Hohenberg-Kohn theorem, electronic kinetic energy

In this work, the electronic kinetic energy is expressed in terms of the potential energy and derivatives of the potential energy with respect to nuclear coordinates, by use of the virial theorem (5-5). Thus, the results are valid for all bound electronic states. However, the functional derived for E does not obey a variational principle with respect to (pg(r)), even though x( ,r co)is in principle a functional of (p (r)), as implied by the Hohenberg-Kohn theorem 9-121... 

The relativistic correction for the kinetic energy in the Dirac equation is naturally applicable to the Kohn-Sham equation. This relativistic Kohn-Sham equation is called the Dirac-KohnSham equation (Rajagopal 1978 MacDonald and Vosko 1979). The Dirac-Kohn-Sham equation is founded on the Rajagopal-Callaway theorem, which is the relativistic expansion of the Hohenberg-Kohn theorem on the basis of QED (Rajagopal and Callaway 1973). In this theorem, two theorems are contained The first theorem proves that the four-component external potential, which is the vector-potential-extended external potential, is determined by the four-component current density, which is the current-density-extended electron density. On the other hand, the second theorem establishes the variational principle for every four-component current density. See Sect. 6.5 for vector potential and current density. Consequently, the solution of the Dirac-Kohn-Sham equation is represented by the four-component orbital. This four-component orbital is often called a molecular spinor. However, this name includes no indication of orbital, which is the solution of one-electron SCF equations moreover, the targets of the calculations are not restricted to molecules. Therefore, in this book, this four-component orbital is called an orbital spinor. The Dirac-Kohn-Sham wavefunction is represented by the Slater determinant of orbital spinors (see Sect. 2.3). Following the Roothaan method (see Sect. 2.5), orbital spinors are represented by a linear combination of the four-component basis spinor functions, Xp, ... 

The Hohenberg-Kohn theorems find a very important application in the derivation of the Kohn-Sham equations, in which the problem of approximating the noninteracting kinetic energy (Ts) is eliminated by introducing single-particle orbitals 9,. The exact electron density is written as the electron density of a Slater determinant,... 

Here T is the kinetic energy functional, U is the electron-electron coulombic interaction and Vext contains the electron-nuclear and (for a molecule) nuclear-nuclear potentials. Equation (19) is a statement of the Hohenberg-Kohn theorem. The problem is that we don t know what the T and U functionals are. However, we know some aspects of them. For example, we can separate out the classical Hartree component of U and write... 

The principal limitation of the earlier used TF and TFD models is the treatment of the kinetic energy which was handled in an approximate way, thereby giving less accurate values for atomic and molecular systems. This was solved by Hohenberg and Kohn in 1964 [19] when they used a variational procedure in the formulation of the modern form of density functional theory, DFT. This theory is based on two theorems which state that the total energy of an electronic system is a unique functional of the electronic density and that the ground state density determines the ground state energy ... 

The proof of this theorem is relatively simple but will not be reproduced here (the interested reader may consult ref. 9). It is based on the variational principle and uses that from the integral of the electron density one knows the total number of electrons and accordingly the kinetic-energy and the electron-electron-interaction parts of the total Hamilton operator. Only the external potential (which above was only the Coulomb potential of the nuclei, but which may contain other parts, too) is unspecified, but assuming that this can be written as a sum of identical single-particle terms, Hohenberg and Kohn proved that also this is uniquely determined within an additive constant. 

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