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(May 16, 2021)

The success of quantitative modern quantum chemistry, relative to its
primitive, qualitative beginnings, can be traced to two sources: better
algorithms and better computers. While the two technologies continue to improve
rapidly, efforts are heavily thwarted by the fact that the total number of ERIs
increases quadratically with the size of the molecular system. Even large
increases in ERI algorithm efficiency yield only moderate increases in
applicability, hindering the more widespread application of *ab initio*
methods to areas of, perhaps, biochemical significance where semi-empirical
techniques
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have already proven so valuable.

Thus, the elimination of quadratic scaling algorithms has been the theme of
many research efforts in quantum chemistry throughout the 1990s and has seen
the construction of many alternative algorithms to alleviate the problem.
Johnson was the first to implement DFT exchange/correlation functionals whose
computational cost scaled linearly with system size.
This paved the way for the most significant breakthrough in the area with the linear
scaling CFMM algorithm
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leading to linear scaling DFT
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Further breakthroughs have been made with
traditional theory in the form of the
QCTC
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algorithms, while more radical
approaches
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may lead to entirely new approaches
to *ab initio*
calculations. Investigations into the quadratic Coulomb problem has not only
yielded linear scaling algorithms, but is also providing large insights into
the significance of many molecular energy components.

Linear scaling Coulomb and SCF exchange/correlation algorithms are not the end
of the story as the $\mathcal{O}({N}^{3})$ diagonalization step has been rate limiting
in semi-empirical techniques and, been predicted to become
rate limiting in *ab initio* approaches in the medium term.
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However, divide-and-conquer techniques
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and the recently developed quadratically convergent SCF algorithm
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show great promise for reducing this problem.