Excitations in time-dependent density-functional theory

Exact Excited-State Functionals of the Asymmetric Hubbard Dimer

The exact functionals associated with the (singlet) ground state and the two singlet excited states of the asymmetric Hubbard dimer at half-filling are calculated using both Levy's constrained search and Lieb's convex formulation. While the ground-state functional is, as is commonly known, a convex function with respect to the density, the functional associated with the doubly excited state is found to be concave. Also, because the density-potential mapping associated with the first excited state is noninvertible, its "functional" is a partial, multivalued function composed of one concave and one convex branch that correspond to two separate domains of the external potential. Remarkably, it is found that, although the one-to-one mapping between density and external potential may not apply (as in the case of the first excited state), each state-specific energy and corresponding universal functional are "functions" whose derivatives are each other's inverse, just as in the ground state formalism.

Curing the Divergence in Time-Dependent Density Functional Quadratic Response Theory

The adiabatic approximation in time-dependent density functional theory is known to give an incorrect pole structure in the quadratic response function, leading to unphysical divergences in excited state-to-state transition probabilities and hyperpolarizabilties. We find the form of the exact quadratic response kernel and derive a practical and accurate approximation that cures the divergence. We demonstrate our results on excited state-to-state transition probabilities of a model system and of the LiH molecule.

Efficient All-Electron Time-Dependent Density Functional Theory Calculations Using an Enriched Finite Element Basis

We present an efficient and systematically convergent approach to all-electron real-time time-dependent density functional theory (TDDFT) calculations using a mixed basis, termed as enriched finite element (EFE) basis. The EFE basis augments the classical finite element basis (CFE) with a compactly supported numerical atom-centered basis, obtained from atomic ground-state DFT calculations. Particularly, we orthogonalize the enrichment functions with respect to the classical finite element basis to ensure good conditioning of the resultant basis. We employ the second-order Magnus propagator in conjunction with an adaptive Krylov subspace method for efficient time evolution of the Kohn-Sham orbitals. We rely on a priori error estimates to guide our choice of an adaptive finite element mesh as well as the time step to be used in the TDDFT calculations. We observe close to optimal rates of convergence of the dipole moment with respect to spatial and temporal discretizations. Notably, we attain a 50-100 times speedup for the EFE basis over the CFE basis. We also demonstrate the efficacy of the EFE basis for both linear and nonlinear responses by studying the absorption spectra in sodium clusters, the linear to nonlinear response transition in the green fluorescence protein chromophore, and the higher harmonic generation in the magnesium dimer. Lastly, we attain good parallel scalability of our numerical implementation of the EFE basis for up to ∼1000 processors, using a benchmark system of a 50-atom sodium nanocluster.

Tension between predicting accurate ground state correlation energies and excitation energies from adiabatic approximations in TDDFT

The connection between the adiabatic excitation energy of time-dependent density functional theory and the ground state correlation energy from the adiabatic connection fluctuation–dissipation theorem (ACFDT) is explored in the limiting case of one excited state. An exact expression is derived for any adiabatic Hartree-exchange–correlation kernel that connects the excitation energy and the potential contribution to correlation. The resulting formula is applied to the asymmetric Hubbard dimer, a system where this limit is exact. Results from a hierarchy of approximations to the kernel, including the random phase approximation (RPA) with and without exchange and the adiabatically exact (AE) approximation, are compared to the exact ones. At full coupling, the numerical results indicate a tension between predicting an accurate excitation energy and an accurate potential contribution to correlation. The AE approximation is capable of making accurate predictions of both quantities, but only in parts of the parameter space that classify as weakly correlated, while RPA tends to be unable to accurately predict these properties simultaneously anywhere. For a strongly correlated dimer, the AE approximation greatly overestimates the excitation energy yet continues to yield an accurate ground state correlation energy due to its accurate prediction of the adiabatic connection integrand. If similar trends hold for real systems, the development of correlation kernels will be important for applications of the ACFDT in systems with large potential contributions to correlation.

Charge-Transfer Excitations within Density Functional Theory: How Accurate Are the Most Recommended Approaches?

The performance of the most recent density functionals is assessed for charge-transfer (CT) excitations using comprehensive intra- and intermolecular CT benchmark sets with high-quality reference values. For this comparison, the state-of-the-art range-separated (RS) and long-range-corrected (LC) double hybrid (DH) approaches are selected, and global DH and LC hybrid functionals are also inspected. The correct long-range behavior of the exchange–correlation (XC) energy is extensively studied, and various CT descriptors are compared as well. Our results show that the most robust performance is attained by RS-PBE-P86/SOS-ADC(2), as it is suitable to describe both types of CT excitations with outstanding accuracy. Furthermore, concerning the intramolecular transitions, unexpectedly excellent results are obtained for most of the global DHs, but their limitations are also demonstrated for bimolecular complexes. Despite the outstanding performance of the LC-DH methods for common intramolecular excitations, serious deficiencies are pointed out for intermolecular CT transitions, and the wrong long-range behavior of the XC energy is revealed. The application of LC hybrids to such transitions is not recommended in any respect.

Accurate Spectral Properties within Double-Hybrid Density Functional Theory: A Spin-Scaled Range-Separated Second-Order Algebraic-Diagrammatic Construction-Based Approach

Our second-order algebraic-diagrammatic construction [ADC(2)]-based double-hybrid (DH) ansatz (J. Chem. Theory Comput. 2019, 15, 4440. DOI: 10.1021/acs.jctc.9b00391) is combined with range-separation techniques. In the present scheme, both the exchange and the correlation contributions are range-separated, while spin-scaling approaches are also applied. The new methods are thoroughly tested for the most popular benchmark sets including 250 singlet and 156 triplet excitations, as well as 80 oscillator strengths. It is demonstrated that the range separation for the correlation contributions is highly recommended for both the genuine and the ADC(2)-based DH approaches. Our results show that the latter scheme slightly but consistently outperforms the former one for single excitation dominated transitions. Furthermore, states with larger fractions of double excitations are assessed as well, and challenging charge-transfer excitations are also discussed, where the recently proposed spin-scaled long-range corrected DHs fail. The suggested iterative fourth-power scaling RS-PBE-P86/SOS-ADC(2) method, using only three adjustable parameters, provides the most robust and accurate excitation energies within the DH theory. In addition, the relative error of the oscillator strengths is reduced by 65% compared to the best genuine DH functionals.

Double and Charge-Transfer Excitations in Time-Dependent Density Functional Theory

Time-dependent density functional theory has emerged as a method of choice for calculations of spectra and response properties in physics, chemistry, and biology, with its system-size scaling enabling computations on systems much larger than otherwise possible. While increasingly complex and interesting systems have been successfully tackled with relatively simple functional approximations, there has also been increasing awareness that these functionals tend to fail for certain classes of approximations. Here I review the fundamental challenges the approximate functionals have in describing double excitations and charge-transfer excitations, which are two of the most common impediments for the theory to be applied in a black-box way. At the same time, I describe the progress made in recent decades in developing functional approximations that give useful predictions for these excitations. Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Self-Consistent Auxiliary Density Perturbation Theory

The working equations for the extension of auxiliary density perturbation theory (ADPT) to hybrid functionals, employing the variational fitting of the Fock potential, are derived. The response equations in the resulting self-consistent ADPT (SC-ADPT) are solved iteratively with an adapted Eirola-Nevanlinna algorithm. As a result, a memory and CPU time efficient implementation of perturbation theory free of four-center electron repulsion integrals (ERIs) is obtained. Our validation calculations of SC-ADPT static and dynamic polarizabilities show quantitative agreement with corresponding coupled perturbed Hartree-Fock and Kohn-Sham results employing four-center ERIs. The comparison of SC-ADPT hybrid functional polarizabilities with coupled cluster reference calculations yield semiquantitative agreement. The presented systematic study of the dynamic polarizabilities of oligothiophenes shows that hybrid functionals can overcome the pathological misplacement of excitation poles by the local density and generalized gradient approximations. Good agreement with experimental dynamic polarizabilities for all studied oligothiophenes is achieved with range-separated hybrid functionals in the framework of SC-ADPT.

Spin-Scaled Range-Separated Double-Hybrid Density Functional Theory for Excited States

Our recently presented range-separated (RS) double-hybrid (DH) time-dependent density functional approach [J. Chem. Theory Comput. 17, 927 (2021)] is combined with spin-scaling techniques. The proposed spin-component-scaled (SCS) and scaled-opposite-spin (SOS) variants are thoroughly tested for almost 500 excitations including the most challenging types. This comprehensive study provides useful information not only about the new approaches but also about the most prominent methods in the DH class. The benchmark calculations confirm the robustness of the RS-DH ansatz, while several tendencies and deficiencies are pointed out for the existing functionals. Our results show that the SCS variant consistently improves the results, while the SOS variant preserves the benefits of the original RS-DH method reducing its computational expenses. It is also demonstrated that, besides our approaches, only the nonempirical functionals provide balanced performance for general applications, while particular methods are only suggested for certain types of excitations.

Density Functional Theory of Highly Excited States of Coulomb Systems

The density functional theory proposed earlier for excited states of Coulomb systems is discussed. The localized Hartree–Fock (LHF) and the Krieger, Li, and Iafrate (KLI) methods combined with correlation are generalized for excited states. Illustrative examples include some highly excited states of Li and Na atoms.

A Simple Range-Separated Double-Hybrid Density Functional Theory for Excited States

A simple and robust range-separated (RS) double-hybrid (DH) time-dependent density functional approach is presented for the accurate calculation of excitation energies of molecules within the Tamm-Dancoff approximation. The scheme can be considered as an excited-state extension of the ansatz proposed by Toulouse and co-workers [J. Chem. Phys. 2018, 148, 164105], which is based on the two-parameter decomposition of the Coulomb potential, for which both the exchange and correlation contributions are range-separated. A flexible and efficient implementation of the new scheme is also presented, which facilitates its extension to any combination of exchange and correlation functionals. The performance of the new approximation is tested for singlet excitations on several benchmark compilations and thoroughly compared to that of representative DH, RS hybrid, and RS DH functionals. The one-electron basis set dependence and computation times are also assessed. Our results show that the new approach improves on standard DHs in most cases, and it can provide a more robust and accurate alternative. In addition, on average, it noticeably surpasses the existing RS hybrid and RS DH functionals.

Time-dependent DFT study of the K-edge spectra of vanadium and titanium complexes: effects of chloride ligands on pre-edge features

X-ray absorption near edge structures (XANES) of vanadium and titanium complexes were investigated with time-dependent density functional theory (TDDFT).

Suppression of Electron-Hole Recombination by Intrinsic Defects in 2D Monoelemental Material

The Shockley-Read-Hall (SRH) model, in which the deep trap defect states in the band gap are proposed as nonradiative electron-hole (e-h) recombination centers, has been widely used to describe the nonradiative e-h recombination through the defects in semiconductor. By using the ab initio nonadiabatic molecular dynamics method, we find that the SRH model fails to describe the e-h recombination behavior for defects in 2D monoelemental material such as monolayer black phosphorus (BP). Through the investigation of three intrinsic defects with shallow and deep defect states in monolayer BP, it is found that, surprisingly, none of these defects significantly accelerates the e-h recombination. Further analysis shows that because monolayer BP is a monoelemental material, the distinct impurity phonon, which often induces fast e-h recombination, is not formed. Moreover, because of the flexibility of 2D material, the defects scatter the phonons present in pristine BP, generating multiple modes with lower frequencies compared with the pristine BP, which further suppresses the e-h recombination. We propose that the conclusion can be extended to other monoelemental 2D materials, which is important guidance for the future design of functional semiconductors.

Combined Density Functional and Algebraic-Diagrammatic Construction Approach for Accurate Excitation Energies and Transition Moments

A composite of time-dependent density functional theory (TDDFT) and the second-order algebraic-diagrammatic construction [ADC(2)] approach is presented for efficient calculation of spectral properties of molecules. Our method can be regarded as a new excited-state double-hybrid (DH) approach or a dressed TDDFT scheme, but it can also be interpreted as an empirically tuned ADC(2) model. Several combinations of exchange-correlation functionals and spin-scaling schemes are explored. Our best-performing method includes the Perdew, Burke, and Ernzerhof exchange and Perdew's 1986 correlation functional and employs the scaled-opposite-spin approximation for the higher-order terms. The computation time of the new method scales as the fourth power of the system size, and an efficient cost-reduction approach is also presented, which further speeds up the calculations. Our benchmark calculations show that the proposed model outperforms not only the existing DH approaches and ADC(2) variants but also the considerably more expensive coupled-cluster methods.

New Near-infrared Fluorescent Probes with Single-photon Anti-Stokes-shift Fluorescence for Sensitive Determination of pH Variances in Lysosomes with a Double-Checked Capability

Two near-infrared luminescent probes with Stokes-shift and single-photon anti-Stokes-shift fluorescence properties for sensitive determination of pH variance in lysosomes have been synthesized. A morpholine residue in probe A which serves as a targeting group for lysosomes in viable cells was attached to the fluorophores via a spirolactam moiety while a mannose residue was ligated to probe B resulting in increased biocompatibility and solubility in water. Probes A and B contain closed spirolactam moieties, and show no Stokes-shift or anti-Stokes-shift fluorescence under neutral or alkali conditions. However, the probes incrementally react to pH variance from 7.22 to 2.76 with measurable increases in both Stokes-shift and anti-Stokes-shift fluorescence at 699 nm and 693 nm under 645 nm and 800 nm excitation, respectively. This acid-activated fluorescence is produced by the breaking of the probe spirolactam moiety, which greatly increased overall π-conjugation in the probes. These probes possess upconversion near-infrared fluorescence imaging advantages including minimum cellular photo-damage, tissue penetration, and minimum biological fluorescence background. They display excellent photostability with low dye photobleaching and show good biocompatibility. They are selective and capable of detecting pH variances in lysosomes at excitation with two different wavelengths, i.e., 645 and 800 nm.

Long-range corrected density functional through the density matrix expansion based semilocal exchange hole

The exchange hole, which is one of the principal constituents of the density functional formalism, can be used to design accurate range-separated hybrid functionals in association with appropriate correlation.

Charge transfer in time-dependent density functional theory

Charge transfer plays a crucial role in many processes of interest in physics, chemistry, and bio-chemistry. In many applications the size of the systems involved calls for time-dependent density functional theory (TDDFT) to be used in their computational modeling, due to its unprecedented balance between accuracy and efficiency. However, although exact in principle, in practise approximations must be made for the exchange-correlation functional in this theory, and the standard functional approximations perform poorly for excitations which have a long-range charge-transfer component. Intense progress has been made in developing more sophisticated functionals for this problem, which we review. We point out an essential difference between the properties of the exchange-correlation kernel needed for an accurate description of charge-transfer between open-shell fragments and between closed-shell fragments. We then turn to charge-transfer dynamics, which, in contrast to the excitation problem, is a highly non-equilibrium, non-perturbative, process involving a transfer of one full electron in space. This turns out to be a much more challenging problem for TDDFT functionals. We describe dynamical step and peak features in the exact functional evolving over time, that are missing in the functionals currently used. The latter underestimate the amount of charge transferred and manifest a spurious shift in the charge transfer resonance position. We discuss some explicit examples.

Dual-Functional Tamm-Dancoff Approximation: A Convenient Density Functional Method that Correctly Describes S1/S0 Conical Intersections

Time-dependent Kohn-Sham density functional theory has been used successfully to compute vertical excitation energies, especially for large molecular systems. However, the lack of double excitation character in the excited amplitudes produced by linear response in the adiabatic approximation holds it back from broader applications in photochemistry; for example, it shows (3N - 7)-dimensional conical intersection seams (where N is the number of atoms) between ground and excited states, although the correct dimensionality is 3N - 8. In this letter, we present a new, conceptually simple, easy-to-implement, and easy-to-use way to employ time-dependent Kohn-Sham density functional theory that has global accuracy comparable with the conventional single-functional version and that recovers the double cone topology of the potential energy surfaces at S₁/S₀ conical intersection seams. The new method is called the dual-functional Tamm-Dancoff approximation (DF-TDA).

Helium Atom Excitations by the GW and Bethe-Salpeter Many-Body Formalism

The helium atom is the simplest many-body electronic system provided by nature. The exact solution to the Schrödinger equation is known for helium ground and excited states, and it represents a benchmark for any many-body methodology. Here, we check the ab initio many-body GW approximation and the Bethe-Salpeter equation (BSE) against the exact solution for helium. Starting from the Hartree-Fock method, we show that the GW and the BSE yield impressively accurate results on excitation energies and oscillator strength, systematically improving the time-dependent Hartree-Fock method. These findings suggest that the accuracy of the BSE and GW approximations is not significantly limited by self-interaction and self-screening problems even in this few electron limit. We further discuss our results in comparison to those obtained by time-dependent density-functional theory.

Optical and vibrational properties of phosphorylcholine-based contact lenses-Experimental and theoretical investigations

The Raman, MIR and UV-vis spectroscopy have been used to characterize Omafilcon A material constructing the one of the Proclear family contact lenses. The Omafilcon A is hydrogel material composed of 2-hydroxyethyl methacrylate (HEMA) and 2-methacryloyloxyethyl phosphorylcholine (PC) polymers crosslinked with ethyleneglycol dimethacrylate (EGDMA). Vibrational and electronic properties of the Omafilcon A material were also investigated by quantum chemical calculations. Experimentally obtained Raman, MIR and optical spectra were compared to the theoretical ones calculated applying RHF and DFT methodology. The quantum chemical calculations were performed for isolated monomers of lenses compounds as well as for their dimers and trimers to elucidate the effect of Omafilcon A polymerization and the role of an individual components.

Near-Infrared Fluorescent Probes with Large Stokes Shifts for Sensing Zn(II) Ions in Living Cells

We report two new near-infrared fluorescent probes based on Rhodol counterpart fluorophore platforms functionalized with dipicolylamine Zn(II)-binding groups. The combinations of the pendant amines and fluorophores provide the probes with an effective three-nitrogen-atom and one-oxygen-atom binding motif. The fluorescent probes with large Stokes shifts offer sensitive and selective florescent responses to Zn(II) ions over other metal ions, allowing a reversible monitoring of Zn(II) concentration changes in living cells, and detecting intracellular Zn(II) ions released from intracellular metalloproteins.

Extension of the Core-Valence-Rydberg B3LYP Functional to Core-Excited-State Calculations of Third-Row Atoms

A modified core-valence-Rydberg Becke's three-parameter exchange (B3) + Lee-Yang-Parr (LYP) correlation (CVR-B3LYP) functional is proposed in order to calculate core-excitation energies of third-row atoms with reasonable accuracy. The assessment of conventional exchange-correlation functionals shows that the appropriate portions of Hartree-Fock (HF) exchange for core-excited-state calculations depend on shells:  70% and 50% for K-shell and L-shell excitations, respectively. Therefore, the modified CVR-B3LYP functional is designed to use the appropriate portions of HF exchange, 70%, 50%, and 20%, for K-shell, L-shell, and valence regions separately. Time-dependent density functional theory calculations with the modified CVR-B3LYP functional yield both K-shell and L-shell excitation energies with reasonable accuracy. The modified CVR-B3LYP also provides valence-excitation energies and standard enthalpies of formation accurately. Thus, the modified CVR-B3LYP describes all of the K-shell, L-shell, and valence electrons appropriately.

Dependence of Spurious Charge-Transfer Excited States on Orbital Exchange in TDDFT: Large Molecules and Clusters

Time-dependent density functional theory (TDDFT) is a powerful tool allowing for accurate description of excited states in many nanoscale molecular systems; however, its application to large molecules may be plagued with difficulties that are not immediately obvious from previous experiences of applying TDDFT to small molecules. In TDDFT, the appearance of spurious charge-transfer states below the first optical excited state is shown to have significant effects on the predicted absorption and emission spectra of several donor-acceptor substituted molecules. The same problem affects the predictions of electronic spectra of molecular aggregates formed from weakly interacting chromophores. For selected benchmark cases, we show that today's popular density functionals, such as purely local (Local Density Approximation, LDA) and semilocal (Generalized Gradient Approximation, GGA) models, are qualitatively wrong. Nonlocal hybrid approximations including both semiempirical (B3LYP) and ab initio (PBE1PBE) containing a small fraction (20-25%) of Fock-like orbital exchange are also susceptible to such problems. Functionals that contain a larger fraction (50%) of orbital exchange like the early hybrid (BHandHLYP) are shown to exhibit far fewer spurious charge-transfer (CT) states at the expense of accuracy. Based on the trends observed in this study and our previous experience we formulate several practical approaches to overcome these difficulties providing a reliable description of electronic excitations in nanosystems.

Synthetic control of spectroscopic and photophysical properties of triarylborane derivatives having peripheral electron-donating groups

The spectroscopic and photophysical properties of triarylborane derivatives were controlled by the nature of the triarylborane core (trixylyl- or trianthrylborane) and peripheral electron-donating groups (N,N-diphenylamino or 9H-carbazolyl groups). The triarylborane derivatives with and without the electron-donating groups showed intramolecular charge-transfer absorption/fluorescence transitions between the π orbital of the aryl group (π(aryl)) and the vacant p orbital on the boron atom (p(B), π(aryl)-p(B) CT), and the fluorescence color was tunable from blue to red by the combination of peripheral electron-donating groups and a triarylborane core. Detailed electrochemical, spectroscopic, and photophysical studies of the derivatives, including solvent dependences of the spectroscopic and photophysical properties, demonstrated that the HOMO and LUMO of each derivative were determined primarily by the nature of the peripheral electron-donating group and the triarylborane core, respectively. The effects of solvent polarity on the fluorescence quantum yield and lifetime of the derivatives were also tunable by the choice of the triarylborane core.

The effects of electron transfer on the energy loss of slow He²⁺, C²⁺, and C⁴⁺ ions penetrating a graphene fragment

Electronic energy loss in the collision processes of slow ions with a graphene fragment is investigated by combining ab initio time-dependent density functional theory calculations for electrons with molecular dynamics simulations for ions in real time and real space. We study the electronic energy loss of slow He²⁺, C²⁺, and C⁴⁺ ions penetrating the graphene fragment as a function of the ion velocity, and establish the velocity-proportional energy loss for low-charged ions down to 0.1 a.u. One mechanism clarified in the simulations for electron transfer is polarization capture, which is effective for bare ions at low velocities. The other one is resonance capture, by which the incident ion can capture electrons from the graphene fragment to its electron affinity levels, which have the same, or nearly the same, energy as those of the electron donor levels. The results demonstrate that the nonlinear behavior of energy loss of C⁴⁺ is attributed to the large number of electrons captured by this multi-charged ion during the collision.

A minimal model for excitons within time-dependent density-functional theory

The accurate description of the optical spectra of insulators and semiconductors remains an important challenge for time-dependent density-functional theory (TDDFT). Evidence has been given in the literature that TDDFT can produce bound as well as continuum excitons for specific systems, but there are still many unresolved basic questions concerning the role of dynamical exchange and correlation (xc). In particular, the roles of the long spatial range and the frequency dependence of the xc kernel f(xc) for excitonic binding are still not very well explored. We present a minimal model for excitons in TDDFT, consisting of two bands from a one-dimensional (1D) Kronig-Penney model and simple approximate xc kernels, providing an easily accessible model system for studying excitonic effects in TDDFT. For the 1D model system, it is found that adiabatic xc kernels can produce at most two bound excitons, confirming that the long spatial range of f(xc) is not a necessary condition. It is shown how the Wannier model, featuring an effective electron-hole interaction, emerges from TDDFT. The collective, many-body nature of excitons is explicitly demonstrated.

A non-self-consistent range-separated time-dependent density functional approach for large-scale simulations

We propose an efficient method for carrying out time-dependent density functional theory (TDDFT) calculations using range-separated hybrid exchange-correlation functionals. Based on a non-self-consistent range-separated Hamiltonian, the method affords large-scale simulations at a fraction of the computational time of conventional hybrid TDDFT approaches. For typical benchmark molecules including N(2), CO, C(6)H(6), H(2)CO and the C(2)H(4)-C(2)F(4) dimer, the method possesses the same level of accuracy as the conventional approaches for the valence, Rydberg, and charge-transfer excitation energies when compared to the experimental results. The method is used to determine π → π* excitations in both disordered and crystalline poly(3-hexylthiophene) (P3HT) conjugated polymers with more than six hundred atoms and it yields excitation energies and charge densities that are in excellent agreement with experiments. The simulation of the crystalline P3HT reveals that the phase of the wavefunctions could have an important effect on the excitation energy; a hypothesis based on π-π stacking is proposed to explain this novel effect in conjugated polymers.

INVESTIGATION OF DOMINANT ELECTRON CONFIGURATIONS IN TIME-DEPENDENT DENSITY FUNCTIONAL THEORY

We investigated the electron configurations that are dominant in excited states of molecules in time-dependent density functional theory (TDDFT). By taking advantage of the discussion on off-diagonal elements in the TDDFT response matrix (Appel et al., Phys Rev Lett, 90, 043005, 2003), we can pick up electron transitions that contribute to an excitation of interest by making use of the diagonal elements of the TDDFT matrix. We can obtain approximate excitation energies by calculating a TDDFT submatrix, which is contracted for a list of collected transitions. This contracted TDDFT was applied to the calculation of excitation energies of the CO molecule adsorbing Pt 10 cluster and some prototype small molecules. Calculated results showed that a TDDFT excitation energy is dominated by a few electron configurations, unless severe degeneracy is involved.

LONG-RANGE-CORRECTED TIME-DEPENDENT DENSITY FUNCTIONAL STUDY ON ELECTRONIC SPECTRA OF FIVE-MEMBERED RING COMPOUNDS AND FREE-BASE PORPHYRIN

Long-range-corrected time-dependent density functional theory (LC-TDDFT) was applied to five-membered ring compounds (cyclopentadiene, pyrrole, and furan molecules) and free-base porphyrin. The vertical π-π* and Rydberg excitation energies and corresponding oscillator strengths were calculated by LC-TDDFT. The LC scheme obviously improved the Rydberg excitation energies and oscillator strengths of these systems, which have been fairly underestimated by TDDFT with conventional pure and hybrid B3LYP functionals. On the whole, LC-TDDFT results were very close to the results of the ab initio symmetry-adapted cluster configuration interaction (SAC-CI) method for most excitations. However, LC-TDDFT is poor in describing doubly excited states such as the 1 1A1 state of five-membered ring compounds.

Excited-State Electronic Structure with Configuration Interaction Singles and Tamm-Dancoff Time-Dependent Density Functional Theory on Graphical Processing Units

Excited-state calculations are implemented in a development version of the GPU-based TeraChem software package using the configuration interaction singles (CIS) and adiabatic linear response Tamm–Dancoff time-dependent density functional theory (TDA-TDDFT) methods. The speedup of the CIS and TDDFT methods using GPU-based electron repulsion integrals and density functional quadrature integration allows full ab initio excited-state calculations on molecules of unprecedented size. CIS/6-31G and TD-BLYP/6-31G benchmark timings are presented for a range of systems, including four generations of oligothiophene dendrimers, photoactive yellow protein (PYP), and the PYP chromophore solvated with 900 quantum mechanical water molecules. The effects of double and single precision integration are discussed, and mixed precision GPU integration is shown to give extremely good numerical accuracy for both CIS and TDDFT excitation energies (excitation energies within 0.0005 eV of extended double precision CPU results).