Local basis states

As has already been discussed above, the local basis states Rsu) play a key role in any local orbital scheme. In the RFPLO method they are calculated as the solutions of an atom-like single particle Dirac equation (33) in a spherical, orbital dependent potential (34) and XC-field (35). The atom-like potential U/ contains the spherically averaged crystal potential V/ around the lattice site s, which ensures that a Bloch sum of the core orbitals and the related core eigenvalues are very good approximations to the solutions of the true crystal Hamiltonian. For the calculation of valence states an additional attractive r -potential, acting on the large components only, is applied. 

In a non-magnetic calculation with = 0, the Hamiltonian Hgl commutes with the operators Jz, and K = /3(SL -I-1) and a suitable ansatz for the eigenstates is 

A substitution of this ansatz into the atom-like Hamiltonian (33) yields a set of four coupled differential equations (except for the case fj, = l + ), 

For the integration of the differential equations a number of numerical standard methods exist, such as Runge-Kutta or multi step methods [31]. In the 

RFPLO code, a predictor corrector method of Adams-Bashford-Moulton type is used. The eigenvalues are found by matching the inward and outward solutions. 

---

In a next step we insert the Bloch ansatz (32) into the Kohn-Sham-Dirac equation (28) and project it onto the local basis states giving rise to a... 

Fig. 2. The total density of states (DOS) for gold and the levels of the optimized local basis states obtained by RFPLO using LDA. All energies are relative to the Fermi energy.

To avoid unnecessarily large matrix problems for the diagonalization of the secular equation (37), the local basis states are divided as usual into two classes, core states and valence states. Core states are defined by the requirement that the overlap between different core states vanishes ... 

In order to have an exact representation of the net densities, the cut-off momentum for the L-expansion of the lattice sum (29) must be two times the maximum momentum I of the local basis states IRst ). The net densities can be further divided into core-core, core-valence and valence-valence contributions. Due to the non-orthogonal local basis scheme, there are also core-core contributions from valence bands, arising from the core-valence orthogonal-ization. 

We define scalar-relativistic local basis states R8pL)... 

The physical meaning of this apparently complex result can be understood by recalling the previous example of two identical oscillators. Let us start from the local basis states = l, 0), 0,1). In manifest... 

The matrix elements of the Hamiltonian operator are then obtained in the local basis (5.35). If we consider the local basis state, given by the array v(i,j), i=l,...,6 j = l,..., ndim, where... 

The localized basis function for the set 0 (Is) are usual frozen-core valence-shell Cl states all the bound states involved in the present calculations are also described at this level. 

With the localized basis set—that is, xe) = et) gm) where ee) ( gm)) is the electronically excited (ground) state of the Bchl Be (Bm)—the total electronic Hamiltonian for the model system is given by [56]... 

Fig. 2. The quantum mechanics of the two-state prpblem provide a paradigm for the much more extensive electronic state space of a real molecular or macromolecular system. The eigenvectors c, of the Hamiltonian are symmetric and antisymmetric linear combinations of the localized basis vectors with an eigenvalue splitting of 2A, where s is the overlap integral and A is the direct coupling (the only kind possible in this case)...

In the next two subsections, we describe collections of calculations that have been used to probe the physical accuracy of plane-wave DFT calculations. An important feature of plane-wave calculations is that they can be applied to bulk materials and other situations where the localized basis set approaches of molecular quantum chemistry are computationally impractical. To develop benchmarks for the performance of plane-wave methods for these properties, they must be compared with accurate experimental data. One of the reasons that benchmarking efforts for molecular quantum chemistry have been so successful is that very large collections of high-precision experimental data are available for small molecules. Data sets of similar size are not always available for the properties of interest in plane-wave DFT calculations, and this has limited the number of studies that have been performed with the aim of comparing predictions from plane-wave DFT with quantitative experimental information from a large number of materials. There are, of course, many hundreds of comparisons that have been made with individual experimental measurements. If you follow our advice and become familiar with the state-of-the-art literature in your particular area of interest, you will find examples of this kind. Below, we collect a number of examples where efforts have been made to compare the accuracy of plane-wave DFT calculations against systematic collections of experimental data. 

The next step is the choice of the basis set of these operators. The choice of the basis set depends on the ensemble. Kadanoff and Swift have considered a grand canonical ensemble with the temperature, chemical potential, and the velocity as the equilibrium parameters. The states that describe a situation where these equilibrium parameters vary slowly are the local equilibrium states. Linear combinations of the densities of the conserved operators acting on the equilibrium state ( ), where L ) =0) provide the desired local... 

In the case of delocalized basis states tpa(r), the main matrix elements are those with 0 = 7 and f3 = 6, because the wave functions of two different states with the same spin are orthogonal in real space and their contribution is small. It is also true for the systems with localized wave functions tpa(r), when the overlap between two different states is weak. In these cases it is enough to replace the interacting part by the Anderson-Hubbard Hamiltonian, describing only density-density interaction... 

---