# Hartree-Fock method

The Hartree-Fock calculation (or Hartree-Fock method , according to Douglas Rayner Hartree and Wladimir Alexandrowitsch Fock ) is a method of quantum mechanics in which systems with several particles of the same type are treated in mean-field approximation . It is used, for example, in atomic physics , theoretical chemistry to describe electrons in molecules and nuclear physics for systems made up of protons and neutrons .

It enables orbital energies and wave functions of quantum mechanical many-body systems to be calculated approximately and is a so-called ab initio method, i.e. H. it works without empirical parameters and only needs natural constants . It is the starting point for post-Hartree-Fock methods , which improve the accuracy of the calculations.

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

## functionality

The Hartree-Fock method is based on the time-independent Schrödinger equation (here in Dirac notation )

${\ displaystyle {\ hat {H}} \, | \ psi \ rangle = E \, | \ psi \ rangle}$

which calculates the energy of a system from the wave function by looking for the eigenvalues ​​of the Hamilton operator for this wave function. The Hamilton operator describes all the energy contributions of the particles and fields in the system and their interactions with one another. In many practically important systems (such as the electrons in a molecule ) the particles are correlated with one another and influence one another. As a result, the Schrödinger equation for such systems can no longer be solved exactly , but only approximately . ${\ displaystyle E}$${\ displaystyle | \ psi \ rangle}$ ${\ displaystyle {\ hat {H}}}$

The Hartree-Fock method simplifies the interactions between the particles in such a way that they no longer interact with each other in pairs, but with a field that is generated by all other particles on average - the so-called mean field . The field still depends on the behavior of the individual particles, but the solution can now be calculated step by step:

• An initial state is selected and the field is generated from it.
• With this the Schrödinger equation is then solved for each individual particle.
• Taken together, the individual solutions then result in a new state and a new field.

This process is repeated until successive solutions differ only slightly, i.e. the field leads to solutions that consistently recreate the field itself. The term self-consistent field is derived from this and is used for this part of the Hartree-Fock method.

A symmetrical (Hartree) product of single-particle wave functions is used as wave functions for the many-body systems discussed for bosons , and an antisymmetrical combination of these products (a so-called Slater determinant ) for fermions (such as electrons, protons and neutrons ). In order to solve the Schrödinger equation, these single-particle wave functions are varied in such a way that the energy resulting from the equation is minimal. Due to the Rayleigh-Ritz principle , this energy is then an upper limit for the actual energy of the system. However, the wave function of the entire system calculated in this way is not necessarily an approximation of the actual wave function.

For some molecules (especially those with unpaired electrons ), instead of a single Slater determinant, a symmetry- adapted linear combination of several Slater determinants is used, the coefficients of which, however, are determined by the ( spin ) symmetry of the system.

## Hartree-Fock equation

The Hartree-Fock equation is a nonlinear eigenvalue problem with a nonlocal integrodifferential operator. It is in Dirac notation

 ${\ displaystyle {\ hat {F}} | \ phi _ {m} \ rangle = \ varepsilon _ {m} | \ phi _ {m} \ rangle \, \!}$

with the jib operator ${\ displaystyle {\ hat {F}}}$

${\ displaystyle {\ hat {F}} = {\ hat {h}} + \ sum \ limits _ {\ gamma} ^ {N} \ left (\ left \ langle \ phi _ {\ gamma} \ right | { \ hat {w}} \ left | \ phi _ {\ gamma} \ right \ rangle - \ left | \ phi _ {\ gamma} \ right \ rangle \ left \ langle \ phi _ {\ gamma} \ right | { \ hat {w}} \ right)}$
${\ displaystyle {\ text {with:}} {\ hat {h}} = - {\ frac {\ Delta (\ mathbf {r})} {2}} - \ sum \ limits _ {k} ^ {N_ {k}} {\ frac {Z_ {k}} {\ left | {\ mathbf {r} - \ mathbf {R_ {k}}} \ right |}} \ qquad {\ text {and}} \ qquad { \ hat {w}} = {\ frac {1} {\ left | {\ mathbf {r_ {1}} - \ mathbf {r_ {2}}} \ right |}}}$

where the one-particle part is the Hamilton operator and the part of the two- particle interaction , as mentioned above for the special case of molecular physics of electrons with Coulomb interaction with each other and in atomic units . ${\ displaystyle {\ hat {h}}}$${\ displaystyle {\ hat {w}}}$

The index runs over the occupied electronic states, i.e. those with the lowest eigenvalues, indicating the number of electrons. The index runs over the atomic nuclei, indicating the number of nuclei. ${\ displaystyle \ gamma}$${\ displaystyle N}$${\ displaystyle N}$${\ displaystyle k}$${\ displaystyle N_ {k}}$

### Matrix display

In order to be able to use efficient solution methods, the equation is also converted into a matrix representation by representing in the base so that . This base is typically not orthogonal. ${\ displaystyle | \ phi _ {m} \ rangle}$ ${\ displaystyle | \ varphi _ {i} \ rangle}$${\ displaystyle | \ phi _ {m} \ rangle = \ sum \ limits _ {i} ^ {n} c_ {im} | \ varphi _ {i} \ rangle}$

${\ displaystyle \ sum \ limits _ {i} {\ hat {F}} c_ {im} | \ varphi _ {i} \ rangle = \ varepsilon _ {m} \ sum \ limits _ {i} c_ {im} | \ varphi _ {i} \ rangle}$

After multiplication with the generalized eigenvalue problem results${\ displaystyle \ langle \ varphi _ {j} |}$

${\ displaystyle \ sum \ limits _ {i} \ langle \ varphi _ {j} | {\ hat {F}} | \ varphi _ {i} \ rangle c_ {im} = \ varepsilon _ {m} \ sum \ limits _ {i} \ langle \ varphi _ {j} | \ varphi _ {i} \ rangle c_ {im}}$
 ${\ displaystyle F \ mathbf {c} _ {m} = \ varepsilon _ {m} S \ mathbf {c} _ {m} \, \!}$

with the Fock matrix , the overlap matrix and the coefficient vectors . This equation is also known as the Roothaan-Hall equation. If the base is diagonalized (e.g., with Löwdin's symmetric orthogonalization ), which makes the overlap matrix an identity matrix, the equation simplifies to a simple single value problem that can be efficiently solved by computers. ${\ displaystyle F_ {ji} = \ langle \ varphi _ {j} | {\ hat {F}} | \ varphi _ {i} \ rangle}$${\ displaystyle S_ {ji} = \ langle \ varphi _ {j} | \ varphi _ {i} \ rangle}$${\ displaystyle \ mathbf {c} _ {m}}$

As a solution, one obtains eigenvalues ​​and eigenvectors, of which the lowest eigenvalues ​​and associated eigenvectors are regarded as occupied states. In many cases, linear combinations of Gaussian Type Orbitals (GTO) or Slater Type Orbitals (STO) are used as basic functions . For calculations on individual atoms and diatomic or linear molecules, the Hartree-Fock equations can also be solved with numerical methods. ${\ displaystyle n}$${\ displaystyle N}$

## Spin

In order to solve the Hartree-Fock equation , the spin orbitals used above have to be split off, so that the pure spatial wave function applies. ${\ displaystyle \ left | \ phi _ {m} \ right \ rangle}$${\ displaystyle \ left | \ chi _ {m} \ right \ rangle}$${\ displaystyle \ left | \ phi _ {m} \ right \ rangle = \ left | \ psi _ {m} \ right \ rangle \ left | \ chi _ {m} \ right \ rangle}$${\ displaystyle \ left | \ psi _ {m} \ right \ rangle}$

### Closed-shell Hartree jib (RHF)

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

${\ displaystyle \ left | \ phi _ {1} \ right \ rangle = \ left | \ psi _ {1} \ right \ rangle \ left | \ alpha \ right \ rangle, \; \ left | \ phi _ {2 } \ right \ rangle = \ left | \ psi _ {1} \ right \ rangle \ left | \ beta \ right \ rangle, \; \ left | \ phi _ {3} \ right \ rangle = \ left | \ psi _ {2} \ right \ rangle \ left | \ alpha \ right \ rangle, \ dots, \ left | \ phi _ {N} \ right \ rangle = \ left | \ psi _ {N / 2} \ right \ rangle \ left | \ beta \ right \ rangle.}$
${\ displaystyle {\ text {with}} {\ hat {s}} _ {z} \ left | \ alpha \ right \ rangle = {\ frac {1} {2}} \ left | \ alpha \ right \ rangle {\ text {and}} {\ hat {s}} _ {z} \ left | \ beta \ right \ rangle = - {\ frac {1} {2}} \ left | \ beta \ right \ rangle}$

Plugging this into the Hartree-Fock equation follows

${\ displaystyle {\ hat {F}} _ {\ text {RHF}} \ left | \ psi _ {m} \ right \ rangle = {\ hat {h}} \ left | \ psi _ {m} \ right \ rangle + \ sum \ limits _ {\ gamma} ^ {N / 2} 2 \ left \ langle \ psi _ {\ gamma} \ right | {\ hat {w}} \ left | \ psi _ {\ gamma} \ right \ rangle \ left | \ psi _ {m} \ right \ rangle - \ left \ langle \ psi _ {\ gamma} \ right | {\ hat {w}} \ left | \ psi _ {m} \ right \ rangle \ left | \ psi _ {\ gamma} \ right \ rangle = \ varepsilon _ {m} \ left | \ psi _ {m} \ right \ rangle.}$

The Coulomb interaction thus occurs between all electrons, whereas the exchange interaction only occurs between electrons with the same spin. Because of the symmetry between spin up and down, the HF equation is the same for both spin configurations, so that only one eigenvalue equation has to be solved, although now only the lowest eigenvalues ​​and eigenvectors have to be used. ${\ displaystyle N / 2}$

### Open-shell Hartree jib (UHF)

In the open-shell Hartree-Fock approach (English Unrestricted Hartree Fock), in comparison to the closed-shell approach (RHF), the requirement that the same number of electrons must be in the state as in the state is dropped . The spin orbitals are therefore set as ${\ displaystyle \ left | \ alpha \ right \ rangle}$${\ displaystyle \ left | \ beta \ right \ rangle}$

${\ displaystyle \ left | \ phi _ {1} \ right \ rangle = \ left | \ psi _ {1} ^ {\ alpha} \ right \ rangle \ left | \ alpha \ right \ rangle, \; \ left | \ phi _ {2} \ right \ rangle = \ left | \ psi _ {2} ^ {\ alpha} \ right \ rangle \ left | \ alpha \ right \ rangle, \ dots, \ left | \ phi _ {N_ {\ alpha}} \ right \ rangle = \ left | \ psi _ {N _ {\ alpha}} ^ {\ alpha} \ right \ rangle \ left | \ alpha \ right \ rangle, \; \ left | \ phi _ {N _ {\ alpha} +1} \ right \ rangle = \ left | \ psi _ {1} ^ {\ beta} \ right \ rangle \ left | \ beta \ right \ rangle, \ dots, \ left | \ phi _ {N} \ right \ rangle = \ left | \ psi _ {N _ {\ beta}} ^ {\ beta} \ right \ rangle \ left | \ beta \ right \ rangle.}$

After inserting it into the original Hartree-Fock equation, two different equations result for and . ${\ displaystyle \ left | \ alpha \ right \ rangle}$${\ displaystyle \ left | \ beta \ right \ rangle}$

${\ displaystyle {\ hat {F}} _ {\ text {UHF}} ^ {\ alpha} \ left | \ psi _ {m} ^ {\ alpha} \ right \ rangle = {\ hat {h}} \ left | \ psi _ {m} ^ {\ alpha} \ right \ rangle + \ sum \ limits _ {\ gamma} ^ {N _ {\ alpha}} \ left \ langle \ psi _ {\ gamma} ^ {\ alpha } \ right | {\ hat {w}} \ left | \ psi _ {\ gamma} ^ {\ alpha} \ right \ rangle \ left | \ psi _ {m} ^ {\ alpha} \ right \ rangle - \ left \ langle \ psi _ {\ gamma} ^ {\ alpha} \ right | {\ hat {w}} \ left | \ psi _ {m} ^ {\ alpha} \ right \ rangle \ left | \ psi _ { \ gamma} ^ {\ alpha} \ right \ rangle + \ sum \ limits _ {\ gamma} ^ {N _ {\ beta}} \ left \ langle \ psi _ {\ gamma} ^ {\ beta} \ right | { \ hat {w}} \ left | \ psi _ {\ gamma} ^ {\ beta} \ right \ rangle \ left | \ psi _ {m} ^ {\ alpha} \ right \ rangle = \ varepsilon _ {m} \ left | \ psi _ {m} ^ {\ alpha} \ right \ rangle.}$

The equation for follows from the substitution and . Here you can see again that electrons with the same spin have Coulomb and exchange interactions, whereas electrons with different spins only interact via the Coulomb term. Since the exchange interaction always reduces the total energy, Hund's second rule can be explained in the context of Hartree-Fock . This means that with other degeneracy or quasi-degeneracy, the spins of two electrons are aligned as parallel as possible. ${\ displaystyle \ left | \ beta \ right \ rangle}$${\ displaystyle \ alpha \ rightarrow \ beta}$${\ displaystyle \ beta \ rightarrow \ alpha}$

## Derivation for fermions

To derive the Hartree-Fock equations, one starts with the stationary Schrödinger equation . Here the special case of a Hamilton operator with Coulomb interaction in the Born-Oppenheimer approximation is considered, as it occurs, for example, for electrons in molecular physics. This means

${\ displaystyle {\ hat {H}} = \ sum \ limits _ {i} ^ {N} \ underbrace {\ left (- {\ frac {\ Delta (\ mathbf {r} _ {i})} {2 }} - \ sum \ limits _ {k} ^ {N_ {k}} {\ frac {Z_ {k}} {\ left | {\ mathbf {r_ {i}} - \ mathbf {R_ {k}}} \ right |}} \ right)} _ {{\ hat {h}} _ {i}} + {\ frac {1} {2}} \ sum \ limits _ {i} ^ {N} \ sum \ limits _ {j \ neq i} ^ {N} \ underbrace {\ frac {1} {\ left | {\ mathbf {r_ {i}} - \ mathbf {r_ {j}}} \ right |}} _ {{ \ hat {w}} _ {ij}}}$

${\ displaystyle \ mathbf {r}}$refers to the electronic coordinates, the number of electrons, and the charge and fixed coordinates of the nuclei. is now a one-particle operator and consists of the kinetic energy and the interaction with all nuclei of the -th electron. is, however, a two-particle operator and represents the Coulomb interaction of the -th with the -th electron. The stationary Schrödinger equation now reads ${\ displaystyle N}$${\ displaystyle Z_ {k}}$${\ displaystyle \ mathbf {R} _ {k}}$${\ displaystyle {\ hat {h}} _ {i}}$${\ displaystyle i}$${\ displaystyle {\ hat {w}} _ {ij}}$${\ displaystyle i}$${\ displaystyle j}$

${\ displaystyle {\ hat {H}} \ left | \ Psi \ right \ rangle = E \ left | \ Psi \ right \ rangle}$

As an approximation of Hartree-Fock one now writes as a Slater determinant of single-particle . The approximation is that one would have to sum over all possible Slater determinants for the exact solution, e.g. B. by replacing with . Thus applies ${\ displaystyle \ left | \ Psi \ right \ rangle}$${\ displaystyle \ left | \ phi _ {i} \ right \ rangle}$${\ displaystyle \ phi _ {N}}$${\ displaystyle \ phi _ {N + 1}}$

${\ displaystyle \ left | \ Psi \ right \ rangle \ approx {\ frac {1} {\ sqrt {N!}}} {\ begin {vmatrix} \ phi _ {1} (1) & \ phi _ {2 } (1) & \ dots & \ phi _ {N} (1) \\\ phi _ {1} (2) & \ phi _ {2} (2) & \ dots & \ phi _ {N} (2 ) \\\ vdots & \ vdots & \ ddots & \ vdots \\\ phi _ {1} (N) & \ phi _ {2} (N) & \ dots & \ phi _ {N} (N) \ end {vmatrix}}}$

and the energy of the system is

${\ displaystyle E = \ left \ langle \ Psi \ right | {\ hat {H}} \ left | \ Psi \ right \ rangle}$

You can do this by taking advantage of the orthogonality of${\ displaystyle \ phi _ {i}}$

${\ displaystyle E [\ {\ phi _ {i} \}] = \ sum \ limits _ {\ alpha} ^ {N} \ int \ phi _ {\ alpha} ^ {*} (\ mathbf {r} _ {i}) {\ hat {h}} _ {i} \ phi _ {\ alpha} (\ mathbf {r} _ {i}) \, \ mathrm {d ^ {3}} \ mathbf {r} _ {i} + {\ frac {1} {2}} \ sum \ limits _ {\ alpha} ^ {N} \ sum \ limits _ {\ gamma \ neq \ alpha} ^ {N} \ left (\ iint \ phi _ {\ alpha} ^ {*} (\ mathbf {r} _ {i}) \ phi _ {\ gamma} ^ {*} (\ mathbf {r} _ {j}) {\ hat {w}} _ {ij} \, \ phi _ {\ gamma} (\ mathbf {r} _ {j}) \ phi _ {\ alpha} (\ mathbf {r} _ {i}) \, d ^ {3} \ mathbf {r} _ {i} \, d ^ {3} \ mathbf {r} _ {j} \ right.}$
${\ displaystyle \ quad - \ left. \ iint \ phi _ {\ alpha} ^ {*} (\ mathbf {r} _ {i}) \ phi _ {\ gamma} ^ {*} (\ mathbf {r} _ {j}) {\ hat {w}} _ {ij} \, \ phi _ {\ gamma} (\ mathbf {r} _ {i}) \ phi _ {\ alpha} (\ mathbf {r} _ {j}) \, d ^ {3} \ mathbf {r} _ {i} \, d ^ {3} \ mathbf {r} _ {j} \ right)}$
${\ displaystyle = \ sum \ limits _ {\ alpha} ^ {N} \ left \ langle \ phi _ {\ alpha} \ right | {\ hat {h}} \ left | \ phi _ {\ alpha} \ right \ rangle + {\ frac {1} {2}} \ sum \ limits _ {\ alpha} ^ {N} \ sum \ limits _ {\ gamma \ neq \ alpha} ^ {N} \ left (\ left \ langle \ phi _ {\ alpha} \ phi _ {\ gamma} \ right | {\ hat {w}} \ left | \ phi _ {\ gamma} \ phi _ {\ alpha} \ right \ rangle - \ left \ langle \ phi _ {\ alpha} \ phi _ {\ gamma} \ right | {\ hat {w}} \ left | \ phi _ {\ alpha} \ phi _ {\ gamma} \ right \ rangle \ right)}$

reshape. Now that is Ritz variational principle used, and as functional according to varied. In order to obtain the orthogonality of the single-particle functions, however , the functional is not minimized directly , but rather the functional using the Lagrange multiplier method${\ displaystyle E}$${\ displaystyle \ phi _ {i}}$${\ displaystyle E}$${\ displaystyle {\ mathcal {L}}}$

${\ displaystyle {\ mathcal {L}} = E [\ {\ phi _ {i} \}] - \ sum \ limits _ {i} ^ {N} \ sum \ limits _ {j} ^ {N} \ varepsilon _ {ij} \ left (\ delta _ {ij} - \ left \ langle \ phi _ {i} \ right | \ left. \ phi _ {j} \ right \ rangle \ \ right).}$

You can now switch to the base , which is diagonal, that is . ${\ displaystyle {\ tilde {\ phi}} _ {i}}$${\ displaystyle \ varepsilon _ {ij}}$${\ displaystyle \ varepsilon _ {ij} = \ varepsilon _ {i} \ delta _ {ij}}$

${\ displaystyle {\ mathcal {L}} = E [\ {{\ tilde {\ phi}} _ {i} \}] - \ sum \ limits _ {i} ^ {N} \ varepsilon _ {i} \ left (1- \ left \ langle {\ tilde {\ phi}} _ {i} \ right | \ left. {\ tilde {\ phi}} _ {i} \ right \ rangle \ right)}$

The tilde is omitted below. Can now regard are minimized. ${\ displaystyle {\ mathcal {L}}}$${\ displaystyle \ phi _ {m}}$

${\ displaystyle {\ frac {\ partial {\ mathcal {L}}} {\ partial \ phi _ {m} ^ {*}}} = {\ frac {\ partial} {\ partial \ phi _ {m} ^ {*}}} \ left \ {\ sum \ limits _ {\ alpha} ^ {N} \ left \ langle \ phi _ {\ alpha} \ right | {\ hat {h}} \ left | \ phi _ { \ alpha} \ right \ rangle + {\ frac {1} {2}} \ sum \ limits _ {\ alpha} ^ {N} \ sum \ limits _ {\ gamma \ neq \ alpha} ^ {N} \ left (\ left \ langle \ phi _ {\ alpha} \ phi _ {\ gamma} \ right | {\ hat {w}} \ left | \ phi _ {\ gamma} \ phi _ {\ alpha} \ right \ rangle - \ left \ langle \ phi _ {\ alpha} \ phi _ {\ gamma} \ right | {\ hat {w}} \ left | \ phi _ {\ alpha} \ phi _ {\ gamma} \ right \ rangle \ right) - \ sum \ limits _ {i} ^ {N} \ varepsilon _ {i} \ left (1- \ left \ langle \ phi _ {i} \ right | \ left. \ phi _ {i} \ right \ rangle \ \ right) \ right \}}$
${\ displaystyle = {\ hat {h}} \ left | \ phi _ {m} \ right \ rangle + \ sum \ limits _ {\ gamma \ neq m} ^ {N} \ int \ phi _ {\ gamma} ^ {*} (\ mathbf {r} _ {1}) {\ hat {w}} _ {12} \ phi _ {\ gamma} (\ mathbf {r} _ {1}) \ phi _ {m} (\ mathbf {r} _ {2}) \, \ mathrm {d ^ {3}} \ mathbf {r} _ {1} - \ int \ phi _ {\ gamma} ^ {*} (\ mathbf {r } _ {1}) {\ hat {w}} _ {12} \ phi _ {m} (\ mathbf {r} _ {1}) \ phi _ {\ gamma} (\ mathbf {r} _ {2 }) \, \ mathrm {d ^ {3}} \ mathbf {r} _ {1} - \ varepsilon _ {m} \ left | \ phi _ {m} \ right \ rangle}$
${\ displaystyle = {\ hat {h}} \ left | \ phi _ {m} \ right \ rangle + \ sum \ limits _ {\ gamma \ neq m} ^ {N} \ underbrace {\ left \ langle \ phi _ {\ gamma} \ right | {\ hat {w}} \ left | \ phi _ {\ gamma} \ right \ rangle \ left | \ phi _ {m} \ right \ rangle} _ {\ text {Coulomb- WW.}} - \ underbrace {\ left \ langle \ phi _ {\ gamma} \ right | {\ hat {w}} \ left | \ phi _ {m} \ right \ rangle \ left | \ phi _ {\ gamma} \ right \ rangle} _ {\ text {exchange WW.}} - \ varepsilon _ {m} \ left | \ phi _ {m} \ right \ rangle}$

Since the summand is equal to zero, it can be added, which means that all equations are identical and thus the index can be omitted. ${\ displaystyle \ gamma = m}$${\ displaystyle N}$${\ displaystyle m}$

${\ displaystyle = {\ hat {h}} \ left | \ phi _ {m} \ right \ rangle + \ sum \ limits _ {\ gamma} ^ {N} \ left \ langle \ phi _ {\ gamma} \ right | {\ hat {w}} \ left | \ phi _ {\ gamma} \ right \ rangle \ left | \ phi _ {m} \ right \ rangle - \ left \ langle \ phi _ {\ gamma} \ right | {\ hat {w}} \ left | \ phi _ {m} \ right \ rangle \ left | \ phi _ {\ gamma} \ right \ rangle - \ varepsilon _ {m} \ left | \ phi _ {m } \ right \ rangle = 0}$

So it follows

${\ displaystyle {\ hat {F}} \ left | \ phi _ {m} \ right \ rangle = {\ hat {h}} \ left | \ phi _ {m} \ right \ rangle + \ sum \ limits _ {\ gamma} ^ {N} \ left (\ left \ langle \ phi _ {\ gamma} \ right | {\ hat {w}} \ left | \ phi _ {\ gamma} \ right \ rangle \ left | \ phi _ {m} \ right \ rangle - \ left \ langle \ phi _ {\ gamma} \ right | {\ hat {w}} \ left | \ phi _ {m} \ right \ rangle \ left | \ phi _ {\ gamma} \ right \ rangle \ right) = \ varepsilon _ {m} \ left | \ phi _ {m} \ right \ rangle,}$

the Hartree-Fock equation with the Fock operator . The first two terms have a classic analogue. contains the kinetic energy and the Coulomb interaction with the nuclei. The second term can be interpreted as the mean Coulomb potential of all other electrons on the th electron. However, the instantaneous correlation of the particles is neglected. The Hartree-Fock method is therefore a mean field approach . The exchange term does not have a classic analogue. The Fock operator for the -th electron contains the wave functions of all other electrons, whereby the Fock equations mostly only with the method of self-consistent fields, i.e. H. iteratively using fixed point iteration . The DIIS method is often used to accelerate convergence . ${\ displaystyle {\ hat {F}}}$${\ displaystyle {\ hat {h}}}$${\ displaystyle m}$${\ displaystyle m}$

## Basic rates

Representation of various STO-NG orbitals and the associated original STO orbital

A direct numerical solution of the Hartree-Fock equation as a differential equation is possible for atoms and linear molecules. As a rule, however, the orbitals are analytically applied as linear combinations of basic functions (basic set), which in turn represents an approximation that becomes better the larger and more intelligent the basic set is chosen. Typically, each atom in the molecule brings with it a number of basis functions, which are determined by the corresponding basis set and centered on it. The analytical solutions of the hydrogen atom, which show a behavior for large distances between the nuclei, serve as a rough starting point for creating such basic sets . Approaches of this type are called Slater Type Orbital (STO). Mostly they have the shape ${\ displaystyle \ exp (- \ zeta r)}$${\ displaystyle r}$

${\ displaystyle \ left | \ varphi _ {i} ^ {k} \ right \ rangle = {\ text {Polynomial}} \ cdot \ exp \ left (- \ zeta _ {i} \ left | \ mathbf {r} - \ mathbf {R_ {k}} \ right | \ right)}$.

An orbital has e.g. B. the shape ${\ displaystyle p_ {z}}$

${\ displaystyle \ left | \ varphi _ {i} ^ {k} \ right \ rangle = (z-z_ {k}) \ cdot \ exp \ left (- \ zeta _ {i} \ left | \ mathbf {r } - \ mathbf {R_ {k}} \ right | \ right)}$.

The major disadvantage of the Slater-Type orbitals, however, is that the required matrix elements cannot generally be calculated analytically. That is why Gaussian Type Orbitals are used almost exclusively . H. Basic function of the form ${\ displaystyle \ left \ langle \ varphi _ {\ tau} \ left | \ langle \ varphi _ {\ nu} \ left | {\ hat {w}} \ right | \ varphi _ {\ eta} \ rangle \ right | \ varphi _ {\ gamma} \ right \ rangle}$

${\ displaystyle \ left | \ varphi _ {i} ^ {k} \ right \ rangle = {\ text {Polynomial}} \ cdot \ exp \ left (- \ zeta _ {i} (\ mathbf {r} - \ mathbf {R_ {k}}) ^ {2} \ right)}$.

The matrix elements can be calculated analytically. It is u. a. exploited the Gaussian Product Theorem , d. This means that the product of two Gaussian functions is again a Gaussian function. In order to better approximate the STOs, a basic function typically consists of several Gaussian functions with fixed parameters defined by the basic set (“contraction”). A simple basic set is e.g. B. the so-called STO-NG , which approximates Slater Type orbitals with Gaussian functions. The solution of the differential equation is reduced to the analytical calculation of integrals over these basic functions and the iterative solution of the generalized eigenvalue problem with the coefficients of the basic functions as parameters to be determined. ${\ displaystyle N}$

Frequently used basic sets are the pople and correlation consistent bases .

The energy calculated with the Hartree-Fock method never reaches the exact value, even if an infinitely large basis set were used. In this borderline case, the so-called Hartree-Fock limit is reached. The reason for this is that by using the averaged potential, the electron correlation , i.e. the exact interaction of the electrons with one another, is not recorded. In order to eliminate this flaw, methods have been developed that are able to record at least part of the electron correlation (see article Correlated calculations ). Coupled cluster methods and the Møller-Plesset perturbation theory , which are based on the solution of the Hartree-Fock method, are particularly important . Another very important method is density functional theory with hybrid functionals , in which the Hartree-Fock exchange is part of the exchange-correlation part of the density functional.

The Hartree-Fock method allows a good determination of their “rough” electronic structure for a large number of molecules. Therefore z. For example, the molecular orbitals can be used for qualitative considerations (e.g. in the case of frontier orbitals ). As a rule, the Hartree-Fock method delivers total electronic energies that correspond to the correct electronic energies up to 0.5% (for calculating energy differences, such as reaction energies, however, it can only be used to a very limited extent, as these are of the order of magnitude of the error), dipole moments which coincide to 20% with the real dipole moments, and very precise distributions of the electron density in the molecule. Because of these properties, Hartree-Fock calculations are often used as a starting point for the more precise calculations mentioned above.

Another advantage of the Hartree-Fock method is that the energy obtained represents an upper limit for the exact ground-state energy according to the principle of variation. By choosing more extensive basic sets, the calculated wave function can be systematically improved up to the so-called "Hartree-Fock limit". Such a systematic approach is not possible with density functional methods.