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Improved density functional calculations including magnetic effects for RfCl4 and its homologues
Abstract
We use the newly developed non-collinear spin polarized density functional method to describe the tetrachlorides of element rutherfordium (Rf) and its homologues. It is the first time that a real three-dimensional molecule is described with this method. Without any additional corrections (used so far for the atomic values) we get nearly complete agreement for all homologues and thus a good prediction for the unknown value for RfCl4.
... 14 Other improvements and the development of the non-collinear spin-polarized (SP) version of the 4c-DFT method provided an even more accurate atomization energy of RfCl 4 of 19.53 eV. 15 The lower value of D e (RfCl 4 ) in comparison with D e (ZrCl 4 ) and D e (HfCl 4 ) was originally explained by a smaller ionic component of bonding. 13 The following relativistic effective core potential (RECP) calculations of Han et al. 16 have given D e (RfCl 4 ) of 18.8 eV. ...
... The calculated D e are expected to be somewhat overestimated, as the use of the functionals dependent on the density rather than on the spin density is likely to result in overbinding. 15 The vertical IPs were obtained by taking the total energy difference between the neutral and the singly charged molecules at the equilibrium geometry of the neutral systems. Projection analysis 49 was used to define the MO composition and the character of bonding. ...
... The calculated X2C DFT R e of MCl 4 (M = Ti, Zr, and Hf) are in good agreement with the experimental values, as well as with the R e from the spin-polarized (SP) 4c-DFT calculations 15 (Table II and Fig. 3). For the closedshell systems both non-SP and SP methods give the same R e . ...
Relativistic, infinite order exact two-component, density functional theory electronic structure calculations were performed for MCl4 and MOCl2 of group-4 elements Ti, Zr, Hf, and element 104, Rf, with the aim to predict their behaviour in gas-phase chromatography experiments. RfCl4 and RfOCl2 were shown to be less stable than their lighter homologs in the group, tetrachlorides and oxychlorides of Zr and Hf, respectively. The oxychlorides turned out to be stable as a bent structure, though the stabilization energy with respect to the flat one (C2v) is very small. The trend in the formation of the tetrachlorides from the oxychlorides in group 4 is shown to be Zr < Hf < Rf, while the one in the formation of the oxychlorides from the chlorides is opposite. All the calculated properties are used to estimate adsorption energy of these species on various surfaces in order to interpret results of gas-phase chromatography experiments, as is shown in Paper II.
... The electronic structure of this ion is non-trivial, as it is affected by remarkable electron correlation and strong relativistic effects [3,[21][22][23]. In spite of this complexity, studies dating back to the 1990s have been devoted to the investigation of its energy levels [3,24,25], atomic radii [26], ionization potentials [26], oxidation states and chemical properties [27][28][29][30]. These studies showed that Rf + features a few well-spaced long-living excited states that can be exploited for the optical-pumping step of the LRC experiment; in this regard, Ramanoantoanina et al. [3] proposed a four-level optical pumping scheme. ...
We propose a theoretically designed laser resonance chromatography (LRC) experiment on Rf + (Z = 104) drifting in He buffer gas. To this end, we first developed a four-level rate equation model that simulates the optical pumping of Rf + from its ground state, 2 D 3/2 (7s 2 6d 1), to the metastable 4 F 3/2 (7s 1 6d 2) state via laser resonant excitation of the intermediate 4 F 3/2 (7s 1 6d 1 7p 1) state prior to electronic state chromatography. This model predicts a 93% pumping efficiency that suffices to enable efficient laser resonance chromatography of this ion. We then performed accurate relativistic Multi-Reference Configuration-Interaction (MRCI) calculations to model the interaction of Rf + with He in the ground 2 D 3/2 (7s 2 6d 1), low-lying 2 D 5/2 (7s 2 6d 1), and metastable 4 F 3/2 (7s 1 6d 2) states. These ion-atom interaction potentials were used to calculate the state-specific ion mobilities. For gas temperatures above 100 K and small applied electric fields, the reduced ion mobilities of the ground and metastable states differ significantly. In particular, at room temperature the difference between the reduced ion mobilities of these states is larger than 11%, and therefore should be sufficiently large to ensure LRC of this ion.
... Nonetheless, Rf has been in the spotlight of many theoretical investigations. Since the early 90s, several studies have been devoted to predictions of its energy levels [11], atomic radii [12], ionisation potentials [12], oxidation states, and chemical properties [13,14,15,16]. For the Rf + ion, to the best of our knowledge, very few theoretical data are found. ...
We report calculation of the energy spectrum and the spectroscopic properties of the superheavy element ion: Rf^+. We use the 4-component relativistic Dirac-Coulomb Hamiltonian and the multireference configuration interaction (MRCI) model to tackle the complex electronic structure problem that combines strong relativistic effects and electron correlation. We determine the energies of the ground and the low-lying excited states of Rf+, which originate from the 7s^26d^1, 7s^16d^2, 7s^27p^1, and 7s^16d^17p^1 configurations. The results are discussed vis-\`a-vis the lighter homologue, Hf^+ ion. We also assess the uncertainties of the predicted energy levels. The main purpose of the presented calculations is to provide a reliable prediction of the energy levels and to identify suitable metastable excited states that are good candidates for the planned ion-mobility-assisted laser spectroscopy studies.
We report calculation of the energy spectrum and the spectroscopic properties of the superheavy element ion: Rf+. We use the four-component relativistic Dirac-Coulomb Hamiltonian and the multireference configuration interaction model to tackle the complex electronic structure problem that combines strong relativistic effects and electron correlation. We determine the energies of the ground and the low-lying excited states of Rf+, which originate from the 7s26d1,7s16d2,7s27p1, and 7s16d17p1 configurations. The results are discussed vis-à-vis the lighter homolog Hf+ ion. We also assess the uncertainties of the predicted energy levels. The main purpose of the presented calculations is to provide a reliable prediction of the energy levels and to identify suitable metastable excited states that are good candidates for the planned ion-mobility-assisted laser spectroscopy studies.
Theoretical chemical studies demonstrated crucial importance of relativistic effects in the physics and chemistry of superheavy elements (SHEs). Performed, with many of them, in a close link to the experimental research, those investigations have shown that relativistic effects determine periodicities in physical and chemical properties of the elements in the chemical groups and rows of the Periodic Table beyond the 6 th one. They could, however, also lead to some deviations from the established trends, so that the predictive power of the Periodic Table in this area may be lost. Results of those studies are overviewed here, with comparison to the recent experimental investigations.
Spectacular developments in the relativistic quantum theory and computational algorithms in the last few decades allowed for accurate calculations of properties of the superheavy elements (SHE) and their compounds. Often conducted in a close link to the experimental research, these investigations helped predict and interpret an outcome of sophisticated and expensive experiments with single atoms. Most of the works, particularly those related to the experimental studies, are overviewed in this publication. The role of relativistic effects being of paramount importance for the heaviest elements is elucidated.
With the aim to interpret results of gas-phase chromatography experiments on volatility of group-4 tetrachlorides and oxychlorides including those of Rf, adsorption enthalpies of these species on neutral, and modified quartz surfaces were estimated on the basis of relativistic, two-component Density Functional Theory calculations of MCl4, MOCl2, MCl6 (-), and MOCl4 (2) with the use of adsorption models. Several mechanisms of adsorption were considered. In the case of physisorption of MCl4, the trend in the adsorption energy in the group should be Zr > Hf > Rf, so that the volatility should change in the opposite direction. The latter trend complies with the one in the sublimation enthalpies, ΔHsub, of the Zr and Hf tetrachlorides, i.e., Zr < Hf. On the basis of a correlation between these quantities, ΔHsub(RfCl4) was predicted as 104.2 kJ/mol. The energy of physisorption of MOCl2 on quartz should increase in the group, Zr < Hf < Rf, as defined by increasing dipole moments of these molecules along the series. In the case of adsorption of MCl4 on quartz by chemical forces, formation of the MOCl2 or MOCl4 (2-) complexes on the surface can take place, so that the sequence in the adsorption energy should be Zr > Hf > Rf, as defined by the complex formation energies. In the case of adsorption of MCl4 on a chlorinated quartz surface, formation of the MCl6 (2-) surface complexes can occur, so that the trend in the adsorption strength should be Zr ≤ Hf < Rf. All the predicted sequences, showing a smooth change of the adsorption energy in the group, are in disagreement with the reversed trend Zr ≈ Rf < Hf, observed in the "one-atom-at-a-time" gas-phase chromatography experiments. Thus, currently no theoretical explanation can be found for the experimental observations.
An analysis of theoretical studies on the electronic structures and chemical properties of superheavy elements has structures and chemical properties of superheavy elements has been performed. It is noted that the specific features of been performed. It is noted that the specific features of calculations of molecular systems including superheavy ele-calculations of molecular systems including superheavy ele-ment atoms are determined by the exceptional role of relativ-ment atoms are determined by the exceptional role of relativ-istic effects that should be taken into account at all stages of istic effects that should be taken into account at all stages of electronic structure modelling. The main approximations for electronic structure modelling. The main approximations for relativistic Hamiltonians and general approaches to the sol-relativistic Hamiltonians and general approaches to the sol-ution of the many-electron problem for both superheavy ution of the many-electron problem for both superheavy element atoms and their compounds have been considered element atoms and their compounds have been considered and compared. The results of modelling the chemical proper-and compared. The results of modelling the chemical proper-ties of superheavy elements by ties of superheavy elements by ab initio ab initio methods and density methods and density functional theory have been analysed. The bibliography functional theory have been analysed. The bibliography includes 143 references includes 143 references. .
In this paper we will calculate the effect of spin-orbit coupling on properties of closed shell molecules, using the zero-order regular approximation to the Dirac equation. Results are obtained using density functionals including density gradient corrections. Close agreement with experiment is obtained for the calculated molecular properties of a number of heavy element diatomic molecules. © 1996 American Institute of Physics.@S0021-9606~96!02138-1# for some ~excited! atomic multiplet energies using this method. In Sec. IV we analyze the spin-orbit effects on the closed shell molecules I2 ,A u 2 ,B i 2, HI, AuH, TlH, IF, TlF, TlI, PbO, and PbTe. These effects have been extensively studied before, especially within a relativistic effective po- tential framework ~see, e.g., Refs. 11 and 12!. In particular Pitzer13,14 analyzed these effects in terms of the bonding be- tween the spin-orbit split jj coupled atomic spinors of the constituent atoms. Here we will proceed slightly differently by treating the spin-orbit interaction as a modification on a scalar relativistic ~LS coupled! starting point. Bond distances, harmonic frequencies, dissociation energies, and dipole mo- ments are discussed. The results for the dissociation energies are very accurate if we include gradient correction ~GGA! terms in the energy. Results for the dissociation energies are then usually within 0.1 eV of experiment, with only a few exceptions. The maximum deviation is 0.28 eV for PbO, and the average deviation is ;0.1 eV for this series of com- pounds. Bond distances and frequencies are not much af- fected by the spin-orbit coupling. Bond distances never change by more than 0.03 Å and for frequencies the effect is less than 10%. Finally we compare our results to other relativistic treat- ments, including Dirac-Fock, relativistic effective potential, and Douglas-Kroll methods.
In this paper we present results of four-component relativistic density-functional calculations for diatomic molecules with heavy constituents. The fully relativistic treatment of the electron kinematics is used for a consistent examination of the importance of gradient and relativistic corrections to the exchange-correlation energy functional. In agreement with recent scalar relativistic calculations, we find that relativistic corrections to exchange-correlation functionals give no significant contribution to the binding properties of the investigated diatomic molecules. On the other hand, the effect of gradient terms is sizable, leading to a clear improvement of dissociation energies over the standard local-density approximation. The usefulness of gradient contributions in the high-Z regime is nevertheless somewhat questioned by the fact that they overcorrect the small errors in bond lengths found with the local-density approximation.
First full-relativistic density functional calculations with the extension of the spin-polarization functional for the relativistic density functional theory in their collinear and noncollinear form are presented here for the molecular system Pt2. The agreement with experiment is very good.
An exact solution is obtained for the Schroedinger equation representing the motions of the nuclei in a diatomic molecule, when the potential energy function is assumed to be of a form similar to those required by Heitler and London and others. The allowed vibrational energy levels are found to be given by the formula E(n)=Ee+hω0(n+12)−hω0x(n+12)2, which is known to express the experimental values quite accurately. The empirical law relating the normal molecular separation r0 and the classical vibration frequency ω0 is shown to be r03ω0=K to within a probable error of 4 percent, where K is the same constant for all diatomic molecules and for all electronic levels. By means of this law, and by means of the solution above, the experimental data for many of the electronic levels of various molecules are analyzed and a table of constants is obtained from which the potential energy curves can be plotted. The changes in the above mentioned vibrational levels due to molecular rotation are found to agree with the Kratzer formula to the first approximation.
Coating of magnetic clusters by gold atoms is becoming an experimental technique of increasing interest for passivation and stabilization of these small metal particles. To computationally investigate the effect of gold coating, we have studied the magnetic clusters Ni6 and Ni13 employing an all-electron scalar-relativistic density functional method. We examine two series of octahedral clusters with increasing gold coverage of up to a monolayer: Ni6Aun (n = 0,8,32) and Ni13Aun (n = 0,6,8,14,24,30,42). Structural features, binding energies, and gold adsorption energies are determined and discussed. The different atomic radii of Au and Ni lead to rather short Au–Au and relatively long Ni–Ni distances in these clusters. The Au–Ni contacts are found to be the longest nearest-neighbor distances; a detailed analysis indicates these bonds to be the strongest in these Au-covered Ni clusters. The atomization energies change only slightly with increasing Au coverage. Also, the effect of gold adsorption on the magnetic properties of the Ni cores is analyzed. For the Ni6Aun series the magnetism decreases with n, while for Ni13Aun a maximum cluster magnetization is calculated for incomplete gold coverage. This different behavior of the two cluster series can be traced to differing numbers of unpaired electrons per atom in the pure Ni clusters and to an increased magnetic moment due to the adsorption of isolated Au atoms. Both series exhibit a residual magnetism at full monolayer coverage of the Ni cores. © 2001 American Institute of Physics.
Ab initio all-electron fully relativistic molecular spinor (RMS) Dirac–Fock (DF) self-consistent field (SCF) and nonrelativistic limit Hartree–Fock (HF) calculations are reported at four Rf–Cl bond distances for the ground state of tetrahedral (Td) rutherfordium tetrachloride (RfCl4) with our universal Gaussian basis set. The optimized Rf–Cl bond distance computed from our relativistic and nonrelativistic SCF wave functions for RfCl4 (Td) is 2.39 and 2.45 Å, respectively. The relativistic correction to the total electronic energy of RfCl4 was calculated as ∼−4355 hartrees (−118 504 eV) at the Dirac–Fock level. The dominant magnetic part of the Breit interaction correction for RfCl4 is estimated by perturbation method as 66.8509 hartrees. Our Hartree–Fock, Dirac–Fock, and Dirac–Fock–Breit calculations predict the tetrahedral RfCl4 to be bound with the calculated dissociation energy of −14.14, −15.56, and −15.53 eV, respectively. Mulliken population analysis of our Dirac–Fock wave function indicates RfCl4 (Td) to be more volatile than that estimated from the corresponding Hartree–Fock wave function. © 1998 American Institute of Physics.
The scalar relativistic variant of the linear combination of Gaussian-type orbitals—fitting functions—density-functional (R-LCGTO-FF-DF) method is extended to a two-component scheme which permits a self-consistent treatment of the spin–orbit interaction. The method is based on the Douglas–Kroll transformation of the four-component Dirac–Kohn–Sham equation. The present implementation in the program PARAGAUSS neglects spin–orbit effects in the electron–electron interaction. This approximation is shown to be satisfactory as long as bonding is restricted to s, p, and d orbitals. The method is applied to the diatomics Au2, Bi2, Pb2, PbO, and TlH using both a local density (LDA) and a gradient-corrected approximation (GGA) of the exchange-correlation functional. At the LDA level, bond lengths and vibrational frequencies are reproduced with high accuracy. For the determination of binding energies the open-shell reference atoms Au, Tl, Pb, Bi have been treated by a jj coupling approach based on a self-consistent noncollinear spin density-functional scheme and with an intermediate coupling procedure. The atomic state energies obtained with the jj coupling scheme agree well with experiment, but they are somewhat too high due to the incomplete inclusion of static correlation. Binding energies of diatomics at the GGA level are considerably improved due to the inclusion of spin–orbit interaction. The jj derived values are somewhat overestimated (by about 10%) compared to experiment, and they compare slightly worse with experiment than results based on the intermediate coupling approximation. © 2001 American Institute of Physics.
Calculations of total and binding energies of group IV tetrachlorides MCl4 (M = Ti, Zr, Hf, and element 104, Rf) were performed using the four-component fully relativistic density functional method. The calculations show the binding energies to be in good agreement with thermochemical dissociation energies obtained via the Born−Haber cycle. The method therefore demonstrates a predictive power which is important for further applications in the area of the heavy element compounds.
The Raman spectra of titanium tetrachloride (at 65°), titanium tetrabromide (at 125°), and the tetrachlorides, tetrabromides, and tetraiodides of zirconium and hafnium (at 380-420°) have been recorded in the vapor phase, and values for all the fundamentals of these molecules are presented. The ν 2(e) and ν 4(t 2) modes have pronounced rotational structure in each case. Although the zirconium and hafnium tetrahalides are polymeric in the solid state, their Raman spectra are interpreted to indicate that all are tetrahedral monomers in the vapor phase and thus isostructural with the titanium tetrahalides (which are tetrahedral in all states of matter). Force constants have been calculated for all the halides, including titanium tetraiodide (by the use of solution data) on the bases of both the modified-valence force field as well as the Urey-Bradley force field. Values for various thermodynamic functions have also been computed for each halide.
Describes a relativistic generalisation of the spin density functional formalism. The full Feynman propagator is presented and discussed. Effective single-particle Dirac equations for the polarised gas are given. Effective one-particle potentials for the magnetised gas are calculated in the local density scheme. These results are displayed graphically and their significance discussed.
The local density theory is developed by Hohenberg, Kohn and Sham is extended to the spin polarized case. A spin dependent one- electron potential pertinent to ground state properties is obtained from calculations of the total energy per electron made with a 'bubble' (or random phase) type of dielectric function. The potential is found to be well represented by an analytic expression corresponding to a shifted and rescaled spin dependent Slater potential. To test this potential the momentum dependent spin susceptibility of an electron gas is calculated. The results compare favourably with available information from other calculations and from experiment. The potential obtained in this paper should be useful for split band calculations of magnetic materials.
The electronic and geometrical structure of neutral and cationic Hg2 and Hg3 molecules are calculated using the all-electron Dirac-Fock-Slater SCF method, with relativistic numerical atomic basis functions. An improved calculation of the direct Coulomb potential has been taken into account in order to get a numerically accurate potential energy surface. The binding, ionization and excitation energies have been compared with experimental results as well as other theoretical results.
A full application of relativistic spin-density functional theory in noncollinear treatment of the exchange and correlation field is given to open-shell atoms and ions of the carbon group. It is shown that the influence of noncollinearity is small as compared with self-interaction corrections. Unfortunately, the defect of the local spin-density approximation not to yield a source-free exchange and correlation field increases in noncollinear treatment. © 1999 John Wiley & Sons, Inc. J Comput Chem 20: 23–30, 1999
The ground states of the halides and oxides containing transactinide elements Rf (element 104), Db (element 105), and Sg (element 106) were calculated at the HF, MP2, QCISD, CCSD, and CCSD(T) levels of theory using one-and two-component relativistic effective core potentials. Spin-orbit effects are rather small for geometries, harmonic vibrational frequencies, charge distributions, overlap populations, and dipole moments, but considerable for atomization energies. Electron correlations are necessary for any accurate determination of the molecular properties, in particular for the evaluation of atomization energies. The bond lengths of Sg compounds are consistently longer than those of the corresponding W compounds by 0.04-0.06 Å. The atomization energies for Sg compounds are slightly smaller than those for the corresponding W compounds due to spin-orbit and correlation effects. The differences tend to increase with the number of oxygen atoms in the compounds. Metal charges and dipole moments are larger for the Sg compounds than for the W compounds, implying that Sg is more ionic than W. The D 3h structures are calculated to be more stable by about 2 kcal/mol than the C 4V ones for TaCl 5 , TaBr 5 , DbCl 5 , and DbBr 5 .
An intra-atomic noncollinear magnetization density has been calculated for the case of ferromagnetic fcc Pu, by means of a newly implemented general-local-spin-density-approximation method which treats the magnetization density as a continuous vector quantity. The presence of noncollinearity is a general effect, not specific to Pu, which is shown to rise due to the interplay of the local exchange and the spin-orbit coupling. The form of the noncollinear part of the magnetization density is very sensitive to the space group symmetry as is demonstrated by calculations with the average spin moment along [001] and [111], respectively. {copyright} {ital 1996 The American Physical Society.}
With present day exchange-correlation functionals, accurate results in nonrelativistic open shell density functional calculations can only be obtained if one uses functionals that do not only depend on the electron density but also on the spin density. We consider the common case where such functionals are applied in relativistic density functional calculations. In scalar-relativistic calculations, the spin density can be defined conventionally, but if spin-orbit coupling is taken into account, spin is no longer a good quantum number and it is not clear what the "spin density" is. In many applications, a fixed quantization axis is used to define the spin density ("collinear approach"), but one can also use the length of the local spin magnetization vector without any reference to an external axis ("noncollinear approach"). These two possibilities are compared in this work both by formal analysis and numerical experiments. It is shown that the (nonrelativistic) exchange-correlation functional should be invariant with respect to rotations in spin space, and this only holds for the noncollinear approach. Total energies of open shell species are higher in the collinear approach because less exchange energy is assigned to a given Kohn-Sham reference function. More importantly, the collinear approach breaks rotational symmetry, that is, in molecular calculations one may find different energies for different orientations of the molecule. Data for the first ionization potentials of Tl, Pb, element 113, and element 114, and for the orientation dependence of the total energy of I+2 and PbF indicate that the error introduced by the collinear approximation is approximately 0.1 eV for valence ionization potentials, but can be much larger if highly ionized open shell states are considered. Rotational invariance is broken by the same amount. This clearly indicates that the collinear approach should not be used, as the full treatment is easily implemented and does not introduce much more computational effort.
The direct adjustment of two-component pseudopotentials (scalar-relativistic + spin-orbit potentials), to atomic total energy valence spectra derived from four-component multiconfiguration Dirac-Hartree-Fock all-electron calculations based on the Dirac-Coulomb-Breit Hamiltonian, has been made a routine tool for an efficient treatment of heavy main-group elements. Both large-core (nsp valence shell) and small-core ((n - 1)spd nsp valence shell) potentials have been generated for all the post-d elements of groups 13-17. At the example of lead and bismuth compounds (PbHal, BiH, BiO, BiHal (Hal = F, Cl, Br, I)), we show how small-core and large-core potentials can be combined in accurate, yet computationally economic, spin-free-state-shifted relativistic electronic structure calculations of molecular ground and excited states.
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