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2.1. THEORETICALDESCRIPTIONSOFMATTER 29 2.1.6 The frozen-core approximation Chemistry is in general associated with interactions between valence orbitals in atoms. Therefore, for many purposes, calculations can be made less computation- ally demanding if the electrons in the core regions of the atoms are frozen to the configurations assumed in the free atom. When this is done, the full system is simulated within a frozen-core approximation. This can be achieved in many dif- ferent ways, for example by using pseudopotentials of different types (e.g., norm- conserving[173] or ultrasoft[174]), or by using projector-augmented wave (PAW) setups. These approximations differ in the accuracy that can be achieved, with PAW setups in general achieving a higher accuracy than e.g., pseudopotentials[175]. However, what is common in all practical implementations of the frozen-core ap- proximation is that only the orbitals that are unimportant for the property under study should be kept frozen. The choice of which electrons to keep in the frozen core must be made based on careful benchmarking versus chemical and physical properties. The PAW description will now be explained in more detail, as it is used for the frozen core in GPAW. Although the PAW method can be extended beyond a frozen- core description[176], it is only used as such in GPAW. A key advantage with the PAW description is that smooth functions (called pseudo wave functions, ψ ̃n (r)) can be used in regions outside the atomic cores, while the orbitals in the atomic core are frozen and described by a pre-generated atomic setup, consisting of core orbitals φ a,core . However, in contrast to pseudopotential descriptions, the all- i electron wave function (through ψn (r)) for the system is still available, and can be reconstructed using a linear transformation ψn(r)=T ̃ψ ̃n(r) (2.27) where the transformation operator T ̃ is given as aaa T ̃ =1+ |φ ⟩−φ ̃ ⟨p ̃ | (2.28) ∑∑i i i ai where φia (r) are atom-centered all-electron wave functions, φ ̃ia (r) are the smooth partial wave functions corresponding to that all-electron wave function, and p ̃ai are projector functions characterizing the transformation from the all-electron to the smooth description. In this notation, n is a certain band index, a is a certain atom index and i is given by the n, l and m quantum numbers. The atomic core all-electron wave functions φia (r) are calculated for the isolated atom, using a certain radial cutoff distance rca (defining the atomic augmentation sphere). The valence states are described by smooth functions (partial waves, φ ̃a), that must i agree with the all-electron (wave) functions at all distances larger than rca. The projector functions are also smooth, and one is needed for each valence state partialPDF Image | Studies of Electrode Processes in Industrial Electrosynthesis
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