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)

Anaytical Gradients

\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}

Density Functional Theory

Please add the basic principles of DFT here

Density Functional Based Tight Binding (DFTB)

Please add the basic principles of DFTB here

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.