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An analytical data inversion method for magnetic circular dichroism spectra dominated by the ‘‘B-term’’† Jeffrey R. Reimers*a and Elmars Krauszb A simple procedure is developed enabling the analytical inversion of an (unpolarized) absorption spectrum combined with a Magnetic Circular Dichroism (MCD) spectrum to resolve two overlapping bands of orthogonal polarization. This method is appropriate when (i) the overlapping transitions are well isolated from other bands, and (ii) when their electronic spacing is large enough so that the ‘‘A-term’’ and ‘‘C-term’’ contributions to the MCD spectrum can be ignored and hence only the ‘‘B-term’’ contribution need be considered. We apply this procedure to assign the Q-band system of chlorophylls, though similar challenges also commonly arise throughout both conventional and X-ray MCD (XMCD) spectroscopy. Analytical data inversion has not previously been possible as the inversion process is two-fold underdetermined. We show that the assumptions of isolated spectra and ‘‘B-term’’ dominance yields one generally valid constraint, leaving only one quantity unspecified by the experimental data. For some systems, an approximation leading to equal but opposite sign B-term magnitudes of the two components
Received 3rd September 2013, Accepted 4th December 2013 DOI: 10.1039/c3cp53730g
may be reasonable, but for chlorophyllides we find this constraint to be inappropriate. Instead, we constrain a bounded variable controlling the relative absorption strengths. Derived spectral bandshapes of the individual components are shown to be insensitive to its particular value, allowing weak spectral components of one polarization overlapped by intense components of the other to be immediately exposed. This is demonstrated for the chlorophylls, molecules for which the failure to detect such weak
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features historically led to incorrect proposals for the Q-band assignments.
1. Introduction Magnetic Circular Dichroism (MCD) offers great potential for the assignment of spectra of overlapping electronic transitions with different polarizations. In general, three types of MCD signals are commonly identified, known as the MCD ‘‘A’’, ‘‘B’’, and ‘‘C’’ terms,1–3 with the ‘‘B’’ term being dominant if the states involved in the transitions are not degenerate. An important application of MCD has been to the Q-band spectra of various porphyrins and chlorophylls critical to natural and artificial photosynthesis,4 as well as to many other spectroscopic systems of technological relevance such as those in acenes, graphene fragments, substituted fullerenes, and inorganic light harvesting, separating, or emitting compounds;5–12 in addition, the emergent technique of X-ray MCD (XMCD) is often applied
a
School of Chemistry, The University of Sydney, Sydney, NSW 2006, Australia. E-mail: Jeff[email protected]; Fax: +61 293513329; Tel: +61 293514417 b Research School of Chemistry, The Australian National University, Canberra, ACT 0200, Australia † Electronic supplementary information (ESI) available: Excel spreadsheet containing all analytical data inversions. See DOI: 10.1039/c3cp53730g
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to characterize nanotubes.13,14 In a symmetric molecule (e.g., zinc porphyrin) having equivalent x and y axes, doubly degenerate electronic transitions will appear that have an ‘‘A’’-type MCD spectrum.15,16 However, chemical substitution usually breaks this degeneracy so that two independent bands emerge, and in porphyrins and chlorophylls these component bands are named Qx and Qy. Typically the induced energy separation DE is larger than the widths of the inhomogeneously broadened electronic origins, changing the MCD spectra to ‘‘B’’ type spectra that may be represented simply as weighted sums of individual spectral components for each state. Overlapping spectroscopic transitions of nearly orthogonal polarization may arise for reasons other than pseudo-symmetry, but result in a similar scenario. The cause of the spectral overlap is not of consequence herein, only the energy gap. Most molecular spectral transitions, including the Q-bands of chlorophylls, have extensive Franck–Condon vibrational sideband structure. We are particularly concerned with situations for which the extent of this sideband structure exceeds the energy gap DE between the two electronic states, leading to complex and entangled spectra. The most important chlorophyllide, chlorophyll-a (Chl-a), indeed displays spectra of this type.
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As a result, its Qx, Qy assignment has been debated for over 50 years,4,17–31 with various assignment options being considered. Each alternative assignment has considerable ramifications, effecting, e.g., the exciton transport and coherence properties of photosystems.4 MCD has been developed as an experimental technique based primarily on its capacity to resolve overlapping spectral transitions.1–3,32,33 It is particularly well suited to this application compared to other methods, such as linear dichroism, as spectra can be measured in liquid, solid, and biological media as a function of temperature and pressure. In particular, many MCD measurements have been made on chlorophyllides, and our recent analysis unifies the deduced polarization information with the (relatively scarce) results obtained alternatively using linear dichroism or polarized fluorescence anisotropy techniques.4 However, there remains a significant technical issue in the unambiguous interpretation of MCD and absorption data. This is because direct data inversion of an (unpolarized) absorption spectrum and its associated MCD spectrum to yield the two individual-component spectra is mathematically underdetermined. A single ‘‘B’’-term contribution to an MCD spectrum of two overlapping states of orthogonal polarization (say x and y) can be expressed as the weighted sum of individual components as1,2 DDA0 ðnÞ ¼
DAðn; HÞ DAðn; 0Þ ¼ Bx sx ðnÞ þ By sy ðnÞ Hn
(1)
where A is the molar extinction, DA(n,H) is the difference in molar absorption between left and right circularly polarized light at applied magnetic-field strength H, DA(n,0) is the (natural) CD spectrum, sx and sy are unit-area band-shape functions for the states, Bx and By are the MCD susceptibilities, and the corresponding absorption spectrum is given by A 0 (n) = A(n)/n = Dxsx(n) + Dysy(n)
(2)
where Dx and Dy are the dipole strengths (squares of the associated transition moments). Of central importance are the relative signs of the two different sets of weightings: in absorption Dx and Dy are both positive whereas in MCD By and Bx have opposite signs (for porphyrinoids including chlorophylls, By is positive whilst Bx is negative).1–3 In principle, combination of the absorption and MCD data should allow the bandshape functions sx and sy to be directly determined, yielding the information required to construct a complete spectral assignment. However, spectroscopic measurements yield data A 0 (ni) and DDA 0 (ni) at n discrete frequencies ni, and eqn (1) and (2) represent this data in terms of 2n + 2 unique parameters, Bx/Dx, By/Dy, Dxsx(ni) and Dysy(ni). Hence a direct determination of these parameters is actually not possible as this approach involves two more variables than the number of available data points. Since its inception, MCD spectroscopy has been limited by this technicality which prohibits direct data inversion. The approach often taken over the last 50 years to get around the data-inversion problem is to interpret the absorption spectra as sums of individual component bands, with each component being assigned an individual B/D ratio to fit the associated MCD spectrum. This approach has the advantage that the MCD
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‘‘A’’ and ‘‘C’’ contributions can also be readily included into a general analysis scheme.2,34–37 Recently we successfully applied a scheme of this type to assign the spectra of a wide range of chlorophyllides using a complex model for vibronically coupled Qy and Qx Franck–Condon band envelopes, fitting 7 non-trivial parameters per sample.4 However, this is not the usual way in which such MCD spectral fitting is performed, with typically Franck–Condon bandshapes being represented instead as the sum of many independent Gaussian functions.2,34–37 Indeed, such Gaussian-fitting procedures have often been applied to the Q-band of chlorophyllides using typically seven independent Gaussian functions.27–29,38,39 For each component Gaussian, four free parameters arise: the centre, height, and width of each function fitted to the absorption data plus B/D fitted to the MCD data, giving a total of twenty eight fitting parameters per sample. Gaussian fitting-based analyses typically fail to recognize the physical constraints necessitated by the Franck–Condon approximation and in particular fail to conserve key physical properties such as relative Huang-Rhys factors when systems are perturbed, e.g., by changing the solvent or by peripheral chemical substitution. B and D are arbitrarily allowed to vary as a function of excitation energy. However, if the two spectral bands of interest do not interact with other (presumably much more distant) transitions, and if B/D is insensitive to inhomogeneous broadening, then these quantities should be constant throughout the band, as specified in eqn (1) and (2). As the analysis is not constrained to reproduce this important characteristic, it becomes very difficult to detect a weak band of one polarization that coincides with a strong band of the other, as all this analysis usually yields is a reduced relative MCD B/D factor for the overlapped Gaussian component of the stronger band. As a result, the assignment of the spectrum of Chl-a based on MCD data27–29 was inconsistent with that deduced considering other properties17–21 including the observed asymmetry between absorption and emission spectra,4,22 as well as with some30 (but not all24–26,31) highresolution analyses. In particular, this approach failed because it focused on identification of the single intense component of x polarization expected when the Born–Oppenheimer and Franck–Condon approximations are applied to the Gouterman spectroscopic model,18 whereas the identification and quantification of two x-polarized components of widely variable intensity, present in every chlorophyllide, was found to be critical to a consistent spectral assignment.4 Our recent spectral-fitting analysis of chlorophyllides did in fact manage to identify such weak components, because it constrained the spectra to only exhibit shapes reasonable for chlorophyllides and explicitly included the effects of vibronic coupling between the Qx and Qy states.4 This coupling, which is outside the normal Born–Oppenheimer approximation, mixes the Q states together, facilitating x polarized absorption within the vibrational sideband of the Qy state and vice versa. The band shapes were constrained to have the form of a Franck–Condon allowed transition based on deduced Huang-Rhys factors for 51 vibrational modes. These were obtained from the high-resolution26 fluorescence excitation spectrum of Chl-a, and, except for a possible overall rescaling, were constrained during the analysis of all
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chlorophyllides in all solvation environments. As a result, the spectral fitting involved only seven non-trivial parameters per sample: (1) the energy difference between the unperturbed origins DE, (2) the fraction of the total absorption that has x polarization fx = Dx/(Dx + Dy), (3) & (4) the Qx and Qy inhomogeneous line widths, (5) & (6) the Qx and Qy ratios of B/D, and (7) the rescaling parameter of the total Huang-Rhys factor. Other parameters were used in the model but were held fixed for every sample. Two describe the vibronic coupling and four describing the shape of the zero-phonon line and its associated phonon side band. While the actual variations of these constrained parameters with sample is interesting, it is not as yet explored. Despite its success, our new analysis is arduous in that the primary qualitative conclusions flow only after spectral fitting of over 40 chlorophyllide systems. It also involves the use of many simplifying assumptions, such as the invariance of the Franck– Condon allowed bandshapes with solvent and chlorophyllide composition. Additionally, the analysis utilised a relatively crude two-parameter universal representation of the vibronic coupling processes. We aspire to an analysis method that allows the primary qualitative conclusions to be drawn independently of the details of the spectrum, and be independent of any numerical data-fitting procedure.
not a useful approach in the present application, though it may prove useful in other contexts. More generally, we note that as sx(ni) and sy(ni) must be both non-negative, Bx/Dx Z max(DDA 0 (ni)/A 0 (ni)).
(5)
Further, as a means to facilitate direct data inversion, we introduce the limiting condition Bx/Dx = max(DDA 0 (ni)/A 0 (ni)).
(6)
If both B/D ratios are known then data inversion follows using Bx By Dx Dy sx ðn i Þ ¼ By DDA0 ðn i Þ A0 ðn i Þ Dy
(7)
By Bx D Dx 0 y ; sy ðn i Þ ¼ Bx DDA0 ðn i Þ A0 ðn i Þ Dx
(8)
0
and
normalized to give sx ðn i Þ ¼
X 0 sx ðn i Þ where Dx ¼ sx ðn i Þ Dx i
(9)
sy ðn i Þ ¼
X 0 sy ðn i Þ where Dy ¼ sy ðn i Þ: Dy i
(10)
0
2. Methods As previously discussed, analytical data inversion is not initially possible because there are two more unknowns than there are experimentally available data values. We may, however, choose to represent the two unspecified variables in any way we please. Consider the lower-energy transition to be y-polarized. Then for eqn (1) to be valid (i.e., for there to be no partial contribution from an MCD ‘‘A’’ term), the ratio By/Dy must satisfy By/Dy = max(DDA 0 (ni)/A 0 (ni))
(3)
and hence one of the missing conditions is actually already known, given previously made assumptions. Note, however, utilization of eqn (3) does demand that the observed absorption and MCD spectra do not exhibit even minor consequences of chromophore aggregation40–42 as this effect would independently distort the bandshapes in the critical region. Chlorophyll Q-bands derive from a doubly degenerate transition of the symmetric parent species (magnesium porphyrin). The Q band MCD of the parent species exhibits an ‘‘A’’ term, which can be considered to be decomposable into two equal and opposite B terms with1–3,43 Bx = By.
(4)
If the chemical asymmetry is small enough and the interaction between the two states of interest and other states are symmetric and/or weak, then this relationship may hold for the asymmetric species as well. For chlorophylls, the magnitude of Bx turns out to be ca. 20% larger than that of By, with this ratio increasing as the Q-Soret band gap decreases.4 Hence this is
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and 0
From these component spectra the polarization function pðn i Þ ¼
Dy sy ðn i Þ Dx sx ðn i Þ ¼ 1 2fx ðn i Þ Dy sy ðn i Þ þ Dx sx ðn i Þ
(11)
may be deduced, where By DDA0 ðn i Þ A0 ðn i Þ Dy Dx sx ðn i Þ ¼ fx ðn i Þ ¼ By Bx Dx sx ðn i Þ þ Dy sy ðn i Þ Dy Dx
(12)
is the fraction of intensity attributed to the x state at each frequency. Integrating this gives the total fraction of the absorption with x polarization as Ð 0 By Ð Q DDA0 ðnÞdn Q A ðnÞdn Dy Dx fx ¼ ¼ : (13) By Bx Dx þ Dy Dy Dx Using the value of Bx/Dx given in eqn (6) which is the lower limit of its feasible range, fx is evaluated as the maximum plausible value for the fraction of x-polarized absorption. If the higher-energy (x) transition is the more intense one then eqn (6) will be a good approximation and the data inversion will be accurate. While visual inspection of MCD spectra of this
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type leads to an unambiguous spectral assignment, analytical data inversion can still be applied to improve quantitative analysis. However, when the higher-energy component is the weaker one, no simple interpretative methods are available. This is the situation for chlorophylls. Eqn (13) then becomes very important as it provides an upper bound to the intensity attributable to the weak component. Because of the simple relationship between Bx/Dx and fx presented in eqn (13), either quantity could be considered to be the imprecisely defined one associated with this direct data inversion procedure. The Qx–Qy band gap can be expressed as
DE ¼ ly lx þ
ð
nsx ðnÞdn Q
ð
nsy ðnÞdn Q
(14)
ð
nðsx ðnÞ sy ðnÞÞdn; Q
where lx and ly are the reorganization energies for the ground state to Qx and ground state to Qy absorptions, respectively, and, as typically |DE| c |ly lx|, these terms can often be neglected. However, this may not be the case if the spectral bands in question have different inherent electronic characteristics. It is possible to rewrite these equations using wavelength instead of frequency but then the resulting parameters would not be transferrable between systems. For example, a change in solvent that did nothing more than uniformly shift the Qx and Qy band origins would, using a frequency-based analysis, not modulate the B/D ratios but only just shift the band. However, if wavelength was used instead, then the B/D ratios would change as well as the bandshape. This is not a desirable feature if the effects of solvent, temperature, or chemical composition, etc. are to be effectively monitored in an MCD study. A generally useful Excel spreadsheet containing these equations is provided in ESI.† It can be adapted to analyse data for general molecular transitions involving two overlapping orthogonally polarized components by simply pasting the appropriately represented absorption and MCD data into the spreadsheet. Some options associated with the treatment of noise in the experimental data appear in the spreadsheet and may need adjustment.
3. Results Fig. 1b shows the x and y components of the absorption and MCD spectra of Chl-a in ether at room temperature27 determined assuming that the maximum-feasible Qx absorption takes place (eqn (6)). These components may be compared to the analogous results from our previous numerical vibronic-coupling analysis4 shown in Fig. 1a. Both associated polarization functions determined using eqn (11) are shown in Fig. 1c. Most striking is the excellent qualitative agreement found between the two methods. At the Qy origin (Dn = 0), the polarization is of pure y character, as is demanded for eqn (3) to be valid, with x-polarized components being easily identified at ca. Dn = 1000 cm1 and 2050 cm1, indicating two x-polarized bands in the spectra. From a quantitative perspective, the analytically deduced polarization becomes of purely x character at Dn = 2050 cm1, a feature resulting from use of the boundary assumption, eqn (6), whereas the polarization changes deduced by fitting are of smaller magnitude. Hence we see that eqn (6) leads to the correct qualitative shape of the spectral polarization function but uniformly overestimates the contribution of the (weak) x band. The determination of an improved analytical estimate for Bx/Dx beyond eqn (6) requires information not contained in the absorption and MCD spectra themselves; the vibronic-coupling-based numerical fitting procedure adds this information implicitly by forcing the deduced bandshape functions to conform to some predetermined shape, that as determined by the observed Franck– Condon factors for Qy, and the use of a universal 2-parameter description of the vibronic-coupling. As no raw experimental data is available that would facilitate data inversion without fitting, we examine the qualitative and quantitative effects that increasing Bx/Dx above its minimum-possible value has on the deduced polarization function. We do this for 21 chlorophyllide samples and the results are given in Fig. 2, where they are compared to those from our previous vibronic-coupling analysis. We utilize Bx/Dx from eqn (6), also examining the situation if Bx/Dx = 2max(DDA 0 (ni)/A 0 (ni)).
(15)
Details of these analyses are provided in the example spreadsheets contained in the ESI.† The results show that rescaling
Fig. 1 Comparison of analyses of the absorption and MCD spectra27 of Chl-a in ether, showing the observed spectra (dots) decomposed into components (black – total, purple – Qy, green – Qx) and the associated polarization p determined using numerical fitting of a vibronic-coupling model and from analytical inversion assuming that Bx/Dx = max(DDA 0 (ni)/A 0 (ni)) (eqn (6)).
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Fig. 2 Comparison of the linear dichroism extracted from absorption and MCD spectra (for sources see Table 1) using either: the vibronic-coupling model (red) and analytical inversion done by assuming the limiting value (eqn (6)) (blue), or twice this value (eqn (15)) (green).
Bx/Dx acts to compress the polarization functions towards pure y polarization (i.e., Dx and thus fx decrease) but does not significantly change their shape. We see that, independent of the value taken for Bx/Dx, key qualitative spectral information is revealed. Table 1 goes further and examines the quantitative reliability of eqn (6) in determining the important spectral parameters fx and DE (eqn (13) and (14)). It also examines the reliability of Bx/By by comparing results to those from our previous vibronic-coupling analysis4 for a variety of chlorophyllides. In addition, the two sets of band gaps are compared in Fig. 3. The values of Bx/By deduced by the two methods are all within 10% of each other for all cases except pyromethylpheophorbide-a.44 Similarly, the fitted and analytically determined maximum values of fx are all consistent except for this molecule and for methylpheophorbide-a.44 These disparities most likely arises from the effects of minor aggregation40–42 on the spectra.
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While both data-fitting techniques and analytical inversion are sensitive to distortion of bandshapes by such experimental effects, data fitting using a minimal parameter set is more constrained and so should be less sensitive. For the band gap DE, Table 1 indicates relatively small differences for BChl-a in ether (the fitted and analytical energy gaps are DE = 4140 and 3870 cm1, respectively) and BChl-c in pyridine (440 and 690 cm1, respectively). However, in some circumstances the differences can be appreciable, as highlighted especially in Fig. 3, with e.g., for chlorin-e6 tri-methylester with deduced band gaps of 3580 and 2200 cm1. Two effects contribute to this. The minor effect is that x intensity is incorrectly assigned to the Qy origin region in the analytical analysis, perhaps due to effects of aggregation distorting the Qy band shape, but more significantly because too much y polarization is attributed for large Dn where vibronic coupling to By starts to
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Table 1 Fitted values of the unperturbed Qx–Qy spacing D E (1000 cm1) for chlorophyllides and related tetrapyrroles deduced from the vibroniccoupling model fit to the ABS and MCD spectra4 compared to those obtained by our approximate analytical method using Bx/Dx = max(DDA 0 (ni)/A 0 (ni)) (eqn (10)), as well as the corresponding fraction fx of absorption attributed to Qx and the associated MCD susceptibility ratio
DE Sample
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b
BChl-a ether Chlorin-e6 TMEc Methylpheophorbide-ac Pheo-a EtOH–MeOH 1.7 Kd Pyromethylpheophorbide-ac BChl-a pyridineb Ni(II)–Chl-a ethere Zn(II)–Chl-a ethere Chl-a ether f Chl-a dry ether 180 Kg BChl-d etherh ChlZ(D1) PS-II 1.7 Ki BChl-c ether f Chl-a pyridine f Chl-d MeOH–EtOH 1.7 Kg BChl-d pyridineh BChl-c pyridine f
Bx/By
fx a
Fit
Anal.
Fit
Anal.
Fit
Anal.
4.14 3.58 3.47 3.43 3.36 3.24 2.60 1.99 1.64 1.58 1.38 1.35 1.15 0.97 0.81 0.71 0.44
3.87 2.20 2.65 2.22 2.58 3.05 2.26 1.73 1.54 1.55 1.30 1.20 1.17 1.01 1.10 0.82 0.69
0.21 0.07 0.12 0.17 0.10 0.22 0.07 0.09 0.10 0.12 0.10 0.10 0.16 0.17 0.23 0.17 0.22
0.21 0.05 0.07 0.15 0.07 0.22 0.07 0.12 0.14 0.15 0.18 0.13 0.26 0.26 0.25 0.30 0.38
1.6 4.4 3.3 1.9 4.7 1.6 2.2 1.7 1.5 1.2 1.5 1.5 1.3 1.2 1.2 1.2 1.1
1.7 3.9 3.0 1.6 2.8 1.7 2.1 1.7 1.4 1.2 1.4 1.4 1.4 1.2 1.1 1.2 1.2
a
Maximum possible value only. b Original spectra from: Umetsu.29 Briat44 in dioxane. d Razeghifard.50 e Nonomura.48 f Umetsu.27 g Reimers.49 h Umetsu.28 i Krausz.51,52 c
data-inversion scheme based on eqn (6), can in itself provide information critical to spectral assignment. For all chlorophyllides, eqn (6) indicates that the Q-band spectrum contains two x-polarized peaks, as shown in Fig. 2. These are sometimes of similar significance, e.g., for BChl-d in pyridine, and sometimes the lower-energy peak is in comparison very weak, e.g., for BChl-a in pyridine. However, the critical information is that all systems have two such peaks. For BChl-a, the weak peak had not been identified in previous29 MCD studies until its discovery in our recent band-fitting analysis.4 Identification through analytical data inversion of two peaks of which one is always located close to Dn B 1200 cm1 independent of DE immediately suggests vibronic coupling as the mechanism dominating the spectrum of Chl-a. Such a vibronic-coupling-dominated assignment of the Qband of Chl-a was identified 50 years ago as a possible option,45–47 but it had never been advanced as a serious contender because only one x-polarized band was generally observed in chlorophyllides. The paired bands revealed in Fig. 2 indeed show all of the expected qualitative features of vibronic coupling: the spectra of chlorin-d6, methylpheophorbide-a, pheophytin-a, and pyromethylpheophorbide-a44 show systematic variations, as do the spectra of Chl-a molecules with substituted metal atoms,48 perturbed chlorophyllides like BChl-c27 and BChl-d28 also fit the same pattern. Further, the analysis can reveal anomalies in data sets, such as that for Chl-a in ether at low temperature.49 Our analytical data-inversion technique thus provides, without the need for extensive fitting, the key qualitative feature required for spectral assignment.
5. Conclusions
Fig. 3 Comparison of the Qx–Qy gaps obtained from a vibronic-coupling model fit4 and from analytical inversion (Table 1). Results shown in red have low results from the analytical fit owing to the presence of more than one chemical species (which distorts lineshapes) and/or the involvement of Qy–Bx vibronic coupling (which adds unexpected y-polarized intensity at high Dn).
dominate the observed spectra. This problem is most noticeable for molecules with large DE and in fact correlates with the By to Qx splitting. Our band-fitting approach is less sensitive to these effects but is also in error owing to neglect of By–Qx coupling in the fitting model.
4. Discussion The qualitative determination of the general shape of the spectral polarization function, as revealed from our analytical
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We have introduced an analytical data inversion technique enabling the interpretation of ‘‘B’’-term MCD spectra which provides a simple solution for one of the most fundamental problems in MCD spectroscopy: how to process unpolarized absorption and MCD data to obtain the implicit bandpolarization information. It allows the separation of two spectral bands of near-orthogonal polarization that themselves overlap but which are otherwise uninfluenced by either spectral overlap or vibronic coupling. However, this technique is only appropriate when the broadened origin lines of the two spectral bands do not overlap, allowing the MCD spectra to be dominated by the ‘‘B’’ term only. In general, MCD data inversion is two-fold underdetermined in these quite common circumstances. We show that the assumption of ‘‘B’’-term spectroscopy immediately yields one constraint, leaving only one quantity undetermined. We define a system variable whose value is bounded. Key features of the inverted spectra are shown to be insensitive to this variable. These include the energy gap between the overlapping transitions and the qualitative shape of the polarization profile. What remains difficult to determine is the absolute magnitude of the polarization fluctuations involved and the partitioning of the total oscillator strength. If the higher-energy transition is much stronger than the lower-energy one, then the identified bound become in fact a
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good approximation. In such circumstances, a qualitative spectral interpretation by eye is possible and the analytical data-inversion procedure provides a method to improved quantitative analysis. However, it is the scenario in which the higher-energy transition is weak enough so that spectral modeling is required in order to extract key qualitative information for which analytical data inversion offers the greatest gains. For the chlorophyllides, the higher-energy transition is indeed the weaker one and we show that analytical data inversion quickly leads to a definitive spectral assignment. For Chl-a, its Q-band is shown to be dominated by strong resonant vibronic coupling between the Qy and Qx states. Had our analytical data inversion been available 50 years ago, the spectrum of Chl-a would have been immediately assigned, a feat not achieved until recently, based on extensive vibronic-coupling simulations of a wide range of chlorophyllide MCD and other spectra.4 While chlorophyll (and porphyrin) spectroscopy is the focus here, the technique is not dependent on details other than the actual band shapes. It can thus be applied to any spectroscopic system in which two bands of nearly orthogonal polarization overlap but are otherwise isolated. In particular, it does not rely on the two transitions having a common parentage, as they do for chlorophylls. The excel spreadsheet provided in ESI† allows the easy application of this technology in order to rapidly analyze and assign absorption and MCD spectra. The procedure is applicable to many chemical systems, including those now commonly being obtained from XMCD.
Acknowledgements We thank the Australian Research Council Discovery Projects scheme for funding this research.
Notes and references 1 A. D. Buckingham and P. J. Stephens, Annu. Rev. Phys. Chem., 1966, 17, 399. 2 P. N. Schatz and A. J. McCaffery, Q. Rev., Chem. Soc., 1969, 23, 552. 3 P. J. Stephens, Annu. Rev. Phys. Chem., 1974, 25, 201. 4 J. R. Reimers, Z.-L. Cai, R. Kobayashi, M. Ratsep, A. Freiberg and E. Krausz, Sci. Rep., 2013, 3, 2761. 5 L. Salem, Molecular Orbital Theory of Conjugated Systems, Benjamin, Reading MA, 1972. 6 X.-N. Li, Z.-J. Wu, Z.-J. Si, H.-J. Zhang, L. Zhou and X.-J. Liu, Inorg. Chem., 2009, 48, 7740. 7 A. R. G. Smith, M. J. Riley, S. C. Lo, P. L. Burn, I. R. Gentle and B. J. Powell, Phys. Rev. B: Condens. Matter Mater. Phys., 2011, 83, 041105. 8 E. W. Thulstrup and J. Michl, J. Mol. Spectrosc., 1976, 61, 203. 9 S. J. Dammer, P. V. Solntsev, J. R. Sabin and V. N. Nemykin, Inorg. Chem., 2013, 52, 9496. ˇpa ´nek, M. Straka, V. Andrushchenko and P. Bourˇ, 10 P. ˇ Ste J. Chem. Phys., 2013, 138, 151103.
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11 J. P. Hare, H. W. Kroto and R. Taylor, Chem. Phys. Lett., 1991, 177, 394. 12 W. Q. Tian, D.-L. Chen, Y.-H. Cui and J.-K. Feng, J. Comput. Theor. Nanosci., 2009, 6, 239. 13 S. Gautam, P. Thakur, S. Augustine, J. K. Kang, J. Y. Kim, N. B. Brookes, K. Asokan and K. H. Chae, arXiv.org, e-Print Arch., Condens. Matter, 2011. 14 A. Briones-Leon, P. Ayala, X. Liu, K. Yanagi, E. Weschke, M. Eisterer, H. Jiang, H. Kataura, T. Pichler and H. Shiozawa, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 87, 195435. 15 W. R. Browett and M. J. Stillman, Inorg. Chim. Acta, 1981, 49, 69. 16 Z. Gasyna, D. H. Metcalf, P. N. Schatz, C. L. McConnell and B. E. Williamson, J. Phys. Chem., 1995, 99, 5865. ¨r, H. Land, E. Schnabel and H. Kuhn, Z. Elektrochem. 17 F. Ba Angew. Phys. Chem., 1961, 65, 346. 18 C. Houssier and K. Sauer, J. Am. Chem. Soc., 1970, 92, 779. 19 W. Haegele, D. Schmid, F. Drissler, J. Naus and H. C. Wolf, Z. Naturforsch., A: Phys. Phys. Chem. Kosmophys., 1978, 33, 1197. 20 M. Gouterman and L. Stryer, J. Chem. Phys., 1962, 37, 2260. 21 C. Weiss Jr, J. Mol. Spectrosc., 1972, 44, 37. 22 M. E. Deroche and J. M. Briantais, Photochem. Photobiol., 1974, 19, 233. 23 T. A. Evans and J. J. Katz, Biochim. Biophys. Acta, Bioenerg., 1975, 396, 414. 24 K. K. Rebane and R. A. Avarmaa, Chem. Phys., 1982, 68, 191. 25 R. A. Avarmaa and A. Suisalu, Opt. Spectrosc., 1984, 56, 54. 26 R. A. Avarmaa and K. K. Rebane, Spectrochim. Acta, Part A, 1985, 41, 1365. 27 M. Umetsu, Z.-Y. Wang, M. Kobayashi and T. Nozawa, Biochim. Biophys. Acta, Bioenerg., 1999, 1410, 19. 28 M. Umetsu, Z.-Y. Wang and T. Nozawa, Recent Res. Dev. Phys. Chem., 2001, 5, 185. 29 M. Umetsu, Z.-Y. Wang, K. Yoza, M. Kobayashi and T. Nozawa, Biochim. Biophys. Acta, Bioenerg., 2000, 1457, 106. 30 J. L. Hughes, B. Conlon, T. Wydrzynski and E. Krausz, Phys. Procedia, 2010, 3, 1591. ¨tsep, J. Linnanto and A. Freiberg, J. Chem. Phys., 2009, 31 M. Ra 130, 194501. 32 D. Caldwell, J. M. Thorne and H. Eyring, Annu. Rev. Phys. Chem., 1971, 22, 259. 33 M. T. Tiedemann and M. J. Stillman, J. Porphyrins Phthalocyanines, 2011, 15, 1134. 34 P. N. Schatz, A. J. McCaffery, W. Suetaka, G. N. Henning, A. B. Ritchie and P. J. Stephens, J. Chem. Phys., 1966, 45, 722. 35 B. R. Hollebone and M. J. Stillman, J. Chem. Soc., Faraday Trans. 2, 1978, 74, 2107. 36 W. R. Browett and M. J. Stillman, Comput. Chem., 1987, 11, 241. 37 J. Mack, Y. Asano, N. Kobayashi and M. J. Stillman, J. Inorg. Biochem., 2008, 102, 472. 38 D. Frackowiak, D. Bauman, H. Manikowski, W. R. Browett and M. J. Stillman, Biophys. Chem., 1987, 28, 101. 39 M. Umetsu, R. Seki, Z.-Y. Wang, I. Kumagai and T. Nozawa, J. Phys. Chem. B, 2002, 106, 3987.
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40 J. L. Hughes, R. J. Pace and E. Krausz, Chem. Phys. Lett., 2004, 385, 116. 41 J. Linnanto, J. A. I. Oksanen and J. E. I. Korppi-Tommola, Phys. Chem. Chem. Phys., 2002, 4, 3061. 42 J. Linnanto, V. M. Helenius, J. A. I. Oksanen, T. Peltola, J. L. Garaud and J. E. I. Korppi-Tommola, J. Phys. Chem. A, 1998, 102, 4337. 43 C. H. Henry, S. E. Schnatterly and C. P. Slichter, Phys. Rev. Lett., 1964, 13, 130. 44 B. Briat, D. A. Schooley, R. Records, E. Bunnenberg and C. Djerassi, J. Am. Chem. Soc., 1967, 89, 6170. 45 M. Gouterman, J. Mol. Spectrosc., 1961, 6, 138. 46 R. L. Fulton and M. Gouterman, J. Chem. Phys., 1964, 41, 2280.
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47 R. L. Fulton and M. Gouterman, J. Chem. Phys., 1961, 35, 1059. 48 Y. Nonomura, S. Igarashi, N. Yoshioka and H. Inoue, Chem. Phys., 1997, 220, 155. ¨tsep, A. Freiberg 49 J. R. Reimers, Z.-L. Cai, R. Kobayashi, M. Ra and E. Krausz, Phys. Chem. Chem. Phys., 2013, DOI: 10.1039/ c3cp53729c. 50 M. R. Razeghifard, M. Chen, J. L. Hughes, J. Freeman, E. Krausz and T. Wydrzynski, Biochemistry, 2005, 44, 11178. ¨ld, in Artificial Photosynthesis: 51 E. Krausz and S. Peterson Årsko From Basic Biology to Industrial Application, ed. A. F. Collings and C. Critchley, Wiley, London, 2005, p. 87. 52 E. Krausz, N. Cox and S. P. Arskold, Photosynth. Res., 2008, 98, 207.
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An analytical data inversion method for magnetic circular dichroism spectra dominated by the ‘‘B-term’’† Jeffrey R. Reimers*a and Elmars Krauszb A simple procedure is developed enabling the analytical inversion of an (unpolarized) absorption spectrum combined with a Magnetic Circular Dichroism (MCD) spectrum to resolve two overlapping bands of orthogonal polarization. This method is appropriate when (i) the overlapping transitions are well isolated from other bands, and (ii) when their electronic spacing is large enough so that the ‘‘A-term’’ and ‘‘C-term’’ contributions to the MCD spectrum can be ignored and hence only the ‘‘B-term’’ contribution need be considered. We apply this procedure to assign the Q-band system of chlorophylls, though similar challenges also commonly arise throughout both conventional and X-ray MCD (XMCD) spectroscopy. Analytical data inversion has not previously been possible as the inversion process is two-fold underdetermined. We show that the assumptions of isolated spectra and ‘‘B-term’’ dominance yields one generally valid constraint, leaving only one quantity unspecified by the experimental data. For some systems, an approximation leading to equal but opposite sign B-term magnitudes of the two components
Received 3rd September 2013, Accepted 4th December 2013 DOI: 10.1039/c3cp53730g
may be reasonable, but for chlorophyllides we find this constraint to be inappropriate. Instead, we constrain a bounded variable controlling the relative absorption strengths. Derived spectral bandshapes of the individual components are shown to be insensitive to its particular value, allowing weak spectral components of one polarization overlapped by intense components of the other to be immediately exposed. This is demonstrated for the chlorophylls, molecules for which the failure to detect such weak
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features historically led to incorrect proposals for the Q-band assignments.
1. Introduction Magnetic Circular Dichroism (MCD) offers great potential for the assignment of spectra of overlapping electronic transitions with different polarizations. In general, three types of MCD signals are commonly identified, known as the MCD ‘‘A’’, ‘‘B’’, and ‘‘C’’ terms,1–3 with the ‘‘B’’ term being dominant if the states involved in the transitions are not degenerate. An important application of MCD has been to the Q-band spectra of various porphyrins and chlorophylls critical to natural and artificial photosynthesis,4 as well as to many other spectroscopic systems of technological relevance such as those in acenes, graphene fragments, substituted fullerenes, and inorganic light harvesting, separating, or emitting compounds;5–12 in addition, the emergent technique of X-ray MCD (XMCD) is often applied
a
School of Chemistry, The University of Sydney, Sydney, NSW 2006, Australia. E-mail: Jeff[email protected]; Fax: +61 293513329; Tel: +61 293514417 b Research School of Chemistry, The Australian National University, Canberra, ACT 0200, Australia † Electronic supplementary information (ESI) available: Excel spreadsheet containing all analytical data inversions. See DOI: 10.1039/c3cp53730g
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to characterize nanotubes.13,14 In a symmetric molecule (e.g., zinc porphyrin) having equivalent x and y axes, doubly degenerate electronic transitions will appear that have an ‘‘A’’-type MCD spectrum.15,16 However, chemical substitution usually breaks this degeneracy so that two independent bands emerge, and in porphyrins and chlorophylls these component bands are named Qx and Qy. Typically the induced energy separation DE is larger than the widths of the inhomogeneously broadened electronic origins, changing the MCD spectra to ‘‘B’’ type spectra that may be represented simply as weighted sums of individual spectral components for each state. Overlapping spectroscopic transitions of nearly orthogonal polarization may arise for reasons other than pseudo-symmetry, but result in a similar scenario. The cause of the spectral overlap is not of consequence herein, only the energy gap. Most molecular spectral transitions, including the Q-bands of chlorophylls, have extensive Franck–Condon vibrational sideband structure. We are particularly concerned with situations for which the extent of this sideband structure exceeds the energy gap DE between the two electronic states, leading to complex and entangled spectra. The most important chlorophyllide, chlorophyll-a (Chl-a), indeed displays spectra of this type.
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As a result, its Qx, Qy assignment has been debated for over 50 years,4,17–31 with various assignment options being considered. Each alternative assignment has considerable ramifications, effecting, e.g., the exciton transport and coherence properties of photosystems.4 MCD has been developed as an experimental technique based primarily on its capacity to resolve overlapping spectral transitions.1–3,32,33 It is particularly well suited to this application compared to other methods, such as linear dichroism, as spectra can be measured in liquid, solid, and biological media as a function of temperature and pressure. In particular, many MCD measurements have been made on chlorophyllides, and our recent analysis unifies the deduced polarization information with the (relatively scarce) results obtained alternatively using linear dichroism or polarized fluorescence anisotropy techniques.4 However, there remains a significant technical issue in the unambiguous interpretation of MCD and absorption data. This is because direct data inversion of an (unpolarized) absorption spectrum and its associated MCD spectrum to yield the two individual-component spectra is mathematically underdetermined. A single ‘‘B’’-term contribution to an MCD spectrum of two overlapping states of orthogonal polarization (say x and y) can be expressed as the weighted sum of individual components as1,2 DDA0 ðnÞ ¼
DAðn; HÞ DAðn; 0Þ ¼ Bx sx ðnÞ þ By sy ðnÞ Hn
(1)
where A is the molar extinction, DA(n,H) is the difference in molar absorption between left and right circularly polarized light at applied magnetic-field strength H, DA(n,0) is the (natural) CD spectrum, sx and sy are unit-area band-shape functions for the states, Bx and By are the MCD susceptibilities, and the corresponding absorption spectrum is given by A 0 (n) = A(n)/n = Dxsx(n) + Dysy(n)
(2)
where Dx and Dy are the dipole strengths (squares of the associated transition moments). Of central importance are the relative signs of the two different sets of weightings: in absorption Dx and Dy are both positive whereas in MCD By and Bx have opposite signs (for porphyrinoids including chlorophylls, By is positive whilst Bx is negative).1–3 In principle, combination of the absorption and MCD data should allow the bandshape functions sx and sy to be directly determined, yielding the information required to construct a complete spectral assignment. However, spectroscopic measurements yield data A 0 (ni) and DDA 0 (ni) at n discrete frequencies ni, and eqn (1) and (2) represent this data in terms of 2n + 2 unique parameters, Bx/Dx, By/Dy, Dxsx(ni) and Dysy(ni). Hence a direct determination of these parameters is actually not possible as this approach involves two more variables than the number of available data points. Since its inception, MCD spectroscopy has been limited by this technicality which prohibits direct data inversion. The approach often taken over the last 50 years to get around the data-inversion problem is to interpret the absorption spectra as sums of individual component bands, with each component being assigned an individual B/D ratio to fit the associated MCD spectrum. This approach has the advantage that the MCD
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‘‘A’’ and ‘‘C’’ contributions can also be readily included into a general analysis scheme.2,34–37 Recently we successfully applied a scheme of this type to assign the spectra of a wide range of chlorophyllides using a complex model for vibronically coupled Qy and Qx Franck–Condon band envelopes, fitting 7 non-trivial parameters per sample.4 However, this is not the usual way in which such MCD spectral fitting is performed, with typically Franck–Condon bandshapes being represented instead as the sum of many independent Gaussian functions.2,34–37 Indeed, such Gaussian-fitting procedures have often been applied to the Q-band of chlorophyllides using typically seven independent Gaussian functions.27–29,38,39 For each component Gaussian, four free parameters arise: the centre, height, and width of each function fitted to the absorption data plus B/D fitted to the MCD data, giving a total of twenty eight fitting parameters per sample. Gaussian fitting-based analyses typically fail to recognize the physical constraints necessitated by the Franck–Condon approximation and in particular fail to conserve key physical properties such as relative Huang-Rhys factors when systems are perturbed, e.g., by changing the solvent or by peripheral chemical substitution. B and D are arbitrarily allowed to vary as a function of excitation energy. However, if the two spectral bands of interest do not interact with other (presumably much more distant) transitions, and if B/D is insensitive to inhomogeneous broadening, then these quantities should be constant throughout the band, as specified in eqn (1) and (2). As the analysis is not constrained to reproduce this important characteristic, it becomes very difficult to detect a weak band of one polarization that coincides with a strong band of the other, as all this analysis usually yields is a reduced relative MCD B/D factor for the overlapped Gaussian component of the stronger band. As a result, the assignment of the spectrum of Chl-a based on MCD data27–29 was inconsistent with that deduced considering other properties17–21 including the observed asymmetry between absorption and emission spectra,4,22 as well as with some30 (but not all24–26,31) highresolution analyses. In particular, this approach failed because it focused on identification of the single intense component of x polarization expected when the Born–Oppenheimer and Franck–Condon approximations are applied to the Gouterman spectroscopic model,18 whereas the identification and quantification of two x-polarized components of widely variable intensity, present in every chlorophyllide, was found to be critical to a consistent spectral assignment.4 Our recent spectral-fitting analysis of chlorophyllides did in fact manage to identify such weak components, because it constrained the spectra to only exhibit shapes reasonable for chlorophyllides and explicitly included the effects of vibronic coupling between the Qx and Qy states.4 This coupling, which is outside the normal Born–Oppenheimer approximation, mixes the Q states together, facilitating x polarized absorption within the vibrational sideband of the Qy state and vice versa. The band shapes were constrained to have the form of a Franck–Condon allowed transition based on deduced Huang-Rhys factors for 51 vibrational modes. These were obtained from the high-resolution26 fluorescence excitation spectrum of Chl-a, and, except for a possible overall rescaling, were constrained during the analysis of all
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chlorophyllides in all solvation environments. As a result, the spectral fitting involved only seven non-trivial parameters per sample: (1) the energy difference between the unperturbed origins DE, (2) the fraction of the total absorption that has x polarization fx = Dx/(Dx + Dy), (3) & (4) the Qx and Qy inhomogeneous line widths, (5) & (6) the Qx and Qy ratios of B/D, and (7) the rescaling parameter of the total Huang-Rhys factor. Other parameters were used in the model but were held fixed for every sample. Two describe the vibronic coupling and four describing the shape of the zero-phonon line and its associated phonon side band. While the actual variations of these constrained parameters with sample is interesting, it is not as yet explored. Despite its success, our new analysis is arduous in that the primary qualitative conclusions flow only after spectral fitting of over 40 chlorophyllide systems. It also involves the use of many simplifying assumptions, such as the invariance of the Franck– Condon allowed bandshapes with solvent and chlorophyllide composition. Additionally, the analysis utilised a relatively crude two-parameter universal representation of the vibronic coupling processes. We aspire to an analysis method that allows the primary qualitative conclusions to be drawn independently of the details of the spectrum, and be independent of any numerical data-fitting procedure.
not a useful approach in the present application, though it may prove useful in other contexts. More generally, we note that as sx(ni) and sy(ni) must be both non-negative, Bx/Dx Z max(DDA 0 (ni)/A 0 (ni)).
(5)
Further, as a means to facilitate direct data inversion, we introduce the limiting condition Bx/Dx = max(DDA 0 (ni)/A 0 (ni)).
(6)
If both B/D ratios are known then data inversion follows using Bx By Dx Dy sx ðn i Þ ¼ By DDA0 ðn i Þ A0 ðn i Þ Dy
(7)
By Bx D Dx 0 y ; sy ðn i Þ ¼ Bx DDA0 ðn i Þ A0 ðn i Þ Dx
(8)
0
and
normalized to give sx ðn i Þ ¼
X 0 sx ðn i Þ where Dx ¼ sx ðn i Þ Dx i
(9)
sy ðn i Þ ¼
X 0 sy ðn i Þ where Dy ¼ sy ðn i Þ: Dy i
(10)
0
2. Methods As previously discussed, analytical data inversion is not initially possible because there are two more unknowns than there are experimentally available data values. We may, however, choose to represent the two unspecified variables in any way we please. Consider the lower-energy transition to be y-polarized. Then for eqn (1) to be valid (i.e., for there to be no partial contribution from an MCD ‘‘A’’ term), the ratio By/Dy must satisfy By/Dy = max(DDA 0 (ni)/A 0 (ni))
(3)
and hence one of the missing conditions is actually already known, given previously made assumptions. Note, however, utilization of eqn (3) does demand that the observed absorption and MCD spectra do not exhibit even minor consequences of chromophore aggregation40–42 as this effect would independently distort the bandshapes in the critical region. Chlorophyll Q-bands derive from a doubly degenerate transition of the symmetric parent species (magnesium porphyrin). The Q band MCD of the parent species exhibits an ‘‘A’’ term, which can be considered to be decomposable into two equal and opposite B terms with1–3,43 Bx = By.
(4)
If the chemical asymmetry is small enough and the interaction between the two states of interest and other states are symmetric and/or weak, then this relationship may hold for the asymmetric species as well. For chlorophylls, the magnitude of Bx turns out to be ca. 20% larger than that of By, with this ratio increasing as the Q-Soret band gap decreases.4 Hence this is
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and 0
From these component spectra the polarization function pðn i Þ ¼
Dy sy ðn i Þ Dx sx ðn i Þ ¼ 1 2fx ðn i Þ Dy sy ðn i Þ þ Dx sx ðn i Þ
(11)
may be deduced, where By DDA0 ðn i Þ A0 ðn i Þ Dy Dx sx ðn i Þ ¼ fx ðn i Þ ¼ By Bx Dx sx ðn i Þ þ Dy sy ðn i Þ Dy Dx
(12)
is the fraction of intensity attributed to the x state at each frequency. Integrating this gives the total fraction of the absorption with x polarization as Ð 0 By Ð Q DDA0 ðnÞdn Q A ðnÞdn Dy Dx fx ¼ ¼ : (13) By Bx Dx þ Dy Dy Dx Using the value of Bx/Dx given in eqn (6) which is the lower limit of its feasible range, fx is evaluated as the maximum plausible value for the fraction of x-polarized absorption. If the higher-energy (x) transition is the more intense one then eqn (6) will be a good approximation and the data inversion will be accurate. While visual inspection of MCD spectra of this
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type leads to an unambiguous spectral assignment, analytical data inversion can still be applied to improve quantitative analysis. However, when the higher-energy component is the weaker one, no simple interpretative methods are available. This is the situation for chlorophylls. Eqn (13) then becomes very important as it provides an upper bound to the intensity attributable to the weak component. Because of the simple relationship between Bx/Dx and fx presented in eqn (13), either quantity could be considered to be the imprecisely defined one associated with this direct data inversion procedure. The Qx–Qy band gap can be expressed as
DE ¼ ly lx þ
ð
nsx ðnÞdn Q
ð
nsy ðnÞdn Q
(14)
ð
nðsx ðnÞ sy ðnÞÞdn; Q
where lx and ly are the reorganization energies for the ground state to Qx and ground state to Qy absorptions, respectively, and, as typically |DE| c |ly lx|, these terms can often be neglected. However, this may not be the case if the spectral bands in question have different inherent electronic characteristics. It is possible to rewrite these equations using wavelength instead of frequency but then the resulting parameters would not be transferrable between systems. For example, a change in solvent that did nothing more than uniformly shift the Qx and Qy band origins would, using a frequency-based analysis, not modulate the B/D ratios but only just shift the band. However, if wavelength was used instead, then the B/D ratios would change as well as the bandshape. This is not a desirable feature if the effects of solvent, temperature, or chemical composition, etc. are to be effectively monitored in an MCD study. A generally useful Excel spreadsheet containing these equations is provided in ESI.† It can be adapted to analyse data for general molecular transitions involving two overlapping orthogonally polarized components by simply pasting the appropriately represented absorption and MCD data into the spreadsheet. Some options associated with the treatment of noise in the experimental data appear in the spreadsheet and may need adjustment.
3. Results Fig. 1b shows the x and y components of the absorption and MCD spectra of Chl-a in ether at room temperature27 determined assuming that the maximum-feasible Qx absorption takes place (eqn (6)). These components may be compared to the analogous results from our previous numerical vibronic-coupling analysis4 shown in Fig. 1a. Both associated polarization functions determined using eqn (11) are shown in Fig. 1c. Most striking is the excellent qualitative agreement found between the two methods. At the Qy origin (Dn = 0), the polarization is of pure y character, as is demanded for eqn (3) to be valid, with x-polarized components being easily identified at ca. Dn = 1000 cm1 and 2050 cm1, indicating two x-polarized bands in the spectra. From a quantitative perspective, the analytically deduced polarization becomes of purely x character at Dn = 2050 cm1, a feature resulting from use of the boundary assumption, eqn (6), whereas the polarization changes deduced by fitting are of smaller magnitude. Hence we see that eqn (6) leads to the correct qualitative shape of the spectral polarization function but uniformly overestimates the contribution of the (weak) x band. The determination of an improved analytical estimate for Bx/Dx beyond eqn (6) requires information not contained in the absorption and MCD spectra themselves; the vibronic-coupling-based numerical fitting procedure adds this information implicitly by forcing the deduced bandshape functions to conform to some predetermined shape, that as determined by the observed Franck– Condon factors for Qy, and the use of a universal 2-parameter description of the vibronic-coupling. As no raw experimental data is available that would facilitate data inversion without fitting, we examine the qualitative and quantitative effects that increasing Bx/Dx above its minimum-possible value has on the deduced polarization function. We do this for 21 chlorophyllide samples and the results are given in Fig. 2, where they are compared to those from our previous vibronic-coupling analysis. We utilize Bx/Dx from eqn (6), also examining the situation if Bx/Dx = 2max(DDA 0 (ni)/A 0 (ni)).
(15)
Details of these analyses are provided in the example spreadsheets contained in the ESI.† The results show that rescaling
Fig. 1 Comparison of analyses of the absorption and MCD spectra27 of Chl-a in ether, showing the observed spectra (dots) decomposed into components (black – total, purple – Qy, green – Qx) and the associated polarization p determined using numerical fitting of a vibronic-coupling model and from analytical inversion assuming that Bx/Dx = max(DDA 0 (ni)/A 0 (ni)) (eqn (6)).
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Fig. 2 Comparison of the linear dichroism extracted from absorption and MCD spectra (for sources see Table 1) using either: the vibronic-coupling model (red) and analytical inversion done by assuming the limiting value (eqn (6)) (blue), or twice this value (eqn (15)) (green).
Bx/Dx acts to compress the polarization functions towards pure y polarization (i.e., Dx and thus fx decrease) but does not significantly change their shape. We see that, independent of the value taken for Bx/Dx, key qualitative spectral information is revealed. Table 1 goes further and examines the quantitative reliability of eqn (6) in determining the important spectral parameters fx and DE (eqn (13) and (14)). It also examines the reliability of Bx/By by comparing results to those from our previous vibronic-coupling analysis4 for a variety of chlorophyllides. In addition, the two sets of band gaps are compared in Fig. 3. The values of Bx/By deduced by the two methods are all within 10% of each other for all cases except pyromethylpheophorbide-a.44 Similarly, the fitted and analytically determined maximum values of fx are all consistent except for this molecule and for methylpheophorbide-a.44 These disparities most likely arises from the effects of minor aggregation40–42 on the spectra.
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While both data-fitting techniques and analytical inversion are sensitive to distortion of bandshapes by such experimental effects, data fitting using a minimal parameter set is more constrained and so should be less sensitive. For the band gap DE, Table 1 indicates relatively small differences for BChl-a in ether (the fitted and analytical energy gaps are DE = 4140 and 3870 cm1, respectively) and BChl-c in pyridine (440 and 690 cm1, respectively). However, in some circumstances the differences can be appreciable, as highlighted especially in Fig. 3, with e.g., for chlorin-e6 tri-methylester with deduced band gaps of 3580 and 2200 cm1. Two effects contribute to this. The minor effect is that x intensity is incorrectly assigned to the Qy origin region in the analytical analysis, perhaps due to effects of aggregation distorting the Qy band shape, but more significantly because too much y polarization is attributed for large Dn where vibronic coupling to By starts to
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Table 1 Fitted values of the unperturbed Qx–Qy spacing D E (1000 cm1) for chlorophyllides and related tetrapyrroles deduced from the vibroniccoupling model fit to the ABS and MCD spectra4 compared to those obtained by our approximate analytical method using Bx/Dx = max(DDA 0 (ni)/A 0 (ni)) (eqn (10)), as well as the corresponding fraction fx of absorption attributed to Qx and the associated MCD susceptibility ratio
DE Sample
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b
BChl-a ether Chlorin-e6 TMEc Methylpheophorbide-ac Pheo-a EtOH–MeOH 1.7 Kd Pyromethylpheophorbide-ac BChl-a pyridineb Ni(II)–Chl-a ethere Zn(II)–Chl-a ethere Chl-a ether f Chl-a dry ether 180 Kg BChl-d etherh ChlZ(D1) PS-II 1.7 Ki BChl-c ether f Chl-a pyridine f Chl-d MeOH–EtOH 1.7 Kg BChl-d pyridineh BChl-c pyridine f
Bx/By
fx a
Fit
Anal.
Fit
Anal.
Fit
Anal.
4.14 3.58 3.47 3.43 3.36 3.24 2.60 1.99 1.64 1.58 1.38 1.35 1.15 0.97 0.81 0.71 0.44
3.87 2.20 2.65 2.22 2.58 3.05 2.26 1.73 1.54 1.55 1.30 1.20 1.17 1.01 1.10 0.82 0.69
0.21 0.07 0.12 0.17 0.10 0.22 0.07 0.09 0.10 0.12 0.10 0.10 0.16 0.17 0.23 0.17 0.22
0.21 0.05 0.07 0.15 0.07 0.22 0.07 0.12 0.14 0.15 0.18 0.13 0.26 0.26 0.25 0.30 0.38
1.6 4.4 3.3 1.9 4.7 1.6 2.2 1.7 1.5 1.2 1.5 1.5 1.3 1.2 1.2 1.2 1.1
1.7 3.9 3.0 1.6 2.8 1.7 2.1 1.7 1.4 1.2 1.4 1.4 1.4 1.2 1.1 1.2 1.2
a
Maximum possible value only. b Original spectra from: Umetsu.29 Briat44 in dioxane. d Razeghifard.50 e Nonomura.48 f Umetsu.27 g Reimers.49 h Umetsu.28 i Krausz.51,52 c
data-inversion scheme based on eqn (6), can in itself provide information critical to spectral assignment. For all chlorophyllides, eqn (6) indicates that the Q-band spectrum contains two x-polarized peaks, as shown in Fig. 2. These are sometimes of similar significance, e.g., for BChl-d in pyridine, and sometimes the lower-energy peak is in comparison very weak, e.g., for BChl-a in pyridine. However, the critical information is that all systems have two such peaks. For BChl-a, the weak peak had not been identified in previous29 MCD studies until its discovery in our recent band-fitting analysis.4 Identification through analytical data inversion of two peaks of which one is always located close to Dn B 1200 cm1 independent of DE immediately suggests vibronic coupling as the mechanism dominating the spectrum of Chl-a. Such a vibronic-coupling-dominated assignment of the Qband of Chl-a was identified 50 years ago as a possible option,45–47 but it had never been advanced as a serious contender because only one x-polarized band was generally observed in chlorophyllides. The paired bands revealed in Fig. 2 indeed show all of the expected qualitative features of vibronic coupling: the spectra of chlorin-d6, methylpheophorbide-a, pheophytin-a, and pyromethylpheophorbide-a44 show systematic variations, as do the spectra of Chl-a molecules with substituted metal atoms,48 perturbed chlorophyllides like BChl-c27 and BChl-d28 also fit the same pattern. Further, the analysis can reveal anomalies in data sets, such as that for Chl-a in ether at low temperature.49 Our analytical data-inversion technique thus provides, without the need for extensive fitting, the key qualitative feature required for spectral assignment.
5. Conclusions
Fig. 3 Comparison of the Qx–Qy gaps obtained from a vibronic-coupling model fit4 and from analytical inversion (Table 1). Results shown in red have low results from the analytical fit owing to the presence of more than one chemical species (which distorts lineshapes) and/or the involvement of Qy–Bx vibronic coupling (which adds unexpected y-polarized intensity at high Dn).
dominate the observed spectra. This problem is most noticeable for molecules with large DE and in fact correlates with the By to Qx splitting. Our band-fitting approach is less sensitive to these effects but is also in error owing to neglect of By–Qx coupling in the fitting model.
4. Discussion The qualitative determination of the general shape of the spectral polarization function, as revealed from our analytical
2320 | Phys. Chem. Chem. Phys., 2014, 16, 2315--2322
We have introduced an analytical data inversion technique enabling the interpretation of ‘‘B’’-term MCD spectra which provides a simple solution for one of the most fundamental problems in MCD spectroscopy: how to process unpolarized absorption and MCD data to obtain the implicit bandpolarization information. It allows the separation of two spectral bands of near-orthogonal polarization that themselves overlap but which are otherwise uninfluenced by either spectral overlap or vibronic coupling. However, this technique is only appropriate when the broadened origin lines of the two spectral bands do not overlap, allowing the MCD spectra to be dominated by the ‘‘B’’ term only. In general, MCD data inversion is two-fold underdetermined in these quite common circumstances. We show that the assumption of ‘‘B’’-term spectroscopy immediately yields one constraint, leaving only one quantity undetermined. We define a system variable whose value is bounded. Key features of the inverted spectra are shown to be insensitive to this variable. These include the energy gap between the overlapping transitions and the qualitative shape of the polarization profile. What remains difficult to determine is the absolute magnitude of the polarization fluctuations involved and the partitioning of the total oscillator strength. If the higher-energy transition is much stronger than the lower-energy one, then the identified bound become in fact a
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good approximation. In such circumstances, a qualitative spectral interpretation by eye is possible and the analytical data-inversion procedure provides a method to improved quantitative analysis. However, it is the scenario in which the higher-energy transition is weak enough so that spectral modeling is required in order to extract key qualitative information for which analytical data inversion offers the greatest gains. For the chlorophyllides, the higher-energy transition is indeed the weaker one and we show that analytical data inversion quickly leads to a definitive spectral assignment. For Chl-a, its Q-band is shown to be dominated by strong resonant vibronic coupling between the Qy and Qx states. Had our analytical data inversion been available 50 years ago, the spectrum of Chl-a would have been immediately assigned, a feat not achieved until recently, based on extensive vibronic-coupling simulations of a wide range of chlorophyllide MCD and other spectra.4 While chlorophyll (and porphyrin) spectroscopy is the focus here, the technique is not dependent on details other than the actual band shapes. It can thus be applied to any spectroscopic system in which two bands of nearly orthogonal polarization overlap but are otherwise isolated. In particular, it does not rely on the two transitions having a common parentage, as they do for chlorophylls. The excel spreadsheet provided in ESI† allows the easy application of this technology in order to rapidly analyze and assign absorption and MCD spectra. The procedure is applicable to many chemical systems, including those now commonly being obtained from XMCD.
Acknowledgements We thank the Australian Research Council Discovery Projects scheme for funding this research.
Notes and references 1 A. D. Buckingham and P. J. Stephens, Annu. Rev. Phys. Chem., 1966, 17, 399. 2 P. N. Schatz and A. J. McCaffery, Q. Rev., Chem. Soc., 1969, 23, 552. 3 P. J. Stephens, Annu. Rev. Phys. Chem., 1974, 25, 201. 4 J. R. Reimers, Z.-L. Cai, R. Kobayashi, M. Ratsep, A. Freiberg and E. Krausz, Sci. Rep., 2013, 3, 2761. 5 L. Salem, Molecular Orbital Theory of Conjugated Systems, Benjamin, Reading MA, 1972. 6 X.-N. Li, Z.-J. Wu, Z.-J. Si, H.-J. Zhang, L. Zhou and X.-J. Liu, Inorg. Chem., 2009, 48, 7740. 7 A. R. G. Smith, M. J. Riley, S. C. Lo, P. L. Burn, I. R. Gentle and B. J. Powell, Phys. Rev. B: Condens. Matter Mater. Phys., 2011, 83, 041105. 8 E. W. Thulstrup and J. Michl, J. Mol. Spectrosc., 1976, 61, 203. 9 S. J. Dammer, P. V. Solntsev, J. R. Sabin and V. N. Nemykin, Inorg. Chem., 2013, 52, 9496. ˇpa ´nek, M. Straka, V. Andrushchenko and P. Bourˇ, 10 P. ˇ Ste J. Chem. Phys., 2013, 138, 151103.
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Paper
11 J. P. Hare, H. W. Kroto and R. Taylor, Chem. Phys. Lett., 1991, 177, 394. 12 W. Q. Tian, D.-L. Chen, Y.-H. Cui and J.-K. Feng, J. Comput. Theor. Nanosci., 2009, 6, 239. 13 S. Gautam, P. Thakur, S. Augustine, J. K. Kang, J. Y. Kim, N. B. Brookes, K. Asokan and K. H. Chae, arXiv.org, e-Print Arch., Condens. Matter, 2011. 14 A. Briones-Leon, P. Ayala, X. Liu, K. Yanagi, E. Weschke, M. Eisterer, H. Jiang, H. Kataura, T. Pichler and H. Shiozawa, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 87, 195435. 15 W. R. Browett and M. J. Stillman, Inorg. Chim. Acta, 1981, 49, 69. 16 Z. Gasyna, D. H. Metcalf, P. N. Schatz, C. L. McConnell and B. E. Williamson, J. Phys. Chem., 1995, 99, 5865. ¨r, H. Land, E. Schnabel and H. Kuhn, Z. Elektrochem. 17 F. Ba Angew. Phys. Chem., 1961, 65, 346. 18 C. Houssier and K. Sauer, J. Am. Chem. Soc., 1970, 92, 779. 19 W. Haegele, D. Schmid, F. Drissler, J. Naus and H. C. Wolf, Z. Naturforsch., A: Phys. Phys. Chem. Kosmophys., 1978, 33, 1197. 20 M. Gouterman and L. Stryer, J. Chem. Phys., 1962, 37, 2260. 21 C. Weiss Jr, J. Mol. Spectrosc., 1972, 44, 37. 22 M. E. Deroche and J. M. Briantais, Photochem. Photobiol., 1974, 19, 233. 23 T. A. Evans and J. J. Katz, Biochim. Biophys. Acta, Bioenerg., 1975, 396, 414. 24 K. K. Rebane and R. A. Avarmaa, Chem. Phys., 1982, 68, 191. 25 R. A. Avarmaa and A. Suisalu, Opt. Spectrosc., 1984, 56, 54. 26 R. A. Avarmaa and K. K. Rebane, Spectrochim. Acta, Part A, 1985, 41, 1365. 27 M. Umetsu, Z.-Y. Wang, M. Kobayashi and T. Nozawa, Biochim. Biophys. Acta, Bioenerg., 1999, 1410, 19. 28 M. Umetsu, Z.-Y. Wang and T. Nozawa, Recent Res. Dev. Phys. Chem., 2001, 5, 185. 29 M. Umetsu, Z.-Y. Wang, K. Yoza, M. Kobayashi and T. Nozawa, Biochim. Biophys. Acta, Bioenerg., 2000, 1457, 106. 30 J. L. Hughes, B. Conlon, T. Wydrzynski and E. Krausz, Phys. Procedia, 2010, 3, 1591. ¨tsep, J. Linnanto and A. Freiberg, J. Chem. Phys., 2009, 31 M. Ra 130, 194501. 32 D. Caldwell, J. M. Thorne and H. Eyring, Annu. Rev. Phys. Chem., 1971, 22, 259. 33 M. T. Tiedemann and M. J. Stillman, J. Porphyrins Phthalocyanines, 2011, 15, 1134. 34 P. N. Schatz, A. J. McCaffery, W. Suetaka, G. N. Henning, A. B. Ritchie and P. J. Stephens, J. Chem. Phys., 1966, 45, 722. 35 B. R. Hollebone and M. J. Stillman, J. Chem. Soc., Faraday Trans. 2, 1978, 74, 2107. 36 W. R. Browett and M. J. Stillman, Comput. Chem., 1987, 11, 241. 37 J. Mack, Y. Asano, N. Kobayashi and M. J. Stillman, J. Inorg. Biochem., 2008, 102, 472. 38 D. Frackowiak, D. Bauman, H. Manikowski, W. R. Browett and M. J. Stillman, Biophys. Chem., 1987, 28, 101. 39 M. Umetsu, R. Seki, Z.-Y. Wang, I. Kumagai and T. Nozawa, J. Phys. Chem. B, 2002, 106, 3987.
Phys. Chem. Chem. Phys., 2014, 16, 2315--2322 | 2321
View Article Online
Published on 04 December 2013. Downloaded by University of Windsor on 25/10/2014 02:05:55.
Paper
40 J. L. Hughes, R. J. Pace and E. Krausz, Chem. Phys. Lett., 2004, 385, 116. 41 J. Linnanto, J. A. I. Oksanen and J. E. I. Korppi-Tommola, Phys. Chem. Chem. Phys., 2002, 4, 3061. 42 J. Linnanto, V. M. Helenius, J. A. I. Oksanen, T. Peltola, J. L. Garaud and J. E. I. Korppi-Tommola, J. Phys. Chem. A, 1998, 102, 4337. 43 C. H. Henry, S. E. Schnatterly and C. P. Slichter, Phys. Rev. Lett., 1964, 13, 130. 44 B. Briat, D. A. Schooley, R. Records, E. Bunnenberg and C. Djerassi, J. Am. Chem. Soc., 1967, 89, 6170. 45 M. Gouterman, J. Mol. Spectrosc., 1961, 6, 138. 46 R. L. Fulton and M. Gouterman, J. Chem. Phys., 1964, 41, 2280.
2322 | Phys. Chem. Chem. Phys., 2014, 16, 2315--2322
PCCP
47 R. L. Fulton and M. Gouterman, J. Chem. Phys., 1961, 35, 1059. 48 Y. Nonomura, S. Igarashi, N. Yoshioka and H. Inoue, Chem. Phys., 1997, 220, 155. ¨tsep, A. Freiberg 49 J. R. Reimers, Z.-L. Cai, R. Kobayashi, M. Ra and E. Krausz, Phys. Chem. Chem. Phys., 2013, DOI: 10.1039/ c3cp53729c. 50 M. R. Razeghifard, M. Chen, J. L. Hughes, J. Freeman, E. Krausz and T. Wydrzynski, Biochemistry, 2005, 44, 11178. ¨ld, in Artificial Photosynthesis: 51 E. Krausz and S. Peterson Årsko From Basic Biology to Industrial Application, ed. A. F. Collings and C. Critchley, Wiley, London, 2005, p. 87. 52 E. Krausz, N. Cox and S. P. Arskold, Photosynth. Res., 2008, 98, 207.
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