Robert Send

First-order nonadiabatic couplings from time-dependent hybrid density functional response theory: Consistent formalism, implementation, and performance

First-order nonadiabatic coupling matrix elements (NACMEs) are key for phenomena such as nonradiative transitions and excited-state decay, yet a consistent and practical first principles treatment has been elusive for molecules with more than a few heavy atoms. Here we present theory, implementation using Gaussian basis sets, and benchmarks of first-order NACMEs between ground and excited states in the framework of time-dependent hybrid density functional theory (TDDFT). A time-dependent response approach to NACMEs which avoids explicit computation of excited-state wave functions is outlined. In contrast to previous approaches, the present treatment produces exact analytical derivative couplings between time-dependent Kohn-Sham (TDKS) determinants in a finite atom-centered basis set. As in analytical gradient theory, derivative molecular orbital coefficients can be eliminated, making the computational cost independent of the number of nuclear degrees of freedom. Our expression reduces to the exact Chernyak-Mukamel formula for first-order NACMEs in the complete basis-set limit, but greatly improves basis-set convergence in finite atom-centered basis sets due to additional Pulay type terms. The Chernyak-Mukamel formula is shown to be equivalent to the Hellmann-Feynman contribution in analytical gradient theory. Our formalism may be implemented in TDDFT analytical excited-state gradient codes with minor modifications. Tests for systems with up to 147 atoms show that evaluation of first-order NACMEs causes total computation times to increase by an insignificant 10% on average. The resolution-of-the-identity approximation for the Coulomb energy (RI-J) reduces the computational cost by an order of magnitude for nonhybrid functionals, while errors are insignificant with standard auxiliary basis sets. We compare the computed NACMEs to full configuration interaction (FCI) in benchmark results for diatomic molecules; hybrid TDDFT and FCI are found to be in agreement for regions of the potential energy curve where the Kohn-Sham ground-state reference is stable and the character of the excitation is properly captured by the present functionals. With these developments, nonadiabatic molecular dynamics simulations of molecular systems in the 100 atoms regime are within reach.

Electronic Excitations of Simple Cyanine Dyes: Reconciling Density Functional and Wave Function Methods

The simplest cyanine dye series [H2N(CH)nNH2](+) with n = 1, 3, 5, 7, and 9 appears to be a challenge for all theoretical excited-state methods since the experimental spectra are difficult to predict and the observed deviations cannot be easily explained with standard arguments. We compute here the lowest vertical excitation energies of these dyes using a variety of approaches, namely, complete active space second-order perturbation theory (CASPT2), quantum Monte Carlo methods (QMC), coupled cluster linear response up to third approximate order (CC3), and various flavors of time-dependent density functional theory (TDDFT), including the recently proposed perturbative correction scheme (B2PLYP). In our calculations, all parameters such as basis set, active space, and geometry dependence are carefully analyzed. We find that all wave function methods give reasonably close excitation energies, with CASPT2 yielding the lowest values, and that the B2PLYP scheme gives excitations in satisfactory agreement with CC3 and DMC, significantly improving on the generalized gradient and hybrid approximations. Finally, to resolve the remaining discrepancy between predicted excitation energies and experimental absorption spectra, we also investigate the effect of excited-state relaxation. Our results indicate that a direct comparison of the experimental absorption maxima and the theoretical vertical excitations is not possible due to the presence of nonvertical transitions. The apparent agreement of earlier CASPT2 calculations with experiments was an artifact of the choice of active space and the use of an older definition of the zero-order Hamiltonian.

Reduction of the virtual space for coupled-cluster excitation energies of large molecules and embedded systems

We investigate how the reduction of the virtual space affects coupled-cluster excitation energies at the approximate singles and doubles coupled-cluster level (CC2). In this reduced-virtual-space (RVS) approach, all virtual orbitals above a certain energy threshold are omitted in the correlation calculation. The effects of the RVS approach are assessed by calculations on the two lowest excitation energies of 11 biochromophores using different sizes of the virtual space. Our set of biochromophores consists of common model systems for the chromophores of the photoactive yellow protein, the green fluorescent protein, and rhodopsin. The RVS calculations show that most of the high-lying virtual orbitals can be neglected without significantly affecting the accuracy of the obtained excitation energies. Omitting all virtual orbitals above 50 eV in the correlation calculation introduces errors in the excitation energies that are smaller than 0.1 eV. By using a RVS energy threshold of 50 eV, the CC2 calculations using triple-ζ basis sets (TZVP) on protonated Schiff base retinal are accelerated by a factor of 6. We demonstrate the applicability of the RVS approach by performing CC2/TZVP calculations on the lowest singlet excitation energy of a rhodopsin model consisting of 165 atoms using RVS thresholds between 20 eV and 120 eV. The calculations on the rhodopsin model show that the RVS errors determined in the gas-phase are a very good approximation to the RVS errors in the protein environment. The RVS approach thus renders purely quantum mechanical treatments of chromophores in protein environments feasible and offers an ab initio alternative to quantum mechanics/molecular mechanics separation schemes.

Benchmarking the Approximate Second-Order Coupled-Cluster Method on Biochromophores.

Extensive benchmarking calculations are presented to assess the accuracy of commonly used quantum chemical methods in studying excited state properties of biochromophores. The first few excited states of 12 common model chromophores of photoactive yellow protein, green fluorescent protein, and rhodopsin have been studied using approximate second-order coupled-cluster (CC2) and linear-response time-dependent density functional theory (TDDFT) calculations. The study comprises investigations of basis-set dependences on CC2 excitation energies as well as comparisons of the CC2 results with excitation energies obtained at other computational levels and with experimental data. The basis-set study shows that the accuracy of the two lowest excitation energies is generally sufficient when triple-ζ basis sets augmented with polarization functions are employed, whereas the third and higher excited states were found to require diffuse basis functions in the basis set. Augmenting the basis set with diffuse functions contributes less than 0.15 eV to the excitation energies of low-lying excited states, except for some of the studied anionic states and for Rydberg states. Calculations at the TDDFT level using the B3LYP functional show the necessity of stabilizing anions with point charges or counterions when aiming at reliable electronic excitation spectra. The two lowest excitation energies of the green fluorescent protein and rhodopsin chromophores calculated at the CC2 level agree within 0.15 eV with experimental excitation energies, whereas the B3LYP values are somewhat less accurate, with a maximum deviation of 0.27 eV. The computed excitation energies for the photoactive yellow protein chromophore models deviate from available experimental values by 0.3-0.4 eV and 0.1-0.5 eV, at the CC2 and B3LYP levels of theory, respectively.

Coupled-cluster and density functional theory studies of the electronic excitation spectra of trans-1,3-butadiene and trans-2-propeniminium

The electronic excitation spectra of trans-1,3-butadiene (CH(2)=CH-CH=CH(2)) and trans-2-propeniminium (CH(2)=CH-CH=NH(2)(+)) have been studied at several coupled-cluster and time-dependent density functional theory levels using the linear response approach. Systematic studies employing large correlation-consistent basis sets show that approximate singles and doubles coupled-cluster calculations yield excitation energies in good agreement with experiment for all states except for the two lowest excited A(g) states of trans-1,3-butadiene which have significant multiconfigurational character. Time-dependent density functional theory calculations employing the generalized gradient approximation and hybrid functionals yield too low excitation energies in the basis set limit. In trans-1,3-butadiene, increasing the basis set size by augmenting multiple diffuse functions is observed to reduce the high-lying excitation energies with most density functionals. The decrease in the energies is connected to the incorrect asymptotic behavior of the exchange-correlation potential. The results also demonstrate that standard density functionals are not capable of providing excitation energies of sufficient accuracy for experimental assignments.

The Effect of Protein Environment on Photoexcitation Properties of Retinal

Retinal is the photon absorbing chromophore of rhodopsin and other visual pigments, enabling the vertebrate vision process. The effects of the protein environment on the primary photoexcitation process of retinal were studied by time-dependent density functional theory (TDDFT) and the algebraic diagrammatic construction through second order (ADC(2)) combined with our recently introduced reduction of virtual space (RVS) approximation method. The calculations were performed on large full quantum chemical cluster models of the bluecone (BC) and rhodopsin (Rh) pigments with 165-171 atoms. Absorption wavelengths of 441 and 491 nm were obtained at the B3LYP level of theory for the respective models, which agree well with the experimental values of 414 and 498 nm. Electrostatic rather than structural strain effects were shown to dominate the spectral tuning properties of the surrounding protein. The Schiff base retinal and a neighboring Glu-113 residue were found to have comparable proton affinities in the ground state of the BC model, whereas in the excited state, the proton affinity of the Schiff base is 5.9 kcal/mol (0.26 eV) higher. For the ground and excited states of the Rh model, the proton affinity of the Schiff base is 3.2 kcal/mol (0.14 eV) and 7.9 kcal/mol (0.34 eV) higher than for Glu-113, respectively. The protein environment was found to enhance the bond length alternation (BLA) of the retinyl chain and blueshift the first absorption maxima of the protonated Schiff base in the BC and Rh models relative to the chromophore in the gas phase. The protein environment was also found to decrease the intensity of the second excited state, thus improving the quantum yield of the photoexcitation process. Relaxation of the BC model on the excited state potential energy surface led to a vanishing BLA around the isomerization center of the conjugated retinyl chain, rendering the retinal accessible for cis-trans isomerization. The energy of the relaxed excited state was found to be 30 kcal/mol (1.3 eV) above the minimum ground state energy, and might be related to the transition state of the thermal activation process.

Electrostatic spectral tuning mechanism of the green fluorescent protein

Understanding the mechanism of spectral tuning of biological chromophores is a major challenge in photobiology. We show here using large-scale full quantum chemical calculations of the green fluorescent protein that state-of-the-art coupled-cluster calculations provide accurate excitation energies and detailed insight about specific environmental effects. We obtain vertical excitation energies of 3.13 eV (396 nm) and 2.68 eV (463 nm), which are in quantitative agreement with the experimental absorption energies of 3.12 eV (397 nm) and 2.61 eV (475 nm) for the A- and B-forms of the protein. We find that the protein environment redshifts the absorption spectra by ∼0.56 eV and ∼0.22 eV for the two states, which can be attributed to ∼80% electrostatic effects and ∼20% steric effects.

Coupled-cluster studies of the lowest excited states of the 11-cis-retinal chromophore.

The first few excited states of the 11-cis-retinal (PSB11) chromophore have been studied at the coupled-cluster approximative singles and doubles (CC2) level using triple-zeta quality basis sets augmented with double sets of polarisation functions. The two lowest vertical excitation energies of 2.14 and 3.21 eV are in good agreement with recently reported experimental values of 2.03 and 3.18 eV obtained in molecular beam measurements. Calculations at the time-dependent density functional theory (TDDFT) level using the B3LYP hybrid functional yield vertical excitation energies of 2.34 and 3.10 eV for the two lowest states. Zero-point vibrational energy (ZPVE) corrections of -0.09 and -0.17 eV were deduced from the harmonic vibrational frequencies for the ground and excited states calculated at the density functional theory (DFT) and TDDFT level, respectively, using the B3LYP hybrid functional.

Coupled-Cluster Studies of Extensive Green Fluorescent Protein Models Using the Reduced Virtual Space Approach

Accurate predictions of photoexcitation properties are a major challenge for modern methods of theoretical chemistry. We show here how approximate coupled-cluster singles and doubles (CC2) calculations in combination with the reduced virtual space (RVS) approach can be employed in studies of excited states of large biomolecular systems. The RVS-CC2 approach is used for accurately predicting optical properties of the p-hydroxybenzylidene-dihydroimidazolinone (p-HBDI) chromophore embedded in green fluorescent protein (GFP) models using quantum mechanical calculations in combination with large basis sets. We study the lowest excited states for the isolated and protein-embedded chromophore in two different protonation states, and show how omitting high-lying virtual orbitals in the RVS calculation of excitation energies renders large-scale CC2 studies computationally feasible. We also discuss how the error introduced by the RVS approach can be systematically estimated and controlled. The obtained CC2 excitation energies of 3.13-3.27 and 2.69-2.77 eV for the two protonation states of different protein models are in excellent agreement with the maxima of the experimental absorption spectra of 3.12-3.14 and 2.61-2.64 eV, respectively. Thus, the calculated energy splitting between the excited states of the two protonation states is 0.44-0.52 eV, which agrees very well with the experimental value of 0.48-0.51 eV. The calculations at the RVS-CC2 level on the protein models show the importance of using large QM regions in studies of biochromophores embedded in proteins.

Protein-Induced Color Shift of Carotenoids in β-Crustacyanin†

β-Crustacyanin (β-CR) is a pigment protein responsible for the blue color of lobsters. We show using correlated ab initio calculations how the protein environment tunes the chromophores of β-CR through electrostatic and steric effects.