Russell-Saunders Terms Research Articles

String-Based Method for Deriving Russell–Saunders Terms

String-Based Method for Deriving Russell–Saunders Terms

Electronic Structure of Ytterbium(III) Solvates-a Combined Spectroscopic and Theoretical Study.

The wide range of optical and magnetic properties of lanthanide(III) ions is associated with their intricate electronic structures which, in contrast to lighter elements, is characterized by strong relativistic effects and spin-orbit coupling. Nevertheless, computational methods are now capable of describing the ladder of electronic energy levels of the simpler trivalent lanthanide ions, as well as the lowest energy term of most of the series. The electronic energy levels result from electron configurations that are first split by spin-orbit coupling into groups of energy levels denoted by the corresponding Russell-Saunders terms. Each of these groups are then split by the ligand field into the actual electronic energy levels known as microstates or sometimes mJ levels. The ligand-field splitting directly informs on the coordination geometry and is a valuable tool for determining the structure and thus correlating the structure and properties of metal complexes in solution. The issue with lanthanide complexes is that the determination of complex structures from ligand-field splitting remains a very challenging task. In this paper, the optical spectra-absorption, luminescence excitation, and luminescence emission-of ytterbium(III) solvates were recorded in water, methanol, dimethyl sulfoxide (DMSO), and N,N-dimethylformamide (DMF). The electronic energy levels, that is, the microstates, were resolved experimentally. Subsequently, density functional theory calculations were used to model the structures of the solvates, and ab initio relativistic complete active space self-consistent field calculations (CASSCF) were employed to obtain the microstates of the possible structures of each solvate. By comparing the experimental and theoretical data, it was possible to determine both the coordination number and solution structure of each solvate. In water, methanol, and N,N-dimethylformamide, the solvates were found to be eight-coordinated and have a square antiprismatic coordination geometry. In DMSO, the speciation was found to be more complicated. The robust methodology developed for comparing experimental spectra and computational results allows the solution structures of homoleptic lanthanide complexes to be determined.

Improved charge transfer multiplet method to simulate M- and L-edge X-ray absorption spectra of metal-centered excited states.

Charge transfer multiplet (CTM) theory is a computationally undemanding and highly mature method for simulating the soft X-ray spectra of first-row transition metal complexes. However, CTM theory has seldom been applied to the simulation of excited-state spectra. In this article, the CTM4XAS software package is extended to simulate M2,3- and L2,3-edge spectra for the excited states of first-row transition metals and also interpret CTM eigenfunctions in terms of Russell-Saunders term symbols. These new programs are used to reinterpret the recently reported excited-state M2,3-edge difference spectra of photogenerated ferrocenium cations and to propose alternative assignments for the electronic state of these cations responsible for the spectroscopic features. These new programs were also used to model the L2,3-edge spectra of FeII compounds during nuclear relaxation following photoinduced spin crossover and to propose spectroscopic signatures for their vibrationally hot states.

Crystal-field study of magnetization and specific heat properties of frustrated pyrochlore Pr2Zr2O7

The experimental results of temperature dependent dc magnetic susceptibility, field dependent isothermal magnetization, magnetic specific heat and entropy of the pyrochlore Pr2Zr2O7 are simulated and analyzed using appropriate D3d crystal-field (CF) and anisotropic molecular field tensors at Pr-sites in the self-consistent mean-field approach involving four magnetically non-equivalent rare-earth spins on the tetrahedral unit of the pyrochlore structure. CF level pattern and wave-functions of the ground 3H4 multiplet of the Pr3+ ions are obtained considering intermediate coupling between different Russell-Saunders terms of the 4f2 electronic configurations of Pr-ion and J-mixing effects. CF analysis shows that the CF ground-state of the Pr3+ ion in Pr2Zr2O7 is a well-isolated doublet, with significant admixtures of terms coming from |MJ=±4〉 and |MJ=±1〉, and the Pr-spins are effectively Ising-like along the local <111> axes. Magnetic specific heat in zero-field is simulated by considering a temperature dependence of the exchange splitting of the ground doublet.

Visualizing, Rather than Deriving, Russell–Saunders Terms: A Classroom Activity with Quantum Numbers

A 1 h classroom activity is presented, aimed at consolidating the concepts of microstates and Russell–Saunders energy terms in transition metal atoms and coordination complexes. The unconventional approach, based on logic and intuition rather than rigorous mathematics, is designed to stimulate discussion and enhance familiarity with quantum numbers in classes of chemistry undergraduate students.

Non-collinear spin DFT for lanthanide ions in doped hexagonal NaYF4

Trivalent lanthanide ions (Ln3+) doped in hexagonal (β)-NaYF4 nanocrystals (Na24Y23Ln1F96, Ln = La, Ce, Pr, Nd, Pm, Sm, Eu, Gd) were systematically studied by density functional theory (DFT) with a perturbative account for spin–orbit coupling. The simulated results, including the optimised molecular structures, electronic and magnetic properties, are compared to previous spin-polarised DFT studies in the same system. The spin–orbit coupling effects become significant with the increase in the number of unpaired 4f electrons in the doped lanthanide ions, particularly for the Sm3+-, Eu3+- and Gd3+-doped nanocrystals. Abnormal behaviour of Eu3+-doped nanocrystals was observed due to the Wybourne–Downer mechanism. A ‘sandwich-like’ 2p–4f–4d,5d electronic structure for Na24Y23Ln1F96 and the energies of the highest occupied 4f electrons from Ce3+ to Gd3+ are consistent with Dorenbos's relationship. The energy difference between the first and second Russell–Saunders terms (2S+1L) of the lanthanide dopant is consistent with Carnall's experimental results and with earlier spin-polarised DFT calculations.

DFT Calculation of Russell–Saunders Splitting for Lanthanide Ions Doped in Hexagonal (β)-NaYF4Nanocrystals

A systematic investigation is reported of the optimized geometry and electronic structure of trivalent lanthanide ions (Ln3+) doped in hexagonal (β)-NaYF4 nanocrystals in the basis of density functional theory with a spin polarization approach. A model Na24Y23Ln1F96 nanocrystal with a single central lanthanide dopant (Ln3+) is used. Electron spins couple to give a total spin, S, and electron orbital angular momenta couple to give total orbital angular momentum, L. Spin–orbit coupling is neglected in this initial study. Several key observables are found to be strongly related to the number of unpaired f-electrons in the model. After geometry optimization, the phenomenon of the lanthanide contraction is observed, and the configurations of 4f-electron-like orbitals satisfy Hund’s Rule under the orbital decomposition. Spin-polarized density functional theory is applied to generate the Russell–Saunders terms (2S+1L) terms of lanthanide ions. The energy differences between the first and the second terms are calculated and show good agreement with experimental measurements. The free-ion-like behavior of Ln3+ ions in the nanocrystal is observed as well.

A combined DFT and restricted open-shell configuration interaction method including spin-orbit coupling: Application to transition metal L-edge X-ray absorption spectroscopy

A novel restricted-open-shell configuration interaction with singles (ROCIS) approach for the calculation of transition metal L-edge X-ray absorption spectra is introduced. In this method, one first calculates the ground state and a number of excited states of the non-relativistic Hamiltonian. By construction, the total spin is a good quantum number in each of these states. For a ground state with total spin S excited states with spin S' = S, S - 1, and S + 1 are constructed. Using Wigner-Eckart algebra, all magnetic sublevels with MS = S,..., -S for each multiplet of spin S are obtained. The spin-orbit operator is represented by a mean-field approximation to the full Breit-Pauli spin-orbit operator and is diagonalized over this N-particle basis. This is equivalent to a quasi-degenerate treatment of the spin-orbit interaction to all orders. Importantly, the excitation space spans all of the molecular multiplets that arise from the atomic Russell-Saunders terms. Hence, the method represents a rigorous first-principles approach to the complicated low-symmetry molecular multiplet problem met in L-edge X-ray absorption spectroscopy. In order to gain computational efficiency, as well as additional accuracy, the excitation space is restricted to single excitations and the configuration interaction matrix is slightly parameterized in order to account for dynamic correlation effects in an average way. To this end, it is advantageous to employ Kohn-Sham rather than Hartree-Fock orbitals thus defining the density functional theory∕ROCIS method. However, the method can also be used in an entirely non-empirical fashion. Only three global empirical parameters are introduced and have been determined here for future application of the method to any system containing any transition metal. The three parameters were carefully calibrated using the L-edge X-ray absorption spectroscopy spectra of a test set of coordination complexes containing first row transition metals. These parameters are universal and transferable. Hence, there are no adjustable parameters that are used to fit experimental X-ray absorption spectra. Thus, the new approach classifies as a predictive first-principles method rather than an analysis tool. A series of calculations on transition metal compounds containing Cu, Ti, Fe, and Ni in various oxidation and spin states is investigated and a detailed comparison to experimental data is reported. In most cases, the approach yields good to excellent agreement with experiment. In addition, the origin of the observed spectral features is discussed in terms of the electronic structure of the investigated compounds.

Atomic term symbols via partitioning techniques

Two techniques for obtaining Russell–Saunders term symbols (terms) for equivalent electrons is presented. Both techniques are based on partitioning the n equivalent electrons into a set of α electrons and a set of β electrons. Fundamental tables are presented that give the number of states for the α or β electrons for all possible values M(M) and Sα(Sβ). One interpretation of these tables leads to the total number of states of the n equivalent electrons for values of L and S. From this the number of terms for L and S is easily determined. Another interpretation of the fundamental tables gives the maximum number of terms for n equivalent electrons with values of L and S. From this the actual number of terms is obtained. Examples of equivalent p, d, f, and g electrons are provided. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Field shifts in hafnium II

By means of classical interference spectroscopy, using enriched isotope samples, the isotope shift between178Hf and180Hf has been measured for 33 transitions in the Hf II spectrum. For the pure Russell-Saunders terms 5d26s4F and2F the parametric analysis yields a field-shift difference of 17(2) mK produced by the second-order interaction of the electrostatic operator and the field-shift operator. Semi-empirical calculations based on the non-relativistic Hartree-Fock method reproduce this value as well as the experimental field shifts if a factor of 1.68(6) is used to scale theab initio electron densities at the nucleus. The corresponding factor for the Hf atom is much smaller. This leads to a re-evaluation of screening ratios for Hf and to a more accurate value of the nuclear parameter λ178,180 (Hf)=0.072(4) fm2.

Optical absorption spectra, crystal-field analysis, and electric dipole intensity parameters for europium in Na 3[En(ODA) 3]-2NaClO 4·6H 2O

Locations and assignments of 61 crystal-field levels are reported for Eu 3+ in the trigonal Na 3[Eu(oxydiacetate) 3]· 2NaClO 46H 2O system. These energy levels span the 0-37400 cm − energy region, and they were located and assigned from optical emission spectra and from axial and orthoaxial (σ and π-polarized) absorption measurements on single crystals. The assigned crystal-field levels span 22 different multiplet manifolds, with principal parentages derived from seven different f 6 Russell-Saunders terms ( 2F, 5D, 5L, 5H, 5F, 5I, and 5K). The assigned levels are analyzed in terms of a 26-parameter electronic Hamiltonian in which six of the parameters are defined to represent the 4f-electron/crystal-field interactions for Eu 3+ ions located at sites with trigonal dihedral (D 3) symmetry. Quantitative line intensities are reported for 39 individual 4f→4f (crystal-field) transitions observed in the low-temperature (10 K) absorption spectra, and these intensity data are analyzed in terms of a general parametric model for 4f→4f transition intensities in lanthanide systems. The energy and intensity parameterizations provide a basis for calculating the 4f→4f absorption spectra of Eu 3+ in Na 3[Eu(oxydiacetate) 3]· 2NaClO 4·6H 2O over a wide spectral range and excellent agreement between calculated and experimentally measured spectra is obtained. Several of the intensity parameters determined to be important in this study carry information of particular significance to understanding the structural and mechanisticbases of lanthanid

A numerical algorithm technique for deriving Russell-Saunders (R-S) terms

Another simple, versatile, and systematic numerical method of deriving the R-S terms as part of the interpretation of atomic and electronic spectra.

Optical spectra, energy levels and crystal-field analysis of Sm3+in Na3[Sm(oxydiacetate)3] · 2NaClO4· 6H2O

Locations and assignments of 144 crystal-field energy levels are reported for Sm3+ in the trigonal Na3[Sm(oxydiacetate)3] · 2NaClO4 · 6H2O system. These energy levels span the 0–36 000 cm-1 energy region, and they were located and assigned from optical emission spectra and from axial and orthoaxial (σ- and π-polarized) absorption measurements on single crystals. The principal SL parentages of the assigned levels span at least 20 Russell-Saunders terms within the 4f 5 electronic configuration of Sm3+, and these assigned levels represent nearly half the total number of crystal-field energy levels predicted to occur within the 0–36 000 cm-1 region. The assigned levels are analysed in terms of a 26-parameter electronic hamiltonian in which six of the parameters are defined to represent the 4f-electron/crystal-field interactions for Sm3+ located at sites with trigonal dihedral (D 3) symmetry. Of the 26 parameters, 22 (including the six crystal-field parameters) are used in fitting calculated energy levels to the experimental data, and excellent calculated-versus-experimental energy level fits are achieved. Fitted values obtained in this study for the Sm3+ ‘free-ion’ energy parameters are compared to those reported previously for Sm3+ in LaF3 and LaCl3. Quantitatively determined line strengths are reported for 50 (4f → 4f) crystal-field transitions observed in the single-crystal absorption spectra of Na3[Sm(oxydiacetate)3] · 2NaClO4 · 6H2O at 10 K.

Vibronic spectra of U4+ in octahedral crystal fields

The absorption spectra of UCl2- 6 diluted into the cubic lattices Cs2ZrCl6 and Cs2SnCl6, and of UBr2- 6 diluted into Cs2ZrBr6 have been measured at liquid helium temperatures. The derived energy levels, together with earlier luminescence measurements have been used to evaluate the crystal-field, spinorbit coupling, inter-electron repulsion and Trees correction parameters and the corresponding eigenvalues. Several of the states are derived from an almost equal admixture of two or more Russell-Saunders terms. Some magnetic properties are also computed and discussed.

Level and line statistic in atomic spectra

The level distribution of an electronic configuration cannot be described in a simple and exact way according to the value of the total angular momentum J. Starting from the calculation of variances of the quantum numbers MJ and ML, and using a Gaussian model, approximate formulae are derived for the number of levels with a fixed J, for the total number of levels, for the number of Russell-Saunders terms of a given (S, L), for the total number of terms with a fixed S and for the number of levels of a jj term. From these results, closed formulae are derived for the total numbers of electric dipole lines in any transition array between two configurations, in intermediate coupling and in Russell-Saunders coupling, with an uncertainty of about 2%. Then, the statistics of the line strengths of an array is studied, and a simple distribution function is presented. When this simple model is used, for example, to calculate how many intense lines are expected to appear on top of an unresolved transition array, the agreement with exact calculations is good, except in very special cases where a sophisticated model is needed.

Energy levels of HoBr 63−

The excitation, electronic absorption and luminescence spectra of cubic Cs 2NaHoBr 6 have been recorded at temperatures down to that of liquid helium. The detailed spectral analyses enable comparisons to be made of the crystal-field splittings of Russell—Saunders terms with those in Cs 2NaHoCl 6. Under intense 647.1 nm laser excitation, luminescence is observed in the neat material in the spectral region between 17800 and 21750 cm −1.

Field shift crossed second-order effects in dNsp configurations

In configurations containing open s subshells, far-configuration mixing ('crossed second-order effects') leads to different electron change densities at the nucleus for the various Russell-Saunders terms and thus to different field isotope shifts. For ndN(n+1)s (n+1)p configurations (n=3, 4, 5; N=1-10) the electron charge densities D of pairs of terms belonging to parents with the same multiplicity calculated using the Hartree-Fock computer code of Froese Fischer (1978). Usually, the term with the higher multiplicity has the larger density. The density difference Delta D increases linearly with the number of d electrons up to the half-filled shell and then decreases. The terms belonging to the parent with the higher multiplicity show larger values of Delta D than those belonging to the parent with the lower multiplicity. The effect increases with the principal quantum number n; it depends on the spins of the parent and grandparent terms but not on their angular momenta. The Hartree-Fock results of Delta D were used to calculate field shift differences between the corresponding terms. The comparisons of calculated and experimental field shift differences shows that the non-relativistic Hartree-Fock method correctly predicts their sign and order of magnitude.

5I7 → 5I8 infrared luminescence spectrum of Cs2NaHoCl6

The 300-15 K luminescence spectrum of Cs2NaHoCl6 is reported for the spectral region 4550–5300 cm−1. In agreement with the calculated magnetic dipole (MD) intensity ratios, twenty MD allowed pure electronic transitions are observed and assigned, corresponding to transitions between the crystal field components of 5I7 → 5I8. The ungerade vibronic origins associated with these transitions are at least an order of magnitude weaker. All of the crystal field levels of the 5I7, 5I8 Russell—Saunders terms are assigned from the spectral analysis. The long-term lifetime of the 5I7 levels is near 50 ms at 85 K.

Crossed-second-order specific-mass isotope shift in the Nickel atom

The crossed-second-order corrections to the specific mass shifts of the lowest terms of the two lowest configurations of the Nickel atom are evaluated ab initio in the Multiconfigurational Hartree-Fock scheme. The excitations towards the nf(l=3) empty subshells play the major role. If the contributions obtained are added to the Hartree-Fock values, the discrepancy between experiment and theory for the 3d8 4s2−3d9 4s (virtual) transition is only reduced by one third. As concerns the differences between the specific shifts of the five Russell-Saunders terms of 3d8 4s2, the crossed-second-order contributions are predicted to be practically as large as the Hartree-Fock values, which makes the total definitely measurable.

Simple method for obtaining Russell-Saunders term symbols

A method is presented here for simplifying the determination of spectroscopic terms obtained from a given electronic configuration by not tabulating all possible microstates.