Hartree-Fock and Kohn-Sham time-dependent response theory in a second-quantization atomic-orbital formalism suitable for linear scaling

We present a second-quantization based atomic-orbital method for the computation of time-dependent response functions within Hartree-Fock and Kohn-Sham density-functional theories. The method is suited for linear scaling. Illustrative results are presented for excitation energies, one- and two-photon transition moments, polarizabilities, and hyperpolarizabilities for hexagonal BN sheets with up to 180 atoms.     

Accurate singlet and triplet excitation energies using the Localized Hartree-Fock Kohn-Sham potential

We assess the accuracy of the LHFX Time-Dependent Density-Functional Theory (TD-DFT) approach, which uses Kohn-Sham orbitals and eigenvalues from the Localized Hartree-Fock (LHF) method and the exchange-only adiabatic local density approximation kernel. We compute 172 singlet and triplet excitation energies of π → π^∗, n → π^∗, σ → π^∗ and Rydberg character, for organic molecules of different size. We find that the LHFX method, which is free from the Self-Interaction-Error (SIE) and from empirical parameters, outperforms the state-of-the-art hybrid TD-DFT approaches, and provides the same accuracy for all different classes of excitations. The SIE-free Kohn-Sham orbitals can be thus considered as starting point for TD-DFT developments.     

On the gauge invariance of the schrodinger equation

The problem of gauge invariance in relation to (approximate) calculations of molecular properties is considered. We show that these difficulties are not avoided by making a unitary transformation to a formalism which is independent of the electromagnetic field potentials.     

A comparative analysis of Hartree-Fock and Kohn-Sham orbital energies

Hartree-Fock and Kohn-Sham orbital energies, the latter computed with several different exchange/correlation functionals, are compared and analyzed for 12 molecules. The Kohn-Sham energies differ significantly from experimental ionization energies, but by amounts that are, for a given molecule and exchange/correlation functional, roughly the same for all of the valence orbitals. With the exchange/correlation functionals used, the energy of the highest occupied Kohn-Sham orbital does not approximate the corresponding ionization potential any better than do the other orbital energies. Received: 24 October 1997 / Accepted 31 October 1997     

Regarding the validity of the time-dependent Kohn-Sham approach for electron-nuclear dynamics via trajectory surface hopping

The implementation of fewest-switches surface-hopping (FSSH) within time-dependent Kohn-Sham (TDKS) theory [Phys. Rev. Lett. 95, 163001 (2005)] has allowed us to study successfully excited state dynamics involving many electronic states in a variety of molecular and nanoscale systems, including chromophore-semiconductor interfaces, semiconductor and metallic quantum dots, carbon nanotubes and graphene nanoribbons, etc. At the same time, a concern has been raised that the KS orbital basis used in the calculation provides only approximate potential energy surfaces [J. Chem. Phys. 125, 014110 (2006)]. While this approximation does exist in our method, we show here that FSSH-TDKS is a viable option for computationally efficient calculations in large systems with straightforward excited state dynamics. We demonstrate that the potential energy surfaces and nonadiabatic transition probabilities obtained within the TDKS and linear response (LR) time-dependent density functional theories (TDDFT) agree semiquantitatively for three different systems, including an organic chromophore ligating a transition metal, a quantum dot, and a small molecule. Further, in the latter case the FSSH-TDKS procedure generates results that are in line with FSSH implemented within LR-TDDFT. The FSSH-TDKS approach is successful for several reasons. First, single-particle KS excitations often give a good representation of LR excitations. In this regard, DFT compares favorably with the Hartree-Fock theory, for which LR excitations are typically combinations of multiple single-particle excitations. Second, the majority of the FSSH-TDKS applications have been performed with large systems involving simple excitations types. Excitation of a single electron in such systems creates a relatively small perturbation to the total electron density summed over all electrons, and it has a small effect on the nuclear dynamics compared, for instance, with thermal nuclear fluctuations. In such cases an additional, classical-path approximation can be made. Third, typical observables measured in time-resolved experiments involve averaging over many initial conditions. Such averaging tends to cancel out random errors that may be encountered in individual simulated trajectories. Finally, if the flow of energy between electronic and nuclear subsystems is insignificant, the ad hoc FSSH procedure is not required, and a straightforward mean-field, Ehrenfest approach is sufficient. Then, the KS representation provides rigorously a convenient and efficient basis for numerically solving the TDDFT equations of motion.     

On correlated electron-nuclear dynamics using time-dependent density functional theory

We discuss possibilities and challenges for describing correlated electron and nuclear dynamics within a surface-hopping framework using time-dependent density functional theory (TDDFT) for the electron dynamics. We discuss the recent surface-hopping method proposed by Craig et al. [Phys. Rev. Lett. 95, 163001 (2005)] that is based on Kohn-Sham potential energy surfaces. Limitations of this approach arise due to the Kohn-Sham surfaces generally having different gradients than the true TDDFT-corrected ones. Two mechanisms of the linear response procedure cause this effect: we illustrate these with examples.     

Modeling ultrafast solvated electronic dynamics using time-dependent density functional theory and polarizable continuum model

A first-principles solvated electronic dynamics method is introduced. Solvent electronic degrees of freedom are coupled to the time-dependent electronic density of a solute molecule by means of the implicit reaction field method, and the entire electronic system is propagated in time. This real-time time-dependent approach, incorporating the polarizable continuum solvation model, is shown to be very effective in describing the dynamical solvation effect in the charge transfer process and yields a consistent absorption spectrum in comparison to the conventional linear response results in solution.     

Dynamics of molecules in strong oscillating electric fields using time-dependent Hartree-Fock theory

Restricted and unrestricted forms of time-dependent Hartree-Fock theory have been implemented and used to study the electronic dynamics of ethene, benzene, and the formaldehyde cation subjected to both weak and strong oscillating electric fields. Absorption spectra and frequency-dependent polarizabilities are calculated via the instantaneous dipole moment and its derivative. In the weak field limit the computed excitation energies agree very well with those obtained using linearized time-dependent Hartree-Fock theory, which is valid only in the low-field perturbation limit. For strong fields the spectra show higher-order excitations, and a shift in the position of the excitations, which is due to the nonadiabatic response of the molecules to the field. For open-shell systems in the presence of strong oscillating electric fields, unrestricted time-dependent Hartree-Fock theory predicts the value of S(2) to vary strongly with time.     

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.     

Exact Kohn-Sham versus Hartree-Fock in momentum space: Examples of two-fermion systems

The question of how density functional theory (DFT) compares with Hartree-Fock (HF) for the computation of momentum-space properties is addressed in relation to systems for which (near) exact Kohn-Sham (KS) and HF one-electron matrices are known. This makes it possible to objectively compare HF and exact KS and hence to assess the potential of DFT for momentum-space studies. The systems considered are the Moshinsky [Am. J. Phys. 36, 52 (1968)] atom, Hooke's atom, and light two-electron ions, for which expressions for correlated density matrices or momentum densities have been derived in closed form. The results obtained show that it is necessary to make a distinction between true and approximate DFTs.     

Quadratic response functions in the time-dependent four-component Hartree-Fock approximation

The second-order response function has been implemented in the time-dependent four-component Hartree-Fock approximation. The implementation is atomic orbital direct and formulated in terms of Fock-type matrices. It employs a quaternion symmetry scheme that provides maximum computational efficiency with consideration made to time-reversal and spatial symmetries. Calculations are presented for the electric dipole first-order hyperpolarizabilities of CsAg and CsAu in the second-harmonic generation optical process beta(-2omega;omega,omega). It is shown that relativistic corrections to property values are substantial in these cases--the orientationally averaged hyperpolarizabilities in the static limit beta(0;0,0) are overestimated in nonrelativistic calculations by 18% and 66% for CsAg and CsAu, respectively. The dispersion displays anomalies in the band gap region due to one- and two-photon resonances with nonrelativistically spin-forbidden states. Although weakly absorbing these states inflict divergences in the quadratic response function, since the response theoretical approach which is used adopts the infinite excited-state lifetime approximation. This fact calls for caution in applications where knowledge of the exact positioning of all excited states in the spectrum is unknown.     

Origin of electronic structure and time-dependent state averaging in the multi-configuration time-dependent Hartree–Fock approach to electron dynamics

In this Letter, we discuss problems that can arise in the interpretation of results obtained by quantum dynamical simulations with the multi-configuration time-dependent Hartree–Fock (MCTDHF) method. In particular, we show that an effect, which can be seen as the time-dependent version of the state averaging known from standard quantum chemistry, can modify the electronic structure as derived from such a simulation. We illustrate our findings with numerical calculations for the laser excitation of a water molecule, with a Gaussian Type Orbital basis set.     

Theoretical studies on electronic spectroscopy and dynamics with the real-time time-dependent density functional theory

Time-dependent density functional theory (TDDFT) has evolved into a general routine to extract the energies of low-lying excited states over the last decades. Driven by the remarkable progress of laser technology, the study of the interaction between matter and intense laser fields with ultrashort pulse duration develops rapidly. A great number of new strong field phenomena emerge. The requirement of a theoretical tool to study the intense field phenomena and dynamical processes of polyatomic systems is urgent. To extend the power of the TDDFT beyond the linear responses, an alternative scheme has been developed by numerically solving the time-dependent Kohn-Sham equations directly in real-time domain. In this article, we summarize the algorithms and capabilities of the real-time TDDFTon studying electron spectroscopy and dynamics of polyatomic systems. The failure of TDDFT with the adiabatic localdensity approximation on some dynamical processes and the possible solutions are synopsized as well. The numerical implementation of algorithms and applications of RT-TDDFT on the linear and nonlinear spectroscopies and electronic dynamics of nano-size nonmetal clusters are displayed.     

Two-photon ionization of helium studied with the multiconfigurational time-dependent Hartree-Fock method

The multiconfigurational time-dependent Hartree-Fock method (MCTDHF) is applied for simulations of the two-photon ionization of helium. We present results for the single and double ionizations from the ground state for photon energies in the nonsequential regime and compare them to direct solutions of the Schro?dinger equation using the time-dependent (full) configuration interaction (TDCI) method. We find that the single ionization is accurately reproduced by MCTDHF, whereas the double ionization results correctly capture the main trends of TDCI.     

Nonvariational time-dependent multiconfiguration self-consistent field equations for electronic dynamics in laser-driven molecules

A time-dependent multiconfiguration self-consistent field (TDMCSCF) scheme is developed to describe the time-resolved electron dynamics of a laser-driven many-electron atomic or molecular system, starting directly from the time-dependent Schrodinger equation for the system. This nonvariational formulation aims at the full exploitations of concepts, tools, and facilities of existing, well-developed quantum chemical MCSCF codes. The theory uses, in particular, a unitary representation of time-dependent configuration mixings and orbital transformations. Within a short-time, or adiabatic approximation, the TDMCSCF scheme amounts to a second-order split-operator algorithm involving generically the two noncommuting one-electron and two-electron parts of the time-dependent electronic Hamiltonian. We implement the scheme to calculate the laser-induced dynamics of the two-electron H2 molecule described within a minimal basis, and show how electron correlation is affected by the interaction of the molecule with a strong laser field.     

The trust-region self-consistent field method: towards a black-box optimization in Hartree-Fock and Kohn-Sham theories

The trust-region self-consistent field (TRSCF) method is presented for optimizing the total energy E(SCF) of Hartree-Fock theory and Kohn-Sham density-functional theory. In the TRSCF method, both the Fock/Kohn-Sham matrix diagonalization step to obtain a new density matrix and the step to determine the optimal density matrix in the subspace of the density matrices of the preceding diagonalization steps have been improved. The improvements follow from the recognition that local models to E(SCF) may be introduced by carrying out a Taylor expansion of the energy about the current density matrix. At the point of expansion, the local models have the same gradient as E(SCF) but only an approximate Hessian. The local models are therefore valid only in a restricted region-the trust region-and steps can only be taken with confidence within this region. By restricting the steps of the TRSCF model to be inside the trust region, a monotonic and significant reduction of the total energy is ensured in each iteration of the TRSCF method. Examples are given where the TRSCF method converges monotonically and smoothly, but where the standard DIIS method diverges.     

Role of Hartree-Fock and Kohn-Sham orbitals in the basis set superposition error for systems linked by hydrogen bonds

In this work the effect of the basis set superposition error (BSSE) is explored with the counterpoise method on the occupied and unoccupied Hartree-Fock (HF) and Kohn-Sham (KS) orbitals. Three different systems linked by hydrogen bonds, H(2)O...FH, H(2)O...H(2)O, and H(2)O...CFH(3), were studied by using the basis set families cc-pVXZ and aug-cc-pVXZ (X = D, T, Q). The basis sets were tested with the HF method and two approximations for the exchange-correlation functional of KS: a generalized gradient approximation and a hybrid approach. In addition to these methods, the second-order M?ller-Plesset perturbation theory, MP2, was considered. It was found that the presence of the "ghost" basis set affects the orbitals in two ways: (1) The occupied KS orbitals are more sensitive to the presence of this "ghost" basis set than the occupied HF orbitals. For this reason the BSSE observed in HF is less than that obtained with KS. (2) The unoccupied HF orbitals are more sensitive to the presence of the "ghost" basis set than their corresponding occupied orbitals. Because the MP2 method uses both, occupied and unoccupied HF orbitals, to compute the total energy, the contribution of the BSSE is bigger than that obtained with HF or KS methodologies.