[Paper Review] A unified treatment of derivative discontinuity, delocalization and static correlation effects in density functional calculations

Fei Zhou, Vidvuds Ozoliņš|arXiv (Cornell University)|Oct 24, 2017

Magnetic and transport properties of perovskites and related materials1 references3 citations

TL;DR

This paper proposes LDA+DMM, a unified method that explicitly incorporates derivative discontinuity, corrects delocalization errors, and treats static correlation in density functional theory. By minimizing the exchange-correlation energy with respect to the Fock space density matrix, LDA+DMM accurately reproduces the Mott-Hubbard gap, magnetic ordering, and Jahn-Teller distortion in KCuF₃—outperforming LDA and LDA+U with modest computational cost.

ABSTRACT

We propose a method that incorporates explicit derivative discontinuity of the total energy with respect to the number of electrons and treats both delocalization and static correlation effects in density functional calculations. Our approach is motivated by the exact behavior of the ground state total energy of electrons and involves minimization of the exchange-correlation energy with respect to the Fock space density matrix. The resulting density matrix minimization (DMM) model is simple to implement and can be solved uniquely and efficiently. In a case study of KCuF$_3$, a prototypical Mott-insulator with strong correlation, LDA+DMM correctly reproduced the Mott-Hubbard gap, magnetic ordering and Jahn-Teller distortion.

Motivation & Objective

• Address the fundamental limitations of LDA and GGA functionals in strongly correlated systems, including delocalization and static correlation errors.
• Correct the missing derivative discontinuity in the Kohn-Sham potential, which leads to underestimated band gaps and poor description of Mott insulators.
• Unify treatment of three major errors—derivative discontinuity, delocalization, and static correlation—within a single, physically motivated framework.
• Develop a method that is both accurate and computationally feasible for implementation in standard DFT codes.
• Demonstrate the method’s ability to correctly describe Mott insulating behavior, magnetic ordering, and Jahn-Teller distortions in KCuF₃

Proposed method

• Formulate the exchange-correlation energy minimization over the Fock space density matrix, enforcing physical constraints on electron number and spin polarization.
• Implement a density matrix minimization (DMM) model that enforces the exact piecewise linearity of the total energy with respect to electron number $N_{ ext{e}}$, ensuring explicit derivative discontinuity.
• Use a semidefinite programming approach to solve the DMM problem uniquely and efficiently, avoiding the multiple minima issues of LDA+U.
• Apply the DMM correction within the LDA framework (LDA+DMM), enabling straightforward integration into existing DFT codes.
• Incorporate both on-site Coulomb repulsion $U$ and exchange $J$ parameters to treat static correlation effects, with $U_{ ext{critical}} = 8.06$ eV at $J = 0.9$ eV.
• Ensure the method respects the exact ground-state behavior: constant energy with respect to fractional spin polarization and linear dependence on $N_{ ext{e}}$

Experimental results

Research questions

• RQ1 Can a single DFT-based method simultaneously correct for derivative discontinuity, delocalization error, and static correlation in strongly correlated materials?
• RQ2 Does the DMM approach reproduce the correct Mott-Hubbard gap in a prototypical Mott insulator like KCuF₃?
• RQ3 How does LDA+DMM compare to LDA+U and GGA+DMFT in predicting magnetic ordering and Jahn-Teller distortions?
• RQ4 Can the DMM model be implemented efficiently and uniquely within standard DFT codes without introducing multiple minima?
• RQ5 What is the role of the critical $U$ parameter in stabilizing the Mott insulating state within the DMM framework?

Key findings

• LDA+DMM correctly reproduces the Mott-Hubbard gap of approximately 7 eV in KCuF₃, consistent with the derivative discontinuity $\mathcal{D}_{\text{xc}} \approx 7$ eV.
• The method predicts a Mott insulator in both the paramagnetic and antiferromagnetic phases, unlike LDA+U, which fails to open a gap in the paramagnetic state.
• Jahn-Teller distortion is stabilized at $\delta_{\text{JT}} = 4.2\%$ in LDA+DMM, in excellent agreement with the experimental value of 4.4%.
• The total energy profile shows that LDA+DMM correctly stabilizes the antiferromagnetic state, with a Néel temperature of 38 K, in qualitative agreement with experiment.
• LDA+DMM predicts a stabilization energy for Jahn-Teller distortion comparable to GGA+DMFT, while GGA alone underestimates it significantly.
• The DMM model avoids the multiple minima problem of LDA+U by solving a convex semidefinite programming problem uniquely and efficiently.