With the availability of reliable full atomic cell orbitals the possibility arises to calculate pressure or stress, restoring or relaxation driving Hellmann -Feynman forces, and total energies (especially of defects) alternatively and directly via stress tensor fields and energy densities, two local quantities. Although quantum mechanical stress field and energy density can not be defined uniquely, there is a recent interest in these quantities, because integrals with physical meaning are gauge invariant.

The mentioned fields can be defined (i) for the full many-body description with the exact one-particle density matrix and pair distribution function as well as (ii) for the Kohn-Sham one-particle description with LDA or beyond (gradient expansion approximation). If the local stress field for a special system once is constructed, then the global stress tensor and /or forces on nuclei can be calculated via the stress theorem and the force theorem by means of unit cell surface integrals. The energy density can be derived from the terms of the stress field by taking the trace and can be used to calculate defect energies without bothering about the thermodynamic limit.