Comment on “Scaling properties of information-theoretic quantities in density functional reactivity theory” by C. Rong, T. Lu, P. W. Ayers, P. K. Chattaraj and S. Liu, Phys. Chem. Chem. Phys., 2015, 17 , 4977–4988
The scaling properties of density functionals are key for fundamentally understanding density functional theory. Accordingly, the dependence of density functionals on the number of particles is of paramount relevance. The numerical exploration by Rong et al. addressed N-scaling for a set of quantum information quantities; they found linear relationships between each one of them and the electronic population for atoms, molecules, and atoms in molecules. The main motivation for their computational work was that the theoretical scaling of these quantities is unknown; however, these scaling properties can be analytically determined. Here I reveal the derivation of the N-scaling rules for the quantities studied by Rong et al. by following the procedure introduced in Comput. Theor. Chem., 2015, 1053, 38. In addition, a new atomic scaling rule explains the linear relationship between atomic populations and atomic values of the same quantum information quantities.
The scaling properties of density functionals are key for fundamentally understanding density functional theory. Accordingly, the dependence of density functionals on the number of particles is of paramount relevance. The numerical exploration by Rong et al. addressed N-scaling for a set of quantum information quantities; they found linear relationships between each one of them and the electronic population for atoms, molecules, and atoms in molecules. The main motivation for their computational work was that the theoretical scaling of these quantities is unknown; however, these scaling properties can be analytically determined. Here I reveal the derivation of the N-scaling rules for the quantities studied by Rong et al. by following the procedure introduced in Comput. Theor. Chem., 2015, 1053, 38. In addition, a new atomic scaling rule explains the linear relationship between atomic populations and atomic values of the same quantum information quantities.
In [Phys. Rev. Lett. 128, 013001 (2022)] a novel ground state method was proposed. It has been suggested that this i-DMFT would be a method within one-particle reduced density matrix functional theory (DMFT), capable of describing accurately molecules at various geometries with an information-theoretical nature. We reassess this work and its suggestions from a conceptual and practical point of view, leading to the following conclusions: i) A method which assigns to each molecule M its own functional FM is not a functional theory (striking violation of "universality") ii) even for the simplest systems i-DMFT yields incorrect one-particle reduced density matrices and iii) the use of an information-theoretical concept to describe molecular dissociation limits was not essential. The latter insight may help to fix the deficiency of i-DMFT to not reproduce correctly the smaller occupation numbers and thus to not recover the important dynamic correlations.
Accurate descriptions of the electronic properties of molecules, solids and other materials are crucial in the development of many modern technologies, and are therefore the subject of much scientific research. In tandem with experimental research, computational simulations play a huge role. Density functional theory (DFT) is the most popular method for performing ground-state electronic structure calculations due to its accuracy in many situations along with its computational efficiency. Its time-dependent extension (TDDFT) allows the modelling of the dynamics of systems of interacting electrons. (TD)DFT is an exact theory in principle, but requires approximations in practice, specifically to the many-body effects of electron exchange and correlation (xc), which are described by some unknown universal functional. Despite much success, it often proves inadequate in many scenarios of increasing interest. Therefore, we aim to contribute to the development of improved approximate density functionals. \n \nIn this thesis, we take the approach of studying small model systems for which we can compute the exact functionals. These systems are designed to exhibit the key features present in realistic quantum systems, such as atoms and molecules, which the exact universal (TD)DFT functional is capable of describing, but are often ill-described by the commonly used approximations. By analysing the nature of these exact functionals, we gain valuable insight into their fundamental properties, together with identifying where the current approximations are lacking, thereby allowing us to suggest how these approximations may be improved.
Time-Independent and Periodic Time-Dependent Potentials ...................... 73 6.3 Formal Foundations of Density Functional Theory for Time-Dependent Electric and Magnetic Fields ........................................... 74 6.4 Practical Schemes for the Calculation of Density and Current Density Variables in TDDFT ......................................................... 77 6.5 Linear Response within TDDFT ................................................................. 79 6.6 Concluding Remarks.................................................................................... 80 Acknowledgments................................................................................................... 80 References ............................................................................................................... 81 The study of behavior of many-electron systems such as atoms, molecules, and solids under the action of time-dependent (TD) external fields, which includes interaction with radiation, has been an important area of research. In the linear response regime, where one considers the external field to cause a small perturbation to the initial ground state of the system, one can obtain many important physical quantities such as polarizabilities, dielectric functions, excitation energies, photoabsorption spectra, van der Waals coefficients, etc. In many situations, for example, in the case of interaction of many-electron systems with strong laser field, however, it is necessary to go beyond linear response for investigation of the properties. Since a full theoretical description based on accurate solution of TD Schrodinger equation is not yet within the reach of computational capabilities, new methods which can efficiently handle the TD many-electron correlations need to be explored, and time-dependent density functional theory (TDDFT) is one such valuable approach.
To avoid confusion, it is necessary to be quite explicit about what is assumed and what is to be proved. A. Klein, Phys. Rev. 121, 950 (1961) Summary The topic of this chapter is functionals that provide a concise formulation for thermodynamic quantities and Green's function methods in interacting many-body systems. Building upon the instructive examples of density functional theory and the Hartree– Fock approximation, functionals of the Green's function are developed to provide a framework for practical methods used extensively in many-body perturbation theory (Chs. 9–15), dynamical mean-field theory (Chs. 16–20), and their combination in Ch. 21. Functional concepts give a firm foundation for those methods, for example, they give the conditions for approximations to obey conservation laws. Additional background and specific aspects are in App. H. Many physical quantities can be expressed as functionals , i.e., they depend on an entire function.1 For example, in the quantum variational method to determine a wavefunction, the total energy of a system of many electrons is a functional of a many-body trial wavefunction ( r 1 , r 2 , …): it is the expectation value of the hamiltonian. At the variational minimum, assuming the trial wavefunction is completely flexible, one finds the many-body Schrödinger equation for the ground state. Many of the properties we are interested in are, formally, simple functionals of the ground- or excited-state wavefunctions. In the search for feasible ways to calculate properties such as excitation spectra, we are led to different types of functional. The idea is to deal only with a physical, measurable quantity Q of interest and a variational functional E [ Q ] of this quantity. The variational character of E allows one to derive equations that determine Q 0 , the value of Q for the system in a desired state, typically the ground state or equilibrium state at T ≠ 0. Other properties should be different functionals F m [ Q ] that can then be evaluated at Q = Q 0 . This strategy is outlined in Fig. 8.1. To appreciate the power of such approaches, one can consider the impact of density functional theory, which is formulated as a functional of the density n ( r ) of electrons; it is defined for a range of densities and the variational minimum for the energy functional E [ n ] leads to equations to determine the actual ground-state density n 0 ( r ) and total energy E 0 .
This note is intended for expanding the details on the derivation and properties of density functional theory, in hope to make them more systematic, better motivated, and step-by-step for readers new to the domain. The note starts with basic concepts in quantum mechanics, then takes the step towards many-body systems using the tools of second quantization and Fock space, with some highlights on properties of the Coulomb system. Given these general technical preparations, the Hartree-Fock method is naturally unrolled, with expressions for various cases and quantities. Density functional theory is then presented, with a motivating reasoning of the Kohn-Sham formulation, and a connection and comparison of the expressions with the Hartree-Fock method. Lieb's celebrated 1983 paper on functional analysis on density functionals is also summarized.