Session B58: DFT and Beyond II

On the Constrained Search-Coordinate Scaling Formulations in DFT

First, a review of the early history of DFT is given, including anecdotes, and the constrained-search formulation is presented within the context of this history. Then it is shown how this formulation leads to the derivation of key constraints for approximating density functionals. These constraints include those that involve the coordinate scaling of the total density, and of the individual up and down spin densities, where the use of spin-free wavefunctions for the derivations has very recently proven to be effective.

The utility of expressing the total energies of ground-states and excited states as simple sums of augmented orbital energies shall also be discussed. Then several open problems in DFT shall be presented.   

Essential difference between the machine learning and artificial Kohn-Sham potentials

The Kohn-Sham density functional theory is widely used to predict physical or chemical properties of various materials with its practical accuracy and computational cost. Its accuracy depends on the exchange-correlation functional. Although there exist many approximations for the functional, its entire structure is yet elusive since the reference databases and physical conditions available are limited. On the other hand, we develop another strategy; the functional structure is represented with a flexible neural network and its parameters are trained with machine-learning algorithms[1]. The NN-based functional becomes transferable enough by training with the density, which contains abundant information in the 3D space.

Here, we analyze the properties of the machine-learning density functionals. We investigate essential differences between the machine-learned functionals and artificial functionals by comparing their performance using accurate densities and ones from artificial functionals as the training data.


[1] R. Nagai, R. Akashi, and O. Sugino, arXiv:1903.00238 (2019).
[2] R. Nagai, K. Burke, R. Akashi, and O. Sugino, in preparation.   

Machine learning accurate exchange and correlation functionals of the electronic density

Density Functional Theory (DFT) is the standard formalism to study the electronic structure of matter at the atomic scale. The balance between accuracy and computational cost that DFT-based simulations provide allows researchers to understand the structural and dynamical properties of increasingly large and complex systems at the quantum mechanical level. In Kohn-Sham DFT, this balance depends on the choice of exchange and correlation functional, which only exists in approximate form. Here we propose a framework to create highly accurate density functionals by using supervised machine learning, termed NeuralXC. These machine-learned functionals are designed to lift the accuracy of local and semilocal functionals to that provided by more accurate methods while maintaining their efficiency. We show that the functionals learn a meaningful representation of the physical information contained in the training data, making them transferable across systems. We further demonstrate how a functional optimized for water can reproduce experimental results when used in molecular dynamics simulations.   

Constrained Machine Learning de-orbitalization of meta-GGA exchange-correlation functionals

Meta-generalized-gradient-approximation (mGGA) exchange-correlation functionals commonly take the orbital kinetic energy density τ(r) as an ingredient in their construction [1]. Such τ(r) dependent functionals have shown impressive performance for diverse problems, but orbital dependence of τ(r) complicates the exchange-correlation potential and increases computational cost. Recently, Rodriguez et.al. constructed a mGGA with laplacian▽²n without significantly degrading accuracy [2]. This suggests an intriguing but unclear relationship between τ(r) and ▽²n . We exploit this relationship using machine learning combined with exact constraints to explore how neural-network models can de-orbitalize functionals and model fundamental components of Kohn-Sham density functional theory.

[1] Jianwei Sun et.al. Phys. Rev. Lett. 115,036402 (2015)
[2] Mejia-Rodriguez et.al. Phys. Rev. A 96, 052512 (2017)   

Semiclassical Approximation to Sums of Eigenvalues with Application to DFT

I present a new mathematical framework developed by Kieron Burke and Michael V. Berry for estimating the sum of the lowest N eigenenergies of a one-dimensional potential. I apply this method to several model onedimensional systems (harmonic oscillator, Poschl-Teller well, quartic oscillator, linear well, and exponential well). This new method gives the sum of the energies as a semiclassical series, which can be shown to reproduce the DFT gradient expansion for slowly varying densities, and also produces a correction to the gradient expansion for finite systems with a discrete spectrum. Explicit corrections to the gradient expansion of the kinetic energy are derived which in simple cases greatly improve accuracy. All work is done assuming a system of non-interacting identical fermions. This research is funded by the NSF (CHE 1856165).
  

Progress in the Development of Advanced Temperature-Dependent Free-Energy Density Functionals Beyond the Generalized Gradient Approximation

Recently, it has been shown that explicit dependence on temperature in the exchange-correlation (XC) free-energy density functional is important in density functional theory studies of warm dense matter [1]. As a first approximation, in the attempt to include thermal effects into XC, the finite-T KSDT local spin-density approximation functional was constructed by the analytical parametrization of the XC free energy of the homogeneous electron gas. Consequently, the non-empirical KDT16 [2] generalized gradient approximation (GGA) functional, which captures nonhomogeneity effects and shows better agreement with experiments, has been constructed. Here, we present recent progress in going beyond the GGA by developing finite-T extensions to the PBE0 and SCAN-L (the de-orbitalized version of SCAN) XC functionals. We present preliminary results for optical reflectivity of shocked CH and warm dense He.

1 V. V. Karasiev, L. Calderín, and S. B. Trickey, Phys. Rev. E 93, 063207 (2016).
2 V. V. Karasiev, J. W. Dufty, and S. B. Trickey, Phys. Rev. Lett. 120, 076401 (2018).   

Symmetry-Breaking Polymorphous Descriptions for Correlated Materials without Interelectronic U

Correlated materials with open-shell d- and f-ions having degenerate band edge states show a rich variety of interesting properties. The textbook view for the electronic structure of these materials is that mean-field approaches are inappropriate, as the interelectronic interaction U is required to open a band gap between the occupied and unoccupied degenerate states while retaining symmetry. We show that the mean-field band theory can lift such degeneracies when nontrivial unit cell representations (polymorphous networks) are allowed to break symmetry, in conjunction with a recently developed non-empirical exchange and correlation density functional without an on-site interelectronic interaction U. We rationalize how density functional theory (DFT) in the polymorphous representation achieves band gap opening in correlated materials through a separate mechanism to the Mott-Hubbard approach. We show the method predicts magnetic moments and gaps for four classical 3d transition-metal monoxides in both the antiferromagnetic and paramagnetic phases, offering a highly-efficient alternative to symmetry-conserving approaches for studying a range of functionalities in open d- and f-shell complex materials.   

A step in the direction of resolving the paradox of Perdew-Zunger self-interaction correction

Self-interaction (SI) error, which results when exchange-correlation contributions to the total energy are approximated, limits the reliability of many density functional approximations. The Perdew-Zunger SI correction (PZSIC), when applied in conjunction with the local spin density approximation (LSDA), improves the description of many properties, but overall, this improvement is limited. Here we propose a modification to PZSIC that uses an iso-orbital indicator to identify regions where local SI corrections should be applied. Using this local-scaling SIC (LSIC) approach with LSDA, we analyze predictions for a wide range of properties including, for atoms, total energies, ionization potentials, and electron affinities, and for molecules, atomization energies, dissociation energy curves, reaction energies, and reaction barrier heights. LSIC preserves the results of PZSIC-LSDA for properties where it is successful and provides dramatic improvements for many of the other properties studied. Atomization energies calculated using LSIC are better than those of the Perdew, Burke, and Ernzerhof (PBE) generalized gradient approximation (GGA). LSIC also restores the uniform gas limit for the exchange energy that is lost in PZSIC-LSDA.   

Kohn-Sham effective potentials from FLOSIC using Ryabinkin-Kohut-Staroverov method

Density functional or beyond methods are often used in combination with photoelectron spectroscopy to obtain physical insights about the electronic structure of molecules and solids. The Kohn-Sham eigenvalues are not electron removal energies except for the highest occupied one, but they often, though not always, provide good approximations to electron binding energies (EBEs). Eigenvalues of the range separated hybrid functionals using tuned separation parameter generally provide good approximations to EBEs due to mitigation of self-interaction (SI) errors. We adapt and implement the Ryabinkin-Kohut-Staroverov method to obtain effective local potentials from the self-interaction corrected Fermi-Lowdin orbitals and density in the FLOSIC code. The density of states and HOMO-LUMO gaps obtained using this approach show much closer agreement with experimental values compared to those obtained with a range of DFA or functionals.   

Performance of scaled self-interaction correction to semilocal functionals

Most semilocal functionals suffer from self-interaction error (SIE). Perdew and Zunger (PZ)¹ applied self-interaction correction (SIC) to each delocalized Kohn-Sham orbital by subtracting the SIE from each occupied orbital. The Fermi-Lowdin orbital self-interaction correction (FLOSIC) uses size-extensive localized Fermi-Lowdin orbitals that improve the results for the properties like barrier heights that involves strong SIE, as in stretched bonds. However, it worsens the equilibrium properties such as atomization energies, as FLOSIC applied to a functional violates the exact constraints and the appropriate norms that the semilocal approximations satisfy². Here, we apply a scaled PZ SIC to LSDA, PBE, and SCAN and calculated different properties such as electron affinity, ionization potential, barrier height, atomization energy, and bond length of some representative test sets. We find that the scaled SIC works well for both the equilibrium properties and for the features that involve stretched bonds or noded densities.

1. J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981)
2. B. Santra and J. P. Perdew, J. Chem. Phys. 150, 174106 (2019)