Degeneracy and size consistency in electronic density functional theory

The Hohenberg-Kohn (HK) theorem is one of the most fundamental theorems of quantum mechanics, and constitutes the basis for the very successful density-functional approach to inhomogeneous interacting many-particle systems. Here we show that in formulations of density-functional theory (DFT) that employ more than one density variable, applied to systems with a degenerate ground state, there is a subtle loophole in the HK theorem, as all mappings between densities, wave functions and potentials can break down. Two weaker theorems which we prove here, the joint-degeneracy theorem and the internal-energy theorem, restore the internal, total and exchange-correlation energy functionals to the extent needed in applications of DFT to atomic, molecular and solid-state physics and quantum chemistry. The joint-degeneracy theorem constrains the nature of possible degeneracies in general many-body systems.

Using simple functionals of the electron density to appreciate and quantify molecular structure and chemical reactivity properties is a recent endeavor in density functional theory (DFT) toward the development of a new chemical reactivity theory. According to the first Hohenberg–Kohn theorem in DFT, the electron density alone should be able to determine any property in the ground state. Exchange and correlation energies are such properties, so are molecular structure and chemical reactivity, and hence they all should accurately be determined by electron density functionals. Quantities such as Shannon entropy, Fisher information, Ghosh–Berkowitz–Parr entropy, information gain, Onicescu information energy, etc., from the information‐theoretic approach are simple electron density functionals, whose analytical forms are exactly known. In this article, we demonstrate their usefulness and validity to quantify regioselectivity, stereoselectivity, and other structure and reactivity properties. We will outline the current understanding of its theoretical framework at first, and then highlight recent applications to chemical problems including isomeric and conformational stability, electrophilicity and nucleophilicity, strong covalent and weak noncovalent interactions, acidity and basicity, aromaticity and antiaromaticity, and numerous other properties. The effort of employing electron density functionals to quantify chemical concepts should open up a new door for us to ultimately develop a chemical reactivity theory using the DFT language.This article is characterized under:Structure and Mechanism > Reaction Mechanisms and CatalysisElectronic Structure Theory > Density Functional TheoryStructure and Mechanism > Molecular StructuresMolecular and Statistical Mechanics > Molecular Interactions

The Jahn-Teller effect is one of the most fascinating phenomena in modern physics and chemistry, providing a general approach to understanding the properties of molecules and crystals and their origins. The effect inspired one of the most important recent scientific discoveries, the concept of high-temperature superconductivity. This comprehensive volume presents the background of the theory and its main applications in physics and chemistry, along with more recent achievements. Full descriptions are presented alongside thorough references to original material. The 2006 book contains over 200 figures to aid visual explanation and avoids bulky mathematical deductions and overly technical language. It is intended for graduate students and academic researchers working in solid state physics, quantum chemistry, crystallography, spectroscopy, and materials science.

There is an ongoing quest to improve on the spectroscopic quality of nuclear energy density functionals (EDFs) of the Skyrme type through extensions of its traditional form. One direction for such activities is the inclusion of terms of higher order in gradients in the EDF. We report on exploratory symmetry-breaking calculations performed for an extension of the Skyrme EDF that includes central terms with four gradients at next-to-next-to-leading order (N2LO) and for which the high-quality parametrization SN2LO1 has been constructed recently [P. Becker et al, Phys. Rev. C 96, 044330 (2017)]. Up to now, the investigation of such functionals with higher-order terms was limited to infinite matter and spherically symmetric configurations of singly- and doubly-magic nuclei. We address here nuclei and phenomena that require us to consider axial and non-axial deformation, both for reflection-symmetric and also reflection-asymmetric shapes, as well as the breaking of time-reversal invariance. Achieving these calculations demanded a number of formal developments. These all resulted from the formulation of the N2LO EDF requiring the introduction of new local densities with additional gradients that are not present in the EDF at NLO. Their choice is not unique, but can differ in the way the gradients are coupled. While designing a numerical implementation of N2LO EDFs in Cartesian 3d coordinate-space representation, we have developed a novel definition and a new unifying notation for normal and pair densities that contain gradients at arbitrary order. The resulting scheme resolves several issues with some of the choices that have been made for local densities in the past, in particular when breaking time-reversal symmetry. Guided by general practical considerations, we propose an alternative form of the N2LO contribution to the Skyrme EDF that is built from a different set of densities.

To begin the topic of DFT, this chapter overviews how the Kohn–Sham method was developed historically and then introduces various extensions of this method. The Thomas–Fermi method, which is the first form of DFT, is first explained, focusing on the local density approximation of kinetic and exchange energy density functionals, in Sect.4.1. Then, the Hohenberg–Kohn theorem, which is the basic theory of DFT, is reviewed, with a mention of the constrained search formulation used to solve the V -representability problem, in Sect. 4.2. The Kohn–Sham method based on this theorem is introduced, along with the corresponding computational methods, in Sect. 4.3. As the extension of the Kohn–Sham method to include general functionals, the generalized Kohn–Sham method is surveyed in Sect. 4.4. The constrained search method, which directly constructs a Kohn–Sham potential from the electron density, is explained, and as a consequence of this method, it is clarified why the Kohn–Sham method can accurately reproduce chemical behavior in Sect. 4.5. Finally, the time-dependent and coupled-perturbed Kohn–Sham methods are reviewed as methods with which to apply the Kohn–Sham method to calculations of photoexcitation spectra and response properties, respectively, in Sects. 4.6 and 4.7.

Since the 1975 Nobel Prize in Physics, nuclear theory has evolved along two main directions. On the one hand, the energy–density functional (EDF) theory was established, which presently encompasses (by enlarging the EDF framework) all the mean-field and beyond-mean-field theories based on energy functionals produced by effective phenomenological interactions. Highly sophisticated structure and reaction models are currently available for the treatment of medium-mass and heavy nuclei. On the other hand, effective field theories (EFTs) have rendered possible the formulation of QCD as a low-energy hadronic theory. Ab initio methods have recently achieved remarkable success in the application of EFT or EFT-inspired potentials to structure analyses of light nuclei. Different but complementary competences have been developed during the past few decades in the EDF and EFT communities. Bridges and connections have in some cases been identified and constructed. We review here some of the developments that have been performed within the EDF theory and the EFT during recent years, with some emphasis on analogies and connections that may one day provide a unified picture of the two theories. Illustrations are given for infinite matter and finite nuclei.

A simple and completely general representation of the exact exchange-correlation functional of density-functional theory is derived from the universal Lieb-Oxford bound, which holds for any Coulomb-interacting system. This representation leads to an alternative point of view on popular hybrid functionals, providing a rationale for why they work and how they can be constructed. A similar representation of the exact correlation functional allows to construct fully nonempirical hyper-generalized-gradient approximations (HGGAs), radically departing from established paradigms of functional construction. Numerical tests of these HGGAs for atomic and molecular correlation energies and molecular atomization energies show that even simple HGGAs match or outperform state-of-the-art correlation functionals currently used in solid-state physics and quantum chemistry.

Both model results and some exact relations concerning kinetic, exchange, and correlation energy density functionals will be presented. As to single-particle kinetic energy, a model of harmonic confinement, motivated by recent experimental activity on Fermion vapors, of an arbitrary number of Fermions will be used to construct exact nonlocal kinetic energy density functionals in both one and three dimensions. Then, some further exact results concerning nuclear cusp conditions for exchange only, and for exchange plus correlation, will be discussed. Finally, in Appendices, the one-shell harmonic confinement is generalized to include Coulomb repulsion in the two-electron Hookean atom and some results on the Pauli potentials for many-Fermion systems are obtained. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

Background: In the framework of nuclear energy density functional (EDF) methods, many nuclear phenomena can be related to the deformation of intrinsic states. Their accurate modeling relies on the correct description of the change of nuclear binding energy with deformation. The two most important contributions to the deformation energy have their origin in shell effects and the surface energy coefficient of nuclear matter.Purpose: It has been pointed out before that the choices made for the center-of-mass (c.m.) correction energy and the effective mass during the parameter adjustment influence the deformation properties of nuclear EDFs. We study the impact of these two properties by means of a set of purpose-built parametrizations of the standard Skyrme EDF at next-to-leading (NLO) order in gradients.Methods: In a first step, we build nine series of parametrizations with a systematically varied surface-energy coefficient ${a}_{\text{surf}}$ for three frequently used options for the c.m. correction (none, one-body term only, full one-body and two-body contributions) combined with three values for the isoscalar effective mass ${m}_{0}^{*}/m$ (0.7, 0.8, 0.85) and analyze how well each of these parametrizations can be adjusted to the properties of spherical nuclei and infinite nuclear matter. In a second step, we performed additional fits without the constraint on surface energy, adding one ``best-fit'' parametrization to each of the nine series. We then benchmark these parametrizations to the deformation properties of heavy nuclei by means of three-dimensional Hartree-Fock-Bogoliubov calculations that allow for nonaxial and/or nonreflection symmetric configurations.Results: We perform a detailed correlation analysis between surface and volume properties of nuclear matter using the nine series of parametrizations. The best fits out of each series are then benchmarked on the fission barriers of $^{240}\mathrm{Pu}$ and $^{180}\mathrm{Hg}$, as well as on the properties of deformed states at normal and superdeformation for actinides and nuclei in the neutron-deficient Hg region.Conclusions: The main conclusions are as follows: (i) Each combination of choices for c.m. correction and ${m}_{0}^{*}/m$ leads to a significantly different optimal value of ${a}_{\text{surf}}$, reason being that the effective interaction has to absorb the contribution of the c.m. correction to the total binding energy. (ii) Many properties of symmetric and asymmetric infinite nuclear matter of Skyrme NLO EDFs are strongly correlated to the value of ${a}_{\text{surf}}$. (iii) Omitting the c.m. correction results in values of ${a}_{\text{surf}}$ that are systematically too small. On the other hand, including the one-body term but neglecting the computationally expensive two-body term means ${a}_{\text{surf}}$ will be too large. Both choices result in unrealistic predictions for fission barriers and superdeformed states of heavy nuclei. Only by incorporating the complete c.m. correction does one obtain quite realistic surface properties from an adjustment protocol that only constrains properties of infinite nuclear matter and spherical nuclei. (iv) Lowering ${a}_{\text{surf}}$ increases the susceptibility of finite nuclei to take an exotic shape.

The effect of solvation on the free energy of reaction intermediates adsorbed on electrocatalyst surfaces can significantly change the thermochemical overpotential, but accurate calculations of this are challenging. Here, we present computational estimates of the solvation energy for reaction intermediates in oxygen reduction reaction (ORR) on a B-doped graphene (BG) model system where the overpotential is found to reduce by up to 0.6 V due to solvation. BG is experimentally reported to be an active ORR catalyst but recent computational estimates using state-of-the-art hybrid density functionals in the absence of solvation effects have indicated low activity. To test whether the inclusion of explicit solvation can bring the calculated activity estimates closer to the experimental reports, up to 4 layers of water molecules are included in the simulations reported here. The calculations are based on classical molecular dynamics and local minimization of energy using atomic forces evaluated from electron density functional theory. Data sets are obtained from regular and coarse-grained dynamics, as well as local minimization of structures resampled from dynamics simulations. The results differ greatly depending on the method used and the solvation energy estimates are deemed untrustworthy. It is concluded that a significantly larger number of water molecules is required to obtain converged results for the solvation energy. As the present system includes up to 139 atoms, it already strains the limits of computational feasibility, so this points to the need for a hybrid simulation approach where efficient simulations of much larger number of solvent molecules is carried out using a lower level of theory while retaining the higher level of theory for the reacting molecules as well as their near neighbors and the catalyst. The results reported here provide a word of caution to the computational catalysis community: activity predictions can be inaccurate if too few solvent molecules are included in the calculations.

Hydrogen tunneling plays a critical role in many biologically and chemically important processes. The nuclear-electronic orbital multistate density functional theory (NEO-MSDFT) method was developed to describe hydrogen transfer systems. In this approach, the transferring proton is treated quantum mechanically on the same level as the electrons within multicomponent DFT, and a nonorthogonal configuration interaction scheme is used to produce delocalized vibronic states from localized vibronic states. The NEO-MSDFT method has been shown to provide accurate hydrogen tunneling splittings for fixed molecular systems. Herein, the NEO-MSDFT analytical gradients for both ground and excited vibronic states are derived and implemented. The analytical gradients and semi-numerical Hessians are used to optimize and characterize equilibrium and transition state geometries and to generate minimum energy paths (MEPs), for proton transfer in the deprotonated acetylene dimer and malonaldehyde. The barriers along the resulting MEPs are lower when the transferring proton is quantized because the NEO-MSDFT method inherently includes the zero-point energy of the transferring proton. Analysis of the proton densities along the MEPs illustrates that the proton density can exhibit symmetric or asymmetric bilobal character associated with symmetric or slightly asymmetric double-well potential energy surfaces and hydrogen tunneling. Analysis of the contributions to the intrinsic reaction coordinate reveals that changes in the C-O bond lengths drive proton transfer in malonaldehyde. This work provides the foundation for future reaction path studies and direct nonadiabatic dynamics simulations of a wide range of hydrogen transfer reactions.

Alkali metal adsorption systems provide important models for chemisorption. Low-energy electron diffraction experiments and density functional theory calculations were carried out for the adsorption of potassium on Pb(1 0 0). The stable structure for all submonolayer coverages was found to be the commensurate c(2 × 2) structure, with potassium atoms located in substitutional sites in the top substrate layer. This structure is temperature activated and occurs for adsorption or annealing of the film above 200 K. This finding is consistent with an earlier theory that proposed that for substrates with low energies of vacancy formation, substitutional structures can be the most stable. The structural and vibrational parameters deduced from the experiment are in agreement with the calculated values, and these values fit well into and add to the database of alkali metal adsorption properties.

The ab initio numerical calculations of the electronic structure of simple oxides BeO and SiO2 and complex oxides Be2SiO4 and Be2Si x Ge1 − x O4 with the phenacite structure have been performed using the electron density functional theory. The calculations indicate that the main feature of the systems under investigation is the presence of oxygen states in both the valence and conduction bands. The splitting of the bottom of the conduction band has been revealed in the electronic structure of the Be2Si x Ge1 − x O4 system. The splitting width is about 1.5 eV. The main contribution to the formation of a narrow subband of the conduction band comes from the 2s and 2p states of oxygen and the 4d state of germanium. Microscopic models of the spatial localization of the electron density on lower energy states of the conduction band of oxide crystals have been developed using the Wannier function technique. The reflection spectra of BeO, SiO2, and Be2SiO4 have been analyzed. The reported calculations of the electronic structure imply the exciton nature of the 9.7-eV reflection peak in the Be2SiO4 crystal.

We have recently completed the preparation of PSI materials (study guides and unit tests) for seven junior–senior level physics courses: Mechanics, Electricity and Magnetism, Atomic Physics, Mathematical Physics, Physics and Computers, Astrophysics, and Relativity. Three additional courses—Solid State Physics, Nuclear Physics, and Quantum Physics—have been prepared by our colleagues. These are all elective courses at our institution and tend to have relatively low enrollments. By preparing them in the PSI format we are able to offer all ten courses every semester for any number of students with efficient use of faculty. A description of our materials, our mode of operation, and the reactions of students, faculty, and administrators are presented.

A comprehensive way of deriving the Hamiltonian of the interacting boson model (IBM) is described. Based on the fact that the multi-nucleon induced surface deformation in finite nucleus is simulated by effective boson degrees of freedom, the potential energy surface calculated with self-consistent mean-field method employing a given energy density functional (EDF) is mapped onto the IBM analog, and thereby the excitation spectra and transition rates with good symmetry quantum numbers are calculated. Recent applications of the proposed approach are reported: (i) an alternative robust interpretation of the γ-soft nuclei and (ii) shape coexistence in lead isotopes.