multiplet states of atoms as constituents of compounds

Atomic Orbitals

Atomic orbitals provide a mathematical description of atomic quantum mechanical states. An atomic orbital is the quantum mechanical wave function describing one electron. It is a solution to the non-relativistic Schrödinger equation, or the relativistic Dirac equation, for one electron. The quantum numbers of the orbitals are the basis to define configurations of atomic states involving several electrons.

Configurations

The configuration specifies the occupation of an atomic shell characterized by the principal quantum number and the angular quantum number of an orbital, in particular the partly filled, open, shells as opposed to the completely filled, closed, shells and the empty shells a consequence of Fermi Dirac statistics applying to electrons.

Slater Determinants

The Pauli principle requires that for atoms with several electrons, valid solutions must form anti-symmetric functions. Such an antisymmetric function can be realized as a Slater determinant formed from the atomic orbitals. Atomic many electron states can be described as wavefunctions composed of usually many such Slater determinants.

Multiplet States

Generally these atomic states are called multiplet states. In the case of free atoms, the multiplet states and the multiplicity of degenerate states can be denoted by their symmetry properties. In the case of light atoms, the wavefunction can be considered as approximately L-S coupled and the ground state can be inferred by Hund's rules. States of atoms forming part of a compound become more or less perturbed by its environment to which it is chemically bonded in any compound. The localized d- and f- electrons of transition metals, lanthanide and actinide atoms are only lightly perturbed by the chemical bonding. The leads to a rich and complex spectrum of low lying atomic states which is at the origin of the interesting and useful magnetic and optical properties of such compounds.

multiX

The multiX [Uldry et al 2012] method is an self-contained, easily accessible ligand-field/crystal-field approach to calculate XAS XMCD XLD RIXS and inelastic neutron spectra (INS) currently. Ground state properties including anisotropic magnetic properties of the ion can also be analyzed using multiX.
multiX works in the restricted active space of the core hole shell and the local orbital valence shell as a self contained first principles code. This aspect calculates the key matrix elements for electron-electron interaction, relativistic effects including spin-orbit splitting and single electron orbitals with minimal input. The atom label and the specification of the core hole shell and the partially occupied valence shell is enough input to define the multi-determinant active space based on Dirac relativistic atomic orbitals from the built in density functional atomic code. This alone produces rough XAS spectra with reasonable excitation energy and SO splitting.

multiX ligand field / crystal field model

Orbitals and atomic states with the proper orientation are conveniently generated by specifying a quasi crystal field as a point charge model of the molecular or crystal environment. The charges are considered as semi-empirical parameters to define the effective ligand field, which can be adjusted semi-empirically to match a corresponding experimental spectrum. An accurate electric crystal field should not be expected from a point charge model with conventional valence charges. And even an accurate crystal field would miss the effects of hybridization on the single electron resonances, which crucially enter the ligand field-model as single electron levels. The crystal field really offers an easily understandable way to parametrize the on-site ligand field with the proper symmetry and orientation. As small number of neighbor shells with their charge parameters is sufficient to model all possible splittings of the localized open valence shell. The simple connection of the Hamiltonian with the Cartesian coordinates of the crystal environment is useful to specify polarization directions. multiX handles the full Hamiltonian in the restricted active space without recourse to symmetry, so any low symmetry environment can be treated with equal ease.
The code automatically switches into Lanczos type spectral calculations for XAS XMCD XLD (RIXS), when this method is expected to be faster.
The finite active space of the open localized shells in the leads to a multiplet line spectrum. For a useful comparision with measured spectra it is unavoidable to include the effect of an intractable number of further states in a semi-empirical fashion. The coupling into states continuum states limiting the core hole lifetime is modeled by Lorentzian lifetime broadening. The energy dependence above the threshold is significant for most comparisons. The coupling with states responsible for screening is modeled semi-empirically by scaling the first principles parameters for electron-electron interaction, S-O splitting and by introducing a threshold correction for the Xray absorption edge. As emphasized above, the coupling of the local valence orbitals with the ligand states tends to become intractable in a seriously first principles manner. So multiX allows to adjust the open-shell single electron resonances by considering point charges as semi-empirical parameters.

multiX history

An early realization that atomic-like multiplets show up prominently in some types of core hole spectroscopies came with the investigation of Moser et al 1984. Model calculations predating the multiX development focus on the interplay of the localized (multiplet) levels with the continuum of band states based on the Anderson model. As summary is in [Delley 1990]. Multifaceted use of multiX is illustrated in a number of articles: An article predating the multiX method publication compares the excitations of the closed shell Ti 4+ ion as calculated with TDDFT to the multi X multiplet calculation. For this case of a non-degenerate ground state, scalar-relativistic TDDFT generates the LS coupled multiplet qualitatively correct and in similarly good agreement with experiment as the LS limit for multiX [Delley, 2010]. A study on iron pnictides shows among other findings, that the 2p XAS and also RIXS is represented well by the crystal/ligand field multiplet [Yang et al 2009]. Combined XAS RIXS studies for the Fe L2/3 edge are presented by [Monney et al 2013]. Mn based magnetism experiments in strained lattices with circularly and linearly 2p XAS and XMCD is analysed with multiX [Heidler et al., 2015]. Magnetic ground-states were studied by circularly polarized XAS [Westerström et al 2015] and with XAS,XMCD on rare earth elements in molecules were studied by [Wäckerlin et al 2015] and [Dreiser et al 2016]. Magnetic groundstates of RE-elements adsorbed on graphene were investigated by XAS, XMCD, XLD [Baltic et al 2018]. A multiX application to inelastic neutron scattering on a rare earth element is given in [Fennel et al 2014], [Gauthier et al 2017] and [Prevost et al 2018]. A recent review on 2p XAS also has a subsection on multiX [DeGroot 2021].

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