Brillouin Zone sampling

Use of different surface models (2D- and 3D-slabs) and basis sets in HF (LCAO) and DFT (PW) calculations makes it difficult to understand the origin of differences in calculated results that may be due both to the model apphed and to the features of the calculation scheme chosen (basis set, integral approximations, Brillouin-zone sampling, and so on). 

There are thus two convergence parameters that need to be fine-tuned for every calculation the Brillouin zone sampling and a cutoff radius in reciprocal space to truncate the sums over reciprocal lattice vectors (we cannot perform infinite summations )... 

A clear and exhaustive description of the method is in Dovesi, R. Civalleri, B. Orlando, R. Roetti, C. Saunders, V. R. Ab initio quantum simulations in solid state chemistry. Revs. Comp. Chem. 2005, 21, 1-125. This review contains a discussion of all aspects of periodic calculations, including differences between conducting and non-conducting systems, Brillouin zone sampling, basis set dependence, etc., and a survey of available computer packages. For the CRYSTAL software package for periodic MO calculations in crystals, developed by the authors, see www.crystal.unito.it. 

Once the phonon frequencies are known it becomes possible to determine various thermodynamic quantities using statistical mechanics (see Appendix 6.1). Here again some slight modifications are required to the standard formulae. These modifications are usually a consequence of the need to sum over the points sampled in the Brillouin zone. For example, the zero-point energy is ... 

In Equation (5.58) the outer summation is over the p points q which are used to sample the Brillouin zone, is the fractional weight associated with each point (related to the volume of Brillouin zone space surrounding q) and vi are the phonon frequencies. In addition to the internal energy due to the vibrational modes it is also possible to calculate the vibrational entropy, and hence the free energy. The Helmholtz free energy at a temperature... 

Equation (2-25) also underscores the fact that the two-electron integrals decay as A--1 with respect to K (the number of wave vector sampling points in the first Brillouin zone). It can also be shown easily that Fock matrix elements display K° dependence. The dependence of the CIS amplitudes is not immediately clear until we consider the normalization condition ... 

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