Similar to how a molecular orbital, also known as a 1-electron wavefunction, can be represented with a linear combination of “basis” functions, e.g., atomic orbitals (LCAO): $$\Phi(\mathbf{r})=\sum_i^N c_i \phi(r).$$
A multi-electron wavefunction can be represented with a linear combination of multi-electron basis functions, also known as determinants, or electron configurations.
In the limit of all possible excited-determinants the solution is exact. The excited-electronic configurations describe both static electronic and dynamic electronic correlation.
However, the main challenge in solving this problem is that the number of determinants scales combinatorially. For even small molecules (e.g. >4 heavy atoms) and modest basis functions this problem is intractable. To make matters worse, solving for the coefficients $c$ requires diagonalizing the Hamiltonian, which scales as $N^3$.
What are some of the best approaches, generally speaking, for working around this problem (within the confines of configuration interaction approaches) while still capturing most of the static and dynamic electronic correlation?
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