Session M01: Density Functional Theory and Beyond III

The Connector Theory Approach: Principles and Development of new Density Functionals

Density Functional Theory (DFT) and Time-Dependent (TD) DFT allow us in principle to express observables as functionals of the ground state density. Relatively efficient approximations exist for the ground state total energy, but major difficulties remain, for example to describe strongly correlated systems, or situations where long range correlation is important. For other observables, even less is known about how to build good density functionals.

One way of designing approximations is to use results of model systems. In this talk we will discuss and compare different strategies to do this [1]. In particular, we will show how model results can be used in an in principle exact way, called “Connector Theory”, in order to describe observables and systems of interest [2]. Within this approach, a quantity of interest is calculated for a parametrized model system once and forever, and the results are stored. The ``connector’’ prescribes the value of the parameter that has to be chosen such that the model result can be used to represent the result of the real system.

We will discuss the principles and general properties of such an approach. In practice, the connector has to be approximated. We will show that formulating in this way the development of functionals is a convenient starting point for approximations, and a strategy to build systematic approximations will be presented.

We will then focus more specifically on the use of the connector approach to design density functionals for DFT and TDDFT. Particular emphasis will be put on effects of non-locality in space and time, which are important to capture, e.g., image potentials or multiple excitations [3], respectively.

[1] A. Aouina, M. Gatti, and L. Reining, Faraday Discussions 224, 27 (2020)

[2] M. Vanzini, A. Aouina, M. Panholzer, M. Gatti, and L. Reining, https://arxiv.org/abs/1903.07930

[3] M Panholzer, M Gatti, L Reining, Phys. Rev. Lett. 120, 166402 (2018).   

Non-adiabatic functionals in TDDFT from the Connector Theory

Time-dependent density functional theory (TDDFT) has become a widely used method for the description of electronic spectra and responses, as well as an important tool for real-time electronic dynamics. However, these calculations use predominantly adiabatic functionals, i.e. with no memory (or frequency) dependence, for the exchange correlation potential (or kernel). With this limitation, accurate description of non-adiabatic phenomena such as resonantly-driven dynamics or dynamics far from the ground-state remains usually out of reach. Recently, a new approach called "Connector Theory" (COT), that makes use of the knowledge obtained on model systems to compute quantities in physical systems, has been developed. COT is in principle exact, but in practice approximations are needed to make it tractable. In this talk, we develop new non-adiabatic functionals for the exchange-correlation potential using the COT approach. We present the various approximations and models chosen to derive our functionals and discuss the quality of the resulting time-dependent quantities obtained in model and realistic systems.   

Exchange-correlation energy from exact and model Green's functions

Density functional theory (DFT) and Green’s function methods represent two large, and seemingly disparate, approaches to modeling strongly correlated physics. Green’s function based methods, such as Dynamical Mean Field Theory (DMFT), have proven effective in modeling properties of real systems with strong correlation. The intersections of DFT and Green’s function methods are still not well understood and the failures and pathologies of each remains a matter of open research. We derive a formula to determine the exchange-correlation (XC) energy associated to exact and model Green’s functions. Using an exactly solvable interacting system, we show that our formula yields the exact XC energy, and we determine the XC energy associated with the Green’s function of DMFT. A formula connecting DFT quantities with Green’s functions provides a promising step towards better approximate forms of Green’s functions and density functionals.   

Memory effects in TDDFT quadratic response theory

TDDFT is routinely used for calculations of excitations and response, achieving an unprecedented balance between accuracy and efficiency, while caution is applied for excitations problematic for the usual adiabatic approximations. In some applications, quantities from quadratic response theory are needed; for example, to obtain coupling matrix elements between excited states in non-adiabatic dynamics simulation. We explore the connection between spurious poles in these matrix elements and the lack of double-excitations in the adiabatic approximation to the kernel. We test the ability of a frequency-dependent kernel that folds in these excitations to eliminate the divergence and produce reliable couplings for quadratic response.    

First-principles, spatially- and temporally-nonlocal exchange-correlation kernel for jellium at all densities

Accurate parameterizations of the ground-state exchange-correlation energy of jellium have been known since Quantum Monte Carlo calculations in the 1980s. However, an equally accurate description of the time-dependent linear response of an electron gas to an external perturbation has been elusive. Many extant models of the jellium exchange-correlation kernel have substantial limitations: restrictions on the range of densities for which the model is accurate, restrictions to purely real or purely imaginary frequencies, etc. In this talk, I’ll motivate and describe a refinement [a] of the MCP07 [b] kernel that modifies its wavevector and frequency dependence, yielding a kernel that extrapolates to a very wide range of densities. The model kernel makes substantial corrections at low densities, where MCP07 is known to be least accurate, and fine tunes MCP07’s already excellent accuracy at typical metallic densities.  Moreover, our model is numerically parameterized at all real and purely imaginary frequencies, a boon to further computational and theoretical applications. Excited-state phenomena predicted with our model will also be discussed.

[a] A.D. Kaplan, N.K. Nepal, A. Ruzsinszky, P. Ballone, and J.P. Perdew, arXiv:2107.02249 (2021) (submitted).

[b] A. Ruzsinszky, N.K. Nepal, J.M. Pitarke, and J.P. Perdew, Phys. Rev. B 101, 245135 (2020).   

Simple Exchange-Correlation Energy Functionals for Strongly Coupled Light-Matter Systems based on the Fluctuation-Dissipation Theorem

Recent experimental advances in strongly coupled light-matter systems has sparked the development of general ab-initio methods capable of describing interacting light-matter systems from first principles. One of these methods, quantum-electrodynamical density-functional theory (QEDFT), promises computationally efficient calculations for large correlated light-matter systems with the quality of the calculation depending on the underlying approximation for the exchange-correlation functional. So far no true density-functional approximation has been introduced limiting the efficient application of the theory. In this paper, we introduce the first gradient-based density functional for the QEDFT exchange-correlation energy derived from the adiabatic-connection fluctuation-dissipation theorem. We benchmark this simple-to-implement approximation on small systems in optical cavities and demonstrate its relatively low computational costs for fullerene molecules up to C₁₈₀ coupled to 400,000 photon modes in a dissipative optical cavity. This work now makes first principle calculations of much larger systems possible within the QEDFT framework effectively combining quantum optics with large-scale electronic structure theory.

[1] J. Flick, arXiv: 2104.06980 (2021)   

Accelerating Time-Dependent Density Functional Theory with Physics-Informed Neural Networks

Time-dependent density functional theory (TDDFT) is an important method for simulating dynamical processes in quantum many-body systems. We explore the feasibility of physics-informed neural networks as a surrogate for TDDFT. We examine the computational efficiency and convergence behaviour of these solvers to state-of-the-art numerical techniques on models and small molecular systems. The method developed here has the potential to accelerate the TDDFT workflow, enabling the simulation of large-scale calculations of electron dynamics in matter exposed to strong electromagnetic fields, high temperatures, and pressures.   

Single-particle-exact density functionals

We introduce a novel kind of density functionals for interacting many-fermion systems in the spirit of the Levy—Lieb constrained search. In our approach all single-particle contributions to the energy are represented by exact functionals, and only the functional for the interaction energy requires an approximation in terms of the single-particle eigenstates. We discuss schemes for constructing the required approximate density matrices and report on benchmarking exercises that deliver ground-state energies, occupation numbers, and particle densities through evolutionary algorithms.   

A reformulation of time-dependent Kohn-Sham theory in terms of the second time derivative of the density

The Kohn-Sham approach to time-dependent density-functional theory (TDDFT) can be formulated, in principle exactly, by invoking the force-balance equation for the density, which leads to an explicit expression for the exchange-correlation potential as an implicit density functional. It is shown that this suggests a reformulation of TDDFT in terms of the second time derivative of the density, rather than the density itself. The result is a time-local Kohn-Sham scheme of second order in time whose causal structure is more transparent than that of the usual Kohn-Sham formalism. The scheme can be used to construct new approximations at the exchange-only level and beyond, and it offers a straightforward definition of the exact adiabatic approximation.