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# Theories and Computational Tools

The theoretical approaches used and developed by the ETSF are based on "quantum mechanics". Quantum mechanics is the theory that describes the behaviour of systems at atomic length scales. In quantum mechanics the key quantity is the "wave function", Ψ(r1,...,rn, t), for a system containing n electrons. The wave function fully describes the physical state of the systems, and gives access to all its physical properties. The wave function can be calculated by solving the "Schrödinger equation", $H \psi = E \psi$

Except for systems containing a small number of electrons, the Schrödinger equation cannot be solved, neither analytically nor numerically. The problem is due to the electron-electron many-body interaction term. If this term were not present, the Hamiltonian could be factorized into n separated single-electron Hamiltonians. One can solve the easier single-electron Schrödinger equation, after which the many-body wave function can be calculated as the antisymmetrized product of n single-electron wavefunctions.

Therefore it is necessary to develop approaches and techniques that simplify the original problem. The knowledge of the full wave function, i.e. complete knowledge of the complete dynamics of each given electron, involves an overwhelming amount of information, which, like in statistical mechanics, is redundant for determining quantities which are of real observer interest. The solution of the problem can be sought by defining new reduced key quantities which contain the essential information needed to provide observables.

The ETSF employs a wide range of theoretical and computational methods to study electrons in nanostructures and materials and their interaction with external fields and light, the principal ones being:

### Density Functional Theory (DFT) • Kohn-Sham DFT for ground-state total energy calculations, structure determination, potential energy surfaces for atomic motion.
• Time-dependent DFT for study of systems excited out of the ground state, e.g. optical absorption.

### Many-Body Perturbation Theory (MBPT)

• GW and GWΓ self-energy approaches for electron addition and removal energies, spectral functions, total energy.
• Bethe-Salpeter approach for neutral electronic excitations.
• Non-equilibrium Green's function theories for Quantum Transport.

### Combined DFT-MBPT approaches

• Generalised Kohn-Sham DFT for total energy calculations, incorporating elements of DFT and MBPT
• GWΓ self-energy, incorporating Density Functional concepts.
• TDDFT approaches for quantum transport.
• TDDFT approaches for total energy calculations, via the fluctuation-dissipation theorem