DFT variational principle

One oft-overlooked facet of the Parr-Bartolotti paper is its mathematical treatment of constrained functional derivatives. The problem of constrained functional derivatives [2,12-16] arises repeatedly in DFT—often in the exactly the same number conserving context considered by Parr and Bartolotti—but their work is rarely cited in that context. Much of the recent work on the shape function is related to its importance for evaluating the constrained functional derivatives associated with the DFT variational principle [13-15]. 

Chattaraj, Cedillo, and Parr formulated the variational principle for the hardness. The function i) that minimizes the functional J/LgI = //g( 7(. g( )drdr subject to the condition that J i)dr= 1 is the Fukui function /(/), and the value of the functional at the minimum, i.e., j/[4, is the chemical hardness of the system. Recently, Ayers and Parr used the basic DFT variational principles in the development of variational principles for Important DFT-based descriptors of chemical reactivity, the Fukui function and the local softness. ... 

A key to the application of DFT in handling the interacting electron gas was given by Kohn and Sham [51] who used the variational principle implied by the minimal properties of the energy functional to derive effective singleparticle Schrodinger equations. The functional F[ ] can be split into four parts ... 

Conventional presentaticsis of DFT start with pure states but sooner w later encounter mixed states and d sities (ensemble densities is the usual formulation in the DFT literature) as well. These arise, for example in formation or breaking of chemical bonds and in treatments of so-called static correlation (situations in which several different one-electron configurations are nearly degenerate). Much of the DFT literature treats these problems by extension and generalization from pure state, closed shell system results. A more inclusively systematic treatment is preferable. Therefore, the first task is to obtain the Time-Dependent Variational Principle (TDVP) in a form which includes mixed states. 

In essentially all of the prior formulations of TDDFT a complex Lagrangian is used, which would amount to using the full expectation value in Eq. (2.9), not just the real part as in our presentation. The form we use is natural for conservative systems and, if not invoked explicitly at the outset, emerges in some fashion when considering such systems. A discussion of the different forms of Frenkel s variational principle, although not in the context of DFT, can be found in (39). 

Among various theories of electronic structure, density functional theory (DFT) [1,2] has been the most successful one. This is because of its richness of concepts and at the same time simplicity of its implementation. The new concept that the theory introduces is that the ground-state density of an electronic system contains all the information about the Hamiltonian and therefore all the properties of the system. Further, the theory introduces a variational principle in terms of the ground-state density that leads to an equation to determine this density. Consider the expectation value (H) of the Hamiltonian (atomic units are used)... 

We discussed mainly some of the possible applications of Fukui function and local softness in this chapter, and described some practical protocols one needs to follow when applying these parameters to a particular problem. We have avoided the deeper but related discussion about the theoretical development for DFT-based descriptors in recent years. Fukui function and chemical hardness can rigorously be defined through the fundamental variational principle of DFT [37,38]. In this section, we wish to briefly mention some related reactivity concepts, known as electrophilicity index (W), spin-philicity, and spin-donicity. 

Equation 24.17 shows that the energy gained by the system when a field E is applied is a function of the electronic density represented by p. According to the variational principle of DFT, the energy in the ground state (in the absence of a field) is minimum [1,2]. 

Density functional theory (DFT) uses the electron density p(r) as the basic source of information of an atomic or molecular system instead of the many-electron wave function T [1-7]. The theory is based on the Hohenberg-Kohn theorems, which establish the one-to-one correspondence between the ground state electron density of the system and the external potential v(r) (for an isolated system, this is the potential due to the nuclei) [6]. The electron density uniquely determines the number of electrons N of the system [6]. These theorems also provide a variational principle, stating that the exact ground state electron density minimizes the exact energy functional F[p(r)]. 

Note that the various T and V terms defined in Eqs. (8.3)—(8.5) are functions of the density, while the density itself is a function of three-dimensional spatial coordinates. A function whose argument is also a function is called a functional , and thus the T and V terms are density functionals . The Thomas-Fermi equations, together with an assumed variational principle, represented the first effort to define a density functional theory (DFT) the energy is computed with no reference to a wave function. However, while these equations are of significant historical interest, the underlying assumptions are sufficiently inaccurate that they find no use in modem chemistry (in Thomas-Fermi DFT, all molecules are unstable relative to dissociation into their constituent atoms...)... 

The local-scaling transformation version of density functional theory (LS-DFT), [1-12] is a constructive approach to DFT which, in contradistinction to the usual Hohenberg-Kohn-Sham version of this theory (HKS-DFT) [13-18], is not based on the IIohenberg-Kohn theorem [13]. Moreover, in the context of LS-DFT it is possible to generate explicit energy density functionals that satisfy the variational principle [8-12]. This is achieved through the use of local-scaling transformations. The latter are coordinate transformations that can be expressed as functions of the one-particle density [19]. 

Another model that describes the electronic structure of a system is provided by density functional theory (DFT). In DFT the electron density p of the system in the ground state plays the role of the many-electron wavefunction T in the wavefunction model because it uniquely defines all ground state properties of a system.An advantage of DFT is that T, which is a function of both spatial and spin coordinates of all electrons in the system, is replaced by a function that depends only on a position in Cartesian space p = p(r). The electron density can be obtained by using the variational principle... 

Density Functional Theory is a ground state theory in principle this is due to the fact that the variational principle is an essential element of DFT. Despite this, determination of excited state energies and properties is still possible in the framework of DFT. There are two main approaches to this problem. 

A totally different point of view is proposed by Time-Dependent Density Functional Theory [211-215] (TD-DFT). This important extension of DFT is based on the Runge-Gross theorem [216]. It extends the Hohenberg-Kohn theorem to time-dependent situations and states that there is a one to one map between the time-dependent external potential t>ea t(r, t) and the time-dependent charge density n(r, t) (provided we know the system wavefunction at t = —oo). Although it is linked to a stationary principle for the system action, its demonstration does not rely on any variational principle but on a step by step construction of the charge current. 

The ADFT/ASCF-DFT scheme has been met with considerable reservation. Thus, ADFT/ASCF-DFT assumes implicitly that a transition can be represented by an excitation involving only two orbitals, an assumption that seems not generally to be satisfied. Also, the variational optimization in ASCF-DFT of the orbitals makes it difficult to ensure orthogonality between different excited state determinants when many transitions are considered, resulting ultimately in a variational collapse. Finally, it has been questioned [110] whether there exists a variational principle for excited states in DFT. In spite of this, some of the first pioneering chemical applications of DFT involved ASCF-DFT calculations on excitation energies [36, 113-116] for transition metal complexes and ASCF-DFT is still widely used [117-121]. 

As mentioned, in order to be able to apply the variational principle in DFT, it is necessary to extend the definition of the functionals beyond the domain of v-representable densities, and the standard procedure is here to apply the Levy constrained-search procedure [17]. This has led to the functionals known as the Levy-Lieb (FL[p ) and Lieb (FL[p ) functionals, respectively, and we shall now investigate the differentiability of these functionals. This will represent the main part of our paper. 

There are many similarities among the various applications of DFT. One of the main characteristics of this parallelism is the existence of variational principles. Thus for electron densities, the electronic energy is a unique functional of the density for a given external potential. For a fluid of atoms or molecules, the intrinsic Helmholtz free-energy is a unique functional of the density for a given interatomic or intermolecular potential. For a nuclear system, the energy of the nuclei can also be regarded as a functional of the... 

In this section we will summarize the main concepts that lead to the formulation of the variational principle for this case. An excellent discussion and review of the several applications within this area has been given recently [3] and we refer to that article and the references therein for the reader desiring more details on this subject. Here we present an introduction to the basic ideas needed for the application of DFT to these systems, in order to establish the analogies and differences with other applications. 

The KS equations are obtained by utilizing the variation principle, which the second Hohenberg-Kohn theorem assures us applies to DFT. We use the fact that the electron density of the reference system, which is the same as that of our real system (see the definition at the beginning of the discussion of the KS energy), is given by [7]... 

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