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

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.

Outstanding improvement in removing the delocalization error by global natural orbital functional

This work assesses the performance of the recently proposed global natural orbital functional (GNOF) against the charge delocalization error. GNOF provides a good balance between static and dynamic electronic correlations leading to accurate total energies while preserving spin, even for systems with a highly multi-configurational character. Several analyses were applied to the functional, namely, (i) how the charge is distributed in super-systems of two fragments, (ii) the stability of ionization potentials while increasing the system size, and (iii) potential energy curves of a neutral and charged diatomic system. GNOF was found to practically eliminate the charge delocalization error in many of the studied systems or greatly improve the results obtained previously with PNOF7.

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.

Electron Correlation in the Iron(II) Porphyrin by Natural Orbital Functional Approximations

The relative stability of the singlet, triplet, and quintet spin states of iron(II) porphyrin (FeP) represents a challenging problem for electronic structure methods. While it is currently accepted that the ground state is a triplet, multiconfigurational wave function-based methods predict a quintet, and density functional approximations vary between triplet and quintet states, leading to a prediction that highly depends on the features of the method employed. The recently proposed Global Natural Orbital Functional (GNOF) aims to provide a balanced treatment between static and dynamic correlation, and together with the previous Piris Natural Orbital Functionals (PNOFs), allowed us to explore the importance of each type of correlation in the stability order of the states of FeP with a method that conserves the spin of the system. It is noteworthy that GNOF correlates all electrons in all available orbitals for a given basis set; in the case of the FeP with a double-ζ basis set as used in this work, this means that GNOF can properly correlate 186 electrons in 465 orbitals, significantly increasing the sizes of systems amenable to multiconfigurational treatment. Results show that PNOF5, PNOF7s, and PNOF7 predict the quintet to have a lower energy than the triplet state; however, the addition of dynamic correlation via second-order Møller-Plesset corrections (NOF-MP2) turns the triplet state to be lower than the quintet state, a prediction also reproduced by GNOF that incorporates much more dynamic correlation than its predecessors.

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.

Functional-Based Description of Electronic Dynamic and Strong Correlation: Old Issues and New Insights

Approximate functionals in Kohn-Sham density functional theory (KS-DFT) and reduced density matrix functional theory (RDMFT) have advantages in dealing with dynamic correlation and strong correlation, respectively; their combination can benefit from complementarity while suffering from the problem of correlation double-counting. Herein, a short-range corrected reduced density matrix (1-RDM) functional is developed to take advantage of the functionals in KS-DFT and RDMFT without double-counting. The resulting functional, denoted as ωP22, outperforms other 1-RDM functionals for the tests of thermochemistry, nonbonded interactions, and bond dissociation energy. In particular, ωP22 shows much less systematic error for systems involving fractional spins, and it can properly predict the energies at both equilibrium and dissociated distances for different single and multiple bonds, which cannot be achieved by commonly used KS-DFT and RDMFT functionals. Therefore, ωP22 is demonstrated effective in balance handling dynamic and strong correlation, and the advances in this work would create new possibilities for the development and application of approximate functionals.

Cumulants as the variables of density cumulant theory: A path to Hermitian triples

We study the combination of orbital-optimized density cumulant theory and a new parameterization of reduced density matrices in which the variables are the particle-hole cumulant elements. We call this combination OλDCT. We find that this new Ansatz solves problems identified in the previous unitary coupled cluster Ansatz for density cumulant theory: the theory is now free of near-zero denominators between occupied and virtual blocks, can correctly describe the dissociation of H₂, and is rigorously size-extensive. In addition, the new Ansatz has fewer terms than the previous unitary Ansatz, and the optimal orbitals delivered by the exact theory are the natural orbitals. Numerical studies on systems amenable to full configuration interaction show that the amplitudes from the previous ODC-12 method approximate the exact amplitudes predicted by this Ansatz. Studies on equilibrium properties of diatomic molecules show that even with the new Ansatz, it is necessary to include triples to improve the accuracy of the method compared to orbital-optimized linearized coupled cluster doubles. With a simple iterative triples correction, OλDCT outperforms other orbital-optimized methods truncated at comparable levels in the amplitudes, as well as coupled cluster single and doubles with perturbative triples [CCSD(T)]. By adding four more terms to the cumulant parameterization, OλDCT outperforms CCSDT while having the same O(V⁵O³) scaling.

Assessing the orbital-optimized unitary Ansatz for density cumulant theory

The previously proposed Ansatz for density cumulant theory that combines orbital-optimization and a parameterization of the 2-electron reduced density matrix cumulant in terms of unitary coupled cluster amplitudes (OUDCT) is carefully examined. Formally, we elucidate the relationship between OUDCT and orbital-optimized unitary coupled cluster theory and show the existence of near-zero denominators in the stationarity conditions for both the exact and some approximate OUDCT methods. We implement methods of the OUDCT Ansatz restricted to double excitations for numerical study, up to the fifth commutator in the Baker-Campbell-Hausdorff expansion. We find that methods derived from the Ansatz beyond the previously known ODC-12 method tend to be less accurate for equilibrium properties and less reliable when attempting to describe H₂ dissociation. New developments are needed to formulate more accurate density cumulant theory variants.

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.