Hartree–Fock method: Difference between revisions

{{shortShort description|Method in quantum physics}}

The Hartree–Fock method often assumes that the exact ''N''-body wave function of the system can be approximated by a single [[Slater determinant]] (in the case where the particles are [[fermion]]s) or by a single [[Permanent (mathematics)|permanent]] (in the case of [[boson]]s) of ''N'' [[spin-orbital]]s. By invoking the [[variational method]], one can derive a set of ''N''-coupled equations for the ''N'' spin orbitals. A solution of these equations yields the Hartree–Fock wave function and energy of the system. Hartree–Fock approximation is an instance of [[mean-field theory]],<ref>{{cite book |last1=Bruus |first1=Henrik |last2=Flensberg |first2=Karsten |title=Many-body quantum theory in condensed matter physics: an introduction |date=2014 |publisher=Oxford University Press |location=Oxford New York |isbn=9780198566335 |edition=Corrected version |url=https://www.phys.lsu.edu/~jarrell/COURSES/ADV_SOLID_HTML/Other_online_texts/Many-body%20quantum%20theory%20in%0Acondensed%20matter%20physics%0AHenrik%20Bruus%20and%20Karsten%20Flensberg.pdf}}</ref> where neglecting higher-order fluctuations in [[Phase_transition#Order_parameters|order parameter]] allows interaction terms to be replaced with quadratic terms, obtaining exactly solvable Hamiltonians.

The Hartree–Fock method finds its typical application in the solution of the Schrödinger equation for atoms, molecules, nanostructures<ref>{{cite journal |first=Mudar A. |last=Abdulsattar |title=SiGe superlattice nanocrystal infrared and Raman spectra: A density functional theory study |journal=[[Journal of Applied Physics|J. Appl. Phys.]] |volume=111 |issue=4 |pages=044306–044306–4 |year=2012 |doi=10.1063/1.3686610 |bibcode = 2012JAP...111d4306A |doi-access=free }}</ref> and solids but it has also found widespread use in [[nuclear physics]]. (See [[Bogoliubov transformation|Hartree–Fock–Bogoliubov method]] for a discussion of its application in [[nuclear structure#Nuclear pairing phenomenon|nuclear structure]] theory). In [[atomic structure]] theory, calculations may be for a spectrum with many excited energy levels, and consequently, the Hartree–Fock method for atoms assumes the wave function is a single [[configuration state function]] with well-defined [[quantum number]]s and that the energy level is not necessarily the [[ground state]].

The discussion here is only for the Restrictedrestricted Hartree–Fock method, where the atom or molecule is a closed-shell system with all orbitals (atomic or molecular) doubly occupied. [[Open shell|Open-shell]] systems, where some of the electrons are not paired, can be dealt with by either the [[Restricted open-shell Hartree–Fock|restricted open-shell]] or the [[Unrestricted Hartree–Fock|unrestricted]] Hartree–Fock methods.

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 |first=D. R. |last=Hartree |title=The Wave Mechanics of an Atom with a Non-Coulomb Central Field |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society|Math. Proc. Camb. Philos. Soc.]] |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|Phys. Rev.]] |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|Math. Proc. Camb. Philos. Soc.]] |volume=24 |issue=2 |pages=328–342 |year=1928 |doi=10.1017/S0305004100015851 |bibcode=1928PCPS...24..328G |s2cid=119685329 }}</ref>

<ref name="Fock1930">{{cite journal |first=V. A. |last=Fock |title=Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems |langlanguage=de |journal=[[Zeitschrift für Physik|Z. Phys.]] |volume=61 |issue=1 |pages=126126–148 |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 |langlanguage=de |journal=[[Zeitschrift für Physik|Z. Phys.]] |volume=62 |issue=11 |pages=795795–805 |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]].

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|Proc. RoyalR. Soc. Lond. 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 }}</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_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.

The Hartree–Fock method is typically used to solve the time-independent Schrödinger equation for a multi-electron atom or molecule as described in the [[Born–Oppenheimer approximation]]. Since there are no known [[Closed-form expression|analytic solutions]] for many-electron systems (there ''are'' solutions for one-electron systems such as [[Hydrogen atom|hydrogenic atoms]] and the [[Dihydrogen_cation|diatomic hydrogen cation]]), the problem is solved numerically. Due to the nonlinearities introduced by the Hartree–Fock approximation, the equations are solved using a nonlinear method such as [[iteration]], which gives rise to the name "self-consistent field method"."

The Hartree–Fock method makes five major simplifications in order to deal with this task:

* The [[mean field theory|mean-field approximation]] is implied. Effects arising from deviations from this assumption are neglected. These effects are often collectively used as a definition of the term [[electron correlation]]. However, the label "electron correlation" strictly spoken encompasses both the Coulomb correlation and Fermi correlation, and the latter is an effect of electron exchange, which is fully accounted for in the Hartree–Fock method.<ref>{{cite book |title=Modelling Molecular Structures |last=Hinchliffe |first=Alan |authorlink= |edition=2nd |year=2000 |publisher=John Wiley & Sons Ltd |location=Baffins Lane, Chichester, West Sussex PO19 1UD, England |isbn=0-471-48993-X |page=186 |url= |accessdate=}}</ref><ref name="Szabo">{{cite book

=== The Fock operatorDerivation ===

: <math>\hat K_j(1)</math>begin{aligned}

[[Numerical stability]] can be a problem with this procedure and there are various ways of combatingcombatting this instability. One of the most basic and generally applicable is called ''F-mixing'' or damping. With F-mixing, once a single-electron wave function is calculated, it is not used directly. Instead, some combination of that calculated wave function and the previous wave functions for that electron is used, the most common being a simple linear combination of the calculated and immediately preceding wave function. A clever dodge, employed by Hartree, for atomic calculations was to increase the nuclear charge, thus pulling all the electrons closer together. As the system stabilised, this was gradually reduced to the correct charge. In molecular calculations a similar approach is sometimes used by first calculating the wave function for a positive ion and then to use these orbitals as the starting point for the neutral molecule. Modern molecular Hartree–Fock computer programs use a variety of methods to ensure convergence of the Roothaan–Hall equations.

* [[DIIS|Direct Inversioninversion of Iterativeiterative Subspacesubspace]]

* [https://doi.org/10.1017/S0305004100011920 The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part II. Some Results and Discussion] by [[Douglas Hartree|D. R. Hartree]], Mathematical Proceedings of the Cambridge Philosophical Society, Volume 24, 111-132111–132, January 1928