Why are time-dependent density functional theory excitations in solids equal to band structure energy gaps for semilocal functionals, and how does nonlocal Hartree-Fock-type exchange introduce excitonic effects?

We examine the time-dependent density functional theory (TD-DFT) equations for calculating excitation energies in solids with Gaussian orbitals and analytically show that for semilocal functionals, their lowest eigenvalue collapses to the minimum band orbital energy difference. With the introduction of nonlocal Hartree-Fock-type exchange (as in hybrid functionals), this result is no longer valid, and the lowest TD-DFT eigenvalue reflects the appearance of excitonic effects. Previously reported "charge-transfer" problems with semilocal TD-DFT excitations in molecules can be deduced from our analysis by taking the limit to infinite lattice constant.     

iVI: An iterative vector interaction method for large eigenvalue problems

Based on the generic “static‐dynamic‐static” framework for strongly coupled basis vectors (Liu and Hoffman, Theor. Chem. Acc. 2014, 133, 1481), an iterative Vector Interaction (iVI) method is proposed for computing multiple exterior or interior eigenpairs of large symmetric/Hermitian matrices. Although it works with a fixed‐dimensional search subspace, iVI can converge quickly and monotonically from above to the exact exterior/interior roots. The efficacy of iVI is demonstrated by taking both mathematical and physical matrices as examples. © 2017 Wiley Periodicals, Inc.     

Calculation of ionization potentials of small molecules: a comparative study of different methods

With the help of various theoretical methods, ionization potentials (IPs) have been computed for a panel of small molecules containing atoms of group 14, 15, or 16 and representing different singly, doubly, or triply bonded systems with or without an interacting heteroatom lone pair. Comparison of experimental IP values to theoretical results indicates that (i) the standard outer valence green function (OVGF), density functional theory (DFT), and DeltaSCF methods lead to rather accurate values, (ii) the CASPT2 method systematically underestimates IPs, (iii) the method of deducing IPs from a shift of some standard DFT eigenvalue spectrum is a straightforward approach leading to rather accurate IPs, (iv) the eigenvalue spectrum obtained with the so-called statistical average of different orbital model potential (SAOP) exchange-correlation model potential is an efficient approach leading directly to quite accurate IPs, and (v) a good prediction of the IP spectrum can be obtained from the shifted excitation spectra of the system calculated by the time-dependent DFT (TD-DFT) method. It is also shown that the TD-DFT calculations of the ionized species bring a significant improvement over the calculations of the neutral molecules, indicating that a great part of the electronic relaxation is already taken into account (in a similar way for all ionizations). Finally, in the case of TD-DFT calculations of neutral molecules, the statistical average of different orbital model potential (SAOP) functional does not lead to significantly better results than the B3LYP functional.     

The relationship between the eigenvalues and eigenvectors of a similarity matrix and its associated Carbó index matrix

The eigenvalues and eigenvectors of a quantum similarity matrix are also generalized eigenvalues and eigenvectors of the associated matrix of Carbó indices. This establishes bounds on the spectrum of the Carbó index matrix; for example, a quantum similarity matrix is positive semidefinite if and only if the associated Carbó index matrix is also positive semidefinite. The generalized eigenvalue problem for the Carbó index matrix has a diagonal metric matrix on the right-hand-side. Every generalized eigenvalue problem can be written in this diagonal form (i.e., this form is not special to this application). This diagonally structure generalized eigenvalue problem is especially convenient because it can be converted to a conventional eigenvalue problem by a particularly simple partial Löwdin transformation.     

Efficient exact-exchange time-dependent density-functional theory methods and their relation to time-dependent Hartree-Fock

A recently introduced time-dependent exact-exchange (TDEXX) method, i.e., a response method based on time-dependent density-functional theory that treats the frequency-dependent exchange kernel exactly, is reformulated. In the reformulated version of the TDEXX method electronic excitation energies can be calculated by solving a linear generalized eigenvalue problem while in the original version of the TDEXX method a laborious frequency iteration is required in the calculation of each excitation energy. The lowest eigenvalues of the new TDEXX eigenvalue equation corresponding to the lowest excitation energies can be efficiently obtained by, e.g., a version of the Davidson algorithm appropriate for generalized eigenvalue problems. Alternatively, with the help of a series expansion of the new TDEXX eigenvalue equation, standard eigensolvers for large regular eigenvalue problems, e.g., the standard Davidson algorithm, can be used to efficiently calculate the lowest excitation energies. With the help of the series expansion as well, the relation between the TDEXX method and time-dependent Hartree-Fock is analyzed. Several ways to take into account correlation in addition to the exact treatment of exchange in the TDEXX method are discussed, e.g., a scaling of the Kohn-Sham eigenvalues, the inclusion of (semi)local approximate correlation potentials, or hybrids of the exact-exchange kernel with kernels within the adiabatic local density approximation. The lowest lying excitations of the molecules ethylene, acetaldehyde, and pyridine are considered as examples.     

Theoretical study of Ln(III) complexes with polyaza-aromatic ligands: geometries of [LnL(H2O)n]3+ complexes and successes and failures of TD-DFT

The accuracy and the usefulness of density functional theory (DFT) and time-dependent density functional theory (TD-DFT) calculations for the theoretical study of Ln (La, Eu, Lu) complexes have been investigated. The geometries calculated at the DFT level for [Ln(H2O)nL]3+ complexes have been successfully compared with crystallographic data. TD-DFT is able to offer valuable insights into VUV spectra of lanthanide complexes. However, the results obtained on the largest ligand (i.e., 2,4,6-tri-(pyridin-2-yl)-1,3,5-triazine (Tptz)) have to be considered as a failure of TD-DFT.     

Comparison of DFT methods for molecular orbital eigenvalue calculations

We report how closely the Kohn-Sham highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) eigenvalues of 11 density functional theory (DFT) functionals, respectively, correspond to the negative ionization potentials (-IPs) and electron affinities (EAs) of a test set of molecules. We also report how accurately the HOMO-LUMO gaps of these methods predict the lowest excitation energies using both time-independent and time-dependent DFT (TD-DFT). The 11 DFT functionals include the local spin density approximation (LSDA), five generalized gradient approximation (GGA) functionals, three hybrid GGA functionals, one hybrid functional, and one hybrid meta GGA functional. We find that the HOMO eigenvalues predicted by KMLYP, BH&HLYP, B3LYP, PW91, PBE, and BLYP predict the -IPs with average absolute errors of 0.73, 1.48, 3.10, 4.27, 4.33, and 4.41 eV, respectively. The LUMOs of all functionals fail to accurately predict the EAs. Although the GGA functionals inaccurately predict both the HOMO and LUMO eigenvalues, they predict the HOMO-LUMO gap relatively accurately (approximately 0.73 eV). On the other hand, the LUMO eigenvalues of the hybrid functionals fail to predict the EA to the extent that they include HF exchange, although increasing HF exchange improves the correspondence between the HOMO eigenvalue and -IP so that the HOMO-LUMO gaps are inaccurately predicted by hybrid DFT functionals. We find that TD-DFT with all functionals accurately predicts the HOMO-LUMO gaps. A linear correlation between the calculated HOMO eigenvalue and the experimental -IP and calculated HOMO-LUMO gap and experimental lowest excitation energy enables us to derive a simple correction formula.     

Assessment of long-range corrected functionals performance for n-->pi* transitions in organic dyes

The first n-->pi* transitions of 18 nitroso and 16 thiocarbonyl dyes have been computed by time-dependent density functional theory (TD-DFT) using pure as well as global and range-separated hybrid functionals. It turns out that the accuracy of all hybrids is relatively similar, i.e., the inclusion of a growing fraction of exact exchange does neither worsen nor improve significantly the raw TD-DFT estimations. However, after a simple linear regression, it appears that the range-separated hybrids provide a better accuracy than global hybrids.     

Spin-adapted open-shell time-dependent density functional theory. III. An even better and simpler formulation

The recently proposed spin-adapted time-dependent density functional theory (S-TD-DFT) [Z. Li and W. Liu, J. Chem. Phys. 133, 064106 (2010)] resolves the spin-contamination problem in describing singly excited states of high spin open-shell systems. It is an extension of the standard restricted open-shell Kohn-Sham-based TD-DFT which can only access those excited states due to singlet-coupled single excitations. It is also far superior over the unrestricted Kohn-Sham-based TD-DFT (U-TD-DFT) which suffers from severe spin contamination for those excited states due to triplet-coupled single excitations. Nonetheless, the accuracy of S-TD-DFT for high spin open-shell systems is still inferior to TD-DFT for well-behaved closed-shell systems. The reason can be traced back to the violation of the spin degeneracy conditions (SDC) by approximate exchange-correlation (XC) functionals. Noticing that spin-adapted random phase approximation (S-RPA) can indeed maintain the SDC by virtue of the Wigner-Eckart theorem, a hybrid ansatz combining the good of S-TD-DFT and S-RPA can immediately be envisaged. The resulting formalism, dubbed as X-TD-DFT, is free of spin contamination and can also be viewed as a S-RPA correction to the XC kernel of U-TD-DFT. Compared with S-TD-DFT, X-TD-DFT leads to much improved results for the low-lying excited states of, e.g., N(2)(+), yet with much reduced computational cost. Therefore, X-TD-DFT can be recommended for routine calculations of excited states of high spin open-shell systems.     

Accurate evaluation of valence and low-lying Rydberg states with standard time-dependent density functional theory

Using a standard exchange-correlation functional, namely, PBE0, the basis set dependence of time-dependent density functional theory (TD-DFT) calculations has been explored using 33 different bases and five organic molecules as test cases. The results obtained show that this functional can provide accurate (i.e., at convergence) results for both valence and low-lying Rydberg excitations if at least one diffuse function for the heavy atoms is included in the basis set. Furthermore, these results are in fairly good agreement with the experimental data and with those delivered by other functionals specifically designed to yield correct asymptotic/long-range behavior. More generally, the PBE0 calculations show that a greater accuracy can be obtained for both Rydberg and valence excitations if they occur at energies below the epsilonHOMO + 1 eV threshold. This latter value is proposed as a thumb rule to verify the accuracy of TD-DFT/PBE0 applications.     

Fundamental photoluminescence properties of pyrene carbonyl compounds through absolute fluorescence quantum yield measurement and density functional theory

We reviewed the photophysical properties of carbonyl-functionalized pyrene derivatives [i.e., pyrene with aldehyde (PA: 1-formylpyrene), ketone (PK: 1-acetylpyrene), carboxylic acid (PCA: 1-pyrenecarboxylic acid), and ester groups (PE: 1-methoxycarbonylpyrene)] using a measurement of absolute fluorescence quantum yield in various solvents and time-dependent density functional theory (TD-DFT) calculations. Here, we obtained new important data that fill in the gaps in existing datasets on these properties and help identify photoluminescence mechanisms. The results of the TD-DFT calculations were in agreement with the experimental results, and indicated that the low fluorescence of PA and PK is derived not only from intersystem crossing but also from internal conversion due to the proximity effect; this inference was also supported by the measurements of the photoluminescence spectra at low temperatures. In addition, factors leading efficiently to non-radiative processes were shown to be absent in PCA and PE. Thus, we successfully revised and systematized the photophysical properties of pyrene modified by carbonyl substitutes, including carboxamide groups, which were previously reported by us. Moreover, we showed that the photoluminescence properties of such compounds might be predictable by using TD-DFT calculations.     

Theoretical study of low-lying excited states of molecular aggregates. I. Development of linear-scaling TD-DFT

The project aims to develop an integrated linear-scaling time-dependent density functional theory (TD-DFT) for studying low-lying excited states of luminescent molecular materials, especially those fluorescence and phosphorescence co-emitting systems. The central idea will be "from fragments to molecule" (FF2M). That is, the fragmental information will be employed to synthesize the molecular wave function, such that the locality (transferability) of the fragments (functional groups) is directly built into the algorithms. Both relativistic and spin-adapted open-shell TD-DFT will be considered. Use of the renormalized exciton method will also be made to further enhance the efficiency and accuracy of TD-DFT. Solvent effects are to be targeted with the fragment-based solvent model. It is expected that the integrated TD-DFT and program will be of great value in rational design of luminescent molecular materials.     

Studies on Optical Properties of Si220 Nanoclusters via Time-dependent Density Functional Theory Calculations

The optical properties of bare and hydrogen passivated Si₂₂₀ nanoclusters(NCs) in four typical motifs(i.e., bulk-like, onion-like, bucky-diamond and icosahedral motifs) were studied via time-dependent density functional theory(TD-DFT) calculations. The calculation results show that there is a significant blue shift in the optical absorption spectra when the Si NCs are passivated with hydrogen. A strong absorption peak in the visible light region appears for the hydrogenated bulk-like, onion-like and bucky-diamond Si NCs.     

Theoretical study of singlet and triplet excitation energies in oligothiophenes

We have analyzed singlet and triplet excitation energies in oligothiophenes (up to five rings) using time-dependent density-functional theory (TD-DFT) with different exchange-correlation functionals and compared them with results from the approximate coupled-cluster singles and doubles model (CC2) and experimental data. The excitation energies have been calculated in geometries obtained by TD-DFT optimization of the lowest excited singlet state and in the ground-state geometries of the neutral and anionic systems. TD-DFT methods underestimate photoluminescence energies but the energy difference between singlet and triplet states shows trends with the chain-length similar to CC2. We find that the second triplet excited state is below the first singlet excited state for long oligomers in contrast with the previous assignment of Rentsch et al. (Phys.Chem. Chem. Phys. 1999, 1, 1707). Their photodetachment photoelectron spectroscopy measurements are better described by considering higher triplet excited states.     

Time-dependent density functional theory as a tool for isomer assignments of hydrogen-bonded solute.solvent clusters

Can isomer structures of hydrogen-bonded solute x solvent clusters be assigned by correlating gas-phase experimental S0 <--> S1 transitions with vertical or adiabatic excitation energies calculated by time-dependent density functional theory (TD-DFT)? We study this question for 7-hydroxyquinoline (7HQ), for which an experimental database of 19 complexes and clusters is available. The main advantage of the adiabatic TD-B3LYP S0 <--> S1 excitations is the small absolute error compared to experiment, while for the calculated vertical excitations, the average offset is +1810 cm(-1). However, the empirically adjusted vertical excitations correlate more closely with the experimental transition energies, with a standard deviation of sigma = 72 cm(-1). For the analogous correlation with calculated adiabatic TD-DFT excitations, the standard deviation is sigma = 157 cm(-1). The vertical and adiabatic TD-DFT correlation methods are applied for the identification of isomers of the 7-hydroxyquinoline.(MeOH) n , n = 1-3 clusters [Matsumoto, Y.; Ebata, T.; Mikami, N. J. Phys. Chem. B 2002, 106, 5591]. These confirm that the vertical TD-DFT/experimental correlation yields more effective isomer assignments.     

On pattern matching of X-ray powder diffraction data

We introduce a novel pattern matching algorithm optimized for X-ray powder diffraction (XRPD) data and useful for data from other types of analytical techniques (e.g., Raman, IR). The algorithm is based on hierarchical clustering with a similarity metric that compares peak positions using the full peak profile. It includes heuristics developed from years of experience manually matching XRPD data, and preprocessing algorithms that reduce the effects of common problems associated with XRPD (e.g., preferred orientation and poor particle statistics). This algorithm can find immediate application in automated polymorph screening and salt selection, common tasks in the development of pharmaceuticals.     

Large blue-shift in the optical spectra of fluorinated polyphenylenevinylenes. A combined theoretical and experimental study

Modifications of the optical properties of poly[2-methoxy-5-(2'-ethyl-hexyloxy)-1,4-phenylene vinylene] induced by fluorination of the vinylene units are investigated by means of time dependent density functional theory (TD-DFT) calculations and spectroscopic measurements in solution. The energy of the main absorption peak is blue-shifted by more than 0.8 eV in the fluorinated polymers. TD-DFT excitation energies for non-fluorinated and fluorinated oligomer structures of increasing number of monomers, employing fully relaxed geometries, are compared to the experimental absorption energies of the polymers. We found that the measured large blue-shift induced by the fluorination of the vinylene units is not caused by the electron-withdrawing effect of the fluorine substituents but it is related to a steric effect. The inter-monomer torsional angle of the fluorinated structures increases above 50 degrees , while in the non-fluorinated systems it is below 20 degrees . Further insight into the origin of the large blue-shift of the excitation energies is gained by a detailed analysis of the torsional potentials of non-fluorinated and fluorinated dihydroxystilbene. While for planar geometries the energy gap increases due to fluorination, it decreases for highly distorted geometries. In addition, we found that the torsional potential of dihydroxystilbene is rather flat, meaning that different isomers might, e.g., in the solid state, coexist.     

Spectral convergence of the quadrature discretization method in the solution of the Schrodinger and Fokker-Planck equations: comparison with sinc methods

Spectral methods based on nonclassical polynomials and Fourier basis functions or sinc interpolation techniques are compared for several eigenvalue problems for the Fokker-Planck and Schrodinger equations. A very rapid spectral convergence of the eigenvalues versus the number of quadrature points is obtained with the quadrature discretization method (QDM) and the appropriate choice of the weight function. The QDM is a pseudospectral method and the rate of convergence is compared with the sinc method reported by Wei [J. Chem. Phys., 110, 8930 (1999)]. In general, sinc methods based on Fourier basis functions with a uniform grid provide a much slower convergence. The paper considers Fokker-Planck equations (and analogous Schrodinger equations) for the thermalization of electrons in atomic moderators and for a quartic potential employed to model chemical reactions. The solution of the Schrodinger equation for the vibrational states of I2 with a Morse potential is also considered.     

Assessment of TD-DFT- and TD-HF-based approaches for the prediction of exciton coupling parameters, potential energy curves, and electronic characters of electronically excited aggregates

The reliability of linear response approaches such as time-dependent Hartree-Fock (TD-HF) and time-dependent density functional theory (TD-DFT) for the prediction of the excited state properties of 3,4;9,10-tetracarboxylic-perylene-bisimide (PBI) aggregates is investigated. A dimer model of PBI is investigated as a function of a torsional motion of the monomers, which was shown before to be an important intermolecular coordinate in these aggregates. The potential energy curves of the ground state and the two energetically lowest neutral excited and charge-transfer (CT) states were obtained with the spin-component scaling modification of the approximate coupled-cluster singles-and-doubles (SCS-CC2) method as a benchmark for dispersion corrected TD-HF and a range of TD-DFT approaches. The highly accurate SCS-CC2 results are used to assess the other, computationally less demanding methods. TD-HF predicts similar potential energy curves and transition dipole moments as SCS-CC2, as well as the correct order of neutral and CT states. This supports an exciton trapping mechanism, which was found on the basis of TD-HF data. However, the investigated TD-DFT methods provide generally the opposite character for the excited states. As a consequence, these TD-DFT results have unacceptably large errors for optical properties of these dye aggregates.