Hartree–Fock method

In Hartree- Fock calculation (or Hartree -Fock method by Douglas Rayner Hartree and Vladimir Aleksandrovich Fock ) refers to a method of quantum mechanics are treated in the problems with multiple identical particles in mean-field approximation. She comes for example in atomic physics and theoretical chemistry (systems of electrons) or nuclear physics (systems of protons and neutrons ) are used.

It allows to calculate orbital energies and wave functions of quantum mechanical many-body systems approximate and is a so-called ab initio method, ie it comes from without empirical parameters and requires only fundamental constants.

The Hartree- Fock method is the basis of molecular orbital theory.

Operation

It can be applied to both many-body systems of fermions ( such as electrons, protons, neutrons ) and for bosons, - for fermions it is an anti- symmetric product of the wavefunctions ( Slater determinant ), in a symmetric bosons ( Hartree ) product. The following is the important for example for the chemical case of electrons is treated.

When setting up the Hartree- Fock equation, the wave function for electrons approximated as antisymmetrisiertes product ( Slater determinant ) is of one-electron wave functions used ( the orbitals, specifically the spin orbitals ) set and then the Rayleigh - Ritz principle. This means that the energy can be calculated with an arbitrary wave function of a system as the expected value over the Hamiltonian of this system is always above the ground state energy of this system. Consequently, the orbitals will be varied so that the energy becomes minimum. However, it is to be noted that not necessarily a corresponding improvement in quality of the wave function is associated with the decrease of the calculated energy. In some (open shell) is a linear combination of several molecules Slater determinant is recognized instead of a single Slater determinant whose coefficients are however determined by the ( spin ) of symmetry of the system.

Hartree- Fock equation

The Hartree- Fock equation is a nonlinear eigenvalue problem with a nonlocal Integrodifferentialoperator. It reads

With the Fockoperator

Where the Einteilchenanteil of the Hamiltonian and the proportion of the two-particle, as mentioned above for the special case of the molecular physics of electrons with Coulomb interaction among themselves and in atomic units.

The sum is carried out via the occupied electronic states, which is the lowest of the N eigenvalues ​​where N is the number of electrons. The sum is carried out over the cores.

Matrix representation

Usually, then proceeds in the matrix representation of the equation, by representing in the base, so that. This base is typically non- orthogonal.

After multiplication with results in the generalized eigenvalue problem

With the Fockmatrix, the Überlappmatrix and the coefficient vectors. This can be converted using eg Löwdins symmetric orthogonalization in a simple eigenvalue problem. This equation is known as Roothaan Hall equation. As a solution, we obtain n eigenvalues ​​and eigenvectors, which we look at the N lowest eigenvalues ​​and corresponding eigenvectors as occupied states. The basis functions are used in many cases linear combinations of Gaussian type orbital ( GTO) or Slater type orbital ( STO ) were used. For calculations of single atoms and diatomic or linear molecules, the Hartree -Fock equations can be solved on a grid.

Spin

In order to solve the Hartree -Fock equation of the spin orbitals used above nor the spin wave function must be eliminated, so that applies to the pure spatial wave function.

In the closed -shell Hartree -Fock approach ( engl. Restricted Hartree Fock ) all the spins are assumed to be paired, which of course is possible only with an even number of electrons. The ground state is thus assumed as a spin- singlet. Thus follows for the wave functions

Substituting this in the Hartree- Fock equation follows

The Coulomb interaction thus occurs between all electrons, the exchange interaction, however, only between electrons with the same spin. Because of the symmetry between spin up and down the HF equation for both spin configurations equal, so still only one eigenvalue equation to be solved, which now, however, only the lowest eigenvalues ​​and eigenvectors must be used.

In the open -shell Hartree -Fock approach (English Unrestricted Hartree Fock ) is compared to the closed -shell approach ( RHF ) dropped the requirement that the same number of electrons in the state, must be like in the state. The spin orbitals are therefore recognized as

After insertion into the original Hartree- Fock equation, there are two different equations for and.

The equation for follows from the replacement and. Here we see again that electrons have the same spin with Coulomb and exchange interaction, electrons with different spins on the other hand, only interact via the Coulombterm. Since the exchange interaction, the total energy always decreases can thus be explained in the framework of Hartree -Fock, the second Hund's rule. This means that for any other degeneracy or quasi- degeneracy of the spins of two electrons are aligned in parallel.

Derivation for fermions

For the derivation of the Hartree-Fock equations are initially assumes that the stationary Schrödinger equation. Here the special case of a Hamiltonian with Coulomb interaction in the Born- Oppenheimer approximation is considered, as occurs, for example, for electrons in molecular physics. that is

Herein, the electronic coordinate, N is the number of electrons, and the charge and fixed coordinates of the cores. is now a single particle and consists of the kinetic energy and the interaction with all of the cores, the i-th electron. however, is a Zweiteilchenoperator and provides coulomb the i-th to the j-th represents the electron stationary SGL. now reads

As an approximation for the Hartree- Fock one now writes as a Slater determinant of wavefunctions. The approximation is that one would have to add up to the exact solution over all possible Slater determinants, eg by replacing by. Thus applies

And the energy of the system is

This can then, by exploiting the orthogonality of the to

Reshape. Now the Ritz variational principle is used as a functional and after varied to obtain the orthogonality of the single particle is, however, not directly minimized, but by the method of Lagrange multipliers, the functional

It is now in the base in the diagonal is change, ie.

The tilde is omitted in the following. Now can be minimized with respect.

As the addend to zero is can be added to taken, whereby all of the N equations are identical and therefore, the index M can be omitted.

It thus follows

The Hartree- Fock equation with the Fockoperator. Here, the first two terms have a classical analog. contains the kinetic energy and the Coulomb interaction with the nuclei. The second term can be interpreted as the mean of all the other electron Coulomb on the m-th electron. The instantaneous correlation of the particles is, however, neglected. The Hartree- Fock method is therefore a mean-field approach. The exchange term has no classical Analagon. The Fockoperator for the m-th electron contains all the wave functions in other electrons whereby the Fockgleichungen usually, only the method of self-consistent field that iteratively by means of fixed point iteration can be solved. On the convergence acceleration this often comes the DIIS method used.

Basis sets

A direct numerical solution of the Hartree -Fock equation as a differential equation is possible for atoms and linear molecules. In general, the orbitals are but analytically recognized as linear combinations of basis functions ( basis set ), which in turn is an approximation that gets better the more is the larger and more intelligent choice of the basis set. Typically brings every atom in the molecule is now a set of appropriate basis set number of basis functions, which are centered on him with. As a rough starting point for the creation of such basis sets are used, the analytical solutions of the hydrogen atom, showing a behavior for large internuclear distances r. Approaches of this type are called Slater type orbital. They usually have the form. A pz orbitals has for example the shape. The great disadvantage of the Slater -type orbital, however, is that the required matrix elements are not in general be calculated analytically. So we use almost exclusively Gaussian type orbitals, ie Basis function of the form. Here, the matrix elements can be calculated analytically. In this case, inter alia, the Gaussian theorem product is used, that is, that the product of two Gaussians is a Gaussian function again. In order to better approximate the STOs, typically there is a basic function of several Gaussian functions with fixed, set by the basis set parameters ( " contraction "). A simple base rate is for example the so-called STO -NG, which approximates Slater type orbitals with N Gaussians. So that the solution of the equation is reduced to the analytical calculation of integrals of these basic functions, and the iterative solving the generalized eigenvalue problem, with the coefficients of the base functions as a parameter to be determined.

Pros and Cons

The calculated with the Hartree- Fock method energy never reaches the exact value even if an infinitely large basis set would be used. In this limit, the so-called Hartree -Fock limit is reached. The reason for this is that is not detected by the use of the averaged potential, the electron correlation, that is the exact interaction between the electrons. To overcome this flaw, methods have been developed that are capable of recording at least a portion of the electron correlation (see item Correlated calculations). Are of particular importance coupled-cluster methods and the Møller- Plesset perturbation theory based on the solution of the Hartree -Fock method. Another very important method is the density functional theory with hybrid functionals, in which the Hartree -Fock exchange rata enters into the exchange-correlation part of the density functional.

The Hartree -Fock method allows but in very many molecules a good determination of their " rough " electronic structure. It provides a rule, total electronic energies, which correspond to 0.5 % with the correct electronic energies ( for the calculation of energy differences, such as reaction energies, but it is not useful ), dipole moments, the 20% with the real dipole moments match, and very accurate distributions of the electron density in the molecule. Because of these properties Hartree- Fock calculations are often used as a starting point for more accurate calculations mentioned above.

Another advantage of the Hartree- Fock method is that the energy obtained according to the variational principle represents an upper bound for the exact ground state energy. By the choice of larger basis sets, the calculated wave function can be systematically until the so-called " Hartree- Fock limit " improved. Such a systematic approach is not possible with density functional methods.