Information entropies of many-electron systems

The Boltzmann–Shannon (BS ) information entropy S_ρ = ∫ ρ(r)log ρ(r)dr measures the spread or extent of the one-electron density ρ(r), which is the basic variable of the density function theory of the many electron systems. This quantity cannot be analytically computed, not even for simple quantum mechanical systems such as, e.g., the harmonic oscillator (HO ) and the hydrogen atom (HA ) in arbitrary excited states. Here, we first review (i) the present knowledge and open problems in the analytical determination of the BS entropies for the HO and HA systems in both position and momentum spaces and (ii) the known rigorous lower and upper bounds to the position and momentum BS entropies of many-electron systems in terms of the radial expectation values in the corresponding space. Then, we find general inequalities which relate the BS entropies and various density functionals. Particular cases of these results are rigorous relationships of the BS entropies and some relevant density functionals (e.g., the Thomas–Fermi kinetic energy, the Dirac–Slater exchange energy, the average electron density) for finite many-electron systems. © 1995 John Wiley & Sons, Inc.     

Effect of the Perdew-Zunger self-interaction correction on the thermochemical performance of approximate density functionals

The Perdew-Zunger self-interaction-corrected density functional theory (SIC-DFT) was implemented self-consistently using a quasi-Newton direct minimization method. We calculated SIC-DFT energies for a number of atoms and molecules using various approximate density functionals, including hybrids. Self-interaction errors (SIE) of these functionals were compared and analyzed in terms of contributions from valence and core orbitals. We also calculated enthalpies of formation of the standard G2-1 set of 55 molecules and found that self-interaction-correction (SIC) improves agreement with experiment only for the LSDA functional, while all other functionals show worse performance upon introducing SIC. This is the first systematic study of the effect of SIC on thermochemical properties. We found no direct connection between the magnitude of the SIE contained in a functional and its performance for thermochemistry. Approximate functionals with large self-interaction errors can accurately reproduce enthalpies of formation. Our results do not support the popular belief that a smaller SIE of hybrid functionals is the main reason for their higher accuracy.     

Many-electron self-interaction and spin polarization errors in local hybrid density functionals

Errors for systems with noninteger occupation have been connected to common failures of density functionals. Previously, global hybrids and pure density functionals have been investigated for systems with noninteger charge and noninteger spin state. Local hybrids have not been investigated for either of those systems to the best of our knowledge. This study intends to close this gap. We investigate systems with noninteger charge to assess the many-electron self-interaction error and systems with noninteger spin state to assess the spin polarization error of recently proposed local hybrids and their range-separated variants. We find that long-range correction is very important to correct for many-electron self-interaction error in cations, whereas most full-range local hybrids seem to be sufficient for anions, where long-range-corrected density functionals tend to overcorrect. On the other hand, while all hitherto proposed long-range-corrected density functionals show large spin polarization errors, the Perdew-Staroverov-Tao-Scuseria (PSTS) functional performs best of all local hybrids in this case and shows an outstanding behavior for the dependence of the energy on the spin polarization.     

Relaxation and dephasing in open quantum systems time-dependent density functional theory: Properties of exact functionals from an exactly-solvable model system

The dissipative dynamics of many-electron systems interacting with a thermal environment has remained a long-standing challenge within time-dependent density functional theory (TDDFT). Recently, the formal foundations of open quantum systems time-dependent density functional theory (OQS-TDDFT) within the master equation approach were established. It was proven that the exact time-dependent density of a many-electron open quantum system evolving under a master equation can be reproduced with a closed (unitarily evolving) and non-interacting Kohn-Sham system. This potentially offers a great advantage over previous approaches to OQS-TDDFT, since with suitable functionals one could obtain the dissipative open-systems dynamics by simply propagating a set of Kohn-Sham orbitals as in usual TDDFT. However, the properties and exact conditions of such open-systems functionals are largely unknown. In the present article, we examine a simple and exactly-solvable model open quantum system: one electron in a harmonic well evolving under the Lindblad master equation. We examine two different representitive limits of the Lindblad equation (relaxation and pure dephasing) and are able to deduce a number of properties of the exact OQS-TDDFT functional. Challenges associated with developing approximate functionals for many-electron open quantum systems are also discussed.     

Hybrid Functional and Plane Waves based Ab Initio Molecular Dynamics Study of the Aqueous Fe2+/Fe3+ Redox Reaction**

Kohn-Sham density functional theory and plane wave basis set based ab initio molecular dynamics (AIMD) simulation is a powerful tool for studying complex reactions in solutions, such as electron transfer (ET) reactions involving Fe²⁺/Fe³⁺ ions in water. In most cases, such simulations are performed using density functionals at the level of Generalized Gradient Approximation (GGA). The challenge in modelling ET reactions is the poor quality of GGA functionals in predicting properties of such open-shell systems due to the inevitable self-interaction error (SIE). While hybrid functionals can minimize SIE, standard plane-wave based AIMD at that level of theory is typically 150 times slower than GGA for systems containing ∼100 atoms. Among several approaches reported to speed-up AIMD simulations with hybrid functionals, the noise-stabilized MD (NSMD) procedure, together with the use of localized orbitals to compute the required exchange integrals, is an attractive option. In this work, we demonstrate the application of the NSMD approach for studying the Fe²⁺/Fe³⁺ redox reaction in water. It is shown here that long AIMD trajectories at the level of hybrid density functionals can be obtained using this approach. Redox properties of the aqueous Fe²⁺/Fe³⁺ system computed from these simulations are compared with the available experimental data for validation.     

Ab initio dynamic correlation effects in density functional theories: a density based study for argon

In an attempt to get more insight into the links between the coverage of dynamic electron correlation effects defined in traditional wave function theories (WFT) by density functional theories (DFT) we have performed comprehensive studies for the Ar atom, for which the dynamic correlation effects play the dominant role. A density-based approach directly hinged on difference radial density (DRD) distributions defined with respect the Hartree-Fock radial density has been employed for analyzing the impact of dynamic correlation effects on the density. The DRD-distributions calculated by ab initio methods have been compared with their DFT counterparts generated for representatives of several generations of broadly used exchange-correlation functionals and for the recently developed orbital-dependent OEP2 exchange-correlation functional (Bartlett et al. in J Chem Phys 122:034104, 2005). For the local, generalized-gradient, and hybrid functionals it has been found that the dynamic WFT correlation effects on the density are to a significant extent accounted for by densities resulting from exchange-only calculations. It has been shown that the removal of self-interaction errors does not change this result. It has been demonstrated that the VWN5 and LYP correlation functionals do not represent any substantial dynamical correlation effects on the electron density, whereas these effects are well represented by the orbital-dependent OEP2 correlation functional. Critical comparison of the results of the present investigations with various published results obtained for more complex many-electron systems has been made. Attention has been paid to bringing into sharper relief the differences between the conclusions reached when using energy- or density-based criteria.     

Some thoughts about the stability and reliability of commonly used exchange–correlation functionals – coverage of dynamic and nondynamic correlation effects

 The self-interaction error (SIE) of commonly used density functional theory (DFT) exchange functionals mimics long-range (nondynamic) pair correlation effects in an unspecified way. Slater exchange suffers from a larger SIE and, therefore, covers more nondynamic correlation effects than Becke exchange, which is the reason why exchange–correlation (XC) functionals based on Slater exchange lead to stabler restricted DFT solutions than those based on Becke exchange. However, the stability of an XC functional does not guarantee higher accuracy. On the contrary, if system-specific nondynamic correlation effects have to be introduced via the form of the wave function, these will be suppressed by nondynamic correlation effects already covered by the exchange functional. Hybrid functionals suffer less from the SIE and, therefore, cover a smaller number of nondynamic electron correlation effects. Accordingly, they are better suited when nondynamic electron correlation has to be introduced by the form of the wave function. It is shown that, for example, broken-symmetry unrestricted DFT calculations are more accurate when carried out with B3LYP than BLYP contrary to claims made in the literature. Received: 8 November 2001 / Accepted: 30 January 2002 / Published online: 8 April 2002     

The self-interaction error and the description of non-dynamic electron correlation in density functional theory

The self-interaction error (SIE) plays a central role in density functional theory (DFT) when carried out with approximate exchange-correlation functionals. Its origin, properties, and consequences for the development of standard DFT to a method that can correctly describe multi-reference electron systems by treating dynamic and non-dynamic electron correlation on an equal footing, is discussed. In this connection, the seminal work of Colle and Salvetti on wave function-based correlation functionals that do no longer suffer from a SIE is essential. It is described how the Colle–Salvetti correlation functional is an anchor point for the derivation of a functional multi-reference DFT method.     

Geminals in Dirac–Coulomb Hamiltonian eigenvalue problem

The applicability of the Dirac–Coulomb model in computational analysis of the properties of many-electron systems has been, since many years, a subject of dispute and controversy. The most common and numerically safe approach, based on the restriction of the variational space to the many-electron spinors spanning a subspace of the positive-energy part of the complete Hilbert space has been challenged by alternative models in which carefully selected both positive and negative energy functions are taken into account. However, these constructions are not possible when one goes beyond the one-electron model, e.g. when geminal-containing trial functions are used. Then the problem becomes particularly difficult and subtle. In this report several aspects specific for the geminal-based variational approach to the Dirac–Coulomb eigenvalue problem are discussed.     

Coupled-perturbed density-matrix functional theory equations. Application to static polarizabilities

Starting from the variational equations for the natural occupation numbers and the recently proposed eigenequations for the natural spin-orbitals, we derive coupled-perturbed density-matrix equations that furnish a linear response of the one-electron reduced density matrix to a static perturbation when the total energy is a functional of the one-electron reduced density matrix. Cases when some occupation numbers achieve exactly 0 or 1 or when the total number of the particles in a system is not preserved are taken into consideration. The scheme is applied to computing static polarizabilities from two simple density-matrix functionals. The behavior of the functionals is erratic and they provide only little or no improvement over the coupled-perturbed Hartree-Fock results.     

Long-range and short-range Coulomb correlation effects as simulated by Hartree–Fock,local density approximation,and generalized gradient approximation exchange functionals

 Exchange functionals used in density functional theory (DFT) are generally considered to simulate long-range electron correlation effects. It is shown that these effects can be traced back to the self-interaction error (SIE) of approximate exchange functionals. An analysis of the SIE with the help of the exchange hole reveals that both short-range (dynamic) and long-range (nondynamic) electron correlation effects are simulated by DFT exchange where the local density approximation (LDA) accounts for stronger effects than the generalized gradient expansion (GGA). This is a result of the fact that the GGA exchange hole describes the exact exchange hole close to the reference electron more accurately than the LDA hole does. The LDA hole is more diffuse, thus leading to an underestimation of exchange and stronger SIE effects, where the magnitude of the SIE energy is primarily due to the contribution of the core orbitals. The GGA exchange hole is more compact, which leads to an exaggeration of exchange in the bond and the nonbonding region and negative SIE contributions. Partitioning of the SIE into intra-/interelectronic and individual orbital contributions makes it possible to test the performance of a given exchange functional in different regions of the molecule. It is shown that Hartree–Fock exchange always covers some long-range effects via interelectronic exchange while self-interaction-corrected DFT is lacking these effects. Received: 25 May 2002 / Accepted: 7 October 2002 / Published online: 21 January 2003 Correspondence to: E. Kraka e-mail: kraka@theoc.gn.se Acknowledgements. This work was supported financially by the Swedish Natural Science Research Council (NFR). Calculations were done on the supercomputers of Nationellt Superdatorcentrum (NSC), Link?ping, Sweden. The authors thank the NSC for a generous allotment of computer time.     

Method of local-scaling transformations and density functional theory in quantum chemistry. II. The procedure for reproducing a many-electron wave function from x-ray diffraction data on one-electron density

The computational procedure for reconstructing a many-electron wave function on the basis of x-ray diffraction data of one-electron density is proposed within the method of local-scaling transformations.     

A simple method to selectively scale down the self-interaction correction

The method of Perdew and Zunger is commonly used to correct the self-interaction error of approximate density functionals. However, it has been shown that this orbital-based self-interaction correction (SIC) tends to overcorrect and often impairs molecular properties. We have recently proposed a method to improve the performance of the SIC by scaling it down in many-electron regions. In this communication, we present a simplification of this scheme. For every occupied orbital, we introduce a scaling factor determined by the ratio of the orbital density to the total spin-density. Thus, the magnitude of the correction is adjusted depending on how much orbital densities overlap with one another. Such a modification of the Perdew-Zunger SIC does not add any appreciable time to the computation, but significantly improves the accuracy for a number of benchmark properties.     

Benchmark calculations for reduced density-matrix functional theory

Reduced density-matrix functional theory (RDMFT) is a promising alternative approach to the problem of electron correlation. Like standard density functional theory, it contains an unknown exchange-correlation functional, for which several approximations have been proposed in the last years. In this article, we benchmark some of these functionals in an extended set of molecules with respect to total and atomization energies. Our results show that the most recent RDMFT functionals give very satisfactory results compared to standard quantum chemistry and density functional approaches.