Gradient Expansions for the Large-Coupling Strength Limit of the Møller-Plesset Adiabatic Connection

Exact Mo̷ller-Plesset Adiabatic Connection Correlation Energy Densities

The Mo̷ller-Plesset adiabatic connection (MPAC) provides a powerful tool for developing density functional theory (DFT)-like approximations that map Hartree-Fock densities to the wave function-based correlation energy, thereby leveraging both wave function and DFT concepts for electronic structure approximations. A key object in this context is the correlation energy density, which represents the local (pointwise) contribution to the total correlation energy. While well-studied in DFT, it remains largely unexplored in the wave function framework. Here, we introduce a rigorous formulation of the wave function-based correlation energy density within MPAC, implement it via full configuration interaction calculations, and analyze its behavior and physically meaningful contributions for representative small (di)atomic systems. We define this quantity by employing a general gauge strategy, from which the conventional DFT correlation energy density gauge also arises. We then discuss the resulting commonalities and differences between correlation energy densities in the DFT and wave function frameworks and derive the small-interaction (MP2) limit of the latter in terms of Hartree-Fock orbitals. Finally, we show how these newly introduced energy densities can serve as new approximation targets in both machine-learning-assisted and traditional electronic structure methods for mapping HF-density-based features to correlation energy within the wave function framework.

Nonempirical Adiabatic Connection Correlation Functional from Hartree-Fock Orbitals

We present a nonempirical strategy to construct a correlation functional rooted in the Møller-Plesset (MP) adiabatic connection (AC) formalism for the strong-interaction regime, which satisfies both the weak- and strong-interaction limits and describes accurately the uniform electron gas (UEG) model. The functional is based on Hartree-Fock (HF) orbitals and employs only the UEG and helium atom as model systems; thus, it can be considered a nonempirical and nonlinear generalization of post-HF approaches based on the second-order perturbation theory (MP2) correlation. The functional describes the correlation of atoms with 1 mHa/electron accuracy, and it is also accurate for jellium surface energies. Accurate tests using a nearly complete basis set on diverse systems and properties (atomization/interaction energies, dispersion forces, and ionization potentials) have shown an excellent performance of the functional that corrects the MP2 overbinding without error cancellation. The present investigation can open the way for the development of a new generation of post-HF functionals based on nonlinear MP2 contributions and strong-correlation ingredients.

Møller-Plesset Adiabatic Connection at Large Coupling Strengths for Open-Shell Systems

We study the adiabatic connection that has as weak-coupling expansion the Møller-Plesset perturbation series, generalizing to the open-shell case previous closed-shell results for the large-coupling limit. We first focus on the hydrogen atom with fractional spins, providing results along the adiabatic connection from small to large coupling strengths. We reveal an intriguing phase diagram and an equation for the large-coupling leading order that has closed-form solutions for specific choices of its relevant quantum numbers. We then show that the hydrogen atom results provide variational estimates for the large-coupling leading terms for the general many-electron open-shell case in terms of functionals of the Hartree-Fock α-spin and β-spin densities.

Investigations of the exchange energy of neutral atoms in the large-Z limit

The non-relativistic large-Z expansion of the exchange energy of neutral atoms provides an important input to modern non-empirical density functional approximations. Recent works report results of fitting the terms beyond the dominant term, given by the local density approximation (LDA), leading to an anomalous Z ln Z term that cannot be predicted from naïve scaling arguments. Here, we provide much more detailed data analysis of the mostly smooth asymptotic trend describing the difference between exact and LDA exchange energy, the nature of oscillations across rows of the Periodic Table, and the behavior of the LDA contribution itself. Special emphasis is given to the successes and difficulties in reproducing the exchange energy and its asymptotics with existing density functional approximations.

Large-Z atoms in the strong-interaction limit of DFT: Implications for gradient expansions and for the Lieb-Oxford bound

We numerically study the strong-interaction limit of the exchange-correlation functional for neutral atoms and Bohr atoms as the number of electrons increases. Using a compact representation, we analyze the second-order gradient expansion, comparing it with the one for exchange (weak interaction limit). The two gradient expansions, at strong and weak interaction, turn out to be very similar in magnitude but with opposite signs. We find that the point-charge plus continuum model is surprisingly accurate for the gradient expansion coefficient at strong coupling, while generalized gradient approximations, such as Perdew-Burke-Ernzerhof (PBE) and PBEsol, severely underestimate it. We then use our results to analyze the Lieb-Oxford bound from the point of view of slowly varying densities, clarifying some aspects on the bound at a fixed number of electrons.

TURBOMOLE: Today and Tomorrow

TURBOMOLE is a highly optimized software suite for large-scale quantum-chemical and materials science simulations of molecules, clusters, extended systems, and periodic solids. TURBOMOLE uses Gaussian basis sets and has been designed with robust and fast quantum-chemical applications in mind, ranging from homogeneous and heterogeneous catalysis to inorganic and organic chemistry and various types of spectroscopy, light-matter interactions, and biochemistry. This Perspective briefly surveys TURBOMOLE's functionality and highlights recent developments that have taken place between 2020 and 2023, comprising new electronic structure methods for molecules and solids, previously unavailable molecular properties, embedding, and molecular dynamics approaches. Select features under development are reviewed to illustrate the continuous growth of the program suite, including nuclear electronic orbital methods, Hartree-Fock-based adiabatic connection models, simplified time-dependent density functional theory, relativistic effects and magnetic properties, and multiscale modeling of optical properties.

Semilocal Meta-GGA Exchange-Correlation Approximation from Adiabatic Connection Formalism: Extent and Limitations

The incorporation of a strong-interaction regime within the approximate semilocal exchange-correlation functionals still remains a very challenging task for density functional theory. One of the promising attempts in this direction is the recently proposed adiabatic connection semilocal correlation (ACSC) approach [Constantin, L. A.; Phys. Rev. B 2019, 99, 085117] allowing one to construct the correlation energy functionals by interpolation of the high and low-density limits for the given semilocal approximation. The current study extends the ACSC method to the meta-generalized gradient approximations (meta-GGA) level of theory, providing some new insights in this context. As an example, we construct the correlation energy functional on the basis of the high- and low-density limits of the Tao-Perdew-Staroverov-Scuseria (TPSS) functional. Arose in this way, the TPSS-ACSC functional is one-electron self-interaction free and accurate for the strictly correlated and quasi-two-dimensional regimes. Based on simple examples, we show the advantages and disadvantages of ACSC semilocal functionals and provide some new guidelines for future developments in this context.

Regularized and Opposite Spin-Scaled Functionals from Møller-Plesset Adiabatic Connection─Higher Accuracy at Lower Cost

Noncovalent interactions (NCIs) play a crucial role in biology, chemistry, material science, and everything in between. To improve pure quantum-chemical simulations of NCIs, we propose a methodology for constructing approximate correlation energies by combining an interpolation along the Møller-Plesset adiabatic connection (MP AC) with a regularization and spin-scaling strategy applied to MP2 correlation energies. This combination yields cosκos-SPL2, which exhibits superior accuracy for NCIs compared to any of the individual strategies. With the N4 formal scaling, cosκos-SPL2 is competitive or often outperforms more expensive dispersion-corrected double hybrids for NCIs. The accuracy of cosκos-SPL2 particularly shines for anionic halogen bonded complexes, where it surpasses standard dispersion-corrected DFT by a factor of 3 to 5.

Møller-Plesset and Density-Fixed Adiabatic Connections for a Model Diatomic System at Different Correlation Regimes

In recent years, adiabatic connection (AC) interpolations developed within density functional theory (DFT) have been found to provide good performances in the calculation of interaction energies when used with Hartree-Fock (HF) ingredients. The physical and mathematical reasons for such unanticipated performance have been clarified, to some extent, by studying the strong-interaction limit of the Møller-Plesset (MP) AC. In this work, we calculate both the MP and the DFT AC integrand for the asymmetric Hubbard dimer, which allows for a systematic investigation of different correlation regimes by varying two simple parameters in the Hamiltonian: the external potential, Δv, and the interaction strength, U. Notably, we find that, while the DFT AC integrand appears to be convex in the full parameter space, the MP integrand may change curvature twice. Furthermore, we discuss different aspects of the second-order expansion of the correlation energy in each AC, and we demonstrate why the derivative of the λ-dependent density in the MP AC at λ = 0 (i.e., at the HF density) is zero in the model. Concerning the strong-interaction limit of both ACs in the Hubbard dimer setting, we show that the asymptotic value of the MP AC, W∞HF, is lower than (or equal to) its DFT analogue, W∞KS, if the two are compared at a given density, just like in real space. However, we also show that this is not always the case if the two quantities are compared at a given external potential.

Leading Correction to the Local Density Approximation for Exchange in Large-Z Atoms

The large-Z asymptotic expansion of atomic energies has been useful in determining exact conditions for corrections to the local density approximation in density functional theory. The correction for exchange is fit well with a leading ZlnZ term, and we find its coefficient numerically. The gradient expansion approximation also has such a term, but with a smaller coefficient. Analytic results in the limit of vanishing interaction with hydrogenic orbitals (a Bohr atom) lead to the conjecture that the coefficients are precisely 2.7 times larger than their gradient expansion counterparts, yielding an analytic expression for the exchange-energy correction which is accurate to ∼5% for all Z.

Assessment of the Second-Order Statically Screened Exchange Correction to the Random Phase Approximation for Correlation Energies

With increasing interelectronic distance, the screening of the electron–electron interaction by the presence of other electrons becomes the dominant source of electron correlation. This effect is described by the random phase approximation (RPA) which is therefore a promising method for the calculation of weak interactions. The success of the RPA relies on the cancellation of errors, which can be traced back to the violation of the crossing symmetry of the 4-point vertex, leading to strongly overestimated total correlation energies. By the addition of second-order screened exchange (SOSEX) to the correlation energy, this issue is substantially reduced. In the adiabatic connection (AC) SOSEX formalism, one of the two electron–electron interaction lines in the second-order exchange term is dynamically screened (SOSEX(W, v_c)). A related SOSEX expression in which both electron–electron interaction lines are statically screened (SOSEX(W(0), W(0))) is obtained from the G3W2 contribution to the electronic self-energy. In contrast to SOSEX(W, v_c), the evaluation of this correlation energy expression does not require an expensive numerical frequency integration and is therefore advantageous from a computational perspective. We compare the accuracy of the statically screened variant to RPA and RPA+SOSEX(W, v_c) for a wide range of chemical reactions. While both methods fail for barrier heights, SOSEX(W(0), W(0)) agrees very well with SOSEX(W, v_c) for charged excitations and noncovalent interactions where they lead to major improvements over RPA.

Comparing correlation components and approximations in Hartree-Fock and Kohn-Sham theories via an analytical test case study

The asymmetric Hubbard dimer is a model that allows for explicit expressions of the Hartree-Fock (HF) and Kohn-Sham (KS) states as analytical functions of the external potential, Δv, and of the interaction strength, U. We use this unique circumstance to establish a rigorous comparison between the individual contributions to the correlation energies stemming from the two theories in the {U, Δv} parameter space. Within this analysis of the Hubbard dimer, we observe a change in the sign of the HF kinetic correlation energy, compare the indirect repulsion energies, and derive an expression for the "traditional" correlation energy, i.e., the one that corrects the HF estimate, in a pure site-occupation function theory spirit [Eq. (45)]. Next, we test the performances of the Liu-Burke and the Seidl-Perdew-Levy functionals, which model the correlation energy based on its weak- and strong-interaction limit expansions and can be used for both the traditional and the KS correlation energies. Our results show that, in the Hubbard dimer setting, they typically work better for the HF reference, despite having been originally devised for KS. These conclusions are somewhat in line with prior assessments of these functionals on various chemical datasets. However, the Hubbard dimer model allows us to show the extent of the error that may occur in using the strong-interaction ingredient for the KS reference in place of the one for the HF reference, as has been carried out in most of the prior assessments.