Time-dependent Schrodinger equation numerical treatment

An accurate numerical description of molecular vibrations in the field of physical chemistry often requires explicit solutions of the time-dependent or time-independent Schrodinger equation. A full quantum mechanical treatment of all involved particles, i.e., all electrons and nuclei, however, is only possible for very small and rather simple systems such as H. For larger systems one must rely an approximations, because the demands on CPU time and memory of a numerically exact treatment quickly exceed today s numerical capacities. 

Time-dependent density-functional theory (TDDFT) extends the basic ideas of ground-state density-functional theory (DFT) to the treatment of excitations and of more general time-dependent phenomena. TDDFT can be viewed as an alternative formulation of time-dependent quantum mechanics but, in contrast to the normal approach that relies on wave-functions and on the many-body Schrodinger equation, its basic variable is the one-body electron density, n(r,t). The advantages are clear The many-body wave-function, a function in a 3A-dimensional space (where N is the number of electrons in the system), is a very complex mathematical object, while the density is a simple function that depends solely on the 3-dimensional vector r. The standard way to obtain n r,t) is with the help of a fictitious system of noninteracting electrons, the Kohn-Sham system. The final equations are simple to tackle numerically, and are routinely solved for systems with a large number of atoms. These electrons feel an effective potential, the time-dependent Kohn-Sham potential. The exact form of this potential is unknown, and has therefore to be approximated. 

---