Trace Minimization with a Nonlinear Term

We consider a nonlinear eigenvalue problem related to the trace minimization of §9.4.3.3: minimize $F(Y) = \frac{1}{2}(\tr(Y^*AY) + g(Y)),$ where $g(Y)$ is some nonlinear term.

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

The only additions to the differentials of the trace minimization are terms from $g(Y)$ which vary from problem to problem. For our example, we have taken $g(Y) = \frac{K}{4} \sum_i \rho_i^2$ for some coupling constant $K$, where $\rho_i = \sum_j \vert Y_{ij}\vert^2$ (the charge density).

In this case

$$dF(Y) = dF_{\mbox{tracemin}} + K \diag(\rho) Y,$$

and

$$\frac{d}{dt} dF(Y(t))\vert _{t=0} = \frac{d}{dt} dF_{\mbox{t... ...} + K \diag(\rho) H + K \diag\left(\frac{d}{dt} \rho\right) Y,$$

where $\dot{Y}(0) = H$ and $\frac{d}{dt} \rho_i \vert _{t=0} = \sum_j Y_{ij} H_{ij}.$