Photoinduced electron transfer plays a key role in various radiation processes in chemistry and biology, for example, in radiation damage of DNA. The most direct way to study ionization processes is based on photoelectron spectroscopy (PES) in the gas phase and solution. The gas-phase studies of chromophores allow one to get valuable information about their properties and dynamics [1, 2]. Liquid jet X-ray [3] and multiphoton UV [4, 5] photoelectron spectroscopy enables to study processes of photoinduced ionization of chromophores in solutions and to determine their vertical electron detachment energies (VDE) and vertical ionization energies (VIE) in case of anionic and neutral chromophores, respectively. However, interpretation of photoelectron spectra is fraught with difficulties, such as inelastic scattering and partial energy loss by electrons upon their interaction with solvent. Modern quantum chemistry methods can be used for calculating VDEs and VIEs as well as for interpreting experimental results; however, the need to account for a large number of water molecules, which interact with a charged/ionized chromophore, and the correct account of the polarization contribution of the solvent (especially when using high-level quantum chemistry methods) represent a serious computational problem.

One of the most common ways to explicitly take into account the environment is the non-empirical effective fragment potential method (EFP). The advantage of the EFP method is that it is a fairly accurate and at the same time computationally inexpensive way to describe the interaction of a solute with solvent molecules in the framework of a hybrid quantum mechanical approach [6, 7]. According to this method, the solute molecule is described at the quantum mechanical level, and the solvent molecules are represented as fragments with a fixed geometry, which are described by a set of parameters that determine the interaction of the solvent molecules with the solute. The influence of the environment on the quantum mechanical subsystem is described by the most important contributions of one-particle operators of intermolecular interaction, namely electrostatic, polarization, and exchange repulsion. The Coulomb interaction operator between two subsystems is represented as a multi-center multipole expansion of the electrostatic potential of each fragment up to the octupole moment. The polarization of fragments in the electric field of a quantum subsystem is described in terms of a self-consistent model. The response to the external field of each localized molecular orbital of the fragment is described by its own polarizability tensor, which is calculated in advance. The repulsive potential is represented as a linear combination of Gaussian functions centered on each atom of the fragment. All potential parameters of effective fragments are pre-calculated using non-empirical methods of quantum chemistry.

The EFP approach in combination with the coupled cluster EOM-IP-CCSD theory has previously been used to study the effect of hydration on the vertical ionization energy of thymine [8] and phenol [9], as well as on the vertical electron detachment energy of the phenolate anion [9] in water environment. The obtained estimate of the first ionization potential of phenol surrounded by a sphere of water molecules with a radius of 35 Å is shown to be in good agreement with the corresponding experimental value; however, the calculated vertical electron detachment energy of the phenolate anion in water differs from the experimental one by 0.6 eV. The difference between the calculated and experimental values is tentatively attributed by the authors to the incomplete consideration of the polarization contribution of the solvent, which has been treated perturbatively. A hybrid approach based on the density functional theory (DFT) and the EFP method has also been used for calculating the first ionization potential of phenol in combination with molecular dynamics sampling [10]. The vertical ionization energy calculated as the difference in total energies between hydrated phenol and its radical cation also agrees well with experimental data. Both approaches (EOM-IP-CCSD/EFP and DFT/EFP) are applicable for quantitative estimates of the ionization potentials of neutral molecules; however, they show a significant discrepancy with experiment for solvated anions.

In this paper, we propose a new theoretical approach for calculating vertical ionization and electron detachment energies of biologically relevant chromophores in an aqueous environment. Our approach considers the effects of specific solvation of the nearest water molecules bound by strong hydrogen bonds to charged chromophores, the effects of microsolvation as well as macrosolvation. The construction of a model system is carried out in several steps, using different methods to describe inner and outer water shells. The calculation of VDEs and VIEs is carried out using an extended version of the multiconfigurational quasi-degenerate perturbation theory XMCQDPT2 [11]. The approach is applied for calculating the VIE of hydrated phenol and the VDE of hydrated phenolate anion, which are typical model fragments of important biological chromophores, such as the chromophore of the green fluorescent protein (GFP) widely used as a fluorescent tag for monitoring biochemical processes in living cells [12].

EXPERIMENTAL

In the present work, we propose a methodology for obtaining equilibrium geometry configurations and for calculating vertical electron detachment and ionization energies of solvated chromophores, which includes the following main steps.

(1) Molecular dynamics simulations of a model system, which consists of a solute molecule placed in the center of a cube of water molecules with a length of 100 Å. Following equilibration in the canonical (NVT) ensemble for 2 ns, the system is gradually cooled down to 20 K with a step of 1 K for 560 ps. The molecular dynamics calculations are carried out using the NAMD software package [13].

(2) Geometry optimization of the internal solvation shell at the DFT/EFP level. The structure, which is obtained at the previous step, is reduced in size by cutting it into a smaller sphere of a radius of 17 Å (~1000 molecules). The quantum mechanical part includes a chromophore and its two nearest water molecules. The remaining water molecules are treated as effective fragments using the EFP method. The geometry of the resulting model system is optimized using the PBE0 functional and the (aug)-cc-pVDZ basis set in combination with the EFP method. Diffuse functions are added to the oxygen atoms to describe the excess electron density. All quantum chemistry calculations are carried out using the Firefly software package [14].

(3) Geometry optimization of the outer solvation shell at the DFT/EFP/MD level of theory. The model system, which is fully optimized using the DFT/EFP method at the previous step, is additionally solvated for molecular dynamics simulations with periodic boundary conditions. At this step, the geometry of the inner water shell is held fixed. The same molecular dynamics protocol is used here as at the first step. Following equilibration, the system is gradually cooled down, and the final geometry of the outer solvation shell of water molecules is optimized using the TIP3P force field parameters.

(4) Calculation of vertical ionization energy (or electron detachment energy) at the DFT/EFP level of theory. The resulting DFT/EFP/MD hybrid structure is cut into smaller-sized systems with a radius of water spheres around the solute ranging from 10 to 40 Å (a maximum of 11 250 water molecules). The vertical electron detachment (or ionization) energy is calculated as the difference between the total energies of the solvated anion (neutral molecule) and the corresponding radical in the geometry of the anion (neutral molecule). The calculations are carried out using the PBE0/(aug)-cc-pVDZ/EFP method.

(5) Hybrid XMCQDPT2/EFP method for calculating the energy of vertical ionization or electron detachment. Structures with a solvation water shell of a radius of 40 Å and an inner solvation shell containing ~1000 water molecules are used for the subsequent high-level calculations (Fig. 1). The zeroth order wave functions used in the second order multiconfigurational perturbation theory are constructed by the complete active space self-consistent field method (CASSCF). To simulate the processes of ionization and electron detachment, we use a modified (aug)-cc-pVDZ+ basis set, which is augmented by diffuse functions of a p type with a very small exponent of 10–10 (IP) and centered outside the solvation shell. In addition to all the π-orbitals of phenol and phenolate, IP orbitals are explicitly included in the active space of the CASSCF method. The active spaces include one, two, or three IP orbitals (giving rise to the CASSCF(8,8), CASSCF(8,9), CASSCF(8,10) levels, respectively). In these calculations, the state averaging procedure over ten SA(10)-CASSCF states is used, which involves both the ionized and valence states described simultaneously. All XMCQDPT2/EFP calculations are performed without taking into account the polarization contribution of the fragments. The polarization contribution of the solvent is estimated separately in a fully self-consistent manner at the DFT/EFP level of theory and is used as a correction to the calculated VDEs and VIEs.

Fig. 1.
figure 1

Model system used to calculate VDE and VIE at the XMCQDPT2/EFP level of theory with IP functions placed outside the water sphere.

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RESULTS AND DISCUSSION

The vertical electron detachment energy of the phenolate anion in aqueous solution is first calculated using the difference method at the DFT/EFP level of theory. The calculated VDE values are found to be strongly dependent on the size of the simulated system, reaching a converged value of 7.3 eV for a system with a solvation shell of a radius of 36 Å (Fig. 2). This is due to the fact that the results of the VDE calculations depend on the accuracy of estimating the absolute energy of long-range interactions between a negatively charged chromophore and polar water molecules.

Fig. 2.
figure 2

Calculated VDEs of aqueous phenolate as a function of the radius of the water sphere around the chromophore anion (r).

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The dependence of the calculated vertical ionization energy on the size of the solvation shell for hydrated phenol is shown in Fig. 3. The converged value of VIE is ~8.0 eV for a system with a solvation shell of a radius of 36 Å. At the same time, it should be noted that the calculated ionization energy of hydrated phenol depends less on the system size compared to the electron detachment energy of hydrated phenolate. This can be rationalized by the fact that the vertical detachment energy depends more strongly on the accuracy of estimating the absolute energy of long-range interactions between negatively charged phenolate and polar water molecules in the geometry of the anion than the vertical ionization energy of a neutral molecule in the corresponding geometry of the neutral.

Fig. 3.
figure 3

Calculated VIEs of aqueous phenol as a function of the radius of the water sphere around the chromophore (r).

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To calculate the vertical detachment energy of hydrated phenolate at the XMCQDPT2/EFP level of theory, we use a structure corresponding to a sphere with a radius of 40 Å. The results obtained are presented in Table 1.

Table 1.   Calculated VDEs of the hydrated phenolate anion at the XMCQDPT2/EFP level of theory using various active spaces of the SA(10)-CASSCF method. Corrected values account for the polarization contribution of the solvent, which equals to 0.43 eV
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As can be seen from Table 1, the VDE calculations using the XMCQDPT2/EFP theory give a value of 6.8–7.0 eV, which is lower than that obtained at the DFT/EFP level (7.3 eV). One obvious reason for this is the solvent polarization effects, which are completely neglected at the XMCQDPT2/EFP level of theory. Taking into account this contribution, which is calculated in a fully self-consistent manner at the DFT/EFP level of theory, the corrected VDE value becomes 7.2–7.4 eV. Therefore, both computational approaches give practically the same VDE for hydrated phenolate. The obtained value (7.3 ± 0.1 eV) is consistent with the experimental data obtained using X-ray (7.1 ± 0.1) [9] as well as multiphoton UV (7.02 ± 0.09) [15] photoelectron spectroscopy. It should be noted that the vertical detachment energy of the phenolate anion is ~5 eV lower in the gas phase compared to that of the hydrated anion in aqueous solution due to the electrostatic stabilization of the anion by the solvent. The polarization contribution of the solvent to the stabilization energy of the molecular anion in solution is significant and reaches 8% of the total solvent-induced shift.

The calculated VDE values of hydrated phenolate are found to be insensitive to the position of the IP orbitals outside the water sphere at distances larger than 10 Å, as well as to the direction along which they are located (along or opposite the direction of the dipole moment of the neutral molecular core). Three p-type IP orbitals that enter the active space give a threefold degenerate energy level for electron detachment. This indicates that the results obtained do not depend on the orientation of the p-type IP orbital relative to the molecular frame, as expected. In these calculations, exact degeneracy ensures that IP orbitals optimized using the SA-CASSCF procedure do not mix with other types of orbitals, which would otherwise mean that some weakly bound states of the non-valence type are formed, inevitably lowering the calculated value of the vertical detachment energy.

When the number of detached states explicitly considered in the framework of various active spaces changes, the number of valence states of the anion also changes if a total number of reference states included in the state averaging procedure is fixed. The nature of the SA(10)-CASSCF states, which span the XMCQDPT2 model space, affects the interaction and therefore mixing of the zeroth order wave functions following the diagonalization of the effective Hamiltonian calculated perturbatively. This can lead to a change in the transition energy between the ground state of the anion, which is of a valence type, and the first detached state. Indeed, the results obtained reveal a tendency for VDE to decrease by 0.2 eV, when the number of valence-type states included in the XMCQDPT2 model space is increased (Table 1).

To calculate the vertical ionization energy of hydrated phenol at the XMCQDPT2/EFP level of theory, as in the case of phenolate, we use a model system with a water shell around the solute of a radius of 40 Å. The largest active space (8,10) is used in these calculations. The calculated VIE of hydrated phenol, corrected for the polarization contribution (–0.39 eV), is 7.9 eV, which is in good agreement with the DFT/EFP value (8.0 eV), as well as with the experimental data obtained using X-ray (7.8 ± 0.1) [9] and multiphoton UV (7.76 ± 0.09) [15] photoelectron spectroscopy. The vertical ionization energy of phenol is 0.6 eV lower in aqueous solution than that in the gas phase. In the case of hydrated phenol, the polarization contribution of the solvent to the stabilization energy of the radical cation in solution reaches ~60% of the total solvent-induced shift. Therefore, the solvent response is significant and must be taken into account in order to obtain accurate quantitative estimates of vertical ionization energies as well as vertical detachment energies of biological chromophores in aqueous solution.

CONCLUSIONS

In this work, we propose a general theoretical approach for calculating vertical electron detachment and ionization energies of chromophores in aqueous solution. The new approach is based on the XMCQDPT2 high-level multiconfigurational quantum chemistry method in combination with the explicit treatment of solvent effects using the effective fragment potential method. The method allows one to predict VDEs and VIEs of hydrated chromophores with an accuracy no worse than 0.2 eV. The developed approach is tested on the hydrated phenolate anion. The calculated vertical detachment energy is shown to be strongly dependent on the size of the model system. The water sphere around the solute with a radius of 40 Å is required for obtaining an accurate VDE. It is also shown that the polarization contribution of the solvent to the VDE reaches a value of 0.4 eV for phenolate in aqueous solution, which is 8% of the total solvent-induced shift. The calculated VDE of the phenolate anion in aqueous solution, when corrected for the solvent response, equals to 7.3 ± 0.1 eV, which is in good agreement with the experimental data obtained using X-ray and multiphoton UV photoelectron spectroscopy. The new approach is also used for calculating the vertical ionization energy of phenol in aqueous solution. In this case, the solvent polarization contribution is found to reach 60% of the total solvent-induced shift. The calculated VIE of hydrated phenol, when corrected for the solvent response (–0.39 eV), equals to 7.9 eV, which agrees well with the experimental data. The developed approach will be useful for studying processes of photoinduced ionization, electron detachment, and electron transfer for both anionic and neutral molecular systems in aqueous solution.