Jib operator

The Fock operator is an effective one-electron operator . The Fock operator is made up of the one-particle Hamiltonian for the -th electron and the two-electron operators (Coulomb and exchange operator). In the case of a closed-shell system (all spins are paired) the Fock operator is: ${\displaystyle i}$

${\displaystyle {\hat {F}}[\{\phi _{j}\}](i)={\hat {H}}^{\text{core}}(i)+\sum _{j=1}^{N/2}[2{\hat {J}}_{j}(i)-{\hat {K}}_{j}(i)]}$

The Fock operator generated from the -orbital is for the -th electron. is the one-particle Hamiltonian for the -th electron: ${\displaystyle {\hat {F}}[\{\phi _{j}\}](i)}$ ${\displaystyle \phi _{j}}$ ${\displaystyle i}$ ${\displaystyle {\hat {H}}^{\text{core}}(i)}$ ${\displaystyle i}$

${\displaystyle {\hat {H}}^{\text{core}}(i)=-{\frac {\hbar ^{2}}{2m}}\nabla _{i}^{2}-{\frac {e^{2}}{4\pi \varepsilon _{0}}}\sum _{k}{\frac {Z_{k}}{r_{ik}}}}$

with the electron mass , Planck's quantum of action , the elementary charge and the electric field constant . ${\displaystyle m}$ ${\displaystyle \hbar }$ ${\displaystyle e}$ ${\displaystyle \varepsilon _{0}}$

In the atomic units commonly used in theoretical chemistry , the Hamilton operator is simplified, since all constants that occur are set equal to one: ${\displaystyle \hbar ,m,e,4\pi \varepsilon _{0}}$

${\displaystyle {\hat {H}}^{\text{core}}(i)=-{\frac {1}{2}}\nabla _{i}^{2}-\sum _{k}{\frac {Z_{k}}{r_{ik}}}}$

The first part of the operator describes the kinetic energy of the -th electron, the second part is the sum of the electron – nucleus Coulomb attraction of the -th electron with the nucleus (which has the charge number ) with the distance of the -th electron from the nucleus . ${\displaystyle i}$ ${\displaystyle i}$ ${\displaystyle k}$ ${\displaystyle Z_{k}}$ ${\displaystyle r_{ik}}$ ${\displaystyle i}$ ${\displaystyle k}$

The Coulomb operator defines the electron-electron repulsion energy of the -th electron with the electron in the j-th orbital. is the exchange operator that defines the electron exchange energy due to the antisymmetry of the many electron wave function, it is an artifact of the Slater determinant . ${\displaystyle {\hat {J}}_{j}(i)}$ ${\displaystyle i}$ ${\displaystyle {\hat {K}}_{j}(i)}$

Calculation of the Hartree-Fock one-electron wave function

Calculating the Hartree-Fock one-electron wave function is now equivalent to solving the eigenvalue equation :

${\displaystyle {\hat {F}}(i)\phi _{n}(i)=\epsilon _{n}\phi _{n}(i)}$

${\displaystyle \phi _{n}\,(i)}$describes the wave function of the -th electron in the -th orbital, they are also referred to as Hartree-Fock molecular orbitals. ${\displaystyle i}$ ${\displaystyle n}$

Since the Fock operator is a one electron operator, it does not contain the electron correlation energy.

Relationship with the total Hamilton operator

The total Hamilton operator can be approximated by a sum of Fock operators:

${\displaystyle {\hat {H}}_{0}=2\sum _{i=1}^{N/2}{\hat {F}}(i)}$

Individual evidence

1. ↑ ^{a b} Ira N. Levine: Quantum Chemistry . 4th ed. Prentice Hall, Englewood Cliffs NJ 1991, p. 403.
2. ↑ ^{a b c d e} Bernd Hartke: I: Quantenchemie. (PDF) In: Theoretical Chemistry. Retrieved July 23, 2018 .