Trivia:
Lets have arbitrary function $\rho$ and set of $N$ orthonormal basis-functions $\{\chi_i\}$
We can approximated $\rho$ (i.e. minimize mean square error) by
$\tilde \rho = \sum_i \tilde \rho_i = \sum_i \lambda_i \psi_i^2 $
$\psi_i = \sum_j \mathrm{U}_{ij} \chi_j$
where $\lambda_i$ are eigenvalues and $\mathrm{U}\equiv\{u_i\}$ matrix of eigenvectors to density matrix
$D_{ij} = \int_i{\rho} \chi_i \chi_j $
The problem:
Assume $\rho$ is sum of $m>N$ spatially localized components (aka peaks) $\{\rho_k\}$ ( $\rho = \sum_k \rho_k$ ). I know $\rho_k$ exactly, but I want to compress them using small number of $\psi_i$.
I want to modify procedure above so that not only $\sum_k \rho_k \approx \sum_i \lambda_i \psi_i^2 $ but also $\rho_k \approx \lambda_i \psi_i^2 $ for first few dominant components $\rho_k$.
In other words I want $\psi_i^2$ to be more localized each on single peak.
Illustration
Consider for simplicity 1D example $\rho \in R^1$ with 3 basisfunctions ($N$=3) e.g.
$\{\chi_0,\chi_1,\chi_2\} = \{ 1, \sin(x),\cos(x)\}$
and
$\rho_k = a_k (cos(x-x_k)-1)^2$
where $a_k$ is random amplitude and $x_k$ random shift
decomposition of $\rho$ to $\rho_k$ looks like this. Main peaks can be attributed to individual dominant components $\rho_k$.
$\rho = \sum_k \rho_k$ ">
Approximation of $\rho$ by $\lambda_i \psi_i^2$ looks like this. you can see that $\psi_i^2$ are very delocalized, the first $\psi_0$ tries to capture all peaks at once. I want something where each $\psi_i^2$ approximately corresponds to single peak. Resp $\rho_0 \approx \lambda_0 \psi_0^2$ and $\rho_1 \approx \lambda_1 \psi_1^2$
$\tilde \rho = \sum_i \tilde \rho_i = \sum_i \lambda_i \psi_i^2 $">
Background and motivation
It comes from computational chemistry. I'm trying to find somehow optimal representation of interaction between an atom $a$ ant $m>4$ other atoms $b$ expressed in $\{s,p_x,p_y,p_y\}$ basiset localized strictly on the atom $a$.