https://www.quanty.org/ 2026-04-14T16:36:36+00:00 https://www.quanty.org/ https://www.quanty.org/_media/wiki/dokuwiki.svg text/html 2025-11-20T01:45:25+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.quanty.org/documentation/standard_operators/atomic_coulomb_repulsion?rev=1763603125&do=diff Atomic coulomb repulsion Note: parts of the functionality of this function will be available in the release of October 2019 NewOperator(“AtomicU”,indices,RadialWavefunctions1,RadialWavefunctions2,options) calculates the full Coulomb interaction operator between all given atomic orbitals.standard_operators index text/html 2025-11-20T01:45:24+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.quanty.org/documentation/standard_operators/coulomb_repulsion?rev=1763603124&do=diff Coulomb repulsion operator (U) The Coulomb interaction is given as: \begin{equation} H = \sum_{i\neq j} \frac{1}{2} \frac{e^2}{|r_i-r_j|}, \end{equation} whereby the sum runs over all electrons and the factor $1/2$ takes care of the double counting as each pair of electrons only repels once. In second quantization this Hamiltonian can be written as:\begin{equation} H = \sum_{\tau_1\tau_2\tau_3\tau_4} U_{\tau_1\tau_2\tau_3\tau_4} a^{\dagger}_{\tau_1}a^{\dagger}_{\tau_2}a^{\phantom{\dagger}}_{\tau… text/html 2025-11-20T01:45:24+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.quanty.org/documentation/standard_operators/crystal_field?rev=1763603124&do=diff Crystal field operator In Crystal-field theory the interactions of a local atom with its environment (covalent bonding) is approximated by an effective potential. These potentials do not really exist in a solid, but should be seen as effective fields describing the (anti-) bonding states of an atom. We assume that the local spin-orbitals are given by a radial equation times a spherical harmonic describing the angular part.\begin{equation} \varphi_{n,l,m}(\vec{r}) = R_{n,l}(r) Y_{l,m}(\theta,\phi… text/html 2025-11-20T01:45:24+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.quanty.org/documentation/standard_operators/ligand_field?rev=1763603124&do=diff Ligand field In ligand field theory the covalent interaction between a correlated (d) shell and a ligand shell is explicitly taken into account by adding hopping between the correlated (d) shell and a ligand shell. This hopping can be seen as an effective potential coupling two different shells. The potential can similar as in the case of crystal field theory be expanded on spherical harmonics and parameterized by $A_{k,m}$$\Delta$$\Delta$$\epsilon_d - \epsilon_p$$U$$U \neq F[0]$$U = F[0] - (2/6… text/html 2025-11-20T01:45:25+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.quanty.org/documentation/standard_operators/spin_orbit_coupling?rev=1763603125&do=diff Spin orbit coupling operator (l.s) The spin-orbit interaction is defined as: \begin{equation} \xi \sum_i l_i \cdot s_i, \end{equation} with $l_i$ and $s_i$ the one electron orbital and spin operators respectively and the sum over $i$ summing over all electrons. The prefactor $\xi$ is an atom dependent constant, which is to a good approximation material independent and given as:\begin{equation} \xi = \left\langle R(r) \left| \frac{1}{2m^2c^2}\frac{1}{r}\frac{\mathrm{d}V(r)}{\mathrm{d} r} \right| … text/html 2025-11-20T01:45:24+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.quanty.org/documentation/standard_operators/start?rev=1763603124&do=diff Standard Operators Several standard operators are defined. Once specific spin-orbitals are grouped in shells and we assigned quantum numbers to them operators can be created. The most obvious example is to relate a set of spin-orbitals to an atomic like shell with a radial wave-function times an angular dependent part that is given by the spherical Harmonics. In this case we can talk about the angular momentum and Coulomb interaction in terms of Slater integrals. Although for real molecules and … text/html 2025-11-20T01:45:24+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.quanty.org/documentation/standard_operators/transition_operators_for_photon_spectroscopy?rev=1763603124&do=diff Transition operators for photon spectroscopy In order to calculate excitation spectra one needs to define a transition operator. In principle any operator can be written in second quantization, making it possible to create many different kinds of excitation spectroscopy. In practice we very often want to calculate absorption spectroscopy due to dipole transitions. In this case one can use the crystal-field operator to quickly create the transition operator wanted.\begin{equation} H = H_0 + \frac…