra with many splitting constants including spin-l species such as nitrogen. The TUNE routine in particular is exceptionally useful for refining the coupling constants once an approximate solution has been found. The correlation approach also provides a good criterion for choosing between two very similar solutions, although isotopic substitution should also be used where feasible. Our best results in finding a solution based on AUTOCORRELATION or MATCH/ SEEK results have been obtained for well-resolved spectra without extensively over-
ANALYSIS
OF ESR SPECTRA ON A SMALL
COMPUTER
51
FIG. 2. Application of the TUNE program on the duroaminophenoxy radical. (A) Experimental spectrum. (B) Simulated spectrum using the hyperfme coupling constants of Table 4, Fit A. (C) Simulated spectrum using the hyperfme coupling constants of Table 4, Fit B.
lapping lines. The product function in our experience is much more sensitive to overlap than to noise, a point which might be considered when acquiring data. For spectra with very broad overlapping lines the TUNE program does not always find a maximum at the correct splittings, even for simulated spectra with no noise. In the special case where one splitting constant is nearly twice another one, we have found particular problems with false maxima in the SEEK program. There is, however, no problem with TUNE in this special case. Within the limitations of excellent resolution (even with noisy spectra) for MATCH/ SEEK and good resolution for TUNE, the correlation approach can be very useful for complex spectra. We have shown that it is practical to run all of the programs needed to solve and refine spectra on a small computer, and the correlation approach should be feasible to use in most laboratories, thus reducing substantially the actual working time spent to solve and/or optimize a spectrum.
52
MOTTEN
AND SCHREIBER TABLE 4
Hypertine Splitting Parameters for Duroaminophenoxy Radical (after TUNE) Fit A’
Fit B
4.03 3.70 3.22 0.34
3.70 4.03 3.22 0.34
aN ah &, ah,
’ Shown to be correct by isot’opic substitution. APPENDIX
1: CALCULATING
MATCH
SPECTRA
The test spectrum, if it consists of more than one line, can be symbolically represented as a superposition of J lines: Y,,(i) = ;; 4.i a)* G(i)
141
j=l
where Z(j, a) represents the stick spectrum and G(n) is a factor derived from the shape function. The first MATCH spectrum using one line can be represented as N/2
MATCHl(n)
C Y&i + n - N/2)*G(i) N,,2 = Pi(n) = i=-N’2
[51
1,=X,2G2(i)11’2
where Nis the number of points in the one-line spectrum. Thus any MATCH spectrum from a multiline test spectrum consisting of J lines can be calculated from MATCH 1 by superposition using J c Pl(4 4.d) MATCH(n) = ‘SW 161 test1 with i running over the width of the test spectrum. This is mathematically equivalent to using Eq. [3]. Calculation of the numerator is thus speeded up substantially and calculation of the denominator need only be performed once. For all practical purposes, MATCH 1 replaces the original experimental spectrum. This is equivalent to using Eq. [5] to transform the spectrum into a more convenient form. Subsequent MATCH spectra and SEEK operations do not require the experimental spectrum. An interesting aspect of the transformation in :Eq. [5] is that it can be used as a noise-filtering process (7). If the linewidth in G(i) in Eq. [5] is chosen slightly lower
ANALYSIS
OF
ESR
SPECTRA
ON
A SMALL
COMPUTER
53
than the intrinsic experimental linewidth and the transformation is done twice, S/N is improved without loss of resolution and without third derivative sidebands. In the case of the duraminophenoxy radical (Fig. 2, intrinsic Lorentzian linewidth 0.065 G), S/N increased by a factor of five (data not shown) when a test spectrum with a single line of 0.03 G was used for the double-MATCH transformation. APPENDIX
2. THE
SEEK
SUBROUTINE
With SEEK, we want to find the maximum correlation between the experimental spectrum and a trial spectrum in which one hyperfine splitting constant is varied, normalized according to the procedure of Jackson. We calculate the hyperfme splitting giving the best product function by adding single points from the current MATCH spectrum in much the same way as the current MATCH spectrum was built up from MATCH 1. For each MATCH/SEEK cycle, the MATCH 1 spectrum is loaded from the tape where it was saved; the trial spectrum is simulated, a MATCH spectrum is calculated from MATCH1 and the trial simulation parameters; and the SEEK subroutine finds a new splitting and multiplicity from the MATCH spectrum. The correlation for SEEK can be represented by
c Y,est(i, 4* Ye&) p(a)= ’ [C Y&(i, u)p2 .
[71
The trial spectrum can be represented as the sum of several trial spectra from the previous level, added using a new coupling constant a and intensity factors Z derived from the desired multiplicity. For example, for a single proton splitting, with two equally intense lines (m = 2, Ii = Z2 = 1) the resultant spectrum contains two components, a trial spectrum from the previous level and the trial spectrum displaced by a. The correlation is then 5 Z(s)*MATCH(center
P(a) = S=l [C YLG>l “2
+ s*a)
PI
where “center” is the highest point in the MATCH spectrum and Z(s) represents the “stick spectrum” for the trial multiplicity. Each term in this expression, correlated with the experimental spectrum, was calculated when the most recent MATCH spectrum was generated. That is, instead of generating a new trial spectrum, and multiplying point-by-point with the experimental spectrum, we can simply add single points from the MATCH spectrum to get the numerator in Eq. [3]. For a single proton splitting, two points are added; one point is the highest point of the MATCH spectrum and the other is either -a or +a from the highest point of the MATCH spectrum. (Both possibilities are calculated and the largest sum is chosen.) The normalization constant must still be calculated for each point P(u). To speed up this calculation, the trial spectrum is represented as a sum of individual lines:
54
MOT-TEN AND SCHREIBER
Y,,(i) = ;: Z(j, a)*G(i)
[91
j=l
as introduced in Eq. [4]. The denominator may then be found by taking the sum of the squares of the individual lines plus the overlap between the lines: Y&(i) = 5 Z*(m, a)*G(i)*
t?l=l
+ 2* ; m=2
mgl Z(m)*l(n)*G(i)*G(i
+ (m - n)*a)
[lo]
n=l
The overlap sums G(i) * G(i + (m - n) * a) = ci Y,(i) * Y, (i + (m - n) * a) are calculated as needed and stored for use with later multiplicity combinations. In practice we only calculate P(u) for every sixth point, then a more exact local maximum is calculated point-by-point around each peak. ACKNOWLEDGMENTS We gratefully acknowledgehelpful discussion with Professor Richard Jackson and thank him for his kindness in sending us copies of his FORTRAN correlation programs. We also thank Dr. Volker Fischer for providing the duroaminophenoxy spectrum and Dr. Paul West for providing the paminophenoxy spectrum. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
J. E. WERTZ AND J. R. BOLTON, “Electron Spin Resonance,” McGraw-Hill, New York, 1972. W. R. DUNHAM, J. A. FEE, L. J. HARDING, AND H. J. GRANDE, J. Magn. Reson. 40, 351 (1980). K. LODERS, Wiss. Z. Karl-Marx-Universit& Leipzig, Math.-Nat. R 33(4), 366 (1984). G. GRAMPP AND C. A. SCHILLER,Anal. Chem. 53,560 (1981). L. C. ALLEN, Nature (London) 196,663 (1962). E. ZIEGLER AND E. G. HOFFMANN, Z. Anal. Chem. 240, 145 (1968). R. A. JACKSON,J. Chem. Sot. Perkin Trans. 2, 523 (1983). P. R. WEST, V. FISCHER, L. S. HARMAN, AND R. P. MASON, presented at the Chemical Institute of Canada 68th Canadian Chemical Conference, Kingston, Ontario, 1985. 9. P. D. JOSEPHY,T. E. ELING, AND R. P. MASON, Mol. Pharmacol. 23,461 (1983). 10. H. B. STEGMANN AND K. SCHEFFLER,Z. Naturforsch. B 19, 537 (1964).