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Correlation energy is generally defined as the difference between the true total energy and the Hartree-Fock limit. There are mainly two reasons for HF not being exact. Firstly, it approximates the many-body wavefunction as a single Slater determinant, while the exact result must be taken as a combination of many Slater determinants. This leads to a ...

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It's thirty-five (!) years since I last did an HF/SCF calculation, but in those days our code worked by minimising the energy: $$E_{HF} = \langle \Psi_{HF} | H | \Psi_{HF} \rangle$$ where $\Psi_{HF}$ is the approximate wavefunction expressed as a sum of some convenient basis set of functions. Once you'd done the HF/SCF calculation you'd do a CI ...

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$$\newcommand{\ket}[1]{| #1 \rangle}$$ I'll try to answer the last two. with an arbitrary superposition, the probability density for the electron could be anything - can we actually find the coefficients of the superposition an electron actually is in? I'm a little confused about what you mean here. If we are given $\ket{\psi}$ as a combination of, ...

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Regarding your first two points: The symmetry axis of an orbital is free for a free atom. If it's bound to some other atom through one of these one-dimensionally elongated orbitals, the orientation of one orbital is fixed. If you take e.g. carbon, silicon or germanium, you have one s orbital and three p orbitals, which are oriented perpendicular to each ...

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