Editing Computational chemistry (section)

=== Linear-scaling CCSD(T) method ===
{{See also|Coupled cluster}}

==== Algorithm ====

An adaptation of the standard CCSD(T) method using local natural orbitals (NOs) to significantly reduce the computational burden and enable application to larger systems.<ref name="Sengupta-2016" />

==== Complexity ====

Achieves linear scaling with the system size, a major improvement over the traditional fifth-power scaling of CCSD. This advancement allows for practical applications to molecules of up to 100 atoms with reasonable basis sets, marking a significant step forward in computational chemistry's capability to handle larger systems with high accuracy.<ref name="Sengupta-2016" />

Proving the complexity classes for algorithms involves a combination of mathematical proof and computational experiments. For example, in the case of the Hartree-Fock method, the proof of NP-hardness is a theoretical result derived from complexity theory, specifically through reductions from known [[NP-hardness|NP-hard]] problems.<ref name="xlink.rsc.org">{{Cite journal |last1=Whitfield |first1=James Daniel |last2=Love |first2=Peter John |last3=Aspuru-Guzik |first3=Alán |date=2013 |title=Computational complexity in electronic structure |url=http://xlink.rsc.org/?DOI=C2CP42695A |journal=Phys. Chem. Chem. Phys. |language=en |volume=15 |issue=2 |pages=397–411 |arxiv=1208.3334 |bibcode=2013PCCP...15..397W |doi=10.1039/C2CP42695A |issn=1463-9076 |pmid=23172634 |s2cid=12351374}}</ref>

For other methods like MD or DFT, the computational complexity is often empirically observed and supported by algorithm analysis. In these cases, the proof of correctness is less about formal mathematical proofs and more about consistently observing the computational behaviour across various systems and implementations.<ref name="xlink.rsc.org" />