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arXiv:2609.34164

Basis Functions for Time-Dependent Kohn-Sham Inversion

Paul Martin Zimmerman, Joshua Kammeraad, Duy-Khoi Dang, Harish S. Bhat

Published Sep 28, 2026• 0 citations
arXivHTMLPDF

Abstract

Floquet theory provides insight into the inversion of time-dependent Kohn-Sham density functional theory. Specifically, mathematical derivations show that the fundamental frequencies of a time-dependent wavefunction solution are the leading-order harmonics for the TD-KS state. Numerical tests of the resulting ansatz in 1D and 3D for atomic and molecular cases demonstrate its utility. In particular, low $L_{2}$ errors in the time-dependent density and longitudinal current were found, even though currents were not an explicit optimization objective. In all, the proposed inversion ansatz provides exchange-correlation potentials from time-dependent wavefunctions, is highly interpretable, and may significantly help in the development of nonadiabatic density functionals.

DOI
10.48550/arxiv.2609.34164

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Floquet theory provides insight into the inversion of time-dependent Kohn-Sham density functional theory. Specifically, mathematical derivations show that the fundamental frequencies of a time-dependent wavefunction solution are the leading-order harmonics for the TD-KS state. Numerical tests of the resulting ansatz in 1D and 3D for atomic and molecular cases demonstrate its utility. In particular, low L2L_{2}L2​ errors in the time-dependent density and longitudinal current were found, even though currents were not an explicit optimization objective. In all, the proposed inversion ansatz provides exchange-correlation potentials from time-dependent wavefunctions, is highly interpretable, and may significantly help in the development of nonadiabatic density functionals.
Given the time-evolution of an electron charge density, the local potential in Kohn-Sham time-dependent density functional theory (KS-TDDFT) can be modeled as a sum of instantaneous and dynamic contributions by assuming a certain form of the time-dependent Kohn-Sham (TD-KS) orbitals. The instantaneous part is obtained numerically using methods from ground-state density functional theory (DFT) and the dynamic part is expressed in terms of a velocity potential that depends on the electron current density. The suggested form of the TD-KS orbitals satisfies several known constraints (orthonormality, N-representability, J-representability), and the domain of validity is shown to depend on the evolution of the instantaneous quantities. Through this decomposition, we can relate time-dependent and ground -state V-representability. The resulting potentials are shown numerically to approximate the exact time-dependent Kohn-Sham potentials for a set of 3 non-singlet two-particle systems (a Kohn-mode, a Coulomb explosion, and a double quantum well) where the exact solutions and reference densities are known or obtained through configuration interaction (CI) approaches.
The Floquet theorem allows one to reformulate periodic time-dependent problems such as the interaction of a many-body system with a laser field in terms of time-independent, field-dressed states, also known as Floquet states. If this was possible for density-functional theory as well, one could reduce in such cases time-dependent density-functional theory to a time-independent Floquet density-functional theory. We analyze under which conditions the Floquet theorem is applicable in a density-functional framework. By employing numerical ab initio solutions of the interacting time-dependent Schr\"odinger equation with time-periodic external potentials we show that the exact effective potential in the corresponding Kohn-Sham equation is not unconditionally periodic. Whenever several Floquet states in the interacting system are involved in a physical process the corresponding Hartree-exchange-correlation potential is not periodic with the external frequency only. Using an analytically solvable example we demonstrate that, in general, the periodicity of the time-dependent Kohn-Sham Hamiltonian cannot be restored by choosing a different initial state. Only if the external periodic potential is sufficiently weak such that the initial state of the interacting system evolves adiabatically to a single, field-dressed state, the resulting Kohn-Sham system admits the application of the Floquet theorem.
The exact time-dependent correlation energy in time-dependent density-functional theory is shown to sometimes become positive, which is impossible with most present TDDFT approximations. Both the correlation potential and energy can be strongly nonlocal in time. A new inequality is derived for the time-dependent exchange-correlation energy. The correlation energy appears to scale to a constant function of scaled time in the high-density limit. In the linear response regime, the correlation energy is shown to become purely adiabatic, but the correlation potential is generally nonadiabatic. The usefulness of the virial theorem as a test of numerical accuracy is demonstrated. All results are found or inspired by exact numerical solution of a simple model system (Hooke’s atom), and inversion of the corresponding Kohn–Sham equations.
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.
We formulate the inverse Kohn–Sham equations based on the density equation theory. The present formulation relies on neither action functionals nor force balance equation, but is constructed by comparing the first-order density equation for the electron interacting real system and that for the electron non-interacting Kohn–Sham system. The derived inverse Kohn–Sham integral equation requires information of an exact ground-state electron density to calculate an exact Kohn–Sham potential for a ground state and it requires an exact time-dependent electron density and a time-derivative of the exact time-dependent first-order density matrix to calculate an exact time-dependent Kohn–Sham potential. A possible algorithm to construct the Kohn–Sham potential for a ground state is suggested as well as a solution algorithm for the time-dependent inverse Kohn–Sham problem without resorting to any approximations. These two solution algorithms, in combination, suggest a way to perform an exact time-dependent density functional calculation as a predictive method.
Abstract The construction of a better exchange-correlation potential in time-dependent density functional theory (TDDFT) can improve the accuracy of TDDFT calculations and provide more accurate predictions of the properties of many-electron systems. Here, we propose a machine learning method to develop the energy functional and the Kohn–Sham potential of a time-dependent Kohn–Sham (TDKS) system is proposed. The method is based on the dynamics of the Kohn–Sham system and does not require any data on the exact Kohn–Sham potential for training the model. We demonstrate the results of our method with a 1D harmonic oscillator example and a 1D two-electron example. We show that the machine-learned Kohn–Sham potential matches the exact Kohn–Sham potential in the absence of memory effect. Our method can still capture the dynamics of the Kohn–Sham system in the presence of memory effects. The machine learning method developed in this article provides insight into making better approximations of the energy functional and the Kohn–Sham potential in the TDKS system.