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61-416: 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 fermions ) or by a single permanent (in the case of bosons ) of N spin-orbitals . 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

122-532: A Slater determinant is an expression that describes the wave function of a multi- fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle , by changing sign upon exchange of two electrons (or other fermions). Only a small subset of all possible fermionic wave functions can be written as a single Slater determinant, but those form an important and useful subset because of their simplicity. The Slater determinant arises from

183-534: A perturbation of the Fock operator. Others expand the true multi-electron wave function in terms of a linear combination of Slater determinants—such as multi-configurational self-consistent field , configuration interaction , quadratic configuration interaction , and complete active space SCF (CASSCF) . Still others (such as variational quantum Monte Carlo ) modify the Hartree–Fock wave function by multiplying it by

244-589: A Slater determinant. The best Slater approximation to a given fermionic wave function can be defined to be the one that maximizes the overlap between the Slater determinant and the target wave function. The maximal overlap is a geometric measure of entanglement between the fermions. A single Slater determinant is used as an approximation to the electronic wavefunction in Hartree–Fock theory . In more accurate theories (such as configuration interaction and MCSCF ),

305-531: A basis in which the Lagrange multiplier matrix λ i j {\displaystyle \lambda _{ij}} becomes diagonal, i.e. λ i j = ϵ i δ i j {\displaystyle \lambda _{ij}=\epsilon _{i}\delta _{ij}} . Performing the variation , we obtain The factor 1/2 in the molecular Hamiltonian drops out before

366-406: 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 is density functional theory , which treats both exchange and correlation energies, albeit approximately. Indeed, it is common to use calculations that are a hybrid of

427-463: A linear combination of Slater determinants is needed. The word " detor " was proposed by S. F. Boys to refer to a Slater determinant of orthonormal orbitals, but this term is rarely used. Unlike fermions that are subject to the Pauli exclusion principle, two or more bosons can occupy the same single-particle quantum state. Wavefunctions describing systems of identical bosons are symmetric under

488-456: A many-particle system is to take the product of properly chosen orthogonal wave functions of the individual particles. For the two-particle case with coordinates x 1 {\displaystyle \mathbf {x} _{1}} and x 2 {\displaystyle \mathbf {x} _{2}} , we have This expression is used in the Hartree method as an ansatz for

549-400: 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,

610-483: A sounder theoretical basis by applying the variational principle to an ansatz (trial wave function) as a product of single-particle functions. In 1930, Slater and V. A. Fock independently pointed out that the Hartree method did not respect the principle of antisymmetry of the wave function. The Hartree method used the Pauli exclusion principle in its older formulation, forbidding the presence of two electrons in

671-465: A wave function that is zero everywhere. The Slater determinant is named for John C. Slater , who introduced the determinant in 1929 as a means of ensuring the antisymmetry of a many-electron wave function, although the wave function in the determinant form first appeared independently in Heisenberg's and Dirac's articles three years earlier. The simplest way to approximate the wave function of

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732-479: Is where h ^ {\displaystyle {\hat {h}}} is the one electron operator including electronic kinetic operators and electron-nucleus Coulombic interaction and To derive Hartree-Fock equation we minimize the energy functional for N electrons with orthonormal constraints. Since the we can choose the basis of ϕ i ( x i ) {\displaystyle \phi _{i}(x_{i})} , we choose

793-483: Is a mixing term. The first contribution is called the "coulomb" term or "coulomb" integral and the second is the "exchange" term or exchange integral. Sometimes different range of index in the summation is used ∑ i j {\textstyle \sum _{ij}} since the Coulomb and exchange contributions exactly cancel each other for i = j {\displaystyle i=j} . It

854-417: Is a single configuration state function with well-defined quantum numbers and that the energy level is not necessarily the ground 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

915-485: 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 . 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. The Hartree–Fock method, despite its physically more accurate picture,

976-425: Is always lower than the electron-electron repulsive energy on the simple Hartree product of the same spin-orbitals. Since exchange bielectronic integrals are different from zero only for spin-orbitals with parallel spins, we link the decrease in energy with the physical fact that electrons with parallel spin are kept apart in real space in Slater determinant states. Most fermionic wavefunctions cannot be represented as

1037-430: 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–electronic Coulombic 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

1098-417: 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

1159-418: Is greater than or equal to the true ground-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 the Hartree–Fock limit ; i.e., the limit of the Hartree–Fock energy as the basis set approaches completeness . (The other

1220-433: Is important to notice explicitly that the exchange term, which is always positive for local spin-orbitals, is absent in simple Hartree product. Hence the electron-electron repulsive energy ⟨ Ψ 0 | G 2 | Ψ 0 ⟩ {\displaystyle \langle \Psi _{0}|G_{2}|\Psi _{0}\rangle } on the antisymmetrized product of spin-orbitals

1281-497: Is now called the Hartree equation as an approximate solution of the Schrödinger 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

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1342-574: Is the full-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 as spin-orbitals . For an atomic orbital calculation, these are typically

1403-477: 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 analytic solutions for many-electron systems (there are solutions for one-electron systems such as hydrogenic atoms and the diatomic hydrogen cation ), the problem is solved numerically. Due to the nonlinearities introduced by

1464-451: The Fock operator below), and hence the terminology continued. The equations are almost universally solved by means of an iterative method , although the fixed-point iteration algorithm does not always converge. 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

1525-552: 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 an identity 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

1586-413: 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 J. C. Slater and J. A. Gaunt independently showed that the Hartree method could be couched on

1647-414: The Pauli principle . Indeed, the Slater determinant vanishes if the set { χ i } {\displaystyle \{\chi _{i}\}} is linearly dependent . In particular, this is the case when two (or more) spin orbitals are the same. In chemistry one expresses this fact by stating that no two electrons with the same spin can occupy the same spatial orbital. Many properties of

1708-417: 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 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

1769-511: The Coulomb and exchange operators respectively, and V nucl {\displaystyle V_{\text{nucl}}} is the total electrostatic repulsion between all the nuclei in the molecule. It should be emphasized that the total energy is not equal to the sum of orbital energies. If the atom or molecule is closed shell , the total energy according to the Hartree-Fock method is Typically, in modern Hartree–Fock calculations,

1830-501: 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 to deal with this task: Relaxation of the last two approximations give rise to many so-called post-Hartree–Fock methods. The variational theorem states that for a time-independent Hamiltonian operator, any trial wave function will have an energy expectation value that

1891-427: The Hartree–Fock wave function and energy of the system. Hartree–Fock approximation is an instance of mean-field theory , where neglecting higher-order fluctuations in order 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 the self-consistent field method ( SCF ). In deriving what

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1952-416: 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 Slater–Condon rules , the expectation value of energy of the molecular electronic Hamiltonian H ^ e {\displaystyle {\hat {H}}^{e}} for a Slater determinant

2013-440: The Pauli principle. This problem can be overcome by taking a linear combination of both Hartree products: where the coefficient is the normalization factor . This wave function is now antisymmetric and no longer distinguishes between fermions (that is, one cannot indicate an ordinal number to a specific particle, and the indices given are interchangeable). Moreover, it also goes to zero if any two spin orbitals of two fermions are

2074-474: The Schrödinger equation for atoms, molecules, nanostructures and solids but it has also found widespread use in nuclear physics . (See Hartree–Fock–Bogoliubov method for a discussion of its application in 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

2135-1293: The Slater determinant come to life with an example in a non-relativistic many electron problem. Starting from a molecular Hamiltonian : H ^ tot = ∑ i p i 2 2 m + ∑ I P I 2 2 M I + ∑ i V nucl ( r i ) + 1 2 ∑ i ≠ j e 2 | r i − r j | + 1 2 ∑ I ≠ J Z I Z J e 2 | R I − R J | {\displaystyle {\hat {H}}_{\text{tot}}=\sum _{i}{\frac {\mathbf {p} _{i}^{2}}{2m}}+\sum _{I}{\frac {\mathbf {P} _{I}^{2}}{2M_{I}}}+\sum _{i}V_{\text{nucl}}(\mathbf {r_{i}} )+{\frac {1}{2}}\sum _{i\neq j}{\frac {e^{2}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}+{\frac {1}{2}}\sum _{I\neq J}{\frac {Z_{I}Z_{J}e^{2}}{|\mathbf {R} _{I}-\mathbf {R} _{J}|}}} where r i {\displaystyle \mathbf {r} _{i}} are

2196-413: The advent of more efficient, often sparse, algorithms for solving the generalized eigenvalue problem , of which the Roothaan–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 called F-mixing or damping. With F-mixing, once a single-electron wave function

2257-424: 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 the restricted open-shell or the unrestricted Hartree–Fock methods. The origin of the Hartree–Fock method dates back to

2318-425: The consideration of a wave function for a collection of electrons, each with a wave function known as the spin-orbital χ ( x ) {\displaystyle \chi (\mathbf {x} )} , where x {\displaystyle \mathbf {x} } denotes the position and spin of a single electron. A Slater determinant containing two electrons with the same spin orbital would correspond to

2379-890: The double integrals due to symmetry and the product rule. We may define the Fock operator to rewrite the equation where the Coulomb operator J ^ ( x k ) {\displaystyle {\hat {J}}(\mathbf {x} _{k})} and the exchange operator K ^ ( x k ) {\displaystyle {\hat {K}}(\mathbf {x} _{k})} are defined as follows The exchange operator has no classical analogue and can only be defined as an integral operator. The solution ϕ k {\displaystyle \phi _{k}} and ϵ k {\displaystyle \epsilon _{k}} are called molecular orbital and orbital energy respectively. Although Hartree-Fock equation appears in

2440-445: The effect of other electrons are accounted for in a mean-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, the Fock operator . At the minimum, the occupied orbitals are eigensolutions to the Fock operator via a unitary transformation between themselves. The Fock operator

2501-506: The electrons and R I {\displaystyle \mathbf {R} _{I}} are the nuclei and For simplicity we freeze the nuclei at equilibrium in one position and we remain with a simplified Hamiltonian where and where we will distinguish in the Hamiltonian between the first set of terms as G ^ 1 {\displaystyle {\hat {G}}_{1}} (the "1" particle terms) and

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2562-546: 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, 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 E = − 1 / n 2 {\displaystyle E=-1/n^{2}} . It

2623-469: The five simplifications outlined in the section "Hartree–Fock algorithm", the fifth is typically the most important. Neglect of 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

2684-548: The form of a eigenvalue problem, the Fock operator itself depends on ϕ {\displaystyle \phi } and must be solved by different technique. The optimal total energy E H F {\displaystyle E_{HF}} can be written in terms of molecular orbitals. J ^ i j {\displaystyle {\hat {J}}_{ij}} and K ^ i j {\displaystyle {\hat {K}}_{ij}} are matrix elements of

2745-452: 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. 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

2806-421: The identical permutation in the determinant in the left part, since all the other N! − 1 permutations would give the same result as the selected one. We can thus cancel N! at the denominator Because of the orthonormality of spin-orbitals it is also evident that only the identical permutation survives in the determinant on the right part of the above matrix element This result shows that the anti-symmetrization of

2867-415: The indices for the fermion coordinates (second shorthand) are written down. All skipped labels are implied to behave in ascending sequence. The linear combination of Hartree products for the two-particle case is identical with the Slater determinant for N = 2. The use of Slater determinants ensures an antisymmetrized function at the outset. In the same way, the use of Slater determinants ensures conformity to

2928-408: The last term G ^ 2 {\displaystyle {\hat {G}}_{2}} (the "2" particle term) which contains exchange term for a Slater determinant. The two parts will behave differently when they have to interact with a Slater determinant wave function. We start to compute the expectation values of one-particle terms In the above expression, we can just select

2989-425: The many-particle wave function and is known as a Hartree product . However, it is not satisfactory for fermions because the wave function above is not antisymmetric under exchange of any two of the fermions, as it must be according to the Pauli exclusion principle . An antisymmetric wave function can be mathematically described as follows: This does not hold for the Hartree product, which therefore does not satisfy

3050-556: The one-electron wave functions 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

3111-400: The orbitals for a hydrogen-like atom (an atom with only one electron, but the appropriate nuclear charge). For a molecular orbital or crystalline calculation, the initial approximate one-electron wave functions are typically a linear 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,

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3172-417: 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 corresponding Fock matrix , the eigenfunctions of the Fock operator are in turn new orbitals, which can be used to construct

3233-424: The product does not have any effect for the one particle terms and it behaves as it would do in the case of the simple Hartree product. And finally we remain with the trace over the one-particle Hamiltonians Which tells us that to the extent of the one-particle terms the wave functions of the electrons are independent of each other and the expectation value of total system is given by the sum of expectation value of

3294-400: 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 antisymmetric property of the exact solution and hence

3355-433: The same. This is equivalent to satisfying the Pauli exclusion principle. The expression can be generalised to any number of fermions by writing it as a determinant . For an N -electron system, the Slater determinant is defined as where the last two expressions use a shorthand for Slater determinants: The normalization constant is implied by noting the number N, and only the one-particle wavefunctions (first shorthand) or

3416-624: 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 of the bare nuclear charge. These early researchers later introduced other potentials containing additional empirical parameters with

3477-1114: The single particles. For the two-particle terms instead If we focus on the action of one term of G 2 {\displaystyle G_{2}} , it will produce only the two terms And finally ⟨ Ψ 0 | G 2 | Ψ 0 ⟩ = 1 2 ∑ i ≠ j [ ⟨ ψ i ψ j | e 2 r i j | ψ i ψ j ⟩ − ⟨ ψ i ψ j | e 2 r i j | ψ j ψ i ⟩ ] {\displaystyle \langle \Psi _{0}|G_{2}|\Psi _{0}\rangle ={\frac {1}{2}}\sum _{i\neq j}\left[\langle \psi _{i}\psi _{j}|{\frac {e^{2}}{r_{ij}}}|\psi _{i}\psi _{j}\rangle -\langle \psi _{i}\psi _{j}|{\frac {e^{2}}{r_{ij}}}|\psi _{j}\psi _{i}\rangle \right]} which instead

3538-460: 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

3599-433: The two methods—the popular B3LYP scheme is one such hybrid functional method. Another option is to use modern valence bond methods. For a list of software packages known to handle Hartree–Fock calculations, particularly for molecules and solids, see the list of quantum chemistry and solid state physics software . Related fields Concepts People Slater determinant In quantum mechanics ,

3660-419: 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. 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

3721-421: 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 E = − 1 / ( n + d ) 2 {\displaystyle E=-1/(n+d)^{2}} , in

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