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This is a pretty big topic, but the basic important qualities of an atom-centered basis set are: The number of separate, unlinked functions it uses at the valence level (the so-called *zeta* count), which allow the modeling of that many different electronic environments, practically speaking. Additional higher angular momentum so-called *polarisation* ...

No, there are several cases where the approximation becomes unphysical and inaccurate. To name a few I'm aware of: excited molecule states, basis functions are typically optimized to describe ground states. Configuration Interaction (CI) methods are rather used here. HF covers ground states best. electron correlation, especially if correlation changes with ...

The Hartree-Fock equations are the result of performing constrained Newton-Raphson minimization of the energy with respect to the parameter space of Slater determinants (I don't have my copy of Szabo-Ostlund at hand, but I believe this is pointed out in the derivation). Hence, HF-SCF will converge if your starting guess is in a convex region around a minimum....

Short answer: For a close-to-optimal representation, instead of Coulomb-wave functions, use spherical harmonics for the angular part and a grid-like representation in radial direction. ("grid-like" means finite-differences, collocation methods, discrete-variable-representations, or similar methods). Long answer: The usual approach to solving the Hartree-...

Density functional theory (DFT) also uses a one-particle approach similar to Hartree-Fock, although the effective potential is a little more involved. To achieve a global minimum, the problem is approached as a non-linear fixed point problem which, as Deathbreath said, can be solved via a constrained Newton-Raphson minimization. A common approach in the DFT ...

Every van-der-Waals bonded molecule like H_2 is not covered by Hartree Fock Theory. Electron Correlation is not considered. So HF is just a good starting point for post HF methods like Møller-Plesset perturbation theory, coupled cluster etc. Statistically the best available method for the ground state in modern quantum chemistry programs is CCSDT, coupled ...

Use the Poisson equation in its differential form to obtain the Hartree potential. It has a nice form in spherical coordinates. See for example the seminal paper of Becke, in which this problem is solved for the general polyatomic case without relying on spherical symmetry.

Most of your answers look good. For the geometry optimization of single-reference organic systems, B3LYP/big should be fine. MPn is bad for open-shell systems in general. MCSCF is the way to go if one has truly pathological multireference effects. CCSD(T)/huge is good any time you want to treat dispersion, but you'll find in the literature that (He)2 ...