Kohn-Sham eigenfunctions

It is a basic consequence of the translational symmetry of a solid that its Kohn-Sham eigenfunctions can be uniquely labeled by four quantum numbers, the band index n and a wavevector k, as in xj/ y.. The diagram s n,k) that represents the n, k dependence of the corresponding eigenenergies is called the band structure. The Bloch theorem asserts that the t>e written in the form of a Fourier... 

Finally, the independent particle susceptibility xo( < ) is defined (see [5]) in terms of the one-particle Kohn-Sham eigenfunctions /(r) and the retarded Green s function G of the Kohn-Sham Hamiltonian,... 

Just as in the unrestricted Hartree-Fock variant, the Slater determinant constructed from the KS orbitals originating from a spin unrestricted exchange-correlation functional is not a spin eigenfunction. Frequently, the resulting (S2) expectation value is used as a probe for the quality of the UKS scheme, similar to what is usually done within UHF. However, we must be careful not to overstress the apparent parallelism between unrestricted Kohn-Sham and Hartree-Fock in the latter, the Slater determinant is in fact the approximate wave function used. The stronger its spin contamination, the more questionable it certainly gets. In... 

How do CMOs and LMOs differ The CMOs are symmetry-adapted eigenfunctions of the Fock (or Kohn-Sham) operator F, necessarily reflecting all the molecular point-group symmetries of F itself,26 whereas the LMOs often lack... 

In a molecular-orbital-type (Hartree-Fock or Kohn-Sham density-functional) treatment of a three-dimensional atomic system, the field-free eigenfunctions ir e can be rigorously separated into radial (r) and angular (9) components, governed by respective quantum numbers n and l. In accordance with Sturm-Liouville theory, each increase of n (for... 

Note that because the E of Eq. (8.14) that we are minimizing is exact, the orbitals / must provide die exact density (i.e., the minimum must correspond to reality). Further note that it is these orbitals that form the Slatcr-dctcrminantal eigenfunction for the separable noninteracting Hamiltonian defined as the sum of the Kohn-Sham operators in Eq. (8.18), i.e.. 

Next, we shall assume that v(r) is the Kohn-Sham potential for the density n(r). In other words, v(r)=v0([n] r) and H0QC=H0[n], As a result, the eigenfunctions and eigenvalues in Eqs.(19) and (28) are the same. By using perturbation theory for small enough a, with Z=A, Ivanov and Levy [26] have developed an expansion for the HF density, i.e. 

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. 

In order to tackle large and complex structures, new methods have recently been developed for solving the eleetronie part of the problem. These are mostly applied to the pseudopotential plane wave method, because of the simplicity of the Hamiltonian matrix elements with plane wave basis functions and the ease with which the Hellmann-Feynman forces can he found. Conventional methods of matrix diagonalization for finding the energy eigenvalues and eigenfunctions of the Kohn-Sham Hamiltonian in (9) can tackle matrices only up to about 1000 x 1000. As a basis set of about 100 plane waves per atom is needed, this restricts the size of problem to... 

Since these orbitals are linear combinations of degenerate orbitals, they are again eigenfunctions of the Kohn-Sham single-particle Hamiltonian for the highest... 

As was the case when solving the Hartree-Fock equations, also the eigenfunctions to the Kohn-Sham equations are expanded in some basis set as in Eq. (13). And since the single-particle operator heff depends on the density, which in turn depends on the orbitals, in this case also the single-particle equations have to be solved self-consistently. 

The overall procedure to achieve self-consistency is very reminiscent of that used in Hartree-Fock theory, involving first an initial guess of the density by superimposing atomic densities, construction of the Kohn-Sham and overlap matrices, and diagonalisation to give the eigenfunctions and eigenvectors from which the Kohn-Sham orbitals can be... 

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