The density equation proposed previously for the direct determination of the density matrix, i.e. for the wave mechanics without wave, is extended to a time- dependent case. The time-dependent density equation has been shown to be equivalent to the time-dependent Schrodinger equation so long as the density matrix, included as a self-contained variable, is N-representable. Formally, it is obtainable from the previous time- independent equation by replacing the energy E with ih @ @t . The perturbation theory formulas for the density equation have also been given for both the time- dependent and time-independent cases.The density equation proposed previously for the direct determination of the density matrix, i.e. for the wave mechanics without wave, is extended to a time- dependent case. The time-dependent density equation has been shown to be equivalent to the time-dependent Schrodinger equation so long as the density matrix, included as a self-contained variable, is N-representable. Formally, it is obtainable from the previous time- independent equation by replacing the energy E with ih @ @t . The perturbation theory formulas for the density equation have also been given for both the time- dependent and time-independent cases.

Jan 1, 19990 citations

Density Matrices and Perturbation Theory

Mitchel Weissbluth

Elsevier eBooksJan 1, 19890 citations

A perturbation solution of the equation of motion for the density matrix

Hernan C. Praddaude

Annals of PhysicsMay 1, 19631 citation

Extension of second-principles density functional theory into the time domain

Toraya Fernández-Ruiz, Jorge Íñiguez, Javier Junquera +1 more

We present an extension of second-principles density functional theory (SPDFT) to perform time-dependent simulations. Our approach calculates the real-time, real-space evolution of the density matrix using the Liouville-von Neumann equation of motion, enabling the determination of optical and transport properties in very large systems, containing tens of thousands of atoms, on modest computational platforms. Unlike other methods, SPDFT is applicable to a wide range of materials, including both metals and insulators. We demonstrate its capabilities by computing the spectra of ${\mathrm{SrTiO}}_{3}$ , diamond, and metallic lithium. While SPDFT results for ${\mathrm{SrTiO}}_{3}$ closely match those obtained from DFT with linear perturbation theory, significant improvements are observed for diamond and metallic lithium. In diamond, the inclusion of electron-electron interactions during the density-matrix evolution yields spectra that more accurately reproduce Bethe-Salpeter equation results than perturbative DFT. In lithium, time-dependent SPDFT captures both interband transitions and the Drude peak, enabling detailed ab initio studies of transport properties beyond usual approximations.We present an extension of second-principles density functional theory (SPDFT) to perform time-dependent simulations. Our approach calculates the real-time, real-space evolution of the density matrix using the Liouville-von Neumann equation of motion, enabling the determination of optical and transport properties in very large systems, containing tens of thousands of atoms, on modest computational platforms. Unlike other methods, SPDFT is applicable to a wide range of materials, including both metals and insulators. We demonstrate its capabilities by computing the spectra of ${\mathrm{SrTiO}}_{3}$, diamond, and metallic lithium. While SPDFT results for ${\mathrm{SrTiO}}_{3}$ closely match those obtained from DFT with linear perturbation theory, significant improvements are observed for diamond and metallic lithium. In diamond, the inclusion of electron-electron interactions during the density-matrix evolution yields spectra that more accurately reproduce Bethe-Salpeter equation results than perturbative DFT. In lithium, time-dependent SPDFT captures both interband transitions and the Drude peak, enabling detailed ab initio studies of transport properties beyond usual approximations.

arXiv:2507.13824Physical review. B./Physical review. BNov 8, 20252 citations

Kinetic Equation for the Density Matrix

É. G. Pestov, Sergey G. Rautian

Journal of Experimental and Theoretical PhysicsJan 1, 19691 citation

Basic Concepts of Quantum Mechanics

M. Nisoli

Basic concepts of quantum mechanics: Schroedinger equation; Dirac notation; the energy representation; expectation value; Hermite operators; coherent superposition of states and motion in the quantum world; perturbation Hamiltonian. Time-dependent perturbation theory: harmonic perturbation. Transition rate: Fermi’s golden rule. The density matrix; pure and mixed states. Temporal dependence of the density operator: von Neuman equation. Randomizing Hamiltonian. Longitudinal and transverse relaxation times. Density matrix and entropy.Basic concepts of quantum mechanics: Schroedinger equation; Dirac notation; the energy representation; expectation value; Hermite operators; coherent superposition of states and motion in the quantum world; perturbation Hamiltonian. Time-dependent perturbation theory: harmonic perturbation. Transition rate: Fermi’s golden rule. The density matrix; pure and mixed states. Temporal dependence of the density operator: von Neuman equation. Randomizing Hamiltonian. Longitudinal and transverse relaxation times. Density matrix and entropy.

Cambridge University Press eBooksNov 10, 20220 citations

{\mathrm{SrTiO}}_{3}

, diamond, and metallic lithium. While SPDFT results for

{\mathrm{SrTiO}}_{3}

closely match those obtained from DFT with linear perturbation theory, significant improvements are observed for diamond and metallic lithium. In diamond, the inclusion of electron-electron interactions during the density-matrix evolution yields spectra that more accurately reproduce Bethe-Salpeter equation results than perturbative DFT. In lithium, time-dependent SPDFT captures both interband transitions and the Drude peak, enabling detailed ab initio studies of transport properties beyond usual approximations.