Hartree-Fock
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In computational physics and computational chemistry, the Hartree-Fock (HF) method is an approximate method for the determination of the ground-state wavefunction and ground-state energy of a quantum many-body system.
The Hartree-Fock method assumes that the exact, N-body wavefunction of the system can be approximated by a single Slater determinant (in the case where the particles are fermions) or by a single permanent (in the case of bosons) of N spin-orbitals. Invoking the variational principle one can derive a set of N coupled equations for the N spin-orbitals. Solution of these equations yields the Hartree-Fock wavefunction and energy of the system, which are approximations of the exact ones.
The Hartree-Fock method finds its typical application in the solution of the electronic Schrödinger equation of atoms, molecules and solids but it has also found widespread use in nuclear physics. See Hartree-Fock-Boloyubov for a discussion of its application in nuclear structure theory. The rest of this article will focus on applications in electronic structure theory.
The Hartree-Fock method is also called, especially in the older literature, the self-consistent field method (SCF) because the resulting equations are almost universally solved by means of an iterative, fixed-point type algorithm (see the following section for more details). This solution scheme is not the only one possible and is not specific of the Hartree-Fock method. Therefore "self-consistent field" is a potentially ambiguous denomination.
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) are doubly occupied. Open shell systems, where some of the electrons are not paired, can be dealt with by one of two Hartree-Fock methods:
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[edit] Brief history
The origin of the HF method dates back to the end of the 1920s, soon after the derivation of the Schrodinger equation in 1926. In 1927 D.R. Hartree introduced a procedure, which he called the self consistent field method, to calculate approximate wavefunctions and energies for atoms and ions. Hartree was guided by some earlier, semi-empirical methods of the early 1920s (by E. Fues, R.B. Lindsay and himself) set in the old quantum theory of Bohr. The basic observation of these methods was that the energy levels of many-electron atoms, as deducted by atomic spectra, could be rather well rationalized applying a slightly modified version of Bohr's formula for the energy levels of atomic hydrogen (which in atomic units reads: E = − 1 / n2 ); writing E = − 1 / (n + d)2 , and treating "d" as an empirical parameter one could reproduce fairly well the observed levels (in particular high-energy levels such those involved in emissions in the X-ray region). It was clear that the physical origin of this parameter "d" was due to the electron-electron repulsion, which partially screened the nucleus. These early researchers thus introduced other potentials containing empirical parameters with the hope of better reproducing the experimental data. Hartree had devised his method along these lines solely on physical reasoning so that it appeared to many people to contain empirical elements and its connection to the solution of the many-body Schrodinger equation was unclear.
Already in 1928 J.C. Slater and, independently, J.A. Gaunt showed that Hartree's method could be couched on a more sound theoretical basis applying the variational principle to a trial wavefunction written as a product of single-particle functions and then introducing some small approximations. In 1930 Slater and (independently) V.A. Fock pointed out that Hartree's method did not respect the principle of antisymmetry of the wavefunction (Hartree's method used Pauli's principle in its older, single-particle form which says that there cannot be two electrons in the same quantum state). Writing the trial wavefunction as a Slater determinant (note however that the idea of writing a wavefunction as a determinant of orbitals is not Slater's but was suggested by Heisenberg and Dirac in 1926) they introduced the Hartree-Fock method. Hartree's original method can be viewed as an approximation to the Hartree-Fock solution which neglects exchange.
Slater's name is not associated with the method probably because he did not explicitely write the Hartree-Fock equations, his main goal being the justification of Hartree's method. Fock's original paper used difficult group-theoretical reasonings and was not well suited to practical implementation. In 1935 Hartree re-formulated it in an equivalent, simpler form more apt to actual calculation.
Both the Hartree and Hartree-Fock methods were applied excusively to atoms, where the spherical symmetry of the system allowed one to greatly simplify the problem. It was (and is) also customary for atoms to use the central field approximation, to use the "restricted" form of the method (see next section) and to impose that electrons in the same shell have the same radial part.
From the practical point of view, Hartree's method for atoms could still be handled by hand computations (possibly with the aid of mechanical calculators), but the amount of work involved was very large. The full Hartree-Fock method adds considerable difficulties and the solution by hand of the equations for a medium sized atom could take weeks of labor. Because of these computational limitation, the HF method was seldomly used before the spread of computers beginning from about 1950. At that time, the solution of the Hartree-Fock equation for even a small molecule was out of the question.
[edit] Hartree-Fock algorithm
The Hartree-Fock method is typically used to solve the time-independent Schrödinger equation for a multi-electron atom or molecule described in the fixed-nuclei approximation by the electronic molecular Hamiltonian. Because of the complexity of the differential equations for any but the smallest systems, the problem is usually impossible to solve analytically, and so the numerical technique of iteration is used. The method makes four major simplifications in order to deal with this task:
- The Born-Oppenheimer approximation is inherently assumed. The true wavefunction is actually a function of the coordinates of each of the nuclei, in addition to those of the electrons.
- Typically, relativistic effects are completely neglected. The momentum operator is assumed to be completely non-relativistic.
- The basis set is composed of a finite number of orthogonal functions. The true wavefunction is a linear combination of functions from a complete basis set.
- The energy eigenfunctions are assumed to be anti-symmetrized linear combinations of products of one-electron wavefunctions. The effects of electron correlation, beyond that of exchange energy resulting from the anti-symmetrization of the wavefunction, are completely neglected.
The variational theorem states that, for a time-independent Hamiltonian operator, any trial wavefunction will have an energy expectation value that is greater than or equal to the true ground state wavefunction 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. The limit of the Hartree-Fock energy as the basis set becomes infinite is called the Hartree-Fock limit. It is a unique set of one-electron orbitals, and their eigenvalues.
The starting point for the Hartree-Fock method is a set of approximate one-electron orbitals. For an atomic calculation, these are typically the orbitals for a hydrogenic atom (an atom with only one electron, but the appropriate nuclear charge). For a molecular or crystalline calculation, the initial approximate one-electron wavefunctions are typically a linear combination of atomic orbitals. This gives a collection of one electron orbitals that, due to the fermionic nature of electrons, must be anti-symmetric. This antisymmetry is achieved through the use of a Slater determinant.
At this point, a new approximate Hamiltonian operator, called the Fock operator, is constructed. The first terms in this Hamiltonian are a sum of kinetic energy operators for each electron, the internuclear repulsion energy, and a sum of nuclear-electronic coulombic attraction terms. The final set of terms models the electronic coulombic repulsion terms between each electron with a sum. The sum is composed of 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 fourth simplification in the above list, (see post-Hartree-Fock).
The newly constructed Fock operator is then used as the Hamiltonian in the time-independent Schrödinger Equation. Solving the equation yields a new set of approximate one-electron orbitals. This new set of orbitals is then used to construct a new Fock operator, as in the preceding paragraph, beginning the cycle again. The procedure is stopped when the change in total electronic energy is negligible between two iterations. In this way, a set of so-called "self-consistent" one-electron orbitals are calculated. The Hartree-Fock electronic wavefunction is then equal to the Slater determinant of these approximate one-electron wavefunctions. From the Hartree-Fock wavefunction, any chemical property of the system in question can be calculated in an approximate manner.
[edit] Mathematical formulation
[edit] The Fock operator
Because the electron-electron repulsion term of the electronic molecular Hamiltonian involves the coordinates of two different electrons, it is necessary to reformulate it in an approximate way. Under this approximation, (outlined under Hartree-Fock algorithm), all of the terms of the exact Hamiltonian except the nuclear-nuclear repulsion term are re-expressed as the sum of one-electron operators outlined below. The "(1)" following each operator symbol simply indicates that the operator is 1-electron in nature.
![\hat F(1) = \hat H^{core}(1)+\sum_{j=1}^{n/2}[2\hat J_j(1)-\hat K_j(1)]](../../../math/8/b/f/8bfb97c6946081da4c2458025deacbe1.png)
where:
is the one-electron Fock operator,
is the one-electron core Hamiltonian,
is the Coulomb operator, defining the electron-electron repulsion energy due to the j-th electron,
is the exchange operator, defining the electron exchange energy. Finding the Hartree-Fock one-electron wavefunctions is now equivalent to solving the eigenfunction equation:
where φi(1) are a set of one-electron wavefunctions, called the Hartree-Fock Molecular Orbitals.
[edit] Linear combination of atomic orbitals
- Main article: basis set
Typically, in modern Hartree-Fock calculations, the one-electron wavefunctions are approximated by a Linear combination of atomic orbitals. These atomic orbitals are called Slater-type orbitals. Furthermore, it is very common for the "atomic orbitals" in use to actually be composed of a linear combination of one or more Gaussian-type orbitals, rather than Slater-type orbitals, in the interests of saving large amounts of computation time.
Various basis sets are used in practice, most of which are composed of Gaussian functions. In some applications, an orthogonalization method such as the Gram-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 the Roothaan-Hall equations by converting the overlap matrix effectively to a identity matrix. However in most modern computer programs for molecular Hartree-Fock calculations this procedure is not followed and the full overlap matrix is used directly.
[edit] Numerical stability
Numerical stability can be a problem with this procedure and there are various ways of combating this instability. One of the most basic and generally applicable is called F-mixing or damping. With F-mixing, once a single electron wavefunction is calculated it is not used directly. Instead, some combination of that calculated wavefunction and the previous wavefunctions for that electron is used - the most common being a simple linear combination of the calculated and immediately preceding wavefunction. 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 wavefunction 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.
[edit] Weaknesses, extensions, and alternatives
Of the four simplifications outlined under Hartree-Fock algorithm, the fourth is typically the most important. Neglecting electron correlation can lead to large deviations from experimental results. A number of approaches to this weakness, collectively called post-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 a perturbation of the Fock operator. Others expand the true multi-electron wavefunction in terms of a linear combination of Slater determinants - such as Multi-configurational self-consistent field, Configuration interaction, Quadratic configuration interaction, Complete active space SCF.
An alternative to Hartree-Fock calculations used in some cases is density functional theory, which gives approximate solutions to both exchange and correlation energies, but is not a purely ab initio method in practice. Indeed, it is common to use calculations that are a hybrid of the two methods - the popular B3LYP schema is one such method.
[edit] Software packages
For a list of software packages known to handle Hartree-Fock calculations, see the Software packages section of Computational Chemistry.
[edit] See also
[edit] Related fields
[edit] Concepts
[edit] People
- Douglas Hartree
- Vladimir Aleksandrovich Fock
- Clemens Roothaan
- George G. Hall
- John Pople
- Reinhart Ahlrichs
