Incomputational physics andchemistry, theHartree–Fock (HF) is used for approximating thewave function and the energy of aquantum many-body system in astationary state. The method is named afterDouglas Hartree andVladimir Fock.
The Hartree–Fock method often assumes that the exactN-body wave function of the system can be approximated by a singleSlater determinant (in the case where the particles arefermions) or by a singlepermanent (in the case ofbosons) ofNspin-orbitals. By invoking thevariational method, one can derive a set ofN-coupled equations for theN spin orbitals. A solution of these equations yields the Hartree–Fock wave function and energy of the system. Hartree–Fock approximation is an instance ofmean-field theory,[1] where neglecting higher-order fluctuations inorder parameter allows interaction terms to be replaced with quadratic terms, obtaining exactly solvable Hamiltonians.
Especially in the older literature, the Hartree–Fock method is also called theself-consistent field method (SCF). In deriving what is now called theHartree equation as an approximate solution of theSchrödinger equation,Hartree required the final field as computed from the charge distribution to be "self-consistent" with the assumed initial field. Thus, self-consistency was a requirement of the solution. The solutions to the non-linear Hartree–Fock equations also behave as if each particle is subjected to the mean field created by all other particles (see theFock operator below), and hence the terminology continued. The equations are almost universally solved by means of aniterative method, although thefixed-point iteration algorithm does not always converge.[2]This solution scheme is not the only one possible and is not an essential feature of the Hartree–Fock method.
The Hartree–Fock method finds its typical application in the solution of the Schrödinger equation for atoms, molecules, nanostructures[3] and solids but it has also found widespread use innuclear physics. (SeeHartree–Fock–Bogoliubov method for a discussion of its application innuclear structure theory). Inatomic 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 singleconfiguration state function with well-definedquantum numbers and that the energy level is not necessarily theground state.
For both atoms and molecules, the Hartree–Fock solution is the central starting point for most methods that describe the many-electron system more accurately.
The rest of this article will focus on applications in electronic structure theory suitable for molecules with the atom as a special case.The discussion here is only for the restricted Hartree–Fock method, where the atom or molecule is a closed-shell system with all orbitals (atomic or molecular) doubly occupied.Open-shell systems, where some of the electrons are not paired, can be dealt with by either therestricted open-shell or theunrestricted Hartree–Fock methods.
The origin of the Hartree–Fock method dates back to the end of the 1920s, soon after the discovery of theSchrödinger equation in 1926. Douglas Hartree's methods were guided by some earlier, semi-empirical methods of the early 1920s (by E. Fues,R. B. Lindsay, and himself) set in theold quantum theory of Bohr.
In theBohr model of the atom, the energy of a state withprincipal quantum numbern is given in atomic units as. 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 thequantum defectd as an empirical parameter, the energy levels of a generic atom were well approximated by the formula, in the sense that one could reproduce fairly well the observed transitions levels observed in theX-ray region (for example, see the empirical discussion and derivation inMoseley'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 partialscreening 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.
In 1927,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.[4] 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. His first proposed method of solution became known as theHartree method, orHartree 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 1928J. C. Slater and J. A. Gaunt independently showed that the Hartree method could be couched on a sounder theoretical basis by applying thevariational principle to anansatz (trial wave function) as a product of single-particle functions.[5][6]
In 1930, Slater andV. A. Fock independently pointed out that the Hartree method did not respect the principle ofantisymmetry of the wave function.[7][8] The Hartree method used thePauli 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 ofquantum statistics.
A solution to the lack of anti-symmetry in the Hartree method came when it was shown that aSlater determinant, adeterminant of one-particle orbitals first used by Heisenberg and Dirac in 1926, trivially satisfies theantisymmetric property of the exact solution and hence is a suitableansatz for applying thevariational principle. The original Hartree method can then be viewed as an approximation to the Hartree–Fock method by neglectingexchange. Fock's original method relied heavily ongroup 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.[9]
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.[10] 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 thecentral 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 aspin 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 theBorn–Oppenheimer approximation. Since there are no knownanalytic solutions for many-electron systems (thereare solutions for one-electron systems such ashydrogenic atoms and thediatomic 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 asiteration, which gives rise to the name "self-consistent field method."
The Hartree–Fock method makes five major simplifications to deal with this task:
Relaxation of the last two approximations give rise to many so-calledpost-Hartree–Fock methods.
Thevariational theorem states that for a time-independent Hamiltonian operator, any trial wave function will have an energyexpectation value that is greater than or equal to the trueground-state wave function corresponding to the given Hamiltonian. Because of this, the Hartree–Fock energy is an upper bound to the true ground-state energy of a given molecule. In the context of the Hartree–Fock method, the best possible solution is at theHartree–Fock limit; i.e., the limit of the Hartree–Fock energy as the basis set approachescompleteness. (The other is thefull-CI limit, where the last two approximations of the Hartree–Fock theory as described above are completely undone. It is only when both limits are attained that the exact solution, up to the Born–Oppenheimer approximation, is obtained.) The Hartree–Fock energy is the minimal energy for a single Slater determinant.
The starting point for the Hartree–Fock method is a set of approximate one-electron wave functions known asspin-orbitals. For anatomic orbital calculation, these are typically the orbitals for ahydrogen-like atom (an atom with only one electron, but the appropriate nuclear charge). For amolecular orbital or crystalline calculation, the initial approximate one-electron wave functions are typically alinear combination of atomic orbitals (LCAO).
The orbitals above only account for the presence of other electrons in an average manner. In the Hartree–Fock method, the effect of other electrons are accounted for in amean-field theory context. The orbitals are optimized by requiring them to minimize the energy of the respective Slater determinant. The resultant variational conditions on the orbitals lead to a new one-electron operator, theFock operator. At the minimum, the occupied orbitals are eigensolutions to the Fock operator via aunitary transformation between themselves. The Fock operator is an effective one-electron Hamiltonian operator being the sum of two terms. The first is a sum of kinetic-energy operators for each electron, the internuclear repulsion energy, and a sum of nuclear–electronicCoulombic attraction terms. The second are Coulombic repulsion terms between electrons in a mean-field theory description; a net repulsion energy for each electron in the system, which is calculated by treating all of the other electrons within the molecule as a smooth distribution of negative charge. This is the major simplification inherent in the Hartree–Fock method and is equivalent to the fifth simplification in the above list.
Since the Fock operator depends on the orbitals used to construct the correspondingFock matrix, the eigenfunctions of the Fock operator are in turn new orbitals, which can be used to construct a new Fock operator. In this way, the Hartree–Fock orbitals are optimized iteratively until the change in total electronic energy falls below a predefined threshold. In this way, a set of self-consistent one-electron orbitals is calculated. The Hartree–Fock electronic wave function is then the Slater determinant constructed from these orbitals. Following the basic postulates of quantum mechanics, the Hartree–Fock wave function can then be used to compute any desired chemical or physical property within the framework of the Hartree–Fock method and the approximations employed.
According to theSlater–Condon rules, the energy expectation value of themolecular electronic Hamiltonian for aSlater determinant is
where is the one electron operator including electronic kinetic energy and electron-nucleus Coulombic interaction, and
To derive the Hartree-Fock equation we minimize the energy functional for N electrons with orthonormal constraints.
We choose a basis set in which theLagrange multiplier matrix becomes diagonal, i.e.. Performing thevariation, we obtain
The factor 1/2 before the double integrals in the molecular Hamiltonian drops out due to symmetry and the product rule. We may define theFock operator to rewrite the equation
where theCoulomb operator and theexchange operator are defined as follows
The exchange operator has no classical analogue and can only be defined as an integral operator.
The solution and are called molecular orbital and orbital energy respectively.
Although Hartree-Fock equation appears in the form of a eigenvalue problem, the Fock operator itself depends on and must be solved by a different technique.
The optimal total energy can be written in terms of molecular orbitals.
and are matrix elements of the Coulomb and exchange operators respectively, and is the total electrostatic repulsion between all the nuclei in the molecule.
The total energy is not equal to the sum of orbital energies.
If the atom or molecule isclosed shell, the total energy according to the Hartree-Fock method is
Typically, in modern Hartree–Fock calculations, the one-electron wave functions are approximated by alinear combination of atomic orbitals. These atomic orbitals are calledSlater-type orbitals. Furthermore, it is very common for the "atomic orbitals" in use to actually be composed of a linear combination of one or moreGaussian-type orbitals, rather than Slater-type orbitals, in the interests of saving large amounts of computation time.
Variousbasis sets are used in practice, most of which are composed of Gaussian functions. In some applications, an orthogonalization method such as theGram–Schmidt process is performed in order to produce a set of orthogonal basis functions. This can in principle save computational time when the computer is solving theRoothaan–Hall equations by converting theoverlap matrix effectively to anidentity matrix. However, in most modern computer programs for molecular Hartree–Fock calculations this procedure is not followed due to the high numerical cost of orthogonalization and the advent of more efficient, often sparse, algorithms for solving thegeneralized eigenvalue problem, of which theRoothaan–Hall equations are an example.
Numerical stability can be a problem with this procedure and there are various ways of combatting this instability. One of the most basic and generally applicable is calledF-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.
Of the five simplifications outlined in the section "Hartree–Fock algorithm", the fifth is typically the most important. Neglect ofelectron correlation can lead to large deviations from experimental results. A number of approaches to this weakness, collectively calledpost-Hartree–Fock methods, have been devised to include electron correlation to the multi-electron wave function. One of these approaches,Møller–Plesset perturbation theory, treats correlation as aperturbation of the Fock operator. Others expand the true multi-electron wave function in terms of a linear combination of Slater determinants—such asmulti-configurational self-consistent field,configuration interaction,quadratic configuration interaction, andcomplete active space SCF (CASSCF). Still others (such asvariational quantum Monte Carlo) modify the Hartree–Fock wave function by multiplying it by a correlation function ("Jastrow" factor), a term which is explicitly a function of multiple electrons that cannot be decomposed into independent single-particle functions.
An alternative to Hartree–Fock calculations used in some cases isdensity functional theory, which treats both exchange and correlation energies, albeit approximately. Indeed, it is common to use calculations that are a hybrid of the two methods—the popular B3LYP scheme is one suchhybrid functional method.Another option is to usemodern valence bond methods.
For a list of software packages known to handle Hartree–Fock calculations, particularly for molecules and solids, see thelist of quantum chemistry and solid state physics software.
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