Locally-Excited (LE) versus Charge-Transfer (CT) Excited State

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Locally-Excited (LE) vs. Charge-Transfer (CT) Excited State Competition in a Series of para-Substituted Neutral Green Fluorescent Protein (GFP) Chromophore Models Seth Olsen J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/jp508723d • Publication Date (Web): 24 Oct 2014 Downloaded from http://pubs.acs.org on October 27, 2014

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The Journal of Physical Chemistry

Locally-Excited (LE) vs. Charge-Transfer (CT) Excited State Competition in a Series of paraSubstituted Neutral Green Fluorescent Protein (GFP) Chromophore Models Seth Olsen* School of Mathematics & Physics, The University of Queensland, QLD 4072, Australia AUTHOR INFORMATION Corresponding Author *[email protected]

Abstract

In this paper, I provide a characterization of the low-energy electronic structure of a series of para-substituted neutral green fluorescent protein (GFP) chromophore models, using a theoretical approach that blends linear free energy relationships (LFERs) with state-averaged complete active space self-consistent field (SA-CASSCF) theory. The substituents are chosen to sample the Hammett σP scale from R=F to NH2, and includes a model of the neutral GFP 1 ACS Paragon Plus Environment


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chromphore (R=OH) structure. I analyze the electronic structure for different members of the series in a common complete active space valence-bond (CASVB) representation, exploiting an isolobal analogy between active space orbitals for different members of the series. I find that the electronic structure of the lowest adiabatic excited state is a strong mixture of weakly-coupled states with charge-transfer (CT) or locally-excited (LE) state character, and that the dominant character changes as the series is traversed. Chromophores with with strongly electrodonating substituents have a CT-like excited state such as expected for a push-pull polyene or asymmetric cyanine. Chromophores with weakly electron donating (or electron withdrawing) substituents have an LE-like excited state with ionic biradicaloid structure localized to the ground-state bridge π-bond.

KEYWORDS (Word Style “BG_Keywords”). Green fluorescent protein (GFP), state-averaged complete active space self-consistent field theory (SA-CASSCF), linear free energy relationships (LFERs), complete active space valence-bond (CASVB) theory, quasi-diabatic representations

Introduction The green fluorescent protein (GFP) and its relatives are important tools of modern biotechnology.1 The chromophore of GFP is a hydroxybenzylidene-imidazolinone motif, which is autocatalytically synthesized post-folding.2

Several GFPs display dual absorption bands

peaked near 397 and 475 nm. The states giving rise to these bands are called “A” (397nm) and

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“B” (475nm), and are assigned to a phenolic neutral form (Scheme 1, middle left) and its conjugate anionic base (Scheme 1, bottom). Scheme 1. pR-BDIs and example fluorescent protein chromophores

Interaction with the protein on its interior significantly alters the optical response of the chromophore motif, relative to solution-state behaviour.

The absorbance of the “A” state of

GFP (thought to have a neutral chromophore3) is at lower energy than the absorbance of pOHBDI (Scheme 1, top, with R=OH) in a range of liquid solutions4.

For the neutral pOH-BDI

model, the solution-state absorbance maximum varies in the small range 3.32 – 3.44 eV (360 – 373 nm) in solvents spanning a broad range of polarities and basicities.4

However, the

absorbance of the A state of GFP has its maximum at 3.12 eV (397 nm) – a position well removed from the absorbance of the same chromophore in a wide range of fluid media.4 A very recent measurement of the electronic gap of pOH-BDI yielded a value of 3.65eV.5 This is 0.23 ACS Paragon Plus Environment


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0.3eV above typical absorbance maxima in solution, and 0.5eV above the absorbance of the A state of GFP. The mechanism by which the protein exerts such a large effect on the absorbance remains a mystery. This mystery is further compounded by the observation that the gas-phase absorbance of anionic pO—BDI anion are actually quite close to the B state absorbance of GFP.6 An important prerequisite to understanding how fluorescent proteins modify the optical properties of their chromophore is to understand the electronic physics of the chromophore itself, identifying key electronic degrees of freedom that can modulate the spectral properties. One of the key parameters determining the spectra of GFPs is the proton occupancy of the titratable para-alcoholic center. In this paper I adopt the perspective of a physical organic chemist, and probe the response of the electronic structure to perturbations at the para site via the identification of substituent linear free energy relationships at that site.

The notion that

substituent LFERs at the para site can model changes in proton occupancy at the para-oxy site is supported by the fact that substituent scales exist that include both protonated (R=OH) and deprotonated (R=O–) substituents7, and that these scales can be correlated with color in dyes8. The electronic structure of pOH-BDI has been a challenge to theory9-11. Ab initio electronic structure models typically predict a gap that is too large by several tenths of an eV, relative to the value indicated by early gas phase experiments.10 A more recent gas-phase estimate appears to offer better agreement with many calculated values.5 The large change in absorbance seen in pOH-BDI in gas-phase, solvent and protein suggests a significant change in the electronic structure of the chromophore is involved. Multi-reference quasi-degenerate perturbation theory calculations by Bravaya et al. indicated significant remixing of the adiabatic SA-CASSCF reference spectrum in response to dynamical correlations9. In particular, remixing between the first excited state with a higher state of the


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reference was observed. This result was observed for multiple active spaces up to and including the full π-electron space9.

The remixing was accompanied by a lowering of the excitation

energy, improving agreement between the calculated excitation energies and gas-phase experiments.12 What electronic structures might one expect for the adiabatic excited state of a pR-BDI molecule? Relevant low-energy electronic structures for the hydrogen-substituted reference (defining the zero of the relevant Hammett substituent scale) are shown in Scheme 2. The states are described approximately using Lewis structures, as well as using a bra notation that is defined in Figure 1. For pH-BDI, one expects the adiabatic ground state to be unambiguously dominated by the unpolarized covalent GS structure at the bottom of Scheme 2. In addition, I consider two alternative excited-state electronic structures: a charge-transfer (CT) electronic structure and a locally-excited (LE) structure. The charge-transfer (CT) excited state is a covalent (bonding) state with opposing charge and bond localization to GS. Such a low-lying excited state is expected of methine (cyanine-like) dyes13. The corresponding CT structure appropriate for pH-BDI is drawn at the top of Scheme 2. Color-constitution theory8 says energy of excitation to the charge transfer transition will to increase as the difference in Brooker basicity13 of the aryl and imidazolol rings in the BDI structure is increased. The Brooker basicity difference is correlated with other measures of local charge stabilization, such as Hammett σ parameters.8 The CT state is the excited state one would expect of an asymmetric monomethine cyanine (or strongly polarized push-pull polyene).13



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Scheme 2. Expected low-lying electronic structures of pH-BDI

The condition of zero Brooker basicity difference is a useful definition of the concept of a “cyanine limit” for asymmetric monomethines, where the electronic structure is approximately symmetric even if the chemical structure is not14. The anionic state of the GFP chromophore motif has been shown to correspond to a state of near-zero Brooker basicity difference, meaning that it exists close to the limit where GS and CT have the same energy14-16. In this case, GS and CT contribute near-equally to the ground electronic state14-19. Protonation of the para-oxy site introduces a diabatic gap of order ~2.5eV, which is of the same order as the electronic resonance coupling15. Substitution with even less electron-donating para-substituents, such as in pH-BDI, will lower the frontier energy of the aryl ring even further20, resulting in a larger diabatic gap between GS and CT. For pH-BDI, where the diabatic gap exceeds the electronic coupling, we


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expect that the ground state should be unambiguously dominated by the GS structure in Scheme 2. The locally excited (LE) state is expected to have ionic biradicaloid character, such as would be expected for a short-chain polyene21. Chemical bonds are composite electronic structures, with internal energy scales.22 If the aryl ring becomes such a poor electron donor that the CT excitation exceeds these scales, then there must come a transition where the lowest electronic excitations will be those associated with internal excitation of an unambiguous ground-state πbond22. Such a locally excited (LE) structure would be dominated by hole-pair configurations of the bonding electrons, such as shown in the middle of Scheme 2.22 The character of this state is as expected for the excited state of a short-chain polyene.23 The CT and LE structures are expected, intuitively, to respond distinctly to the stabilization of charge on the benzyl ring. Specifically, CT should be destabilized with respect to both LE and GS as electronic charge on the benzyl ring is destabilized. This will occur as the Hammett σp parameter of the para substituent becomes more negative24. On the other hand, the excitation to LE from GS would be expected to show weak dependence on σp, because its formation does not require removal of charge from the aryl ring. The term “locally excited” has been applied in different contexts in the photochemical literature, and it is not always clear that it always means the same thing, in the sense of having a similar electronic structure for all cases where the term is used25-26. I could also have referred to the LE state in Scheme 2 as a “bond-excited” state (BE?), and this would have specifically highlighted its content. I decided to stick with the term “locally-excited” because, firstly, it is already known and vague enough to apply, and, secondly, it is correct in the sense that LE is an excitation of bond-pairing degrees of freedom local to the GS state.



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Theoretical Approach & Computational Models In order to examine the interplay of CT and LE states in pR-BDIs, I adopt a novel theoretical approach that blends linear free energy relationship (LFER) analysis with state-averaged complete active space self-consistent field (SA-CASSCF) electronic structure theory27-28. All quantum chemistry calculations used a cc-pvdz basis set29 and were performed using geometries obtained by optimization on the ground state potential energy surface of a MøllerPlesset 2nd order perturbation theory (MP2) model30, obtained with the same basis set. All calculations were done using the Molpro software package31. I adopt a SA-CASSCF model with four-electrons in three orbitals. This active space was chosen because it is the minimal complete quantum many-body model that can describe coherent transfer of an excess electronic charge and a covalent bond-pair. This level of active space descriptions has been shown to be reasonable for other bridged conjugated dyes32. In a previous study of a series of related dyes, I showed that the same 3-orbital, 4-electron CAS provided a more balanced description of the low-energy electronic structure than possible with larger active spaces short of the full π-electron space32. The reason is that the aromaticity of the rings changes for different structures involved, and these correlations are deep. They are sufficiently deep to break symmetry if they are not treated completely for both rings. The 4electron, 3-orbital CASSCF approximation was shown to constitute a “Pauling point”, where further increases in the size of the CAS made things worse before making them better. We would expect similar behavior here, since similar considerations apply (cf. Scheme 2). A key aspect of the theoretical approach is the use of a self-consistent thermostatistical equilibration of the classical probabilities (weights) in the SA-CASSCF.

This means that the

converged ensemble is a canonical thermostatistical (i.e. Boltzmann) state with the converged


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effective Hamiltonian as sufficient (quantum) statistic.

The advantages of weighting SA-

CASSCF in this way, when a diabatic representation of the results is desired, have been discussed in an earlier paper33. The calculations are parameterized by a Lagrange multiplier whose reciprocal β-1 has energy units and acts like an effective “temperature” for the SACASSCF CI density matrix33.

I use quotes throughout to distinguish this “temperature”

parameter from the physical temperature, which it is not. At equilibrium, the SA-CASSCF ensemble maximizes the von Neumann entropy consistent with a well-defined expected energy34. This implies that the relevant variational objective is no longer the average energy (as in microcanonical SA-CASSCF), but the free energy35. The free energy is a scalar at equilibrium, so that it is invariant to unitary transformations of the SA-CASSCF ensemble support. This gives a greater practical flexibility to self-consistently represent the SA-CASSCF results in an alternate representation33, which is what I am going to do. It is worth saying a few things about how to understand the “temperature” in these calculations, as this is a subtle thing that often confuses me. The Boltzmann CAS-CI density matrix is the least biased distribution consistent with an energy scale, given a sample CAS-CI Hilbert space of states.34 Instead of thinking in the usual way about state-averaging, as seeking to “balance” evenly over a set number of states, the weighting scheme used here balances according to an energy scale, instead of over a number of states. All of the states generated by the active space are included in the support of the ensemble, so the constraint on state number is as relaxed as it can be. The choice of energy scale should reflect the physics of interest (in this case, electronic excitation).

The energy scale used in the Figures in this paper was β-1=2.75±0.25eV, which

spans most of the emission band of the pH-BDI reference in benzene.36 The perspective can be



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compared to heat-engine models of lasers, where the relevant reservoir temperatures are set according to the photon energy of the relevant transitions in the lasing material.37

This

description is not unreasonable, because the electronic degrees of freedom implicated in Figure 1 should be among those with the strongest dipolar coupling to the field. For dyes with electron-donating substituents (R=OH, NH2), the CASSCF was stable for “temperatures” spanning the visible spectrum. For R=CH3, H and F, I noted that the CASSCF began to break down (in the sense of an abrupt change of orbital structure) for “temperatures” greater than 3.0eV, with the onset of failure getting “cooler” for more electron-withdrawing substituents. This is consistent with my principle result, which is that there is a polyene-like (ionic) excited state contributing to the low-energy electronic structure of these systems. This can be attributed to the known tendency for SA-CASSCF to fall over when covalent and ionic states are included with comparable weighting in the same self-consistent field, because the orbital structures of covalent and ionic states are different38. Thermostatistical equilibration of the SA-CASSCF classical probabilities was implemented as an iterative loop, using Molpro’s internal scripting language. The weights for each step were obtained from a Boltzmann CAS-CI density matrix using by the effective Hamiltonian from the previous step, and the process was iterated to convergence. The use of a canonical (i.e. constant particle number) ensemble is justified because the number of active electrons and orbitals is constant for all molecules in the series. The equilibrations were seeded with an initial guess generated using an evenly weighted 2-state calculation. Equilibrium was obtained to less than an nanohartree in the free energy, usually in less than 10 iterations. At convergence, the ensemble CAS-CI density matrix

has the Boltzmann form, with the effective Hamiltonian as sufficient

statistic:


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(0.1)

,

where f is the corresponding free energy (Lagrange multiplier conjugate to the identity matrix I, in units of β−1), and β is the Lagrange multiplier conjugate to the Hamiltonian matrix H. The reciprocal β-1 has units of energy and plays the role of an effective electronic “temperature” distinct from the physical temperature. The SA-CASSCF ensembles I use here have a rigorous interpretation as least-biased and minimal estimates of the structure of a charge/bond-transfer system embedded within pR-BDI, given only a priori information in the molecular structure, the one-particle basis set, the active space partition and an energy scale. The previous sentence is important, because it specifies exactly what questions my calculations have answered. The Boltzmann density matrix Equation (0.1) is the least-biased density matrix consistent with a given expected energy on the Hilbert space defined by the identity I34. A four-electron, three-orbital active space is the minimal active space that can completely describe coherent transfer of an excess electronic charge and a bondpair. The ensembles I use have support on the entire CAS-CI Hilbert space, so there is no constraint on the dimensionality apart from the identity on the CAS-CI space. In the calculations reported here, the electronic “temperature” was varied over a 0.5 eV range centered at 2.75 eV (451 nm). This range was chosen because it covers most of the emission envelope reported for the pH-BDI reference in benzene solution39. The thermostatistical ensembles we use in this paper are equilibrated with support on the complete CAS-CI Hilbert space. The state-averaging is over all states in the CAS-CI Hilbert space. For a four-electron, three-orbital CAS, this Hilbert space is six-dimensional, so there are six states in the support.

This allows transformation with the full range of unitary

transformations that can be defined on the CAS-CI Hilbert space, and allows representation of 11 ACS Paragon Plus Environment


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the converged ensemble in any electronic basis that can be defined on the CAS-CI Hilbert space33. In this paper, will use a complete active space valence-bond (CASVB) representation of singlet configurations over state-averaged Boys-localized40 orbitals. 41-42

I will refer to this basis as the

“CASVB basis”. The Boys orbital basis is useful because it has a clear and simple physical interpretation as the orbital set with maximally separated charge centers. A very accessible review of the nature and history of the Boys localization procedure has been given by Subotnik and co-authors43. We can make a physically meaningful association between the electronic structures shown in Scheme 1 and states in the CASVB Hilbert space. This association must be somewhat arbitrary, because of orbital delocalization due to orthogonality and also to coarse-graining of different charge-accomodating sites in the rings by the three-orbital SA-CASSCF model. Below, I rely heavily on the following (loose, but chemically defensible) association (which is even simpler than that suggested in Scheme 2):

CT ↔ 112 (0.2)

LE ↔ 220

.

GS ↔ 211 This association will be adequate for making the qualitative point that CT and LE are in close competition in this series of pR-BDIs, and is broadly consistent with the structures in Scheme 2. A more quantitative analysis would require a physically defensible specific representation of the electronic structures of Scheme 1 on the CAS-CI Hilbert space. The state-averaged Boys orbitals obtained for our series of pR-BDIs, calculated at “temperature” β−1=2.75 eV, are shown in Figure 1.

Orbitals calculated with other

“temperatures” in the range 2.5-3.0 eV were visually indistinguishable.

The orbitals are 12

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localized on the aryl ring, the bridge, and the imidazoloxy ring for each molecule.

This

electronic structure has been described before in similar calculations on arylmethane dyes32, 44-45, azomethine dyes44, 46 and other fluorescent protein chromophore models14-17, 19, 47.

Figure 1. (Top) Ensemble-averaged Boys-localized SA-CASSCF active space orbitals for pRBDIs. The orbitals were obtained by a CAS(4,3)SCF with “temperature” set to β−1=2.75eV. The orbitals have a qualitatively identical structure for all dyes in the set. This is the isolobal analogy referred to in the text. Orbitals are localized on the benzyl ring, the bridge, or the imidazoloxy ring, and a similar nodal structure. (Bottom) Ordering and labelling of CASVB states over the localized active orbitals, as discussed in the text, are shown. The analogy between the orbitals induces an analogy on the CASVB states, so that it is physically meaningful to associate the CASVB states for different dyes with a single abstract set of states spanning a shared electronic Hilbert space. This allows the classification of the SA-CASSCF solutions using the theory of linear free energy relationships (LFERs).



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It is clear from Figure 1 that an isolobal analogy48-49 exists between the active space orbitals of different pR-BDIs. The isolobal analogy allows unambiguous use of the same set of orbital labels for different molecules in the series. The product structure of the CAS-CI Hilbert space induces an isoconfigurational analogy on the CASVB basis, which I will exploit.

The

isoconfigurationally analogous CASVB states are shown at the bottom of Figure 1. In this paper, I will leverage the isolobal and isoconfigurational analogies between SACASSCF solutions to facilitate their analysis using linear free energy relationships (LFERs) from physical organic chemistry24, 50. This is because the analogies make it meaningful to discuss the state of any member of the series in the same abstract Hilbert space. Tbe identity on this Hilbert space is understood as (0.3)

,

where I R is the identity matrix on the CAS-CI Hilbert space calculated for the molecule with substituent R, and I is the identity on an abstract CSF Hilbert space common to all dyes and spanned as by the abstract CASVB states at the bottom of Figure 1. At a given “temperature”, the states of the molecules in this common Hilbert space obey independently their own thermodynamic identities, (0.4)

fR I = H R − β −1S R

,

for all substituents R. Here f R is the free energy for substituent R, H R is its self-consistent Hamiltonian,

is a surprisal whose expectation gives the von Neumann entropy.

Representing the states of different molecules on a common Hilbert space naturally defines concepts of sums and differences of the states of differently substituted molecules. This allows us to compare and discuss SA-CASSCF results for different molecules in the series using lineaer free energy relationships (LFERs), as traditionally used in physical organic chemistry24. To see 14 ACS Paragon Plus Environment

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this, take the difference of thermodynamic identities (0.5) for two molecules with substituents R and H to obtain

( fR − fH )I = (H R − H H ) − β −1 ( S R − S H )

(0.6)

Equation (0.6) defines a substituent parameter scale with the same origin as the Hammett σP parameter scale24,

50

.

The Hammett σP parameter for substituent R is the difference in

dissociation free energies of para-R substituted benzoic acid relative to unsubstituted (i.e. R=H) benzoic acid in water at 250C.24, 50 It is interpreted as a measure of the ability of the substituent to stabilize charge at the reaction centre via an inductive electronic interaction.24, 50 I found that the free energy scale obtained by applying (0.6) to the thermostatistical SACASSCF ensembles for the molecules in Scheme 1 gives strong linear correlations with the Hammett parameter σp. This underlies the analytical approach used throughout the paper. SA-CASSCF is well known to overestimate excitation energies, providing errors (with respect to more accurate calculations or experiments) as large as a few eV51. We would not expect the SA-CASSCF excitation energies to quantitatively match experimental absorbance peaks in any case. However, we can more reasonably evaluate the trend in the predicted excitation energies. This should be accurate as long as the “dynamical” correlations51 not described in the SACASSCF effect different dyes similarly. A typical protocol for the treatment of dynamical correlation effects would be to use a multi-reference perturbation theory correction to a zerothorder Hamiltonian constructed from the SA-CASSCF reference52. However, most available implementations do not preserve the invariance that allows transformation to the CASVB basis53. The results below support significant potential for adiabatic remixing in the low-energy spectra of these dyes, so the question of what perturbation theory is appropriate may be subtle. I avoid it 15 ACS Paragon Plus Environment


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here by sticking to the SA-CASSCF results, and accept that quantitative comparison is limited to substitution trends in the energetics, not the energetics themselves. Figure 2 shows a comparison of the predicted trend for the S1-S0 gap against a basket of absorbance maxima available for the models in different environments4, 36. Experimental data for synthetic models are taken from published studies studies by Tolbert and co-workers4, and by Jye-Shane and co-workers54, as well as from unpublished absorbance spectra provided generously by the authors of reference 36. Values for chromophores in the GFP protein55-56 are also shown. In this case the value for R=H was taken from the value for GFPY66H published by Heim and co-workers, while the remainder are for variants of the cycle 3 GFP used by Schultz and co-workers56 which has similar spectra to wild type when the chromophore is the same. In each of the environments for which experimental data is shown, there are at least two data points, so that a substitution trend can be defined and compared with the SA-CASSCF results. The black line is a linear fit (coefficient of determination R2=0.99) to the SA-CASSCF adiabatic first excitation energy, with y intercept set at the absorbance maximum of the pH-BDI reference in benzene36. A recent measurement of the gas-phase pOH-BDI falls just above the range of the plot.5 The experimental trend appears overall to be captured by the SA-CASSCF model. More accurate comparison would require more experimental data than is currently available in the literature. Comparisons should be conditioned by the recognition that I am not using a solvation model, while significant solvent effects occur for dyes with low-lying charge-transfer behaviour57.

Solvatochromic effects in this series of pR-BDIs are apparently subtle, with

significant solvatochromism reported for pNH2-BDI54, but only weak solvatochromism for pOHBDI4. Weaker solvatochromism would be expected for excitation to a state with LE character,


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since the solvent does not need to reorganize to accommodate a electronic charge transfer over several Å. A recent measurement of the gas-phase electronic gap of pOH-BDI yielded a value of 3.65eV.5 This is fortuitously close to an estimate that I have previously reported, using perturbation theory calculations on a SA-CASSCF model that was identical in structure to the one used here.17 This further supports the applicability of this SA-CASSCF model for these cases.

Figure 2. Comparison of the trend in the first SA-CASSCF adiabatic excitation for a series of pR-BDIs against absorbance maxima reported in the literature. Solid dots represent solution state absorbance maxima reported in the literature. For R=H,F,CH3 in benzene and in organic host-guest complexes, spectral data for comparison were generously provided by the authors of reference 3636. For each medium, there are at least two data points shown. Fit lines for different media are shown; in most cases these are fit to only two points. The slope of the solid black line was obtained by linear fit (coefficient of determination R2=0.9) to the lowest excitation energy predicted by the SA-CASSCF model equilibrated at β-1=2.75eV. The y-intercept of the line was set equal to the absorbance maximum of the pH-BDI reference in benzene.



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Results The diabatic energies of the CASVB states (cf. Figure 1, bottom) are shown in Figure 3, plotted against the Hammett substituent parameter σP. The variation in the diabatic energies is described quite well using (at most) linear functions of σP. The dependence of the diabatic energies on σP reflects the charge localization in the diabatic states. CASVB states with double electron occupation in the aryl ring orbital

( 211 , 220 ) are

destabilized by para substituents with progressively greater electrodonation power (lower σP ), while those with double occupation of the imidazolone ring with symmetrical occupation ( 121 , 202

( 112 , 022 )

are lowered. CSFs

) are nearly independent of σP.

The lowest-energy CASVB state for every molecule is 211 , which is associated with the GS state (cf. Scheme 2). The GS state is separated from the next higher by a significant gap. This indicates dominance, in the ground state, of the of the GS valence-bond structure (cf. Scheme 2). There is a large amount of diabatic congestion near the median of the diabatic spectrum. Four diabatic CASVB states clustered within a 2 eV window above the lowest diabatic gap for pHBDI, including , 121

220 , 202 and 112 in that order. The ordering of these diabatic

states changes for other dyes in the series. Figure 3 shows that the CASVB states 112 and 220 cross as σP is varied in the series. For pOH-BDI, the ordering of 202 and 112 have interchanged relative to the R=H reference. For pNH2-BDI, the ordering of 220 and 112 have interchanged, relative to R=OH. The 1,3-diradical state 121 is the lowest diabatic excited state for all dyes. However, it does not appreciably contribute to the adiabatic S1 or S0 states. The reason is that it is high in energy and more weakly coupled to the low-energy GS state than other CASVB states. As it happens, 18 ACS Paragon Plus Environment

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the low-energy electronic physics in these molecules turns out to be dominated by the larger dispersion of 202 (cf. Figure 4). The 1,3-biradicaloid 121 plays only a minor role in the space relevant to the low-energy absorbance band. In longer-chain polyenes, such a biradicaloid state can lead to fast depopulation of the optically excited state.58 I do not expect that here, because the bridge is short. I found in the calculations that 121 mixed significantly only into adiabatic states S2 and above. This was also upheld in larger active space calculations where the corresponding state could be reliably identified for comparison. Significant couplings between CASVB states are shown in Figure 4. These can be classified into two distinguishable groups. The first of these includes the couplings that stabilize the pairbonds in the left- and right-polarized covalent CSFs: 211 H 202 and 211 H 220 stabilize the covalent bond between the singly-occupied sites in

211 , while

202 H 112

and

022 H 112 do the same thing with 112 . These couplings are associated with the formation of stable chemical bonds by the singlet pairs in the low-energy polarized covalent states (cf. Figure 1). The other grouping includes matrix elements connecting the left and right covalent states

211 and 112 with the 1,3-diradical state 121 .

These couplings can be associated with

charge delocalization within the covalent CASVB states (cf. Figure 1). Figure 4 shows that the most significant (>1.0eV) couplings between CASVB states correspond to one-electron transfers between nearest neighbour orbitals. The direct inter-ring coupling was significantly less than this ( 211 H 112 ~0.4eV at β-1=2.75eV), which that approximations neglecting this coupling may be useful models. All other couplings were even smaller, with most CT transition should respond strongly to perturbation by polar environments, while the response of a GS->LE transition should respond weakly at most. I suspect now that the low range of solvatochromism observed in pOH-BDI (relative to, for example pNH2-BDI) can be qualitatively explained by supposing that the adiabatic electronic excited state has significant LE character in these solvents.4, 59 I also suspect now that the large red shift observed in GFP relative to both gas-phase and solution may be attributed to selective stabilization of the CT state, leading to depression of the adiabatic excited state energy and an increased weight of CT in the adiabatic excited state. This hypothesis is qualitatively supported by the existence of excited-state proton transfer reactions in GFPs, because the CT state would be expected to have lower basicity than the LE state for the pOH-BDI system.3,

60-61

A

mechanism for the selective stabilization of a CT state is readily available, because there is a conserved, presumed positively charged, arginine residue coordinating the imidazolone oxygen, which would coordinate negative partial charge in a CT state.2 However, there is also evidence that contraindicates CT character in the excited state of A GFPs: the conspicuous absence of a strong linear Stark effect, which would indicate a charge-transfer excitation.62



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The notion of competing CT and LE states provides context to the multi-state perturbation theory results obtained by Bravaya et al.9 for the pOH-BDI system. In these works the authors report significant remixing of the reference SA-CASSCF spectrum upon application of a multistate perturbation theory treatment. This result did not depend on the size of the active space chosen, and persisted even for active spaces spanning the π electron system9. Our own results are consistent with those results, because we find closely-lying ionic and covalent CASVB diabatic excited states. A CT state is expected to be dominated by the covalent configuration

211 , while the LE configuration will be dominated by 220 .

Dynamical correlation

treatments change the relative weights of covalent and ionic states in SA-CASSCF reference states51, and would be expected to cause remixing of CT and LE states in the adiabatic spectrum. Our results provide support and elaboration to the interesting results reported by Bravaya et al. for the pOH-BDI system9. Significant remixing of the low-energy spectra of pOH-BDI (and other molecules in Scheme 1) by dynamical correlations is plausible. The notion of competition between LE and CT states in highly asymmetric monomethines explains readily an earlier observation14, that SA-CASSCF results in these cases were not simultaneously consistent with diabatic representations defined using Platt’s electrochromic model63 or a generalized Mulliken-Hush (GMH) model64, while the two diabatizations were equivalent for near-resonant dyes. The problem, now clear, was that mixing of LE electronic structure in the SA-CASSCF adiabatic excited state was introducing spurious diabatic dipole coupling in the states defined using Platt’s model63. Since the states of a GMH model are defined by vanishing diabatic dipole coupling43, the constraints of both representations could not be simultaneously fulfilled.


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In symmetric cyanines or asymmetric cyanines near the cyanine limit, the charge-resonance in the ground and low-energy excited states is mediated via superexchange via “bridge” states, the more important of which might be associated with my CASVB state 202 .45, 65-66 It is not clear to what extent this state can be adiabatically correlated with the LE state I describe. It seems likely that the states would overlap in any reasonable explicit representation. For substituents with moderate electrodonation capacity, it seems plausible that the LE state may be understood as a deformed version of the bridge state of cyanines, appropriate to the limit of large detuning from the limit. In symmetric systems, the bridge state gives rise to an excitation at higher energies above the low-energy absorbance band, with orthogonal polarization.67 In the (strongly asymmetric) systems we are discussing here, this is not the case: I expect the transition dipole connecting the GS and LE states in Scheme 2 to lie along the bond.68 The application of the Hammett equation to charged substituent groups is subtle.69 However, a value of σp = -0.81 has been published for the oxide anion (R=O–) substituent7, which would correspond to a model of the anionic GFP chromophore. The idea that originally bore this paper was the notion that the Hammett σp parameter might be a proxy reaction coordinate for understanding photoinduced proton transfers to and from a titratable para-oxy site. I established in preliminary calculations that the analogous CASVB diabatic energies for R=O– could not be fit on the lines in Figure 3 using the published values of σp. This is consistent with the idea that the change in electronic structure between the neutral and anionic cases in Scheme 1 is not entirely inductive.



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Conclusion I have analyzed the low-energy electronic structure of a series of para-substituted GFP chromophore models, using a theoretical approach that blends linear free energy relationships (LFERs) with state-averaged complete active space self-consistent field (SA-CASSCF) theory. The approach involved self-consistent thermostatistical weighting of the SA-CASSCF ensemble state, followed by analysis of the SA-CASSCF effective Hamiltonians using the framework of linear free energy relationships, as taught and used in physical organic chemistry24. The LFER framework was elaborated using an isolobal analogy between actives spaces for the different members of the molecular series, at given SA-CASSCF “temperature”. The results I report suggest that BDIs with electron-donating para substituents are characterized by close competition between weakly coupled diabatic excited states of chargetransfer (CT) or locally-excited (LE) character. A CT-like excitation is expected of a push-pull polyene57,

70-71

or asymmetric Brooker (cyanine) dye63. The locally excited (LE) state is an

internal excitation of the bridge π-bond, and more characteristic of a polyene. I find that the dominant character of the adiabatic excited state turns over between R=NH2 and R=F. This region includes the case of the neutral GFP chromophore (R=OH), indicating particularly close competition in this case. The neutral GFP chromophore itself appears poised on the edge of where polyenes become polymethines.

ASSOCIATED CONTENT Cartesian coordinates (Å) for all molecular structures used, adiabatic state energies (hartree), state-averaged natural orbitals and occupations, such as would assist in reproduction of


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computations, and complete references 12 and 65 are available in a supplement. This material is available free of charge via the Internet at http://pubs.acs.org. Notes The authors declare no competing financial interests. ACKNOWLEDGMENT This work was supported by Australian Research Council Discovery Project Grant DP110101580.

Computations were carried out using the NCI facility at ANU using time

generously provided under Merit Allocation Scheme Grant m03. I thank R.H. McKenzie and T.J. Martínez for comments on an early version of the manuscript. I thank K. Solntsev, L.M. Tolbert, V. Ramamurthy and S.R. Samanta for providing absorbance data for the R=F,H and CH3 derivatives in benzene and in organic host-guest complexes, and for helpful discussion. REFERENCES 1.

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12. Rajput, J.; Rahbek, D. B.; Andersen, L. H.; Rocha-Rinza, T.; Christiansen, O.; Bravaya, K. B.; Erokhin, A. V.; Bochenkova, A. V.; Solntsev, K. M.; Dong, J., et al., Photoabsorption Studies of Neutral Green Fluorescent Protein Model Chromophores In Vacuo. Phys. Chem. Chem. Phys. 2009, 11, 9996-10002. 13. Brooker, L. G. S., Absorption and resonance in dyes. Rev. Mod. Phys. 1942, 14 (2-3), 275-293. 14. Olsen, S.; McKenzie, R. H., Bond Alternation, Polarizability, and Resonance Detuning in Methine Dyes. J. Chem. Phys. 2011, 134 (11), 114520. 15. Olsen, S., A Modified Resonance-Theoretic Framework for Structure−Property Relationships in a Halochromic Oxonol Dye. J. Chem. Theory Comput. 2010, 6 (4), 1089-1103. 16. Olsen, S.; McKenzie, R. H., A Dark Excited State of Fluorescent Protein Chromophores, Considered as Brooker Dyes. Chem. Phys. Lett. 2010, 492, 150-156. 17. Olsen, S.; McKenzie, R. H., A Two-State Model of Twisted Intramolecular ChargeTransfer in Monomethine Dyes. J. Chem. Phys. 2012, 137, 164319. 18. Olsen, S.; Lamothe, K.; Martínez, T. J., Protonic Gating of Excited-State Twisting and Charge Localization in GFP Chromophores: A Mechanistic Hypothesis for Reversible Photoswitching. J. Am. Chem. Soc. 2010, 132 (4), 1192-1193. 19. Olsen,

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Scheme 1. pR-BDIs (top) and green fluorescent protein (GFP) chromophores (bottom). 94x114mm (300 x 300 DPI)



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pR-BDIs and exemplary green fluorescent protein chromophores 85x106mm (300 x 300 DPI)


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Figure 1. (Top) Ensemble-averaged Boys-localized SA-CASSCF active space orbitals for pR-BDIs. The orbitals were obtained by a CAS(4,3)SCF with “temperature” set to β−1=2.75eV. The orbitals have a qualitatively identical structure for all dyes in the set. This is the isolobal analogy referred to in the text. Orbitals are localized on the benzyl ring, the bridge, or the imidazoloxy ring, and a similar nodal structure. (Bottom) Ordering and labelling of diabatic configurations over the localized active orbitals are shown. The analogy between the orbitals induces an analogy on the CASVB states, so that it is physically meaningful to associate the CASVB states for different dyes with a single abstract set of states spanning a shared electronic Hilbert space. This allows the classification of the SA-CASSCF solutions using the theory of linear free energy relationships (LFERs). 94x150mm (300 x 300 DPI)



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Comparison of the trend in the first SA-CASSCF adiabatic excitation for a series of pR-BDIs against absorbance maxima reported in the literature. Solid dots represent solution state absorbance maxima reported in the literature. For R=H,F,CH3 in benzene and in organic host-guest complexes, spectral data for comparison were generously provided by the authors of reference 3636. For each medium, there are at least two data points shown. Fit lines for different media are shown; in most cases these are fit to only two points. The slope of the solid black line was obtained by linear fit (coefficient of determination R2=0.9) to the lowest excitation energy predicted by the SA-CASSCF model equilibrated at β-1=2.75eV. The yintercept of the line was set equal to the absorbance maximum of the pH-BDI reference in benzene. 94x101mm (300 x 300 DPI)


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CASVB diabatic state energies (eV) for a series of pR-BDIs, obtained with the thermostatistical SA-CASSCF model (c.f. Figure 1) with “temperature” β-1=2.75eV. The diabatic basis is the basis of singlet configurations over Boys localized orbitals (c.f. Figure 1). Error bars show the range of variation as the “temperature” β−1 is scanned in the range 2.5-3.0eV. The trend with respect to σp reflects the physical identity of the CASVB states (c.f. Figure 1). 95x141mm (300 x 300 DPI)



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CASVB diabatic couplings (eV) for a series of pR-BDIs, obtained with the thermostatistical SA-CASSCF model (c.f. Figure 1) at electronic “temperature” β-1=2.75eV. The CASVB diabatic basis is the basis of singlet configurations over Boys-localized active orbitals (c.f. Figure 1). Error bars show the range of variation as the energy scale kT is varied in the range 2.5-3.0eV. Only couplings of magnitude are shown. The couplings are at most weakly dependent on the Hammett σp parameter. The couplings shown can be separated into distinct groups corresponding to charge delocalization (covalent-covalent) couplings and chemical bonding (covalent-ionic) couplings. Couplings corresponding to direct inter-ring transfer (i.e. not through the bridge) were of order 0.5eV at β-1=2.75eV, and are not shown. All other couplings were smaller. 95x153mm (300 x 300 DPI)


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Locally-Excited (LE) versus Charge-Transfer (CT) Excited State

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