Energy Kohn-Sham theory, physical

H. A. Physical interpretation of Kohn-Sham theory exchange-correlation energy functional and its derivative... 

H. A. PHYSICAL INTERPRETATION OF KOHN-SHAM THEORY EXCHANGE-CORRELATION ENERGY FUNCTIONAL AND ITS DERIVATIVE... 

There is, of course, much that remains to be understood with regard to the physical interpretation. For example, the correlation-kinetic-energy field Z, (r) and potential W, (r) need to be investigated further. However, since accurate wavefunctions and the Kohn-Sham theory orbitals derived from the resulting density now exist for light atoms [40] and molecules [54], it is possible to determine, as for the Helium atom, the structure of the fields P(r), < P(r), and Zt (r), and the potentials WjP(r), W (r), W (r), and W (r) derived from them, respectively. A study of these results should lead to insights into the correlation and correlation-kinetic-energy components, and to the numerical determination of the asymptotic power-law structure of these fields and potentials. The analytical determination of the asymptotic structure of either [Z, (r), W, (r)] or [if (r), WP(r)] would then lead to the structure of the other. 

Since the electron-interaction energy functional E [p] of Kohn-Sham theory is representative of Pauli and Coulomb correlations as well as the correlation contribution to the kinetic energy, so is the corresponding local potential v (r) obtained from it through functional differentiation. In the physical interpretation of the potential v (r), however, it is possible to distinguish between the purely quantum-mechanical (Pauli and Coulomb) electron-correlation component Wee(r), and the correlation-kinetic-energy component W, (r). We begin this section with a description of the physical interpretation of Vee (r), and then discuss its components Wee(r) and W, (r) more fully. 

Are there any remedies in sight within approximate Kohn-Sham density functional theory to get correct energies connected with physically reasonable densities, i. e., without having to use wrong, that is symmetry broken, densities In many cases the answer is indeed yes. But before we consider the answer further, we should point out that the question only needs to be asked in the context of the approximate functionals for degenerate states and related problems outlined above, an exact density functional in principle also exists. The real-life solution is to employ the non-interacting ensemble-Vs representable densities p intro-... 

By the way, through ensemble theory with unequal weights, Ref. [68] identifies an effective potential derivative discontinuity that links physical excitation energies to excited Kohn-Sham orbital energies from a ground-state calculation.)... 

The Kohn-Sham operator / KS is defined by Eq. 7.23 the significance of these orbitals and energy levels is considered later, but we note here that in practice they can be interpreted in a similar way to the corresponding wavefunction entities. Pure DFT theory has no orbitals or wavefunctions these were introduced by Kohn and Sham only as a way to turn Eq. 7.11 into a useful computational tool, via the artifice of noninteracting electrons, but if we can interpret the KS orbitals and energies in some physically useful way, so much the better. 

A good first approach to a quantum mechanical system is often to consider one-electron functions only, associating one such function, a spin-orbital , with one electron. Most popular are the one-electron functions which minimize the energy in the sense of Hartree-Fock theory. Alternatively one can start from a post-HF wave function and consider the strongly occupied natural spin orbitals (i.e. the eigenfunctions of the one-particle density matrix with occupation numbers close to 1) as the best one-electron functions. Another possibility is to use the Kohn-Sham orbitals, although their physical meaning is not so clear. 

Car and Parrinello [97,98] proposed a scheme to combine density functional theory [99] with molecular dynamics in a paper that has stimulated a field of research and provided a means to explore a wide range of physical applications. In this scheme, the energy functional [ (/, , / , ] of the Kohn-Sham orbitals, (/(, nuclear positions, Ri, and external parameters such as volume or strain, is minimized, subject to the ortho-normalization constraint on the orbitals, to determine the Born-Oppenheimer potential energy surface. The Lagrangian,... 

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