Coupled-perturbed Kohn-Sham Research Articles

Application of the Adiabatic Connection Random Phase Approximation to Electron-Nucleus Hyperfine Coupling Constants.

The electron-nucleus hyperfine coupling constant is a challenging property for density functional methods. For accurate results, hybrid functionals with a large amount of exact exchange are often needed and there is no clear "one-for-all" functional which describes the hyperfine coupling interaction for a large set of nuclei. To alleviate this unfavorable situation, we apply the adiabatic connection random phase approximation (RPA) in its post-Kohn-Sham fashion to this property as a first test. For simplicity, only the Fermi-contact and spin-dipole terms are calculated within the nonrelativistic and the scalar-relativistic exact two-component framework. This requires to solve a single coupled-perturbed Kohn-Sham equation to evaluate the relaxed density matrix, which comes with a modest increase in computational demands. RPA performs remarkably well and substantially improves upon its Kohn-Sham (KS) starting point while also reducing the dependence on the KS reference. For main-group systems, RPA outperforms global, range-separated, and local hybrid functionals─at similar computational costs. For transition-metal compounds and lanthanide complexes, a similar performance as for hybrid functionals is observed. In contrast, related post-Hartree-Fock methods such as Møller-Plesset perturbation theory or CC2 perform worse than semilocal density functionals.

Implementation and First Evaluation of Strong-Correlation-Corrected Local Hybrid Functionals for the Calculation of NMR Shieldings and Shifts.

Local hybrid functionals containing strong-correlation factors (scLHs) and range-separated local hybrids (RSLHs) have been integrated into an efficient coupled-perturbed Kohn-Sham implementation for the calculation of nuclear shielding constants. Several scLHs and the ωLH22t RSLH have then been evaluated for the first time for the extended NS372 benchmark set of main-group shieldings and shifts and the TM70 benchmark of 3d transition-metal shifts. The effects of the strong-correlation corrections have been analyzed with respect to the spatial distribution of the sc-factors, which locally diminish exact-exchange admixture at certain regions in a molecule. The scLH22t, scLH23t-mBR, and scLH23t-mBR-P functionals, which contain a "damped" strong-correlation factor to retain the excellent performance of the underlying LH20t functional for weakly correlated situations, tend to make smaller corrections to shieldings and shifts than the "undamped" scLH22ta functional. While the latter functional can also deteriorate agreement with the reference data in certain weakly correlated cases, it provides overall better performance, in particular for systems where static correlation is appreciable. This pertains only to a minority of systems in the NS372 main-group test set but to many more systems in the TM70 transition-metal test set, in particular for high-oxidation-state complexes, e.g., Cr(+VI) complexes and other systems with stretched bonds. Another undamped scLH, the simpler LDA-based scLH21ct-SVWN-m, also tends to provide significant improvements in many cases. The differences between the functionals and species can be rationalized on the basis of one-dimensional plots of the strong-correlation factors, augmented by isosurface plots of the fractional orbital density (FOD). Position-dependent exact-exchange admixture is thus shown to provide substantial flexibility in treating response properties like NMR shifts for both weakly and strongly correlated systems.

Zero-field splitting parameters within exact two-component theory and modern density functional theory using seminumerical integration.

An efficient implementation of zero-field splitting parameters based on the work of Schmitt et al. [J. Chem. Phys. 134, 194113 (2011)] is presented. Seminumerical integration techniques are used for the two-electron spin-dipole contribution and the response equations of the spin-orbit perturbation. The original formulation is further generalized. First, it is extended to meta-generalized gradient approximations and local hybrid functionals. For these functional classes, the response of the paramagnetic current density is considered in the coupled-perturbed Kohn-Sham equations for the spin-orbit perturbation term. Second, the spin-orbit perturbation is formulated within relativistic exact two-component theory and the screened nuclear spin-orbit (SNSO) approximation. The accuracy of the implementation is demonstrated for transition-metal and diatomic main-group compounds. The efficiency is assessed for Mn and Mo complexes. Here, it is found that coarse integration grids for the seminumerical schemes lead to drastic speedups while introducing clearly negligible errors. In addition, the SNSO approximation substantially reduces the computational demands and leads to very similar results as the spin-orbit mean field Ansatz.

Investigating the Linear Response Function under Approximations Following the Coupled-Perturbed Approach for Atoms and Molecules.

The linear response kernel also referred to as linear response function (LRF) in the framework of conceptual density functional theory has gained tremendous success in time-dependent density functional theory. Comparatively less attention has been devoted to the LRF from a chemical reactivity perspective in its time- or frequency-independent context, although it has recently been used to qualitatively describe electron delocalization, (anti-)aromaticity, inductive and mesomeric effects, etc. Despite these successes, which were obtained by approximating the LRF using the independent particle approximation deriving from a coupled-perturbed Kohn-Sham computation, the robustness of this LRF approach needs to be assessed. In this work, we compute the LRF at four levels of approximation (independent particle approximation, random phase approximation, Hartree-Fock approximation, and the (exact) DFT (density functional theory) expression) using functionals from the first four rungs of Jacob's ladder of exchange-correlation energy functionals. To scrutinize the impact of these approximations, new visualization strategies are discussed and systematized. The overall conclusion is that the independent particle approximation yields qualitatively correct results (ergo previous conceptual applications of the LRF are trustworthy), but for quantitative results, LRF expressions including coulomb and exchange(-correlation) terms should be included. With respect to functionals, density-gradient contributions to the exchange-correlation kernel are less than 10% and may be omitted safely where that is preferable computationally.

First-principles evaluation of the second harmonic generation response of reference organic and inorganic crystals.

Using the CRYSTAL17 package at the coupled-perturbed Kohn-Sham (CPKS) level, periodic boundary conditions first-principles calculations are enacted to predict the second harmonic generation second-order nonlinear optical (NLO) susceptibility, χ(2), values of six historical NLO crystals. This selection allowed the comparison between state-of-the-art calculations and experiment. Several computational aspects are tackled to define conditions where the results are converged with respect to the range of lattice summations, to the number of k-points in the first Brillouin zone, to the order of the multipole expansions for evaluating the long-range part of the electrostatic interactions, as well as to the atomic basis set size. A valence triple zeta basis set supplemented with polarization functions has been selected. Then, χ(2) calculations have been performed using a range of exchange-correlation functionals (XCFs). Results show the large impact of the amount of Hartree-Fock (HF) exchange on the amplitude but also on the signon the χ(2) tensor components. To a given extent, these amplitude effects are consistent with results on molecules, but the signreversal effects and the non-monotonic behavior of the χ(2) tensor components as a function of the amount of HF exchange are scarcely found for molecules. Then, using the recommended range-separated hybrid XCFs, the CPKS scheme leads to good agreement with experimental data for potassium dihydrogenophosphate, urea, and χZXX (2) of LiNbO3. The agreement is more questionable for χZZZ (2) of LiNbO3 whereas it remains poor for ammonium dihydrogenophosphate and 2-methyl-4-nitroaniline, with large underestimations by about a factor of 3, opening a path to further fine-tuning of the ranges of inclusion of HF exchange.

Theoretical studies on nonlinear optical properties of BaGa4Q7 (Q = S, Se) crystals.

The ever-growing power of theoretical calculations has made it possible to accurately predict physical properties of condensed matter systems. In this Letter, using a linear combination of atomic orbitals (LCAO)-type quantum chemistry calculation package and the coupled perturbed Kohn-Sham (CPKS) method, it is shown that the bandgaps, static refractive indices, and full matrix elements of the nonlinear optical coefficients of recently found chalcogenides, which are potential mid-IR nonlinear optical crystals, can be obtained simultaneously with unprecedented accuracy. The results can serve as the foundation for future applications of those materials with optimum conditions.

Implementation and Validation of Local Hybrid Functionals with Calibrated Exchange-Energy Densities for Nuclear Shielding Constants.

A recently reported coupled-perturbed Kohn-Sham implementation to compute nuclear shielding constants with gauge-including atomic orbitals and local hybrid functionals has been extended to cover higher derivatives of the density in the local mixing function (LMF) of the local hybrid as well as the calibration function (CF) needed to deal with the ambiguity of exchange-energy densities. This allowed the first evaluation of state-of-the-art local hybrids with "calibrated" exchange-energy densities for nuclear shieldings. Compared to previously evaluated simpler local hybrids without a CF, appreciable improvements are found for proton shieldings. Furthermore, the recent LH20t functional is still competitive with the outstanding performance of the uncalibrated LH12ct-SsirSVWN and LH12ct-SsifSVWN LHs for heavier nuclei, suggesting that LH20t is possibly the most robust choice of any rung-four functional for computing the nuclear shieldings of main-group nuclei so far. Interestingly, the presence of a CF in the functional significantly reduces the number of artifacts introduced by the widely used Maximoff-Scuseria framework to treat the local kinetic energy τ. The latter occurs in so-called t-LMFs used in many of the present local hybrids. In any case, the use of Dobson's current-density functional framework is also recommended with more advanced calibrated τ-dependent local hybrid functionals.

Nonlinear Optical and Spectroscopical Properties of Functionalized Oligoanilines

AbstractFunctionalized octamers of polyaniline neutral forms (leucoemeraldine, emeraldine, pernigraniline) were studied in order to optimize the first hyperpolarizabilities (β). Remarkable static β values were found for the pernigraniline and emeraldine derivatives by using the coupled perturbed Kohn‐Sham (CPKS) approach at the PBE1PBE/6‐31G** level. The enhanced nonlinear response was achieved through the introduction of donor (D) and/or acceptor (A) groups into the oligoanilines structures. The nonlinear optical (NLO) properties of the octamers missing the substituent groups were also investigated. The largest β values were obtained for the pernigraniline and emeraldine derivatives having the tert‐butyl (D) and dicyanoethenyl (A) substituents at opposite ends, as well as the ones with only the acceptor group. These oligomers show charge transfer transitions with large oscillator strengths according to the time dependent density functional theory (TD‐DFT). Predicted β values for the majority of functionalized oligoanilines strongly indicate them as potential building blocks for molecular‐based nonlinear materials with second‐order response.

An Efficient Coupled-Perturbed Kohn-Sham Implementation of NMR Chemical Shift Computations with Local Hybrid Functionals and Gauge-Including Atomic Orbitals.

Nuclear shielding calculations for local hybrid (LH) functionals with position-dependent exact-exchange admixtures within a coupled-perturbed Kohn-Sham (CPKS) framework have been implemented into the Turbomole code using efficient seminumerical integration techniques to deal with two-electron integrals. When using gauge-including atomic orbitals, LHs generate additional terms within the "pre-loop" section of the CPKS scheme compared to global hybrid (GH) functionals, related to perturbed electron-repulsion integrals. These terms have been implemented and tested in detail, together with dependencies on grid sizes and integral screening procedures. Even with relatively small grids, a seminumerical treatment of GHs reproduces analytical GH results with high accuracy while improving scaling with system and basis-set sizes significantly. The extra terms generated by LHs in the pre-loop part increase the scaling of that contribution slightly, but the advantages compared to the analytical scheme are largely retained, in particular for the typically large basis sets used in NMR shift calculations, allowing for a very efficient computational scheme. An initial comparison of four first-generation LHs based on LDA exchange for a shielding test set of 15 small main-group molecules against high-level CCSD(T) benchmark data indicates a substantial reduction of the systematically underestimated shieldings compared to semilocal functionals or GHs for non-hydrogen nuclei when a so-called t-LMF is used to control the position dependence of the exact-exchange admixture. In contrast, proton shieldings are underestimated with this LMF, while an LH with a so-called s-LMF performs much better. These results are discussed in the context of experience for other properties, and they suggest directions for further improvements of LHs regarding nuclear shieldings.

Potential Application of h-BNC Structures in SERS and SEHRS Spectroscopies: A Theoretical Perspective.

In this work, the electronic and optical properties of hybrid boron-nitrogen-carbon structures (h-BNCs) with embedded graphene nanodisks are investigated. Their molecular affinity is explored using pyridine as model system and comparing the results with the corresponding isolated graphene nanodisks. Time-dependent density functional theory (TDDFT) analysis of the electronic excited states was performed in the complexes in order to characterize possible surface and charge transfer resonances in the UV region. Static and dynamic (hyper)polarizabilities were calculated with coupled-perturbed Kohn-Sham theory (CPKS) and the linear and nonlinear optical responses of the complexes were analyzed in detail using laser excitation wavelengths available for (Hyper)Raman experiments and near-to-resonance excitation wavelengths. Enhancement factors around 103 and 108 were found for the polarizability and first order hyperpolarizability, respectively. The quantum chemical simulations performed in this work point out that nanographenes embedded within hybrid h-BNC structures may serve as good platforms for enhancing the (Hyper)Raman activity of organic molecules immobilized on their surfaces and for being employed as substrates in surface enhanced (Hyper)Raman scattering (SERS and SEHRS). Besides the better selectivity and improved signal-to-noise ratio of pristine graphene with respect to metallic surfaces, the confinement of the optical response in these hybrid h-BNC systems leads to strong localized surface resonances in the UV region. Matching these resonances with laser excitation wavelengths would solve the problem of the small enhancement factors reported in Raman experiments using pristine graphene. This may be achieved by tuning the size/shape of the embedded nanographene structure.

Density perturbation theory.

Despite the fundamental importance of electron density in density functional theory, perturbations are still usually dealt with using Hartree-Fock-like orbital equations known as coupled-perturbed Kohn-Sham (CPKS). As an alternative, we develop a perturbation theory that solves for the perturbed density directly, removing the need for CPKS. This replaces CPKS with a true Hohenberg-Kohn density perturbation theory. In CPKS, the perturbed density is found in the basis of products of occupied and virtual orbitals, which becomes ever more over-complete as the size of the orbital basis set increases. In our method, the perturbation to the density is expanded in terms of a series of density basis functions and found directly. It is possible to solve for the density in such a way that it makes the total energy stationary even if the density basis is incomplete.

Analytic Gradient for Density Functional Theory Based on the Fragment Molecular Orbital Method.

The equations for the response terms for the fragment molecular orbital (FMO) method interfaced with the density functional theory (DFT) gradient are derived and implemented. Compared to the previous FMO-DFT gradient, which lacks response terms, the FMO-DFT analytic gradient has improved accuracy for a variety of functionals, when compared to numerical gradients. The FMO-DFT gradient agrees with the fully ab initio DFT gradient in which no fragmentation is performed, while reducing the nonlinear scaling associated with standard DFT. Solving for the response terms requires the solution of the coupled perturbed Kohn-Sham (CPKS) equations, where the CPKS equations are solved through a decoupled Z-vector procedure called the self-consistent Z-vector method. FMO-DFT is a nonvariational method and the FMO-DFT gradient is unique compared to standard DFT gradients in that the FMO-DFT gradient requires terms from both DFT and time-dependent density functional theory (TDDFT) theories.

How does hybrid bridging core modification enhance the nonlinear optical properties in donor-π-acceptor configuration? A case study of dinitrophenol derivatives.

This study spotlights the fundamental insights about the structure and static first hyperpolarizability (β) of a series of 2,4-dinitrophenol derivatives (1-5), which are designed by novel bridging core modifications. The central bridging core modifications show noteworthy effects to modulate the optical and nonlinear optical properties in these derivatives. The derivative systems show significantly large amplitudes of first hyperpolarizability as compared to parent system 1, which are 4, 46, 66, and 90% larger for systems 2, 3, 4, and 5, respectively, at Moller-Plesset (MP2)/6-31G* level of theory. The static first hyperpolarizability and frequency dependent coupled-perturbed Kohn-Sham first hyperpolarizability are calculated by means of MP2 and density functional theory methods and compared with respective experimental values wherever possible. Using two-level model with full-set of parameters dependence of transition energy (ΔΕ), transition dipole moment (μ(0)) as well as change in dipole moment from ground to excited state (Δμ), the origin of increase in β amplitudes is traced from the change in dipole moment from ground to excited state. The causes of change in dipole moments are further discovered through sum of Mulliken atomic charges and intermolecular charge transfer spotted in frontier molecular orbitals analysis. Additionally, analysis of conformational isomers and UV-Visible spectra has been also performed for all designed derivatives. Thus, our present investigation provides novel and explanatory insights on the chemical nature and origin of intrinsic nonlinear optical (NLO) properties of 2,4-dinitrophenol derivatives.

Ab initio investigation of electronic and vibrational contributions to linear and nonlinear dielectric properties of ice.

Electronic and vibrational contributions to the static and dynamic (hyper)polarizability tensors of ice XI and model structures of ordinary hexagonal ice have been theoretically investigated. Calculations were carried out by the finite field nuclear relaxation method for periodic systems (FF-NR) recently implemented in the CRYSTAL code, using the coupled-perturbed Kohn-Sham approach (CPKS) for evaluating the required electronic properties. The effect of structure on the static electronic polarizabilities (dielectric constants) and second-hyperpolarizabilities is minimal. On the other hand, the vibrational contributions to the polarizabilities were found to be significant. A reliable evaluation of these (ionic) contributions allows one to discriminate amongst ice phases characterized by different degrees of proton-order, primarily through differences caused by librational motions. Transverse static and dynamic vibrational (hyper)polarizabilities were found by extrapolating calculations for slabs of increasing size, in order to eliminate substantial surface contributions.

Analytical First-Order Molecular Properties and Forces within the Adiabatic Connection Random Phase Approximation.

The random phase approximation (RPA) is an increasingly popular method for computing molecular ground-state correlation energies within the adiabatic connection fluctuation-dissipation theorem framework of density functional theory. We present an efficient analytical implementation of first-order RPA molecular properties and nuclear forces using the resolution-of-the-identity (RI) approximation and imaginary frequency integration. The centerpiece of our approach is a variational RPA energy Lagrangian invariant under unitary transformations of occupied and virtual reference orbitals, respectively. Its construction requires the solution of a single coupled-perturbed Kohn-Sham equation independent of the number of perturbations. Energy gradients with respect to nuclear displacements and other first-order properties such as one-particle densities or dipole moments are obtained from partial derivatives of the Lagrangian. Our RPA energy gradient implementation exhibits the same [Formula: see text] scaling with system size N as a single-point RPA energy calculation. In typical applications, the cost for computing the entire gradient vector with respect to nuclear displacements is ∼5 times that of a single-point RPA energy calculation. Derivatives of the quadrature nodes and weights used for frequency integration are essential for RPA gradients with an accuracy consistent with RPA energies and can be included in our approach. The quality of RPA equilibrium structures is assessed by comparison to accurate theoretical and experimental data for covalent main group compounds, weakly bonded dimers, and transition metal complexes. RPA outperforms semilocal functionals as well as second-order Møller-Plesset (MP2) theory, which fails badly for the transition metal compounds. Dipole moments of polarizable molecules and weakly bound dimers show a similar trend. RPA harmonic vibrational frequencies are nearly of coupled cluster singles, doubles, and perturbative triples quality for a set of main group compounds. Compared to the ring-coupled cluster based implementation of Rekkedal et al. [J. Chem. Phys. 2013, 139, 081101.], our method scales better by two powers of N and supports a semilocal Kohn-Sham reference. The latter is essential for the good performance of RPA in small-gap systems.

Prediction of electronic (hyper)polarizabilities of titania nanotubes: A DFT periodic study

In this work, the static electronic (hyper)polarizabilities of (0,n) and (n,0) lepidocrocite nanotubes have been computed for the first time using the coupled perturbed Kohn Sham (CPKS) scheme recently implemented in the CRYSTAL code. The dependence of the response properties of the 1D-systems to electric field on the choice of the basis set and density functional has been critically analyzed. Results obtained by fit at the infinite diameter limit are compared to those of the lepidocrocite flat sheet.

Exact two-component relativistic theory for NMR parameters: General formulation and pilot application

The previously proposed exact two-component (X2C) relativistic theory of nuclear magnetic resonance (NMR) parameters [Q. Sun, W. Liu, Y. Xiao, and L. Cheng, J. Chem. Phys. 131, 081101 (2009)] is reformulated to accommodate two schemes for kinetic balance, five schemes for magnetic balance, and three schemes for decoupling in a unified manner, at both matrix and operator levels. In addition, three definitions of spin magnetization are considered in the coupled-perturbed Kohn-Sham equation. Apart from its simplicity, the most salient feature of X2C-NMR lies in that its diamagnetic and paramagnetic terms agree individually with the corresponding four-component counterparts for any finite basis. For practical applications, five approximate schemes for the first order coupling matrix X(10) and four approximate schemes for the treatment of two-electron integrals are introduced, which render the computations of X2C-NMR very much the same as those of approximate two-component approaches.

Vibrational contribution to static and dynamic (Hyper)polarizabilities of zigzag BN nanotubes calculated by the finite field nuclear relaxation method

AbstractThe vibrational contribution to static and dynamic (hyper)polarizabilities for the zigzag (n,0) family of BN nanotubes, with n ranging from (6,0) to (36,0), has been obtained. Calculations were done by the finite field nuclear relaxation (FF‐NR) method for periodic systems, newly implemented in the CRYSTAL code, using the Coupled Perturbed Kohn‐Sham (CPKS) scheme at the B3LYP/6‐31G* level for the required electronic properties. Both transverse and transverse‐longitudinal tensor components are determined by applying finite, i.e. static, fields in the transverse direction. The magnitude of the vibrational term increases with the radius of the nanotube as determined by the increase in the field‐induced geometric deformation. The resulting vibrational (hyper)polarizability varies from being dominant to negligible, when compared with the corresponding static electronic contribution. This depends upon the radius, as well as the property and the component, in a systematic manner. The extension to longitudinal components, not yet available, will be implemented next. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

The First and Second Static Electronic Hyperpolarizabilities of Zigzag Boron Nitride Nanotubes. An ab Initio Approach through the Coupled Perturbed Kohn–Sham Scheme

The coupled perturbed Kohn-Sham (CPKS) computational scheme for the evaluation of electric susceptibility tensors in periodic systems, recently implemented in the CRYSTAL code, has been extended to third-order. It is, then, used to obtain static electronic hyperpolarizabilities of zigzag BN nanotubes for the first time. This procedure, which is fully analytic in all key steps, requires a double self-consistent treatment for taking into account the first- and second-order response of the system to the applied field. The performance of different functionals is compared and the B3LYP hybrid is ultimately chosen for calculations on nanotubes having radii as large as R = 20 Å (6-200 atoms in the unit cell). Such large radii are sufficient to give the pure longitudinal component of the (hyper)polarizability tensors to within 1% of the "exact" hexagonal BN monolayer limit. Other tensor components involving the transverse direction converge more slowly. They can, however, be extrapolated to the monolayer limit to within 4% accuracy except for the pure transverse second hyperpolarizability, which has an error of 13% in that limit.

Polarizabilities of carbon nanotubes: Importance of the crystalline orbitals relaxation in presence of an electric field

AbstractThe static polarizabilities of a series of zigzag single walled carbon nanotubes [(m, 0) with m ≠ 3n for m = 7–25] were computed by the coupled perturbed Kohn‐Sham (CPKS) coupled method recently implemented in the periodic CRYSTAL code. This method makes it possible to compute both the longitudinal and transverse polarizabilities with the same approach including the local field depolarization effects for the whole series of the investigated nanotubes. To quantify these effects, the unscreened longitudinal and transverse polarizabilities were also calculated within the linear response theory using the Sum Over States (SOS) method. Our results confirmed the inverse‐square dependence on the bandgap of the longitudinal polarizabilities which are found weakly affected by the crystalline orbital relaxation. On the other hand, the comparison between the screened (CPKS) and unscreened (Sum Over States) transverse polarizabilities permit to calibrate more precisely the classical correction of Benedict et al. (Phys Rev B, 1995, 52, 8541) which must be applied to include the local field effect at the SOS level. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011