Which properties can be directly obtained from Electron Charge Density (obtained from CHGCAR in VASP, etc.) and no other info?

I've never used VASP, but I'll answer this as best I can. There are essentially two parts to this:

1. What properties are computable from the charge density in principle
2. What properties are computable from the charge density in practice (with reasonable accuracy)

It isn't clear to me which you're interested in, so I'll attempt to tackle both.

1. In theory...

In principle, the ground state properties of a system may be determined from its charge density, but this does require some minimal information about the system itself. The Hohenberg-Kohn theorems allow the ground state energy $E_0$ of a quantum system to be expressed as: $$ E_0[\rho] = F[\rho] + \iiint V_\mathrm{ext}(\vec{r})\rho(\vec{r})d^3 \vec{r}, \tag{1} $$ where $\vec{r}$ is the position in 3D space, $\rho=\rho(\vec{r})$ is the ground state particle density of the system, $V_\mathrm{ext}(\vec{r})$ is the external potential applied to the system and $F[\rho]$ is a universal functional of the density. (For simplicity, we will ignore the fact that this assumes a local external potential.)

Note that this depends on the external potential, $V_\mathrm{ext}(\vec{r})$, which means that to evaluate the ground state energy you need not only the density, $\rho(\vec{r})$, but also the atomic coordinates, lattice vectors and (pseudo)potentials.

Any property which depends on this ground state energy can in principle be computed from $\rho(\vec{r})$ and $V_\mathrm{ext}(\vec{r})$, and how they change under perturbations. Such properties would include forces, unit cell stresses, force constant tensors (hence phonons), polarisation etc.

From your list, we can see that these are computable in principle:

5. Elastic Tensor
6. Magnetic Moment (OSZICAR & OUTCAR)
8. Bulk Modulus
9. Shear Modulus
10. Poisson's Ratio

Knowing the number of atoms would also let us calculate:

3. Total (Relaxed) Energy per Atom

If we also compute related systems and take appropriate energy differences, then we can further compute:

2. Formation Energy per Atom
4. BandGap
7. Energy above Hull

This leaves three properties:

1. SPG
Assuming you mean the symmetry group then you can get some strong clues from the symmetry of the density; however, the density has the symmetry of the potential (Bloch's theorem), which is not necessarily exactly the same as the symmetry of the atoms.

11. Superconducting transition temperature (Tc)
The electron-phonon coupling is computable and you should be able to get the DOS as well, so at least for BCS superconductors you might be able to compute $T_c$.

12. mBJ BandGap
I'm struggling to see how this would make any sense at all. The potential is computable from $\rho(\vec{r})$, but there is no energy functional so I don't see how you could formally define its band-gap. Nevertheless, you could do it if you really wanted -- see section 2.2 for details.

2. In practice...

So far I've discussed what density functional theory can give you in principle, i.e. assuming that we know the universal functional $F[\rho]$ and $V_\mathrm{ext}$. Your question asks what we can compute from a CHGCAR; if CHGCAR contains only the charge density, then we also need the atomic positions, lattice and (pseudo)potentials in order to determine $V_\mathrm{ext}$.

Assuming that we do have enough information to also determine $V_\mathrm{ext}$, we next need to address the fact that $F[\rho]$ is unknown. There are two main options here: orbital-free DFT; and Kohn-Sham DFT.

2.1 Orbital-free DFT

Orbital-free DFT methods approximate $F[\rho]$ directly, as a functional of the density. All of the properties from section 1 are computable in this approach, but it is very difficult to incorporate all of the required physical and mathematical behaviour into orbital-free functionals, and they are rarely as accurate as the Kohn-Sham approach. Using an orbital-free functional would also mean that the CHGCAR density was no longer the ground state of the functional.

2.2 Kohn-Sham DFT

The most common approximation to $F[\rho]$ is that of Kohn and Sham, who introduced a set of auxiliary single-particle wavefunctions, $\psi_n(\vec{r})$. At first sight, these wavefunctions seem a problem, since they aren't stored in the CHGCAR file (I assume); however, they are also uniquely determined from the ground state density, for any given choice of exchange-correlation functional. To regenerate the wavefunctions, you simply compute the Kohn-Sham potential from $\rho(\vec{r})$, and then solve the Kohn-Sham equations non-self-consistently, i.e. solve:

$$ \hat{H}[\rho]\psi_n(\vec{r}) = \epsilon_n \psi_n(\vec{r}), \tag{2} $$ for the fixed density $\rho(\vec{r})$ from the CHGCAR file. Note that here $n$ labels the Kohn-Sham state; the integral of $\rho(\vec{r})$ tells you how many Kohn-Sham particles to use, and therefore the minimum number of states to include.

Once you have $\{\psi_n\}$ then you can compute anything you can usually compute in VASP, with the usual accuracy.

3. Specific questions

There are a few parts in the question text that I don't understand:

• "when we talk about the Total Energy (relaxed) of the system, it is dependent on a bunch of other factors other than the charge density like the Exchange correlation potential"

The exchange-correlation (XC) potential is directly computable from the density, so once we know the choice of exchange-correlation functional which was made, e.g. LDA or PBE, then we can generate all the extra information we need. The choice of XC functional is not immediately clear from the density, but it can be determined provided you have:

1. The cell vectors, atom locations and pseudopotentials, so that we can define $V_\mathrm{ext}$
2. A sufficiently good ground state density

If both of these conditions are satisfied, then you could choose a candidate functional, solve equation (2) (non-self-consistently) to obtain the Kohn-Sham states, and then compute $$ \rho_\mathrm{out}(\vec{r}) = \sum_{bk} f_{bk}\left\vert\psi_{bk}(\vec{r})\right\vert^2, \tag{3} $$ where $f_{bk}$ are the band-occupancies for band $b$ at k-point $k$. $\rho_\mathrm{out}(\vec{r})$ is the "output density", exactly as would be computed in a self-consistent field method.

The difference between $\rho_\mathrm{out}(\vec{r})$ and $\rho(\vec{r})$ should be zero, since $\rho(\vec{r})$ is supposed to be the ground state density; in practice, it won't be exactly zero, because you will have terminated the original optimisation procedure after a finite number of optimisation steps, but it should be very small everywhere.

If the density difference is not small everywhere, and you are convinced that the original calculation was well-converged to the ground state, then that means that $\rho(\vec{r})$ is not the ground state density for that choice of XC and, therefore, that XC was not the one used in the original calculation. You can now repeat the calculation with a different XC functional, and see whether that gives a smaller density difference.

A more sophisticated method would be to define your own mixed-XC functional, comprising all the XC terms you think could have been used in the original calculation, solve equation (2) as before, but then use perturbation theory to compute the optimal change in the XC mixture to reduce the density difference. This method would allow automatic optimisation of the XC choice to match the original, but I highly doubt that VASP (or any other DFT software packages) have this implemented already.

Once you have found the correct XC choice, you have also regenerated the Kohn-Sham states and eigenvalues, so all the usual properties are available to you.

• "Note: I am not talking about the CHGCAR difference. I have just one converged CHGCAR for every material."

Many properties are defined as the difference between the densities (or derived properties) of related materials systems, so this is automatically excluding a lot of properties. The equation of state, for example, is determined by how the properties change as you change the simulation cell.

4. No really, just the CHGCAR

• "Considering the CHGCAR file as the only piece of information I have, what are the properties that can be directly connected to CHGCAR?"

If you really, really only want to compute things from the CHGCAR, without knowing $V_\mathrm{ext}(\vec{r})$, then you cannot use density functional theory and you're extremely restricted in what you can compute -- essentially, you're left with things which are trivially defined by the density itself. Total charge (and therefore number of particles) and charge density are the obvious ones, as well as charge multipoles. Charge density differences would give you much more information, but you've already stated that you don't want to do that.

I'm struggling to think of anything you could compute that isn't trivial. You could try some basic charge decompositions, such as Bader or Hirshfeld analysis, perhaps or electron localisation functions (ELF), although even there it would be difficult to draw too many conclusions without knowing where the nuclei were. Orbital-free methods might open up more options in terms of energy decomposition, but you would still need to know $V_\mathrm{ext}(\vec{r})$ for anything more interesting.

5. Summary

If the question is literally "what can I compute from just the CHGCAR", without any other information at all, the answer is not very much, and nothing very interesting. If you also know the external potential, then you can compute a lot of properties, almost all of the ones on your list, albeit some of them would require you to regenerate the Kohn-Sham states (but they are also functionals of the density).