Constraints upon Functionals of the 1-Matrix, Universal Properties of Natural Orbitals, and the Fallacy of the Collins "Conjecture".

Reliability of quantum-chemical calculations based upon the density functional theory and its 1-matrix counterpart hinges upon minimizing the extent of empirical parameterization in the approximate energy expressions of these formalisms while imposing as many rigorous constraints upon them as possible. The recently uncovered universal properties of the natural orbitals facilitate the construction of such constraints for the 1-matrix functionals. The benefits of their employment in the validation of these functionals are vividly demonstrated by a critical review of the three incarnations of the so-called Collins conjecture. Although the incorporation of rigorous definitions of the correlation energy and entropy, and the identification of individual potential energy hypersurfaces as probable domains of its applicability turn the originally published unsubstantiated claim into a proper conjecture, the resulting formalism is found to be merely a conduit for incorporation of static correlation effects in electronic structure calculations that is unlikely to allow attaining chemical accuracy.

Symmetry Equiincidence of Natural Orbitals.

The symmetry equiincidence principle quantifies the apportionment of the natural orbitals (NOs), ordered according to their nonascending occupation numbers, among the irreducible representations (irreps) of the point group pertaining to the underlying on-top two-electron density. This principle, which is rigorously proven for the resolvable Cs, C2v, C3v, C4v, C6v, D2h, D3h, D4h, D6h, and Oh point groups, states that the symmetry incidences, i.e., the asymptotic probabilities with which the NOs belonging to different irreps occur, are proportional to the squares of irreps' dimensions. Since its proof hinges upon a sufficient number of planes of symmetry among the elements of a given point group, it yields only linear combinations of the symmetry incidences for the quasiresolvable groups with too few such planes and fails for the unresolvable C1, Ci, Cn, Dn, S2n, T, O, and I groups whose nontrivial elements comprise only symmetry axes and/or the center of inversion.

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.

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.

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.

A Universal Power Law Governing the Accuracy of Wave Function-Based Electronic Structure Calculations.

A universal power law governing the accuracy of wave function-based electronic structure calculations is derived from first principles. The resulting expression ΔE(N,N)N≳19π2gNN, where g is a system-specific factor assuming values between zero and one and ≳ stands for asymptotic inequality at the limit of N→∞, allows facile estimation of the error ΔE(N,N) in the electronic energy of a singlet state of an N-electron system computed with a basis set of N one-electron functions. Several approaches to the estimation of the factor g, which depends on the on-top two-electron density, are presented.

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.

From Fredholm to Schrödinger via Eikonal: A New Formalism for Revealing Unknown Properties of Natural Orbitals.

Previously unknown properties of the natural orbitals (NOs) pertaining to singlet states (with natural parity, if present) of electronic systems with even numbers of electrons are revealed upon the demonstration that, at the limit of n → ∞, the NO ψn(r⃗) with the nth largest occupation number νn approaches the solution ψ̃n(r⃗) of the zero-energy Schrödinger equation that reads T̂([ρ2(r⃗, r⃗)]-1/8 ψ̃n(r⃗)) - (π2/vn)1/4 [ρ2(r⃗, r⃗)]1/4 ([ρ2(r⃗, r⃗)]-1/8 ψ̃n(r⃗)) = 0 (where T̂ is the kinetic energy operator), whereas νn approaches ν̃n. The resulting formalism, in which the "on-top" two-electron density ρ2(r⃗, r⃗) solely controls the asymptotic behavior of both ψn(r⃗) and νn at the limit of the latter becoming infinitesimally small, produces surprisingly accurate values of both quantities even for small n. It opens entirely new vistas in the elucidation of their properties, including single-line derivations of the power laws governing the asymptotic decays of νn and ⟨ψn(r⃗)|T̂|ψn(r⃗)⟩ with n, some of which have been obtained previously with tedious algebra and arcane mathematical arguments. These laws imply a very unfavorable asymptotics of the truncation error in the total energy computed with finite numbers of natural orbitals that severely affects the accuracy of certain quantum-chemical approaches such as the density matrix functional theory. The new formalism is also shown to provide a complete and accurate elucidation of both the observed order (according to decreasing magnitudes of the respective occupation numbers) and the shapes of the natural orbitals pertaining to the 1Σg+ ground state of the H2 molecule. In light of these examples of its versatility, the above Schrödinger equation is expected to have its predictive and interpretive powers harnessed in many facets of the electronic structure theory.

Angular-Momentum Extrapolations to the Complete Basis Set Limit: Why and When They Work.

The leading L-3 dependence of the errors in the energies computed with nuclei-centered basis sets comprising functions with angular momenta not exceeding L is rigorously proven for the 1Σ states of linear molecules and molecular ions with arbitrary even numbers of electrons. This major expansion of the domain of applicability over that offered by the routinely cited Hill asymptotic expression, which is valid only for the helium isoelectronic series, is accomplished with a formalism in which the off-diagonal cusp conditions for the one- and two-electron reduced density matrices play the central role. Despite being provided by these results with theoretical foundations more solid than ever before, the angular-momentum extrapolations to the complete basis set limit appear to work more by happenstance than mathematical rigor due to the poorly predictable variability in the prefactor multiplying the L-3 term and the far from negligible contributions from the terms involving higher powers of L-1.

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.

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.

One-Electron Reduced Density Matrix Functional Theory of Spin-Polarized Systems.

The formulation of the density matrix functional theory (DMFT) in which the correlation component U of the electron-electron repulsion energy is expressed in terms of a model two-electron density cumulant matrix (a.k.a. the 2-cumulant) [Formula: see text] and the tensor of two-electron integrals g is critically analyzed. The dependence of [Formula: see text] on both the vector of occupation numbers n and g derived from the corresponding vector [Formula: see text](1) of natural spinorbitals, and its concomitant invariance to the nature of the spin-independent interparticle interaction potential that enters the definition of g are emphasized. In the case of spin-polarized systems, [Formula: see text] (and thus U) is found to be a function of not only n and g, but also of the vector [Formula: see text] and the matrix [Formula: see text], where [Formula: see text] is the spin-flip operator. The presence of spin polarization imposes additional constraints upon [Formula: see text], including a simple condition that, when satisfied, assures the underlying wavefunction being an eigenstate of both the [Formula: see text] and [Formula: see text] operators with the eigenvalues [Formula: see text] and [Formula: see text], respectively. This feature allows targeting electronic states with definite multiplicities, which is virtually impossible in the case of the Kohn-Sham implementation of density functional theory. Among the four possible pairing schemes for the natural spinorbitals that give rise to approximations employing [Formula: see text] with only two independent indices, three are found to result in unphysical constraints even for spin-unpolarized systems, whereas the failure of the fourth one turns out to be precipitated by the presence of spin polarization. Consequently, any implementation of DMFT based upon "two-index" [Formula: see text] is shown to be generally unsuitable for spin-polarized systems (and incapable of yielding the spin-parallel components of U for the spin-unpolarized ones). A clear distinction is made between the genuine 1-matrix functionals that are defined for arbitrary N-representable 1-matrices and general energy expressions that depend on auxiliary quantities playing the role of fictitious 1-matrices subject to additional (often unphysical) constraints.

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.

Bilinear Constraints upon the Correlation Contribution to the Electron-Electron Repulsion Energy as a Functional of the One-Electron Reduced Density Matrix.

Perturbative analysis of the functional U[n, ψ] that yields the correlation component U of the electron-electron repulsion energy in terms of the vectors ψ(1) and n of the natural spinorbitals and their occupation numbers (the 1-matrix functional) facilitates examination of the flaws inherent to the present implementations of the density matrix functional theory. Recognizing that the practical usefulness of any approximate 1-matrix functional hinges upon its capability of exactly reproducing the leading contribution to U at the limit of vanishing electron-electron interactions gives rise to asymptotic bilinear constraints for the (exact or model) 2-cumulant G2 that enters the expression for U. The asymptotic behavior of certain blocks of G2 is found to be equally important. These identities, which are obtained for both the single-determinantal and a model multideterminantal cases, take precedence over the linear constraints commonly enforced in the course of approximate construction of such functionals. This observation reveals the futility of designing sophisticated approximations tailored for the second-order contribution to G2 while neglecting proper formulation of the respective first-order contribution that in the case of the so-called JKL-only functionals requires abandoning the JK-dependence altogether. It has its repercussions not only for the functionals of the PNOF family but also for the expressions involving only the L-type two-electron repulsion integrals (in the guise of their exchange counterparts) that account only for the correlation effects due to electrons with antiparallel spins and are well-defined only for spin-unpolarized and high-spin systems (yielding vanishing U for the latter).

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.

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.

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.