Ab Initio Model Potential Research Articles

An embedded cluster CASPT2 study of the Ce:YVO4 spectrum.

Multiconfigurational theory, in combination with the embedded cluster approach, is a precise and abinitio approach to describe the electronic structure of solids. In this work, the spectrum of a Ce(III) dopant in YVO4 has been studied by complete active space perturbation theory of the second order (CASPT2), with the host material represented as a set of abinitio model potentials and point-charges. We assess the sensitivity of the spectrum to the size of both the embedded cluster size as well as the size of the electronic basis set. A comparison of our best computational model with experimental results shows that the embedding approach is robust and can accurately model the spectrum of low-concentration dopants in complex host materials.

Electron Trap Depths in Cubic Lutetium Oxide Doped with Pr and Ti, Zr or Hf─From Ab Initio Multiconfigurational Calculations.

We propose a universal approach to model intervalence charge transfer (IVCT) and metal-to-metal charge transfer (MMCT) transitions between ions in solids. The approach relies on already well-known and reliable ab initio RASSCF/CASPT2/RASSI-SO calculations for a series of emission center coordination geometries (restricted active space self-consistent field, complete active space second-order perturbation theory, and restricted active space state interaction with spin-orbit coupling). Embedding with ab initio model potentials (AIMPs) is used to represent the crystal lattice. We propose a way to construct the geometries via interpolation of the coordinates obtained using solid-state density functional theory (DFT) calculations for the structures where the activator metal is at specific oxidation (charge) states of interest. The approach thus takes the best of two worlds: the precision of the embedded cluster calculations (including localized excited states) and the geometries from DFT, where the effects of ionic radii mismatch (and eventual nearby defects) can be modeled explicitly. The method is applied to the Pr activator and Ti, Zr, Hf codopants in cubic Lu2O3, in which the said ions are used to obtain energy storage and thermoluminescence properties. Electron trap charging and discharging mechanisms (not involving a conduction band) are discussed in the context of the IVCT and MMCT role in them. Trap depths and trap quenching pathways are analyzed.

Convergence of Electronic Structure Properties in Ionic Oxides Within a Fragment Approach

Embedded-cluster models of crystalline solids are important to allow accurate wave function methods to be applicable to solids. The ab-initio model potential method, in which the crystal is divided into three different fragments, one quantum fragment, one ab-initio model potential fragment and one point-charge fragment, has historically been shown to be a viable tool for describing the electronic structure in ionic solids. The optimal size of these regions is, of course, individual for each crystal. In this study we analyzed the convergence of the electronic structure properties with respect to an increase of the size of the quantum part and the layer of potentials. MgO crystal and Ni: MgO were used for this purpose as examples of an ideal crystal and a crystal with a point defect. We demonstrated that with an increase of the cluster size, the electron density in the inner part of the cluster becomes very similar to the electron density in the periodic model. Clusters, embedded into a layer of model potential and electrostatic field, are a good alternative to periodic description.

A program system for self-consistent embedded potentials for ionic crystals

Embedded-cluster models of crystalline solids are important to allow accurate wave function methods to be applicable to solids. The ab initio model potential method for embedding ionic solids has historically been shown to be a viable tool. While useful, the method has been limited by the need to generate such potentials for each crystal structure and the lack of a freely available program for generating ab initio model potentials. Herein, this is remedied by showcasing a new, AFL licensed, program, SCEPIC, which can be used in combination with Molcas or OpenMolcas codes to derive ab initio model potentials for ionic crystals.The applicability of ab initio model potentials derived via SCEPIC is evaluated for three simple ionic solids: MgO, CaO and CaF2. The following questions are addressed: (i) the capability of the method to reproduce the density matrix from periodic density functional theory calculations, (ii) the feasibility of performing geometry optimisations, (iii) the possibility to model band gaps of insulators and (iv) the ligand-field splitting of Ni-doped MgO. Going beyond the classical restriction of parametrising ab initio model potentials only at the Hartree–Fock level-of-theory, this work additionally address the sensitivity of results to the underlying Hamiltonian used to derive the potentials. The results demonstrate that good agreement with periodic density functional theory calculations can be achieved, geometry optimisations are feasible and accurate band gaps and ligand-field splittings can be computed.

Exploring the Rich Potential Energy Surface of (H2O)11 and Its Physical Implications.

The rich potential energy surface of the water undecamer (H2O)11 was explored with a basin hopping algorithm using a TIP4P potential and other methods followed by extensive ab initio MP2 minimizations and CCSD(T) corrections. This protocol yielded 17, 66, and 125 distinct isomers within 0.5, 1.0, and 2.0 kcal mol-1 of the complete basis set CCSD(T) global minimum, respectively. These isomers were categorized into 15 different families based on their oxygen framework and hydrogen bonding topology. Determination of the global minimum proved challenging because of the presence of many nearly isoenergetic isomers. The predicted global minimum varied among ab initio methods, density functionals, and model potentials, and it was sensitive to the choice of energy extrapolation schemes, higher-order CCSD(T) corrections, and inclusion of zero-point vibrational energy. The presence of a large number of nearly degenerate structures and the isomerization between them has manifested itself in the anomalous broadening of the heat capacity curve of the undecamer in simulations around the melting region.

Ab initio theoretical study of the [formula omitted] and 4f75d manifolds of Tb3+-doped BaF2 cubic sites

An ab initio model potential embedded cluster study on the 4f8 and 4f7 5d manifolds of the long range compensated, cubic, substitutional Tb3+ defects of Tb-doped BaF2 is presented here. 4f–4f and 4f–5d transition energies and 4f–5d absorption electric dipole oscillator strengths for the dipole allowed transitions are reported and comparisons with emission and excitation experimental spectra are presented. We find good agreement with the energies of the experimental 4f–4f green and blue emissions. The characteristic features of the 4f–5d excitation spectrum (spin allowed vs. spin forbidden transitions) are well reproduced by the calculations. The transition energies from the ground state to the different spin nonet and spin septet levels of the 4f7 5d configuration of this ion in the center of the lanthanide series show an error of around 10%, which is in line with previous calculations on other lanthanide ions with lower spin multiplicities and simpler manifolds at the beginning and at the end of the series. The calculation supports a recent assignment of experimental excitation bands between 54,000 and 66,000cm−1 as 4f→5d(t2g) transitions. On the other hand, a new assignment is made of the highest band in the excitation spectrum at around 76,000cm−1: it corresponds to transitions to levels of the 4f7 5d configuration with the 4f7 subshell in the excited 6P term, rather than to a Tb-bound exciton. Preliminary studies on excitonic states of Tb-doped BaF2 indicate that energy levels that can be characterized as Tb-trapped excitons are found in the same spectral region as the spin forbidden 4f→5d(t2g) transitions at ground state Tb–F distances, and their energy lowers very much at shorter Tb–F distances.

Ab initio and DFT analysis of the low-lying electronic states of metal dihalides: quantum chemical calculations on the neutral BrMCl (M = Cu, Ag, Au)

The electronic configuration of the electronic ground and low-lying doublet excited states of neutral metal dihalides BrMCl (M = Cu, Ag, Au) has been investigated on the basis of CASSCF/CASPT2 methods taking into account scalar relativistic effects. A preliminary study of the electronic problem in BrAgCl, based on DFT and CASSCF/CASPT2 approaches and using various basis sets, namely relativistic all-electron basis sets, effective core potentials and ab initio model potentials (AIMP), as well as non-relativistic AIMP is discussed. It is shown that single-determinant methods are not flexible enough to describe the bonding of the neutral species in the electronic ground state regardless of the basis set. The failure to allocate the single electron of BrAgCl correlates with a wrong charge distribution within the complex, which is more accentuated when using pseudopotential basis sets. The inclusion of static and dynamic correlation effects by means of CASSCF/CASPT2 methods using large relativistic all-electron basis sets provides a correct qualitative picture of the electronic structure of the BrMCl series (M = Cu, Ag, Au). The spin unrestricted KS-DFT approach leads to a reasonable description of the degenerate electronic ground state ((2)Σ/(2)Π) bonding in these complexes with negligible spin contamination providing comparative spin densities in the series of molecules under investigation.

Charge Transfer in Uranyl(VI) Halides [UO2X4]2− (X = F, Cl, Br, and I). A Quantum Chemical Study of the Absorption Spectra

The electronic spectra of uranyl(VI) coordinated with four equatorial halide ligands, [UO2X4]2- (X = F, Cl, Br, and I), have been calculated at the all-electron level using the multiconfigurational CASPT2 method, with spin-orbit coupling included through the variational-perturbational method. The halide-to-uranyl charge-transfer states were taken into account in the calculation by including ligand orbitals in the active space. In order to do that, it is assumed that the charge transfer takes place from only one of the four ligands. Two models, which in principle can describe this, were investigated: the first one makes use of a localizing technique and the second one replaces three ligands by ab initio model potentials (AIMPs). The basis set dependence was investigated by using two different basis sets for the halides, of triple-zeta and quadruple-zeta quality. The localization procedure turned out to be strongly basis set dependent, and the most stable results were obtained with ab initio model potentials. The ground state is a closed shell singlet state, and the first excitation is from the bonding sigma(u) orbital on uranyl to the nonbonding delta(u) orbitals, except for the [UO2I4]2- complex, where the first excited state has a mixed character of charge transfer from the I- and the sigma(u)(1)phi(u)(1) configuration. In [UO2F4]2- there is no charge transfer excitation below 50,000 cm(-1) , while in [UO2Cl4]2- it appears around 33,000 cm(-1) and in [UO2Br4]2- around 23,000 cm(-1) . A blueshift of the spectra, from F- to I-, is observed. The calculations compare reasonably well with available experimental results.

Improved Embedding Ab Initio Model Potentials for Embedded Cluster Calculations

An improvement in the method of production of embedding ab initio model potentials (AIMP) for embedded cluster calculations in ionic solids is proposed and applied to the oxides CeAlO(3), CeO(2), and UO(2). The improvement affects the calculation of one of the AIMP components, the Pauli repulsion operator, which prevents the cluster electrons from collapsing onto the occupied orbitals of the host in embedded cluster calculations and, so, their over occupancy. The linear constants involved in such operator are proposed to be obtained in embedded cluster calculations in the perfect host, with the requirement that local structures calculated with working embedded clusters of relatively small size agree with those calculated with reference embedded clusters of much larger size.

Penning ionization electron spectra of pyrene, chrysene, and coronene in collision with metastable He(2 3S) atoms in the gas phase

Penning ionization electron spectra (PIES) of pyrene (C(16)H(10)), chrysene (C(18)H(12)), and coronene (C(24)H(12)) in the gas phase are recorded using metastable He(*)(2 (3)S) atoms. The assignments of PIES are presented based on the outer valence Green's function calculations with the correlation consistent polarized valence triple-zeta basis sets and the exterior electron density calculations of contributing molecular orbitals. The definite positions of all of the pi bands in the PIES are identified making use of the large PIES cross sections. Broad bands are observed in low-electron-energy regions for chrysene and coronene and are ascribed to ionization processes of non-Koopmans types from sigma orbitals. The anisotropic interaction potential energy surfaces for the colliding systems are obtained from ab initio model potentials for the related systems with similar outer valences Li(2 (2)S) + C(16)H(10), C(18)H(12), and C(24)H(12), respectively. The attractive well depths in the out-of-plane directions are found to be similar between these molecules, and the repulsive walls embrace the in-plane perimeters uniformly. Collision energy dependencies for partial Penning ionization cross sections and negative peak shifts in PIES for chrysene support these anisotropic interactions. Effects from thermal populations in low-frequency vibrational modes are estimated to be minor in one-electron ionization processes.

Embedding Fragment ab Initio Model Potentials in CASSCF/CASPT2 Calculations of Doped Solids: Implementation and Applications.

In this article, we present a fragment model potential approach for the description of the crystalline environment as an extension of the use of embedding ab initio model potentials (AIMPs). The biggest limitation of the embedding AIMP method is the spherical nature of its model potentials. This poses problems as soon as the method is applied to crystals containing strongly covalently bonded structures with highly nonspherical electron densities. The newly proposed method addresses this problem by keeping the full electron density as its model potential, thus allowing one to group sets of covalently bonded atoms into fragments. The implementation in the MOLCAS 7.0 quantum chemistry package of the new method, which we call the embedding fragment ab inito model potential method (embedding FAIMP), is reported here, together with results of CASSCF/CASPT2 calculations. The developed methodology is applied for two test problems: (i) the investigation of the lowest ligand field states (2)A1 and (2)B1 of the Cr(V) defect in the YVO4 crystal and (ii) the investigation of the lowest ligand field and ligand-metal charge transfer (LMCT) states at the Mn(II) substitutional impurity doped into CaCO3. Comparison with similar calculations involving AIMPs for all environmental atoms, including those from covalently bounded units, shows that the FAIMP treatment of the YVO4 units surrounding the CrO4(3-) cluster increases the excitation energy (2)B1 → (2)A1 by ca. 1000 cm(-1) at the CASSCF level of calculation. In the case of the Mn(CO3)6(10-) cluster, the FAIMP treatment of the CO3(2-) units of the environment give smaller corrections, of ca. 100 cm(-1), for the ligand-field excitation energies, which is explained by the larger ligands of this cluster. However, the correction for the energy of the lowest LMCT transition is found to be ca. 600 cm(-1) for the CASSCF and ca. 1300 cm(-1) for the CASPT2 calculation.

Geometry and electronic structure of impurity-trapped excitons in Cs2GeF6:U4+ crystals. The 5f17s1 manifold

Excitons trapped at impurity centers in highly ionic crystals were first described by McClure and Pedrini [Phys. Rev. B 32, 8465 (1985)] as excited states consisting of a bound electron-hole pair with the hole localized on the impurity and the electron on nearby lattice sites, and a very short impurity-ligand bond length. In this work the authors present a detailed microscopic characterization of impurity-trapped excitons in U(4+)-doped Cs(2)GeF(6). Their electronic structure has been studied by means of relativistic ab initio model potential embedded cluster calculations on (UF(6))(2-) and (UF(6)Cs(8))(6+) clusters embedded in Cs(2)GeF(6), in combination with correlation methods based on multireference wave functions. The local geometry of the impurity-trapped excitons, their potential energy curves, and their multielectronic wave functions have been obtained as direct, nonempirical results of the methods. The calculated excited states appear to be significantly delocalized outside the UF(6) volume and their U-F bond length turns out to be very short, closer to that of a pentavalent uranium defect than to that of a tetravalent uranium defect. The wave functions of these excited states show a dominant U 5f(1)7s(1) configuration character. This result has never been anticipated by simpler models and reveals the unprecedented ability of diffuse orbitals of f-element impurities to act as electron traps in ionic crystals.

The [formula omitted] luminescence spectrum of [formula omitted] in [formula omitted]2GeF 6 crystals—A quantum chemical study

The 5 f 1 6 d ( t 2 g ) 1 emission spectrum of Cs 2GeF 6:U 4+ single crystals has been calculated using the relativistic ab initio model potential (AIMP) embedded cluster method. It consists of a series of broad, intense bands in the UV followed by broad and increasingly weaker bands in the visible and near infrared spectral range, which appear to be associated with emissions from the lowest 5 f 1 6 d ( t 2 g ) 1 excited state, 1 E u , to a number of levels of the 5 f 2 manifold. The electric dipole allowed emission of highest energy, 1 E u → 1 T 1 g , is found to have very low oscillator strength, - 0.115 × 10 - 2 ; this value is only 4% of the most intense electronic transition: 1 E u → 1 T 2 g whose electronic origin is found to be at 34 200 cm - 1 . A very large Stokes shift of 6200 cm - 1 is obtained, which should result in low luminescence quenching due to reabsorption and should favourably contribute to the emission quantum efficiency. The lifetime of the spontaneous emission from 1 E u has been calculated to be in the nanoseconds range: 33 ns.

Large anomalies due to insufficiency of Madelung embedding inab initiocalculations of4f−5dand4f−6sexcitations of lanthanides in ionic crystals: TheBaF2:Ce3+crystal

Ab initio embedded-cluster calculations are performed on the ${(\mathrm{Ce}{\mathrm{F}}_{8})}^{5\ensuremath{-}}$ cluster embedded in $\mathrm{Ba}{\mathrm{F}}_{2}$. The local structure around the ${\mathrm{Ce}}^{3+}$ impurity and the transition energies are calculated in the states of main character $4{f}^{1}$, $5{d}^{1}$, and $6{s}^{1}$, without and with spin-orbit coupling. When Madelung embedding is used, large anomalies are observed in the $5{d}^{1}$ and $6{s}^{1}$ manifolds. They are shown to be due to an artificial electron leak out of the cluster, at a regime of short Ce-F distances, when it occupies diffuse orbitals like the molecular orbitals of main character $5d$ and $6s$. The main reason for the leak is the lack of linear-independency conditions between the wave functions of the cluster and its environment in Madelung embedding. When these conditions are added by means of ab initio model potential embedding (AIMP), the anomalies disappear. The $4f\text{\ensuremath{-}}5d$ absorption transitions calculated with spin-orbit coupling, AIMP embedding, and correlating the electrons with main character $\mathrm{F}\text{\ensuremath{-}}2s2p$ and $\mathrm{Ce}\text{\ensuremath{-}}4d5s5p4f5d$, show good agreement with experiment. The present results support previous claims that state-of-the-art ab initio methods of quantum chemistry as the ones used here are nowadays in a position to reliably calculate $4f\text{\ensuremath{-}}5d$ transitions of lanthanide-doped solids. A proper embedding is crucial for this.

The 5f2→5f16d1 absorption spectrum of Cs2GeF6:U4+ crystals: A quantum chemical and experimental study

Single crystals of U(4+)-doped Cs2GeF6 with 1% U4+ concentration have been obtained by the modified Bridgman-Stockbarger method in spite of the large difference in ionic radii between Ge4+ and U4+ in octahedral coordination. Their UV absorption spectrum has been recorded at 7 K, between 190 and 350 nm; it consists of a first broad and intense band peaking at about 38,000 cm(-1) followed by a number of broad bands of lower intensity from 39,000 to 45,000 cm(-1). None of the bands observed shows appreciable fine vibronic structure, so that the energies of experimental electronic origins cannot be deduced and the assignment of the experimental spectrum using empirical methods based on crystal field theory cannot be attempted. Alternatively, the profile of the absorption spectrum has been obtained theoretically using the U-F bond lengths and totally symmetric vibrational frequencies of the ground 5f2 - 1A(1g) and 5f16d(t(2g))1 - iT(1u) excited states, their energy differences, and their corresponding electric dipole transition moments calculated using the relativistic ab initio model potential embedded cluster method. The calculations suggest that the observed bands are associated with the lowest five 5f2 - 1A(1g)-->5f16d(t(2g))1 - iT(1u) (i = 1-5) dipole allowed electronic origins and their vibrational progressions. In particular, the first broad and intense band peaking at about 38,000 cm(-1) can be safely assigned to the 0-0 and 0-1 members of the a(1g) progression of the 5f2 - 1A(1g)-->5f16d(t(2g))1 - 1T(1u) electronic origin. The electronic structure of all the states with main configurational character 5f16d(t(2g))1 has been calculated as well. The results show that the lowest crystal level of this manifold is 5f16d(t(2g))1 - 1E(u) and lies about 6200 cm(-1) above the 5f2 level closest in energy, which amounts to some 11 vibrational quanta. This large energy gap could result in low nonradiative decay and efficient UV emission, which suggest the interest of investigating further this new material as a potential UV solid state laser.

Probing anisotropic interaction potentials of unsaturated hydrocarbons with He*(2S3) metastable atom: Attractive-site preference of σ-direction in C2H2 and π-direction in C2H4

State-resolved collision energy dependence of Penning ionization cross sections of acetylene (C2H2) and ethylene (C2H4) with He*(2 3S) metastable atoms was observed in a wide collision energy range from 20 to 350 meV. A recently developed discharge nozzle source with a liquid N2 circulator was employed for the measurements in the low-energy range from 20 to 80 meV. Based on classical trajectory calculations for the energy dependence of the partial ionization cross sections, anisotropic potential energy surfaces for the present systems were obtained by optimizing ab initio model potentials for the chemically related systems Li+C2H2 and C2H4. In the case of C2H2, the global minimum was found to be located around the H atom along the molecular axis with a well depth of 48 meV (ca. 1.1 kcal/mol). On the other hand, a dominant attractive well with a depth of 62 meV (ca. 1.4 kcal/mol) was found in the piCC electron region of C2H4. These findings were discussed in connection with orbital interactions between molecular orbitals of the target molecules and atomic orbitals of the metastable atom. It is concluded that sigma-type unoccupied molecular orbitals of C2H2 and a piCC-type highest occupied molecular orbital of C2H4 play a significant role for the attractive-site preference of sigma direction in C2H2 and pi direction in C2H4, respectively.

Development of a cooled He*(2S3) beam source for measurements of state-resolved collision energy dependence of Penning ionization cross sections: Evidence for a stereospecific attractive well around methyl group in CH3CN

A low-temperature discharge nozzle source with a liquid-N(2) circulator for He*(2(3)S) metastable atoms has been developed in order to obtain the state-resolved collision energy dependence of Penning ionization cross sections in a low collision energy range from 20 to 80 meV. By controlling the discharge condition, we have made it possible to measure the collision energy dependence of partial ionization cross sections (CEDPICS) for a well-studied system of CH(3)CN+He*(2(3)S) in a wide energy range from 20 to 350 meV. The anisotropic interaction potential energy surface for the present system was obtained starting from an ab initio model potential via an optimization procedure based on classical trajectory calculations for the observed CEDPICS. A dominant attractive well depth was found to be 423 meV (ca. 10 kcal/mol) at a distance of 3.20 A from the center of mass of CH(3)CN in the N-atom side along the CCN axis. In addition, a weak attractive well (ca. 0.9 kcal/mol) surrounding the methyl group (-CH(3)) has been found and ascribed to the interaction between an unoccupied molecular orbital of CH(3)CN and 2s atomic orbital of He*(2(3)S).

Spin–orbit effects on the electronic spectroscopy of transition metal dihydrides H2M(CO)4 (M=Fe, Os)

The influence of spin–orbit (SO) effects on the transition energies to the low-lying singlet and triplet electronic states of H2M(CO)4 (M = Fe, Os) are calculated by means of a two-step procedure in which CASSCF/MS-CASPT2 method using relativistic effective core potentials in the ab initio model potential (AIMP) approach was used to obtain the spin-free vertical transitions in a first step. Spin–orbit effects are taken into account in the second step by building a determinantal model space which includes a set of reference configurations able to represent all the desired states along with singly excited configurations selected with respect to the SO operator within the atomic mean-field approximation. The results of the first step calculation are exploited in order to include correlation effects via an effective Hamiltonian technique and diagonalize the full matrix on the determinantal basis. It is shown that while the SO has little influence on the absorption spectrum of H2Fe(CO)4 it does affect significantly the electronic spectroscopy of the third row transition metal complex H2Os(CO)4 leading to a large mixing between the low-lying states and to a splitting exceeding 3000 cm−1 for the lowest 3B2 state.

Two-dimensional Penning ionization electron spectroscopic study on outer characteristics of molecules

Collision-energy/electron-energy-resolved two-dimensional Penning ionization electron spectroscopy (2D-PIES) has been used to study outer characteristics of molecules. Anisotropic potential energy surfaces for collisional ionization of molecules with a metastable helium atom He*(2 3S) have been determined starting from ab initio model potentials via optimization based on trajectory calculations. Since ionization widths in the theoretical model of Penning ionization are functionals of target molecular orbitals, outer electron distributions of molecular orbitals can be determined via optimization procedures of calculated ionization cross-sections by trajectory calculations with respect to observed 2D-PIES data.

Electronic absorption spectra ofCu2+inMgO:Ab initiotheory and experiment

Ab initio model potential embedded cluster calculations on ${({\mathrm{CuO}}_{6})}^{10\ensuremath{-}}$ units were used to model the electronic absorption spectra of ${\mathrm{Cu}}^{2+}$ doped $\mathrm{MgO}$ single crystals. These theoretical results were compared with optical absorption spectra determined experimentally. The major interactions present in the system, including intracluster electron correlation and scalar relativistic effects, were incorporated, providing a good description of both ground and excited states. The first step in calculating the electronic spectrum was to optimize the geometry of the cluster by taking into account the distortion produced by the Jahn-Teller effect. As a result a ${\mathrm{D}}_{4h}$ geometry was obtained, which is associated with a compression of the original octahedron along the $z$ axis. Comparison of this geometry with extended x-ray absorption fine structure measurements is satisfactory. In clusters with ${\mathrm{D}}_{4h}$ geometry, vertical absorptions were calculated for $d$-$d$ transitions as well as for ligand-to-metal charge transfer transitions. Optical absorption measurements showed two broad absorption bands at about 4.5 and $5.5\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, which are attributed to charge-transfer transitions from the main ligand $2p$ ${t}_{1u}$ $\ensuremath{\sigma}$ and $\ensuremath{\pi}$ molecular orbitals to the metal $3d$ shell.