Analytical Hessian of electronic excited states in time-dependent density functional theory with Tamm-Dancoff approximation

We present the analytical expression and computer implementation for the second-order energy derivatives of the electronic excited state with respect to the nuclear coordinates in the time-dependent density functional theory (TDDFT) with Gaussian atomic orbital basis sets. Here, the Tamm-Dancoff approximation to the full TDDFT is adopted, and therefore the formulation process of TDDFT excited-state Hessian is similar to that of configuration interaction singles (CIS) Hessian. However, due to the replacement of the Hartree-Fock exchange integrals in CIS with the exchange-correlation kernels in TDDFT, many quantitative changes in the derived equations are arisen. The replacement also causes additional technical difficulties associated with the calculation of a large number of multiple-order functional derivatives with respect to the density variables and the nuclear coordinates. Numerical tests on a set of test molecules are performed. The simulated excited-state vibrational frequencies by the analytical Hessian approach are compared with those computed by CIS and the finite-difference method. It is found that the analytical Hessian method is superior to the finite-difference method in terms of the computational accuracy and efficiency. The numerical differentiation can be difficult due to root flipping for excited states that are close in energy. TDDFT yields more exact excited-state vibrational frequencies than CIS, which usually overestimates the values.     

Analytical approach for the excited-state Hessian in time-dependent density functional theory: formalism, implementation, and performance

The paper presents the formalism, implementation, and performance of the analytical approach for the excited-state Hessian in the time-dependent density functional theory (TDDFT) that extends our previous work [J. Liu and W. Z. Liang, J. Chem. Phys. 135, 014113 (2011)] on the analytical Hessian in TDDFT within Tamm-Dancoff approximation (TDA) to full TDDFT. In contrast to TDA-TDDFT, an appreciable advantage of full TDDFT is that it maintains the oscillator strength sum rule, and therefore yields more precise results for the oscillator strength and other related physical quantities. For the excited-state harmonic vibrational frequency calculation, however, full TDDFT does not seem to be advantageous since the numerical tests demonstrate that the accuracy of TDDFT with and without TDA are comparable to each other. As a common practice, the computed harmonic vibrational frequencies are scaled by a suitable scale factor to yield good agreement with the experimental fundamental frequencies. Here we apply both the optimized ground-state and excited-state scale factors to scale the calculated excited-state harmonic frequencies and find that the scaling decreases the root-mean-square errors. The optimized scale factors derived from the excited-state calculations are slightly smaller than those from the ground-state calculations.     

Combined theoretical and experimental deep-UV resonance raman studies of substituted pyrenes

The results of time-dependent density functional theory (TDDFT) calculations of resonance Raman intensities are combined with experimental deep-ultraviolet resonance Raman measurements at a single wavelength, i.e., 244 nm, in order to test the possibility to distinguish several very similar compounds. Pyrene and three of its substituted derivatives, in which a single hydrogen atom has been replaced by a halogen atom, are compared. The fixed 244 nm excitation wavelength overlapped with the same electronic transition of the four pyrenes. Ground-state calculations using the BP86 exchange-correlation functional were used to predict the Raman frequencies, whereas excited-state calculations have been carried out employing the "statistical averaging of (model) orbital potentials" (SAOP) potential within a linear-response TDDFT framework in combination with the short-time approximation of resonance Raman intensities. In view of the simplistic theoretical approach, we find a surprisingly good agreement between the simulated and measured resonance Raman spectra of pyrene and its substituted analogues in terms of frequencies and intensities, which shows that the calculations can be used reliably to interpret the experimental spectra. With this combined information, it is possible to find criteria to distinguish the compounds under investigation, although many features of their vibrational spectra are similar.     

Resonance Raman spectra of uracil based on Kramers-Kronig relations using time-dependent density functional calculations and multireference perturbation theory

The use of time-dependent density functional calculations for the optimization of excited-state structures and the subsequent calculation of resonance Raman intensities within the transform-theory framework is compared to calculations of Hartree-Fock/configuration interaction singles-type (CIS). The transform theory of resonance Raman scattering is based on Kramers-Kronig relations between polarizability tensor components and the optical absorption. Stationary points for the two lowest excited singlet states of uracil are optimized and characterized by means of numerical differentiation of analytical excited-state gradients. It is shown that the effect of electron correlation leads to substantial modifications of the relative intensities. Calculations of vibrational frequencies for ground and excited states are carried out, which show that the neglect of Duschinsky mixing and the assumption of equal wave numbers for ground and excited state are not in all cases good approximations. We also compare the transform-theory resonance Raman intensities with those obtained within a simple approximation from excited-state gradients at the ground-state equilibrium position, and find that they are in qualitative agreement in the case of CIS, but show some important differences in calculations based on density functional theory. Since the results from CIS calculations are in better agreement with experiment, we also present approximate resonance Raman spectra obtained using excited-state gradients from multireference perturbation theory calculations, which confirm the CIS gradients.     

Theory and algorithms for the excited states of large molecules and molecular aggregates

This project aims to attack the frontiers of electronic structure calculations on the excited states of large molecules and molecular aggregates by developing novel theoretical and computational methods. The methodology development is especially based on the time-dependent density functional theory (TDDFT) and valence bond (VB) theory, and is expected to be computationally effective and accurate as well. Research works on the following related subjects will be performed: (1) The analytical energy-derivative approaches for electronically excited state within TDDFT will be developed to reduce bypass the computational costs in the calculation of molecular excited-state properties. (2) The ab initio methods for electronically excited state based on VB theory and hybrid TDDFT-VB method will be developed to overcome the limitations of current TDDFT in simulating photophysics and photochemistry. (3) For larger aggregates, neither ab initio methods nor TDDFT is applicable. We intend to build the effective model Hamiltonian by developing novel theoretical and computational methods to calculate the involved microscopic physical parameters from the first-principles methods. The constructed effective Hamiltonian is then used to describe the excitonic states and excitonic dynamics of the natural or artificial photosynthesized systems, organic or inorganic photovoltaic cell. (4) The condensed phase environment is taken into account by combining the developed theories and algorithms based on TDDFT and VB with the polarizable continuum solvent models (PCM), molecular mechanism (MM), classical electrodynamics (ED) or molecular dynamics (MD) theory. (5) Highly efficient software packages will be designed and developed.     

Rotational symmetry of the molecular potential energy in the Cartesian coordinates

We consider the molecular Born-Oppenheimer potential energy as a function of atomic Cartesian coordinates and discuss the non-stationary Hessian properties arising due to rotational symmetry. A connection with the extended Hessian theory is included. New applications of Cartesian representation for examining and correcting raw numerical Hessian data and a simple formulation of harmonic vibrational analysis of partially optimized systems are proposed. Exemplary calculations for the porphyrin molecule with an internal proton transfer are reported. We also develop the normal transformation method to incorporate the rotational symmetry into the approximate analytical potentials, which are parametrized in the Cartesian coordinates. The transformation converts the coordinates from the space fixed frame to the frame which translates and rotates with the molecule and is determined by the Eckart conditions. New simple analytical formulas for the first and second derivatives of the transformed potential are derived. This fast method can be used to calculate the potential and its derivatives in the simulations of chemical reaction dynamics in the space fixed Cartesian frame without the need to constrain the molecular rotation or to define the local non-redundant internal coordinates.     

Lagrangian approach to molecular vibrational Raman intensities using time-dependent hybrid density functional theory

The authors propose a new route to vibrational Raman intensities based on analytical derivatives of a fully variational polarizability Lagrangian. The Lagrangian is constructed to recover the negative frequency-dependent polarizability of time-dependent Hartree-Fock or adiabatic (hybrid) density functional theory at its stationary point. By virtue of the variational principle, first-order polarizability derivatives can be computed without using derivative molecular orbital coefficients. As a result, the intensities of all Raman-active modes within the double harmonic approximation are obtained at approximately the same cost as the frequency-dependent polarizability itself. This corresponds to a reduction of the scaling of computational expense by one power of the system size compared to a force constant calculation and to previous implementations. Since the Raman intensity calculation is independent of the harmonic force constant calculation more, computationally demanding density functionals or basis sets may be used to compute the polarizability gradient without much affecting the total time required to compute a Raman spectrum. As illustrated for fullerene C60, the present approach considerably extends the domain of molecular vibrational Raman calculations at the (hybrid) density functional level. The accuracy of absolute and relative Raman intensities of benzene obtained using the PBE0 hybrid functional is assessed by comparison with experiment.     

Mechanism of ligand photodissociation in photoactivable [Ru(bpy)2L2]2+ complexes: a density functional theory study

A series of four photodissociable Ru polypyridyl complexes of general formula [Ru(bpy)2L2](2+), where bpy = 2,2'-bipyridine and L = 4-aminopyridine (1), pyridine (2), butylamine (3), and gamma-aminobutyric acid (4), was studied by density functional theory (DFT) and time-dependent density functional theory (TDDFT). DFT calculations (B3LYP/LanL2DZ) were able to predict and elucidate singlet and triplet excited-state properties of 1-4 and describe the photodissociation mechanism of one monodentate ligand. All derivatives display a Ru --> bpy metal-to-ligand charge transfer (MLCT) absorption band in the visible spectrum and a corresponding emitting triplet (3)MLCT state (Ru --> bpy). 1-4 have three singlet metal-centered (MC) states 0.4 eV above the major (1)MLCT states. The energy gap between the MC states and lower-energy MLCT states is significantly diminished by intersystem crossing and consequent triplet formation. Relaxed potential energy surface scans along the Ru-L stretching coordinate were performed on singlet and triplet excited states for all derivatives employing DFT and TDDFT. Excited-state evolution along the reaction coordinate allowed identification and characterization of the triplet state responsible for the photodissociation process in 1-4; moreover, calculation showed that no singlet state is able to cause dissociation of monodentate ligands. Two antibonding MC orbitals contribute to the (3)MC state responsible for the release of one of the two monodentate ligands in each complex. Comparison of theoretical triplet excited-state energy diagrams from TDDFT and unrestricted Kohn-Sham data reveals the experimental photodissociation yields as well as other structural and spectroscopic features.     

The calculation of the vibrational frequencies of crystalline compounds and its implementation in the CRYSTAL code

The problem of numerical accuracy in the calculation of vibrational frequencies of crystalline compounds from the hessian matrix is discussed with reference to alpha-quartz (SiO(2)) as a case study and to the specific implementation in the CRYSTAL code. The Hessian matrix is obtained by numerical differentiation of the analytical gradient of the energy with respect to the atomic positions. The process of calculating vibrational frequencies involves two steps: the determination of the equilibrium geometry, and the calculation of the frequencies themselves. The parameters controlling the truncation of the Coulomb and exchange series in Hartree-Fock, the quality of the grid used for the numerical integration of the Exchange-correlation potential in Density Functional Theory, the SCF convergence criteria, the parameters controlling the convergence of the optimization process as well as those controlling the accuracy of the numerical calculation of the Hessian matrix can influence the obtained vibrational frequencies to some extent. The effect of all these parameters is discussed and documented. It is concluded that with relatively economical computational conditions the uncertainty related to these parameters is smaller than 2-4 cm(-1). In the case of the Local Density Approximation scheme, comparison is possible with recent calculations performed with a Density Functional Perturbation Theory method and a plane-wave basis set.     

Ground- and excited-state properties of the mixed-valence complex [(NH3)5Ru(III)NCRu(II)(CN)5]-

A computational study of the ground- and excited-state properties of the mixed-valence complex [(NH 3) 5Ru (III)NCRu (II)(CN) 5] (-) is presented. Employing DFT and TDDFT calculations for the complex in the gas phase and in aqueous solution, we investigate the vibrational and electronic structure of the complex in the electronic ground state as well as the character of the electronically excited states. The relevance of the various excited states for the intervalence metal-metal charge-transfer process in the complex is analyzed based on the change of charge density, spin density, and dipole moment upon photoexcitation as well as by a Mulliken-Hush analysis. Furthermore, those intramolecular modes, which are important for the charge-transfer process, are identified and characterized.     

Photoinduced intramolecular charge transfer in 4-(dimethyl)aminobenzonitrile--a theoretical perspective

Recent advances in time-dependent density functional theory (TDDFT) have led to computational methods that can predict properties of photoexcited molecules with satisfactory accuracy at comparably moderate cost. We apply these methods to study the photophysics and photochemistry of 4-(dimethyl)aminobenzonitrile (DMABN). DMABN is considered the paradigm of photoinduced intramolecular charge transfer (ICT), leading to dual fluorescence in polar solvents. By comparison of calculated emission energies, dipole moments, and vibrational frequencies with recent results from transient spectroscopy measurements, a definitive assignment of the electronic and geometric structure of the two lowest singlet excited states of DMABN is possible for the first time. We investigate the mechanism of the ICT reaction by means of minimum energy path calculations. The results confirm existing state-crossing models of dual fluorescence. Our study suggests that analytical TDDFT derivative methods will be useful to predict and classify emissive properties of other donor-acceptor systems as well.     

Rapid anharmonic vibrational corrections derived from partial Hessian analysis

Vibrational analysis within a partial Hessian framework can successfully describe the vibrational properties of a variety of systems where the vibrational modes of interest are localized within a specific region of the system. We have developed a new approach to calculating anharmonic frequencies based on vibrational frequencies and normal modes obtained from a partial Hessian analysis using second-order vibrational perturbation theory and the transition optimized shifted Hermite method. This allows anharmonic frequencies for vibrational modes that are spatially localized to be determined at a significantly reduced computational cost. Several molecular systems are examined in order to demonstrate the effectiveness of this method including organic molecules adsorbed on the Si(100)-2×1 surface, model peptides in solution, and the C-H stretching region of polycyclic aromatic hydrocarbons. Overall, for a range of systems, anharmonic frequencies calculated using the partial Hessian approach are found to be in close agreement with the results obtained using full anharmonic calculations while providing a significant reduction in computational cost.     

Troubleshooting time-dependent density-functional theory for photochemical applications: oxirane

The development of analytic-gradient methodology for excited states within conventional time-dependent density-functional theory (TDDFT) would seem to offer a relatively inexpensive alternative to better established quantum-chemical approaches for the modeling of photochemical reactions. However, even though TDDFT is formally exact, practical calculations involve the use of approximate functional, in particular the TDDFT adiabatic approximation, the use of which in photochemical applications must be further validated. Here, we investigate the prototypical case of the symmetric CC ring opening of oxirane. We demonstrate by direct comparison with the results of high-quality quantum Monte Carlo calculations that, far from being an approximation on TDDFT, the Tamm-Dancoff approximation is a practical necessity for avoiding triplet instabilities and singlet near instabilities, thus helping maintain energetically reasonable excited-state potential energy surfaces during bond breaking. Other difficulties one would encounter in modeling oxirane photodynamics are pointed out.     

Dcdftbmd: Divide-and-Conquer Density Functional Tight-Binding Program for Huge-System Quantum Mechanical Molecular Dynamics Simulations

Dcdftbmd is a Fortran 90/95 program that enables efficient quantum mechanical molecular dynamics (MD) simulations using divide-and-conquer density functional tight-binding (DC-DFTB) method. Based on the remarkable performance of previous massively parallel DC-DFTB energy and gradient calculations for huge systems, the code has been specialized to MD simulations. Recent implementations and modifications including DFTB extensions, improved computational speed in the DC-DFTB computational steps, algorithms for efficient initial guess charge prediction, and free energy calculations via metadynamics technique have enhanced the capability to obtain atomistic insights in novel applications to nanomaterials and biomolecules. The energy, structure, and other molecular properties are also accessible through the single-point calculation, geometry optimization, and vibrational frequency analysis. The available functionalities are outlined together with efficiency tests and simulation examples. © 2019 Wiley Periodicals, Inc.     

Efficient time-dependent density functional theory approximations for hybrid density functionals: analytical gradients and parallelization

In this paper, we present the implementation of efficient approximations to time-dependent density functional theory (TDDFT) within the Tamm-Dancoff approximation (TDA) for hybrid density functionals. For the calculation of the TDDFT/TDA excitation energies and analytical gradients, we combine the resolution of identity (RI-J) algorithm for the computation of the Coulomb terms and the recently introduced "chain of spheres exchange" (COSX) algorithm for the calculation of the exchange terms. It is shown that for extended basis sets, the RIJCOSX approximation leads to speedups of up to 2 orders of magnitude compared to traditional methods, as demonstrated for hydrocarbon chains. The accuracy of the adiabatic transition energies, excited state structures, and vibrational frequencies is assessed on a set of 27 excited states for 25 molecules with the configuration interaction singles and hybrid TDDFT/TDA methods using various basis sets. Compared to the canonical values, the typical error in transition energies is of the order of 0.01 eV. Similar to the ground-state results, excited state equilibrium geometries differ by less than 0.3 pm in the bond distances and 0.5° in the bond angles from the canonical values. The typical error in the calculated excited state normal coordinate displacements is of the order of 0.01, and relative error in the calculated excited state vibrational frequencies is less than 1%. The errors introduced by the RIJCOSX approximation are, thus, insignificant compared to the errors related to the approximate nature of the TDDFT methods and basis set truncation. For TDDFT/TDA energy and gradient calculations on Ag-TB2-helicate (156 atoms, 2732 basis functions), it is demonstrated that the COSX algorithm parallelizes almost perfectly (speedup ~26-29 for 30 processors). The exchange-correlation terms also parallelize well (speedup ~27-29 for 30 processors). The solution of the Z-vector equations shows a speedup of ~24 on 30 processors. The parallelization efficiency for the Coulomb terms can be somewhat smaller (speedup ~15-25 for 30 processors), but their contribution to the total calculation time is small. Thus, the parallel program completes a Becke3-Lee-Yang-Parr energy and gradient calculation on the Ag-TB2-helicate in less than 4 h on 30 processors. We also present the necessary extension of the Lagrangian formalism, which enables the calculation of the TDDFT excited state properties in the frozen-core approximation. The algorithms described in this work are implemented into the ORCA electronic structure system.     

Relativistically corrected geometries obtained with analytical gradients: normalized elimination of the small component using an effective potential

For the quasi-relativistic normalized elimination of small component using an effective potential (NESC-EP) method, analytical energy gradients were developed, programmed, and implemented in a standard quantum chemical program package. NESC-EP with analytical gradients was applied to determine geometry, vibrational frequencies, and dissociation enthalpies of ferrocene, tungsten hexafluoride, and tungsten hexacarbonyle. Contrary to non-relativistic calculations and calculations carried out with RECPs for the same compounds, NESC-EP provided reliable molecular properties in good agreement with experiment. The computational power of NESC-EP results from the fact that reliable relativistic corrections are obtained at a cost level only slightly larger than that of a non-relativistic calculation.     

FTIR and FT-Raman spectra, assignments, ab initio HF and DFT analysis of xanthine

The molecular vibrations of xanthine were investigated in polycrystalline sample, at room temperature by Fourier transform infrared (FTIR) and FT-Raman spectroscopies. The spectra of the molecule have been recorded in the regions 4000-50 cm(-1) and 3500-100 cm(-1), respectively. Theoretical information on the optimized geometry, harmonic vibrational frequencies, infrared and Raman intensities were obtained by means of ab initio Hartree-Fock (HF) and density functional theory (DFT) gradient calculations with complete relaxation in the potential energy surface using 6-311++G(d,p) basis set. The vibrational frequencies which were determined experimentally from the spectral data are compared with those obtained theoretically from ab initio and DFT calculations. A close agreement was achieved between the observed and calculated frequencies by refinement of the scale factors. The infrared and Raman spectra were also predicted from the calculated intensities. Thermodynamic properties like entropy, heat capacity, zero point energy have been calculated for the molecule. Unambiguous vibrational assignment of all the fundamentals was made using the potential energy distribution (PED).     

Investigation of the structure, the optical properties, and the photophysics of some indolocarbazoles having terminal aromatic rings

Structures, optical properties, and photophysics of ladder indolo[3,2-b]carbazoles substituted symmetrically by phenylene and thiophene rings have been investigated theoretically and experimentally. The ground state optimized structures were obtained using the density functional theory (DFT) as approximated by the B3LYP functional and employing the 6-31G^* basis set. All derivatives were found nonplanar in their electronic ground states. The character and the energy of the singlet–singlet electronic transitions have been investigated by applying the time-dependent density functional theory (TDDFT) to the correspondingly optimized-ground-state geometries. The ab initio restricted configuration interaction (singles) method (RCIS/6-31G^*) was adopted to obtain the first singlet excited-state structures (S₁) of the molecule. TDDFT calculations performed on the S₁ optimized geometries was used to obtain emission energies. UV–vis and fluorescence spectroscopies were analyzed in conjunction with theoretical calculations. The computed excitation and emission energies were found in reasonable agreement with the experimental absorption and fluorescence spectra. Finally, the photophysical behavior of the indolocarbazoles have been studied by means of steady state and time resolved fluorescence. The overall data have allowed the determination of the rate constants for the radiative and nonradiative decay processes. Both theoretical and experimental data show that the replacement of phenylene rings by thiophene units induces a red shift in the absorption and fluorescence spectra. This behavior is interpreted in terms of the electron donor properties of the thiophene ring. On the other hand, the change of the substitutional pattern, from 2,8 to 3,9, causes a significant hypsochromic shift of the absorption and fluorescence bands.