Self-consistent field method

The self-consistent field method (SCF) is an iterative solution method in computational chemistry . SCF is one of many ways to solve molecular calculations. The self-consistent-field is a set of functions that solves an eigenvalue equation .

HF-SCF

If the self-consistent field method (SCF) is used to solve the Hartree-Fock equation in the Hartree-Fock method (HF), the functions obtained are referred to as SCF orbitals.

The Hartree-Fock equation is:

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

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

${\displaystyle {\hat {F}}_{i}={\hat {h}}_{i}+\sum \limits _{j=1}^{N_{\text{Elektr.}}}\left(2{\hat {J}}_{j}-{\hat {K}}_{j}\right)}$

The Fock operator is an effective one-electron operator that describes the kinetic energy of an electron , the Coulomb attraction with the nucleus ( ) and the Coulomb repulsion with all other electrons ( and ). and describe the mean value-like interaction with all other particles as a local Coulomb operator and exchange operator. The Fock operator thus depends on its own eigenfunctions (since all occupied orbitals must be known), the Hartree-Fock equation is thus non-linear, which makes an iterative solution necessary. For further mathematics, please refer to the article on the Hartree-Fock method. ${\displaystyle i}$ ${\displaystyle {\hat {h}}_{i}}$ ${\displaystyle j}$ ${\displaystyle {\hat {J}}_{j}}$ ${\displaystyle {\hat {K}}_{j}}$ ${\displaystyle {\hat {J}}_{j}}$ ${\displaystyle {\hat {K}}_{j}}$

SCF is not limited to use in the Hartree-Fock method.

The solutions of the SCF run in HF are the optimized orbitals as eigenfunctions of the Fock operator with the associated eigenenergies as orbital energies . ${\displaystyle \phi _{m}}$ ${\displaystyle \varepsilon _{m}}$

SCF process in HF

Since the Fock matrix depends on its own solutions, it has to be solved iteratively, this is done with the SCF method. A typical SCF process looks like this:

1. Calculation of all one and two electron integrals.
2. Generation of suitable MO coefficients.
3. Formation of the initial density matrix.
4. Formation of the Fock matrix as one-electron integrals and the density matrix times the two-electron integrals.
5. Diagonalization of the Fock matrix. As a result, the eigenvectors contain the new MO coefficients.
6. Formation of a new density matrix. If the new density matrix and the initial density matrix are the same, the SCF run can be ended; if not, the cycle starts again at step 4 with the new density matrix.

Generation of suitable MO coefficients

The type of MO coefficients initially generated is also responsible for the convergence of SCF runs. The source of suitable MO coefficients are often the results from simpler methods (e.g. semiempirism (e.g. Extended Hückel method )) or the results of previous calculations using the same method but a smaller base set.

convergence

In computational chemistry, convergence is the term used to describe the calculated energies remaining almost constant with each new calculation run. There is no guarantee that SCF passes will converge. For small basis sets and molecular geometries near the minimum, SCF converges relatively unproblematically. Geometries of molecules in transition states, large basis sets with diffuse functions and metal complexes often make convergence considerably more difficult.

There are several possibilities for SCF to converge. For example, a new Fock matrix can be extrapolated from older Fock matrices (usually the last three) (English: extrapolation ). The formation of the density matrix is ​​skipped here. Other methods are:

• Damping - A strategy for oscillating SCF results. A weighted average from previously calculated density matrices is used to overcome the oscillation.
• Level shifting - use of excited orbitals in the Fock matrix.
• Direct inversion in the iterative subspace (DIIS) - generation of an error function from "old" Fock / density matrices, which forces new cycles to converge.
• ...

Individual evidence

1. ^ ^{A b c d} Jensen, Frank: Introduction to computational chemistry . Third ed. Chichester West Sussex, UK, ISBN 978-1-118-82599-0 , pp. 91-92 .
2. ^ Jensen, Frank: Introduction to computational chemistry . Third ed. Chichester West Sussex, UK, ISBN 978-1-118-82599-0 , pp. 100 .
3. ^ ^{A b c} Jensen, Frank: Introduction to computational chemistry . Third ed. Chichester West Sussex, UK, ISBN 978-1-118-82599-0 , pp. 101-105 .