Dynamical optimization for partition theory

How to partition a chemical system into its constituent parts is a classic problem of theoretical chemistry. A formally exact solution has recently been developed, partition theory (PT), based on density functional theory [Cohen, M. H.; Wasserman, A. J. Phys. Chem. A 2007, 111, 2229]. PT presents a constrained optimization problem to which the Car-Parrinello (CP) method of electronic structure theory is well suited. We propose here a generalization of the CP method suitable for PT and thereby make way for its practical numerical implementation. We demonstrate that this CP implementation of PT need not increase the complexity of the computation of the system's electronic structure. The scheme provides an exact DFT formulation of, e.g., atoms in molecules theory that is amenable to numerical implementation.     

The devil in the details: A tutorial review on some undervalued aspects of density functional theory calculations

Density functional theory (DFT) has become ubiquitous for chemical applications in research and in education. The exact functional at the foundation of DFT is unfortunately unknown, and issues arise when choosing an approximation for a specific application. With this tutorial review, we tackle the selection problem and many related ones, such as the choices of a basis set and of an integration grid, that are often overlooked by occasional practitioners and by more experienced users as well. We offer a practical approach in the form of a commented notebook containing 12 experiences that can be run on a simple computer in just a few hours. We propose this review as a primary source for those who are willing to include DFT in their everyday research or teaching activities in a way that reflects the research advances of the field in the last couple of decades.     

Self-interaction and strong correlation in DFTB

The density functional based tight-binding (DFTB) method can benefit substantially from a number of developments in density functional theory (DFT) while also providing a simple analytical proving ground for new extensions. This contribution begins by demonstrating the variational nature of charge-self-consistent DFTB (SCC-DFTB), proving the presence of a defined ground-state in this class of methods. Because the ground state of the SCC-DFTB method itself can be qualitatively incorrect for some systems, suitable forms of the recent LDA+U functionals for SCC-DFTB are also presented. This leads to both a new semilocal self-interaction correction scheme and a new physical argument for the choice of parameters in the LDA+U method. The locality of these corrections can only partly repair the HOMO-LUMO gap and chemical potential discontinuity, hence a novel method for introducing this further physics into the method is also presented, leading to exact derivative discontinuities in this theory at low computational cost. The prototypical system NiO is used as an illustration for these developments.     

Exact density functionals for two-electron systems in an external magnetic field

In principle, the extension of density functional theory (DFT) to Coulombic systems in a nonvanishing magnetic field is via current DFT (CDFT). Though CDFT is long established formally, relatively little is known with respect to any generally applicable, reliable approximate E(XC) and A(XC) functionals analogous with the workhorse approximate functionals (local density approximation and generalized gradient approximation) of ordinary DFT. Progress can be aided by having benchmark studies on a solvable correlated system. At zero field, the best-known finite system for such purposes is Hooke's atom. Recently we extended the exact ground state solutions for this two-electron system to certain combinations of nonzero external magnetic fields and confinement strengths. From those exact solutions, as well as high-accuracy numerical results for other field and confinement combinations, we construct the correlated electron density and paramagnetic current density, the exact Kohn-Sham orbitals, and the exact DFT and CDFT exchange-correlation energies and potentials. We compare with results from several widely used approximate functionals, all of which exhibit major qualitative failures, whether in CDFT or in naive application of ordinary DFT. We also illustrate how the CDFT vorticity variable nu is a computationally difficult quantity which may not be appropriate in practice to describe the external B field effects on E(XC) and A(XC).     

Modeling the adiabatic connection in H2

Full configuration interaction (FCI) data are used to quantify the accuracy of approximate adiabatic connection (AC) forms in describing the ground state potential energy curve of H2, within spin-restricted density functional theory (DFT). For each internuclear separation R, accurate properties of the AC are determined from large basis set FCI calculations. The parameters in the approximate AC form are then determined so as to reproduce these FCI values exactly, yielding an exchange-correlation energy expressed entirely in terms of FCI-derived quantities. This is combined with other FCI-derived energy components to give the total electronic energy; comparison with the FCI energy quantifies the accuracy of the AC form. Initial calculations focus on a [1/1]-Padé-based form. The potential energy curve determined using the procedure is a notable improvement over those from existing DFT functionals. The accuracy near equilibrium is quantified by calculating the bond length and vibrational wave numbers; errors in the latter are below 0.5%. The molecule dissociates correctly, which can be traced to the use of virtual orbital eigenvalues in the slope in the noninteracting limit, capturing static correlation. At intermediate R, the potential energy curve exhibits an unphysical barrier, similar to that noted previously using the random phase approximation. Alternative forms of the AC are also considered, paying attention to size extensivity and the behavior in the strong-interaction limit; none provide an accurate potential energy curve for all R, although good accuracy can be achieved near equilibrium. The study demonstrates how data from correlated ab initio calculations can provide valuable information about AC forms and highlight areas where further theoretical progress is required.     

Intracule densities in the strong-interaction limit of density functional theory

The correlation energy in density functional theory can be expressed exactly in terms of the change in the probability of finding two electrons at a given distance r(12) (intracule density) when the electron-electron interaction is multiplied by a real parameter lambda varying between 0 (Kohn-Sham system) and 1 (physical system). In this process, usually called adiabatic connection, the one-electron density is (ideally) kept fixed by a suitable local one-body potential. While an accurate intracule density of the physical system can only be obtained from expensive wavefunction-based calculations, being able to construct good models starting from Kohn-Sham ingredients would highly improve the accuracy of density functional calculations. To this purpose, we investigate the intracule density in the lambda --> infinity limit of the adiabatic connection. This strong-interaction limit of density functional theory turns out to be, like the opposite non-interacting Kohn-Sham limit, mathematically simple and can be entirely constructed from the knowledge of the one-electron density. We develop here the theoretical framework and, using accurate correlated one-electron densities, we calculate the intracule densities in the strong interaction limit for few atoms. Comparison of our results with the corresponding Kohn-Sham and physical quantities provides useful hints for building approximate intracule densities along the adiabatic connection of density functional theory.     

Exact-exchange density functional theory for hyperpolarizabilities

Time-dependent density functional theory (TDDFT) employing the exact-exchange functional (TDDFTx) has been formulated using the optimized effective potential method for the beta static hyperpolarizabilities, where it reduces to coupled-perturbed Kohn-Sham theory. A diagrammatic technique is used to take the functional derivatives for the derivation of the adiabatic second kernel, which is required for the analytical calculation of the beta static hyperpolarizabilities with DFT. The derived formulas have been implemented using Gaussian basis sets. The structure of the adiabatic exact-exchange second kernel is described and numerical examples are presented. It is shown that no current DFT functional satisfies the correct properties of the second kernel. Not surprisingly, TDDFTx, which corrects the self-interaction error in standard DFT methods and has the correct long-range behavior, provides results close to those of time-dependent Hartree-Fock in the static limit.     

Meta-generalized gradient approximation: explanation of a realistic nonempirical density functional

Tao, Perdew, Staroverov, and Scuseria (TPSS) have constructed a nonempirical meta-generalized gradient approximation (meta-GGA) [Phys. Rev. Lett. 91, 146401 (2003)] for the exchange-correlation energy, imposing exact constraints relevant to the paradigm densities of condensed matter physics and quantum chemistry. Results of their extensive tests on molecules, solids, and solid surfaces are encouraging, suggesting that this density functional achieves uniform accuracy for diverse properties and systems. In the present work, this functional is explained and details of its construction are presented. In particular, the functional is constructed to yield accurate energies under uniform coordinate scaling to the low-density or strong-interaction limit. Its nonlocality is displayed by plotting the factor F(xc) that gives the enhancement relative to the local density approximation for exchange. We also discuss an apparently harmless order-of-limits problem in the meta-GGA. The performance of this functional is investigated for exchange and correlation energies and shell-removal energies of atoms and ions. Non-self-consistent molecular atomization energies and bond lengths of the TPSS meta-GGA, calculated with GGA orbitals and densities, agree well with those calculated self-consistently. We suggest that satisfaction of additional exact constraints on higher rungs of a ladder of density functional approximations can lead to further progress.     

ABEEM方法预测响应函数

在密度泛函理论和原子-键电负性均衡模型基础上,定义了与化学键有关的响应函数以及化学键区域的Fukui函数,建立了一套快速确定分子中各区域(包括原子区域和化学键区域)响应函数的新方法.对大量分子的响应函数的计算结果表明,该方法得到的响应函数可以较好地预测分子中各点的反应活性,并更加快捷省时,展示了原子-键电负性均衡模型的广阔应用前景.     

Orbital-corrected orbital-free density functional theory

A new implementation of density functional theory (DFT), namely orbital-corrected orbital-free (OO) DFT, has been developed. With at most two non-self-consistent iterations, OO-DFT accomplishes the accuracy comparable to fully self-consistent Kohn-Sham DFT as demonstrated by its application on the cubic-diamond Si and the face-centered-cubic Ag systems. Our work provides a new impetus to further improve orbital-free DFT method and presents a robust means to significantly lower the cost associated with general applications of linear-scaling Kohn-Sham DFT methods on large systems of thousands of atoms within different chemical bonding environment.     

Two-electron valence indices from the Kohn-Sham orbitals

The recent Hartree-Fock (HF) difference approach to the chemical valence indices (ionic and covalent), formulated in the framework of the pair-density matrix, is implemented within the Kohn-Sham (KS) density functional theory (DFT). The valence numbers are quadratic in terms of displacements of the molecular spin-resolved charge-and-bond-order (CBO) matrix elements, relative to values in the separated atoms limit (SAL). It is shown that the global valence represents a generalized “distance” quantity measuring a degree of similarity between the two CBO matrices: the molecular and SAL. Numerical values for typical molecules exhibiting single and multiple bonds demonstrate that the KS orbitals give rise to these new bond valences in good agreement with both chemical and HF predictions. This KS bond multiplicity analysis is applied to the chemisorption system including the allyl radical and a model surface cluster of molybdenum oxide. It is concluded that the quadratic valence analysis represents a valuable procedure for extracting useful chemical information from standard DFT calculations. © 1997 John Wiley & Sons, Inc.     

Improved accuracy with medium cost computational methods for the evaluation of bond length alternation of increasingly long oligoacetylenes

The difference between the length of the central carbon-carbon bond and that of the adjacent flanked double bonds in polymers such as polyacetylene is closely related to their electronic properties and plays a central role in their conductivity upon doping. Simple as it seems, this bond length alternation (BLA) is a difficult test for many theoretical methods. Accurate coupled-cluster (CC) benchmark values are difficult to obtain even for small- and medium-sized oligoacetylenes due to their intrinsic computational limitations. Here we present a computationally much cheaper alternative to obtain accurate benchmark BLA values, even for large polyacetylene oligomers, by using the so-called spin-component scaled M?ller-Plesset perturbation theory up to second order (SCS-MP2). Comparison between these new benchmark BLA with those provided by density functional theory (DFT) calculations shows a large dispersion of the results depending on the amount of exact exchange used in the exchange-correlation functional. We find that the percentage of exact exchange needed to accurately reproduce the new benchmark BLA is much larger than previously thought when comparison was made with values obtained using the MP2 method.     

General performance of density functionals

The density functional theory (DFT) foundations date from the 1920s with the work of Thomas and Fermi, but it was after the work of Hohenberg, Kohn, and Sham in the 1960s, and particularly with the appearance of the B3LYP functional in the early 1990s, that the widespread application of DFT has become a reality. DFT is less computationally demanding than other computational methods with a similar accuracy, being able to include electron correlation in the calculations at a fraction of time of post-Hartree-Fock methodologies. In this review we provide a brief outline of the density functional theory and of the historic development of the field, focusing later on the several types of density functionals currently available, and finishing with a detailed analysis of the performance of DFT across a wide range of chemical properties and system types, reviewed from the most recent benchmarking studies, which encompass several well-established density functionals together with the most recent efforts in the field. Globally, an overall picture of the level of performance of the plethora of currently available density functionals for each chemical property is drawn, with particular attention being dedicated to the relative performance of the popular B3LYP density functional.     

Rotationally invariant ab initio evaluation of Coulomb and exchange parameters for DFT+U calculations

Conventional density functional theory (DFT) fails for strongly correlated electron systems due to large intra-atomic self-interaction errors. The DFT+U method provides a means of overcoming these errors through the use of a parametrized potential that employs an exact treatment of quantum mechanical exchange interactions. The parameters that enter into this potential correspond to the spherically averaged intra-atomic Coulomb (U) and exchange (J) interactions. Recently, we developed an ab initio approach for evaluating these parameters on the basis of unrestricted Hartree-Fock (UHF) theory, which has the advantage of being free of self-interaction errors and does not require experimental input [Mosey and Carter, Phys. Rev. B 76, 155123 (2007)]. In this work, we build on that method to develop a more robust and convenient ab initio approach for evaluating U and J. The new technique employs a relationship between U and J and the Coulomb and exchange integrals evaluated using the entire set of UHF molecular orbitals (MOs) for the system. Employing the entire set of UHF MOs renders the method rotationally invariant and eliminates the difficulty in selecting unambiguously the MOs that correspond to localized states. These aspects overcome two significant deficiencies of our earlier method. The new technique is used to evaluate U and J for Cr(2)O(3), FeO, and Fe(2)O(3). The resulting values of U-J are close to empirical estimates of this quantity for each of these materials and are also similar to results of constrained DFT calculations. DFT+U calculations using the ab initio parameters yield results that are in good agreement with experiment. As such, this method offers a means of performing accurate and fully predictive DFT+U calculations of strongly correlated electron materials.     

A density functional theory for Lennard-Jones fluids in cylindrical pores and its applications to adsorption of nitrogen on MCM-41 materials

A density functional theory (DFT) constructed from the modified fundamental-measure theory and the modified Benedict-Webb-Rubin equation of state is presented. The Helmholtz free energy functional due to attractive interaction is expressed as a functional of attractive weighted-density in which the weight function is a mean-field-like type. An obvious advantage of the present theory is that it reproduces accurate bulk properties such as chemical potential, bulk pressure, vapor-liquid interfacial tension, and so forth when compared with molecular simulations and experiments with the same set of molecular parameters. Capabilities of the present DFT are demonstrated by its applicability to adsorption of argon and nitrogen on, respectively, a model cylindrical pore and mesoporous MCM-41 materials. Comparison of the theoretical results of argon in the model cylindrical pore with those from the newly published molecular simulations indicates that the present DFT predicts accurate average densities in the pore, slightly overestimates the pore pressure, and correctly describes the effect of the fluid-pore wall interaction on average densities and pressures in the pore. Application to adsorption of nitrogen on MCM-41 at 77.4 K shows that the present DFT predicts density profiles and adsorption isotherms in good agreement with those from molecular simulations and experiments. In contrast, the hysteresis loop of adsorption calculated from the mean-field theory shifts toward the low pressure region because a low bulk saturated pressure is produced from the mean-field equation of state. The present DFT offers a good way to describe the adsorption isotherms of porous materials as a function of temperature and pressure.     

Local hybrid functionals based on density matrix products

We present a novel similarity metric comparing exact and semilocal density functional theory (DFT) exchange holes in real space. This metric is obtained from the product of the one-particle density matrix and the uniform electron gas model density matrix. The metric is bound between 0 and 1, 1 in the uniform electron gas, 0 in regions asymptotically far from finite systems, and can detect delocalization of the exact exchange hole and effective fractional occupations. We also present a parameter-free local hybrid functional that uses this similarity metric to locally mix exact and semilocal DFT exchange energy densities. The resulting functional gives better thermochemistry and reaction barrier heights than our original local hybrids [Jaramillo et al., J. Chem. Phys. 118, 1068 (2003)], while retaining moderate accuracy for symmetric radical cation dimers.     

A New Metric for Evaluating DFT Calculated Proton and Carbon Chemical Shifts of Natural Products and Organic Compounds

The calculation of DFT (density functional theory) chemical shifts have become an important technique for the verification of a proposed structure. An easily calculated metric developed for proton and carbon chemical shifts of natural products and organic compounds, the calculated chemical shift index (CCSI), has been developed, which uses the deviation of each pair of calculated and experimental chemical shifts. The mean absolute deviation (MAD), which is commonly used as the goodness of fit metric for DFT calculated chemical shifts, can conceal large deviations in the calculated data. A classification strategy is also proposed for the CCSI to highlight when further assessment of the NMR data is required.