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arXiv:2403.04458 · Physical Review Letters

Extended Time-Dependent Density Functional Theory for Multibody Densities

Jiong-Hang Liang, Tian-Xing Hu, Dong Wu, Zheng-Mao Sheng, Jian Zhang

Published Dec 23, 2024• 1 citation
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Abstract

Time-dependent density functional theory (TDDFT) is widely used for understanding and predicting properties and behaviors of matter. As one of the fundamental theorems in TDDFT, Van Leeuwen theorem [Phys. Rev. Lett. 82, 3863 (1999)PRLTAO0031-900710.1103/PhysRevLett.82.3863] guarantees how to construct a unique potential with the same one-body density evolution. Here we extend Van Leeuwen theorem by exploring truncation criteria in Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Our generalized theorem demonstrates the existence of a unique nonlocal potential to accurately reconstruct the multibody density evolution in binary interacting systems. Under nonstringent conditions, truncation of the BBGKY-hierarchy equations aligns with the behavior of multibody density evolution and maintains consistency in the reduced equations. Based on this theorem, we further construct a static many-body potential and extend the well-known Casida equation for multiple-excitation energy. The extended TDDFT provides an accurate and first-principles framework capable of describing the kinetic processes of correlated systems, including strongly coupled particle transport, multiple-excitation, and ionization processes.

DOI
10.1103/physrevlett.133.263001

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Time-dependent density functional theory (TDDFT) is widely used for understanding and predicting properties and behaviors of matter. As one of the fundamental theorems in TDDFT, Van Leeuwen theorem [Phys. Rev. Lett. 82, 3863 (1999)PRLTAO0031-900710.1103/PhysRevLett.82.3863] guarantees how to construct a unique potential with the same one-body density evolution. Here we extend Van Leeuwen theorem by exploring truncation criteria in Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Our generalized theorem demonstrates the existence of a unique nonlocal potential to accurately reconstruct the multibody density evolution in binary interacting systems. Under nonstringent conditions, truncation of the BBGKY-hierarchy equations aligns with the behavior of multibody density evolution and maintains consistency in the reduced equations. Based on this theorem, we further construct a static many-body potential and extend the well-known Casida equation for multiple-excitation energy. The extended TDDFT provides an accurate and first-principles framework capable of describing the kinetic processes of correlated systems, including strongly coupled particle transport, multiple-excitation, and ionization processes.
The equations of motion for reduced density matrices form a coupled chain known as the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. To close the coupled chain at the two-body level, approximations for a three-body density matrix with one-body and two-body density matrices are needed. The time-dependent density-matrix theory (TDDM) assumes that the three-body density matrix is given by the antisymmetrized products of the one-body and two-body density matrices. In this review the truncation schemes of the BBGKY hierarchy beyond TDDM are discussed and a formulation for the study of excited states which is derived from the time-dependent approach is explained. The truncation schemes and the formulation for excited states are applied to the Lipkin model and the Hubbard model to corroborate their validity. Two realistic applications of the TDDM approaches are also presented. One is the dipole and quadrupole excitations of 40^{40}40Ca and 48^{48}48Ca and the other the fusion reactions of 16^{16}16O + 16^{16}16O.
The Bogoliubov-Born-Green-Kirkwood-Yvon or time-dependent density matrix (TDDM) hierarchy of equations for higher density matrices is truncated at the three-body level in approximating the three-body correlation function by a quadratic form of two-body ones, closing the equations in this way. The procedure is discussed in detail and it is shown in nontrivial model cases that the approximate inclusion of three-body correlation functions is very important to obtain precise results. A small amplitude approximation of this time-dependent nonlinear equation for the two-body correlation function is performed (STDDM*-b) and it is shown that the one-body sector of this generalized nonlinear second random phase approximation (RPA) equation is equivalent to the self-consistent RPA (SCRPA) approach which had been derived previously by different techniques. It is discussed in which way SCRPA also contains the three-body correlations. TDDM and SCRPA are tested versus exactly solvable model cases.
The general expectation that, in principle, the time-dependent density-functional theory (TDDFT) is an exact formulation of the time evolution of an interacting NNN-electron system is critically reexamined. It is demonstrated that the previous TDDFT foundation, resting on four theorems by Runge and Gross (RG) [Phys. Rev. Lett. 52, 997 (1984)], is invalid because undefined phase factors corrupt the RG action integral functionals. Our finding confirms much of a previous analysis by van Leeuwen [Int. J. Mod. Phys. B 15, 1969 (2001)]. To analyze the RG theorems and other aspects of TDDFT, an utmost simplification of the Kohn-Sham (KS) concept has been introduced, in which the ground-state density is obtained from a single KS equation for one spatial (spinless) orbital. The time-dependent (TD) form of this radical Kohn-Sham (rKS) scheme, which has the same validity status as the ordinary KS version, has proved to be a valuable tool for analysis. The rKS concept is used to clarify also the alternative nonvariational formulation of TD KS theory. We argue that it is just a formal theory, allowing one to reproduce but not predict the time development of the exact density of the interacting NNN-electron system. Besides the issue of the formal exactness of TDDFT, it is shown that both the static and time-dependent KS linear response equations neglect the particle-particle (ppp-ppp) and hole-hole (hhh-hhh) matrix elements of the perturbing operator. For a local (multiplicative) operator this does not lead to a loss of information due to a remarkable general property of local operators. Accordingly, no logical inconsistency arises with respect to DFT, because DFT requires any external potential to be local. For a general nonlocal operator the error resulting from the neglected matrix elements is of second order in the electronic repulsion.
We reformulate the uniqueness and existence proofs of time-dependent density-functional theory. The central idea is to restate the fundamental one-to-one correspondence between densities and potentials as a global fixed-point question for potentials on a given time interval. We show that the unique fixed point, i.e. the unique potential generating a given density, is reached as the limiting point of an iterative procedure. The one-to-one correspondence between densities and potentials is a straightforward result provided that the response function of the divergence of the internal forces is bounded. The existence, i.e. the v -representability of a density, can be proven as well provided that the operator norms of the response functions of the members of the iterative sequence of potentials have an upper bound. The densities under consideration have second time-derivatives that are required to satisfy a condition slightly weaker than being square-integrable. This approach avoids the usual restrictions of Taylor-expandability in time of the uniqueness theorem by Runge and Gross ( Phys. Rev. Lett. , 52 (1984) 997 ) and of the existence theorem by van Leeuwen ( Phys. Rev. Lett. , 82 (1999) 3863 ). Owing to its generality, the proof not only answers basic questions in density-functional theory but also has potential implications in other fields of physics.
Time-dependent density functional theory (TDDFT) for quantum many-body systems on a lattice is formulated rigorously. We prove the uniqueness of the density-to-potential mapping and demonstrate that a given density is vvv representable if the initial many-body state and the density satisfy certain well-defined conditions. In particular, we show that for a system evolving from its ground state any density with a continuous second time derivative is locally in time vvv representable and therefore the lattice TDDFT is guaranteed to exist. The TDDFT existence and uniqueness theorem is valid for any connected lattice, independently of its size, geometry, and/or spatial dimensionality. General statements of the existence theorem are illustrated on a pedagogical exactly solvable example, which displays all the details and subtleties of the proof in a transparent form. In conclusion we briefly discuss remaining open problems and directions for future research.
This thesis is devoted to the formulation and implications of the time-dependent density functional theory (TDDFT). The work is divided into two main parts. In the first part we develop rigorous theorems for the density-potential mapping in quantum many-body systems on a lattice. We prove the uniqueness of the TDDFT map and demonstrate that a given density is v-representable if the initial many-body state and the density satisfy certain well defined conditions. In particular, we show that for a system evolving from its ground state any density with a continuous second time derivative is v-representable. Then the lattice TDDFT formulation is extended to cover system of interacting lattice electrons strongly coupled to cavity photons.We prove that under some mathematical conditions the electron-photon wave function is a unique functional of the electronic density and the expectation value of the photonic coordinate. Then we further generalize the ground state v representability theorem to include the ground state of a general lattice electron-photon Hamiltonian.