## Chapter 9. From Atoms to Solids: The Pseudopotential Method.

Development note Development of this chapter has been postponed.

## 9.1. The basic idea behind the pseudopotential method.

The inner core of electrons is tightly bound to the nucleus and have a relatively high kinetic energy. Indeed, for atoms with high atomic number relativistic effects can be substantial. Tightly bound electrons with large velocities translates into wavefunctions with a high degree of curvature and a broad spectrum in momentum space. To represent these wavefunctions would require a large number of basis functions (especially in the case of plane-waves), but at the same time these states have only an indirect role in chemical processes and are insensitive to the chemical environment. The valence state wavefunctions also have a high degree of curvature in the core region, which manifests itself as oscillatory behavior. Unlike the core states, however, the amplitude of the valence wavefunctions decrease rapidly near the core. A key assumption of the pseudopotential method is that the core states in a chemical environment are the same as in a free atom. The main objective is to replace the need for calculating the core electron states with a pseudopotential that reproduces the effects of the core electrons.

### 9.1.1. Key terms and points to keep in mind in implementing the pseudopotential method.

Before we start a general discussion we would like to mention a few key terms and points to keep in mind, which we hope will become clearer as one progresses. These are:

• Transferability: the idea that a pseudopotential should reproduce all-electron results in a variety of chemical environments.

• Norm-conserving [132]: the notion that outside the core region the pseudo valence wavefunctions and the all-electron wavefunctions should be identical; and the integrated charge within the "core" region calculated with the pseudo wavefunctions should equal the real charge within that region. An idea introduced by Topp and Hopfield [131] to improve transferability.

• Consistency: pseudopotentials constructed in the framework of the local-density approximation (LDA) should not be used in calculations using the generalized gradient approximation (GGA) and vice versa.

• Local, semilocal and nonlocal potentials: A local potential is only a function of the distance from the nucleus, r. Both the nonlocal and the semilocal pseudopotentials are angular momentum dependent, but a semilocal pseudopotential is only nonlocal in angular coordinates, not in the radial coordinate [134].

In most formulations both the pseudopotential and the pseudo wavefunctions are equal to their all-electron counterparts beyond some cut-off radius  ${r}_{c}$. At radial distances less the cut-off radius  ${r}_{c}$ , the oscillatory behavior of the all-electron (AE) valence wavefunction is replaced by a smoothly varying function. Thus,  ${r}_{c}$  must be large enough that all the nodes found in the AE valence wavefunction lie within this radius. At the same time, however, with a smaller  ${r}_{c}$  it is easier to replicate the behavior of the AE valence wavefunction and therefore improve transferability.