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Trace Minimization with a Nonlinear Term

We consider a nonlinear eigenvalue problem related
to the trace minimization of
§9.4.3.3:
minimize
where is some nonlinear term.

This sort of minimization occurs in electronic structures computations
such as local density approximations (LDAs), where the columns of
represent electron wave functions, is a fixed Hamiltonian, and
represents the energy of the electron-electron interactions
(see, for example, [151,428]).
In this case, the optimal represents the state of lowest energy of the
system.

The only additions to the differentials of the trace minimization are
terms from which vary from problem to problem. For our
example, we have taken
for some
coupling constant , where
(the charge density).

In this case

and

where
and

Susan Blackford
2000-11-20