Erratum: “Calculating frequency-dependent hyperpolarizabilities using time-dependent density functional theory” [J. Chem. Phys. 109, 10644 (1998)]

An accurate determination of frequency-dependent molecular hyperpolarizabilities is at the same time of possible technological importance and theoretically challenging. For large molecules, Hartree–Fock theory was until recently the only available ab initio approach. However, correlation effects are usually very important for this property, which makes it desirable to have a computationally efficient approach in which those effects are (approximately) taken into account. We have recently shown that frequency-dependent hyperpolarizabilities can be efficiently obtained using time-dependent density functional theory. Here, we shall present the necessary theoretical framework and the details of our implementation in the Amsterdam Density Functional program. Special attention will be paid to the use of fit functions for the density and to numerical integration, which are typical of density functional codes. Numerical examples for He, CO, and para-nitroaniline are presented, as evidence for the correctness of the equations and the implementation.<br/

Accurate density functional calculations on frequency-dependent hyperpolarizabilities of small molecules

In this paper we present time-dependent density functional calculations on frequency-dependent first (β) and second (γ) hyperpolarizabilities for the set of small molecules,

Improved density functional theory results for frequency-dependent polarizabilities, by the use of an exchange-correlation potential with correct asymptotic behavior.

The exchange‐correlation potentials vxc which are currently fashionable in density functional theory (DFT), such as those obtained from the local density approximation (LDA) or generalized gradient approximations (GGAs), all suffer from incorrect asymptotic behavior. In atomic calculations, this leads to substantial overestimations of both the static polarizability and the frequency dependence of this property. In the present paper, it is shown that the errors in atomic static dipole and quadrupole polarizabilities are reduced by almost an order of magnitude, if a recently proposed model potential with correct Coulombic long‐range behavior is used. The frequency dependence is improved similarly. The model potential also removes the overestimation in molecular polarizabilities, leading to slight improvements for average molecular polarizabilities and their frequency dependence. For the polarizability anisotropy we find that the model potential results do not improve over the LDA and GGA results. Our method for calculating frequency‐dependent molecular response properties within time‐dependent DFT, which we described in more detail elsewhere, is summarized

Exchange and Correlation Kernels at the Resonance Frequency -- Implications for Excitation Energies in Density-Functional Theory

Specific matrix elements of exchange and correlation kernels in time-dependent density-functional theory are computed. The knowledge of these matrix elements not only constraints approximate time-dependent functionals, but also allows to link different practical approaches to excited states, either based on density-functional theory, or on many-body perturbation theory, despite the approximations that have been performed to derive them.Comment: Submitted to Phys. Rev. Lett. (February 4, 1999). Other related publications can be found at http://www.fhi-berlin.mpg.de/th/paper.htm

Ferromagnetism in Mn doped GaAs due to substitutional-interstitial complexes

While most calculations on the properties of the ferromagnetic semiconductor GaAs:Mn have focussed on isolated Mn substituting the Ga site (Mn$_{Ga}$), we investigate here whether alternate lattice sites are favored and what the magnetic consequences of this might be. Under As-rich (Ga-poor) conditions prevalent at growth, we find that the formation energies are lower for Mn$_{Ga}$ over interstitial Mn (Mn$_i$).As the Fermi energy is shifted towards the valence band maximum via external $p$-doping, the formation energy of Mn$_i$ is reduced relative to Mn$_{Ga}$. Furthermore, under epitaxial growth conditions, the solubility of both substitutional and interstitial Mn are strongly enhanced over what is possible under bulk growth conditions. The high concentration of Mn attained under epitaxial growth of p-type material opens the possibility of Mn atoms forming small clusters. We consider various types of clusters, including the Coulomb-stabilized clusters involving two Mn$_{Ga}$ and one Mn$_i$. While isolated Mn$_i$ are hole killers (donors), and therefore destroy ferromagnetism,complexes such as Mn$_{Ga}$-Mn$_i$-Mn$_{Ga}$) are found to be more stable than complexes involving Mn$_{Ga}$-Mn$_{Ga}$-Mn$_{Ga}$. The former complexes exhibit partial or total quenching of holes, yet Mn$_i$ in these complexes provide a channel for a ferromagnetic arrangement of the spins on the two Mn$_{Ga}$ within the complex. This suggests that ferromagnetism in Mn doped GaAs arises both from holes due to isolated Mn$_{Ga}$ as well as from strongly Coulomb stabilized Mn$_{Ga}$-Mn$_i$-Mn$_{Ga}$ clusters.Comment: 7 figure

Initial-state dependence in time-dependent density functional theory

Time-dependent density functionals in principle depend on the initial state of the system, but this is ignored in functional approximations presently in use. For one electron it is shown there is no initial-state dependence: for any density, only one initial state produces a well-behaved potential. For two non-interacting electrons with the same spin in one-dimension, an initial potential that makes an alternative initial wavefunction evolve with the same density and current as a ground state is calculated. This potential is well-behaved and can be made arbitrarily different from the original potential

Ab initio studies of structures and properties of small potassium clusters

We have studied the structure and properties of potassium clusters containing even number of atoms ranging from 2 to 20 at the ab initio level. The geometry optimization calculations are performed using all-electron density functional theory with gradient corrected exchange-correlation functional. Using these optimized geometries we investigate the evolution of binding energy, ionization potential, and static polarizability with the increasing size of the clusters. The polarizabilities are calculated by employing Moller-Plesset perturbation theory and time dependent density functional theory. The polarizabilities of dimer and tetramer are also calculated by employing large basis set coupled cluster theory with single and double excitations and perturbative triple excitations. The time dependent density functional theory calculations of polarizabilities are carried out with two different exchange-correlation potentials: (i) an asymptotically correct model potential and (ii) within the local density approximation. A systematic comparison with the other available theoretical and experimental data for various properties of small potassium clusters mentioned above has been performed. These comparisons reveal that both the binding energy and the ionization potential obtained with gradient corrected potential match quite well with the already published data. Similarly, the polarizabilities obtained with Moller-Plesset perturbation theory and with model potential are quite close to each other and also close to experimental data.Comment: 33 pages including 10 figure

Optical response of small silver clusters

The time-dependent local density approximation is applied to the optical response of the silver clusters, Ag_2, Ag_3, Ag_8 and Ag_9^+. The calculation includes all the electrons beyond the closed-shell Ag^{+11} ionic core, thus including for the first time explicitly the filled d-shell in the response. The excitation energy of the strong surface plasmon near 4 eV agrees well with experiment. The theoretical transition strength is quenched by a factor of 4 with respect to the pure s-electron sum rule in Ag_8 due to the d-electrons. A comparable amount of strength lies in complex states below 6 eV excitation. The total below 6 eV, about 50% of the s sum rule, is consistent with published experiments.Comment: 13 pages RevTex and 9 Postscript figure

Linear ensemble-coding in midbrain superior colliculus specifies the saccade kinematics

Recently, we proposed an ensemble-coding scheme of the midbrain superior colliculus (SC) in which, during a saccade, each spike emitted by each recruited SC neuron contributes a fixed minivector to the gaze-control motor output. The size and direction of this ‘spike vector’ depend exclusively on a cell’s location within the SC motor map (Goossens and Van Opstal, in J Neurophysiol 95: 2326–2341, 2006). According to this simple scheme, the planned saccade trajectory results from instantaneous linear summation of all spike vectors across the motor map. In our simulations with this model, the brainstem saccade generator was simplified by a linear feedback system, rendering the total model (which has only three free parameters) essentially linear. Interestingly, when this scheme was applied to actually recorded spike trains from 139 saccade-related SC neurons, measured during thousands of eye movements to single visual targets, straight saccades resulted with the correct velocity profiles and nonlinear kinematic relations (‘main sequence properties– and ‘component stretching’) Hence, we concluded that the kinematic nonlinearity of saccades resides in the spatial-temporal distribution of SC activity, rather than in the brainstem burst generator. The latter is generally assumed in models of the saccadic system. Here we analyze how this behaviour might emerge from this simple scheme. In addition, we will show new experimental evidence in support of the proposed mechanism