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A simplified approach to multiline ESR spectra resolution using a digital computer
JOURNAL
OF MOLECULAR
A Simplified
SPECTROSCOPY
20,
141-152
(1966)
Approach to Multiline ESR Spectra Using a Digital Computer*
J. R. REEnER,j’
G. M. ODE&t
Department of Chemistry,
R. E. SIODA, AND
The Johns Hopkins University,
Resolution
W. S. KOSKI
Baltimore,
Maryhnd
A simplified method for the analysis of multiline isotropic ESR spectra is presented which provides significant computational advantages over some of the standard analysis techniques. The method utilizes the positions of the apparent maxima of the partially resolved lines in a multiline spectrum to determine the splitting constants characterizing the spectrum. The relative intensities of the apparent maxima are used as a check on the resolution provided. Application of the method to the resolution of a spectrum of acenaphthylene demonstrates its usefulness in refining the estimates of a set of splitting constants. INTRODUCTION
The electron spin resonance spectra of organic free radicals resulting from the isotropic or anisotropic interaction of the unpaired electron with the magnetic fields may be represented in terms of a superposition of elementary waveforms (or lines) whose shape is approximately known (1) . The relative weights and positions of the lines may be specified by a set of splitting constants representing a set of convolution operations upon the elementary waveforms. The determination of the splitting con&ants from experimental spectra may be easily accomplished provided the number of lines is small and there is no appreciable overlapping of the lines. Such is the case, for instance, in some simple isotropic intcractions exhibited by some free radicals in the liquid phase. The interpretation of more complex spectra can be, however, a difficult task. The modern high-speed digital computer provides a valuable tool for the interpretat)ion of such spectra. Chen, Sane, Walter, and Weil (2) have devised a program for the determination of the ratio of t’he nitrogen splitting constants in p-substitut’ed l , l-diphenyl-2-picryl-hydrazyl. Spectral simulation programs which reconstruct a spectrum from a trial set of splitting constants have been prepared by Stone and Maki (3) for isotropic ESR spectra and by Lefebvre and Maruani (4,5) for ESR spectra of amorphous solid samples. Marquardt (6)) * This investigation was supported in part by research grant GM 5144 from the Divisiotl of General Medical Sciences, Public Health Service, National Institutes of Healt,h. t Department of Electrical Engineering, The Johns Hopkins University. 141
142
REEDER,
ODELL,
SIODA,
AND
KOSKI
Welch (7)) Gladney (8)) and Marshall (9) have presented methods of leastsquares approximation which are applicable to the interpretation of many types of spectra by the digital computer. Kaplan prepared a digital computer program for least-squares analysis of multiline spectra using tentative splitting constants (10). The amount of computation required by most of the techniques cited above becomes prohibitive when the number of lines comprising the spectrum becomes very large. In this paper we present a simplified method for the analysis of multiline isotropic ESR spectra of free radicals which provides significant computational advantages over some of the standard analysis techniques. The method utilizes only the positions of the apparent maxima of the partially resolved lines in a multi-line spectrum to determine the splitting constants characterizing the spectrum. The relative intensities of the apparent maxima are used as a check on the resolution provided. Application of the method to the resolution of a spectrum of acenaphthylene demonstrates its usefulness in refining t,he estimates of a set of splitting constants in an efficient manner. I. FORMULATION
OF THE METHOD
An ESR spectrum S(o) may be represented as the superposition of elementary waveforms (or lines) whose shape is approximately A mathematical model which is appropriate for many applications
of a number known (1). is of the form
(1) in which q(o/a) is the line shape function. This function is normally a symmetric “bell-shaped” function such as the Gaussian, Lorentzian, or Voigt functions. LJ is the independent variable; pi is an epoch parameter establishing the location of the ith line; A; is the weight parameter of the ith line; ui is a structural parameter establishing the scale of the line; N is the number of lines comprising the spectrum. In those instances in which the structural parameters ui for all the lines may be regarded as identical, the spectrum S(w) may be considered to be the result of convolving an ideal line spectrum B( CO)with the elementary lineshape (p( W/U) ; thus
NW) =
11B(7) (; - 7)d7 = B(o)*‘p ($) ‘$0
)
where
and S(o -
pi) is a Dirac delta function
located
at w = pi _
COMPUTER
RESOLUTION
OF ESR
SPECTKA
143
A method for specifying the Iine spectrum B(w) given by Eq. (3) involves the use of a set of spect.roscopic splitting constants a1 . . a, . Each splitt,ing constant oli is representable by an operator hi(w) which causes each line in the parent line speckurn to “split” into two or more lines whose locations and intensities depend upon the associated nuclear spins as discussed in Appendix II. The ideal line spectrum B(w) is t’he result of the successive convolution of these operators. B(w)
* 6dwj *
= bl(w) * b,jw)
. . .
b,(W).
(4)
A graphical representation for the generation of the ideal spectrum corresponding to Eq. (2) is provided by Fig. 1 for the simplified case of t’wo splitting constants. The determination of the splitting constants cyi . . . a, is of primary importance to research scientists in many branches of physics and chemistry. In many instances it1 is possible to determine these constants from a visual inspection of the spectrum. For more complex spectra the use of a computer programmed to implement one of the least-squares approximation tec,hniques of Marquardt (6), Welch (7), or Gladney (8) may be required. These t,echniques require the eonstruction of bhe model spectrum S( LO,(Y~. . a,) at a number (NSP) of discrete points Wl ’ ’ ’ aN8P . The splitting constants are varied until the error E’ between the experimental spectrum SX(w) measured at the discrete points w1 . . . wNSp , and the corresponding model spectrum 8(w; CQ . . . a,) is minimized according t,o the criterion NBP E
=
C i=l
[fix(~)
-
S(w;,
a1
. . .
CU,)]~
=
minimum.
The above procedures provide excellent results when the number of lines comprising the model spectrum is small. As the number of lines comprising the spectrum becomes large, however, the computation required to construct the model spectrum becomes increasingly prohibitive. To reduce the computational burden for large multiline spectra, we have developed the following approximate technique which utilizes the ideal line spectrum R(W) given by Eq. (4) instead of the more complex model spectrum S(o) as given by Eq. (2). In t.he simplified analysis, the experimental spectrum 8X(w) is represent.ed by a set of weighted impulses located at the apparent maxima of the partially reb, (~1
b&d
FIG.
1.
Generation
of
ideal
spectrum
144
REEDER,
ODELL,
SIODA,
AND KOSKI
solved lines of the spectrum as shown in Fig. 2. This approach has great practical appeal since line positions may be determined from an experimental spectrum with some precision, while the intensities of the lines may often be obscure, In this method we ignore the detailed shape of the partially resolved lines. Each such line is assumed to be the result of combining several basic line shapes (D(w/u) as specified in Eqs. (1) and (2). The number of lines comprising the weighted impulse representation of the experimental spectrum 8X(,) is normally less than the number of lines comprising the ideal line spectrum as given by the model of Eq. (3). This is caused by the overlapping of the lines comprising the experimental spectrum and by the limited resolving power of the apparatus used to measure the spectrum. Thus to compare the ideal line spectrum with the experimental line spectrum, a set of rules must be formulated for combining the lines of the ideal spectrum to form the lines of the experimental spectrum. Perhaps the only rigorous manner for accomplishing this task is to associate a known shape to each line and superpose the lines as suggested by Eq. (1). Unfortunately, this approach presents a heavy computational burden when the number of lines comprising the ideal spectrum becomes large. This approach has been abandoned in favor of the following approximate method which combines computational simplicity with some intuitive appeal. Two lines of the ideal line spectrum such as shown in Fig. 3 are combined to form a single line by requiring that the weight and first moment of the single line conserve the weights and first moments of the original lines. The resultant weight A and location liT of the single line are given by A = aj + aj+1,
(6)
u = (ajclj + ~i+Ni+l>l(aj where ui and The number desired value proceeding in
+ aj+l> 7
(7)
pj are the weights and locations of the jth uncombined lines. of lines comprising the ideal line spectrum may be reduced to any by beginning the clustering procedure with nearest neighbors and an iterative fashion unt’il the desired value is obtained. In the
a) Observed FIG.
Spectrum 2. Representation
b) Weighted of experimental
Impulse
absorption
Representation
spectrum
COMPUTER
RESOLUTIOK
OF ESR
SPECTRA
143
A : J
i
I Pj FIG. 3. Clustering
principle
I
aj+i
i
T
I
I
u
Pj+l
used to combine
lines of ideal spectrum
,_‘_“_______‘--,
II
Supplied by Experimenter
i ,
Alter Estimate
I
FIG. 4. Iterative
scheme for determining
splitting
I
constants
following procedures we cluster the lines in the ideal line spectrum until their number equals the number of lines in the experimental line spectrum. The splitting constants characterizing the experimental spectrum XX(w) are determined in accordance with the following iterative procedure which is shown graphically in Fig. 4. (1) The experimental spectrum XX(w) is reduced to its weighted impulse response BX(w) by the experimenter. (2) A trial set of splitting constants is chosen based upon theoretical considerations of t’he chemical problem being treated and a visual examination of the spectrum. (3) The ideal line spectrum B(w, al* . . . a,*) is constructed from the trial splitting constants al* - . . a, * in accordance with Eq. (4) and Appendix II. (4) The lines of the ideal line spectrum B(w, (YI* . . . am*) are clustered until their number equals the number of lines in the experimental line spectrum BX(w) . (5) The clustered ideal line spectrum is compared to the experimental line spectrum in accordance with the error criterion discussed below. (6) If the ideal and experimental line spectra agree to within acceptable
146
REEDER,
ODELL,
SIODA,
AND
KOSKI
limits, the trial splitting constants (or* . . . CL* are accepted as the estimates of the true splitting constants and the process is terminated. Otherwise the trial splitting constants are altered as discussed below and the procedure is repeated from step 3. Many choices are possible for the error criterion indicated in step 5 of the iterative procedure. Again in the interest of economy of computational effort, we have chosen a very simple criterion. In this criterion, the error E(cu) associated with the set of trial splitting constants CY= cyl . . - a, is defined as NSP
in which BXK(w) denotes the position of the kth line of the experimental spectrum, which contains NW lines, and BCi(w) is the position of the ith line of the clustered ideal line spectrum. The summation on i runs over the set of indices for which the lines BC;(o) are nearest neighbors of the line BX,(w). The “nearest neighbors” are determined by constructing bisectors between the various lines BXK(w) comprising the experimental line spectrum BX(w). The clustered ideal line spectrum is superimposed upon this spectrum as shown in Fig. 5 and each of the clustered lines BCi(w) is associated with the experimental line BXK(w) lying within the set of bisectors bounding the clustered line. In the event that no clustered line falls within a set of bisectors, the error function is increased by an amount equal to the distance between the bisectors. This step is included to express in the error function the necessity that the set of splitting constants explain every line in the experimental line spectrum.
!
I
i
A ?
i
I I
i
! ! !
4
i i i i
t I
I !
I I
i
I
-
Experimental
-----
Ideal
---.-
Bisectors
Line
Nearest FIG. 5. Determination
Line
4 I I I
I
Spectrum
Spectrum used to
4
(Clustered) Establish
Neighbors of nearest
neighbors
in error criterion
COMPUTER
RESOLUTION
OF ESR
147
SPECTRA
The criterion of acceptability which we have established fol step 6 of the iterative procedure is that the error defined by Eq. (8) be a minimum. The error criterion which we have established can admit discontinuities in its first partial derivatives with respect to the trial splitting con&ants. We have therefore chosen an optimization scheme for altering the set of trial splitting constants which does not directly utilize information concerning the first partial derivatives of t,he error function. The method chosen is a modification of the method of “direetsearch” due to Hooke and Jeeves (11). The method has been modified to utilize the information concerning the angle between successive correction vectors in the splitting constant parameter space to aid in fut,ure alteration of these parameters as shown in Appendix I. This modification of the method was found to materially aid the speed of convergence of the algorithm to the minimum value of the error function. As with all parameter optimization schemes, the question of local versus global minima arises. The strategy which we have employed is to begin the search from another point in the parameter spare once a minimum has been reached. The true minimum is chosen as the lowest of the minima encountered after a reasonable number of trials. Fortunately, most of the false local minima are easily detected when the intensities of the lines in the ideal and experimental line spectra are compared. II. AN EXPERIMENTAL
VERIFICATION
OF THE
METHOD
The method has been applied for improving the resolution of several experimental ESR spectra containing approximately one hundred theoretical lines. The most interesting results of resolution were obtained for the speckurn of the free radical anion of acenaphthylene shown in Fig. 6. (Note this is a derivative of the spectra we have been discussing. The former discussion applies to both absorption and derivative curves.) The spectrum exhibits 49 partially resolved
FIG. 6. Experimental
derivative
spectrum
of the free radical
anion of acenaphthylene
148
REEDER,
ODELL,
SIODA,
AND KOSKI
lines having an average peak-to-peak linewidth of approximately 0.25 gauss. The positions of the middle points of the derivative lines of the spectrum and the relative intensities of the lines were obt,ained as the mathematical average of the two sides of the experiment.al spectrum. The spectrum is the result of the interaction of the magnetic field with the magnetic moments of the four pairs of equivalent protons in the acenaphthylene chemical structure shown in Fig. 7. The ideal line spectrum is representable by four splitting constants of type I and degeneracy 2 (see Appendix II) and consists of 81 possible lines. The application of Hiickel molecular orbital spin densities (12) and the application of McConnell’s formula (13) to this problem with Q set equal to 30 gauss yields theoretical splitting constants of al = 3.12 G,
a4 = 0.42 G,
a3 = 4.53 G,
a6 = 5.34 G.
The results of using the McConnell estimates and a second arbitrary set of trial splitting constants as the initial estimates to the optimization program are shown in Table I. Both estimates converged to the same set of splitting constants, also shown in Table I. The intensities and positions of the experimental and ideal line spectra showed excellent agreement for this choice of the splitting constants. The final set of splitting constants also showed excellent agreement with the results of Iwaizumi and Isobe (14) which was obtained in an independent manner. The initial error for the McConnell estimate of the splitting constants was 3,654 gauss compared to 5.462 for the second choice for the constants. The final error of 0.378 gauss required 14 seconds and 149 evaluations of the error function
FIG. 7. Chemical
structure
of the radical
anion
of acenaphthylene
COMPUTER
RESOLUTION
149
OF ESR SPECTRA
TABLE 1 ESR SPLITTINQ CONSTANTS IN ACENAPHTHYLENE Position in acenaphthylene
McConnell estimate
Arbitrary estimate
Optimum values found by search
Literature values*
la2 3,8 4,7 5,6
3.12 4.53 0.42 5.34
3.00 4.30 0.52 5.00
3.020 4.457 0.476 5.549
3.06 4.41 0.46 5.60
* See Reference 13. for the McConnell estimate and 22 seconds and 250 estimates of the error function for the second estimate using an IBM 7094 computer to find the splitting c0nstant.s to an accuracy of 0.001 gauss. Less than 10 seconds were required for both initial estimates to determine the splitting constants to an accuracy of 0.01 gauss. III. SUMMARY AND CONCLUSIONS A method for improving the estimates of a set of splitting constants for an ESR spectrum has been devised which utilizes a weighted impulse representation (line spectrum) for the experimental data and an ideal line spectrum in the optimization procedure. The procedure minimizes the error between the positions of the lines in the experimental and ideal line spect’ra by a modified method of direct search minimization using the weights of the lines in the experimental and ideal line spectra as a check on the process. The number of lines in the ideal spectrum is made equal t’o the number of lines in the experimental spectrum by a clustering procedure which prese.rves the weights and first moments of t.he nearest neighbors in the ideal spectrum. The method has been tested on a number of multiline ESR spectra including a spectrum of the free radical anion of acenaphthylene with good results. The method is characterized by its computational simplicity resulting in significant savings in time for t’he analysis of large multiline spectra. The method may find application to t#he improvement of the accuracy of the splitting const’ants for complex multiline ESR spectra. APPENDIX
I. A MODIFIED
METHOD OF DIRECT-SEARCH
MINIMIZATION
The method of direct-search minimization proposed by Hooke and Jeeves (10) provides a simple but highly effective iterative procedure for the minimization of error functions such as those specified by Eqs. (5) and (8) in the text. The method presented here is a modification of this technique designed to improve its rate of convergence to the minimum. The basic iterative t,echnique proposed by Hooke and Jeeves for the minimiza-
150
REEDER,
ODELL,
SIODA,
AND
KOSKI
tion of an error function E( @) where @is an N dimensional vector of the parameters to be estimated is as follows: (1) Choose an initial estimate @*for the parameter vector @.The estimate @* will be denoted by the term “base point.” (2) Evaluate the error function E(@*) for the base point @*. (3) Perform a series of univariate exploratory moves around the current base point @*as follows : (a) Set an index i = 1. Define an exploratory point 8 = @*and the corresponding error E( @) = ,Y( @*) . (b) Define @+ = /j + A@; w h ere A& represents a positive increment in the ith parameter of @. (c) Evaluate the error E( @+). (d) If E(@+) < E(o), set 0 = IJ+ and E( @) = E( Q+) and proceed to step 3h. Otherwise proceed to step 3e. in the (e) Define lj- = 0 - A& w h ere A@ represents a positive increment ith parameter of @. (f) Evaluate the error E( @-). (g) If E(K) < E(o), set 0 = @- and E(@) = E(Q-), otherwise proceed to step 3h. (h) Increment the index parameter i by 1. Proceed to step 3b if i 6 N. If i > N proceed to step 4. (4) If E(e) < E(@*) P er f orm a pattern move as defined by step 5, otherwise proceed to step 7. (5) (a) Define a pattern point @” = e* + 2( ,$ - @*). (b) Define a new base point @* = @. (c) Perform the series of univariate exploratory moves given in step 3 around the pattern point @”to obtain a new exploratory point 8. (6) If E(6) < E(@*) P roceed to step 5a, otherwise proceed to step 2. (7) Test to see if all the parameter increments A& are less than the allowable minimum tolerances Aei . If all the parameter increments are less than the allowed tolerances, the estimate of the parameters is taken as @*and the process is terminated, otherwise the parameter increments are reduced by a constant multiplicative factor p less than 1.0 and the process is repeated from step 2. A flow chart for the basic direct-search minimization scheme is given in Fig. 8. The above iterative procedure has been modified to use the angle @ between successive pattern moves to modify the definition of the pattern point 0” given in step 5 of the procedure. The new pattern point is defined by the expression @” = @* +
(C + D cos”@+(~
-
@*).
(I-1)
The selection of the empirical constants C, D, and n given in Eq. (I-l) and the selection of the parameter increment reduction factor p used in step 7 of the iterative procedure is largely a matter of trial and error. We have found satis-
151
COMPUTER RESOLUTION OF ESR SPECTRA Yes
Perform
Establish
IS
--+o
aYes
Perform IS
Terminote
+@
Scorch
FIG. 8. Flow chart for basic direct-search minimization procedure
FIG. 9. Modification of the direct-search procedure
factory values for these constants c = 1.7,
to be
D = 34,
n = 3,
P = 540.
These values have been found to allow the search program to successfully negotiate “sharp corners” in the error surface by effecting a series of parameter increment reductions and exploratory moves without seriously impairing Dhe ability to accelerate toward the minimum along a “long straight trough” by a series of successive exploratory moves. Provisions have also been incorporated in the iterative procedure to permit the individual adjustment of the parameter increments following a successful pattern move. This provides a different search sensitivity for the various parameters on each iteration. The individual parameter increments are increased or decreased in accordance with the relative magnitudes of the changes of the parameter on successive pattern moves. The flow chart for the modified direct search procedure is identical with Fig. 8 with the exception that step 6 is expanded as shown in Fig. 9 and cos 9 is set t,o 0.1 in step 2.
152
REEDER, ODELL, SIODA, AND KOSKI APPENDIXII.
GENERATION OF THE IDEAL LINE SPECTRUM
The splitting constants QC;may be represented by various basic types of convolution operators b ( w, a) utilized in Eq. (4). We have considered two such operators. The first operat,or is associated with a nucleus which possesses a nuclear spin of &x. This operator causes each parent line in a line spectrum to split into two lines each having $5 the intensity of the parent line. The second operator is associated with a nuclear spin of fl. It causes each parent line to split into three lines having g the intensity of the parent line. The magnitude of the splitting constant is defined as the distance between the split lines for both classes of operators. A type one splitting constant is defined by a number of successive convolutions of the first operator, and a type two splitting constant is defined by a number of successive convolutions of the second operator. The number of successive convolutions of the basic operator is determined by the degeneracy of the splitting constant. A degeneracy of 2, for example, indicates that the basic operator is applied two times in succession. Each splitting constant of a certain type and degeneracy is representable by an equivalent single convolution operator. A table of these equivalent, operators is stored for use by the computer program. The ideal line spectrum B ( CO)defined by Eq. (4) is generated by retrieving and convolving these equivalent operators for each of the splitting constants whose number, types, and degeneracies are supplied as input to the computer program. RECEIVED : February 7,1966 REFERENCES 1. I>. J. E. INGRAM “Free Butterworths, B. 8. 4. 6. 6. 7. 8. 9.
10. 11.
18. 19. 14.
London,
Radicals 1958.
as Studied
by Electron
Spin Resonance,”
p. 121.
M. M. CHEN, K. V. SANE, R. I. WALTER, AND J. A. WEIL, J. Phys. Chem. 65,713 (1961). E. W. STONE AND A. H. MAKI, J. Chem. Phys. 38, 1999 (1963). R. LEFEBVRE AND J. MARUANI, J. Chem. Phys. 42,148O (1965). R. LEFEBVRE AND J. MARUANI, J. Chem. Phys. 42, 1496 (1965). D. W. MARQUARDT, R. G. BENNETT, AND E. J. BURRELL, J. Mol. Spectry. 7, 269 (1961). M. WELCH, “Line Shape Fitting by Variable Metric Minimization,” Argonne National Laboratory Program Library 112O/PhY 220. H. M. GLADNEY AND J. D. SWALEN, IBM J. Res. Develop. 8,515 (1964). S. W. MARSHALL, J. A. NELSON, AND R. M. WILENZICK, Least-Squares Analysis of Resonance Spectra on Small Computers, Communications of Association for Computing Machines 8, 313 (1965). J. R. B~LTON AND G. K. FRAENKEL, J. Chem. Phys. 40,3307 (1964), Reference 85. R. HOUSE AND T. A. JEEVES, Assoc. Computing Machinery J. 8,212 (1961). E. DE BOER AND S. I. WEISSMAN, J. Am. Chem. Sot. 80,4549 (1953). H. M. MCCONNELL, J. Chem. Phya. 24, 632 (1956). M. IWAIZUMI AND T. ISOBE, Bull. Chem. Sot. Japan 37, 1651 (1964).
OF MOLECULAR
A Simplified
SPECTROSCOPY
20,
141-152
(1966)
Approach to Multiline ESR Spectra Using a Digital Computer*
J. R. REEnER,j’
G. M. ODE&t
Department of Chemistry,
R. E. SIODA, AND
The Johns Hopkins University,
Resolution
W. S. KOSKI
Baltimore,
Maryhnd
A simplified method for the analysis of multiline isotropic ESR spectra is presented which provides significant computational advantages over some of the standard analysis techniques. The method utilizes the positions of the apparent maxima of the partially resolved lines in a multiline spectrum to determine the splitting constants characterizing the spectrum. The relative intensities of the apparent maxima are used as a check on the resolution provided. Application of the method to the resolution of a spectrum of acenaphthylene demonstrates its usefulness in refining the estimates of a set of splitting constants. INTRODUCTION
The electron spin resonance spectra of organic free radicals resulting from the isotropic or anisotropic interaction of the unpaired electron with the magnetic fields may be represented in terms of a superposition of elementary waveforms (or lines) whose shape is approximately known (1) . The relative weights and positions of the lines may be specified by a set of splitting constants representing a set of convolution operations upon the elementary waveforms. The determination of the splitting con&ants from experimental spectra may be easily accomplished provided the number of lines is small and there is no appreciable overlapping of the lines. Such is the case, for instance, in some simple isotropic intcractions exhibited by some free radicals in the liquid phase. The interpretation of more complex spectra can be, however, a difficult task. The modern high-speed digital computer provides a valuable tool for the interpretat)ion of such spectra. Chen, Sane, Walter, and Weil (2) have devised a program for the determination of the ratio of t’he nitrogen splitting constants in p-substitut’ed l , l-diphenyl-2-picryl-hydrazyl. Spectral simulation programs which reconstruct a spectrum from a trial set of splitting constants have been prepared by Stone and Maki (3) for isotropic ESR spectra and by Lefebvre and Maruani (4,5) for ESR spectra of amorphous solid samples. Marquardt (6)) * This investigation was supported in part by research grant GM 5144 from the Divisiotl of General Medical Sciences, Public Health Service, National Institutes of Healt,h. t Department of Electrical Engineering, The Johns Hopkins University. 141
142
REEDER,
ODELL,
SIODA,
AND
KOSKI
Welch (7)) Gladney (8)) and Marshall (9) have presented methods of leastsquares approximation which are applicable to the interpretation of many types of spectra by the digital computer. Kaplan prepared a digital computer program for least-squares analysis of multiline spectra using tentative splitting constants (10). The amount of computation required by most of the techniques cited above becomes prohibitive when the number of lines comprising the spectrum becomes very large. In this paper we present a simplified method for the analysis of multiline isotropic ESR spectra of free radicals which provides significant computational advantages over some of the standard analysis techniques. The method utilizes only the positions of the apparent maxima of the partially resolved lines in a multi-line spectrum to determine the splitting constants characterizing the spectrum. The relative intensities of the apparent maxima are used as a check on the resolution provided. Application of the method to the resolution of a spectrum of acenaphthylene demonstrates its usefulness in refining t,he estimates of a set of splitting constants in an efficient manner. I. FORMULATION
OF THE METHOD
An ESR spectrum S(o) may be represented as the superposition of elementary waveforms (or lines) whose shape is approximately A mathematical model which is appropriate for many applications
of a number known (1). is of the form
(1) in which q(o/a) is the line shape function. This function is normally a symmetric “bell-shaped” function such as the Gaussian, Lorentzian, or Voigt functions. LJ is the independent variable; pi is an epoch parameter establishing the location of the ith line; A; is the weight parameter of the ith line; ui is a structural parameter establishing the scale of the line; N is the number of lines comprising the spectrum. In those instances in which the structural parameters ui for all the lines may be regarded as identical, the spectrum S(w) may be considered to be the result of convolving an ideal line spectrum B( CO)with the elementary lineshape (p( W/U) ; thus
NW) =
11B(7) (; - 7)d7 = B(o)*‘p ($) ‘$0
)
where
and S(o -
pi) is a Dirac delta function
located
at w = pi _
COMPUTER
RESOLUTION
OF ESR
SPECTKA
143
A method for specifying the Iine spectrum B(w) given by Eq. (3) involves the use of a set of spect.roscopic splitting constants a1 . . a, . Each splitt,ing constant oli is representable by an operator hi(w) which causes each line in the parent line speckurn to “split” into two or more lines whose locations and intensities depend upon the associated nuclear spins as discussed in Appendix II. The ideal line spectrum B(w) is t’he result of the successive convolution of these operators. B(w)
* 6dwj *
= bl(w) * b,jw)
. . .
b,(W).
(4)
A graphical representation for the generation of the ideal spectrum corresponding to Eq. (2) is provided by Fig. 1 for the simplified case of t’wo splitting constants. The determination of the splitting constants cyi . . . a, is of primary importance to research scientists in many branches of physics and chemistry. In many instances it1 is possible to determine these constants from a visual inspection of the spectrum. For more complex spectra the use of a computer programmed to implement one of the least-squares approximation tec,hniques of Marquardt (6), Welch (7), or Gladney (8) may be required. These t,echniques require the eonstruction of bhe model spectrum S( LO,(Y~. . a,) at a number (NSP) of discrete points Wl ’ ’ ’ aN8P . The splitting constants are varied until the error E’ between the experimental spectrum SX(w) measured at the discrete points w1 . . . wNSp , and the corresponding model spectrum 8(w; CQ . . . a,) is minimized according t,o the criterion NBP E
=
C i=l
[fix(~)
-
S(w;,
a1
. . .
CU,)]~
=
minimum.
The above procedures provide excellent results when the number of lines comprising the model spectrum is small. As the number of lines comprising the spectrum becomes large, however, the computation required to construct the model spectrum becomes increasingly prohibitive. To reduce the computational burden for large multiline spectra, we have developed the following approximate technique which utilizes the ideal line spectrum R(W) given by Eq. (4) instead of the more complex model spectrum S(o) as given by Eq. (2). In t.he simplified analysis, the experimental spectrum 8X(w) is represent.ed by a set of weighted impulses located at the apparent maxima of the partially reb, (~1
b&d
FIG.
1.
Generation
of
ideal
spectrum
144
REEDER,
ODELL,
SIODA,
AND KOSKI
solved lines of the spectrum as shown in Fig. 2. This approach has great practical appeal since line positions may be determined from an experimental spectrum with some precision, while the intensities of the lines may often be obscure, In this method we ignore the detailed shape of the partially resolved lines. Each such line is assumed to be the result of combining several basic line shapes (D(w/u) as specified in Eqs. (1) and (2). The number of lines comprising the weighted impulse representation of the experimental spectrum 8X(,) is normally less than the number of lines comprising the ideal line spectrum as given by the model of Eq. (3). This is caused by the overlapping of the lines comprising the experimental spectrum and by the limited resolving power of the apparatus used to measure the spectrum. Thus to compare the ideal line spectrum with the experimental line spectrum, a set of rules must be formulated for combining the lines of the ideal spectrum to form the lines of the experimental spectrum. Perhaps the only rigorous manner for accomplishing this task is to associate a known shape to each line and superpose the lines as suggested by Eq. (1). Unfortunately, this approach presents a heavy computational burden when the number of lines comprising the ideal spectrum becomes large. This approach has been abandoned in favor of the following approximate method which combines computational simplicity with some intuitive appeal. Two lines of the ideal line spectrum such as shown in Fig. 3 are combined to form a single line by requiring that the weight and first moment of the single line conserve the weights and first moments of the original lines. The resultant weight A and location liT of the single line are given by A = aj + aj+1,
(6)
u = (ajclj + ~i+Ni+l>l(aj where ui and The number desired value proceeding in
+ aj+l> 7
(7)
pj are the weights and locations of the jth uncombined lines. of lines comprising the ideal line spectrum may be reduced to any by beginning the clustering procedure with nearest neighbors and an iterative fashion unt’il the desired value is obtained. In the
a) Observed FIG.
Spectrum 2. Representation
b) Weighted of experimental
Impulse
absorption
Representation
spectrum
COMPUTER
RESOLUTIOK
OF ESR
SPECTRA
143
A : J
i
I Pj FIG. 3. Clustering
principle
I
aj+i
i
T
I
I
u
Pj+l
used to combine
lines of ideal spectrum
,_‘_“_______‘--,
II
Supplied by Experimenter
i ,
Alter Estimate
I
FIG. 4. Iterative
scheme for determining
splitting
I
constants
following procedures we cluster the lines in the ideal line spectrum until their number equals the number of lines in the experimental line spectrum. The splitting constants characterizing the experimental spectrum XX(w) are determined in accordance with the following iterative procedure which is shown graphically in Fig. 4. (1) The experimental spectrum XX(w) is reduced to its weighted impulse response BX(w) by the experimenter. (2) A trial set of splitting constants is chosen based upon theoretical considerations of t’he chemical problem being treated and a visual examination of the spectrum. (3) The ideal line spectrum B(w, al* . . . a,*) is constructed from the trial splitting constants al* - . . a, * in accordance with Eq. (4) and Appendix II. (4) The lines of the ideal line spectrum B(w, (YI* . . . am*) are clustered until their number equals the number of lines in the experimental line spectrum BX(w) . (5) The clustered ideal line spectrum is compared to the experimental line spectrum in accordance with the error criterion discussed below. (6) If the ideal and experimental line spectra agree to within acceptable
146
REEDER,
ODELL,
SIODA,
AND
KOSKI
limits, the trial splitting constants (or* . . . CL* are accepted as the estimates of the true splitting constants and the process is terminated. Otherwise the trial splitting constants are altered as discussed below and the procedure is repeated from step 3. Many choices are possible for the error criterion indicated in step 5 of the iterative procedure. Again in the interest of economy of computational effort, we have chosen a very simple criterion. In this criterion, the error E(cu) associated with the set of trial splitting constants CY= cyl . . - a, is defined as NSP
in which BXK(w) denotes the position of the kth line of the experimental spectrum, which contains NW lines, and BCi(w) is the position of the ith line of the clustered ideal line spectrum. The summation on i runs over the set of indices for which the lines BC;(o) are nearest neighbors of the line BX,(w). The “nearest neighbors” are determined by constructing bisectors between the various lines BXK(w) comprising the experimental line spectrum BX(w). The clustered ideal line spectrum is superimposed upon this spectrum as shown in Fig. 5 and each of the clustered lines BCi(w) is associated with the experimental line BXK(w) lying within the set of bisectors bounding the clustered line. In the event that no clustered line falls within a set of bisectors, the error function is increased by an amount equal to the distance between the bisectors. This step is included to express in the error function the necessity that the set of splitting constants explain every line in the experimental line spectrum.
!
I
i
A ?
i
I I
i
! ! !
4
i i i i
t I
I !
I I
i
I
-
Experimental
-----
Ideal
---.-
Bisectors
Line
Nearest FIG. 5. Determination
Line
4 I I I
I
Spectrum
Spectrum used to
4
(Clustered) Establish
Neighbors of nearest
neighbors
in error criterion
COMPUTER
RESOLUTION
OF ESR
147
SPECTRA
The criterion of acceptability which we have established fol step 6 of the iterative procedure is that the error defined by Eq. (8) be a minimum. The error criterion which we have established can admit discontinuities in its first partial derivatives with respect to the trial splitting con&ants. We have therefore chosen an optimization scheme for altering the set of trial splitting constants which does not directly utilize information concerning the first partial derivatives of t,he error function. The method chosen is a modification of the method of “direetsearch” due to Hooke and Jeeves (11). The method has been modified to utilize the information concerning the angle between successive correction vectors in the splitting constant parameter space to aid in fut,ure alteration of these parameters as shown in Appendix I. This modification of the method was found to materially aid the speed of convergence of the algorithm to the minimum value of the error function. As with all parameter optimization schemes, the question of local versus global minima arises. The strategy which we have employed is to begin the search from another point in the parameter spare once a minimum has been reached. The true minimum is chosen as the lowest of the minima encountered after a reasonable number of trials. Fortunately, most of the false local minima are easily detected when the intensities of the lines in the ideal and experimental line spectra are compared. II. AN EXPERIMENTAL
VERIFICATION
OF THE
METHOD
The method has been applied for improving the resolution of several experimental ESR spectra containing approximately one hundred theoretical lines. The most interesting results of resolution were obtained for the speckurn of the free radical anion of acenaphthylene shown in Fig. 6. (Note this is a derivative of the spectra we have been discussing. The former discussion applies to both absorption and derivative curves.) The spectrum exhibits 49 partially resolved
FIG. 6. Experimental
derivative
spectrum
of the free radical
anion of acenaphthylene
148
REEDER,
ODELL,
SIODA,
AND KOSKI
lines having an average peak-to-peak linewidth of approximately 0.25 gauss. The positions of the middle points of the derivative lines of the spectrum and the relative intensities of the lines were obt,ained as the mathematical average of the two sides of the experiment.al spectrum. The spectrum is the result of the interaction of the magnetic field with the magnetic moments of the four pairs of equivalent protons in the acenaphthylene chemical structure shown in Fig. 7. The ideal line spectrum is representable by four splitting constants of type I and degeneracy 2 (see Appendix II) and consists of 81 possible lines. The application of Hiickel molecular orbital spin densities (12) and the application of McConnell’s formula (13) to this problem with Q set equal to 30 gauss yields theoretical splitting constants of al = 3.12 G,
a4 = 0.42 G,
a3 = 4.53 G,
a6 = 5.34 G.
The results of using the McConnell estimates and a second arbitrary set of trial splitting constants as the initial estimates to the optimization program are shown in Table I. Both estimates converged to the same set of splitting constants, also shown in Table I. The intensities and positions of the experimental and ideal line spectra showed excellent agreement for this choice of the splitting constants. The final set of splitting constants also showed excellent agreement with the results of Iwaizumi and Isobe (14) which was obtained in an independent manner. The initial error for the McConnell estimate of the splitting constants was 3,654 gauss compared to 5.462 for the second choice for the constants. The final error of 0.378 gauss required 14 seconds and 149 evaluations of the error function
FIG. 7. Chemical
structure
of the radical
anion
of acenaphthylene
COMPUTER
RESOLUTION
149
OF ESR SPECTRA
TABLE 1 ESR SPLITTINQ CONSTANTS IN ACENAPHTHYLENE Position in acenaphthylene
McConnell estimate
Arbitrary estimate
Optimum values found by search
Literature values*
la2 3,8 4,7 5,6
3.12 4.53 0.42 5.34
3.00 4.30 0.52 5.00
3.020 4.457 0.476 5.549
3.06 4.41 0.46 5.60
* See Reference 13. for the McConnell estimate and 22 seconds and 250 estimates of the error function for the second estimate using an IBM 7094 computer to find the splitting c0nstant.s to an accuracy of 0.001 gauss. Less than 10 seconds were required for both initial estimates to determine the splitting constants to an accuracy of 0.01 gauss. III. SUMMARY AND CONCLUSIONS A method for improving the estimates of a set of splitting constants for an ESR spectrum has been devised which utilizes a weighted impulse representation (line spectrum) for the experimental data and an ideal line spectrum in the optimization procedure. The procedure minimizes the error between the positions of the lines in the experimental and ideal line spect’ra by a modified method of direct search minimization using the weights of the lines in the experimental and ideal line spectra as a check on the process. The number of lines in the ideal spectrum is made equal t’o the number of lines in the experimental spectrum by a clustering procedure which prese.rves the weights and first moments of t.he nearest neighbors in the ideal spectrum. The method has been tested on a number of multiline ESR spectra including a spectrum of the free radical anion of acenaphthylene with good results. The method is characterized by its computational simplicity resulting in significant savings in time for t’he analysis of large multiline spectra. The method may find application to t#he improvement of the accuracy of the splitting const’ants for complex multiline ESR spectra. APPENDIX
I. A MODIFIED
METHOD OF DIRECT-SEARCH
MINIMIZATION
The method of direct-search minimization proposed by Hooke and Jeeves (10) provides a simple but highly effective iterative procedure for the minimization of error functions such as those specified by Eqs. (5) and (8) in the text. The method presented here is a modification of this technique designed to improve its rate of convergence to the minimum. The basic iterative t,echnique proposed by Hooke and Jeeves for the minimiza-
150
REEDER,
ODELL,
SIODA,
AND
KOSKI
tion of an error function E( @) where @is an N dimensional vector of the parameters to be estimated is as follows: (1) Choose an initial estimate @*for the parameter vector @.The estimate @* will be denoted by the term “base point.” (2) Evaluate the error function E(@*) for the base point @*. (3) Perform a series of univariate exploratory moves around the current base point @*as follows : (a) Set an index i = 1. Define an exploratory point 8 = @*and the corresponding error E( @) = ,Y( @*) . (b) Define @+ = /j + A@; w h ere A& represents a positive increment in the ith parameter of @. (c) Evaluate the error E( @+). (d) If E(@+) < E(o), set 0 = IJ+ and E( @) = E( Q+) and proceed to step 3h. Otherwise proceed to step 3e. in the (e) Define lj- = 0 - A& w h ere A@ represents a positive increment ith parameter of @. (f) Evaluate the error E( @-). (g) If E(K) < E(o), set 0 = @- and E(@) = E(Q-), otherwise proceed to step 3h. (h) Increment the index parameter i by 1. Proceed to step 3b if i 6 N. If i > N proceed to step 4. (4) If E(e) < E(@*) P er f orm a pattern move as defined by step 5, otherwise proceed to step 7. (5) (a) Define a pattern point @” = e* + 2( ,$ - @*). (b) Define a new base point @* = @. (c) Perform the series of univariate exploratory moves given in step 3 around the pattern point @”to obtain a new exploratory point 8. (6) If E(6) < E(@*) P roceed to step 5a, otherwise proceed to step 2. (7) Test to see if all the parameter increments A& are less than the allowable minimum tolerances Aei . If all the parameter increments are less than the allowed tolerances, the estimate of the parameters is taken as @*and the process is terminated, otherwise the parameter increments are reduced by a constant multiplicative factor p less than 1.0 and the process is repeated from step 2. A flow chart for the basic direct-search minimization scheme is given in Fig. 8. The above iterative procedure has been modified to use the angle @ between successive pattern moves to modify the definition of the pattern point 0” given in step 5 of the procedure. The new pattern point is defined by the expression @” = @* +
(C + D cos”@+(~
-
@*).
(I-1)
The selection of the empirical constants C, D, and n given in Eq. (I-l) and the selection of the parameter increment reduction factor p used in step 7 of the iterative procedure is largely a matter of trial and error. We have found satis-
151
COMPUTER RESOLUTION OF ESR SPECTRA Yes
Perform
Establish
IS
--+o
aYes
Perform IS
Terminote
+@
Scorch
FIG. 8. Flow chart for basic direct-search minimization procedure
FIG. 9. Modification of the direct-search procedure
factory values for these constants c = 1.7,
to be
D = 34,
n = 3,
P = 540.
These values have been found to allow the search program to successfully negotiate “sharp corners” in the error surface by effecting a series of parameter increment reductions and exploratory moves without seriously impairing Dhe ability to accelerate toward the minimum along a “long straight trough” by a series of successive exploratory moves. Provisions have also been incorporated in the iterative procedure to permit the individual adjustment of the parameter increments following a successful pattern move. This provides a different search sensitivity for the various parameters on each iteration. The individual parameter increments are increased or decreased in accordance with the relative magnitudes of the changes of the parameter on successive pattern moves. The flow chart for the modified direct search procedure is identical with Fig. 8 with the exception that step 6 is expanded as shown in Fig. 9 and cos 9 is set t,o 0.1 in step 2.
152
REEDER, ODELL, SIODA, AND KOSKI APPENDIXII.
GENERATION OF THE IDEAL LINE SPECTRUM
The splitting constants QC;may be represented by various basic types of convolution operators b ( w, a) utilized in Eq. (4). We have considered two such operators. The first operat,or is associated with a nucleus which possesses a nuclear spin of &x. This operator causes each parent line in a line spectrum to split into two lines each having $5 the intensity of the parent line. The second operator is associated with a nuclear spin of fl. It causes each parent line to split into three lines having g the intensity of the parent line. The magnitude of the splitting constant is defined as the distance between the split lines for both classes of operators. A type one splitting constant is defined by a number of successive convolutions of the first operator, and a type two splitting constant is defined by a number of successive convolutions of the second operator. The number of successive convolutions of the basic operator is determined by the degeneracy of the splitting constant. A degeneracy of 2, for example, indicates that the basic operator is applied two times in succession. Each splitting constant of a certain type and degeneracy is representable by an equivalent single convolution operator. A table of these equivalent, operators is stored for use by the computer program. The ideal line spectrum B ( CO)defined by Eq. (4) is generated by retrieving and convolving these equivalent operators for each of the splitting constants whose number, types, and degeneracies are supplied as input to the computer program. RECEIVED : February 7,1966 REFERENCES 1. I>. J. E. INGRAM “Free Butterworths, B. 8. 4. 6. 6. 7. 8. 9.
10. 11.
18. 19. 14.
London,
Radicals 1958.
as Studied
by Electron
Spin Resonance,”
p. 121.
M. M. CHEN, K. V. SANE, R. I. WALTER, AND J. A. WEIL, J. Phys. Chem. 65,713 (1961). E. W. STONE AND A. H. MAKI, J. Chem. Phys. 38, 1999 (1963). R. LEFEBVRE AND J. MARUANI, J. Chem. Phys. 42,148O (1965). R. LEFEBVRE AND J. MARUANI, J. Chem. Phys. 42, 1496 (1965). D. W. MARQUARDT, R. G. BENNETT, AND E. J. BURRELL, J. Mol. Spectry. 7, 269 (1961). M. WELCH, “Line Shape Fitting by Variable Metric Minimization,” Argonne National Laboratory Program Library 112O/PhY 220. H. M. GLADNEY AND J. D. SWALEN, IBM J. Res. Develop. 8,515 (1964). S. W. MARSHALL, J. A. NELSON, AND R. M. WILENZICK, Least-Squares Analysis of Resonance Spectra on Small Computers, Communications of Association for Computing Machines 8, 313 (1965). J. R. B~LTON AND G. K. FRAENKEL, J. Chem. Phys. 40,3307 (1964), Reference 85. R. HOUSE AND T. A. JEEVES, Assoc. Computing Machinery J. 8,212 (1961). E. DE BOER AND S. I. WEISSMAN, J. Am. Chem. Sot. 80,4549 (1953). H. M. MCCONNELL, J. Chem. Phya. 24, 632 (1956). M. IWAIZUMI AND T. ISOBE, Bull. Chem. Sot. Japan 37, 1651 (1964).