Ab initio characterization of electron transfer coupling in photoinduced systems: generalized Mulliken-Hush with configuration-interaction singles

To calculate electronic couplings for photoinduced electron transfer (ET) reactions, we propose and test the use of ab initio quantum chemistry calculation for excited states with the generalized Mulliken-Hush (GMH) method. Configuration-interaction singles (CIS) is proposed to model the locally excited (LE) and charge-transfer (CT) states. When the CT state couples with other high lying LE states, affecting coupling values, the image charge approximation (ICA), as a simple solvent model, can lower the energy of the CT state and decouple the undesired high-lying local excitations. We found that coupling strength is weakly dependent on many details of the solvent model, indicating the validity of the Condon approximation. Therefore, a trustworthy value can be obtained via this CIS-GMH scheme, with ICA used as a tool to improve and monitor the quality of the results. Systems we tested included a series of rigid, sigma-linked donor-bridge-acceptor compounds where "through-bond" coupling has been previously investigated, and a pair of molecules where "through-space" coupling was experimentally demonstrated. The calculated results agree well with experimentally inferred values in the coupling magnitudes (for both systems studied) and in the exponential distance dependence (for the through-bond series). Our results indicate that this new scheme can properly account for ET coupling arising from both through-bond and through-space mechanisms.     

Distance dependence analysis of the electron transfer reactivity for the metastable FeOH2/FeOH2 system: an ab initio investigation

A novel theoretical scheme and ab initio application in discussing the electron transfer (ET) reactivity are presented in this paper, and are also calibrated in terms of the mono-hydrated iron ion system, Fe²⁺–OH₂/Fe³⁺–OH₂. The detailed geometry optimizations have been made at UMP2(full)/6-311+G^* level, and the activation geometrical configuration and the energy have been obtained at this level of theory using the activation model and the ab initio potential energy surface fitted from MP2(full)/6-311+G^* single point energies. The corresponding energy quantities (such as the activation energy, and dissociation energy) have also been obtained at different levels of theory (HF, MP2, MP3, MP4, QCISD and PUHF, PMP2 and PMP3 with the spin-projection) and a same basis set (6-311+G^*). The electron correlation calculations include the all electron correlation and the valence electron correlation. The electronic transmission coefficient is calculated using the ab initio potential energy surface slopes and the coupling matrix element determined from the two-state model and the Slater-type d-electron wave functions. The pair distribution function is calculated using two different schemes. Taking the pair distribution function and the local ET rate into account, a statistically averaged overall observed ET rate scheme and a spherically averaged local ET rate scheme are proposed. The relevant kinetic parameters are obtained in terms of these new schemes at different ab initio calculational levels. The contact distance dependence of these parameters and the applicability of the presented models and ab initio calculational method are also discussed.     

Electron spin-spin coupling from multireference configuration interaction wave functions

We present the implementation of two-electron spin-spin coupling as a quasidegenerate perturbative treatment of the Breit-Pauli spin-spin Hamiltonian. The evaluation is based on a multireference CI treatment and constitutes one of the first efforts in the calculation of this effect within a highly sophisticated consideration of both nondynamical and dynamical correlation. The extension of existing schemes for efficient calculation, in particular, of the spin-coupling elements necessitated some involved derivations, the outline of which is presented herein. Application of the program to calculations of diagonal as well as off-diagonal spin-coupling elements is illustrated with the test cases O(2) and NH.     

Triplet-triplet energy-transfer coupling: theory and calculation

Triplet-triplet (TT) energy transfer requires two molecular fragments to exchange electrons that carry different spin and energy. In this paper, we analyze and report values of the electronic coupling strengths for TT energy transfer. Two different methods were proposed and tested: (1) Directly calculating the off-diagonal Hamiltonian matrix element. This direct coupling scheme was generalized from the one used for electron transfer coupling, where two spin-localized unrestricted Hartree-Fock wave functions are used as the zero-order reactant and product states, and the off-diagonal Hamiltonian matrix elements are calculated directly. (2) From energy gaps derived from configuration-interaction-singles (CIS) scheme. Both methods yielded very similar results for the systems tested. For TT coupling between a pair of face-to-face ethylene molecules, the exponential attenuation factor is 2.59 A(-1)(CIS6-311+G(**)), which is about twice as large as typical values for electron transfer. With a series of fully stacked polyene pairs, we found that the TT coupling magnitudes and attenuation rates are very similar irrespective of their molecular size. If the polyenes were partially stacked, TT couplings were much reduced, and they decay more rapidly with distance than those of full-stacked systems. Our results showed that the TT coupling arises mainly from the region of close contact between the donor and acceptor frontier orbitals, and the exponential decay of the coupling with separation depends on the details of the molecular contacts. With our calculated results, nanosecond or picosecond time scales for TT energy-transfer rates are possible.     

Water-mediated electron transfer between protein redox centers

Recent experimental and theoretical investigations show that water molecules between or near redox partners can significantly affect their electron-transfer (ET) properties. Here we study the effects of intervening water molecules on the electron self-exchange reaction of azurin (Az), by performing a conformational sampling on the water medium and by using a newly developed ab initio method to calculate transfer integrals between molecular redox sites. We show that the insertion of water molecules at the interface between the copper active sites of Az dimers slightly increases the overall ET rate, while some favorable water conformations can considerably enhance the ET kinetics. These features are traced back to the interplay of two competing factors: the electrostatic interaction between the water and protein subsystems (mainly opposing the ET process for the water arrangements drawn from MD simulations) and the effectiveness of water in mediating ET coupling pathways. Such an interplay provides a physical basis for the found absence of correlation between the electronic couplings derived through ab initio electronic structure calculations and the related quantities obtained through the Empirical Pathways (EP) method. In fact, the latter does not account for electrostatic effects on the transfer integrals. Thus, we conclude that the water-mediated electron tunneling is not controlled by the geometry of a single physical pathway. We discuss the results in terms of the interplay between different ET pathways controlled by the conformational changes of one of the water molecules via its electrostatic influence. Finally, we examine the dynamical effects of the interfacial water and check the validity of the Condon approximation.     

About the calculation of exchange coupling constants using density-functional theory: the role of the self-interaction error

The effect of the correction of the self-interaction error on the calculation of exchange coupling constants with methods based on density-functional theory has been tested in simple model systems. The inclusion of the self-interaction correction cancels the nondynamical correlation energy contributions simulated by the commonly used functionals. Hence, such correction should be important in the accurate determination of exchange coupling constants. We have also tested several recent functionals to calculate exchange coupling constants in transition-metal complexes, such as meta-GGA functionals or new formulations of hybrid functionals. The influence of the basis set and of the use of pseudopotentials on the calculated J values has also been evaluated for a Fe(III) dinuclear complex in which the paramagnetic centers bear several unpaired electrons.     

Improving the calculation of Electron Paramagnetic Resonance hyperfine coupling tensors for d-block metals

Calculation of hyperfine coupling constants (HFCs) of Electron Paramagnetic Resonance from first principles can be a beneficial complement to experimental data in cases where the molecular structure is unknown. We have recently investigated basis set convergence of HFCs in d-block complexes and obtained a set of basis functions for the elements Sc-Zn, which were saturated with respect to both the Fermi contact and spin-dipolar components of the hyperfine coupling tensor [Hedeg?rd et al., J. Chem. Theory Comput., 2011, 7, 4077-4087]. Furthermore, a contraction scheme was proposed leading to very accurate, yet efficient basis sets for the elements Sc-Zn. Here this scheme is tested against a larger test set of molecules and a wider range of DFT functionals. We further investigate the regular aug-cc-pVTZ and core-valence correlation aug-cc-pCVTZ basis sets as well as another core-property basis set, CP(PPP). While aug-cc-pVTZ-J provides hyperfine coupling constants that are almost identical to the converged series (aug-cc-pVTZ-Juc), we observe that not only the regular but also the core-valence correlation basis sets provide results far from the converged results. The usage of specialized core-basis sets leads to a large and highly significant improvement of the calculated hyperfine couplings in comparison with experimental data.     

MS-CASPT2 calculation of excess electron transfer in stacked DNA nucleobases

Calculations using the complete active space self-consistent field (CASSCF) and complete active space second-order perturbation (CASPT2) methods, and the multistate formulation of CASPT2 (MS-CASPT2), are performed for the ground and excited states of radical anions consisting of two pi-stacked nucleobases. The electronic couplings for excess electron transfer (EET) in the pi-stacks are estimated by using the generalized Mulliken-Hush approach. We compare results obtained within the different methods with data derived using Koopmans' theorem approximation at the Hartree-Fock level. The results suggest that although the one-electron scheme cannot be applied to calculate electron affinities of nucleobases, it provides reasonable estimates for EET energies. The electronic couplings calculated with KTA lie between the CASPT2 and the MS-CASPT2 based values in almost all cases.     

Pathway analysis of super-exchange electronic couplings in electron transfer reactions using a multi-configuration self-consistent field method

We present a novel pathway analysis of super-exchange electronic couplings in electron transfer reactions using localized molecular orbitals from multi-configuration self-consistent field (MCSCF) calculations. In our analysis, the electronic coupling and the tunneling pathways can be calculated in terms of the configuration interaction (CI) Hamiltonian matrix obtained from the localized MCSCF wave function. Making use of the occupation restricted multiple active spaces (ORMAS) method can effectively produce the donor, acceptor, and intermediate configuration state functions (CSFs) and CIs among these CSFs. In order to express the electronic coupling as a sum of individual tunneling pathways contributions, we employed two perturbative methods: L?wdin projection-iteration method and higher-order super-exchange method. We applied them to anion couplings of butane-1,4-diyl and pentane-1,5-diyl. The results were (1) the electronic couplings calculated from the two perturbative methods were in reasonable agreement with those from a non-perturbative method (one-half value of the energy difference between the ground and first excited states), (2) the main tunneling pathways consisted of a small number of lower-order super-exchange pathways where bonding, anti-bonding, or extra-valence-shell orbitals were used once or twice, and (3) the interference among a huge number of higher-order super-exchange pathways significantly contributed to the overall electronic coupling, whereas each of them contributed only fractionally. Our method can adequately take into account both effects of non-dynamical electron correlation and orbital relaxation. Comparing with the analyses based on the Koopmans' theorem (ignoring both effects) and the ORMAS-CIs from frozen localized reference orbitals (ignoring the effect of orbital relaxation), we discuss these effects.     

Estimates of electronic coupling for excess electron transfer in DNA

Electronic coupling V(da) is one of the key parameters that determine the rate of charge transfer through DNA. While there have been several computational studies of V(da) for hole transfer, estimates of electronic couplings for excess electron transfer (ET) in DNA remain unavailable. In the paper, an efficient strategy is established for calculating the ET matrix elements between base pairs in a pi stack. Two approaches are considered. First, we employ the diabatic-state (DS) method in which donor and acceptor are represented with radical anions of the canonical base pairs adenine-thymine (AT) and guanine-cytosine (GC). In this approach, similar values of V(da) are obtained with the standard 6-31G(*) and extended 6-31+ +G(**) basis sets. Second, the electronic couplings are derived from lowest unoccupied molecular orbitals (LUMOs) of neutral systems by using the generalized Mulliken-Hush or fragment charge methods. Because the radical-anion states of AT and GC are well reproduced by LUMOs of the neutral base pairs calculated without diffuse functions, the estimated values of V(da) are in good agreement with the couplings obtained for radical-anion states using the DS method. However, when the calculation of a neutral stack is carried out with diffuse functions, LUMOs of the system exhibit the dipole-bound character and cannot be used for estimating electronic couplings. Our calculations suggest that the ET matrix elements V(da) for models containing intrastrand thymine and cytosine bases are essentially larger than the couplings in complexes with interstrand pyrimidine bases. The matrix elements for excess electron transfer are found to be considerably smaller than the corresponding values for hole transfer and to be very responsive to structural changes in a DNA stack.     

The electronic structure of the photoexcited triplet state of free-base (tetraphenyl)porphyrin by time-resolved electron-nuclear double resonance and density functional theory

The photoexcited triplet states of free-base porphyrin (H(2)P) and free-base tetraphenylporphyrin (H(2)TPP) have been investigated by time-resolved electron paramagnetic resonance and electron-nuclear double resonance in a toluene glass at 80 K. Both the zero-field splitting parameters, D and E, and the proton A(zz) hyperfine coupling tensor components could be determined. D is about 13% larger in H(2)P than in H(2)TPP. In contrast, however, the A(zz) hyperfine coupling tensor components showed differences of less than 2%. To aid the understanding of these results, the electronic structures of H(2)P and H(2)TPP have been modeled using density functional theory. The geometrical structures of both molecules in their lowest triplet states were calculated using the Becke3 Lee-Yang-Parr composite exchange correlation functional and the 6-31G* basis set. Hyperfine couplings for these structures were calculated using the same functional but with the extended EPR-II basis set. These allow unambiguous assignment of the experimentally determined couplings. The theoretical values for H(2)P and H(2)TPP agree with the experimental values in that the presence of the phenyl groups has only a small effect on the unpaired electron spin-density distribution. The difference in sensitivity of the zero-field splitting parameters and the hyperfine couplings to mesophenyl substitution is discussed in terms of the wave functions of the four frontier orbitals of porphyrins introduced by Gouterman.     

Localized orbital corrections for the calculation of ionization potentials and electron affinities in density functional theory

This paper describes the extension of a previously reported empirical localized orbital correction model to the correction of ionization potential energies (IP) and electron affinities (EA) for atoms and molecules of first and second row elements. The B3LYP localized orbital correction version of the model (B3LYP-LOC) uses 22 heuristically determined parameters that improve B3LYP DFT IP and EA energy calculations on the G2 data set of 134 molecules from a mean absolute deviation (MAD) from experiment of 0.137 to 0.039 eV. The method significantly reduces the number of outliers and overall MAD to error levels below that achieved with G2 wave function based theory; furthermore, the new model has zero additional computational cost beyond standard DFT calculations. Although the model is heuristic and is based on a multiple linear regression to experimental errors, each of the parameters is justified on physical grounds, and each provides insight into the fundamental limitations of DFT, most importantly the failure of current DFT methods to accurately account for nondynamical electron correlation.     

Reaction barrier heights from an exact-exchange-based density-functional correlation model

A recent exact-exchange-based density-functional model of nondynamical and dynamical correlation [A.D. Becke, J. Chem. Phys. 122, 064101 (2005)] is tested on 70 barrier heights for a variety of reaction types: hydrogen transfer reactions, heavy-atom transfer reactions, nucleophilic substitutions, association reactions, and unimolecular rearrangements, including both even- and odd-electron systems. The mean absolute error with respect to accurate reference data is 1.4 kcal/mol. This is achieved without any refitting of the parameters of the model to the barrier height data.     

Linking the historical and chemical definitions of diabatic states for charge and excitation energy transfer reactions in condensed phase

Marcus theory of electron transfer (ET) and Fo?rster theory of excitation energy transfer (EET) rely on the Condon approximation and the theoretical availability of initial and final states of ET and EET reactions, often called diabatic states. Recently [Subotnik et al., J. Chem. Phys. 130, 234102 (2009)], diabatic states for practical calculations of ET and EET reactions were defined in terms of their interactions with the surrounding environment. However, from a purely theoretical standpoint, the definition of diabatic states must arise from the minimization of the dynamic couplings between the trial diabatic states. In this work, we show that if the Condon approximation is valid, then a minimization of the derived dynamic couplings leads to corresponding diabatic states for ET reactions taking place in solution by diagonalization of the dipole moment matrix, which is equivalent to a Boys localization algorithm; while for EET reactions in solution, diabatic states are found through the Edmiston-Ruedenberg localization algorithm. In the derivation, we find interesting expressions for the environmental contribution to the dynamic coupling of the adiabatic states in condensed-phase processes. In one of the cases considered, we find that such a contribution is trivially evaluable as a scalar product of the transition dipole moment with a quantity directly derivable from the geometry arrangement of the nuclei in the molecular environment. Possibly, this has applications in the evaluation of dynamic couplings for large scale simulations.     

Hartree-Fock-Heitler-London method. 2. First and second row diatomic hydrides

The Hartree-Fock-Heitler-London, HF-HL, method is a new ab initio approach which variationally combines the Hartree-Fock, HF, and the Heitler-London, HL, approximations, yielding correct dissociation products. Furthermore, the new method accounts for nondynamical correlation and explicitly considers avoided crossing. With the HF-HL model we compute the ground-state potential energy curves for H2 [1Sigma+g], LiH [X 1Sigma+], BeH [2Sigma+], BH [1Sigma+], CH [2Pi], NH [3Sigma-], OH [2Pi], and FH [1Sigma+], obtaining in average 80% of the experimental binding energy with a correct representation of bond breaking. Inclusion of ionic configurations improves the computed binding energy. The computed dipole moment is in agreement with laboratory data. The dynamical and nondynamical correlation energies for atomic and molecular systems with 2-10 electrons are analyzed. For BeH the avoided crossing of the two lowest [2Sigma+] states is considered in detail. The HF-HL function is proposed as the zero-order reference wave function for molecular systems. To account for the dynamical correlation energy a post-HF-HL technique based on multiconfiguration expansions is presented. We have computed the potential energy curves for H2 [1Sigma+g], HeH [2Sigma+], LiH [X1Sigma+], LiH [A1Sigma+], and BeH [2Sigma+]. The corresponding computed binding energies are 109.26 (109.48), 0.01 (0.01), 57.68 (58.00), 24.19 (24.82), and 49.61 (49.83) kcal/mol, with the experimental values given in parentheses. The corresponding total energies are -1.1741, -3.4035, -8.0695, -7.9446, and -15.2452 hartrees, respectively, the best ab initio variational published calculations, H2 excluded.     

Mechanism of intramolecular electron transfer in the photoexcited Zn-substituted cytochrome c: theoretical and experimental perspective

Photoinduced electron transfer (ET) in zinc-substituted cytochrome c (Zn-cyt c) has been utilized in many studies on the long-range ET in protein. Attempting to understand its ET mechanism in terms of electronic structure of the molecule, we have calculated an all-electron wave function for the ground-state of Zn-cyt c on the basis of density functional theory (DFT). The four molecular orbitals (MOs) responsible for excitation by UV-vis light (Gouterman's 4-orbitals) are assigned on the basis of the excited states of chromophore model for Zn-porphine complex calculated with the time-dependent DFT method. ET rates between each Gouterman's 4-orbitals and other MOs were estimated using Fermi's golden rule. It appeared that the two occupied MOs of the 4-orbitals show exclusively higher ET rate from/to particular MOs that localize on outermost amino acid residues (Lys 7 or Asn 54), respectively, whereas ET rates involving the two unoccupied MOs of the 4-orbitals are much slower. These results imply that the intramolecular ET in photoexcited Zn-cyt c is governed by the hole transfer through occupied MOs. The couplings of MOs between zinc porphyrin core and specific amino acid residues on the protein surface have been demonstrated in Zn-cyt c immobilized on an Au electrode via carboxylic acid group-terminated self-assembled monolayer. The Zn-cyt c-modified electrode showed photocurrents responsible for photoillumination. The action spectrum of the photocurrent was identical with the absorption spectrum of Zn-cyt c, indicating photoinduced electron conduction via occupied MOs. The voltage dependence of the photocurrent appeared to be linear and bidirectional like a photoconductor, which strongly supports the intramolecular ET mechanism in Zn-cyt c proposed on the basis of the theoretical calculations.     

Analysis of electron correlation effects and contributions of NMR J-couplings from occupied localized molecular orbitals

NMR J-coupling calculations at the second-order of polarization propagator approach, SOPPA, are among the most reliable. They include a high percentage of the total electron correlation effects in saturated and unsaturated molecular systems. Furthermore, J-couplings are quite sensitive to the whole electronic molecular framework. We present in this article the first study of all three response mechanisms, Fermi contact, FC, spin-dipolar, SD and paramagnetic spin-orbital, PSO, for J-couplings with occupied localized molecular orbitals at the SOPPA level of approach. Even though SOPPA results are not invariant under unitary transformations, the difference between results obtained with canonical and localized molecular orbitals, LMOs, are small enough to permit its application with confidence. The following small-size saturated and unsaturated compounds were analyzed: CH(4), CH(3)F, C(2)H(6), NH(3), C(2)H(4), CH(2)NH, H(2)C═CHF, and FHC═CHF. The local character of the FC mechanism that appears in J-couplings of these molecular models is shown through the analysis of contributions from LMOs. The importance of including the electron correlation on the engaged bonding orbitals for one-bond couplings is emphasized. Almost all electron correlation effects are included in such orbitals. Interesting findings were the large contributions by s-type LMOs to the C-H and C-C J-couplings; they are responsible for the variation of (1)J(C-C) when going from ethane to ethene and to 1,2-difluoroethene. The previously proposed hyperconjugative transfer mechanism has been tested. Among other tests we found the difference anti-syn of one-bond (1)J(C-H) in imine as due to both the corresponding σ(C-H) and the lone-pair, LP, contribution. Geminal and vicinal J-couplings were also analyzed. Our findings are in accord with a previous work by Pople and Bothner-by, who considered results taken from calculations or empirical data. For all geminal couplings the pattern of J-couplings, like the change of sign, is originated in the main bondings that participate in the coupling pathways. The finding of asymmetric contributions of LP to vicinal H-H couplings in imine is highlighted. The analysis of J-couplings by contributions from LMOs to the noncontact mechanisms, SD and PSO, show that the π electronic framework makes both terms grow in the specific case of the model compounds studied here. The PSO mechanism is more efficient when a σ bond is vicinal to a π bond. We found in this way an efficient and powerful scheme to get a deeper insight on the electronic molecular framework on which J-couplings are transmitted.