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A computational approach to the spectrophotometric determination of stability constants—II
Talanra, Vol. 30, No. 8, pp. 579-586, 1983 Printed in Great Britain. All rights reserved
Copyright
0
0039-9140/83 %3.00+0.00 1983 Pergamon Press Ltd
A COMPUTATIONAL APPROACH TO THE SPECTROPHOTOMETRIC DETERMINATION OF STABILITY CONSTANTS-II APPLICATION TO METALLOPORPHY RIN-AXIAL LIGAND INTERACTIONS IN NON-AQUEOUS SOLVENTS D. J. LEGGETT,* S. L. KELLY, L. R. SHIUE, Y. T. Wu, D. CHANG and K. M. RADISH Department
of Chemistry, University of Houston, Houston, TX 77004, U.S.A. (Received 29 September
1982. Accepted
14 February
1983)
Summar-The ability of the computer program SQUAD to deduce a plausible equilibrium model, associated stability constants and spectra of individual species is described. The original version of SQUAD has been extensively modified and these changes are detailed. In particular a “user-friendly” method of data input has been implemented that simplifies familiarization with the program. Brevity
of program code has been sacrificed in favour of the new data input and error-checking features of SQUAD, with beneficial results. The application of SQUAD to five non-aqueous metalloporphyrin-axial ligand interactions exemplifies the program’s ability to handle widely different types of equilibrium systems.
The Benesi-Hildebrand method’ has been extensively used to evaluate the stability constants of metalloporphyrin-axial ligand complexes. For the general reaction M + L=ML
(1)
where M is any metal ion or metalloporphyrin and L is any ligand, the following equations may be written: C, = [M] + [ML]
(2)
C,. = [L] + [ML]
(3)
Blot =
[MLIWIV-I
(4)
The subscripts for j refer to the number of metal ions, hydroxide or hydrogen ions, and ligand ions, respectively, associated with that stability constant. The studies are most often performed in a nonaqueous, non-donor solvent, in which case the middle subscript is then always zero. Making use of Beer’s law: A = +,,[M] + &JL] + cML[ML]
(5)
it can be shown’ that log
(A - 4,) = log CL + log j&o, ~ (A, - A)
where A is the absorbance,
(6)
at a preselected wave-
*Author for correspondence. Present address: Dow Chemical USA, Texas Division, Freeport, TX 77541, U.S.A.
length, of a solution having a known molarity of ligand, C, and known molarity of metal, C, , A, is the absorbance, measured at the same wavelength, of a solution where Cr = 0, and A, is the measured absorbance of a solution for which CL>>&, so that the absorbance is constant with increasing CL. Thus a plot of the left-hand side of equation (6) against log CL should give a straight line which intersects the abscissa at -log CL, which will be equal to log fl,,,, . The Benesi-Hildebrand (B-H) method may be generalized to handle equilibrium systems in which the stoichiometry of the complex formed is unknown. In this situation equation (4) becomes BW =
[MW[Ml[Ll
(7)
and equation (6) becomes log
(A - 4) ~ = n log CL + log j&). W-A)
(8)
Thus the slope of the straight line in the log-log plot provides the stoichiometric coefficient for the complex. Inherent in this method is the assumption that only one complex is formed. This situation is not too frequently encountered and when there is more than one complex present the B-H plots of log [(A - A,)/(A, -A)] vs. log CL will show considerable curvature. It has been argued’ that the plots may still yield information concerning the stoichiometry and stability constants of the complexes, but except under 579
580
D. J. LEGGETT et al.
the most favourable conditions, encountered when complexes of widely differing stability are formed, this claim is not valid. This method, and many similar approaches,3 have the common drawback that they are only applicable if one and only one complex exists under the conditions of measurement. However, in general, more than one complex will exist in solution unless C, >>Cr,, or C,>> Cr. Moreover, when several complexes are present, their nature may not be readily determinable from simple continuous-variation or molar-ratio experiments. Therefore, more sophisticated and rigorous data-processing techniques need to be adopted. Several general computer programs are currently available4” that permit various equilibrium models to be fitted to the experimental data, by change of one or more data car&. This type of program is clearly more desirable than one where the program code needs to be rewritten for each new model tested. The majority of programs developed to date require potentiometric data as input, which may either be obtained from measurements of pH us. volume of base, or of ion-sensing electrode potentials. Spectrophotometry is a widely used technique for the study and determination of equilibrium constants but only a few programs are available to process the data.4.5 Spectrophotometric measurements are generally less precise than potentiometric ones, both inherently and because of the need to prepare several separate solutions. A further complication arises from the fact that for fitting an equilibrium model to absorbance measurements, not only are the stability constants to be determined, but the molar absorptivities for each species, at each wavelength, are also to be evaluated. The situation is not as bad as it might
seem, since the evaluation of molar absorptivities and stability constants is done at two levels within most programs, patterned after an approach suggested by Sillen The program used in the studies described here is SQUAD.‘,’ It is designed to calculate the best values for the stability constants of the proposed equilibrium model by employing a non-linear leastsquares approach. The program is completely general in scope, having the capability to refine stability constants for the general complex M,M; H,L, L;, where m, 1, n, q 2 0 and j is positive (for protons), negative (for hydroxide ions) or zero. Therefore, the same program may be used to study acid-base equilibria for ligands that are weak acids (or bases); metal-ion hydrolysis; complexes of the type ML,, or M,L,; mixed-ligand (or mixed-metal) complexes; protonated or hydroxo complexes. SQUAD was originally designed to process absorbance data from aqueous solutions. Recently extensive modifications have been made to the program that now permit the analysis of data from any type of solution. The algorithms employed in SQUAD and their relationships to each other are shown in Fig. 1. The overall modus operandi will be described briefly and then greater attention will be paid to the novel data-input method. The input data consist of: (a) the absorbance values A+, for each spectrum, giving a total of I for each wavelength, and a grand total of NW; (6) the total metal and ligand concentrations, C,,,, &, pH, (pH for aqueous and mixed solvents only) and pathlength b,, for each spectrum; (c) any known or previously determined molar absorptivities (c,,~) for the jth species at the k th wavelength; (d) the stoichi-
‘-_____-__-___--_----~ PREPRO
INPUT
I
I
User aliases converted to internal parameters
ERR -
REFINE
of GN 1 & Solve equations for shifts on constants
Numerical differentiation
I Calculate
sum
I
SIGMAE
ECOEF
Standard deviation of E
SOlW A=E.C
I
Fig. 1. Block diagram
of SQUAD,
showing
the inter-relationships
among
the various
algorithms.
581
Spectrophotometric determination of stability constants-II ometry and stability constant for the jth species together with an indication of which constants are to be refined. The data are read in by subroutines PREPRO and INOUT. Once data input is complete the refinement process is commenced. The Gauss-Newton non-linear least-squares algorithm is used to minimize CJ, where I NW u=CC(A$-A;,b”.)2 I I
will now be described.
END: SPECIES:
(9)
(10)
is solved, within the subroutines ECOEF, SOLVE and/or NNLS, for the molar absorptivities of each species, j, wavelength by wavelength. ECOEF provides the book-keeping for SOLVE. Two possible algorithms may be selected within SOLVE. Either a conventional multiple regression (MR) technique is used to solve the system of linear equations, or NNLS, a non-negative linear least-squares algorithm, is chosen. NNLS is employed when molar absorptivities are found to be negative when the MR algorithm is used. Full details of MR and NNLS have already been published.* Once the cj,k values have been calculated, the A$ matrix is evaluated for the current set of constants. Control returns to REFINE and then proceeds to subroutine SEARCH where the shifts for the refining constants are calculated. Assuming that the refinement process is working smoothly, the new values of the constants will provide a better numerical fit to the data, i.e., lower U, than the previous set of constants. This is checked in the third step by repeating the sequence of calls, starting at subroutine RESID. If the changes in the refining constants are all less than 0.001, refinement is considered to be complete. Otherwise, a new refinement cycle is started. It is most important with computer programs of this size that the method of data-input should be as straightforward as possible. This is particularly true for first-time users and potential users with little or no programming experience. Considerable attention has been devoted to achieving this end and the method of data-input
Section 1
Section 2
Subroutine REFINE controls the minimization in two major steps. First the Jacobian and Hessian matrices are built up by numerical differentiation using Sterling’s central difference method. This involves, for the current set of stability constants, calculating the concentrations of all species contributing to each spectrum. In essence, the massbalance equations are solved, given CM,i, CL,iand /Ii, yielding [speciesljj. These calculations are performed by subroutines CCSCC and COGSNR. The first of these performs “book-keeping” functions for the second, which solves the mass-balance equations by using a controlled Newton-Raphson algorithm. Once the [species]i,j values have been found, the set of linear equations, derived from Beer’s law, &+ = [species]i,,6j,kbi
Table 1 A SUITABLE TITLE A SUITABLE SUBTITLE DICTIONARY:
END: OTHER: Section 3
DATA: Section 4
MOL. ABS.: Section 5
END:
BASELINE: Section 6
SPECTRA: Section 7
-AMAX The data deck consists of seven major sections, some of which may be. omitted. Each section starts with a key word and most finish with the key word END. The basic layout of the data deck is shown in Table 1. The DICTIONARY section provides SQUAD with the “aliases” (4-character identification words) that the user will employ to identify the components of the complexes. Consider, as an example, a study involving the interactions between zinc, imidazole and a porphyrin, designated TBDH. To begin communication between SQUAD and the user, “aliases” must be supplied for the programrecognizable variables MTLl (first metal in complex), MTL2 (second metal), LIGl (first ligand in complex), LIG2 (second ligand), PROT (the proton as ligand), HYDR (hydroxide ion as ligand). The second section of the data deck is used to describe the model and to indicate whether stability constants are to be refined (VB) or held constant (FB) and whether molar absorptivities are to be calculated (VE) or not (FE). SQUAD will use the dictionary to construct its internal vectors and arrays that describe the stoichiometry of each complex in the model being
D. J. LEGGETTet al.
582
Table 2. The set-up of the dictionary and species data for SQUAD DICTIONARY: MTLl = ZN; LIGI = IMID; LIG2 = TBDH:
END: SPECIES: ZN(l) IMID( 3.30; FB; FE: ZN(1) IMID(2); 5.87; FB; FE: ZN(1) TBDH(l); 4.31; VB; VE: ZN(l) TBDH(2); 7.44; VB; VE: ZN(l) IMID(1) TBDH(I); 6.91; VB; VE: END:
fitted to the data. For example, the interactions of zinc, imidazole and TBDH are believed to give rise to five complexes, Zn(Imid), Zn(Imid), , Zn (TBDH), Zn(TBDH), and Zn(Imid) (TBDH). The zincimidazole system has already been studied and the relevant stability constants and molar absorptivities are known. The first two sections of the data deck are shown in Table 2. Since the molar absorptivities of the zinc-imidazole system had been previously determined they are included in the data deck at Section 5. The layout of these values is shown in Table 3. The examples given above of three of the sections of the data deck illustrate the simplicity by which models can be specified to SQUAD and also the ease with which a model description may be changed. Although the method of data input is straightforward, key-punching errors can occur, such as mis-spelling TBDH or omitting a semicolon or parenthesis. Extensive error checks are performed throughout the data-input phase and the conversion of user-specified descriptions into internal data for SQUAD. Thus, an incorrectly constructed data deck will be detected by the program rather than by the compiler. Section 3 (OTHER), which is optional, is used to indicate whether the molar absorptivity of any of the components is to be calculated, or whether the molar
absorptivities of more than one species are the same. Section 4 (DATA) contains information such as the starting and finishing wavelengths and the wavelength increment; whether the p or the log p values
are to be refined; whether the multiple regression or the non-negative linear least-squares method is to be used to solve the system of linear equations arising from Beer’s law; and so on. Section 5 (BASELINE), which is optional, allows for baseline corrections to be performed. Section 6 (SPECTRA) will contain the recipe concentrations used for each spectrum, the pH (if applicable) of each solution, the path-length for each spectral measurement and the absorbance readings at each incremental wavelength. The final card in the data deck is labelled minus one and is used to indicate the physical end of data to be processed.
EXPERIMENTAL The various metalloporphyrins, discussed below, were purified by previously published procedures.’ Spectra of solutions containing various porphyrin:axial ligand ratios (C,:C,) were recorded with a Cary 14 spectrophotometer at 21.0 + 1.O” or a Tracer Northern Rapid Scan Spectrophotometer. In all instances non-bonding solvents were used, due precautions being taken to avoid evaporation throughout the solution preparation and data acquisition. The spectral traces were manually digitized at 1.25, 2.5 or 5.0nm intervals, 20-45 absorbance readings per spectrum being taken. Absorbance values were read from the top of the trace, to within kO.003 absorbance units. Although it may be instructive to record several spectra, one on top of another, this practice is to be discouraged when obtaining data that will be processed by SQUAD. Difficulties in aligning the starting position of the pen on the paper, and possible confusion of traces, especially where cross-over points exist, far outweigh the benefits gained from being able to view trends in peak maxima shifts as a function of hgand concentration. The absorbance values, so obtained, are the major part of the data for SQUAD. Additional data required have been specified and described earlier.
RESULTS
The reactions of imidazole and pyridine with TPPCd, TPPMg and TPPTPPMn(III)ClO,, Ru(C0); pyridine with TPPCu; and DMSO with TPPMn(III)ClO, and TPPRu(C0) will be used to illustrate the applicability of SQUAD and to unravel
Table 3. Section 5 for inputting molar absorptivities to SQUAD MOL.ABS.: ZN(l) IMID(1); ZN(l) IMID(2): END: 6~“lrnKl 6&d? 6&1d ~Z”llIlld2
c;llim1d c &ll,d* tEnnlrnid and t~~,,,,ld2are the molar absorptivities of Zn(Imid) and Zn(Imid), at the kth wavelength.
Fig. 2. Representative spectra obtained during the titration of 5.0 x lO~‘M TPPMn(III)CIO, with pyridine, in O.lM TBAP, C2H4CIZ. Arrows indicate spectral trends observed when changing C,,,:C, from I:0 to 1:4000.
Spectrophotometric
determination
of stability constants-II
Table 4. Stability constants for metalloporphyrin-axial Equilibrium system
log BlOlf clog B,o,
583
ligand interactions 1 start; 1 stop; number of wavelengths; number of spectra
coata
log BlOZ* clog 8102
TPPMn(III)CIO,
Imidazole Pyridine DMSO
0.006 0.008 0.003
360; 500; 29; I2 360; 500; 29; IO 360; 500; 29; I3
0.009 0.007
555; 640; 18; I6 555; 640; 18; I6
0.003 0.006
555; 640; 18; I6 555; 640; 18; I6
4.96 + 0.02, 4.63 f 0.01, 4.53 f 0.02,
0.004 0.003 0.003
510; 567; 5; 24; I4 510; 567; 5; 24; I4 510; 567; 5; 24; I5
- I .28 k 0.00,
0.002
525; 580; 45; 7
4.35 + 0.43, 4.08 k 0.47, 3.49 f 0.01,
7.45 * 0.04, 6.99 f 0.07, 5.74 + 0.02,
TPPCd
Imidazole Pyridine
4.23 k 0.08, 3.37 f 0.06,
TPPMg
Imidazole Pyridine
4.98 + 0.03, 3.63 k 0.04,
5.19 * 0.10, 2.90 & 0.03,
TPPRu (CO)
Imidazole Pyridine DMSO TPPCu
Pyridine
equilibria that in several instances could not be interpreted by using conventional techniques. TPPMn(III)ClO, interactions with imidazole, pyridine and DMSO give rise to relatively small spectral changes,‘O Fig. 2. Previous studies on (pCH,)TPPMnCl complexes with pyridine” noted the absence of clear isosbestic points, and attempts to determine the nature of the species in solution proved inconclusive. Our attempts to interpret the data in the manner suggested by Walker et af.* for iron porphyrins did not alleviate the situation. A qualitative examination of the spectral changes over the Cu : C, range from 1:0 to 1:4000 indicated the formation of more than two species in equilibrium. No wavelength (in the 36&500 nm region) could be found that would provide the appropriate slopes for the B-H plots. In contrast, analysis of the absorbance data by SQUAD provided clear evidence of existence of the ML and ML, complexes. The difficulties encountered when using the B-H method are traceable to the close similarities of the spectra for M and ML, particularly for L = pyridine (Fig. 2) and the fact that overlapping equilibria exist. Table 4 lists the results of the refinement process for these three ligands. The TPPMg equilibrium systems with pyridine and imidizole’ illustrate a situation where reliable equilibrium constants were unobtainable without the use of SQUAD. The magnitudes of the stability constants, Table 4, and the observed spectral changes, together pose considerable problems. The addition of the first axial ligand, forming ML, is characterized by a very small red shift (-3 nm) and only a slight increase in absorbance, Fig. 3A, but /I, is about 104-10’. In direct contrast, the formation of ML, is accompanied by large red shifts (N 25 nm), dramatic absorbance changes, Fig. 3B, but a very small K, ( =&02/fl,o,). The equilibria for both ligands may be considered as two distinct steps. B-H analysis of the data, considering formation of ML and ML, sepa-
rately, did not provide any useful answers. The minimal spectral changes associated with the formation of ML would not be expected to provide any suitable wavelength for such data assessment. However, it is somewhat surprising that little conclusive evidence could be gleaned by the B-H method from the data relevant to the formation of ML*, other than that ML2 is a weak complex. Previous studies of TPPMg with pyridine and other axial ligands have (A)
540
t
560
580
600
620
640
A (nm)
Fig. 3A. Representative spectra obtained during the titration of 5.0 x IO-‘M TPPMg with pyridine in O.lM TBAP, CH,C&. Arrows indicate spectral trends observed when changing C,:C, from I:0 to l:lOOO. Fig. 3B. As for Fig. 3A but illustrating spectral trends observed when changing C,:C, from I:4000 to I :223000.
D. J. LEGGETTet al.
584
I
I 490
I
I
I
I
510
530 A (nm)
550
570
Fig. 5. Representative spectra obtained during the titration of 5.0 x lo-‘M TPPRu(C0) with pyridine in O.lM TBAP, CH,Cl,. Arrows indicate spectral trends observed when changing CM:C, from 1:O to I : I.
I
550
600 X tnm)
I
650
fig. 4. Representative spectra obtained during the titration of 5 x 10-5M TPPCd with pyridine in O.lM TBAP, CHCI,. Arrows indicate spectral trends observed when changing C,:C, from I:0 to 1:lOO. reported only approximate values or a single value for M+ML+ML,.‘2.‘3 Interactions of TPPCd with pyridine and imidazole
were investigated by use of B-H plots and SQUAD. These two approaches provided values of log B,,,, that were in reasonable agreement for each equilibrium system. The results shown in Table 4 are taken from the SQUAD analysis of the data. These may be compared with, for example, log &,, = 3.28 and 4.3 1 for TPPCd with pyridine and imidazole, respectively, obtained by using absorbance data at 625 nm and the B-H approach. Other selected wavelengths gave a spread of values for log plO, that spanned the values obtained from SQUAD. These systems are inherently amenable to the B-H methodology since only ML is formed and there are significant differences in the spectra of complexed and uncomplexed TPPCd, Fig. 4. The addition of DMSO, pyridine and imidazole to TPPRu(CO)‘~ results in spectral shifts of less than 8 nm and absorbance changes of no more than 15x, Fig. 5. Although only ML is formed, B-H analysis of the data yielded slopes ranging from 0.8 to 1.2, leading to considerable uncertainty in the value of log B101as determined from B-H plots. However, analysis by SQUAD of the absorbance data from the interactions of the 20 ligands with TPPRu(C0) permitted meaningful deductions to be made concerning the nature of the bonding in TPPRu(C0) axial interactions with nitrogenous bases. The reaction of pyridine with TPPCu exemplifies
the ability of SQUAD to process data arising from extremely weak interactions. For this study the reference cell contained the same molarity of ligand as was used in the sample cell, for each spectrum. This precaution was taken to minimize refractive index changes due to the high concentration of pyridine (l-8M) required to achieve any significant degree of complexation. Fig. 6. No evidence was found, during data processing, for any complexes other than ML. The stability constant found was in excellent agreement with previously reported results.12 DISCUSSION
The results presented above demonstrate that SQUAD is capable of providing reliable estimates of stability constants that best describe the available
I
I
530
I
550 A (nm)
I
570
Fig. 6. Representative spectra obtained during the titration of 5 x 10-5M TPPCu with pyridine in 0.1M TBAP, CH,Cl,. Arrows indicate spectral trends observed when changing C, :C, from 1:0 to 1:200,000.
Spectrophotometric determination of stability constants-II data. However the question “how do we know that the answers are correct?’ must be dealt with. Although the numerical reliability of SQUAD is not in doubt, the values of stability constants obtained from 15 spectra and 20 absorbance values per spectrum, for example, can only be considered to be applicable to that set of data. Assuming that no gross errors have been made in solution preparation etc., it may be safe to assume that these refined values, from one set of data, are descriptive of the metal-ligand equilibrium system. It is wiser though, to repeat the experiment, making fresh stock solutions of metal and ligand and using different C, :C, for the spectra. Values of constants from different data sets that are in agreement, within experimental uncertainty, lend credence to the reliability of the numbers and the proposed model. Initial acceptance of the stability-constant values derived from any set of absorbance data, may be governed by a number of statistical parameters calculated by SQUAD. The “standard deviation in the absorbance data” QnATA,calculated from the set of stability constants’at the end of each refinement cycle is, in essence, an overall measure of the fit of the model to the data. Realistically, we may expect that any single absorbance value will be accurate to +0.003 (for a Cary 14 spectrophotometer). Consequently, we would expect that if the fit between data and model were perfect there would be a residual, due to experimental error and uncertainty, of between about 4 0.003 and &-0.005. Thus inspection ofa nATA,based on the refined stability constants, will indicate how closely we have come to a “perfect” fit. Although there is no hard and fast rule, values of > -0.01 indicate that a good fit has nor been uDATA obtained. SQUAD will also provide estimates of the standard deviation of each refined constant, cr,,,,. Generally, values of oCONST of the order of 1% of the particular constant are indicative of a good fit. Clearly the lower acoNsT is (below 1%) the better the fit. These remarks pose the question “what if gnATA and gcONST are greater than the recommended levels?’ Several answers are possible. If the equilibrium model being fitted to the data considers only ML but in reality both ML and ML, are formed, the fit may worsen as the concentration of hgand, relative to metal increases, i.e., as ML2 starts to predominate. This may result in high cIDATA and ~~~~~~~The trend will be visible most clearly in the values of espECT(the standard deviation of each spectrum), increasingly bad fits being observed for higher ligand concentrations. The reverse situation would be one where the model was ML and ML, but the data were representative of a system consisting of only ML. Again GnATAwould be unacceptably high, but uSpEC. would not necessarily indicate bad fits for low ligand concentrations, relative to the metal, compared to higher concentrations. SQUAD would attempt to “remove” ML, from numerical consideration by
585
successively reducing the value of log &Z until [ML,], for all spectra taken, was effectively zero. This would be shown by a rapidly increasing ~eorusr for log &M and te~ination of the refinement process if ~+~,.,~r became greater than 200. The second major cause for large values of vDATA and ~CONST arises from gross experimental error in making stock solutions or preparing solutions for a specific spectrum. The former error is sometimes difficult to detect until a new experiment is performed. The latter problem is readily seen from inspection of each rrSpECT value. An anomalously high value indicates a problem with that particular spectrum. If the data for that spectrum are removed from the data-set, aDATArucoNSr and aspEcT should be reduced when the data are reprocessed but the values of the refined constants should no? be substantially different. Sloppy technique will cause random errors to be introduced at each stage of the experiment, with the result that all cr values will be high and the refinement will take a large number of cycles, if it can be achieved at all. Care should be taken, however, to distinguish poor manip~ative technique from impurities in metal or ligand. Obviously, samples of metal and ligand should be purified as carefully as possible without resorting, in the first instance, to extreme measures. Nothing is to be gained by inordinate time spent on purifications followed by inadequate precautions taken in preparing solutions. Assessment of the quality of the data and the model may also be obtained, indirectly, from inspection of the calculated molar absorptivities. One particularly useful feature of SQUAD is the ability to calculate molar absorptivities of species even though the stability constant is not being refined. In the studies described earlier, irrespective of whether the model was ML alone or ML and ML,, the molar absorptivities for the complex(es) and the uncomplexed metalloporphyrin were calculated. The values obtained for free metalloporphyrin were then compared with those obtained from absorbance studies of solutions containing various con~ntrations of only the porphyrin. Close agreement between the results from the simple Beer’s law calculations and from SQUAD provided extra assurance for the validity of the proposed model as well as the purity of the metalloporphyrin. A second diagnostic indicator arises from the calculated molar absorptivities for each species, which when plotted provide spectra for individual species. Clearly, within the ultraviolet-visible region, the shape of each species spectrum should be a smooth continuous curve bearing some resemblance to portions of the observed spectra for various C, : C, ratios. Severely disjointed plots are strongly indicative of a less than adequate fit of the model to the data. It is also possible that negative values for the molar absorptivities are returned. Although this situation may often be traced to poor quality data or an
586
D. J. LEGGETTef al.
model, there is another cause. Consider two species in solution, having concentrations of 1.753 x 10m5and 3.14 x lo-‘M. At some wavelength the absorbance is observed to be 0.141. By use of the multiple regression algorithm, values of 1.11 x lo4 and - 1.73 x lo4 l.mole-‘.cm-’ are found for the molar absorptivities of these two species, at the wavelength concerned. However, solving the same linear equations by the non-negative linear leastsquares algorithm yields values of 8.5 x lo3 and 1.0 x lo2 l.molee’.cm-’ for the same species. It should be noted that within SQUAD, the linear equations arising from Beer’s law are solved wavelength by wavelength, from absorbance data and the appropriate species concentrations, for each spectrum. Therefore, we are dealing with an overdetermined set of simultaneous equations. Since a negative molar absorptivity has no chemical significance, the cause of these negative values must be sought. It is assumed that the equilibrium model is correct and the data are as reliable as the spectrophotometer can produce, i.e., f 0.003 absorbance units. That a multiple regression treatment of the data gives rise to a negative molar absorptivity is attributable to the very small contribution (0.2%) of the second species to the observed absorbance at that wavelength. In this particular instance, the molar absorptivity is in fact close to zero. Non-negative linear least-squares analysis precludes the calculation of negative values for molar absorptivities and provides numbers that are in agreement with the qualitative observations-namely that the second species does not absorb radiation at this wavelength. A more detailed discussion of this point has already been published,’ where the validity and use of NNLS have been amply demonstrated. Clearly, there will be times when the multiple regression algorithm will calculate negative molar absorptivities, but the correctness of the model will still be in doubt. In these situations switching to the non-negative least squares algorithm should be done only as a last resort and efforts should be made to establish independently whether or not one species has negligible interaction with radiation at the particular wavelength(s). incorrect
Conclusions
The usefulness of SQUAD for determining stability constants from metalloporphyrin-axial ligand
interactions has been clearly demonstrated. For most of the examples chosen the constants were not obtainable by conventional graphical or singlewavelength techniques. The program is completely general and no “simplifying” assumptions are required such as [free ligand] = [total ligand] or selection of “appropriate” wavelengths. In passing it should be noted that simplifying assumptions, used to obtain stability constant values, may well complicate theories that have been based on the values obtained from such methodologies. SQUAD also produces individual species spectra for each complex present in solution. Careful examination of the output from SQUAD can act as a valuable guide to remedying shortcomings in experimental technique. The program is easy to use and requires no knowledge of FORTRAN. The arrangement of the data deck has been designed to be “user-friendly” and many instructive error messages are available to assist the user in rectifying any mistakes that may have occurred. Listings of SQUAD, user’s manual, sample data and outputs are available (from DJL) and specific arrangements can be made for obtaining a tape copy of program and data. Acknowledgements-We
gratefully acknowledge the sup-
port of the National Science Foundation (Grant No. CHE7921536, to KMK) and the Robert A. Welch Foundation (Grant No. E-680, to KMK, and E-755, to DJL). REFERENCES 1. R. W. Ramette, J. Chem. Educ., 1967, 44, 647.
2. F. Walker, M.-W. Lo and M. T. Ree, J. Am. Chem. Sot., 1976, 98, 5552. 3. W. A. E. McBryde, Talanta, 1974, 21, 979. 4. D. J. Leggett, Am. Lab., 1982, 14, (No. l), 29. 5. F. Gaizer, Coord. Chem. Rev., 1979, 27, 195. 6. L. G. Sillin and G. Wamquist, Arkiv Kemi, 1968, 31, 315.
7. D. J. Leggett and W. A. E. McBryde, Anal. Chem., 1975, 47, 1065. 8. D. J. Leggett, ibid., 1978, SO, 718. 9. K. M. Kadish and L. R. Shiue, Inorg. Chem., 1982, 21, 1112.
10. S. L. Kelly and K. M. Kadish, ibid., 1982, 21, 3631. II. G. N. La Mar and F. A. Walker, J. Am. Chem. Sot., 1975, 97, 5103.
12. J. R. Miller and G. D. Dorough, ibid., 1952, 74, 3977. 13. C. B. Storm, A. H. Corwin, R. R. Arellano, M. Martz and R. Weintraub, ibid., 1966, 88, 2525. 14. K. M. Kadish, D. J. Leggett and D. Chang, Inorg. Chem., 1982, 21, 3618.
Copyright
0
0039-9140/83 %3.00+0.00 1983 Pergamon Press Ltd
A COMPUTATIONAL APPROACH TO THE SPECTROPHOTOMETRIC DETERMINATION OF STABILITY CONSTANTS-II APPLICATION TO METALLOPORPHY RIN-AXIAL LIGAND INTERACTIONS IN NON-AQUEOUS SOLVENTS D. J. LEGGETT,* S. L. KELLY, L. R. SHIUE, Y. T. Wu, D. CHANG and K. M. RADISH Department
of Chemistry, University of Houston, Houston, TX 77004, U.S.A. (Received 29 September
1982. Accepted
14 February
1983)
Summar-The ability of the computer program SQUAD to deduce a plausible equilibrium model, associated stability constants and spectra of individual species is described. The original version of SQUAD has been extensively modified and these changes are detailed. In particular a “user-friendly” method of data input has been implemented that simplifies familiarization with the program. Brevity
of program code has been sacrificed in favour of the new data input and error-checking features of SQUAD, with beneficial results. The application of SQUAD to five non-aqueous metalloporphyrin-axial ligand interactions exemplifies the program’s ability to handle widely different types of equilibrium systems.
The Benesi-Hildebrand method’ has been extensively used to evaluate the stability constants of metalloporphyrin-axial ligand complexes. For the general reaction M + L=ML
(1)
where M is any metal ion or metalloporphyrin and L is any ligand, the following equations may be written: C, = [M] + [ML]
(2)
C,. = [L] + [ML]
(3)
Blot =
[MLIWIV-I
(4)
The subscripts for j refer to the number of metal ions, hydroxide or hydrogen ions, and ligand ions, respectively, associated with that stability constant. The studies are most often performed in a nonaqueous, non-donor solvent, in which case the middle subscript is then always zero. Making use of Beer’s law: A = +,,[M] + &JL] + cML[ML]
(5)
it can be shown’ that log
(A - 4,) = log CL + log j&o, ~ (A, - A)
where A is the absorbance,
(6)
at a preselected wave-
*Author for correspondence. Present address: Dow Chemical USA, Texas Division, Freeport, TX 77541, U.S.A.
length, of a solution having a known molarity of ligand, C, and known molarity of metal, C, , A, is the absorbance, measured at the same wavelength, of a solution where Cr = 0, and A, is the measured absorbance of a solution for which CL>>&, so that the absorbance is constant with increasing CL. Thus a plot of the left-hand side of equation (6) against log CL should give a straight line which intersects the abscissa at -log CL, which will be equal to log fl,,,, . The Benesi-Hildebrand (B-H) method may be generalized to handle equilibrium systems in which the stoichiometry of the complex formed is unknown. In this situation equation (4) becomes BW =
[MW[Ml[Ll
(7)
and equation (6) becomes log
(A - 4) ~ = n log CL + log j&). W-A)
(8)
Thus the slope of the straight line in the log-log plot provides the stoichiometric coefficient for the complex. Inherent in this method is the assumption that only one complex is formed. This situation is not too frequently encountered and when there is more than one complex present the B-H plots of log [(A - A,)/(A, -A)] vs. log CL will show considerable curvature. It has been argued’ that the plots may still yield information concerning the stoichiometry and stability constants of the complexes, but except under 579
580
D. J. LEGGETT et al.
the most favourable conditions, encountered when complexes of widely differing stability are formed, this claim is not valid. This method, and many similar approaches,3 have the common drawback that they are only applicable if one and only one complex exists under the conditions of measurement. However, in general, more than one complex will exist in solution unless C, >>Cr,, or C,>> Cr. Moreover, when several complexes are present, their nature may not be readily determinable from simple continuous-variation or molar-ratio experiments. Therefore, more sophisticated and rigorous data-processing techniques need to be adopted. Several general computer programs are currently available4” that permit various equilibrium models to be fitted to the experimental data, by change of one or more data car&. This type of program is clearly more desirable than one where the program code needs to be rewritten for each new model tested. The majority of programs developed to date require potentiometric data as input, which may either be obtained from measurements of pH us. volume of base, or of ion-sensing electrode potentials. Spectrophotometry is a widely used technique for the study and determination of equilibrium constants but only a few programs are available to process the data.4.5 Spectrophotometric measurements are generally less precise than potentiometric ones, both inherently and because of the need to prepare several separate solutions. A further complication arises from the fact that for fitting an equilibrium model to absorbance measurements, not only are the stability constants to be determined, but the molar absorptivities for each species, at each wavelength, are also to be evaluated. The situation is not as bad as it might
seem, since the evaluation of molar absorptivities and stability constants is done at two levels within most programs, patterned after an approach suggested by Sillen The program used in the studies described here is SQUAD.‘,’ It is designed to calculate the best values for the stability constants of the proposed equilibrium model by employing a non-linear leastsquares approach. The program is completely general in scope, having the capability to refine stability constants for the general complex M,M; H,L, L;, where m, 1, n, q 2 0 and j is positive (for protons), negative (for hydroxide ions) or zero. Therefore, the same program may be used to study acid-base equilibria for ligands that are weak acids (or bases); metal-ion hydrolysis; complexes of the type ML,, or M,L,; mixed-ligand (or mixed-metal) complexes; protonated or hydroxo complexes. SQUAD was originally designed to process absorbance data from aqueous solutions. Recently extensive modifications have been made to the program that now permit the analysis of data from any type of solution. The algorithms employed in SQUAD and their relationships to each other are shown in Fig. 1. The overall modus operandi will be described briefly and then greater attention will be paid to the novel data-input method. The input data consist of: (a) the absorbance values A+, for each spectrum, giving a total of I for each wavelength, and a grand total of NW; (6) the total metal and ligand concentrations, C,,,, &, pH, (pH for aqueous and mixed solvents only) and pathlength b,, for each spectrum; (c) any known or previously determined molar absorptivities (c,,~) for the jth species at the k th wavelength; (d) the stoichi-
‘-_____-__-___--_----~ PREPRO
INPUT
I
I
User aliases converted to internal parameters
ERR -
REFINE
of GN 1 & Solve equations for shifts on constants
Numerical differentiation
I Calculate
sum
I
SIGMAE
ECOEF
Standard deviation of E
SOlW A=E.C
I
Fig. 1. Block diagram
of SQUAD,
showing
the inter-relationships
among
the various
algorithms.
581
Spectrophotometric determination of stability constants-II ometry and stability constant for the jth species together with an indication of which constants are to be refined. The data are read in by subroutines PREPRO and INOUT. Once data input is complete the refinement process is commenced. The Gauss-Newton non-linear least-squares algorithm is used to minimize CJ, where I NW u=CC(A$-A;,b”.)2 I I
will now be described.
END: SPECIES:
(9)
(10)
is solved, within the subroutines ECOEF, SOLVE and/or NNLS, for the molar absorptivities of each species, j, wavelength by wavelength. ECOEF provides the book-keeping for SOLVE. Two possible algorithms may be selected within SOLVE. Either a conventional multiple regression (MR) technique is used to solve the system of linear equations, or NNLS, a non-negative linear least-squares algorithm, is chosen. NNLS is employed when molar absorptivities are found to be negative when the MR algorithm is used. Full details of MR and NNLS have already been published.* Once the cj,k values have been calculated, the A$ matrix is evaluated for the current set of constants. Control returns to REFINE and then proceeds to subroutine SEARCH where the shifts for the refining constants are calculated. Assuming that the refinement process is working smoothly, the new values of the constants will provide a better numerical fit to the data, i.e., lower U, than the previous set of constants. This is checked in the third step by repeating the sequence of calls, starting at subroutine RESID. If the changes in the refining constants are all less than 0.001, refinement is considered to be complete. Otherwise, a new refinement cycle is started. It is most important with computer programs of this size that the method of data-input should be as straightforward as possible. This is particularly true for first-time users and potential users with little or no programming experience. Considerable attention has been devoted to achieving this end and the method of data-input
Section 1
Section 2
Subroutine REFINE controls the minimization in two major steps. First the Jacobian and Hessian matrices are built up by numerical differentiation using Sterling’s central difference method. This involves, for the current set of stability constants, calculating the concentrations of all species contributing to each spectrum. In essence, the massbalance equations are solved, given CM,i, CL,iand /Ii, yielding [speciesljj. These calculations are performed by subroutines CCSCC and COGSNR. The first of these performs “book-keeping” functions for the second, which solves the mass-balance equations by using a controlled Newton-Raphson algorithm. Once the [species]i,j values have been found, the set of linear equations, derived from Beer’s law, &+ = [species]i,,6j,kbi
Table 1 A SUITABLE TITLE A SUITABLE SUBTITLE DICTIONARY:
END: OTHER: Section 3
DATA: Section 4
MOL. ABS.: Section 5
END:
BASELINE: Section 6
SPECTRA: Section 7
-AMAX The data deck consists of seven major sections, some of which may be. omitted. Each section starts with a key word and most finish with the key word END. The basic layout of the data deck is shown in Table 1. The DICTIONARY section provides SQUAD with the “aliases” (4-character identification words) that the user will employ to identify the components of the complexes. Consider, as an example, a study involving the interactions between zinc, imidazole and a porphyrin, designated TBDH. To begin communication between SQUAD and the user, “aliases” must be supplied for the programrecognizable variables MTLl (first metal in complex), MTL2 (second metal), LIGl (first ligand in complex), LIG2 (second ligand), PROT (the proton as ligand), HYDR (hydroxide ion as ligand). The second section of the data deck is used to describe the model and to indicate whether stability constants are to be refined (VB) or held constant (FB) and whether molar absorptivities are to be calculated (VE) or not (FE). SQUAD will use the dictionary to construct its internal vectors and arrays that describe the stoichiometry of each complex in the model being
D. J. LEGGETTet al.
582
Table 2. The set-up of the dictionary and species data for SQUAD DICTIONARY: MTLl = ZN; LIGI = IMID; LIG2 = TBDH:
END: SPECIES: ZN(l) IMID( 3.30; FB; FE: ZN(1) IMID(2); 5.87; FB; FE: ZN(1) TBDH(l); 4.31; VB; VE: ZN(l) TBDH(2); 7.44; VB; VE: ZN(l) IMID(1) TBDH(I); 6.91; VB; VE: END:
fitted to the data. For example, the interactions of zinc, imidazole and TBDH are believed to give rise to five complexes, Zn(Imid), Zn(Imid), , Zn (TBDH), Zn(TBDH), and Zn(Imid) (TBDH). The zincimidazole system has already been studied and the relevant stability constants and molar absorptivities are known. The first two sections of the data deck are shown in Table 2. Since the molar absorptivities of the zinc-imidazole system had been previously determined they are included in the data deck at Section 5. The layout of these values is shown in Table 3. The examples given above of three of the sections of the data deck illustrate the simplicity by which models can be specified to SQUAD and also the ease with which a model description may be changed. Although the method of data input is straightforward, key-punching errors can occur, such as mis-spelling TBDH or omitting a semicolon or parenthesis. Extensive error checks are performed throughout the data-input phase and the conversion of user-specified descriptions into internal data for SQUAD. Thus, an incorrectly constructed data deck will be detected by the program rather than by the compiler. Section 3 (OTHER), which is optional, is used to indicate whether the molar absorptivity of any of the components is to be calculated, or whether the molar
absorptivities of more than one species are the same. Section 4 (DATA) contains information such as the starting and finishing wavelengths and the wavelength increment; whether the p or the log p values
are to be refined; whether the multiple regression or the non-negative linear least-squares method is to be used to solve the system of linear equations arising from Beer’s law; and so on. Section 5 (BASELINE), which is optional, allows for baseline corrections to be performed. Section 6 (SPECTRA) will contain the recipe concentrations used for each spectrum, the pH (if applicable) of each solution, the path-length for each spectral measurement and the absorbance readings at each incremental wavelength. The final card in the data deck is labelled minus one and is used to indicate the physical end of data to be processed.
EXPERIMENTAL The various metalloporphyrins, discussed below, were purified by previously published procedures.’ Spectra of solutions containing various porphyrin:axial ligand ratios (C,:C,) were recorded with a Cary 14 spectrophotometer at 21.0 + 1.O” or a Tracer Northern Rapid Scan Spectrophotometer. In all instances non-bonding solvents were used, due precautions being taken to avoid evaporation throughout the solution preparation and data acquisition. The spectral traces were manually digitized at 1.25, 2.5 or 5.0nm intervals, 20-45 absorbance readings per spectrum being taken. Absorbance values were read from the top of the trace, to within kO.003 absorbance units. Although it may be instructive to record several spectra, one on top of another, this practice is to be discouraged when obtaining data that will be processed by SQUAD. Difficulties in aligning the starting position of the pen on the paper, and possible confusion of traces, especially where cross-over points exist, far outweigh the benefits gained from being able to view trends in peak maxima shifts as a function of hgand concentration. The absorbance values, so obtained, are the major part of the data for SQUAD. Additional data required have been specified and described earlier.
RESULTS
The reactions of imidazole and pyridine with TPPCd, TPPMg and TPPTPPMn(III)ClO,, Ru(C0); pyridine with TPPCu; and DMSO with TPPMn(III)ClO, and TPPRu(C0) will be used to illustrate the applicability of SQUAD and to unravel
Table 3. Section 5 for inputting molar absorptivities to SQUAD MOL.ABS.: ZN(l) IMID(1); ZN(l) IMID(2): END: 6~“lrnKl 6&d? 6&1d ~Z”llIlld2
c;llim1d c &ll,d* tEnnlrnid and t~~,,,,ld2are the molar absorptivities of Zn(Imid) and Zn(Imid), at the kth wavelength.
Fig. 2. Representative spectra obtained during the titration of 5.0 x lO~‘M TPPMn(III)CIO, with pyridine, in O.lM TBAP, C2H4CIZ. Arrows indicate spectral trends observed when changing C,,,:C, from I:0 to 1:4000.
Spectrophotometric
determination
of stability constants-II
Table 4. Stability constants for metalloporphyrin-axial Equilibrium system
log BlOlf clog B,o,
583
ligand interactions 1 start; 1 stop; number of wavelengths; number of spectra
coata
log BlOZ* clog 8102
TPPMn(III)CIO,
Imidazole Pyridine DMSO
0.006 0.008 0.003
360; 500; 29; I2 360; 500; 29; IO 360; 500; 29; I3
0.009 0.007
555; 640; 18; I6 555; 640; 18; I6
0.003 0.006
555; 640; 18; I6 555; 640; 18; I6
4.96 + 0.02, 4.63 f 0.01, 4.53 f 0.02,
0.004 0.003 0.003
510; 567; 5; 24; I4 510; 567; 5; 24; I4 510; 567; 5; 24; I5
- I .28 k 0.00,
0.002
525; 580; 45; 7
4.35 + 0.43, 4.08 k 0.47, 3.49 f 0.01,
7.45 * 0.04, 6.99 f 0.07, 5.74 + 0.02,
TPPCd
Imidazole Pyridine
4.23 k 0.08, 3.37 f 0.06,
TPPMg
Imidazole Pyridine
4.98 + 0.03, 3.63 k 0.04,
5.19 * 0.10, 2.90 & 0.03,
TPPRu (CO)
Imidazole Pyridine DMSO TPPCu
Pyridine
equilibria that in several instances could not be interpreted by using conventional techniques. TPPMn(III)ClO, interactions with imidazole, pyridine and DMSO give rise to relatively small spectral changes,‘O Fig. 2. Previous studies on (pCH,)TPPMnCl complexes with pyridine” noted the absence of clear isosbestic points, and attempts to determine the nature of the species in solution proved inconclusive. Our attempts to interpret the data in the manner suggested by Walker et af.* for iron porphyrins did not alleviate the situation. A qualitative examination of the spectral changes over the Cu : C, range from 1:0 to 1:4000 indicated the formation of more than two species in equilibrium. No wavelength (in the 36&500 nm region) could be found that would provide the appropriate slopes for the B-H plots. In contrast, analysis of the absorbance data by SQUAD provided clear evidence of existence of the ML and ML, complexes. The difficulties encountered when using the B-H method are traceable to the close similarities of the spectra for M and ML, particularly for L = pyridine (Fig. 2) and the fact that overlapping equilibria exist. Table 4 lists the results of the refinement process for these three ligands. The TPPMg equilibrium systems with pyridine and imidizole’ illustrate a situation where reliable equilibrium constants were unobtainable without the use of SQUAD. The magnitudes of the stability constants, Table 4, and the observed spectral changes, together pose considerable problems. The addition of the first axial ligand, forming ML, is characterized by a very small red shift (-3 nm) and only a slight increase in absorbance, Fig. 3A, but /I, is about 104-10’. In direct contrast, the formation of ML, is accompanied by large red shifts (N 25 nm), dramatic absorbance changes, Fig. 3B, but a very small K, ( =&02/fl,o,). The equilibria for both ligands may be considered as two distinct steps. B-H analysis of the data, considering formation of ML and ML, sepa-
rately, did not provide any useful answers. The minimal spectral changes associated with the formation of ML would not be expected to provide any suitable wavelength for such data assessment. However, it is somewhat surprising that little conclusive evidence could be gleaned by the B-H method from the data relevant to the formation of ML*, other than that ML2 is a weak complex. Previous studies of TPPMg with pyridine and other axial ligands have (A)
540
t
560
580
600
620
640
A (nm)
Fig. 3A. Representative spectra obtained during the titration of 5.0 x IO-‘M TPPMg with pyridine in O.lM TBAP, CH,C&. Arrows indicate spectral trends observed when changing C,:C, from I:0 to l:lOOO. Fig. 3B. As for Fig. 3A but illustrating spectral trends observed when changing C,:C, from I:4000 to I :223000.
D. J. LEGGETTet al.
584
I
I 490
I
I
I
I
510
530 A (nm)
550
570
Fig. 5. Representative spectra obtained during the titration of 5.0 x lo-‘M TPPRu(C0) with pyridine in O.lM TBAP, CH,Cl,. Arrows indicate spectral trends observed when changing CM:C, from 1:O to I : I.
I
550
600 X tnm)
I
650
fig. 4. Representative spectra obtained during the titration of 5 x 10-5M TPPCd with pyridine in O.lM TBAP, CHCI,. Arrows indicate spectral trends observed when changing C,:C, from I:0 to 1:lOO. reported only approximate values or a single value for M+ML+ML,.‘2.‘3 Interactions of TPPCd with pyridine and imidazole
were investigated by use of B-H plots and SQUAD. These two approaches provided values of log B,,,, that were in reasonable agreement for each equilibrium system. The results shown in Table 4 are taken from the SQUAD analysis of the data. These may be compared with, for example, log &,, = 3.28 and 4.3 1 for TPPCd with pyridine and imidazole, respectively, obtained by using absorbance data at 625 nm and the B-H approach. Other selected wavelengths gave a spread of values for log plO, that spanned the values obtained from SQUAD. These systems are inherently amenable to the B-H methodology since only ML is formed and there are significant differences in the spectra of complexed and uncomplexed TPPCd, Fig. 4. The addition of DMSO, pyridine and imidazole to TPPRu(CO)‘~ results in spectral shifts of less than 8 nm and absorbance changes of no more than 15x, Fig. 5. Although only ML is formed, B-H analysis of the data yielded slopes ranging from 0.8 to 1.2, leading to considerable uncertainty in the value of log B101as determined from B-H plots. However, analysis by SQUAD of the absorbance data from the interactions of the 20 ligands with TPPRu(C0) permitted meaningful deductions to be made concerning the nature of the bonding in TPPRu(C0) axial interactions with nitrogenous bases. The reaction of pyridine with TPPCu exemplifies
the ability of SQUAD to process data arising from extremely weak interactions. For this study the reference cell contained the same molarity of ligand as was used in the sample cell, for each spectrum. This precaution was taken to minimize refractive index changes due to the high concentration of pyridine (l-8M) required to achieve any significant degree of complexation. Fig. 6. No evidence was found, during data processing, for any complexes other than ML. The stability constant found was in excellent agreement with previously reported results.12 DISCUSSION
The results presented above demonstrate that SQUAD is capable of providing reliable estimates of stability constants that best describe the available
I
I
530
I
550 A (nm)
I
570
Fig. 6. Representative spectra obtained during the titration of 5 x 10-5M TPPCu with pyridine in 0.1M TBAP, CH,Cl,. Arrows indicate spectral trends observed when changing C, :C, from 1:0 to 1:200,000.
Spectrophotometric determination of stability constants-II data. However the question “how do we know that the answers are correct?’ must be dealt with. Although the numerical reliability of SQUAD is not in doubt, the values of stability constants obtained from 15 spectra and 20 absorbance values per spectrum, for example, can only be considered to be applicable to that set of data. Assuming that no gross errors have been made in solution preparation etc., it may be safe to assume that these refined values, from one set of data, are descriptive of the metal-ligand equilibrium system. It is wiser though, to repeat the experiment, making fresh stock solutions of metal and ligand and using different C, :C, for the spectra. Values of constants from different data sets that are in agreement, within experimental uncertainty, lend credence to the reliability of the numbers and the proposed model. Initial acceptance of the stability-constant values derived from any set of absorbance data, may be governed by a number of statistical parameters calculated by SQUAD. The “standard deviation in the absorbance data” QnATA,calculated from the set of stability constants’at the end of each refinement cycle is, in essence, an overall measure of the fit of the model to the data. Realistically, we may expect that any single absorbance value will be accurate to +0.003 (for a Cary 14 spectrophotometer). Consequently, we would expect that if the fit between data and model were perfect there would be a residual, due to experimental error and uncertainty, of between about 4 0.003 and &-0.005. Thus inspection ofa nATA,based on the refined stability constants, will indicate how closely we have come to a “perfect” fit. Although there is no hard and fast rule, values of > -0.01 indicate that a good fit has nor been uDATA obtained. SQUAD will also provide estimates of the standard deviation of each refined constant, cr,,,,. Generally, values of oCONST of the order of 1% of the particular constant are indicative of a good fit. Clearly the lower acoNsT is (below 1%) the better the fit. These remarks pose the question “what if gnATA and gcONST are greater than the recommended levels?’ Several answers are possible. If the equilibrium model being fitted to the data considers only ML but in reality both ML and ML, are formed, the fit may worsen as the concentration of hgand, relative to metal increases, i.e., as ML2 starts to predominate. This may result in high cIDATA and ~~~~~~~The trend will be visible most clearly in the values of espECT(the standard deviation of each spectrum), increasingly bad fits being observed for higher ligand concentrations. The reverse situation would be one where the model was ML and ML, but the data were representative of a system consisting of only ML. Again GnATAwould be unacceptably high, but uSpEC. would not necessarily indicate bad fits for low ligand concentrations, relative to the metal, compared to higher concentrations. SQUAD would attempt to “remove” ML, from numerical consideration by
585
successively reducing the value of log &Z until [ML,], for all spectra taken, was effectively zero. This would be shown by a rapidly increasing ~eorusr for log &M and te~ination of the refinement process if ~+~,.,~r became greater than 200. The second major cause for large values of vDATA and ~CONST arises from gross experimental error in making stock solutions or preparing solutions for a specific spectrum. The former error is sometimes difficult to detect until a new experiment is performed. The latter problem is readily seen from inspection of each rrSpECT value. An anomalously high value indicates a problem with that particular spectrum. If the data for that spectrum are removed from the data-set, aDATArucoNSr and aspEcT should be reduced when the data are reprocessed but the values of the refined constants should no? be substantially different. Sloppy technique will cause random errors to be introduced at each stage of the experiment, with the result that all cr values will be high and the refinement will take a large number of cycles, if it can be achieved at all. Care should be taken, however, to distinguish poor manip~ative technique from impurities in metal or ligand. Obviously, samples of metal and ligand should be purified as carefully as possible without resorting, in the first instance, to extreme measures. Nothing is to be gained by inordinate time spent on purifications followed by inadequate precautions taken in preparing solutions. Assessment of the quality of the data and the model may also be obtained, indirectly, from inspection of the calculated molar absorptivities. One particularly useful feature of SQUAD is the ability to calculate molar absorptivities of species even though the stability constant is not being refined. In the studies described earlier, irrespective of whether the model was ML alone or ML and ML,, the molar absorptivities for the complex(es) and the uncomplexed metalloporphyrin were calculated. The values obtained for free metalloporphyrin were then compared with those obtained from absorbance studies of solutions containing various con~ntrations of only the porphyrin. Close agreement between the results from the simple Beer’s law calculations and from SQUAD provided extra assurance for the validity of the proposed model as well as the purity of the metalloporphyrin. A second diagnostic indicator arises from the calculated molar absorptivities for each species, which when plotted provide spectra for individual species. Clearly, within the ultraviolet-visible region, the shape of each species spectrum should be a smooth continuous curve bearing some resemblance to portions of the observed spectra for various C, : C, ratios. Severely disjointed plots are strongly indicative of a less than adequate fit of the model to the data. It is also possible that negative values for the molar absorptivities are returned. Although this situation may often be traced to poor quality data or an
586
D. J. LEGGETTef al.
model, there is another cause. Consider two species in solution, having concentrations of 1.753 x 10m5and 3.14 x lo-‘M. At some wavelength the absorbance is observed to be 0.141. By use of the multiple regression algorithm, values of 1.11 x lo4 and - 1.73 x lo4 l.mole-‘.cm-’ are found for the molar absorptivities of these two species, at the wavelength concerned. However, solving the same linear equations by the non-negative linear leastsquares algorithm yields values of 8.5 x lo3 and 1.0 x lo2 l.molee’.cm-’ for the same species. It should be noted that within SQUAD, the linear equations arising from Beer’s law are solved wavelength by wavelength, from absorbance data and the appropriate species concentrations, for each spectrum. Therefore, we are dealing with an overdetermined set of simultaneous equations. Since a negative molar absorptivity has no chemical significance, the cause of these negative values must be sought. It is assumed that the equilibrium model is correct and the data are as reliable as the spectrophotometer can produce, i.e., f 0.003 absorbance units. That a multiple regression treatment of the data gives rise to a negative molar absorptivity is attributable to the very small contribution (0.2%) of the second species to the observed absorbance at that wavelength. In this particular instance, the molar absorptivity is in fact close to zero. Non-negative linear least-squares analysis precludes the calculation of negative values for molar absorptivities and provides numbers that are in agreement with the qualitative observations-namely that the second species does not absorb radiation at this wavelength. A more detailed discussion of this point has already been published,’ where the validity and use of NNLS have been amply demonstrated. Clearly, there will be times when the multiple regression algorithm will calculate negative molar absorptivities, but the correctness of the model will still be in doubt. In these situations switching to the non-negative least squares algorithm should be done only as a last resort and efforts should be made to establish independently whether or not one species has negligible interaction with radiation at the particular wavelength(s). incorrect
Conclusions
The usefulness of SQUAD for determining stability constants from metalloporphyrin-axial ligand
interactions has been clearly demonstrated. For most of the examples chosen the constants were not obtainable by conventional graphical or singlewavelength techniques. The program is completely general and no “simplifying” assumptions are required such as [free ligand] = [total ligand] or selection of “appropriate” wavelengths. In passing it should be noted that simplifying assumptions, used to obtain stability constant values, may well complicate theories that have been based on the values obtained from such methodologies. SQUAD also produces individual species spectra for each complex present in solution. Careful examination of the output from SQUAD can act as a valuable guide to remedying shortcomings in experimental technique. The program is easy to use and requires no knowledge of FORTRAN. The arrangement of the data deck has been designed to be “user-friendly” and many instructive error messages are available to assist the user in rectifying any mistakes that may have occurred. Listings of SQUAD, user’s manual, sample data and outputs are available (from DJL) and specific arrangements can be made for obtaining a tape copy of program and data. Acknowledgements-We
gratefully acknowledge the sup-
port of the National Science Foundation (Grant No. CHE7921536, to KMK) and the Robert A. Welch Foundation (Grant No. E-680, to KMK, and E-755, to DJL). REFERENCES 1. R. W. Ramette, J. Chem. Educ., 1967, 44, 647.
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