(Time-dependent) Density Functional Theory

University, 13 rue Pierre et Marie Curie, 75005 Paris, France Abstract Electronic circular dichroism is one of the most used spectroscopic techniques for peptide and protein structural characterization. However, while valuable experimental spectra exist for α- helix, β-sheet and random coil secondary structures, previous studies showed important discrepancies for β-turns, limiting their use as reference for structural studies. In this paper, we simulated circular dichroism spectra for the most encountered β-turns in peptides, namely types I, II, I’ and II’. In particular, by combining classical molecular dynamics simulations and state- of-the-art quantum time-dependent density functional theory (with polarizable embedding multiscale model) computations, two common electronic circular dichroism patterns were found for couples of β-turn types (type I/type II’ and type II/type I’). These patterns were subsequently rationalized based on the exciton coupling theory, paving the way for a possible experimental characterization of β-turns based on circular dichroism spectroscopy.

Concerning the TDDFT/ELMO method, it was also proved that, due to the strictly localized treatment of the region involved in the excitation, the novel method intrinsically avoids the presence of spurious low-lying charge-transfer states typical of Time-Dependent Density Functional Theory, even when one uses an exchange- correlation functional that is not long-range corrected. Furthermore, from the application to a reduced model of the Green Fluorescent Protein, it was shown that the new approach can be potentially applied to large systems and that more accurate results can be obtained when a sufficient amount of crucial fragments/residues for the electronic transition under exam is included in the fully quantum mechanical subsystem. Finally, the performed test calculations also revealed that the TDDFT/ELMO strategy can be successfully used to assess the contribution of chemical subunits to the global excitation.

Time Dependent Density Functional Theory for X-ray Absorption 8 broadening at the two edges (1.7 and 2.2 eV, respectively, for KMnO 4 , 1 and 1.5 eV for Fe 3 O 4 ) to account for these effects. Generally, the spectral weight of the TDDFT calculations is shifted by some tenths of eV to higher energies from their ground state (LSDA) parent. This shift has been suppressed from the figures and all energy scales were adjusted to the experimental one.

During the twenty five years since the introduction of the CIS(D) method in 1994, time- dependent Density Functional Theory (TD-DFT) has emerged as the method of choice for the study of excited states, by providing a convenient compromise between accuracy and computational cost. In its full form, TD-DFT is a method related to the random phase approximation (RPA), and as such it involves both single excitations and deexcitations. However, following the Tamm-Dancoff approximation, TD(A)-DFT is equivalent to perform a CIS calculation starting from the Kohn-Sham ground state determinant, involving only single excitations. On the other hand, while TD(A)-DFT was being recognized as the method of choice for the study of excited states, Grimme 6 proposed an approach to apply the usual MP2 perturbative correction to a Kohn-Sham ground state determinant. This idea led to a new class of density functional approximations (DFA) termed double hybrids, which are characterized by the energy expression

We present the valence electron energy-loss spectrum and the dielectric function of monoclinic hafnia (m-HfO 2 ) obtained from time-dependent density-functional theory (TDDFT) predictions and compared to energy-filtered spectroscopic imaging measurements in a high-resolution transmission- electron microscope. Fermi’s Golden Rule density-functional theory (DFT) calculations can capture the qualitative features of the energy-loss spectrum, but we find that TDDFT, which accounts for local-field effects, provides nearly quantitative agreement with experiment. Using the DFT density of states and TDDFT dielectric functions, we characterize the excitations that result in the m-HfO 2

S = S 0 + S 0 KS. The kernel K of this Dyson-like screening equation, that links the response function S 0 , of a system of independent (Kohn-Sham or quasi-) particles, to the full response S, is composed of two terms: 1) a bare Coulomb interaction, which is the same in both approaches. Its role will be investigated in chapter 6; 2) the exchange-correlation (in TD- DFT) or electron-hole (in BSE) contribution, which is, instead, different in time depen- dent density functional theory and Bethe-Salpeter. The importance of this contribution will be analyzed in chapter 7, showing, in particular, that very simple approximations can lead to satisfying spectra for both semiconductors and insulators. In chapter 8 we combine the knowledge gained from BSE, with the advantages of TDDFT, in order to ob- tain exchange-correlation kernels, to be used in time dependent density functional theory. They are fully ab initio, and parameter-free, and they are able to describe the optical spec- tra of solids. The results are in fact extremely promising. We show the optical absorption of bulk semiconductors (chapter 9) and insulators (chapter 10) in good agreement with the experiments, almost indistinguishable from those of the BSE approach. This is, to our opinion, an important step towards the solution of the long-standing problem of how to calculate ab initio realistic absorption spectra of materials, without solving the BSE.

In the current paper, we extend our recent formulation of real-time time-dependent Kohn-Sham method based on localized basis set functions and developed within the Psi4NumPy framework to the FDE scheme. The latter has been implemented in its uncoupled avor (in which the time evolution is only carried out for the active sub- system, while the environment subsystems remain at their ground state), using and adapting the FDE implementation already available in the PyEmbed module of the scripting framework PyADF. The implementation was facilitated by the fact that both Psi4NumPy and PyADF, being native Python API, provided an ideal framework of development using the Python advantages in terms of code readability and reusability. We employed this new implementation to investigate the stability of the time propaga- tion procedure, which is based an ecient predictor/corrector second-order midpoint Magnus propagator employing an exact diagonalization, in combination with the FDE scheme. We demonstrate that the inclusion of the FDE potential does not introduce any numerical instability in time propagation of the density matrix of the active subsystem and in the limit of weak external eld, the numerical results for low-lying transition energies are consistent with those obtained using the reference FDE calculations based on the linear response TDDFT. The method is found to give stable numerical results also in the presence of strong external eld inducing non-linear eects. Preliminary re- sults are reported for high harmonic generation (HHG) of a water molecule embedded in a small water cluster. The eect of the embedding potential is evident in the HHG spectrum reducing the number of the well resolved high harmonics at high energy with respect to the free water. This is consistent with a shift towards lower ionization energy

To make a direct comparison between the PADs, we cannot avoid discussing the difference between the experimental and theoretical pulse durations. Indeed, the experimental pulse lasts several picoseconds, which implies a true coupling of ions to the laser. In addition, the ionic temperature, and again the long pulse duration, makes the ions exploring a large variety of positions and thus induces large shape fluctuations around the ground-state configuration. While the first effect may tend to align emission, the second one may render it more isotropic. As indicated above, we have no direct possibility to explore the balance between these two effects from the present theoretical point of view. The best we can do is to estimate theoretical error bars due to the ionic temperature and associated shape fluctuations. To that end, we have computed by means of simulated annealing an ensemble of ionic configurations at the experimental temperature of 800 K. We have picked stochastically 50 samples and computed the PAD for each one. Incoherent summation of the resulting PADs yields the error band shown in Fig. 2 . The experimental error bar is that measured on β 2 . Theory and experiments qualitatively agree in the sense that, in both cases, positive values are found, denoting that the ionization is preferably along the polarization axis. The anisotropy parameter is also higher for the HOMO orbital as compared to that for the HOMO-1 orbital. Quantitatively, the theoretical β 2 values are significantly smaller than the experimental ones. One should, however, keep in mind that the laser frequency is 20 eV, precisely at the C60 plasmon peak, and that β2 strongly depends on ωlas around the plasmon peak [ 48 ]. And once again, the estimated temperature of C 60 in the experiment is 800 K and the pulse length (some picoseconds) is orders of magnitude longer than the value used in the calculations (60 fs). So, the effect of ionic motion certainly plays a significant role in this PAD.

have been increasingly used. The range-separated TDDFT approach that was first developed is based on the long-range correction (LC) scheme [23], which combines long-range Hartree-Fock (HF) exchange with a short-range exchange density functional and a stan- dard full-range correlation density functional. It has been demonstrated that the LC scheme corrects the underestimation of Rydberg excitation energies of small molecules [23] and the overestimation of (hy- per)polarizabilities of long conjugated molecules [24– 31] usually obtained with standard (semi)local density- functional approximations. A variety of other similar range-separated TDDFT schemes have also been em- ployed, which for example use an empirically modified correlation density functional depending on the range- separation parameter [32], or introduce a fraction of HF exchange at shorter range as well [33–45], such as in the CAM-B3LYP approximation [33].

The present SOC implementation is based on the multiple-interface idea of MolSOC, but benefits from the flexibility of a script language. In the computational chemistry community, such script languages are often used, for example Tcl/Tk in Chemshell 53 , Perl in Newton-X 54 , and Python in SHARC 55 . We have adopted Python scripting to deal with the I/O interface in PySOC. The formally exact RP theory is not well suited for our multiple-interface philosophy, and therefore, to balance flexibility, accuracy, and efficiency, we have adopted the wave function approach to describe the excited states, including all CI terms in TDDFT via Casida’s Ansatz, which also eliminates the need of using the alteration procedure. Reconstructed Casida-type wavefunctions are in principle capable of producing the exact LR-TDDFT matrix elements for one-body operators between ground and excited states 56 and between pairs of excited states 57 .

ee the RSH+RPAx reduces to the RSH+MP2 method [26, 31]. Computational details. Equations (14)-(17) have been implemented in the time-dependent DFT development module [32] of MOLPRO 2008 [33]. We perform a self- consistent RSH calculation with the short-range PBE xc functional of Ref. 24 and add the long-range RPAx cor- relation energy calculated with RSH orbitals. The range separation parameter is taken at µ = 0.5, in agreement with previous studies [28], without trying to fit it. The λ-integration in Eq. (14) is done by a 7-point Gauss- Legendre quadrature [6]. We use large Dunning basis sets [34] and remove the basis set superposition error (BSSE) by the counterpoise method. The full-range RPA and RPAx calculations have been done with PBE [35] and HF orbitals, respectively. The computational cost of RSH+RPAx is essentially identical to that of full-range ACFDT RPA.

(Dated: January 28, 2011) We present a local density approximation (LDA) for one-dimensional (1D) systems interacting via the soft-Coulomb interaction based on quantum Monte-Carlo calculations. Results for the ground- state energies and ionization potentials of finite 1D systems show excellent agreement with exact calculations, obtained by exploiting the mapping of an N -electron system in d dimensions, onto a single electron in N ×d dimensions properly symmetrized by the Young diagrams. We conclude that 1D LDA is of the same quality as its three-dimensional (3D) counterpart, and we infer conclusions about 3D LDA. The linear and non-linear time-dependent responses of 1D model systems using LDA, exact exchange, and the exact solution are investigated and show very good agreement in both cases, except for the well known problem of missing double excitations. Consequently, the 3D LDA is expected to be of good quality beyond linear response. In addition, the 1D LDA should prove useful in modeling the interaction of atoms with strong laser fields, where this specific 1D model is often used.

In order to realize the difficulty in mimicking photo- striction, let us start by recalling that the Kohn-Sham (KS) implementation of DFT [13] reformulates the many-body problem of interacting electrons into many single-body problems, and “only” guarantees that the model noninter- acting KS Hamiltonian yields the same ground state density and energy as the real interacting Hamiltonian. Such a fact, therefore, leaves the description of unoccupied states within traditional DFT an unanswered question, and the determination of excitation energies remains the privi- lege of rather costly techniques, such as time-dependent DFT [14] or the GW approximation [15] . However, an alternative formulation of DFT that treats ground and excited states on the same footing has been proposed [16] . In particular, Ref. [16] connected each eigenstate of a real interacting Hamiltonian with the eigenstate of a model noninteracting Hamiltonian through a generalized adiabatic connection (GAC) scheme. The so-called ΔSCF method [17] takes advantage of this GAC scheme, and assumes an one-to-one correspondence between the excited states of a single Kohn-Sham system and the real system [16] . This ΔSCF scheme has proved successful and computationally

Abstract. We report onsemilocal and hybriddensity functional theory (DFT) studyof strainedwurtzitecrystals of InAs and InP. The crystal-field splitting has a large and non-linear dependence on strain for both crystals.Moreover, the study of the electronic deformation potentials reveals that the well-known quasi-cubic (QC) approximation fails to reproduce the non-ideal c/a ratio. This theoretical study is of crucial importance for the simulation of self-assembled InAs/InP nanowires.

This work is a rst step towards the study of ET reaction in water and at electrode/water interfaces based on MDFT. The solvent effect sometimes called outer-sphere contribution is not the only mechanism playing a role in the ET reaction. The rearrangement of the electron cloud of the solute entering the so-called inner-sphere contribution may also play an impor- tant role. This effect is well taken into account in QM/MM calculation. There are mainly two approaches to deal with the MM part in such calculations. The rst one is to use MD, which takes into account the molecular nature of the solvent, but remains computationally costly. The second one is to use PCM-like models in which the solvent is described as a dielectric continuum. This approach neglects the molecular nature of solvent. As a consequence, it always assumes the validity of the linear response approximation and cannot properly describe systems violating Marcus theory. The strength of this method is its numerical efficiency: calcula- tions are almost instantaneous. MDFT is thus a promising alternative to those two approaches to account for solvation in

8 the solvation free energy can be computed by minimising the functional F [⇢ e , ⇢] = E QM [⇢ e ] + F id [⇢] + F ext [⇢ e , ⇢] + F exc [⇢] (9) with respect to the electronic density ⇢ e (r) and the solvent density ⇢(r, ⌦). Instead of carrying the joint minimisation we adopt a simpler strategy. First, the elec- tronic functional is minimised in vacuum. The equilibrium electronic density is then used in the MDFT calculation to compute the electrostatic contribution to external term using equa- tion 4. After minimisation of the MDFT functional, the equilibrium solvent charge density is used to compute the electrostatic external potential acting on the electronic density using equation 4. The electronic functional is minimised and a new electronic density is obtained. This process is repeated until both the electronic energy of equation 8 and the solvation free energy of equation 1 are converged to a given threshold. Using this procedure, the electrostatic energy of equation 4 is computed twice, once in the electronic DFT calculation and once in the MDFT one. These two values can be compared as a sanity check to verify convergence.

The hydroxyl ( OH) and ether oxide ( O ) groups of saccharides interact preferentially with water molecules. Each hydroxyl group can participate in up to 3 hydrogen bonds and intracyclic oxygen can par ticipate in 2 hydrogen bonds ( Molteni & Parrinello, 1998 ). The enthalpy optimization calculations are thus carried out by adding one water molecule at time in a stepwise manner, in order to construct the sac charide hydration shell. The optimization is performed at each step, without any symmetry constraints. The saccharide hydration enthalpy, H S/W , is calculated as the difference between the enthalpy of the

and reproduce BR2 exchange energy density for (x > 5). The curve obtained by BR2 recovers the exact exchange energy density for (x ≤ 5) . We do not expect any of the two models to recover the exact exchange energy density when (x > 5). In this range, the contribution of the exchange energy density to the total exchange energy is very small due to the low-density limit. The oscillations observed in figure 3.7 arise from small un- dulations in Kohn-Sham orbitals[62] that are amplified by any derivative of the orbitals or the density. In this regard the biharmonic condition, which contains the biharmonic of the density and other high derivatives of orbitals, transfers these oscillations into the BR4 model during the its parameterization causing oscillations in the exchange energy density. It is important to mention that models constructed to satisfy the short-range quartic behavior (the on-top, the curvature and the biharmonic conditions) along with the normalization condition could present unexpected behaviors related to these oscilla- tions. By "unexpected", we mean for instance the impossibility to satisfy all constraints imposed on the model therefore leading to unphysical total exchange energies.

indeed weaker in the t orientation, while at the same time the exo-anomeric effect is more important in the g conformation. These two observations are both in favor of more stable g conformers, even if the energy gap between t and g conformers is rather small (Table 4). Finally, in the case of alkoxy4 anions, the most favorable orientation of OH(2) is trans when the anomeric hydroxyl is gauche (T4, Tg x t t g ). This can be attributed to the lack of endo- anomeric effect, which seems to play a key role in the energetically favored gauche orientation of the T4c structure.

■ INTRODUCTION Gold is one of the most widely used metal substrate/electrode materials in research studies and technological applications. Bulk gold has a face centered cubic (fcc) crystal structure, it is a highly inert metal that rarely forms oxides, and it does not strongly react with the atmosphere or with a large number of substances. This high tolerance to the environment is of practical convenience for conducting experiments and for industrial applications. 1,2 By virtue of these characteristics, it is relatively simple to prepare a clean, flat, and stable gold surface. The most common method for doing so it is by thin film growth on silicon wafers, glass, mica, or plastic substrates. Gold surfaces are regularly prepared by vapor deposition methods, electrodeposition, or electroless deposition. 2−4 After annealing, the surface is nearly flat and exhibits terraces of the most stable (111) surface. The crystalline quality, the number of non- (111)-oriented crystallites, the density of defects, etc., can vary substantially, depending on the evaporation conditions, the thermal treatment, and other parameters. 5