Density Variational Principle

We seek a density functional analog of (1.30). Instead of the original derivation of Hohenberg, Kohn and Sham [25,6], which was based upon reductio ad absurdum , we follow the constrained search approach of Levy [28], which is in some respects simpler and more constructive. 

Equation (1.30) tells us that the ground state energy can be found by minimizing ( F. H F) over all normalized, antisymmetric W-particle wavefunctions  

We now separate the minimization of (1.53) into two steps. First we consider all wavefunctions which yield a given density n(r), and minimize over those wavefunctions  

The constraint of fixed N can be handled formally through introduction of a Lagrange multiplier fi  

H is to be adjusted imtil (1.5) is satisfied. Equation (1.58) shows that the external potential w(r) is miiquely determined by the groimd state density (or by any one of them, if the groimd state is degenerate). 

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Part I of the paper develops an exact variational principle for the ground-state energy, in which the density (r) is the variable function (i.e. the one allowed to vary). The authors introduce a universal functional F[n(r)] which applies to all electronic systems in their ground states no matter what the external potential is. This functional is used to develop a variational principle. 

Assume now that two different external potentials (which may be from nuclei), Vext and Vgjjj, result in the same electron density, p. Two different potentials imply that the two Hamilton operators are different, H and H, and the corresponding lowest energy wave functions are different, and Taking as an approximate wave function for H and using the variational principle yields... 

Frieden s theory is that any physical measurement induces a transformation of Fisher information J I connecting the phenomenon being measured to intrinsic data. What we call physics - i.e. our objective description of phenomenologically observed behavior - thus derives from the Extreme Physical Information (EPI) principle, which is a variational principle. EPI asserts that, if we define K = I — J as the net physical information, K is an extremum. If one accepts this EPI principle as the foundation, the status of a Lagrangian is immediately elevated from that of a largely ad-hoc construction that yields a desired differential equation to a measure of physical information density that has a definite prior significance. 

Time-Dependent Variational Principle in Density Functional Theory... 

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. 

We shall illustrate the applicability of the GvdW(S) functional above by considering the case of gas-liquid surface tension for the Lennard-Jones fluid. This will also introduce the variational principle by which equilibrium properties are most efficiently found in a density functional theory. Suppose we assume the profile to be of step function shape, i.e., changing abruptly from liquid to gas density at a plane. In this case the binding energy integrals in Ey can be done analytically and we get for the surface tension [9]... 

Stated in still other words this means that for any trial density p(r) - which satisfies the necessary boundary conditions such as p( ) - 0, J p( ) dr = N, and which is associated with some external potential Vext - the energy obtained from the functional given in equation (4-6) represents an upper bound to the true ground state energy E0. E0 results if and only if the exact ground state density is inserted into equation (4-8). The proof of the inequality (4-11) is simple since it makes use of the variational principle established for wave functions as detailed in Chapter 1. We recall that any trial density p(r) defines its own Hamiltonian H and hence its own wave function. This wave function can now be taken as the trial wave function for the Hamiltonian generated from the true external potential Vext. Thus, we arrive at... 

In order to connect this variational principle to density functional theory we perform the search defined in equation (4-13) in two separate steps first, we search over the subset of all the infinitely many antisymmetric wave functions Px that upon quadrature yield a particular density px (under the constraint that the density integrates to the correct number of electrons). The result of this search is the wave function vFxin that yields the lowest... 

Applying the variational principle to the energy given by Eq. 1, Kohn and Sham reformulated the density functional theory by deriving a set of one-electron Hartree-like equations leading to the Kohn-Sham orbitals v().(r) involved in the calculation of p(r)15. The Kohn-Sham (KS) equations are written as follows ... 

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)... 

This functional satisfy a variational principle [3,5] EVo o[p0, m0 < EVo>Bo[p, m. EVoJ3o [p0, hi0] denote the ground state energy with density p0(r), and magnetization m0(r) of a particular system characterized by the external fields (v0(r), Bt>(r)). One of the main differences between the spin-restricted and spin-polarized cases is that the one-to-one relation between the external potential and the density cannot be extrapolated to the set of quantities (v0(r), B0(r)) and (p0(r), m0(r)) [3]. 

The utility of the Fukui function for predicting chemical reactivity can also be described using the variational principle for the Fukui function [61,62], The Fukui function from the above discussion, /v (r), represents the best way to add an infinitesimal fraction of an electron to a system in the sense that the electron density pv/v(r) I has lower energy than any other N I -electron density... 

Notice that how the shape function naturally enters this discussion. Because the number of electrons is fixed, the variational procedure for the electron density is actually a variational procedure for the shape function. So it is simpler to restate the equations associated with the variational principle in terms of the shape function. Parr and Bartolotti have done this, and note that because the normalization of the shape function is fixed,... 

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)]. 

But there is a more basic difficulty in the Hohenberg-Kohn formulation [19-21], which has to do with the fact that the functional iV-representability condition on the energy is not properly incorporated. This condition arises when the many-body problem is presented in terms of the reduced second-order density matrix in that case it takes the form of the JV-representability problem for the reduced 2-matrix [19, 22-24] (a problem that has not yet been solved). When this condition is not met, an energy functional is not in one-to-one correspondence with either the Schrodinger equation or its equivalent variational principle therefore, it can lead to energy values lower than the experimental ones. 

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