Cubic response functions in time-dependent density functional theory

We present density-functional theory for time-dependent response functions up to and including cubic response. The working expressions are derived from an explicit exponential parametrization of the density operator and the Ehrenfest principle, alternatively, the quasienergy ansatz. While the theory retains the adiabatic approximation, implying that the time-dependency of the functional is obtained only implicitly-through the time dependence of the density itself rather than through the form of the exchange-correlation functionals-it generalizes previous time-dependent implementations in that arbitrary functionals can be chosen for the perturbed densities (energy derivatives or response functions). In particular, general density functionals beyond the local density approximation can be applied, such as hybrid functionals with exchange correlation at the generalized-gradient approximation level and fractional exact Hartree-Fock exchange. With our implementation the response of the density can always be obtained using the stated density functional, or optionally different functionals can be applied for the unperturbed and perturbed densities, even different functionals for different response order. As illustration we explore the use of various combinations of functionals for applications of nonlinear optical hyperpolarizabilities of a few centrosymmetric systems; molecular nitrogen, benzene, and the C(60) fullerene. Considering that vibrational, solvent, and local field factors effects are left out, we find in general that very good experimental agreement can be obtained for the second dynamic hyperpolarizability of these systems. It is shown that a treatment of the response of the density beyond the local density approximation gives a significant effect. The use of different functional combinations are motivated and discussed, and it is concluded that the choice of higher order kernels can be of similar importance as the choice of the potential which governs the Kohn-Sham orbitals.     

On the test of different atomic exchange functionals

The yet unknown exchange-correlation energy functional and the lack of a systematic way to find such a functional has led to propose different expressions for exchange, on the one hand, and correlation on the other, by several authors. A common factor in the design of exchange functionals is to reproduce different kinds of energies rather than the important atomic shell structure. We present a strategy for testing the behavior of exchange functionals based on the reproduction of atomic shell structure. A variety of well-known exchange functionals are tested on He (closed shell), Li (open shell), and Be (closed subshell). Important differences are found among them. Finally, we propose an exchange energy functional that includes a Becke-88-type expression and, in addition, a nongradient correction term that leads to improved shell structure behavior. © 1995 John Wiley & Sons, Inc.     

Efficient evaluation of analytic vibrational frequencies in Hartree-Fock and density functional theory for periodic nonconducting systems

We report a method for the efficient evaluation of analytic energy second derivatives with respect to in-phase nuclear coordinate displacements within Hartree-Fock and Kohn-Sham density functional theories using Gaussian orbitals and periodic boundary conditions. The use of an atomic orbital formulation for all computationally challenging steps allows us to adapt the direct space fast multipole method for the Coulomb-type infinite summations. Our implementation also exploits the local character of the exact Hartree-Fock exchange in nonconducting systems. Exchange-correlation contributions are computed using extensive screening and fast numerical quadratures. We benchmark our scheme for in-phase vibrational frequencies of a trans-polyacetylene chain, a two-dimensional boron nitride sheet, and bulk diamond with the 6-31G** basis set and various density functionals. A study of computational scaling with the size of the unit cell for trans-polyacetylene reveals subquadratic scaling for our scheme.     

Calculation of two-photon absorption strengths with the approximate coupled cluster singles and doubles model CC2 using the resolution-of-identity approximation

An implementation of two-photon absorption matrix elements using the approximate second-order coupled-cluster singles and doubles model CC2 is presented. In this implementation we use the resolution-of-the-identity approximation for the two-electron repulsion integrals to reduce the computational cost. To avoid storage of large arrays we introduce in addition a numerical Laplace transformation of orbital energy denominators for the response of the doubles amplitudes. The error due to the numerical Laplace transformation is found to be negligible. Using this new implementation, we performed a series of benchmark calculations on substituted benzene and azobenzene derivatives to get reference values for TD-DFT results. We show that results obtained with the Coulomb-attenuated B3LYP functional are in reasonable agreement with the coupled-cluster results, whereas other density functionals which do not have a long-range correction give considerably less accurate results. Applications to the AF240 dye molecule and a weakly bound molecular tweezer complex demonstrate that this new RI-CC2 implementation allows for the first time to compute two-photon absorption cross sections with a correlated wave function method for molecules with more than 70 atoms and to apply this method for benchmarking TD-DFT calculations on molecules which are of particular relevance for experimental studies of two-photon absorption.     

Analytic derivatives for the XYG3 type of doubly hybrid density functionals: Theory,implementation, and assessment

We present a theoretical development of the equations required to perform an analytic geometry optimization of a molecular system using the XYG3 type of doubly hybrid (xDH) functionals. In contrast to the well‐established B2PLYP type of DH functionals, the energy expressions in the xDH functionals are constructed by using density and orbital information from another standard Kohn–Sham (KS) functional (e.g., B3LYP) for doing the self‐consistent field calculations. Thus, the xDH functionals are nonvariational in both the hybrid density functional part and the second‐order perturbation part, each of which requires formally to solve a coupled‐perturbed KS equation. An implementation is reported here which combines the two parts by defining a total Lagrangian such that only a single set of the Z‐vector equations need to be solved. The computational cost with our implementation is of the same order as those for the conventional Møller–Plesset theory to the second order (MP2) and B2PLYP. Systematic test calculations are provided for covalently bonded molecules as well as compounds involving the intramolecular nonbonded interactions for the main group elements. Satisfactory performance of the xDH functionals demonstrates that the extra computer time on top of the conventional KS procedure is well‐invested, in particular, when the standard KS functionals and MP2 as well, are problematic. © 2013 Wiley Periodicals, Inc.     

Systematic studies on the computation of nuclear magnetic resonance shielding constants and chemical shifts: the density functional models

We present a systematic density functional investigation on the prediction of the 13C, 15N, 17O, and 19F NMR properties of 23 molecules with 21 density functionals. Extensive comparisons are made for both 13C magnetic shieldings and chemical shifts with respect to the gas phase experimental data and the best CCSD(T) results. We find that the OPBE and OPW91 exchange-correlation functionals perform significantly better than some popular functionals such as B3LYP and PBE1PBE, even surpassing, in many cases, the standard wavefunction-based method MP2. Further analysis has been performed to explore the individual role played by various exchange and correlation functionals. We find that the B88 and PBE exchange functionals have a too strong tendency of deshielding, leading to too deshielded magnetic shielding constants; whereas the OPTX exchange functional performs remarkably well. We claim that the main source of error arises from the exchange functional, but correlation functional also makes important contribution. We find that the correlation functionals may be grouped into two classes. class A, such as LYP and B98, leads to deshielded NMR values, deteriorating the overall performance; whereas class B, such as PW91 and PBE, generally increases the absolute shieldings, which complements the exchange functionals, leading to improved results in the calculation of NMR data.     

PSIXAS: A Psi4 plugin for efficient simulations of X-ray absorption spectra based on the transition-potential and Δ-Kohn–Sham method

Near edge X-ray absorption fine structure (NEXAFS) spectra and their pump-probe extension (PP-NEXAFS) offer insights into valence- and core-excited states. We present PSIXAS, a recent implementation for simulating NEXAFS and PP-NEXAFS spectra by means of the transition-potential and the Δ-Kohn–Sham method. The approach is implemented in form of a software plugin for the Psi4 code, which provides access to a wide selection of basis sets as well as density functionals. We briefly outline the theoretical foundation and the key aspects of the plugin. Then, we use the plugin to simulate PP-NEXAFS spectra of thymine, a system already investigated by others and us. It is found that larger, extended basis sets are needed to obtain more accurate absolute resonance positions. We further demonstrate that, in contrast to ordinary NEXAFS simulations, where the choice of the density functional plays a minor role for the shape of the spectrum, for PP-NEXAFS simulations the choice of the density functional is important. Especially hybrid functionals (which could not be used straightforwardly before to simulate PP-NEXAFS spectra) and their amount of “Hartree-Fock like” exact exchange affects relative resonance positions in the spectrum.     

香豆素衍生物的荧光发射能计算及XC泛函的合理选择

采用含时密度泛函理论(TDDFT)与单激发组态相互作用(CIS)处理相结合的计算方案对香豆素系列15种已知荧光化合物的发射能进行了系统考察. 结果表明, 发射能与吸收能一样, 其计算值主要取决于交换-相关(XC)泛函的选择. 只要泛函选用得当, 在使用较小基组的TDDFT/6-31G(d)//CIS/3-21G(d)理论水平上即可使绝大部分化合物的实验发射能在精度达0.16 eV以内得以重现. 与吸收能计算不同的是, 无法选用单一的一种泛函来对全系列化合物的发射能作出满意的理论预测. 激发态无明显电荷转移的、7位上有氨(或胺)基取代或有氮原子相连的化合物, 其适用泛函为不含Hartree-Fock(HF)交换能的纯泛函OLYP和BLYP. 而激发态发生较大程度电荷转移的、3 位上有共轭取代基的衍生物, 其适用泛函则为含20%的HF交换成分的混合泛函B3LYP. 因此, 发射能计算中的XC泛函选择, 应同时考虑取代基团效应以及激发态的电子结构特征. 其中, 发射能计算值受XC泛函中HF交换能比例的影响十分敏感. 文中还对激发能计算中的溶剂效应校正方案和激发态几何优化精度的影响进行了讨论.     

新一代密度泛函方法XYG3

大量的应用研究测试表明,以B3LYP为代表的传统密度泛函方法在反应能垒以及非键相互作用等重要性质的预测上存在困难,并且预测精度随研究体系的增大而不断变差。开发越来越精确的交换相关泛函是现代密度泛函理论发展的主线。近年来,引入未占轨道信息的新一代双杂化密度泛函方法的研究受到越来越多的关注。本篇综述回顾了这一领域已取得的一些进展;推导了双杂化泛函方法在Kohn-Sham密度泛函理论框架下的理论基础;依据各自特点,将现已提出的双杂化泛函划分为三种类别。本文着重测评了以XYG3为代表的一类双杂化泛函的表现。最后,就双杂化泛函的未来发展方向,提出了一些设想与建议。     

Kinetic energy density study of some representative semilocal kinetic energy functionals

There is a number of explicit kinetic energy density functionals for noninteracting electron systems that are obtained in terms of the electron density and its derivatives. These semilocal functionals have been widely used in the literature. In this work, we present a comparative study of the kinetic energy density of these semilocal functionals, stressing the importance of the local behavior to assess the quality of the functionals. We propose a quality factor that measures the local differences between the usual orbital-based kinetic energy density distributions and the approximated ones, allowing us to ensure if the good results obtained for the total kinetic energies with these semilocal functionals are due to their correct local performance or to error cancellations. We have also included contributions coming from the Laplacian of the electron density to work with an infinite set of kinetic energy densities. For all but one of the functionals, we have found that their success in the evaluation of the total kinetic energy is due to global error cancellations, whereas the local behavior of their kinetic energy density becomes worse than that corresponding to the Thomas-Fermi functional.     

Embedded density functional theory for covalently bonded and strongly interacting subsystems

Embedded density functional theory (e-DFT) is used to describe the electronic structure of strongly interacting molecular subsystems. We present a general implementation of the Exact Embedding (EE) method [J. Chem. Phys. 133, 084103 (2010)] to calculate the large contributions of the nonadditive kinetic potential (NAKP) in such applications. Potential energy curves are computed for the dissociation of Li(+)-Be, CH(3)-CF(3), and hydrogen-bonded water clusters, and e-DFT results obtained using the EE method are compared with those obtained using approximate kinetic energy functionals. In all cases, the EE method preserves excellent agreement with reference Kohn-Sham calculations, whereas the approximate functionals lead to qualitative failures in the calculated energies and equilibrium structures. We also demonstrate an accurate pairwise approximation to the NAKP that allows for efficient parallelization of the EE method in large systems; benchmark calculations on molecular crystals reveal ideal, size-independent scaling of wall-clock time with increasing system size.     

Simple implementation of complex functionals: scaled self-consistency

We explore and compare three approximate schemes allowing simple implementation of complex density functionals by making use of self-consistent implementation of simpler functionals: (i) post-local-density approximation (LDA) evaluation of complex functionals at the LDA densities (or those of other simple functionals) (ii) application of a global scaling factor to the potential of the simple functional, and (iii) application of a local scaling factor to that potential. Option (i) is a common choice in density-functional calculations. Option (ii) was recently proposed by Cafiero and Gonzalez [Phys. Rev. A 71, 042505 (2005)]. We here put their proposal on a more rigorous basis, by deriving it, and explaining why it works, directly from the theorems of density-functional theory. Option (iii) is proposed here for the first time. We provide detailed comparisons of the three approaches among each other and with fully self-consistent implementations for Hartree, local-density, generalized-gradient, self-interaction corrected, and meta-generalized-gradient approximations, for atoms, ions, quantum wells, and model Hamiltonians. Scaled approaches turn out to be, on average, better than post approaches, and unlike these also provide corrections to eigenvalues and orbitals. Scaled self-consistency thus opens the possibility of efficient and reliable implementation of density functionals of hitherto unprecedented complexity.     

Orbital-dependent correlation energy in density-functional theory based on a second-order perturbation approach: success and failure

We have developed a second-order perturbation theory (PT) energy functional within density-functional theory (DFT). Based on PT with the Kohn-Sham (KS) determinant as a reference, this new ab initio exchange-correlation functional includes an exact exchange (EXX) energy in the first order and a correlation energy including all single and double excitations from the KS reference in the second order. The explicit dependence of the exchange and correlation energy on the KS orbitals in the functional fits well into our direct minimization approach for the optimized effective potential, which is a very efficient method to perform fully self-consistent calculations for any orbital-dependent functionals. To investigate the quality of the correlation functional, we have applied the method to selected atoms and molecules. For two-electron atoms and small molecules described with small basis sets, this new method provides excellent results, improving both second-order Moller-Plesset expression and any conventional DFT results significantly. For larger systems, however, it performs poorly, converging to very low unphysical total energies. The failure of PT based energy functionals is analyzed, and its origin is traced back to near degeneracy problems due to the orbital- and eigenvalue-dependent algebraic structure of the correlation functional. The failure emerges in the self-consistent approach but not in perturbative post-EXX calculations, emphasizing the crucial importance of self-consistency in testing new orbital-dependent energy functionals.     

Application of the DFT Theory to Study Cobalamin Complexes

The aim of present study is to select the best methodology in the frame of the Density Functional Theory (DFT), which may be employed to study the cobalamin complexes. Our discussion is limited to two approaches, one in which hybrid B3LYP and UB3LYP functionals are used, and the second in which geometry parameters are calculated within LDA-VWN functional, and energies of the investigated systems are computed within RPBE functional. Results of performed calculations show that both methodologies can be successfully applied to study cobalamin derivatives. Probably because of the small ligand binding energies in the studied complexes, the B3LYP and UB3LYP functionals may be used only to predict the pattern of changes in the binding energies. The use of the RPBE functional, originally parameterized to reproduce in a proper way the chemisorption energies of the small molecules on the metallic surfaces, allows to improve their values so as they fit into experimental data. Geometry parameters of the investigated complexes computed within both approaches are in good agreement with the experimental values. Interatomic distances are a little overestimated while calculated within both hybrid functionals, what is in contrast to VWN functional results. The latter, in general, gives shorter distances as observed experimentally.     

Double hybrids and time-dependent density functional theory: An implementation and benchmark on charge transfer excited states

In this paper we present the implementation and benchmarking of a Time Dependent Density Functional Theory approach in conjunction with Double Hybrid (DH) functionals. We focused on the analysis of their performance for through space charge-transfer (CT) excitations which are well known to be very problematic for commonly used functionals, such as global hybrids.Two different families of functionals were compared, each of them containing pure, hybrid and double-hybrid functionals.The results obtained show that, beside the robustness of the implementation, these functionals provide results with an accuracy comparable to that of adjusted range-separated functionals, with the relevant difference that for DHs no parameter is tuned on specific compounds thus making them more appealing for a general use. Furthermore, the algorithm described and implemented is characterized by the same computational cost scaling as that of the ground state algorithm employed for MP2 and double hybrids.     

Time-dependent density functional theory benchmarking for the calculations of atomic spectra: efficiency of exc-ETDZ basis set

We present a time-dependent density functional theory (TD-DFT) benchmarking of recently constructed basis set, namely exc-ETDZ (Guevara et al. in J Chem Phys 131: 064104, 2009) for predicting the atomic spectra of the first-row atoms. A systematic testing with 31 density functional methods has been performed to see whether convincing performance of this basis set carries over the TD-DFT formalism. The efficiency of exc-ETDZ basis set for reproducing atomic spectra has been compared with Pople- and Dunning-style basis sets. We focused on the atomic low-lying valence excited states with single excitation character for our benchmarking, and the calculated excitation energies were compared to experimental data. On average, the functionals providing the best match with exc-ETDZ basis are BMK, BH&HLYP and ωB97. Moreover, on the basis of comparison between the results of these superior functionals with CIS(D) estimates, it turned out that TD-DFT and CIS(D) errors are of the same order of magnitude, once the exc-ETDZ basis set is used. Finally, the results of present study indicate that different functionals show results that are highly dependent on the atomic configuration as well as the basis set.     

Proper Choice of XC Functionals and Calculations of Fluorescence-Emitting Energies for Coumarin Derivatives

A scheme of time-dependent density functional theory (TDDFT) combined with single-excitation configuration interaction (CIS) approach was employed to make a detailed investigation of the emitting energy for fifteen well-known coumarin derivatives. The results showed that the predicted emitting energies as well as the absorption ones were dominated mainly by the exchange-correlation (XC) functional to be used. So long as a functional is properly chosen, the experimental emitting energy of most derivatives can be accurately reproduced within 0.16 eV by a calculation at the TDDFT/6-31G(d)//CIS/3-21G(d) theoretical level. It was found that, nevertheless, the hybrid functional, B3LYP, well predicted the absorption energies for all the fifteen coumarin derivatives but none of the functionals could work equally well for the emitting energy calculations. Two pure functionals, OLYP and BLYP, yield good emitting energies for the 7-aminocoumarins or derivatives with a N atom connected to 7-position, which exhibit inconspicuous charge transfer (CT) in their excited states, whereas the B3LYP hybrid functional, with 20% Hartree-Fock (HF) exchange energy, performs significantly better than OLYP and BLYP for those 3-substituted coumarins with larger CT in excited states. Thus, in comparison with the absorption energies, the selection of proper functionals for the emitting energy calculations becomes more complex. In all probability, it is effective and doable to choose an XC-functional with alterable fraction of HF exchange energy according to the composition and structure characteristics of molecule.     

Calculation of origin-independent optical rotation tensor components in approximate time-dependent density functional theory

We outline an implementation of the origin-independent optical rotation tensor, which includes electric dipole-magnetic dipole and electric dipole-electric quadrupole polarizability. The method is based on approximate time-dependent density functional theory. We utilize time-periodic magnetic-field-dependent basis functions as well as a modified velocity-gauge formulation of dynamic polarizability tensors in order to obtain a gauge-origin independence. To ensure gauge-origin independence of the results within a given numerical accuracy, density fit coefficient derivatives are employed. A damping constant has been introduced into the linear response equations to treat both resonance and nonresonance regions of optical activity. We present calculations for trans-2,3-dimethyloxirane and derivatives thereof as well as calculations for androst-4,17-dien-3-one. In the Appendix, we derive the equivalence between the common-gauge origin and gauge-including atomic orbitals formulations for the optical rotation tensor in time-dependent DFT.