Relativistic Effects in the Electronic Structure of Atoms.

Periodic trends in relativistic effects are investigated from 1H through 103Lr using Dirac-Hartree-Fock and nonrelativistic Hartree-Fock calculations. Except for 46Pd (4d10) (5s0), all atoms have as outermost shell the ns or n'p spinors/orbitals. We have compared the relativistic spinor energies with the corresponding nonrelativistic orbital energies. Apart from 24Cr (3d5) (4s1), 41Nb (4d4) (5s1), and 42Mo (4d5) (5s1), the ns+ spinor energies are lower than the corresponding ns orbital energies for all atoms having ns spinor (ns+) as the outermost shell, as some preceding works suggested. This indicates that kinematical effects are larger than indirect relativistic effects (the shielding effects of the ionic core plus those due to electron-electron interactions among the valence electrons). For all atoms having np+ spinors as their outermost shell, in contrast, the np+ spinor energies are higher than the corresponding np orbital energies as again the preceding workers suggested. This implies that indirect relativistic effects are greater than kinematical effects. In the neutral light atoms, the np- spinor energies are close to the np+ spinor energies, but for the neutral heavy atoms, the np- spinor energies are considerably lower than the np+ spinor energies (similarly, the np- spinors are considerably tighter than the np+ spinors), indicating the importance of the direct relativistic effects in np-. In the valence nd and nf shells, the spinor energies are always higher than the corresponding orbital energies, except for 46Pd (4d10) (5s0). Correspondingly, the nd and nf spinors are more diffuse than the nd and nf orbitals, except for 46Pd.

Electronic spectra of DyF studied by four-component relativistic configuration interaction methods.

The electronic states of the DyF molecule below 3.0 eV are studied using 4-component relativistic CI methods. Spinors generated by the average-of-configuration Hartree-Fock method with the Dirac-Coulomb Hamiltonian were used in CI calculations by the KRCI (Kramers-restricted configuration interaction) program. The CI reference space was generated by distributing 11 electrons among the 11 Kramers pairs composed mainly of Dy [4f], [6s], [6p] atomic spinors, and double excitations are allowed from this space to the virtual molecular spinors. The CI calculations indicate that the ground state has the dominant configuration (4f(9))(6s(2))(Ω = 7.5). Above this ground state, 4 low-lying excited states (Ω = 8.5, 7.5, 7.5, 7.5) are found with dominant configurations (4f(10))(6s). These results are consistent with the experimental studies of McCarthy et al. Above these 5 states, 2 states were observed at T0 = 2.39 eV, 2.52 eV by McCarthy et al. and were named as [19.3]8.5 and [20.3]8.5. McCarthy et al. proposed that both states have dominant configurations (4f(9))(6s)(6p), but these configurations are not consistent with the large Re's (∼3.9 a.u.) estimated from the observed rotational constants. The present CI calculations provide near-degenerate states of (4f(10))(6p3/2,1/2), (4f(10))(6p3/2,3/2), and (4f(9))(6s)(6p3/2,1/2) at around 3 eV. The former two states have larger Re (3.88 a.u.) than the third, so that it is reasonable to assign (4f(10))(6p3/2,1/2) to [19.3]8.5 and (4f(10))(6p3/2,3/2) to [20.3]8.5.

Electronic structure of CeO studied by a four-component relativistic configuration interaction method.

We studied the ground and excited states of CeO using the restricted active space CI method in the energy range below 25,000 cm(-1). Energy levels are computed to within errors of 2700 cm(-1). Electron correlation effects arising from the ionic core composed of Ce5s, 5p, 4f(*), 5d(*), and O2s, 2p spinors play crucial role to CeO spectra, as well as correlation effects of electrons distributed in the valence Ce 4f, 5d, 6s, and 6p spinors. Here, 4f(*) and 5d(*) denote spinors expanded to describe electron polarization between Ce and O. A bonding mechanism is proposed for CeO. As the two separate atoms in their ground states, Ce(4f(1)5d(1)6s(2))(1)G4 and O(2s(2)2p(4))(3)P2, approach each other, a CeO(2+) core is formed by two-electron transfer from Ce5d, 6s to O2p. Inside this ellipsoidal ion, a valence bond between Ce5p and O2s and an ionic bond between O2p and Ce5p are formed with back-donation through Ce 4f(*) and 5d(*).

Cu 4s → 4p atomic like excitations in the Ne matrix.

The lowest three or four excited states (the triplet or quartet states) of the Cu atom in a neon (Ne) matrix have been studied experimentally, and have been presumed to have the electronic configuration of Cu 4p(1). The origins of the triplet and the quartet are not yet fully clear, although many models have been proposed. It has been argued, for example, that the existence of different trapping sites would give rise to two partly overlapping triplets, leading to spectra having three or four lines or more. Below, the electronic structures of the ground state and lowest excited states of the Cu atom in the neon matrix are clarified by means of ab initio molecular orbital calculations, using the cluster model. It was found that a rather large vacancy (hollow) with residual Ne atoms is vital for explaining the observed spectra having three or more lines; the Cu atom occupies the center of the substitutional site of a face-centered cubic (fcc)-like cluster comprising 66 Ne atoms, in which the first shell composed of 12 Ne atoms is empty. The presence of the residual Ne atoms in the first shell gives rise to more than three excited states, explaining the experimental spectra. Electron-electron interaction (including the crystal field) and spin-orbit interaction are both important in explaining the experimental spectra.

Electronic spectra of GdF reanalyzed by decomposing state functions according to f-shell angular momentum.

The electronic structure of the GdF molecule was studied by means of four-component relativistic configuration interaction (CI) calculations [S. Yamamoto, H. Tatewaki, and T. Saue, J. Chem. Phys. 129, 244505 (2008)]. To analyze the electronic spectra more accurately, the CI wave function is decomposed according to the angular momentum (Ω(f)) generated from the (4f)(7) electrons. The weight of a specified Ω(f) is referred to as the "f-shell Omega component weight." This Ω(f) plays a crucial role in classifying the strong electronic transitions from the upper states (0.7 eV-3.0 eV) to the lower states (~0.55 eV). For these transitions, the upper and lower states have almost identical Ω(f) weights. This appears to be a necessary condition for a transition to be strong. The same condition is expected to hold for other lanthanide linear molecules. A point charge model is also studied, acting as a simplified model of GdF; it successfully reproduces the spectra of GdF, justifying studies based on ligand field theory.

Effect of removing the no-virtual pair approximation on the correlation energy of the He isoelectronic sequence. II. Point nuclear charge model.

The correlation energies (CEs) of the He isoelectronic sequence Z=2-116 with a point nuclear charge model were investigated with the four component relativistic configuration interaction method. We obtained CEs with and without the virtual pair approximation which are close to the values from Pestka et al.'s Hylleraas-type configuration interaction calculation. We also found that the uniform charge and point charge models for the nucleus differ substantially for Z > or = 100.

Electronic structure of LaO based on frozen-core four-component relativistic multiconfigurational quasidegenerate perturbation theory.

The electronic structure of the LaO molecule is studied using frozen-core four-component multiconfigurational quasidegenerate perturbation theory. The ground state and nine experimentally observed excited states are examined. The ground state is (2)Sigma(1/2)(+) and its gross atomic orbital population is La(5p(5.76)6s(0.83)6p(0.14)p(*(0.21) )d(*(1.17) )f(*(0.26) )) O(2p(4.63)), where p*, d*, and f* are the polarization functions of La that form molecular spinors with O 2ps. We found that it is not necessary to consider the excitation from the O 2p electrons when analyzing the experimental spectra. This validates the foundation of the ligand field theory on diatomic molecules, including the La atom where only one electron is considered. The spectroscopic constants R(e), omega(e), and T(0) calculated for the ground state and low-lying excited states A'((2)Delta(3/2)), A'((2)Delta(5/2)) A((2)Pi(1/2)), and A((2)Pi(3/2)) are in good agreement with the experimental values.

Excited states of PbF: a four-component relativistic study.

The electronic states of lead monofluoride (PbF) are studied from the (Pb 6s)(2) (F 2p-pi)(4) (F 2p-sigma)(2) (Pb 6p-pi)(1) X(1) ground state up to the F state, using the four-component relativistic configuration interaction and Fock-space coupled-cluster singles and doubles methods. Difficulties arising from the valence-Rydberg mixing are overcome by using a flexible basis set including Rydberg-type diffuse functions and by large-scale correlation calculations. The excited states are successfully characterized with the help of computed transition dipole moments. The three lowest-lying states (X(1), X(2), and A) are confirmed to be valence states arising from the (Pb 6p) spinors. The B state is assigned to the lowest Rydberg state (Omega=1/2), represented by a single excitation from the (Pb 6p-pi) spinor to the (F 3s) Rydberg spinor. Its calculated excitation energy (4.30 eV) is comparable to the observed one (4.42 eV). The C state is a multiconfigurational valence state whose dominant configuration is represented by (Pb 6s)(2) (F 2p-pi)(4) (F 2p-sigma)(1) (Pb 6p-pi)(2). Its calculated excitation energy (4.71 eV) is in good agreement with experiment (4.72 eV). The remaining D, E, and F states are assigned as Rydberg states. The calculated ionization potential (7.44 eV) is also close to the value (7.55 eV) determined recently by multiphoton ionization experiments.

Electronic structure of CeF from frozen-core four-component relativistic multiconfigurational quasidegenerate perturbation theory.

We have investigated the ground state and the two lowest excited states of the CeF molecule using four-component relativistic multiconfigurational quasidegenerate perturbation theory calculations, assuming the reduced frozen-core approximation. The ground state is found to be (4f(1))(5d(1))(6s(1)), with Omega = 3.5, where Omega is the total electronic angular momentum around the molecular axis. The lowest excited state with Omega = 4.5 is calculated to be 0.104 eV above the ground state and corresponds to the state experimentally found at 0.087 eV. The second lowest excited state is experimentally found at 0.186 eV above the ground state, with Omega = 3.5 based on ligand field theory calculations. The corresponding state having Omega = 3.5 is calculated to be 0.314 eV above the ground state. Around this state, we also have the state with Omega = 4.5. The spectroscopic constants R(e), omega(e), and nu(1-0) calculated for the ground and first excited states are in almost perfect agreement with the experimental values. The characteristics of the CeF ground state are discussed, making comparison with the LaF(+) and LaF molecules. We denote the d- and f-like polarization functions as d(*) and f(*). The chemical bond of CeF is constructed via {Ce(3.6+)(5p(6)d(*0.3)f(*0.1))F(0.6-)(2p(5.6))}(3+) formation, which causes the three valence electrons to be localized at Ce(3.6+).

Experimental and theoretical approaches toward anion-responsive tripod-lanthanide complexes: mixed-donor ligand effects on lanthanide complexation and luminescence sensing profiles.

A new series of tripods were designed to form anion-responsive, luminescent lanthanide complexes. These tripods contain pyridine, thiazole, pyrazine, or quinoline chromophores combined with amide carbonyl oxygen and tertiary nitrogen atoms. Crystallographic and EXAFS studies of the 10-coordinated tripod-La(NO(3))(3) complexes revealed that each La(3+) cation was cooperatively coordinated by one tetradentate tripod and three bidentate NO(3)(-) anions in the crystal and in CH(3)CN. Quantum chemical calculations indicated that the aromatic nitrogen plays a significant role in lanthanide complexation. The experimentally determined stability constants of complexes of the tripod with La(NO(3))(3), Eu(NO(3))(3), and Tb(NO(3))(3) were in good agreement with the theoretically calculated interaction energies. Complexation of each tripod with lanthanide triflate gave a mixture of several lanthanide complex species. Interestingly, the addition of a coordinative NO(3)(-) or Cl(-) anion to the mixture significantly influenced the lanthanide complexation profiles. The particular combination of tripod and a luminescent Eu(3+) center gave anion-selective luminescence enhancements. Pyridine-containing tripods exhibited the highest NO(3)(-) anion-selective luminescence and thus permit naked-eye detection of the NO(3)(-) anion.

Electronic structure of LaF+ and LaF from frozen-core four-component relativistic multiconfigurational quasidegenerate perturbation theory.

The electronic structure of the molecules LaF+ and LaF was studied using frozen-core four-component multiconfigurational quasidegenerate perturbation theory. To obtain proper excitation energies for LaF+, it was essential to include electronic correlations between the outermost valence electrons (4f, 5d, and 6s) and ionic core electrons composed of (4s, 4p, 4d, 5s, and 5p). The lowest-lying 16 excited states were examined for LaF+, and the lowest 30 states were examined for LaF. The excitation energies calculated for LaF+ agree with the available experimental values, as well as with values from ligand field theory. Errors are within 0.4 eV; for example, the highest observed state 2Pi is 3.77 eV above the ground state, and the present value is 4.09 eV. For LaF, agreement between the experimental and theoretical state assignments and between the experimental and calculated excitation energies was generally good, except for the electron configurations of certain states. Errors are within 0.4 eV except for a single anomaly; for example, the highest observed excited-state discussed in this work is 2.80 eV above the ground state, and the present value is 2.42 eV. We discuss the characteristics of the bonding in LaF+ and LaF.

Dipole allowed transitions in GdF: A four-component relativistic general open-shell configuration interaction study.

A four-component relativistic study of electronic transitions in the gadolinium monofluoride molecule (GdF) is presented. The electronic spectra of GdF have been investigated with a general open-shell configuration interaction method, where active electrons are distributed among molecular spinors mainly consisting of the Gd 4f, 5d, and 6s atomic spinors. The near-degeneracy effects of these spinors on the molecular electronic structure are considered by the valence full-CI-like approach. By the magnitudes of calculated transition dipole moments, the candidates for the observable transitions were selected. The present result is complementary to our previous study based on multireference configuration interaction singles and doubles calculations, which identified the electronic excited states of GdF by comparing the calculated excitation energies and angular momenta with those given by the laser spectroscopy. The spectra of the excited states less than 3.0 eV have been refined with the help of the calculated transition probabilities. The transitions between the excited states are newly analyzed and a rearrangement is proposed.

Inversion of 4f-states in CeB6 thermally excited at 430 K.

The 4f states of Ce in a typical Kondo crystal, CeB(6), are split into an excited state Gamma(7) and the ground state Gamma(8), with an excitation energy at 560 K. The electron-density distribution of the thermally excited state was measured at 430 K using a four-circle diffractometer equipped with a small furnace. In contrast to the previous results at lower temperature, electrons are transferred from B(6) to Ce at 430 K. X-ray atomic-orbital analysis revealed that the 5d-Gamma(8) orbitals (the energy level of which is similar to that of the B-2p orbitals) are fully occupied and the 4f-Gamma(7) orbitals are more populated than the 4f-Gamma(8) orbitals. Fully occupied 5d-Gamma(8) makes the 4f-Gamma(8) states unstable and the energy levels of 4f-Gamma(7) and 4f-Gamma(8) are inverted.

Electronic structures and bonding of CeF: a frozen-core four-component relativistic configuration interaction study.

We study the electronic structure of the ground and several low-lying states of the CeF molecule using Dirac-Fock-Roothaan (DFR) and four-component relativistic single and double excitation configuration interaction (SDCI) calculations in the reduced frozen-core approximation (RFCA). The ground state and two low-lying excited states are calculated to have (4f)1(5d)1(6s)1 configurations with Omega = 3.5, 4.5, and 3.5, and the resulting excitation energies, T0, are, respectively, 0.319 and 0.518 eV. The experimental configurations for these states are the same, although the experimental T0 values are approximately 0.3 eV smaller than those calculated. Experimentally, the red-degraded band was observed to be 2.181 eV above the ground state, having the configuration (4f)1(5d)1(6p)1 with Omega = 4.5. The calculation for this state gives 2.197 eV and configuration (4f)1.0(5d)1.7(6p)0.3 with Omega = 4.5. We found that Omega, Re, and nu(1-0) obtained by CI agree well with experiment. Bonding between the Ce and the F is highly ionic. The 4f, 5d, and 6s valence electrons are localized at the Ce+ ion, because they are attracted by the Ce4+ ion core, and are excluded from the bonding region because of the electronic cloud around the negatively charged fluoride anion. The bonding in the ground and excited states of the CeF molecule is significantly influenced by the 6s and 5d electron distributions between the Ce and the F.

Effect of removing the no-virtual-pair approximation on the correlation energy of the He isoelectronic sequence.

The correlation energies (CEs) for the He-like ions are studied with the virtual-pair approximation (VPA) and with the no-virtual-pair approximation (NVPA). In contrast to the nonrelativistic CEs, the CEs calculated with relativity fell sharply as the nuclear charge Z increased, although the CE calculated with the NVPA was considerably lower than with the VPA for the heavier atoms. It is shown that CE calculated with a Hylleraas-type function implicitly includes the effects of the excitations into negative-energy states, which corresponds to the VPA. The present results verify that the strong dependence on Z of the CE of He-like ions is an essential effect of the relativity.

Electronic structure of the GdF molecule by frozen-core four-component relativistic configuration interaction calculations.

The electronic structure of GdF is calculated based on frozen-core four-component relativistic configuration interactions. The resulting excitation energies are fairly close to experiment and correctly designate the excited states. For instance, the existence of the experimentally inferred state at 0.55 eV above the ground state is confirmed, having Omega=132 with (4f(7)5d(+) (1)6s(+) (1)); it is 0.58 eV above the ground state according to the present calculation.

Gaussian-type function set without prolapse for the Dirac-Fock-Roothaan equation (II): 80Hg through 103Lr.

We present prolapse-free universal Gaussian-type basis sets for 80Hg through 103Lr. The basis set is determined so that the Dirac-Fock-Roothaan total energy should decrease monotonically toward the numerical Dirac-Fock total energy. The difference between the Dirac-Fock-Roothaan total energy and the numerical Dirac-Fock total energy is less than 3 x 10(-6) hartree for 1H through 102No, and less than 5 x 10(-6) hartree for 103Lr. The exponents of the present sets are determined in an even-tempered manner, aiming to give total energy closer to the numerical Dirac-Fock value as the expansion term increases. The recommended set is expanded by (64, 64, 64, 46, 46, 46, 46) terms for (s+, p-, p+, d-, d+, f-, f+) symmetries, respectively. A practical set with (56, 48, 48, 36, 36, 36, 36) terms is also presented.

A study of the ground state of manganese dimer using quasidegenerate perturbation theory.

We study the electronic structure of the ground state of the manganese dimer using the state-averaged complete active space self-consistent field method, followed by second-order quasidegenerate perturbation theory. Overall potential energy curves are calculated for the 1Sigmag+, 11Sigmau+, and 11Piu states, which are candidates for the ground state. Of these states, the 1Sigmag+ state has the lowest energy and we therefore identify it as the ground state. We find values of 3.29 A, 0.14 eV, and 53.46 cm(-1) for the bond length, dissociation energy, and vibrational frequency, in good agreement with the observed values of 3.4 A, 0.1 eV, and 68.1 cm(-1) in rare-gas matrices. These values show that the manganese dimer is a van der Waals molecule with antiferromagnetic coupling.

Correlation energies for He isoelectronic sequence with Z=2-116 from four-component relativistic configuration interactions.

The relativistic correlation energies (CEs) for the He isoelectronic sequence from 2He to 116Uuh were investigated using configuration-interaction (CI) calculations. We used a large universal-type Gaussian basis set, which gives accurate Dirac-Fock total energies for the ions under consideration. In contrast to nonrelativistic CEs, the relativistic CEs decrease monotonically with increasing nuclear charge, but the p-, d-, and f-partial CEs have a hump like the relativistic Hylleraas CI.

Characterization of molecular orbitals by counting nodal regions.

The number of nodal regions can be used as an index for characterizing molecular orbitals. A computer program has been developed to count the number of nodal regions, based on the labeling and contraction algorithms. This program is applied to the water molecule, the hydrogen sulfide molecule, the hydrogen atomic orbitals, the Rydberg excited states of ethylene, dissociation of carbon monoxide, and CASSCF calculations of formaldehyde. Because the number of nodal regions is independent of the coordinate system, the method is applicable even when the molecular structure changes drastically as in bond rotation or bond elongation. Changes of nodal regions with bond elongation are investigated for carbon monoxide. A prescription for problems arising with basis set expansion techniques is also given.