# Theoretical Framework¶

## Introduction¶

### Notation for Electronic Integrals¶

For one- and two-electron integrals we use chemists’ notation over spin orbitals χ and spatial Ψ orbitals:

(ab|cd) = \int d r_1 r_2 \Psi

## Hartree-Fock-Method¶

The Hartree-Fock total energy is given by:

E = \sum_{\mu\nu} P_{\nu\mu} H_{\mu\nu}^{core} + \frac{1}{2} \sum_{\mu\nu\lambda\sigma} (\mu\nu||\sigma\lambda) + V_{NN}

### The Canonical Hartree-Fock Equations¶

The electronic Hartree-Fock energy E^{\rm RHF} of a n-electron system is given by: E^{\rm RHF} = \sum_{i}^{n} h_{ii} + \frac{1}{2} \sum_{i}^{n} \sum_{j}^{n} \left( 2 J_{ij} - K_{ij} \right)

The operator h_{ii} consists of the kinetic energy of all electrons and the interactions of each electron with all nuclei.

The Coulomb integral $J_{ij}$ describes the electron repulsion:

J_{ij} = \int \psi_i^* (1)\psi_j^* (2) \frac{1}{r_{12}} \psi_i (1) \psi_j (2) dr_1 dr_2 = ( \psi_i \psi_i | \psi_j \psi_j)

The non-classical Exchange integral K_{ij} and arises from the anti-symmetry of the wave function:

K_{ij} = \int \psi_i^*(1)\psi_j^*(2)\frac{1}{r_{12}}\psi_j(1)\psi_i(2) dr_1 dr_2 = (\psi_i\psi_j\vert\psi_j\psi_i)

Through unitary transformation a set of orbitals can be derived $\rangle{\psi_i^{\prime}}$, that diagonalizes the Lagrange-multipliers. The eigenvalues are orbital energies:

F \vert {\psi^{\prime}_{i}} \rangle = \epsilon_i \vert {\psi^{\prime}_{j}} \rangle

These equations are the so called canonical Hartree-Fock equations. The canonical orbitals make a physical interpretetation of orbital energies possible.

The so called Fock operator F is given by:

F = \hat{h}_{i} + \sum_{j=1}^{n/2} \left( 2 J_{j} - K_{j} \right)

The sum of all kinetic energy and electron-nucei attraction operators \hat{h}_{i}

The Coulomb operator J und der Exchange operator K are defined as:

J_j \psi_i(2) = \int \psi_j^{*}(1) \frac{1}{r_{12}} \psi_j(1) dr_1 \psi_i(2)

and

K_j \psi_i(2) = \int \psi_j^{*}(1) \frac{1}{r_{12}} \psi_i(1) dr_1 \psi_j(2)

\frac{\partial E}{\partial X_A} = \sum_{\mu\nu} P_{\nu\mu} \frac{\partial H^{core}}{\partial X_A} + \frac{1}{2} \sum_{\mu\nu\lambda\sigma} P_{\mu\nu} P_{\lambda\sigma} + \frac{\partial (\mu\nu||\sigma\lambda)}{\partial X_A} - \sum_{\mu\nu} Q_{\nu\mu} \frac{\partial S_{\mu\nu}}{\partial X_A} + \frac{\partial V_{NN}}{\partial X_A}

## The Cyclic Cluster Model¶

The Cyclic Cluster Model (CCM) directly applies periodic boundary conditions (PBC) to a finite free cluster corresponding to a non-primitive unit cell. The interaction range of every atom within the cluster is defined by its Wigner-Seitz Cell (WSC) defined by the translation vectors.

In contrast to conventional periodic models, the CCM is a Γ point approach and integration is carried out in real space.

The challenge for the development of the CCM at ab initio level is the treatment of three- and four-center integrals.Special attention has to be paid to atoms at the border of WSCs, interactions have to be weighted correctly.