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The Journal of Chemical Physics

Strong electron correlation from partition density functional theory

Yi Shi, Yuming Shi, Adam Wasserman

Published Dec 12, 2023• 3 citations
Publisher

Abstract

Standard approximations for the exchange-correlation functional in Kohn-Sham density functional theory (KS-DFT) typically lead to unacceptably large errors when applied to strongly correlated electronic systems. Partition-DFT (PDFT) is a formally exact reformulation of KS-DFT in which the ground-state density and energy of a system are obtained through self-consistent calculations on isolated fragments, with a partition energy representing inter-fragment interactions. Here, we show how typical errors of the local density approximation (LDA) in KS-DFT can be largely suppressed through a simple approximation, the multi-fragment overlap approximation (MFOA), for the partition energy in PDFT. Our method is illustrated on simple models of one-dimensional strongly correlated linear hydrogen chains. The MFOA, when used in combination with the LDA for the fragments, improves LDA dissociation curves of hydrogen chains and produces results that are comparable to those of spin-unrestricted LDA, but without breaking the spin symmetry. MFOA also induces a correction to the LDA electron density that partially captures the correct density dimerization in strongly correlated hydrogen chains. Moreover, with an additional correction to the partition energy that is specific to the one-dimensional LDA, the approximation is shown to produce dissociation energies in quantitative agreement with calculations based on the density matrix renormalization group method.

DOI
10.1063/5.0175538

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Standard approximations for the exchange-correlation functional in Kohn-Sham density functional theory (KS-DFT) typically lead to unacceptably large errors when applied to strongly correlated electronic systems. Partition-DFT (PDFT) is a formally exact reformulation of KS-DFT in which the ground-state density and energy of a system are obtained through self-consistent calculations on isolated fragments, with a partition energy representing inter-fragment interactions. Here, we show how typical errors of the local density approximation (LDA) in KS-DFT can be largely suppressed through a simple approximation, the multi-fragment overlap approximation (MFOA), for the partition energy in PDFT. Our method is illustrated on simple models of one-dimensional strongly correlated linear hydrogen chains. The MFOA, when used in combination with the LDA for the fragments, improves LDA dissociation curves of hydrogen chains and produces results that are comparable to those of spin-unrestricted LDA, but without breaking the spin symmetry. MFOA also induces a correction to the LDA electron density that partially captures the correct density dimerization in strongly correlated hydrogen chains. Moreover, with an additional correction to the partition energy that is specific to the one-dimensional LDA, the approximation is shown to produce dissociation energies in quantitative agreement with calculations based on the density matrix renormalization group method.
Standard approximations for the exchange-correlation (XC) functional in Kohn-Sham density functional theory (KS-DFT) typically lead to unacceptably large errors when applied to strongly-correlated electronic systems. Partition-DFT (PDFT) is a formally exact reformulation of KS-DFT in which the ground-state density and energy of a system are obtained through self-consistent calculations on isolated fragments, with a partition energy representing the \textit{inter}-fragment interactions. Here we show how typical errors of the local density approximation (LDA) in KS-DFT can be largely suppressed through a simple approximation, the generalized overlap approximation (GOA), for the partition energy in PDFT. Our method is illustrated on simple models of one-dimensional strongly-correlated linear hydrogen chains. The GOA, when used in combination with the LDA for the fragments, improves the LDA dissociation curves of hydrogen chains and produces results that are comparable to those of spin-unrestricted LDA, but without breaking the spin symmetry. GOA also induces a correction to the LDA electron density that partially captures the correct density dimerization in strongly-correlated hydrogen chains. Moreover, with an additional correction to the partition energy, the approximation is shown to produce dissociation energies in quantitative agreement to calculations based on the Density Matrix Renormalization Group method.
Most of the basic ideas of the density-functional theory (DFT) are shown to be unrelated to the fact that the particle density is used as the basic variable. After presenting the general formalism and a simple example unrelated to densities, we discuss various approaches to density-functional theory on a lattice. One has proven useful for the understanding of various fundamental issues of the DFT. The exact Kohn-Sham band gap and the band gap in the local-density approximation (LDA) are compared to the exact results for a one-dimensional model. As the interacting homogeneous system can be solved exactly no further approximation is needed to formulate the LDA.
The projection-based wave function in density functional theory (WF-in-DFT) embedding enables an efficient description of both the energetics and properties of large and complex chemical systems, with accuracy exceeding that of pure DFT. Recently, we have proposed using the density matrix renormalization group (DMRG) as the WF method for molecules containing strongly correlated fragments [ Beran, P. J. Phys. Chem. Lett. 2023, 14, 716–722]. In this work, we demonstrate that the accuracy of the DMRG-in-DFT approach is primarily limited by the approximate treatment of the coupling between the active component and its environment through nonadditive exchange-correlation functionals. To address this issue, we combine exact exchange to reduce the nonadditive exchange error with a multireference adiabatic connection (AC) scheme to recover nonadditive correlation. The performance of the improved DMRG-in-DFT embedding is illustrated on two prototypical strongly correlated systems: the dissociation of the H 20 chain and the cleavage of a triple CN bond in propionitrile.
Strongly correlated electrons have been the subject of substantial theoretical interest for many years. Most work has focused on obtaining the energy in a low-cost fashion. Here, we show that even methods with good energies can yield significant "delocalization errors" that affect the orbitals and density, leading to large errors in predicting other important properties such as dipole moments. We illustrate this point by comparing existing state-of-art approaches with an accurate exchange correlation functional based on a generalised valence bond ansatz, in which orbitals and fractional occupations are treated as variational parameters via a common optimized effective potential (OEP). We show that the OEP exhibits step and peak features which, similar to the exact Kohn-Sham (KS) potential of DFT, are crucial to prevent charge delocalization. We further show that the step is missing in common approximations within reduced density matrix functional theory resulting in delocalization errors comparable to those found in DFT approximations. Finally, we explain the delocalisation error as coming from an artificial mixing of the ground state with a charge-transfer excited state which is avoided if occupation numbers exhibit discontinuities.
We propose to expand the territory of density functional theory to strongly correlated electrons by reformulating the Kohn-Sham scheme in the representation of fractionalized particles. We call it the "KS^{*} scheme." Using inhomogeneous t-J chains as a test bed, we show that the KS^{*} scheme with simple local density approximation is able to achieve accurate ground-state energy and density distribution comparable to the density matrix renormalization group method, while the computational complexity is much lower.
Aiming to combine density functional theory (DFT) and wave-function theory, we study a mapping from the many-body interacting system to an effectively interacting Kohn-Sham system instead of a noninteracting Kohn-Sham system. Because a ground state of effectively interacting systems requires having a solution for the correlated many-body wave functions, this provides a natural framework to many-body wave-function theories such as the configuration interaction and the coupled-cluster method in the formal theoretical framework of DFT. Employing simple one-dimensional two-electron systems---namely, the one-dimensional helium atom, the hydrogen molecule, and the heteronuclear diatomic molecule---we investigate properties of many-body wave functions and exact exchange-correlation potentials of effectively interacting Kohn-Sham systems. As a result, we find that the asymptotic behavior of the exact exchange-correlation potential can be controlled by optimizing that of the effective interaction. Furthermore, the typical features of the exact noninteracting Kohn-Sham system, namely, a spiky feature and a step feature in the exchange-correlation potential for the molecular dissociation limit, can be suppressed by a proper choice of the effective interaction. These findings open a possibility to construct numerically robust and efficient exchange-correlation potentials and functionals based on the effectively interacting Kohn-Sham scheme.