Editing Hartree–Fock method (section)

== Brief history ==

===Early semi-empirical methods===
The origin of the Hartree–Fock method dates back to the end of the 1920s, soon after the discovery of the [[Schrödinger equation]] in 1926. Douglas Hartree's methods were guided by some earlier, semi-empirical methods of the early 1920s (by E. Fues, [[Robert Bruce Lindsay|R. B. Lindsay]], and himself) set in the [[old quantum theory]] of Bohr.

In the [[Bohr model]] of the atom, the energy of a state with [[principal quantum number]] ''n'' is given in atomic units as <math>E = -1 / n^2</math>. It was observed from atomic spectra that the energy levels of many-electron atoms are well described by applying a modified version of Bohr's formula. By introducing the [[quantum defect]] ''d'' as an empirical parameter, the energy levels of a generic atom were well approximated by the formula <math>E = -1 / (n + d)^2</math>, in the sense that one could reproduce fairly well the observed transitions levels observed in the [[X-ray]] region (for example, see the empirical discussion and derivation in [[Moseley's law]]). The existence of a non-zero quantum defect was attributed to electron–electron repulsion, which clearly does not exist in the isolated hydrogen atom. This repulsion resulted in partial [[screening effect|screening]] of the bare nuclear charge. These early researchers later introduced other potentials containing additional empirical parameters with the hope of better reproducing the experimental data.

===Hartree method===
{{main|Hartree equation}}
In 1927, [[Douglas Hartree|D. R. Hartree]] introduced a procedure, which he called the self-consistent field method, to calculate approximate wave functions and energies for atoms and ions.<ref name="Hartree1928">{{cite journal |first1=D. R. |last1=Hartree |authorlink1=Douglas Hartree |title=The Wave Mechanics of an Atom with a Non-Coulomb Central Field |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] |volume=24 |issue=1 |pages=111 |year=1928 |doi=10.1017/S0305004100011920 |bibcode=1928PCPS...24..111H |s2cid=121520012 }}</ref> Hartree sought to do away with empirical parameters and solve the many-body time-independent Schrödinger equation from fundamental physical principles, i.e., [[ab initio quantum chemistry methods|ab initio]]. His first proposed method of solution became known as the ''Hartree method'', or ''[[Hartree product]]''. However, many of Hartree's contemporaries did not understand the physical reasoning behind the Hartree method: it appeared to many people to contain empirical elements, and its connection to the solution of the many-body Schrödinger equation was unclear. However, in 1928 [[John C. Slater|J. C. Slater]] and J. A. Gaunt independently showed that the Hartree method could be couched on a sounder theoretical basis by applying the [[variational principle]] to an [[ansatz]] (trial wave function) as a product of single-particle functions.<ref name="Slater1928">{{cite journal |first=J. C. |last=Slater |title=The Self Consistent Field and the Structure of Atoms |journal=[[Physical Review]] |volume=32 |issue=3 |pages=339–348 |year=1928 |doi=10.1103/PhysRev.32.339 |bibcode=1928PhRv...32..339S }}</ref><ref name="Gaunt1928">{{cite journal |first=J. A. |last=Gaunt |title=A Theory of Hartree's Atomic Fields |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] |volume=24 |issue=2 |pages=328–342 |year=1928 |doi=10.1017/S0305004100015851 |bibcode=1928PCPS...24..328G |s2cid=119685329 }}</ref>

In 1930, Slater and [[Vladimir Fock|V. A. Fock]] independently pointed out that the Hartree method did not respect the principle of [[exchange symmetry|antisymmetry]] of the wave function.<ref name="Slater1930">{{cite journal |first=J. C. |last=Slater |title=Note on Hartree's Method |journal=[[Physical Review]] |volume=35 |issue=2 |pages=210–211 |year=1930 |doi=10.1103/PhysRev.35.210.2 |bibcode=1930PhRv...35..210S }}</ref>
<ref name="Fock1930">{{cite journal |first=V. A. |last=Fock |title=Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems |language=de |journal=[[Zeitschrift für Physik]] |volume=61 |issue=1 |pages=126–148 |year=1930 |doi=10.1007/BF01340294 |bibcode=1930ZPhy...61..126F |s2cid=125419115 }} {{cite journal |first=V. A. |last=Fock |title="Selfconsistent field" mit Austausch für Natrium |language=de |journal=[[Zeitschrift für Physik]] |volume=62 |issue=11 |pages=795–805 |year=1930 |doi=10.1007/BF01330439 |bibcode=1930ZPhy...62..795F |s2cid=120921212 }}</ref> The Hartree method used the [[Pauli exclusion principle]] in its older formulation, forbidding the presence of two electrons in the same quantum state. However, this was shown to be fundamentally incomplete in its neglect of [[quantum statistics]].

===Hartree–Fock===
A solution to the lack of anti-symmetry in the Hartree method came when it was shown that a [[Slater determinant]], a [[determinant]] of one-particle orbitals first used by Heisenberg and Dirac in 1926, trivially satisfies the [[exchange symmetry|antisymmetric]] property of the exact solution and hence is a suitable [[ansatz]] for applying the [[variational principle]]. The original Hartree method can then be viewed as an approximation to the Hartree–Fock method by neglecting [[exchange symmetry|exchange]]. Fock's original method relied heavily on [[group theory]] and was too abstract for contemporary physicists to understand and implement. In 1935, Hartree reformulated the method to be more suitable for the purposes of calculation.<ref name="Hartree1935">{{cite journal |first1=D. R. |last1=Hartree |first2=W. |last2=Hartree |title=Self-consistent field, with exchange, for beryllium |journal=[[Proceedings of the Royal Society A]] |volume=150 |issue=869 |pages=9 |year=1935 |doi=10.1098/rspa.1935.0085 |bibcode=1935RSPSA.150....9H |doi-access=free }}</ref>

The Hartree–Fock method, despite its physically more accurate picture, was little used until the advent of electronic computers in the 1950s due to the much greater computational demands over the early Hartree method and empirical models.<ref>{{cite journal | url=https://link.aps.org/doi/10.1103/PhysRev.81.385 | doi=10.1103/PhysRev.81.385 | title=A Simplification of the Hartree-Fock Method | year=1951 | last1=Slater | first1=J. C. | journal=[[Physical Review]] | volume=81 | issue=3 | pages=385–390 | bibcode=1951PhRv...81..385S | url-access=subscription }}</ref> Initially, both the Hartree method and the Hartree–Fock method were applied exclusively to atoms, where the spherical symmetry of the system allowed one to greatly simplify the problem. These approximate methods were (and are) often used together with the [[central field approximation]] to impose the condition that electrons in the same shell have the same radial part and to restrict the variational solution to be a [[Spin (physics)#Mathematical formulation|spin eigenfunction]]. Even so, calculating a solution by hand using the Hartree–Fock equations for a medium-sized atom was laborious; small molecules required computational resources far beyond what was available before 1950.