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: .. math:: (ab|cd) = \int d r_1 r_2 \Psi Hartree-Fock-Method ******************* The Hartree-Fock total energy is given by: .. math:: 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 :math:E^{\rm RHF} of a n-electron system is given by: :math: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 :math:h_{ii} consists of the kinetic energy of all electrons and the interactions of each electron with all nuclei. The Coulomb integral :math:$J_{ij}$ describes the electron repulsion: .. math:: 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 :math:K_{ij} and arises from the anti-symmetry of the wave function: .. math:: 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 :math:$\rangle{\psi_i^{\prime}}$, that diagonalizes the Lagrange-multipliers. The eigenvalues are orbital energies: .. math:: 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 :math:F is given by: .. math:: 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 :math:\hat{h}_{i} The Coulomb operator :math:J und der Exchange operator :math:K are defined as: .. math:: J_j \psi_i(2) = \int \psi_j^{*}(1) \frac{1}{r_{12}} \psi_j(1) dr_1 \psi_i(2) and .. math:: K_j \psi_i(2) = \int \psi_j^{*}(1) \frac{1}{r_{12}} \psi_i(1) dr_1 \psi_j(2) Anaytical Gradients =================== .. math:: \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.