**Quadratic forms in the calculation of time dependent electronic structures
**

**Abstract:**

The calculation
of the time dependent (TD) electronic structure requires the study of the
wave function Y(t). We use an approach Keller and Weinberger 2007)
which considers the electronic density as a quadratic form r = å_{i}r_{i}=|Y|^{2} with the corresponding linear form Y = å_{i}y_{i} . In the study of N-identical-fermion systems we have
the additional feature that Y is a function of the 3N configuration
space coordinates and r is defined in 3-D real space. For
many-electron systems the wave function and the Hamiltonian are to be
expressed in terms of the configuration space (CS), a replica of real space
for each electron. Here we present the time dependent geometric formulation
of the CS, of the wave function, of the density, and of the Hamiltonian to
compute the electronic structure of the system. Then, using the new
geometric notation and the indistinguishability and equivalence of the
electrons, we obtain an alternative computational method for the TD state of
the system. For the general TD formulation the additional energy-momentum
eV(x,t)+(e/c)A(x,t) will have a space (and time) symmetry different from the
starting potential. The original states, classified according to the
original potentials symmetry, will evolve into combinations induced by the
total potential new symmetry. The equation of motion for Y(t) is given.

J. Keller and P. Weinberger The use of quadratic forms in the calculation of ground state electronic structures, Journal of Mathematical Physics, 47 (8) 2006.

Joint work with *Peter Weinberger*.