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I am interested in representing an arbitrary charge density (say, of atoms in a molecule) $\rho(r), \; r\in \mathbb{R}^3$ by a finite linear combination of basis functions

$\rho(r) = \sum_{i=1}^N q_i \phi_i (r)$

where $\phi_i (r)$ is normalized to $\int_{\mathbb{R}^3} \phi_i (r) dr = 1$ and has the interpretation of being the shape of some charge distribution (shape) of a unit charge. $\rho$ and the $\phi_i$s are real-valued but may be positive in some regions and negative in others. The basis functions are nonorthogonal and local in space but not strictly compact. Let's say for now that we use spherical Gaussians of the form $\phi_i (r) \propto \exp (-\alpha_i |r-R_i|^2)$, where $R_i$ is where the basis function is centered around. The number of basis functions chosen scales approximately as the number of atoms, as we expect charge to concentrate around atomic nuclei. (We may add additional basis function per atom of different shapes until we achieve a reasonable approximation to the desired shape of the charge distribution around an atom.)

The energy of the system can then be given by

$E = \frac 1 2 \sum_{i,j=1}^N q_i q_j J_{ij}$

where the matrix $J$ has elements

$J_{ij} = \int_{\mathbb{R}^{3\times 2}} \frac{\phi_i(r_1) \phi_j(r_2)}{|r_1 - r_2|} dr_1 dr_2$

and represents the Coulomb interaction between the unit charge distributions $\phi_i$ and $\phi_j$.

One way to look at the matrix $J$ is as a finite dimensional (approximate) representation of the Coulomb operator $\hat J = 1 / {|r_1 -r_2|}$. We know that $\hat J$ has certain nice properties such as positivity, so we expect a "good" representation of $\hat J$ should be a symmetric positive definite matrix.

My question is this: are there conditions on the discrete representation (possibly expressible as conditions on the {$\phi_i$} basis) to detect whether or not a given claimed representation $J$ is "good" in that it preserves such properties? Or asked another way, if I have some matrix $J$ which is claimed to represent $\hat J$, what are necessary and sufficient conditions on its matrix elements for it to be a "good" representation of $\hat J$?

I hope the question makes sense, and that I am not misusing too much terminology.

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# When is a finite matrix a "good" approximate representation of an operator?

I am interested in representing an arbitrary charge density (say, of atoms in a molecule) $\rho(r), \; r\in \mathbb{R}^3$ by a finite linear combination of basis functions

$\rho(r) = \sum_{i=1}^N q_i \phi_i (r)$

where $\phi_i (r)$ is normalized to $\int_{\mathbb{R}^3} \phi_i (r) dr = 1$ and has the interpretation of being the shape of some charge distribution (shape) of a unit charge. $\rho$ and the $\phi_i$s are real-valued but may be positive in some regions and negative in others.

The energy of the system can then be given by

$E = \frac 1 2 \sum_{i,j=1}^N q_i q_j J_{ij}$

where the matrix $J$ has elements

$J_{ij} = \int_{\mathbb{R}^{3\times 2}} \frac{\phi_i(r_1) \phi_j(r_2)}{|r_1 - r_2|} dr_1 dr_2$

and represents the Coulomb interaction between the unit charge distributions $\phi_i$ and $\phi_j$.

One way to look at the matrix $J$ is as a finite dimensional (approximate) representation of the Coulomb operator $\hat J = 1 / {|r_1 -r_2|}$. We know that $\hat J$ has certain nice properties such as positivity, so we expect a "good" representation of $\hat J$ should be a symmetric positive definite matrix.

My question is this: are there conditions on the discrete representation (possibly expressible as conditions on the {$\phi_i$} basis) to detect whether or not a given claimed representation $J$ is "good" in that it preserves such properties? Or asked another way, if I have some matrix $J$ which is claimed to represent $\hat J$, what are necessary and sufficient conditions on its matrix elements for it to be a "good" representation of $\hat J$?

I hope the question makes sense, and that I am not misusing too much terminology.