BO approximation波近似.pptVIP

Fundamentals of DFTR. WentzcovitchU of MinnesotaVLab Tutorial Hohemberg-Kohn and Kohn-Sham theorems Self-consistency cycle Extensions of DFT BO approximation Pseudopotentials BO approximation Proof of theorem I Assume Vext(1)(r) and Vext(2)(r) differ by more than a constant and produce the same n(r). Vext(1)(r) and Vext(2)(r) produce H(1) and H(2) , which have different ground state wavefunctions, Ψ(1) and Ψ(2) which are hypothesized to have the same charge density n(r). It follows that Then and Adding both which is an absurd! Proof of theorem II Each Vext(r) has its Ψ(R) and n(r). Therefore the energy Eel(r) can be viewed as a functional of the density. Consider and a different n(2)(r) corresponding to a different It follows that The Kohn-Sham Ansatz Meaning of the eigenvalues and eigenfunctions: Eigenvalues and eigenfunctions have only mathematical meaning in the KS approach. However, they are useful quantities and often have good correspondence to experimental excitation energies and real charge densities. There is, however, one important formal identity These eigenvalues and eigenfunctions are used for more accurate calculations of total energies and excitation energy. The Hohemberg-Kohn-Sham functional concerns only ground state properties. The Kohn-Sham equations must be solved self-consistently Self consistency cycle Extensions of the HKS functional Spin density functional theory The HK theorem can be generalized to several types of particles. The most important example is given by spin polarized systems. Finite T and ensemble density functional theory The HK theorem has been generalized to finite temperatures. This is the Mermin functional. This is an even stronger generalization