1-Matrix functional for long-range interaction energy of two hydrogen atoms.

The leading terms in the large-R asymptotics of the functional of the one-electron reduced density matrix for the ground-state energy of the H2 molecule with the internuclear separation R are derived thanks to the solution of the phase dilemma at the R → ∞ limit. At this limit, the respective natural orbitals (NOs) are given by symmetric and antisymmetric combinations of "half-space" orbitals with the corresponding natural amplitudes having the same amplitudes but opposite signs. Minimization of the resulting explicit functional yields the large-R asymptotics for the occupation numbers of the weakly occupied NOs and the C6 dispersion coefficient. The highly accurate approximates for the radial components of the p-type "half-space" orbitals and the corresponding occupation numbers (that decay like R-6), which are available for the first time thanks to the development of the present formalism, have some unexpected properties.

Contactium: A strongly correlated model system.

At the limit of an infinite confinement strength ω, the ground state of a system that comprises two fermions or bosons in harmonic confinement interacting through the Fermi-Huang pseudopotential remains strongly correlated. A detailed analysis of the one-particle description of this "contactium" reveals several peculiarities that are not encountered in conventional model systems (such as the two-electron harmonium atom, ballium, and spherium) involving Coulombic interparticle interactions. First of all, none of the natural orbitals (NOs) {ψn(ω;r)} of the contactium is unoccupied, which implies nonzero collective occupancies for all the angular momenta. Second, the NOs and their non-ascendingly ordered occupation numbers {νn} turn out to be related to the eigenfunctions and eigenvalues of a zero-energy Schrödinger equation with an attractive Gaussian potential. This observation enables the derivation of their properties, such as the n-4/3 asymptotic decay of νn at the n→∞ limit (which differs from that of n-8/3 in the Coulombic systems), the independence of the confinement energy vn=⟨ψn(ω;r)|12ω2r2|ψn(ω;r)⟩ of n, and the n-2/3 asymptotic decay of the respective contribution νntn to the kinetic energy. Upon suitable scaling, the weakly occupied NOs of the contactium turn out to be virtually identical to those of the two-electron harmonium atom at the ω → ∞ limit, despite the entirely different interparticle interactions in these systems.

Electronic Structure Calculations on Endohedral Complexes of Fullerenes: Reminiscences and Prospects.

The history of electronic structure calculations on the endohedral complexes of fullerenes is reviewed. First, the long road to the isolation of new allotropes of carbon that commenced with the seminal organic syntheses involving simple inorganic substrates is discussed. Next, the focus is switched to author's involvement with fullerene research that has led to the in silico discovery of endohedral complexes. The predictions of these pioneering theoretical studies are juxtaposed against the data afforded by subsequent experimental developments. The successes and failures of the old and modern quantum-chemical calculations on endohedral complexes are summarized and their remaining deficiencies requiring further attention are identified.

Errors in approximate ionization energies due to the one-electron space truncation of the EKT eigenproblem.

Unless the approximate wavefunction of the parent system is expressed in terms of explicitly correlated basis functions, the finite size of the generalized Fock matrix is unlikely to be the leading source of the truncation error in the ionization energy E produced by the EKT (extended Koopmans' theorem) formalism. This conclusion is drawn from a rigorous analysis that involves error partitioning into the parent- and ionized-system contributions, the former being governed by asymptotic power laws when the underlying wavefunction is assembled from a large number of spinorbitals and the latter arising from the truncation of the infinite-dimensional matrix V whose elements involve the 1-, 2-, and 3-matrices of the parent system. Quite surprisingly, the decay of the second contribution with the number n of the natural spinorbitals (NOs) employed in the construction of the truncated V turns out to be strongly system-dependent even in the simplest case of the 1S states of two-electron systems, following the n-5 power law for the helium atom while exhibiting an erratic behavior for the H- anion. This phenomenon, which stems from the presence of the so-called solitonic natural spinorbitals among the NOs, renders the extrapolation of the EKT approximates of E to the complete-basis-set limit generally unfeasible. However, attaining that limit is not contingent upon attempted reproduction of the ill-defined one-electron function known as "the removal orbital," which does not have to be invoked in the derivation of EKT and whose expansion in terms of the NOs diverges.

Solitonic natural orbitals in Coulombic systems.

High-accuracy electronic structure calculations on the members of the helium isoelectronic series and the H2 molecule with a stretched bond reveal that the ground-state wavefunctions of these Coulombic systems give rise to natural orbitals (NO) with unusual properties. These solitonic NOs (SoNOs) possess fewer nodes than expected from their small occupation numbers, exhibit substantial spatial localization, and respond (with approximate retention of their shapes) in a paradoxical manner (e.g., by moving away from nuclei upon an increase in the nuclear charge or decrease in the internuclear distance) to changes in the underlying Hamiltonian. An efficient tool for the identification of the SoNOs is provided by an index constructed from two expectation values pertaining to a given NO and the corresponding occupation number. In the case of the helium-like species, the rapid decay of the occupation numbers of the SoNOs with increasing nuclear charge Z is governed by an asymptotic expression that involves the radial positions and spreads of the orbitals. Three s-type SoNOs (with the occupation numbers amounting to only ∼7.9·10-67, 6.8·10-92, and 9.0·10-113 in the case of the helium atom) are predicted to turn into unoccupied NOs at Z equal to ∼2.673, 2.587, and 2.536, respectively. The persistence of the analogous p-type orbital beyond Z = 2 is consistent with the computed properties of the σu-type SoNO of the H2 molecule with a stretched bond. In particular, the profiles of this SoNO along two perpendicular lines bear great resemblance to the radial profiles of its p-type counterpart.

Construction of explicitly correlated one-electron reduced density matrices.

A general construction of an ensemble N-representable one-electron reduced density matrix Γ1(r1→';r→1) is presented. Unlike the conventional spectral representation, it explicitly incorporates the recently derived discontinuity in the fifth derivative of Γ1(r1→';r→1) with respect to |r1→'-r→1|. Its practical relevance in the context of the density-matrix functional theory is discussed.

Off-diagonal derivative discontinuities in the reduced density matrices of electronic systems.

An explicit expression relating the magnitude of the fifth-order off-diagonal cusp in the real part of the one-electron reduced density matrix to the "on-top" two-electron density is derived in a rigorous manner from the behavior of the underlying electronic wavefunction at the electron-electron coalescence. The implications of the presence of this cusp upon electronic structure calculations of quantum chemistry and solid-state physics, including the limits imposed upon their accuracy, are elucidated. In particular, the power-law decay of the occupation numbers of the natural orbitals is demonstrated for 1S states of systems composed of arbitrary even numbers of electrons. The practical importance of analogous off-diagonal cusps in many-electron reduced density matrices is briefly discussed.

Uniform description of the helium isoelectronic series down to the critical nuclear charge with explicitly correlated basis sets derived from regularized Krylov sequences.

An efficient computational scheme for the calculation of highly accurate ground-state electronic properties of the helium isoelectronic series, permitting uniform description of its members down to the critical nuclear charge Zc, is described. It is based upon explicitly correlated basis functions derived from the regularized Krylov sequences (which constitute the core of the free iterative CI/free complement method of Nakatsuji) involving a term that introduces split length scales. For the nuclear charge Z approaching Zc, the inclusion of this term greatly reduces the error in the variational estimate for the ground-state energy, restores the correct large-r asymptotics of the one-electron density ρ(Z; r), and dramatically alters the manifold of the pertinent natural amplitudes and natural orbitals. The advantages of this scheme are illustrated with test calculations for Z = 1 and Z = Zc carried out with a moderate-size 12th-generation basis set of 2354 functions. For Z = Zc, the augmentation is found to produce a ca. 5000-fold improvement in the accuracy of the approximate ground-state energy, yielding values of various electronic properties with between seven and eleven significant digits. Some of these values, such as those of the norms of the partial-wave contributions to the wavefunction and the Hill constant, have not been reported in the literature thus far. The same is true for the natural amplitudes at Z = Zc, whereas the published data for those at Z = 1 are revealed by the present calculations to be grossly inaccurate. Approximants that yield correctly normalized ρ(1; r) and ρ(Zc; r) conforming to their asymptotics at both r → 0 and r → ∞ are constructed.

Nine questions on energy decomposition analysis.

The paper collects the answers of the authors to the following questions: Is the lack of precision in the definition of many chemical concepts one of the reasons for the coexistence of many partition schemes? Does the adoption of a given partition scheme imply a set of more precise definitions of the underlying chemical concepts? How can one use the results of a partition scheme to improve the clarity of definitions of concepts? Are partition schemes subject to scientific Darwinism? If so, what is the influence of a community's sociological pressure in the "natural selection" process? To what extent does/can/should investigated systems influence the choice of a particular partition scheme? Do we need more focused chemical validation of Energy Decomposition Analysis (EDA) methodology and descriptors/terms in general? Is there any interest in developing common benchmarks and test sets for cross-validation of methods? Is it possible to contemplate a unified partition scheme (let us call it the "standard model" of partitioning), that is proper for all applications in chemistry, in the foreseeable future or even in principle? In the end, science is about experiments and the real world. Can one, therefore, use any experiment or experimental data be used to favor one partition scheme over another? © 2019 Wiley Periodicals, Inc.

Natural amplitudes of the ground state of the helium atom: Benchmark calculations and their relevance to the issue of unoccupied natural orbitals in the H2 molecule.

Employment of exact numerical quadratures in the evaluation of matrix elements involving highly accurate wavefunctions of helium (and its isoelectronic congeners) generated with the help of the regularized Krylov sequences of Nakatsuji results in an efficient algorithm for the calculation of natural orbitals and the corresponding natural amplitudes {λnl}. The results of such calculations are presented for the total of 600 natural orbitals pertaining to the ground state of the helium atom. The benchmark-quality values of {λnl} computed for 1 ≤ n ≤ 100 and 0 ≤ l ≤ 5 reveal gross inaccuracies in the previously published data. In particular, the dependence of λnl on n is found to follow very closely a simple power-scaling law λnl≈-Al (n+Bl)-4 with Al that, contrary to previous claims, varies only weakly with l. Even more importantly, the numerical trends observed in the present calculations strongly suggest that in the case of the ground state of the helium atom, the only positive-valued natural amplitude is that pertaining to the strongly occupied orbital, i.e., λ10. The relevance of this finding to the existence of unoccupied natural orbitals pertaining to the ground state wavefunction of the H2 molecule is discussed.

Universalities among natural orbitals and occupation numbers pertaining to ground states of two electrons in central potentials.

Although both the natural orbitals (NOs) {ψnlm(r→)} and their occupation numbers {νnl} pertaining to the ground state of two electrons confined by a central potential are completely determined by the spatial component Ψ(r→1,r→2) of the underlying wavefunction through a homogeneous Fredholm equation of the second kind in which Ψ(r→1,r→2) plays the role of the kernel, for the species with a single positive-valued natural amplitude that corresponds to the strongly occupied NO ("the normal sign pattern"), these quantities turn out to depend almost entirely on the "on-top" wavefunction Ψ(r→,r→). For such species, for which the occupation numbers are found to have the large-n asymptotics of n-8, a universal expression involving only Ψ(r→,r→) that reproduces the weakly occupied NOs with remarkable accuracy is inferred from the electron-electron coalescence cusp in Ψ(r→1,r→2). These theoretical predictions are fully confirmed by comparisons among the benchmark-quality NOs computed for the helium atom, the isoelectronic cations with the nuclear charges ranging from 3 to 5, and the two-electron harmonium atom at the limit of an infinitely strong confinement.

Solitonic natural orbitals.

The dependence of the natural amplitudes of the harmonium atom in its ground state on the confinement strength ω is thoroughly investigated. A combination of rigorous analysis and extensive, highly accurate numerical calculations reveals the presence of only one positive-valued natural amplitude ("the normal sign pattern") for all ω≥12. More importantly, it is shown that unusual, weakly occupied natural orbitals (NOs) corresponding to additional positive-valued natural amplitudes emerge upon sufficient weakening of the confinement. These solitonic NOs, whose shapes remain almost invariant as their radial positions drift toward infinity upon the critical values of ω being approached from below, exhibit strong radial localization. Their asymptotic properties are extracted from the numerical data and their relevance to calculations on fully Coulombic systems is discussed.

Simpler is often better: Computational efficiency of explicitly correlated two-electron basis sets generated by the regularized Krylov sequences of Nakatsuji.

A measure Δ of computational efficiency of the explicitly correlated basis sets (XCBSs) generated by the regularized Krylov sequences of Nakatsuji (also known as "the free complement" or "the free iterative CI" method) is derived from the convergence characteristics of the underlying iterative process. A complete mathematical definition of this process, which includes a crucial projection operator undefined in previous publications, is provided. Comparison of the values of Δ pertaining to several XCBSs designed for the helium isoelectronic series (for which Δ = -K -1/3 ln ϵ, where ϵ is the error in the computed energy and K is the number of the basis functions comprising the basis set) leads to a surprising conclusion that, among sufficiently large XCBSs, those stemming from the seed basis sets (SBSs) independent of the interelectron distance are the most efficient ones. Consequently, taking into account the simplicity of the resulting matrix elements of diverse quantum-mechanical operators, the XCBSs generated from the {exp(-ζs), s 1/2 exp(-ζs)} SBS (where s is the first of the Hylleraas coordinates) appear to be best suited for extremely accurate electronic structure calculations on helium-like species.

Five- and six-electron harmonium atoms: Highly accurate electronic properties and their application to benchmarking of approximate 1-matrix functionals.

Electronic properties of several states of the five- and six-electron harmonium atoms are obtained from large-scale calculations employing explicitly correlated basis functions. The high accuracy of the computed energies (including their components), natural spinorbitals, and their occupation numbers makes them suitable for testing, calibration, and benchmarking of approximate formalisms of quantum chemistry and solid state physics. In the case of the five-electron species, the availability of the new data for a wide range of the confinement strengths ω allows for confirmation and generalization of the previously reached conclusions concerning the performance of the presently known approximations for the electron-electron repulsion energy in terms of the 1-matrix that are at heart of the density matrix functional theory (DMFT). On the other hand, the properties of the three low-lying states of the six-electron harmonium atom, computed at ω = 500 and ω = 1000, uncover deficiencies of the 1-matrix functionals not revealed by previous studies. In general, the previously published assessment of the present implementations of DMFT being of poor accuracy is found to hold. Extending the present work to harmonically confined systems with even more electrons is most likely counterproductive as the steep increase in computational cost required to maintain sufficient accuracy of the calculated properties is not expected to be matched by the benefits of additional information gathered from the resulting benchmarks.

Harmonium atoms at weak confinements: The formation of the Wigner molecules.

The formation of the Wigner molecules in three-dimensional assemblies of equicharged particles upon weakening of harmonic confinements with spherical symmetry is investigated using five electronic states of the four-electron harmonium atom as a representative example. Numerical results of accurate explicitly-correlated calculations, carried out for a wide range of confinement strengths ω, reveal nonmonotonic convergence of the total energies and one-electron densities to those predicted by the formalism asymptotically exact at the ω→0 limit. The convergence rates are strongly state- and property-dependent, which implies the nonexistence of a single measure quantifying the extent of the formation of the Wigner molecule in a particular state of the harmonium atom. In light of these observations, it is concluded that, like in two dimensions, the emergence of the Wigner molecules in Coulombic systems confined by spherically symmetric harmonic potentials is a complex and gradual process that takes place over a range of confinement strengths spanning several orders of magnitude.

Rovibrational states of Wigner molecules in spherically symmetric confining potentials.

The strong-localization limit of three-dimensional Wigner molecules, in which repulsively interacting particles are confined by a weak spherically symmetric potential, is investigated. An explicit prescription for computation of rovibrational wavefunctions and energies that are asymptotically exact at this limit is presented. The prescription is valid for systems with arbitrary angularly-independent interparticle and confining potentials, including those involving Coulombic and screened (i.e., Yukawa/Debye) interactions. The necessary derivations are greatly simplified by explicit constructions of the Eckart frame and the parity-adapted primitive wavefunctions. The performance of the new formalism is illustrated with the three- and four-electron harmonium atoms at their strong-correlation limits. In particular, the involvement of vibrational modes with the E symmetry is readily pinpointed as the origin of the "anomalous" weak-confinement behavior of the (1)S+ state of the four-electron species that is absent in its (1)D+ companion of the strong-confinement regime.

Note: An explicit solution of the optimal superposition and Eckart frame problems.

Attention is called to an explicit solution of both the optimal superposition and Eckart frame problems that requires neither matrix diagonalization nor quaternion algebra. A simple change in one variable that enters the expression for the solution matrix T allows for selection of T representing either a proper rotation or a more general orthogonal transformation. The issues concerning the use of these alternative selections and the equivalence of the two problems are addressed.

One-electron reduced density matrices of strongly correlated harmonium atoms.

Explicit asymptotic expressions are derived for the reduced one-electron density matrices (the 1-matrices) of strongly correlated two- and three-electron harmonium atoms in the ground and first excited states. These expressions, which are valid at the limit of small confinement strength ω, yield electron densities and kinetic energies in agreement with the published values. In addition, they reveal the ω(5/6) asymptotic scaling of the exchange components of the electron-electron repulsion energies that differs from the ω(2/3) scaling of their Coulomb and correlation counterparts. The natural orbitals of the totally symmetric ground state of the two-electron harmonium atom are found to possess collective occupancies that follow a mixed power/Gaussian dependence on the angular momentum in variance with the simple power-law prediction of Hill's asymptotics. Providing rigorous constraints on energies as functionals of 1-matrices, these results are expected to facilitate development of approximate implementations of the density matrix functional theory and ensure their proper description of strongly correlated systems.

The Coulomb, exchange, and correlation components of the electron-electron repulsion in harmonium atoms.

Highly accurate Coulomb, exchange, and correlation components of the electron-electron repulsion energies of the three-electron harmonium atoms in the (2)P- and (4)P+ states are obtained for 19 values of the confinement strength ω ranging from 10(-3) to 10(3). The computed data are consistent with their ω → 0 and ω → ∞ asymptotics that are given by closed-form algebraic expressions. Robust approximants that accurately reproduce the actual values of the energy components while strictly conforming to these limits are constructed, opening an avenue to stringent tests capable of predicting the performance of electronic structure methods for systems with varying extents of the dynamical and nondynamical electron correlation. The values of the correlation components, paired with the computed 1-matrices are expected to be particularly useful in the context of benchmarking of approximate density matrix functionals.

Robust validation of approximate 1-matrix functionals with few-electron harmonium atoms.

A simple comparison between the exact and approximate correlation components U of the electron-electron repulsion energy of several states of few-electron harmonium atoms with varying confinement strengths provides a stringent validation tool for 1-matrix functionals. The robustness of this tool is clearly demonstrated in a survey of 14 known functionals, which reveals their substandard performance within different electron correlation regimes. Unlike spot-testing that employs dissociation curves of diatomic molecules or more extensive benchmarking against experimental atomization energies of molecules comprising some standard set, the present approach not only uncovers the flaws and patent failures of the functionals but, even more importantly, also allows for pinpointing their root causes. Since the approximate values of U are computed at exact 1-densities, the testing requires minimal programming and thus is particularly suitable for rapid screening of new functionals.