^{1,a)}and Ross H. McKenzie

^{1}

### Abstract

We give a quantum chemical description of the photoisomerization reaction of green fluorescent protein (GFP) chromophores using a representation over three diabatic states. Photoisomerization leads to nonradiative decay, and competes with fluorescence in these systems. In the protein, this pathway is suppressed, leading to fluorescence. Understanding the electronic states relevant to photoisomerization is a prerequisite to understanding how the protein suppresses it, and preserves the emitting state of the chromophore. We present a solution to the state-averaged complete active space problem, which is spanned at convergence by three fragment-localized orbitals. We generate the diabatic-state representation by block diagonalization transformation of the Hamiltonian calculated for the anionic chromophore model HBDI with multireference, multistate perturbation theory. The diabatic states are charge localized and admit a natural valence-bond interpretation. At planar geometries, the diabatic picture of the optical excitation reduces to the canonical two-state charge-transfer resonance of the anion. Extension to a three-state model is necessary to describe decay via *two* possible pathways associated with photoisomerization of the (methine) bridge. Parametric Hamiltonians based on the three-state ansatz can be fit directly to data generated using the underlying active space. We provide an illustrative example of such a parametric Hamiltonian.

This work was supported by the Australian Research Council (ARC) Discovery under Project No. DP0877875. All computations were carried out at the National Computational Infrastructure (NCI) National Facility in Canberra. We thank the NCI staff for their assistance and expertise. Time on the NCI machines was generously provided through a Merit Allocation Scheme (MAS) Grant (Project No. M03). Some of the figures were generated using VMD,^{101}MATHEMATICA,^{102} and CHEMBIODRAW ULTRA.^{103} We thank N.S. Hush, A. Jacko, T.J. Martínez, J. Michl, B. Powell, L. Radom, H.F. Schaefer III, S. Shaik, and M. Smith, for helpful discussions. We also thank J.R. Reimers and M.J.T. Jordan for many constructive suggestions on a draft manuscript.

I. INTRODUCTION

II. QUANTUM CHEMISTRY CALCULATIONS

III. THE BLOCK-DIAGONALIZATION TRANSFORMATION

IV. STRUCTURE OF THE DIABATIC STATES

V. ENERGY EIGENSTATES IN THE DIABATIC REPRESENTATION

VI. THE LOWER-BLOCK EFFECTIVE HAMILTONIAN

VII. IMPLICATIONS FOR MODEL DEVELOPMENT

VIII. RELATIONSHIP TO OTHER MODELS

IX. LIMITS OF APPLICABILITY

X. CONCLUSION

### Key Topics

- Proteins
- 29.0
- Excited states
- 11.0
- Chemical bonds
- 10.0
- Electronic structure
- 10.0
- Charge transfer
- 6.0

## Figures

The charge-transfer resonance of HBDI anion at two isomeric geometries. The resonance superposes structures wherein location of a formal anionic charge is correlated with double bond alternation.

The charge-transfer resonance of HBDI anion at two isomeric geometries. The resonance superposes structures wherein location of a formal anionic charge is correlated with double bond alternation.

The electronic structure solution that motivates the ansatz. There is a solution to the three-state-averaged four-electron-in-three-orbital variational problem which, at convergence, yields an active space spanned by localized fragment orbitals (top) on the phenoxy, methine bridge, and imidazolinone fragments. Over this orbital space are built six singlet CSFs (bottom). The covalent configurations, each of which supports one doubly occupied orbital and one singlet pair, correlate location of the charge with bond alternation (as in the resonance shown in Fig. 1). The ionic configurations can be created by polarizing the singlet pairs of the covalent configurations. Double headed arrows are used to highlight the relationship. Note the analogy to carbogenic orbitals and valence-bond structures of an allyl anion.

The electronic structure solution that motivates the ansatz. There is a solution to the three-state-averaged four-electron-in-three-orbital variational problem which, at convergence, yields an active space spanned by localized fragment orbitals (top) on the phenoxy, methine bridge, and imidazolinone fragments. Over this orbital space are built six singlet CSFs (bottom). The covalent configurations, each of which supports one doubly occupied orbital and one singlet pair, correlate location of the charge with bond alternation (as in the resonance shown in Fig. 1). The ionic configurations can be created by polarizing the singlet pairs of the covalent configurations. Double headed arrows are used to highlight the relationship. Note the analogy to carbogenic orbitals and valence-bond structures of an allyl anion.

The electronic structure of the diabatic states of HBI anion at representative geometries: the minimum of the isomer , the phenoxy-twisted minimum of the isomer , an imidazolinone-twisted minimum , and a disrotatory (hula) twisted structure relaxed on under bridge torsion constraints . At each geometry, the one-electron basis has been localized using the Boys procedure (left). The Hamiltonian is block diagonalized using a unitary transformation to yield three diabatic states, whose density matrices in the localized orbital CSF basis are shown (matrix element labels at bottom). Diabatic states labeled , , and are dominated by CSFs with double occupation of the phenoxy , bridge , and imidazolinone orbitals. The figure shows that the diabatic representation is transferrable across geometries with substantially different bridge torsion.

The electronic structure of the diabatic states of HBI anion at representative geometries: the minimum of the isomer , the phenoxy-twisted minimum of the isomer , an imidazolinone-twisted minimum , and a disrotatory (hula) twisted structure relaxed on under bridge torsion constraints . At each geometry, the one-electron basis has been localized using the Boys procedure (left). The Hamiltonian is block diagonalized using a unitary transformation to yield three diabatic states, whose density matrices in the localized orbital CSF basis are shown (matrix element labels at bottom). Diabatic states labeled , , and are dominated by CSFs with double occupation of the phenoxy , bridge , and imidazolinone orbitals. The figure shows that the diabatic representation is transferrable across geometries with substantially different bridge torsion.

Inferring appropriate Lewis structures for the diabatic states, from the structure of their density matrices in the configuration state basis. (Top) At a planar geometry, using as an example the ground state geometry of the isomer . At planar geometries, both the and diabats contain ionic contributions indicative of chemical bonding. The state contains substantially less ionic contribution, which contraindicates the presence of a chemical bond. (Bottom) At a twisted geometry, using as an example the imidazolinone-twisted excited-state minimum . At twisted geometries, the states with singlet pairing across the twisted bond ( and in this case) contain virtually no ionic contribution, indicating a diradical structure. The state with singlet pairing across a planar bond ( in this case) maintains ionicity appropriate for a chemical bond.

Inferring appropriate Lewis structures for the diabatic states, from the structure of their density matrices in the configuration state basis. (Top) At a planar geometry, using as an example the ground state geometry of the isomer . At planar geometries, both the and diabats contain ionic contributions indicative of chemical bonding. The state contains substantially less ionic contribution, which contraindicates the presence of a chemical bond. (Bottom) At a twisted geometry, using as an example the imidazolinone-twisted excited-state minimum . At twisted geometries, the states with singlet pairing across the twisted bond ( and in this case) contain virtually no ionic contribution, indicating a diradical structure. The state with singlet pairing across a planar bond ( in this case) maintains ionicity appropriate for a chemical bond.

Application of the twin state model to the planar and twisted configurations of HBDI, using the ground state minimum of the isomer and the imidazolinone-twisted minimum as examples. The twin state picture is usually invoked for two-state systems, such as the resonating Kekulé structures of benzene. The model can be applied in its normal sense to describe the first excitation of HBDI, but cannot be applied within the same set of structures to the geometries appropriate to an excited-state diradical. At these configurations, the and states are no longer twins, but the and states are.

Application of the twin state model to the planar and twisted configurations of HBDI, using the ground state minimum of the isomer and the imidazolinone-twisted minimum as examples. The twin state picture is usually invoked for two-state systems, such as the resonating Kekulé structures of benzene. The model can be applied in its normal sense to describe the first excitation of HBDI, but cannot be applied within the same set of structures to the geometries appropriate to an excited-state diradical. At these configurations, the and states are no longer twins, but the and states are.

Potential energy surfaces of the three lowest singlet states yielded by fitting a parametric functional form for the effective Hamiltonian to a collection of geometries obtained by minimization on the first excited adiabatic state under constraint of the bridge torsion angles. The resulting surfaces (top) show favorable twisting of both bonds on the surface. Furthermore, the charge localization that accompanies excited-state twisting is described by the distribution of diabatic populations in the state. This is possible because the diabatic states have charge-localized character which is maintained over the surface. Details of the fitting procedure are described in the Supplement.

Potential energy surfaces of the three lowest singlet states yielded by fitting a parametric functional form for the effective Hamiltonian to a collection of geometries obtained by minimization on the first excited adiabatic state under constraint of the bridge torsion angles. The resulting surfaces (top) show favorable twisting of both bonds on the surface. Furthermore, the charge localization that accompanies excited-state twisting is described by the distribution of diabatic populations in the state. This is possible because the diabatic states have charge-localized character which is maintained over the surface. Details of the fitting procedure are described in the Supplement.

## Tables

Comparison of published estimates of the splitting of HBDI and HBI anion obtained with SA-CASSCF and MR-RSPT2 methods, and the peak of the gas-phase absorption spectrum of HBDI. Sources are indicated in the far right column.

Comparison of published estimates of the splitting of HBDI and HBI anion obtained with SA-CASSCF and MR-RSPT2 methods, and the peak of the gas-phase absorption spectrum of HBDI. Sources are indicated in the far right column.

Density matrix elements of the diabatic state, represented in the basis of CSFs over fragment orbitals on the phenoxy , methine bridge , and imidazolinone , at representative geometries of HBDI anion. Geometries include ground state minima of the and isomers , phenoxy-twisted minima of the and isomers , an imidazolinone-twisted minimum and a constrained, -relaxed hula-twist geometry . Regardless of geometry, is dominated by the CSF with a doubly occupied phenoxy fragment orbital.

Density matrix elements of the diabatic state, represented in the basis of CSFs over fragment orbitals on the phenoxy , methine bridge , and imidazolinone , at representative geometries of HBDI anion. Geometries include ground state minima of the and isomers , phenoxy-twisted minima of the and isomers , an imidazolinone-twisted minimum and a constrained, -relaxed hula-twist geometry . Regardless of geometry, is dominated by the CSF with a doubly occupied phenoxy fragment orbital.

Density matrix elements of the diabatic state, represented in the basis of CSFs over the fragment orbitals on the phenoxy , methine bridge , and imidazolinone , at representative geometries of HBDI anion. Geometries include ground state minima of the and isomers , phenoxy-twisted minima of the and isomers , an imidazolinone-twisted minimum and a constrained, -relaxed hula-twist geometry . Regardless of geometry, is dominated by the CSF with a doubly occupied methine bridge fragment orbital.

Density matrix elements of the diabatic state, represented in the basis of CSFs over the fragment orbitals on the phenoxy , methine bridge , and imidazolinone , at representative geometries of HBDI anion. Geometries include ground state minima of the and isomers , phenoxy-twisted minima of the and isomers , an imidazolinone-twisted minimum and a constrained, -relaxed hula-twist geometry . Regardless of geometry, is dominated by the CSF with a doubly occupied methine bridge fragment orbital.

Density matrix elements of the diabatic state, represented in the basis of CSFs over the fragment orbitals on the phenoxy , methine bridge , and imidazolinone , at representative geometries of HBD I anion. Geometries include ground state minima of the and isomers , phenoxy-twisted minima of the and isomers , an imidazolinone-twisted minimum and a constrained, -relaxed hula-twist geometry . Regardless of geometry, is dominated by the CSF with a doubly occupied imidazolinone fragment orbital.

Density matrix elements of the diabatic state, represented in the basis of CSFs over the fragment orbitals on the phenoxy , methine bridge , and imidazolinone , at representative geometries of HBD I anion. Geometries include ground state minima of the and isomers , phenoxy-twisted minima of the and isomers , an imidazolinone-twisted minimum and a constrained, -relaxed hula-twist geometry . Regardless of geometry, is dominated by the CSF with a doubly occupied imidazolinone fragment orbital.

Density matrix elements of the lowest adiabatic state at representative geometries of HBDI anion. Geometries include ground state minima of the and isomers , phenoxy-twisted minima of the and isomers , an imidazolinone-twisted minimum and a constrained, -relaxed hula-twist geometry .

Density matrix elements of the lowest adiabatic state at representative geometries of HBDI anion. Geometries include ground state minima of the and isomers , phenoxy-twisted minima of the and isomers , an imidazolinone-twisted minimum and a constrained, -relaxed hula-twist geometry .

Density matrix elements of the first excited adiabatic state at representative geometries of HBDI anion. Geometries include ground state minima of the and isomers , phenoxy-twisted minima of the and isomers , an imidazolinone-twisted minimum and a constrained, -relaxed hula-twist geometry .

Density matrix elements of the first excited adiabatic state at representative geometries of HBDI anion. Geometries include ground state minima of the and isomers , phenoxy-twisted minima of the and isomers , an imidazolinone-twisted minimum and a constrained, -relaxed hula-twist geometry .

Density matrix elements of the second excited adiabatic state at representative geometries of HBDI anion. Geometries include ground state minima of the and isomers , phenoxy-twisted minima of the and isomers , an imidazolinone-twisted minimum and a constrained, -relaxed hula-twist geometry .

Density matrix elements of the second excited adiabatic state at representative geometries of HBDI anion. Geometries include ground state minima of the and isomers , phenoxy-twisted minima of the and isomers , an imidazolinone-twisted minimum and a constrained, -relaxed hula-twist geometry .

Elements of the lower block effective Hamiltonian at representative geometries of HBDI anion. Geometries include ground state minima of the and isomers , phenoxy-twisted minima of the and isomers , an imidazolinone-twisted minimum and a constrained, -relaxed hula-twist geometry . All energies are in kcal/mol and are referenced to the mean along the diagonal evaluated at the geometry.

Elements of the lower block effective Hamiltonian at representative geometries of HBDI anion. Geometries include ground state minima of the and isomers , phenoxy-twisted minima of the and isomers , an imidazolinone-twisted minimum and a constrained, -relaxed hula-twist geometry . All energies are in kcal/mol and are referenced to the mean along the diagonal evaluated at the geometry.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content