Ground-state density functional theory

P. Schmitteckert, F. Evers, Exact Ground State Density-Functional Theory for Impurity Models Coupled to External Reservoirs and Transport Calculations, Phys. Rev. Lett. 100 (2008) 08640. 

Ghosh, S. K. Berkowitz, M. Parr, R. G. Transcription of ground-state density-functional theory into a local thermodynamics. Proc. Natl. Acad. Sci. 1984, 81, 8028-8031. 

Virial theorem has an important role in quantum mechanics. It proved to be also very useful in density functional theory (see e.g. Parr and Yang, 1989). Several forms of virial theorem have been proposed in ground-state density functional theory, for example, local virial theorem (Nagy and Parr 1990), differential virial theorem (Holas and March 1995), regional virial theorem (Nagy 1992), and spin virial theorem (Nagy 1994b). In this chapter, local and differential virial theorems are extended to excited states in the frame of density functional theory. 

A second distinct approximation to /xc is in terms of its (usually) dominant exchange contribution. A highly accurate approximation to the exact exchange-only equations of ground-state density functional theory (the optimized effective potential equations) was introduced by Krieger, Li, and lafrate. This approximation has been extended to the time-dependent case ... 

Time-dependent density-functional theory (TDDFT) extends the basic ideas of ground-state density-functional theory (DFT) to the treatment of excitations and of more general time-dependent phenomena. TDDFT can be viewed as an alternative formulation of time-dependent quantum mechanics but, in contrast to the normal approach that relies on wave-functions and on the many-body Schrodinger equation, its basic variable is the one-body electron density, n(r,t). The advantages are clear The many-body wave-function, a function in a 3A-dimensional space (where N is the number of electrons in the system), is a very complex mathematical object, while the density is a simple function that depends solely on the 3-dimensional vector r. The standard way to obtain n r,t) is with the help of a fictitious system of noninteracting electrons, the Kohn-Sham system. The final equations are simple to tackle numerically, and are routinely solved for systems with a large number of atoms. These electrons feel an effective potential, the time-dependent Kohn-Sham potential. The exact form of this potential is unknown, and has therefore to be approximated. 

Perdew, J. P., Ruzsinszky, A., Constantin, L. A., Sun, J. W., 8c Csonka, G. I. (2009). Some fundamental issues in ground-state density functional theory A guide for the perplexed. Journal of Chemical Theory and Computation, 5, 902-908. 

The Levy construction [222] can be used to prove Hohenberg-Kohn theorems for the ground state of any such theory. It should be noted that any explicit model of the Hohenberg-Kohn functional F[p] implies a corresponding orbital functional theory. The relevant density function p(r) is that constructed from an OFT ground state. This has the orbital decomposition , as postulated by Kohn and Sham [205]. Unlike the density p,, for an exact A-electron wave function T, which cannot be determined for most systems of interest, the OFT ground-state density function is constructed from explicit solutions of the orbital Euler-Lagrange equations, and the theory is self-contained. 

The genesis of the failure of TDDFT can be traced to the fact that TDDFT is a linear response theory. When an excitation moves charge from one area in a molecule to another, both ends of that molecule will geometrically relax. While charge transfer between molecules can be well approximated by ground-state density functional calculations of the total energies of the species involved, TDDFT must deduce the correct transitions by... 

Lejaeghere K, Van Speybroeck V, Van Qost G, Cottenier S (2014) Error estimates for solid-state density-functional theory predictions an overview by means of the ground-state elemental crystals. Crit Rev Solid State Mater Sci 39 1-24... 

To use direct dynamics for the study of non-adiabatic systems it is necessary to be able to efficiently and accurately calculate electronic wave functions for excited states. In recent years, density functional theory (DFT) has been gaining ground over traditional Hartree-Fock based SCF calculations for the treatment of the ground state of large molecules. Recent advances mean that so-called time-dependent DFT methods are now also being applied to excited states. Even so, at present, the best general methods for the treatment of the photochemistry of polyatomic organic molecules are MCSCF methods, of which the CASSCF method is particularly powerful. 

Ah initio methods are applicable to the widest variety of property calculations. Many typical organic molecules can now be modeled with ah initio methods, such as Flartree-Fock, density functional theory, and Moller Plesset perturbation theory. Organic molecule calculations are made easier by the fact that most organic molecules have singlet spin ground states. Organics are the systems for which sophisticated properties, such as NMR chemical shifts and nonlinear optical properties, can be calculated most accurately. 

Addition of these two inequalities gives Eq + Eo>Eq + Eo, showing that the assumption was wrong. In other words, for the ground state there is a one-to-one correspondence between the electron density and the nuclear potential, and thereby also with the Hamilton operator and tlie energy. In the language of Density Functional Theory, the energy is a unique functional of the electron density, [p]. 

The ab initio methods used by most investigators include Hartree-Fock (FFF) and Density Functional Theory (DFT) [6, 7]. An ab initio method typically uses one of many basis sets for the solution of a particular problem. These basis sets are discussed in considerable detail in references [1] and [8]. DFT is based on the proof that the ground state electronic energy is determined completely by the electron density [9]. Thus, there is a direct relationship between electron density and the energy of a system. DFT calculations are extremely popular, as they provide reliable molecular structures and are considerably faster than FFF methods where correlation corrections (MP2) are included. Although intermolecular interactions in ion-pairs are dominated by dispersion interactions, DFT (B3LYP) theory lacks this term [10-14]. FFowever, DFT theory is quite successful in representing molecular structure, which is usually a primary concern. 

It is clear that an ah initio calculation of the ground state of AF Cr, based on actual experimental data on the magnetic structure, would be at the moment absolutely unfeasible. That is why most calculations are performed for a vector Q = 2ir/a (1,0,0). In this case Cr has a CsCl unit cell. The local magnetic moments at different atoms are equal in magnitude but opposite in direction. Such an approach is used, in particular, in papers [2, 3, 4], in which the electronic structure of Cr is calculated within the framework of spin density functional theory. Our paper [6] is devoted to the study of the influence of relativistic effects on the electronic structure of chromium. The results of calculations demonstrate that the relativistic effects completely change the structure of the Or electron spectrum, which leads to its anisotropy for the directions being identical in the non-relativistic approach. 

An alternative stream came from the valence bond (VB) theory. Ovchinnikov judged the ground-state spin for the alternant diradicals by half the difference between the number of starred and unstarred ir-sites, i.e., S = (n -n)l2 [72]. It is the simplest way to predict the spin preference of ground states just on the basis of the molecular graph theory, and in many cases its results are parallel to those obtained from the NBMO analysis and from the sophisticated MO or DFT (density functional theory) calculations. However, this simple VB rule cannot be applied to the non-alternate diradicals. The exact solutions of semi-empirical VB, Hubbard, and PPP models shed light on the nature of spin correlation [37, 73-77]. 

In the Hartree-Fock approach, the many-body wave function in form of a Slater determinant plays the key role in the theory. For instance, the Hartree-Fock equations are derived by minimization of the total energy expressed in terms of this determinantal wave function. In density functional theory (3,4), the fundamental role is taken over by an observable quantity, the electron density. An important theorem of density functional theory states that the correct ground state density, n(r), determines rigorously all electronic properties of the system, in particular its total energy. The totd energy of a system can be expressed as a functional of the density n (r) and this functional, E[n (r)], is minimized by the ground state density. 

Suzumura, T., Nakajima, T. and Hirao, K. (1999) Ground-state properties of MH, MCI, and M2 (M—Cu, Ag, and Au) calculated by a scalar relativistic density functional theory International Journal of Quantum Chemistry, 75, lVJ-1. ... 

Do We Know the Ground State Wave Function in Density Functional Theory ... 

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