Time-dependent density functional theory for calculating origin-independent optical rotation and rotatory strength tensors

An approach to calculate origin-independent electronic chiroptical property tensors using time-dependent density functional theory (TDDFT) and gauge-including atomic orbital (GIAO) basis sets is evaluated. Computations of origin-dependent optical rotation tensors and of rotatory strengths needed to simulate circular dichroism spectra are presented. The optical rotation tensor computations employ solutions of coupled perturbed Kohn-Sham equations for a dynamic electric field and a static magnetic field. Because the magnetic field is time independent, the GIAO treatment is somewhat simplified compared to a previously reported method, at some added computational cost if hybrid functionals are employed. GIAO rotatory strengths are also calculated, using transition density matrices from a standard TDDFT excitation energy module. A new implementation in the NWChem quantum chemistry package is employed for representative computations of origin-invariant chiroptical response tensors for methyloxirane, norbornenone, and the ketosteroid androstadienone. For the steroid molecule the vibrational structure of the CD spectrum is modeled explicitly by using calculated Franck-Condon factors. The agreement with experiment is favorable.     

Time-dependent density functional theory of open quantum systems in the linear-response regime

Time-dependent density functional theory (TDDFT) has recently been extended to describe many-body open quantum systems evolving under nonunitary dynamics according to a quantum master equation. In the master equation approach, electronic excitation spectra are broadened and shifted due to relaxation and dephasing of the electronic degrees of freedom by the surrounding environment. In this paper, we develop a formulation of TDDFT linear-response theory (LR-TDDFT) for many-body electronic systems evolving under a master equation, yielding broadened excitation spectra. This is done by mapping an interacting open quantum system onto a noninteracting open Kohn-Sham system yielding the correct nonequilibrium density evolution. A pseudoeigenvalue equation analogous to the Casida equations of the usual LR-TDDFT is derived for the Redfield master equation, yielding complex energies and Lamb shifts. As a simple demonstration, we calculate the spectrum of a C(2 +) atom including natural linewidths, by treating the electromagnetic field vacuum as a photon bath. The performance of an adiabatic exchange-correlation kernel is analyzed and a first-order frequency-dependent correction to the bare Kohn-Sham linewidth based on the Go?rling-Levy perturbation theory is calculated.     

Response calculations based on an independent particle system with the exact one-particle density matrix: excitation energies

Adiabatic response time-dependent density functional theory (TDDFT) suffers from the restriction to basically an occupied → virtual single excitation formulation. Adiabatic time-dependent density matrix functional theory allows to break away from this restriction. Problematic excitations for TDDFT, viz. bonding-antibonding, double, charge transfer, and higher excitations, are calculated along the bond-dissociation coordinate of the prototype molecules H(2) and HeH(+) using the recently developed adiabatic linear response phase-including (PI) natural orbital theory (PINO). The possibility to systematically increase the scope of the calculation from excitations out of (strongly) occupied into weakly occupied ("virtual") natural orbitals to larger ranges of excitations is explored. The quality of the PINO response calculations is already much improved over TDDFT even when the severest restriction is made, to virtually the size of the TDDFT diagonalization problem (only single excitation out of occupied orbitals plus all diagonal doubles). Further marked improvement is obtained with moderate extension to allow for excitation out of the lumo and lumo+1, which become fractionally occupied in particular at longer distances due to left-right correlation effects. In the second place the interpretation of density matrix response calculations is elucidated. The one-particle reduced density matrix response for an excitation is related to the transition density matrix to the corresponding excited state. The interpretation of the transition density matrix in terms of the familiar excitation character (single excitations, double excitations of various types, etc.) is detailed. The adiabatic PINO theory is shown to successfully resolve the problematic cases of adiabatic TDDFT when it uses a proper PI orbital functional such as the PILS functional.     

Adiabatic approximation of time-dependent density matrix functional response theory

Time-dependent density matrix functional theory can be formulated in terms of coupled-perturbed response equations, in which a coupling matrix K(omega) features, analogous to the well-known time-dependent density functional theory (TDDFT) case. An adiabatic approximation is needed to solve these equations, but the adiabatic approximation is much more critical since there is not a good "zero order" as in TDDFT, in which the virtual-occupied Kohn-Sham orbital energy differences serve this purpose. We discuss a simple approximation proposed earlier which uses only results from static calculations, called the static approximation (SA), and show that it is deficient, since it leads to zero response of the natural orbital occupation numbers. This leads to wrong behavior in the omega-->0 limit. An improved adiabatic approximation (AA) is formulated. The two-electron system affords a derivation of exact coupled-perturbed equations for the density matrix response, permitting analytical comparison of the adiabatic approximation with the exact equations. For the two-electron system also, the exact density matrix functional (2-matrix in terms of 1-matrix) is known, enabling testing of the static and adiabatic approximations unobscured by approximations in the functional. The two-electron HeH(+) molecule shows that at the equilibrium distance, SA consistently underestimates the frequency-dependent polarizability alpha(omega), the adiabatic TDDFT overestimates alpha(omega), while AA improves upon SA and, indeed, AA produces the correct alpha(0). For stretched HeH(+), adiabatic density matrix functional theory corrects the too low first excitation energy and overpolarization of adiabatic TDDFT methods and exhibits excellent agreement with high-quality CCSD ("exact") results over a large omega range.     

Time-dependent density functional theory based on a noncollinear formulation of the exchange-correlation potential

In this study we have introduced a formulation of time-dependent density functional theory (TDDFT) based on a noncollinear exchange-correlation potential. This formulation is a generalization of conventional TDDFT. The form of this formulation is exactly the same as that of the conventional TDDFT for the excitation energies of transitions that do not involve spin flips. In addition, this noncollinear TDDFT formulation allows for spin-flip transitions. This feature makes it possible to resolve more fully excited state spin multiplets, while for closed-shell systems, the spin-flip transitions will result in singlet-triplet excitations and this excitation energy calculated from this formulation of TDDFT is exactly the same as that from ordinary TDDFT. This formulation is applied to the dissociation of H(2) in its (1)Sigma(g) (+) ground state and (1)Sigma(u) (+) and (3)Sigma(u) (-) excited states with (3)Sigma(u) (-) (M(s)=+1) as the reference state and the multiplets splitting of some atoms.     

Analytic techniques for the coupled multiconfiguration SCF perturbation theory

The perturbation theory based on the paired excitation multiconfiguration self-consistent field approach of Clementi and Veillard is considered. The coupled first-order perturbed orbital equations are analysed and an appropriate computational scheme for their solution is discussed. The proposed computational scheme is analogous to the technique employed for the solution of the coupled Hartree–Fock equations in the one-configuration approximation. However, because of the presence ofnondiagonal Lagrangian multipliers and the use of different one-electron operators for different orbitals, the present scheme raises some new computational problems. In this context a new technique for the solution of the unperturbed multiconfiguration self-consistent field equations is proposed. A simple illustration of the superiority of the multiconfiguration perturbation approach with respect to the ordinary coupled Hartree–Fock scheme is given. Also the validity of the variation formulation of the presented scheme and its relation to the finite-field approach are discussed.     

Couplings between electronic transitions in a subsystem formulation of time-dependent density functional theory

A subsystem formulation of time-dependent density functional theory (TDDFT) within the frozen-density embedding (FDE) framework and its practical implementation are presented, based on the formal TDDFT generalization of the FDE approach by Casida and Wesolowski [Int. J. Quantum Chem. 96, 577 (2004)]. It is shown how couplings between electronic transitions on different subsystems can be seamlessly incorporated into the formalism to overcome some of the shortcomings of the approximate TDDFT-FDE approach in use so far, which was only applicable for local subsystem excitations. In contrast to that, the approach presented here allows to include couplings between excitations on different subsystems, which become very important in aggregates composed of several similar chromophores, e.g., in biological or biomimetic light-harvesting systems. A connection to Forster- and Dexter-type excitation energy coupling expressions is established. A hybrid approach is presented and tested, in which excitation energy couplings are selectively included between different chromophore fragments, but neglected for inactive parts of the environment. It is furthermore demonstrated that the coupled TDDFT-FDE approach can cure the inability of the uncoupled FDE approach to describe induced circular dichroism in dimeric chromophores, a feature known as a "couplet," which is also related to couplings between (nearly) degenerate electronic transitions.     

Tailoring transition metal complexes for nonlinear optics applications. 2. A theoretical investigation of the second-order nonlinear optical properties of M(CO)(5)L complexes (M = Cr, W; L = Py, PyCHO, Pyz, PyzBF(3), BPE, BPEBF(3))

In this work, we report an ab initio investigation of second-order nonlinear optical (NLO) properties and absorption electronic spectra of push-pull transition metal chromophores of the formula [M(CO)(5)L] (M = Cr, W; L = pyridine (Py), 4-formyl-pyridine (PyCHO), pyrazine (Pyz), trans-1,2-bis(4-pyridyl)ethylene (BPE)). Pyz and BPE are considered either with one nitrogen atom free or interacting with the strong acceptor BF(3). All of the molecular properties have been calculated using two different and methodologically independent approaches: the time dependent and coupled perturbed density functional theories (TDDFT and CPDFT) and the sum-over-states (SOS) approach, where the excited states are obtained via the single configuration interaction (SCI) ab initio method. DFT results are in acceptable agreement with the experimental energy values of electronic transitions (with the exception of chromophores with the large pi-delocalization, like BPE); SCI calculations overestimate excitation energies and produce an inversion in the order of d(M) --> pi(L) and d(M) --> pi(CO) transitions. The SCI-SOS approach gives first-order hyperpolarizabilities, basically in agreement as trend and values with the experiments and seems to be a tool generally suitable for the evaluation of these properties also for transition metal complexes. On the other hand, the first-order hyperpolarizabilities computed using the CPDFT approach are consistently overestimated in comparison with the experimental results, especially in the case of a ligand with large pi-delocalization. We also show that the "two-level" approximation taking into account only the lowest energy charge transfer excitation (e.g., d(M) --> pi(L)) is not applicable to chromophores with the extended pi-delocalized ligand (BPE) coordinated to a transition metal, due to significant contributions originating from intraligand pi(L) --> pi(L) transitions. This study reports a detailed analysis and comparison of electronic NLO effects of transition metal complexes computed with DFT and ab initio SCI-SOS methodology.     

Optoelectronic Properties of Carbon Nanorings: Excitonic Effects from Time-Dependent Density Functional Theory

The electronic structure and size-scaling of optoelectronic properties in cycloparaphenylene carbon nanorings are investigated using time-dependent density functional theory (TDDFT). The TDDFT calculations on these molecular nanostructures indicate that the lowest excitation energy surprisingly becomes larger as the carbon nanoring size is increased, in contradiction with typical quantum confinement effects. In order to understand their unusual electronic properties, I performed an extensive investigation of excitonic effects by analyzing electron-hole transition density matrices and exciton binding energies as a function of size in these nanoring systems. The transition density matrices allow a global view of electronic coherence during an electronic excitation, and the exciton binding energies give a quantitative measure of electron-hole interaction energies in the nanorings. Based on overall trends in exciton binding energies and their spatial delocalization, I find that excitonic effects play a vital role in understanding the unique photoinduced dynamics in these carbon nanoring systems.     

Spin-multiplet energies from time-dependent density functional theory

Starting from a formally exact density-functional representation of the frequency-dependent linear density response and exploiting the fact that the latter has poles at the true excitation energies, we develop a density-functional method for the calculation of excitation energies. Simple additive corrections to the Kohn-Sham single-particle transition energies are derived whose actual computation only requires the ordinary static Kohn-Sham orbitals and the corresponding eigenvalues. Numerical results are presented for spin-singlet and triplet energies. © 1996 John Wiley & Sons, Inc.     

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.     

Energy levels for perturbed Morse oscillators

The energy levels and perturbation expansions for the expectation values of arbitrary powers of position for a perturbed Morse oscillator are obtained by application of the hypervirial and Hellmann–Feynman theorems, solely in terms of the unperturbed energy. We obtain expressions for the first-order corrections for (1 ? e^?aq )^m for 4 ? m ? 8 and the expressions to second and third order for the quartic perturbation. A numerical application to the CO molecule is made.     

Inexpensive and accurate predictions of optical excitations in transition-metal complexes: the TDDFT/PBE0 route

Time-dependent density functional theory (TDDFT) is applied to calculate vertical excitation energies of three representative transition metal complexes. The computational model (PBE0) is obtained by combining the Perdew-Burke-Erzenrhof (PBE) generalized gradient functional with a predetermined amount of exact exchange. Our results show that the TDDFT/PBE0 model represents a cheap and reliable tool for the computation of optical excitations for transition metal complexes. Received: 8 August 2000 / Accepted: 7 September 2000 / Published online: 23 November 2000     

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.     

Thermodynamics of interface elasticity

We generalize and extend a series of articles on the thermodynamics of interface elasticity. The whole formulation is based on taking into account a finite elastic deformation. This leads to generalized forms of the equations for the surface excess of the internal energy, the Gibbs adsorption equation and the Shuttleworth equation. In all cases correction terms occur containing the strain tensor. Although the deformation might be small, the corrections are critical in performing derivatives with respect to the strain which are needed to obtain equations connecting measurable quantities. These consequences are demonstrated in detail for the simple case of the spherical electrode. In the traditional electrochemical treatment, the densities of extensive quantities are related to the deformed surface, which is usually not mentioned. In analogy to the theory of volume elasticity, where the densities are related to the unstrained solid, one can also relate the densities of surface excess quantities to the unstrained surface. This formulation gives at first an additional proof for some of the generalized equations. Moreover, within this formulation one can express the surface excess of the energy density by the superficial work of the unstrained surface and three surface elastic constants, two of them being the surface Lamé constants. In the alternative formulation, the Shuttleworth equation appears simply as the relation between a generalized force and a generalized potential. Published in Elektrokhimiya in Russian, 2009, Vol. 45, No. 1, pp. 78–86. The text was submitted by the authors in English. This study was prepared for the special issue devoted to the 100th anniversary of B.V. Ershler.     

Time-dependent density functional theory description of on-site electron repulsion and ligand field effects in the optical spectrum of hexaaquoruthenium(II) in solution

Time-dependent density functional theory (TDDFT) and density functional-based molecular dynamics were used to simulate the finite temperature t(2g)5 e(g) <-- t(2g)6 absorption band of the Ru2+ hexahydrate coordination complex in aqueous solution. The (1)T1 <-- (1)A1 and (1)T2 <-- (1)A1 molecular term splitting of this transition, which is not accounted for by the Kohn-Sham excitation spectrum, is shown to be satisfactorily reproduced by TDDFT at the BLYP/ALDA level of theory. Comparison to the spectrum of the Ru2+ (H2O)6 complex in vacuo computed by similar density functional classical molecular dynamics methods leads to the observation that bulk solvation has a negligible effect on the position and the shape of the absorption profile.     

Failure of time-dependent density functional theory for long-range charge-transfer excited states: the zincbacteriochlorin-bacteriochlorin and bacteriochlorophyll-spheroidene complexes

It is well-known that time-dependent density functional theory (TDDFT) yields substantial errors for the excitation energies of charge-transfer (CT) excited states, when approximate standard exchange-correlation (xc) functionals are used, for example, SVWN, BLYP, or B3LYP. Also, the correct 1/R asymptotic behavior of CT states with respect to a distance coordinate R between the separated charges of the CT state is not reproduced by TDDFT employing these xc-functionals. Here, we demonstrate by analysis of the TDDFT equations that the first failure is due to the self-interaction error in the orbital energies from the ground-state DFT calculation, while the latter is a similar self-interaction error in TDDFT arising through the electron transfer in the CT state. Possible correction schemes, such as inclusion of exact Hartree-Fock or exact Kohn-Sham exchange, as well as aspects of the exact xc-functional are discussed in this context. Furthermore, a practical approach is proposed which combines the benefits of TDDFT and configuration interaction singles (CIS) and which does not suffer from electron-transfer self-interaction. The latter approach is applied to a (1,4)-phenylene-linked zincbacteriochlorin-bacteriochlorin complex and to a bacteriochlorophyll-spheroidene complex, in which CT states may play important roles in energy and electron-transfer processes. The errors of TDDFT alone for the CT states are demonstrated, and reasonable estimates for the true excitation energies of these states are given.     

Role of polyols (erythritol, xylitol and sorbitol) on the structural stabilization of collagen

A state-specific scheme for time-dependent density functional theory (SS-TDDFT) based on the Davidson algorithm is presented. SS-TDDFT is a method devised for speeding up TDDFT calculations by screening transitions that contribute to a specific excitation. By applying this method to calculations of the low-lying excitation energies of test molecules (N₂, CO, H₂CO, C₂H₄ and C₆H₆), water clusters and polyenes, we found that SS-TDDFT accurately reproduced the excitation energies of standard TDDFT while drastically reducing the rank of the TDDFT response matrix without loss of accuracy. We have thus formulated TDDFT that works more efficiently and economically for memory storage.     

Frequency dependent nonlinear optical properties of molecules: Formulation and implementation in the HONDO program

This article summarizes the detailed equations for the time-dependent Hartree–Fock treatment of nonlinear properties for perturbations made up of a static electric field and an oscillating field. Explicit expressions for all nonlinear processes up to third order are obtained in terms of the density matrices at the same order. For processes at second and third order in perturbation, expressions in terms of lower order quantities are also obtained by applying the (2n + 1) theorem of perturbation theory. The corresponding computer implementation in the HONDO program is described.