Spin-orbit Configuration Interaction Calculations Research Articles

On the Accuracy of Mean-Field Spin-Orbit Operators for 3d Transition-Metal Systems.

We present an extensive study of the performance of mean-field approximations to the spin-orbit operators on realistic molecular systems, as widely used in applications like single-molecule magnets, molecular quantum bits, and molecular spintronic devices. The test systems feature a 3d transition-metal center ion (V, Cr, Mn, Fe, Co, and Ni) in various coordinations and a multitude of energetically close-lying open-shell configurations that can couple via the spin-orbit operator. We performed complete active space spin-orbit configuration interaction calculations and compared the full two-electron Breit-Pauli spin-orbit operator to different approximations: the one-center approximation, the spin-orbit mean-field approach with electron densities from different state-averaging procedures, and the atomic mean-field integral approximation. We show that the mean-field approaches can lead to significant errors in the spin-orbital coupling matrix elements, which becomes particularly visible for the computed zero-field splittings. The one-center approximation, keeping all relevant two-electron terms, seems to be a significantly more accurate choice for the examples from our test set.

Time-resolved inner-shell photoelectron spectroscopy: From a bound molecule to an isolated atom

Due to its element- and site-specificity, inner-shell photoelectron spectroscopy is a widely used technique to probe the chemical structure of matter. Here we show that time-resolved inner-shell photoelectron spectroscopy can be employed to observe ultrafast chemical reactions and the electronic response to the nuclear motion with high sensitivity. The ultraviolet dissociation of iodomethane (CH$_3$I) is investigated by ionization above the iodine 4d edge, using time-resolved inner-shell photoelectron and photoion spectroscopy. The dynamics observed in the photoelectron spectra appear earlier and are faster than those seen in the iodine fragments. The experimental results are interpreted using crystal field and spin-orbit configuration interaction calculations, and demonstrate that time-resolved inner-shell photoelectron spectroscopy is a powerful tool to directly track ultrafast structural and electronic transformations in gas-phase molecules.

Effective bond orders from two-step spin-orbit coupling approaches: the I2, At2, IO(+), and AtO(+) case studies.

The nature of chemical bonds in heavy main-group diatomics is discussed from the viewpoint of effective bond orders, which are computed from spin-orbit wave functions resulting from spin-orbit configuration interaction calculations. The reliability of the relativistic correlated wave functions obtained in such two-step spin-orbit coupling frameworks is assessed by benchmark studies of the spectroscopic constants with respect to either experimental data, or state-of-the-art fully relativistic correlated calculations. The I2, At2, IO(+), and AtO(+) species are considered, and differences and similarities between the astatine and iodine elements are highlighted. In particular, we demonstrate that spin-orbit coupling weakens the covalent character of the bond in At2 even more than electron correlation, making the consideration of spin-orbit coupling compulsory for discussing chemical bonding in heavy (6p) main group element systems.

Electronic spectrum of the UO and UO(+) molecules.

Electronic theory calculations are applied to the study of the UO molecule and the UO(+) ion. Relativistic effective core potentials are used along with the accompanying valence spin-orbit operators. Polarized double-ς and triple-ς basis sets are used. Molecular orbitals are obtained from state-averaged multiconfiguration self-consistent field calculations and then used in multireference spin-orbit configuration interaction calculations with a number of millions of terms. The ground state of UO has open shells of 5f(3)7s(1), angular momentum Ω = 4, and a spin-orbit-induced avoided crossing near the equilibrium internuclear distance. Many UO excited states are studied with rotational constants, intensities, and experimental comparisons. The ground state of UO(+) is of 5f(3) nature with Ω = 9/2. Many UO(+) excited states are also studied. The open-shell nature of both UO and UO(+) leads to many low-lying excited states.

Magnetic anisotropy through cooperativity in multinuclear transition metal complexes: theoretical investigation of an anisotropic exchange mechanism

To be a candidate for a good single-molecule magnet, a compound should have a reasonably large zero-field splitting, a reasonably large total spin, and preferably axial symmetry. We investigate how this can be achieved in multi-heteronuclear d-Block transition metal complexes with both 3d and 5d centres. While spin-orbit coupling is large at the 5d centre, the 3d centres contribute many unpaired electrons. If the heavy-element building block does not have a single-ion anisotropy it might still transfer its spin-orbit effects to the 3d centre(s) through anisotropic exchange. We present spin-orbit configuration interaction calculations on a bi- and trinuclear compound with one OsIII and one or two MnII centres to demonstrate the mechanism. The interaction between the metal centres brings about zero-field splitting that is entirely due to anisotropic exchange. Because of strong spin-orbit coupling, the Os centre behaves as a pseudospin with that shows antiferromagnetic and anisotropic exchange coupling with the Mn centres. The ground state of the binuclear complex has a pseudo S = 2 ground state with an axial zero-field splitting parameter D = +0.59 cm−1, while the trinuclear complex has a pseudo ground state with a negative D = −0.32 cm−1. Coordination of the second Mn centre thus increases total spin and magnetic anisotropy, and reverses the sign of D.

Ab initioconfiguration interaction study of theB- andC-band photodissociation of methyl iodide

Multireference spin-orbit configuration interaction calculations have been carried out for the valence and low-lying Rydberg states of CH(3)I. Potential energy surfaces along the C-I dissociation coordinate (minimal energy paths with respect to the umbrella angle) have been obtained as well as transition moments for excitation of the Rydberg states. It is shown that the B and C absorption bands of CH(3)I are dominated by the perpendicular (3)R(1),(1)R (E)←X̃ A(1) transitions, while the (3)R(2)(E), (3)R(0(+) )(A(1))←X̃ A(1) transitions are very weak. It is demonstrated that the bound Rydberg states of the B and C bands are predissociated due to the interaction with the repulsive E and A(2) components of the (3)A(1) state, with the (3)A(1)(E) state being the main decay channel. It is predicted that the only possibility to obtain the I((2)P(3/2)) ground state atoms from the CH(3)I photodissociation in the B band is by interaction of the (3)R(1)(E) state with the repulsive (1)Q(E) valence state at excitation energies above 55,000 cm(-1). The calculated ab initio data are used to analyze the influence of the Rydberg state vibrational excitation on the decay process. It is shown that, in contrast to intuition, excitation of the ν(3) C-I stretching mode supresses the predissociation, whereas the ν(6) rocking vibration enhances the predissociation rate.

Anisotropy of the static dipole polarizability induced by the spin–orbit interaction: the S-state atoms N–Bi, Cr, Mo and Re

A systematic ab initio study is performed on the ground state static dipole polarizabilities of the 2 S +1 S, S >1/2, atoms from N to Bi, Cr, Mo, Mn, Tc and Re. The benchmark scalar-relativistic values of the scalar polarizability components are obtained using the coupled cluster method. The spin–orbit configuration interaction calculations are carried out for the anisotropic (tensor) polarizability components of these atoms (except Mn and Tc) that arise from the second-order spin–orbit interaction. The tensor polarizabilities are calculated for the first time and found to increase from 10 −5 (N) to 3.8 atomic units (Bi) approximately as the fourth power of the nuclear charge. The simple correlations and implication to magnetic trapping of cold atoms are discussed.

Energy level shifts in two-step spin–orbit coupling ab initio calculations

We point out a problem with two-step spin–orbit ab initio calculations in which the energy levels of spin–orbit free Hamiltonians are shifted as a means to including dynamic correlations at low cost in small spin–orbit configuration interaction calculations. The usual shifts driven by the energy order of the states can lead to anomalous results when avoided crossings exist with significant change of wave function character, which take place at different nuclear positions in the configurational spaces of the first and the second steps. In these cases, the shifts of the spin–orbit free energy levels must be assigned according to the characters of the wave functions.

Assessment of the Accuracy of Shape-Consistent Relativistic Effective Core Potentials Using Multireference Spin−Orbit Configuration Interaction Singles and Doubles Calculations of the Ground and Low-Lying Excited States of U4+and U5+

Multireference spin-orbit configuration interaction calculations were used to determine the accuracy of 60-, 68-, and 78-electron shape-consistent relativistic effective core potentials (RECPs) for uranium V and VI ground and low-lying excited states. Both 5f(n) and (5f6d)(n), (n = 1, 2) reference spaces were investigated using correlation-consistent double-zeta quality basis sets. Accuracy was assessed against gas-phase experimental spectra. The 68-electron RECP calculations yielded low relative and rms errors and predicted the empirical ordering of states most consistently.

An ab initio study of the CH3I photodissociation. II. Transition moments and vibrational state control of the I* quantum yields

Multireference spin-orbit configuration interaction calculations of transition moments from the X A1 ground state to the 3Q0+, 3Q1, and 1Q excited states responsible for the A absorption band of CH3I are reported and employed for an analysis of the photofragmentation in this system. Contrary to what is usually assumed, the 3Q0+(A1), 3Q1(E), and 1Q(E)<--X A1 transition moments are found to be strongly dependent on the C-I fragmentation coordinate. The sign of this dependence is opposite for the parallel and perpendicular transitions, which opens an opportunity for vibrational state control of the photodissociation product yields. The computed absorption intensity distribution and the I* quantum yield as a function of excitation energy are analyzed in comparison with existing experimental data, and good agreement between theory and experiment is found. It is predicted that significantly higher I* quantum yield values (>0.9) may be achieved when vibrationally hot CH3I molecules are excited in the appropriate spectral range. It is shown that vibrational state control of the I*/I branching ratio in the alkyl (hydrogen) iodide photodissociation has an electronic rather than a dynamic nature: Due to a different electron density distribution at various molecular geometries, one achieves a more efficient excitation of a particular fragmentation channel rather than influences the dynamics of the decay process.

Ab initio study of the BiSe and BiTe electronic spectra: what happens with X2-X1 emission in the heavier Bi chalcogenides?

A series of spin-orbit configuration interaction calculations has been carried out for the BiSe and BiTe molecules and analyzed in comparison with data obtained earlier for the isovalent BiO and BiS systems. An avoided crossing caused by the spin-orbit interaction between the X2Pi and A4Pi electronic states is shown to have a decisive effect on the lower-energy spectrum in each case. Irregularities in the X2 3/2 state vibrational manifold occur as a consequence of this nonadiabatic interaction, and the v vibrational number for the onset of these perturbations is found to gradually decrease in going from BiO to BiSe, in agreement with experiment. In BiTe the shape of the X2 potential curve is so altered by the avoided crossing that its minimum becomes shifted to a significantly larger distance than for the X1 state, unlike the case for BiSe or the lighter Bi chalcogenides. This characteristic appears to be the root cause for the fact that the X2 state has not yet been found experimentally in the BiTe spectrum, despite careful searches in the expected energy range. Radiative lifetimes have also been calculated for the low-lying states of both the BiSe and BiTe molecules, and these results are found to be consistent with experimental observations.

An ab initio study of the ionization potentials and f–f spectroscopy of europium atoms and ions

The first three ionization potentials of europium and the f–f spectroscopy of the two lowest multiplets of Eu+3 have been calculated using ab initio spin–orbit configuration interaction techniques. To accomplish this, a new averaged relativistic effective core potential has been developed which leaves only the 5s, 5p, and 4f in the valence space. A series of configuration interaction calculations were carried out up through single and partial double excitations with a double-zeta quality basis set. The computed ionization values have an absolute error of about 0.1 eV from the experimental values. The computed f–f spectroscopy for the lowest F7 multiplet of Eu+3 has a RMS error with experiment of about 100 cm−1. The computed f–f spectroscopy for the first excited D5 multiplet has a higher RMS error of about 350 cm−1. The computed center of gravity separation between the D5–F7 multiplet is underestimated by 750 cm−1. Comparisons between non-spin–orbit and spin–orbit configuration interaction calculations for the separations of the centers of gravity of multiplets are very favorable up through single and double excitations with differences of a tenth of an eV or less. The spin–orbit configuration interaction calculations are among the largest ever performed for lanthanides, with expansion lengths in excess of 1.9 million double-group-adapted functions. The calculations were achieved by application of a new parallel spin–orbit configuration interaction component in the COLUMBUS Program System.

Ab initio spin-orbit CI calculations of the potential curves and radiative lifetimes of low-lying states of lead monofluoride

The electronic structure of the lead monofluoride molecule is studied by means of ab initio configuration interaction (CI) calculations including the spin-orbit interaction. Potential-energy curves are generated for a large number of electronic states, of which only the X1 2Π1/2 ground and X2 2Π3/2 and A 2Σ+ excited states have been observed experimentally. Two different methods are compared for the inclusion of spin-orbit effects in the theoretical treatment, a contracted CI which employs a basis of large-scale Λ–S eigenfunctions to form a rather small matrix representation of the full relativistic Hamiltonian (two-step approach), and a more computationally laborious technique which involves solution of a secular equation of order 250 000 S2 eigenfunctions of different spin and spatial symmetry to achieve a potentially more evenly balanced description of both relativistic and electron correlation effects (one-step approach). In the present application, it is found that both methods achieve quite good agreement with measured spectroscopic constants for the X1, X2, and A states. The simpler of these methods is also employed to predict the radiative lifetimes of the latter two states. The key A 2Σ+–X 2Π transition moment in these calculations is found to vary strongly with internuclear distance and to vanish in the neighborhood of the respective equilibrium distances of both participating states. The computed lifetime for the A, v′=0 state of 16 μs overestimates the corresponding measured value by a factor of three, but those of higher vibrational states are found to decrease rather sharply with increasing v′, suggesting that only a slight displacement of the theoretical A–X transition moment curve is needed to explain the above discrepancy.

Wood-Boring ab initio model potential relativistic treatment of Ce and CeO

A simple, approximate method of simultaneously handling the 5 s and 6 s valence orbitals of the lanthanide elements in Wood-Boring based relativistic ab initio core model potential calculations (WB-AIMP), which had previously been formulated with the restriction of having only one valence orbital with a given l, is presented. Its applicability is shown in spin-orbit configuration interaction calculations on Ce and spin-free calculations on CeO. The good quality of the Wood-Boring spin-orbit operators of Ce is shown and a recommended contraction of its valence basis set is established. These results support the production of WB-AIMP data for the lanthanide series.

Pseudopotential study of the ground and excited states of Yb 2

The ground state of the van der Waals-type lanthanide dimer Yb2 has been studied by means of relativistic energy-consistent ab initio pseudopotentials using three different core definitions. Electron correlation was treated by coupled-cluster theory, whereby core-valence correlation effects have been accounted for either explicitly by correlating the energetically highest coreorbitals or implicitly by means of an effective core-polarization potential. Results for the first and second atomic ionization potentials, the atomic dipole polarizability, and the spectroscopic constants of the molecular ground state are reported. Low-lying excited states have been investigated with spin-orbit configuration interaction calculations. It is also demonstrated for the whole lanthanide series that correlation effects due to the atomic-like, possibly open 4f-shell in lanthanides can be modeled effectively by adding a core-polarization potential to pseudopotentials attributing the 4f-shell to the core.

Low-lying electronic states of lanthanocenes and actinocenes M(C8H8)2 (M=Nd, Tb, Yb, U)

Large-scale state-averaged multiconfiguration self-consistent field, multireference configuration interaction, and averaged coupled-pair functional calculations, including relativistic effects by means of energy-consistent quasirelativistic pseudopotentials, have been carried out for the ground and low-lying excited states of the di-π-cyclooctatetraene (or bis[8]annulene) metal sandwich complexes M(C8H8)2 (M=Nd, Tb, Yb, U). It is found that the ground state configurations for the lanthanocenes are 4fnπ3, while for the actinocenes they are 5fn−1π4. The ground states for the lighter and heavier lanthanocenes are, respectively, the lower and higher multiplicity states resulting from the coupling between the highest possible spin–multiplicity of the central metal 4fn subshell and the unpaired ligand π electron, whereas they always have the highest possible multiplicity of 5fn−1π4 for the actinocenes. The metal–ring distances and symmetric metal–ring stretching frequencies are reported. The special characteristics of the uranium 5f orbitals in uranocene are described. Extensive spin–orbit configuration interaction calculations were performed for uranocene and confirm the assignment of the ground state and first excited state of uranocene made previously by other authors. However, a different ordering is obtained for the higher states. The calculated term energy for the first excited state of uranocene is in excellent agreement with the experimental value.

Formally tetravalent cerium and thorium compounds: a configuration interaction study of cerocene Ce(C8H8)2 and thorocene Th(C8H8)2 using energy-adjusted quasirelativistic ab initio pseudopotentials

Abstract Large-scale state-averaged multi-configuration self-consistent field, configuration interaction, averaged coupled-pair functional and spin-orbit configuration interaction calculations have been carried out for the ground states and low-lying excited states of the bis(cyclooctatetraene)f-element sandwich complexes cerocene Ce(C 8 H 8 ) 2 and thorocene Th(C 8 H 8 ) 2 . Whereas for a single-determinant wavefunction thorocene may be pictured as a Th IV compound, i.e. a Th 4+ closed-shell ion complexed by two aromatic C 8 H 8 2− ligands, cerocene is to a first approximation a Ce III compound, i.e. a Ce 3+ ion with a 4f 1 configuration and two C 8 H 8 1.5− ligands. When cerocene is described by a multi-determinant wavefunction admixture of the Ce 4+ (C 8 H 8 2− ) 2 configuration leads to a 1 A 1g ground state of the same symmetry as in thorocene. For both compounds results for the metal-ring distance, the symmetric metal-ring stretching frequency, the photoelectron spectrum and the optical spectrum are compared to experimental data.

Electronic structure of actinocenes and actinofullerenes

Spin-orbit configuration-interaction calculations have been carried out on actinide complexes using relativistic core potentials and contracted Gaussian atomic orbitals. At present, this type of calculation gives moderate-accuracy results on ground states and excited states and on a number of molecular properties. Our previous work on uranocene has been continued with protactinocene, neptunocene, and plutonocene, giving the patterns of low-lying f→f states and higher-lying f→d and π→d states. Accurate magnetic moment values are obtained from these wavefunctions. The U@C 60 and U@C 28 complexes show extensive mixing of π orbitals with both 6d and 5f orbitals. The lower states of U@C 28 are of π ∗ f and f 2 character with the ground state appearing to be a π ∗ f state with mainly singlet character.

Energy-adjusted pseudopotentials for the actinides. Parameter sets and test calculations for thorium and thorium monoxide

We present nonrelativistic and quasirelativistic energy-adjusted pseudopotentials, the latter augmented by spin–orbit operators, as well as optimized (12s11p10d8f)/ [8s7p6d4f]-Gaussian-type orbitals (GTO) valence basis sets for the actinide elements actinium through lawrencium. Atomic excitation and ionization energies obtained by the use of these pseudopotentials and basis sets in self-consistent field (SCF) calculations differ by less than 0.2 eV from corresponding finite-difference all-electron results. Large-scale multiconfiguration self-consistent field (MCSCF), multireference configuration interaction (MRCI), and multireference averaged coupled-pair functional (MRACPF) calculations for thorium and thorium monoxide yield results in satisfactory agreement with available experimental data. Preliminary results from spin–orbit configuration interaction calculations for the low-lying electronic states of thorium monoxide are also reported.

Ab initio potential energy surfaces and trajectory studies of A-band photodissociation dynamics: ICN*→I+CN and I*+CN

The photodissociation reaction of ICN in the A continuum has been theoretically studied based on ab initio potential energy surfaces and classical trajectories. Ab initio contracted spin–orbit configuration interaction calculations have been carried out to obtain potential energy surfaces (PES’s) of 3Π1, 3Π0+ and 1Π1 excited states, where results are fit to five diabatic potential functions and their couplings as functions of all three internal degrees of freedom. The transition dipoles at the Franck–Condon region have also been calculated. All the PES’s involved in photodissociation are bent near the Franck–Condon region. Classical trajectory calculations performed on these potential surfaces have produced results that are in agreement with various experimental findings and provide a basis for their interpretation. The calculations indicate that the absorption is a mixture of parallel and perpendicular transition. A reasonable I/I* branching ratio can be obtained by considering the effect of initial bending vibrations in addition to the character of mixed transitions. The I/I* channel selectivity of the CN rotation can be compared to the shape of PES’s with respect to the bending angle. The rotational excitation of the CN fragment is determined by the shape of PES’s on which trajectories travel before and after the transition. The higher rotational component in the I channel is attributed to the energy gradient of 1Π1 with respect to the bending angle at the transition region where the C–I distance is between 5.0 and 8.0 a.u. The lower component in the I channel emerges from 3Π1. The average rotational distribution obtained with the proper weight of Boltzmann populations and transition intensities is in agreement with the experiment. This interpretation can also be applied to the rotational quantum number dependence of anisotropy parameters. Trajectory calculations on the 3Π1 surface alone, give a single Boltzmann rotational distribution. Reflecting the shape of PES’s with respect to the CN distance, the product CN vibration on 3Π0+ and 1Π becomes suppressed while that on 3Π1 becomes slightly more excited. The anisotropic parameter was also analyzed. Some comments on the femtosecond transition spectroscopy are also made.