Why Pseudopotentials ?

• Pseudopotentials should be additive and transferable. Additivity can most easily be achieved by building pseudopotentials for atoms in reference states. Transferability means that one and the same pseudopotential should be adequate for an atom in all possible chemical environments. This is especially important when a change of the environment is expected during a simulation, like in chemical reactions or for phase transitions.
• Pseudopotentials replace electronic degrees of freedom in the Hamiltonian by an effective potential. They lead to a reduction of the number of electrons in the system and thereby allow for faster calculation or the treatment of bigger systems.
• Pseudopotentials allow for a considerable reduction of the basis set size. Valence states are smoother than core states and need therefore less basis functions for an accurate description. The pseudized valence wavefunctions are nodeless (in the here considered type of pseudopotentials) functions and allow for an additional reduction of the basis. This is especially important for plane waves.
  Consider the 1s function of an atom
  $$\varphi_{\rm 1s}({\bf r}) \sim e^{-Z^\star r}$$
  with $Z^\star \approx Z$ , the nuclear charge. The Fourier transform of the orbital is
  $$\varphi_{\rm 1s}({\bf G}) \sim 16 \pi { Z^{5/2} \over {\bf G}^2 + Z^2 } \enspace .$$
  From this formula we can estimate the relative cutoffs needed for different elements in the periodic table (see table 7.12).
• Most relativistic effects are connected to core electrons. These effects can be incorporated in the pseudopotentials without complicating the calculations of the final system.

Table 3: Relative cutoffs (in energy units) and number of plane waves for several atoms.

Atom	Z	Cutoff	Plane Waves
H	1	1	1
Li	3	4	8
C	6	9	27
Si	14	27	140
Ge	32	76	663
Sn	50	133	1534