Open-shell Atoms Research Articles

Avoiding spin contamination and spatial symmetry breaking by exact-exchange-only optimized-effective-potential methods within the symmetrized Kohn-Sham framework.

For open-shell atoms and molecules, Kohn-Sham (KS) methods typically resort to spin-polarized approaches that exhibit spin-contamination and often break spatial symmetries. As a result, the KS Hamiltonian operator and the KS orbitals do not exhibit the space and spin symmetry of the physical electron system. The KS formalism can be symmetrized in a rigorous way only in real space, only in spin space, or both in real and spin space. Within such symmetrized KS frameworks, we present exact-exchange-only optimized-effective-potential (OEP) methods that are free of spin contamination and/or spatial symmetry breaking. The effect of symmetrizations on the total energy and its parts and on the exchange potential is analyzed. The presented exact-exchange-only OEP methods may serve as a starting point for high-level symmetrized KS methods based, e.g., on the adiabatic-connection fluctuation-dissipation theorem.

Photoionization cross-sections of valence ns and np (n = 2,3) atomic shells in soft x-ray region

Theoretical photoionization cross-sections for average of configuration terms of valence ns- and np- shells of atoms C, N, O, F, Ne, Si, P, S, Cl and Ar are calculated with account for intrachannel and interchannel interactions for the energy region from threshold to 400 eV. The many-electron effects were accounted for within the framework of RPAE (random phase approximation with exchange) method, which is extended for average terms of unfilled shells. Calculations demonstrate for open-shell atoms similar effects as that were known for noble gas atoms Ne and Ar, namely a strong decrease of photoionization cross-sections of ns- subshell due to the interaction of transitions ns→εp and np→εd.

Quadrupole Effects in the Photoionisation of Sodium 3s in the Vicinity of the Dipole Cooper Minimum

A procedure to obtain relativistic expressions for photoionisation angular distribution parameters using the helicity formulation is discussed for open-shell atoms. Electric dipole and quadrupole transition matrix elements were considered in the present work, to study the photoionisation dynamics of the 3s electron of the sodium atom in the vicinity of the dipole Cooper minimum. We studied dipole–quadrupole interference effects on the photoelectron angular distribution in the region of the dipole Cooper minimum. Interference with quadrupole transitions was found to alter the photoelectron angular distribution, even at rather low photon energies. The initial ground and final ionised state discrete wavefunctions of the atom were obtained in the present work using GRASP, and we employed RATIP with discrete wavefunctions, to construct continuum wavefunctions and to calculate transition amplitudes, total cross-sections and angular distribution asymmetry parameters.

The selective role of the orbital angular momentum on the reaction stereo-dynamics

This paper reports on the characterization of the stereo-dynamic controlling three different chemi-ionization reactions, recent objective of our study, since they participate to the balance of phenomena occurring in plasma, interstellar medium, planetary atmospheres, flames and lasers. The optical potential, obtained by a phenomenological method and defined in the whole space of the relative configurations of reagents, has been formulated in an accurate and internally consistent way for three different systems. Some cuts of the multidimensional potential, that asymptotically correlate with a specific fine level of the open shell atom and/or with a defined orientation of the molecular reagent, have been exploited in the present study to emphasize crucial features of the collision dynamics along selected entrance channels of the reactions. Consistently, basic quantities determining the topology of the reaction stereo-dynamics have been properly defined, emphasizing in the three cases relevant changes in the microscopic reaction evolution. Much attention focused on the selectivity of the orbital angular momentum, affecting each collision event at any chosen collision energy. It controls the relative weight of two different reaction mechanisms. The direct reaction mechanism is driven by short-range chemical forces, promoting, by direct electron transfer between reagents, a prototypical elementary oxidation reaction. The indirect mechanism, controlled by the combination of long-range chemical and physical forces, can be triggered by a virtual photon exchanged between reagents, promoting a sort of photo-ionization process. Obtained results and emphasized differences appear to be of general interest for many other elementary processes, more difficult to characterize at this level of detail.Graphic abstract

Long-range interactions of aromatic molecules with alkali-metal and alkaline-earth-metal atoms.

The isotropic and anisotropic coefficients Cn l,m of the long-range spherical expansion ∼1/Rn (R-the intermolecular distance) of the dispersion and induction intermolecular energies are calculated using the first principles for the complexes containing an aromatic molecule (benzene, pyridine, furan, and pyrrole) and alkali-metal (Li, Na, K, Rb, and Cs) or alkaline-earth-metal (Be, Mg, Ca, Sr, and Ba) atoms in their electronic ground states. The values of the first- and second-order properties of the aromatic molecules are calculated using the response theory with the asymptotically corrected LPBE0 functional. The second-order properties of the closed-shell alkaline-earth-metal atoms are obtained using the expectation-value coupled cluster theory and of the open-shell alkali-metal atoms using analytical wavefunctions. These properties are used for the calculation of the dispersion Cn,disp l,m and induction Cn,ind l,m coefficients (Cn l,m=Cn,disp l,m+Cn,ind l,m) with n up to 12 using the available implemented analytical formulas. It is shown that the inclusion of the coefficients with n > 6 is important for reproducing the interaction energy in the van der Waals region at R ≈ 6 Å. The reported long-range potentials should be useful for constructing the analytical potentials valid for the whole intermolecular interaction range, which are needed for spectroscopic and scattering studies.

Downward quantum learning from element 118: Automated generation of Fermi-Löwdin orbitals for all atoms.

A new algorithm based on a rigorous theorem and quantum data computationally mined from element 118 guarantees automated construction of initial Fermi-Löwdin-Orbital (FLO) starting points for all elements in the Periodic Table. It defines a means for constructing a small library of scalable FLOs for universal use in molecular and solid-state calculations. The method can be systematically improved for greater efficiency and for applications to excited states such as x-ray excitations and optically silent excitations. FLOs were introduced to recast the Perdew-Zunger self-interaction correction (PZSIC) into an explicit unitarily invariant form. The FLOs are generated from a set of N quasi-classical electron positions, referred to as Fermi-Orbital descriptors (FODs), and a set of N-orthonormal single-electron orbitals. FOD positions, when optimized, minimize the PZSIC total energy. However, creating sets of starting FODs that lead to a positive definite Fermi orbital overlap matrix has proven to be challenging for systems composed of open-shell atoms and ions. The proof herein guarantees the existence of a FLOSIC solution and further guarantees that if a solution for N electrons is found, it can be used to generate a minimum of N - 1 and a maximum of 2N - 2 initial starting points for systems composed of a smaller number of electrons. Applications to heavy and super-heavy atoms are presented. All starting solutions reported here were obtained from a solution for element 118, Oganesson.

On the origins of spontaneous spherical symmetry-breaking in open-shell atoms through polymer self-consistent field theory.

An alternative approach to density functional theory based on self-consistent field theory for ring polymers is applied to neutral atoms hydrogen to neon in their ground-states. The spontaneous emergence of an atomic shell structure and spherical symmetry-breaking of the total electron density are predicted by the model using the ideas of polymer excluded-volume between pairs of electrons to enforce the Pauli-exclusion principle and an exact electron self-interaction correction. The Pauli potential is approximated by neglecting inter-atomic correlations along with other types of correlations, and comparisons to Hartree-Fock theory are made, which also ignores correlations. The model shows excellent agreement with Hartree-Fock theory to within the standards of orbital-free density functional theory for the atomic binding energies and density profiles of the first six elements, providing exact matches for the elements hydrogen and helium. The predicted shell structure starts to deviate significantly past the element neon, and spherical symmetry-breaking is first predicted to occur at carbon instead of boron. The self-consistent field theory energy functional that describes the model is decomposed into thermodynamic components to trace the origin of spherical symmetry-breaking. It is found to arise from the electron density approaching closer to the nucleus in non-spherical distributions, which lowers the energy despite resulting in frustration between the quantum kinetic energy, electron-electron interaction, and the Pauli exclusion interaction. The symmetry-breaking effect is found to have a minimal impact on the binding energies, which suggests that the spherical-averaging approximation used in previous work is physically reasonable when investigating atomic systems. The pair density contour plots display behavior similar to polymer macro-phase separation, where individual electron pairs occupy single lobe structures that together form a dumbbell shape analogous to the 2p orbital shape. It is further shown that the predicted densities satisfy known constraints and produce the same total electronic density profile that is predicted by other formulations of quantum mechanics.

Single Metal Atoms on Oxide Surfaces: Assessing the Chemical Bond through 17O Electron Paramagnetic Resonance.

ConspectusEven in the gas phase single atoms possess catalytic properties, which can be crucially enhanced and modulated by the chemical interaction with a solid support. This effect, known as electronic metal-support interaction, encompasses charge transfer, orbital overlap, coordination structure, etc., in other words, all the crucial features of the chemical bond. These very features are the object of this Account, with specific reference to open-shell (paramagnetic) single metal atoms or ions on oxide supports. Such atomically dispersed species are part of the emerging class of heterogeneous catalysts known as single-atom catalysts (SACs). In these materials, atomic dispersion ensures maximum atom utilization and uniform active sites, whereby the nature of the chemical interaction between the metal and the oxide surface modulates the catalytic activity of the metal active site by tuning the energy of the frontier orbitals. A comprehensive set of examples includes fourth period metal atoms and ions in zeolites on insulating (e.g., MgO) or reducible (e.g., TiO2) oxides and are among the most relevant catalysts for a wealth of key processes of industrial and environmental relevance, from the abatement of NOx to the selective oxidation of hydrocarbons and the conversion of methane to methanol.There exist several spectroscopic techniques able to inform on the geometric and electronic structure of isolated single metal ion sites, but either they yield information averaged over the bulk or they lack description of the intimate features of chemical bonding, which include covalency, ionicity, electron and spin delocalization. All of these can be recovered at once by measuring the magnetic interactions between open-shell metals and the surrounding nuclei with Electron Paramagnetic Resonance (EPR) spectroscopy. In the case of oxides, this entails the synthesis of 17O isotopically enriched materials. We have established 17O EPR as a unique source of information about the local binding environment around oxygen of magnetic atoms or ions on different oxidic supports to rationalize structure-property relationships. Here, we will describe strategies for 17O surface enrichments and approaches to monitor the state of charge and spin delocalization of atoms or ions from K to Zn dispersed on oxide surfaces characterized by different chemical properties (i.e., basicity or reducibility). Emphasis is placed on chemical insight at the atomic-scale level achieved by 17O EPR, which is a crucial step in understanding the structure-property relationships of single metal atom catalysts and in enabling efficient design of future materials for a range of end uses.

Application of Symmetry-Adapted Atomic Amplitudes

Following the work of Giulio Racah and others from the 1940s onward, the rotational symmetry of atoms and ions, e.g., the conservation of angular momentum, has been utilized in order to efficiently predict atomic behavior, from their level structure to the interaction with external fields, and up to the angular distribution and polarization of either emitted or scattered photons and electrons, while this rotational symmetry becomes apparent first of all in the block-diagonal structure of the Hamiltonian matrix, it also suggests a straight and consequent use of symmetry-adapted interaction amplitudes in expressing the observables of most atomic properties and processes. We here emphasize and discuss how atomic structure theory benefits from exploiting this symmetry, especially if open-shell atoms and ions in different charge states need to be combined with electrons in the continuum. By making use of symmetry-adapted amplitudes, a large number of excitation, ionization, recombination or even cascade processes can be formulated rather independently of the atomic shell structure and in a language close to the formal theory. The consequent use of these amplitudes in existing codes such as Grasp will therefore qualify them to deal with the recently emerging demands for developing general-purpose tools for atomic computations.

Kinetic energy density for open-shell systems: analysis and development of a novel technique

The quest for an approximate yet accurate kinetic energy density (KED) functional is central to the development of orbital-free density functional theory. While a recipe for closed-shell systems has been proposed earlier, we have shown that it cannot be naïvely extended to open-shell atoms. In this present work, we investigated the efficacy of an ad-hoc recipe to compute the kinetic energy densities for open-shell atoms by extending the methodology used for closed-shell systems. We have also analysed the spin-dependent features of Pauli potentials derived from two previously devised enhancement factors. Further, we have proposed an alternate but exact methodology to systematically compute the KED for atoms of arbitrary spin multiplicity.

Reactive Solubilization of Heterometallic Clusters by Treatment of (TrBi3 )2- Anions (Tr=Ga, In, Tl) with [Mn{N(SiMe3 )2 }2

Lowering the charge of Zintl anions by (element-)organic substituents allows their use as sources of (semi)metal nanostructures in common organic solvents, as realized for group 15 anions or Ge9 4- and Sn9 4- . We developed a new strategy for other anions, using low-coordinate 3d metal complexes as electrophiles. [K(crypt-222)]+ salts of (TrBi3 )2- anions dissolved in situ in Et2 O and/or THF when reacted with [Mn(hmds)2 ]. Work-up afforded soluble [K(crypt-222)]+ salts of [{(hmds)2 Mn}2 (TlBi3 )]2- (in 1), [{(hmds)2 Mn}2 (Bi2 )]2- (in 2), and [{(hmds)Mn}4 (Bi2 )2 ]2- (in 3) (crypt-222=4,7,13,16,21,24-hexaoxa-1,10-diazabicyclo[8.8.8]hexacosane; Tr=Ga, In, Tl; hmds=N(SiMe3 )2 ), representing rare cases of Zintl clusters with open-shell metal atoms. 1 comprises the first coordination compound of the (TlBi3 )2- anion, 2 features a diamond-shaped {Pn2 M2 } unit, and 3 is a mixed-valent MnI /MnII compound. The uncommon electronic structures in 1-3 and magnetic coupling were studied by comprehensive DFT calculations.

Density Functional Response Calculations of Dispersion Coefficients C6 and C9 of Closed- and Open-Shell Systems.

Dipole polarizabilities and C6 and C9 dispersion coefficients are computed for closed- and open-shell atoms and molecules, using dynamic (time-dependent) density functional (TD-DFT) linear response theory as implemented in the response module of the NWChem quantum chemistry package. The response module is capable of accurate calculations of these properties, based on spin-restricted and spin-unrestricted formalisms. The calculated static polarizabilities and dispersion coefficients are compared to available experimental and other theoretical data. The behavior of the dynamic polarizability at imaginary frequencies is analyzed for differently sized closed- and open-shell systems. An interpolation method enforcing the monotonic decrease of the polarizability with increasing imaginary frequency is beneficial for the integration used to obtain C6 and C9. Scaling of the TD-DFT data by ratios of the static polarizability, which can be calculated with a variety of methods, including highly accurate theories, may be used as a leading-order correction.

Interaction of free-base and 3d metal(II) phthalocyanines with open-shell endohedral fullerenes species N@C60 and P@C60

We studied theoretically the interactions of phthalocyanines (Pcs) of general formula MPc (M = 2H, Mn, Fe, Co, Ni, Cu, Zn) with endohedral fullerenes (EFs) N@C60 and P@C60, at the PBE-D/DNP theoretical level. All MPcs can form stable noncovalent complexes with both N@C60 and P@C60. The binding energies vary between −17.7 and −31.1 kcal/mol. Complexation pattern depends on the central atom of MPc, but not on the endohedral species (N or P): for H2Pc, two central H atoms approach the carbon atoms of C60 of adjacent 5:6:6 junctions; for MnPc, manganese atom coordinates to two carbon atoms of 6,6 C-C bond; for other MPcs, the central metal coordinates to 5:6:6 junction of fullerene. In MPc+N@C60 HOMO is localized exclusively on phthalocyanine, with LUMO found mostly on fullerene. In MPc+P@C60 complexes, HOMO has essentially the same localization and shape, but LUMO distribution is variable. After complexation with MPcs, no tangible changes are observed in the N and P spin state. MnPc, CoPc and CuPc having open-shell metal atoms behave differently: while the complexation has almost no effect on Cu spin, and causes only a moderate decrease in Co spin, Mn atom transfers roughly two-thirds of its unpaired electrons to fullerene.

Spherical vs non-spherical and symmetry-preserving vs symmetry-breaking densities of open-shell atoms in density functional theory.

The atomization energies of molecules from first-principles density functional approximations improve from the local spin-density approximation to the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) to the strongly constrained and appropriately normed (SCAN) meta-GGA, and their sensitivities to non-spherical components of the density increase in the same order. Thus, these functional advances increase density sensitivity and imitate the exact constrained search over correlated wavefunctions better than that over ensembles. The diatomic molecules studied here, singlet C2 and F2 plus triplet B2 and O2, have cylindrically symmetric densities. Because the densities of the corresponding atoms are non-spherical, the approximate Kohn-Sham potentials for the atoms have a lower symmetry than that of the external (nuclear) potential so that the non-interacting wavefunctions are not eigenstates of the square of total orbital angular momentum, breaking a symmetry that yields a feature of the exact ground-state density. That spatial symmetry can be preserved by a non-self-consistent approach in which a self-consistent equilibrium-ensemble calculation is followed by integer re-occupation of the Kohn-Sham orbitals as the first of several steps. The symmetry-preserving approach is different from symmetry restoration based on projection. First-step space- (and space-spin-) symmetry preservation in atoms is shown to have a small effect on the atomization energies of molecules, quantifying earlier observations by Fertig and Kohn. Thus, the standard Kohn-Sham way of calculating atomization energies, with self-consistent symmetry breaking to minimize the energy, is justified at least for the common cases where the molecules cannot break symmetry. Unless symmetry breaking is allowed in the molecule, SCAN strongly underestimates the atomization energy of strongly correlated singlet C2.

Trapping and migration of P-state atoms in rare gas solids: effect of angular momentum anisotropy for model O(3P) and C(3P) atoms

Classical simulations of model O(P) atom isolated in Ar, Kr and Xe crystals with the variable magnitude and sign of the angular momentum anisotropy are performed. The guest–host interaction anisotropy remarkably affects the structure of the atomic trapping sites and the barriers to the temperature-induced atomic mobility. For interstitial trapping, inversion of anisotropy induces a kind of critical phenomenon, which transforms the migration path saddle point to a minimum and another saddle. As a result, a new peculiar axially symmetric interstitial site is formed with the migration barrier reduced. Preliminary ab initio-based study of the C(P) atom, whose interactions with Ar, Kr and Xe have the anisotropy opposite to oxygen, fully confirms the model analysis. Our results emphasise profound effect of the angular momentum anisotropy on the structure and dynamics of the open-shell atoms at matrix isolation.

Selectivity of weak intermolecular forces and precursor state of elementary oxidation reactions, a new insight on Ne*\u2009+\u2009N2 chemiionization

This paper reports on the collision dynamics of N2 with metastable Ne* promoting chemiionizations, prototype of barrier-less oxidation reactions of great interest for fundamental and applied research. Extending guidelines presented in previous papers for the atom–atom case, an innovative treatment of the reaction stereodynamics involving molecules in a quantum state-to-state resolution conditions is proposed that emphasizes the role of structure and stability of the precursor that is here the reaction transition state. A critical test of such treatment, carried out exploiting a new formulation both of real and imaginary parts of the optical potential driving the reaction dynamics, is represented by the detailed-combined description of all relevant findings, provided by high resolution molecular beam scattering experiments carried out in our and other laboratories. The present analysis casts light on basic electronic rearrangements of such prototype oxidation reaction which are expected to be of fundamental interest for many other reactions involving open shell atoms and free radicals.

On deformability of atoms—comparative study between atoms and atomic nuclei

Atomic nuclei can be spontaneously deformed into non-spherical shapes as many-nucleon systems. We discuss to what extent a similar deformation takes place in many-electron systems. To this end, we employ several many-body methods, such as the unrestricted Hartree–Fock method, post-Hartree–Fock methods, and the density functional theory, to compute the electron density distribution in atoms. We show that the electron density distribution of open-shell atoms is deformed due solely to the single-particle valence orbitals, while the core part remains spherical. This is in contrast to atomic nuclei, which can be deformed collectively. We qualitatively discuss the origin for this apparent difference between atoms and nuclei by estimating the energy change due to deformation. We find that nature of the interaction plays an essential role for the collective deformation.

Absolute NMR shielding scales in methyl halides obtained from experimental and calculated nuclear spin-rotation constants

The nonrelativistic ``Ramsey-Flygare relationship'' is the most used procedure to obtain semiexperimental nuclear magnetic resonance (NMR) absolute shieldings by a correspondence between NMR shieldings ($\mathbit{\ensuremath{\sigma}}$) and nuclear spin-rotation constants ($\mathbit{M}$). One of its generalizations to the relativistic framework is known as the M-V model, which was proposed few year ago by some of the authors of the present work and right now is only applied to linear molecules. This model includes terms that do not have nonrelativistic counterparts and also include the paramagnetic contribution to the NMR shielding of nuclei in free atoms. All this ensures that its results fit quite well with those of four-component (4c) calculations. The first application of the M-V model to nonlinear molecules, like methyl halides or ${\mathrm{CH}}_{3}X$ molecules ($X=\mathrm{F}$, Cl, Br, and I), is given here. The analysis of each electronic mechanism of $\mathbit{\ensuremath{\sigma}}$ shows that most of their electron correlation effects are strongly related with the same effects in $\mathbit{M}$. By including experimental data of $M$ in the M-V model most of the correlation effects are accurately taken into account for the absolute values of $\mathbit{\ensuremath{\sigma}}$. Calculations of ${\mathbit{M}}_{Y}$ and ${\mathbit{\ensuremath{\sigma}}}_{Y}$ ($Y=\mathrm{H}$, C, and $X$) were carried out within the linear response formalism at the random-phase level of approach and density functional theory in both 4c and nonrelativistic frameworks. The best fits between calculations of $\mathbit{M}$ and experimental data are obtained from calculations at 4c-PBE0 level of theory in all cases, but not for ${M}_{\ensuremath{\parallel},\mathrm{Cl}}$, which suggests that a revision of the available experimental data may be necessary. There is an additional advantage of using the M-V model: one can indirectly calculate shieldings of open-shell free atoms, which cannot be obtained at the moment by applying 4c methods.

Density functional study of the variants of inter-Coulombic decay resonances in the photoionization of Cl@C60

Inter-Coulombic decay (ICD) resonances in the photoionization of Cl@C60 endofullerene molecule are calculated using a perturbative density functional theory (DFT) method. This is the first ICD study of an open shell atom in a fullerene cage. Three classes of resonances are probed: (i) Cl inner vacancies decaying through C60 outer continua, (ii) C60 inner vacancies decaying through Cl outer continua, and (iii) inner vacancies of either system decaying through the continua of Cl-C60 hybrid levels, the hybrid Auger-ICD resonances. Comparisons with Ar@C60 results reveal that the properties of hybrid Auger-ICD resonances are affected by the extent of level hybridization.

Ionization of many-electron atoms by the action of two plasma models

The Hartree-Fock equations for many-electron atoms embedded in a plasma medium are solved using two different plasma models: (a) Debye-Hückel screening (DHS) potential and (b) exponential cosine screened Coulomb (ECSC) potential. Roothaan's approach is implemented for these models after solving the inherent difficulties to evaluate integrals where screening appears explicitly. A corresponding computer code was developed using the method of global basis sets (GBS). The reliability of this approach was verified by solving the Hartree-Fock equations through implementation of the finite-differences and finite-element grid methods and applied to two-electron atoms, yielding excellent agreement with the Roothaan-GBS (RGBS) method. The RGBS method was used to study the energy evolution and ionization threshold of several closed- and open-shell many-electron atoms embedded either in weak or strong DHS or ECSC plasma conditions. In all cases, a critical value of the screening length is obtained for which ionization is achieved, being systematically larger for DHS conditions, indicating the effect of a more repulsive ECSC potential. For He-like atoms in the ground state, we report a comprehensive set of accurate total energy data as a function of the screening constant using the Lagrange mesh method, which includes the electron correlation effects. The electron correlation energy is estimated using this data with reference to the RGBS estimates of energy as the Hartree-Fock energy. The variation of correlation energy as a function of screening constant under the different plasma potentials is rationalized in terms of a conjectured comparison theorem. Finally, a discussion on the effect of plasma strength on localization or delocalization of the electronic density derived from the RGBS method is presented in terms of changes in the Shannon entropy, yielding consistent results for delocalization close to the ionization threshold.