Relativistic Effective Potentials Research Articles

A new generation of effective core potentials: Selected lanthanides and heavy elements.

We construct correlation-consistent effective core potentials (ccECPs) for a selected set of heavy atoms and f elements that are currently of significant interest in materials and chemical applications, including Y, Zr, Nb, Rh, Ta, Re, Pt, Gd, and Tb. As is customary, ccECPs consist of spin-orbit (SO) averaged relativistic effective potential (AREP) and effective SO terms. For the AREP part, our constructions are carried out within a relativistic coupled-cluster framework while also taking into account objective function one-particle characteristics for improved convergence in optimizations. The transferability is adjusted using binding curves of hydride and oxide molecules. We address the difficulties encountered with f elements, such as the presence of large cores and multiple near-degeneracies of excited levels. For these elements, we construct ccECPs with core-valence partitioning that includes 4f subshell in the valence space. The developed ccECPs achieve an excellent balance between accuracy, size of the valence space, and transferability and are also suitable to be used in plane wave codes with reasonable energy cutoffs.

The Space-Time Properties of Three Static Black Holes

In the curved space-time, the neutral test particle is not affected by any other force except for the influence of the curved space-time. Similar to the free sub in the flat space, the Lagrangian of the test particle only contains the kinetic energy term—the kinetic energy term of the four-dimensional curved space-time. In the case of small space-time curvature, linear approximation can be made. That is, under the weak field approximation, the Lagrangian quantity degenerates into the Lagrangian quantity in the axisymmetric gravitational field in Newtonian mechanics. In this paper, the curved space-time composed of axisymmetric equidistant black holes is taken as a model. We study the geodesic motion of the test particles around three black holes with equal mass and static axisymmetric distribution, including time-like particles and photons. The three extreme Reissner–Nordstrom black holes are balanced by electrostatic and gravitational forces. We first give the geodesic motion equation of particles in Three black holes space-time, give the relativistic effective potential, discuss the possible motion state of particles, and classify their motion trajectories. Then, the particle motion of the special plane (equatorial plane) is studied. The circular orbits of the two types of particles in the symmetric plane are studied, respectively. The circular orbits outside the symmetric plane are also studied, and their stability is also discussed. We will show the influence of the separation distance of the three black holes on the geodesic motion and explore the change of the relativistic effective potential. Then, the relationship between the inherent quantity and the coordinate quantity in space-time is analyzed. Finally, the chaos of the test particle orbit is explored.

A new generation of effective core potentials from correlated and spin-orbit calculations: Selected heavy elements.

We introduce new correlation consistent effective core potentials (ccECPs) for the elements I, Te, Bi, Ag, Au, Pd, Ir, Mo, and W with 4d, 5d, 6s, and 6p valence spaces. These ccECPs are given as a sum of spin-orbit averaged relativistic effective potential (AREP) and effective spin-orbit (SO) terms. The construction involves several steps with increasing refinements from more simple to fully correlated methods. The optimizations are carried out with objective functions that include weighted many-body atomic spectra, norm-conservation criteria, and SO splittings. Transferability tests involve molecular binding curves of corresponding hydride and oxide dimers. The constructed ccECPs are systematically better and in a few cases on par with previous effective core potential (ECP) tables on all tested criteria and provide a significant increase in accuracy for valence-only calculations with these elements. Our study confirms the importance of the AREP part in determining the overall quality of the ECP even in the presence of sizable spin-orbit effects. The subsequent quantum Monte Carlo calculations point out the importance of accurate trial wave functions that, in some cases (mid-series transition elements), require treatment well beyond a single-reference.

Theoretical investigations of structural, thermal properties and stability of the group 12 metal M(XH) isomers in atmosphere: M = (Zn, Cd, Hg) and XH = (OH, SH)

Electronic structure calculations have been performed of the group 12 metal (Zn, Cd, Hg) reaction between hydroxyl OH and hydrosulfide SH to understand the processes of transformation in the atmosphere. DFT/B3LYP methods are employed to calculate the structure of reactants, intermediates, transition states and products as well as their equilibrium geometries, electronic properties, dissociation energy. The scalar relativistic effects have been accurately included by making use of the relativistic effective potentials. The computing results obtained at B3LYP/MWB level of M(OH) and M(SH) for doublet and quartet (2,4∑+, 2A′, 4A″ and 2,4∏) state suggest that the doublet states are most stable than the quartet. The computed potential hypersurface 2A′ indicates four stationary points; two of them correspond to the stable structures with symmetries: MOH/MSH(2A′) and OMH/SMH(2∏), and two points correspond to transition states with the symmetries: MOH/MSH(2∑+) and MOH/MSH(2A′). The isomerization process MXH(2A′) → XMH(2∏) via the transition state and the atmospheric stability of M(XH) are also discussed. Examination of dissociation energy, enthalpy of reaction and wavelength from linear form OMH/SMH(2∏) estimate that the molecule has appreciable stability. As for the angled form MOH/MSH(2A′), the values obtained suggest that it is easily dissociable in the presence of solar radiation. Our results are compared with previous work, when available.

Samarium doped silicon clusters SmSin (n = 3–10) and their anions: Structures, thermochemistry, electron affinities, and magnetic moments

The structures, adiabatic electron affinities (AEAs), dissociation energies (DEs), hardness, and magnetic moments of SmSin (n=3–10) and their anions were examined by means of B2PLYP, TPSSh, PBE0 and B3LYP methods. Basis sets adopted are of segmented (SEG) Gaussian valence basis sets and relativistic small-core effective potentials (ECP) with additional diffuse 2dfg functions, denoted aug-SEG/ECP for Sm atoms and aug-cc-pVTZ for Si atoms. The most stable structures can be viewed as replacing a Si of the ground-state structure of Sin+1 with a Sm atom. The theoretical AEAs are in excellent agreement with experimental data, especially the AEAs at B2PLYP and TPSSh levels. The average absolute errors from experiment are by 0.04, 0.06, 0.10 and 0.11eV at the B2PLYP, the TPSSh, the PBE0 and the B3LYP levels, respectively. The DEs of Sm from SmSin (n=3–10) and their anions were estimated to examine their relative stabilities. Natural population analysis (NPA) reveals that the Sm atom in all the neutral and anionic clusters acts as an electron donor. Hardness analysis reveals that doping Sm atom to Sin (n=3–10) clusters raises the photochemical sensitivity. Analysis of magnetic moments reveals that the total magnetic moments are contributed by Samarium atom.

An ab Initio Benchmark and DFT Validation Study on Gold(I)-Catalyzed Hydroamination of Alkynes.

High level ab initio calculations have been carried out on an archetypal gold(I)-catalyzed reaction: hydroamination of ethyne. We studied up to 12 structures of possible gold(I)-coordinated species modeling different intermediates potentially present in a catalytic cycle for the addition of a protic nucleophile to an alkyne. The benchmark is used to evaluate the performances of some popular density functionals for describing geometries and relative energies of stationary points along the reaction profile. Most functionals (including hybrid or meta-hybrid) give accurate structures but large nonsystematic errors (4-12 kcal/mol) along the reaction energy profile. The double hybrid functional B2PLYP outperforms all considered functionals and compares very nicely with our reference ab initio benchmark energies. Moreover, we present an assessment of the accuracy of commonly used approaches to include relativistic effects, such as relativistic effective potentials and a scalar ZORA Hamiltonian, by a comparison with the results obtained using a relativistic all-electron four-component Dirac-Kohn-Sham method. The contribution of nonscalar relativistic effects in gold(I)-catalyzed reactions, as we investigated here, is expected to be on the order of 1 kcal/mol.

Electron trapping in the electrosound solitary wave for propagation of high intensity laser in a relativistic plasma

Using a set of relativistic equations for plasmas with warm electrons and cold ions, we have investigated the effects of trapped electrons in the propagation of an electrosound wave and discussed the possibility of the formation of electromagnetic solitons in a plasma. The effective potential energy and deviations of the electron and ion number densities in this relativistic model have been found. We have obtained the governing equations for the amplitude of the HF field with relativistic corrections. In order to show the destructive impact of the trapped electrons on the solitary wave, a relativistic effective potential and the governing equation have been found. It is shown that for certain values of the parameters the condition of localization of the HF amplitude is violated. In addition, it is shown that as the flow velocity of the plasma changes, the shape of the solitary wave shows two opposing behaviours, depending on whether the solitary wave velocity is larger than the flow velocity or smaller. Also, the existence of stationary solitary waves which are prohibited for nonrelativistic plasma has been predicted. Finally, we have obtained the Korteweg–de Vries equation showing the relativistic, trapping and nonlinearity effects.

A quantum chemical study of the structure of fluorinated silver acetate(I) monomers and dimers

Quantum chemical calculations of the equilibrium structure, potential energy surface cross-sections along the nonrigid degrees of freedom of (CF3COOAg)2, (CHF2COOAg)2, (CH2FCOOAg)2, and (CH3COOAg)2 dimers and the corresponding monomers are presented. The B3LYP method with the augmented cc-pVTZ basis set for C, O, F atoms and the Stuttgart 1997 RSC basis set together with the relativistic effective potential for Ag atoms is used. It is shown that in all dimers the eight-membered ring is a rather rigid planar fragment, and the Ag atoms can be bonded with the bond order of 0.2 in dimers. Almost free rotation of acyclic groups around the C-C bond in (CH3COOAg)2 and (CF3COOAg)2 dimers transforms into a hindered one in the (CHF2COOAg)2 dimer, and further into the existence of syn- and anti-structures divided by a high rotation barrier in the (CH2FCOOAg)2 dimer. In monomers, the ratio of the internal rotation barriers is similar. With increasing number of fluorine substitution for hydrogen atoms in dimers the Ag-Ag bond length is found to increase (2.79 A, 2.81 A, 2.83 A, 2.84 A) and the ring rigidity to simultaneously decrease in the acetate-triflouroacetate series.

Theoretical study of substituted methyl mercury in gas phase and in solution

Quantum chemistry calculations were carried out, using density functional theory, in order to investigate the thermodynamic properties, enthalpy (ΔH), Gibbs free energy (ΔG) and equilibrium constant (K$_{eq}$), for the dissociation of a series of methylmercury series compounds. Two kind of pseudopotentials were studied, the relativistic compact effective potentials and the Stuttgart pseudopotential. The B3LYP, BLYP, BP86 and BMK functionals were employed. The calculations were performed in gas phase and in presence of solvent. The results reveal that, none of the studied functionals was able to reproduce the experimental C-Hg and Hg-X, X=Cl, Br and I bond distances. The presence of solvent, water, acetonitrile and dimethylsulfoxide, has a strong influence on the ΔH and ΔG values. On the other hand, the results show that there is a lineal correlation between the logarithm of the experimental K$_{eq}$(exp) and the calculated K$_{eq}$(theo). This type of correlation could be useful in the resolution of problems related with the contamination by methylmercury compounds.

The Paczyński-Wiita potential. A step-by-step “derivation”

Paczynski realized that a properly chosen gravitational potential may accurately model (in a "pseudo Newtonian" theory) general relativistic effects that determine motion of matter near a non-rotating black hole. Paczynski's choice, known today as the "Paczynski-Wiita potential", proved to be very practical. It was used by numerous researchers in the black hole accretion theory, and became a standard tool in relativistic astrophysics. The model is an example of Paczynski's admired ability to invent "out of nowhere" simple ideas that were brilliant, deep and useful. Paczynski has guessed intuitively the form of the potential. However, it could be also derived by a a step-by-step formal procedure. I show the derivation here. My derivation is based on a standard definition of the relativistic "effective potential" in the Schwarzschild spacetime. The relativistic effective potential may be uniquely divided into its "gravitational" and "centrifugal" part. The gravitational part differs from the Paczynski-Wiita potential only by a constant.

BLOCK-LOCALIZED WAVEFUNCTION ENERGY DECOMPOSITION (BLW-ED) ANALYSIS OF σ/π INTERACTIONS IN METAL-CARBONYL BONDING

The bonding features in metal-carbonyls including neutral M CO (M = Ni , Pd , Pt ) and M CO + (M+ = Cu +, Ag +, Au +) complexes have been elucidated at the DFT level with relativistic compact effective potentials for transition metals and 6-311+G(d) basis sets for C and O by the block-localized wavefunction (BLW) method. The BLW method can decompose the intermolecular interactions in terms of Heitler–London, polarization, and charge transfer energy contributions. Since the metal– CO bonding involves two synergic interactions, namely the σ-dative bond from the carbon lone electron pair to an empty d σ orbital on the metal, and the π back-donation from filled d π orbitals to the empty 2π* orbital on CO , the present BLW-ED analyses quantitatively demonstrated that in neutral M CO complexes the π-bonding dominates over the σ-bonding, whereas in cationic M CO + complexes, the σ-bonding plays a major role. But in both neutral and cationic species, the CO polarization induced by the metals enhances the C–O bond and increases the C–O vibrational frequencies, while the π back-donation tends to weaken the C–O bond and decrease the C–O vibrational frequencies. For neutral complexes, the latter is more prominent than the former, and consequently, there is a red-shifting of the C–O vibrational frequencies. In contrast, the π back-donation is insignificant in M CO + cations, and the C–O eventually vibrates at higher frequencies than the free CO frequency.

Interplay Between Quark-Antiquark and DiquarkCondensates in Vacuum in a Two-Flavor Nambu–Jona–LasinioModel

By means of a relativistic effective potential, weanalytically research competition between the quark-antiquarkcondensates ⟨q̄q⟩ and the diquark condensates⟨qq⟩ in vacuum in ground state of a two-flavorNambu–Jona–Lasinio (NJL) model and obtain the GS-HS phasediagram, where GS and HS are the respective four-fermioncoupling constants in scalar quark-antiquark channel and scalarcolor anti-triplet diquark channel. The results show that, in thechiral limit, there is only the pure ⟨q̄q⟩phase when GS/HS>2/3, and as GS/HS decreases to2/3>GS/HS⩾0 one will first have a coexistence phase of thecondensates ⟨q̄q⟩ and ⟨qq⟩ andthen a pure ⟨qq⟩ phase. In non-zero bare quarkmass case, the critical value of GS/HS at which the pure⟨q̄q⟩ phase will transfer to the coexistencephase of the condensates ⟨q̄q⟩ and ⟨qq⟩ will be less than 2/3. Our theoretical results,combined with present phenomenological fact that there is nodiquark condensates in the vacuum of QCD, will also impose a realrestriction to any given two-flavor NJL model which is intended tosimulate QCD, i.e. in such model the resulting smallest ratioGS/HS after the Fierz transformations in the Hartreeapproximation must be larger than 2/3. A few phenomenologicalQCD-like NJL models are checked and analyzed.

High-order electron-correlation methods with scalar relativistic and spin-orbit corrections

An assortment of computer-generated, parallel-executable programs of ab initio electron-correlation methods has been fitted with the ability to use relativistic reference wave functions. This has been done on the basis of scalar relativistic and spin-orbit effective potentials and by allowing the computer-generated programs to handle complex-valued, spinless orbitals determined by these potentials. The electron-correlation methods that benefit from this extension are high-order coupled-cluster methods (up to quadruple excitation operators) for closed- and open-shell species, coupled-cluster methods for excited and ionized states (up to quadruples), second-order perturbation corrections to coupled-cluster methods (up to triples), high-order perturbation corrections to configuration-interaction singles, and active-space (multireference) coupled-cluster methods for the ground, excited, and ionized states (up to active-space quadruples). A subset of these methods is used jointly such that the dynamical correlation energies and scalar relativistic effects are computed by a lower-order electron-correlation method with more extensive basis sets and all-electron relativistic treatment, whereas the nondynamical correlation energies and spin-orbit effects are treated by a higher-order electron-correlation method with smaller basis sets and relativistic effective potentials. The authors demonstrate the utility and efficiency of this composite scheme in chemical simulation wherein the consideration of spin-orbit effects is essential: ionization energies of rare gases, spectroscopic constants of protonated rare gases, and photoelectron spectra of hydrogen halides.

A Computational Study of the Thermochemistry of Bromine- and Iodine-Containing Methanes and Methyl Radicals

Bromo- and iodomethanes and the corresponding halogenated methyl radicals have been investigated by ab initio methods. Geometries and vibrational frequencies were derived with quadratic configuration interaction methods at the QCISD/6-311G(d,p) level of theory, and energies via QCISD(T)/6-311+G(3df,2p). Core electrons were represented with relativistic effective potentials. Anharmonicity of the out-of-plane bending modes in the methyl radicals was taken into account by numerical integration of the Schrödinger equation with potentials derived from relaxed scans of these modes. The results are in good accord with experimental data where available. Thermochemistry derived via isodesmic reactions referenced to CH3, CH4, and monohalomethanes yields excellent accord with new experiments on dihalomethanes and provides recommendations for the more poorly characterized tri- and tetrahalomethanes and halomethyl radicals. For the methanes CH2Br2, CHBr3, CBr4, CH2I2, CHI3, CI4, CH2BrI, CHBr2I, and CHBrI2 we compute DeltafH degrees (298) values of 4.3, 51.6, 110.6, 108.1, 208.5, 321.3, 56.8, 104.8, and 157.1 kJ mol(-1), respectively. For the methyl radicals CH2Br, CHBr2, CBr3, CH2I, CHI2, CI3, CHBrI, CBr2I, and CBrI2 we compute DeltafH degrees (298) values of 166.6, 191.7, 224.0, 217.2, 290.4, 369.1, 241.6, 320.8, and 272.3 kJ mol(-1), respectively. Recommended confidence limits are +/-3 kJ mol(-1) per Br or I atom. Trends in these values and the corresponding C-H bond strengths are discussed and compared with prior experiments, empirical estimation schemes, and ab initio calculations.

Smooth relativistic Hartree–Fock pseudopotentials for H to Ba and Lu to Hg

We report smooth relativistic Hartree-Fock pseudopotentials (also known as averaged relativistic effective potentials) and spin-orbit operators for the atoms H to Ba and Lu to Hg. We remove the unphysical extremely nonlocal behavior resulting from the exchange interaction in a controlled manner, and represent the resulting pseudopotentials in an analytic form suitable for use within standard quantum chemistry codes. These pseudopotentials are suitable for use within Hartree-Fock and correlated wave function methods, including diffusion quantum Monte Carlo calculations.

Formation and stability of free charged lanthanum hexaboride clusters at field evaporation

Field evaporation of LaB6 single crystals resulting in a large variety of ionized clusters of LaBnm+ type, where n=1,2,…,6 and m=1, 2, 3, 4, is investigated in a collaborative experimental and theoretical study. The aim of the present work was to compare the measured appearance probability of various clusters in a flux with the calculated energetic stability of corresponding free clusters. The experiments were carried out with a time-of-flight atom probe. The appearance probability of cluster ions in the atom probe spectra was estimated by calculating its rating in many selected spectra. The methods for determining such ratings have been developed and discussed. The theoretical calculations are based on ab initio quantum chemical Hartree–Fock self-consistent field and configuration interaction methods. A double ζ basis set was used to determine the cluster geometries, energies, and relative stability. For the inner-shell electrons of lanthanum we have used relativistic compact effective potentials. The geometry optimization was carried out using the analytical gradient method. The experimentally obtained ratings of some ionic species were compared with the energetic stability of free clusters calculated as described previously. The main conclusion of the present study is that unstable clusters, e.g., LaB3+, or almost unstable LaBnm+ clusters were preferably observed in the ion flux. The physical explanation of this phenomenon lies in the possibility of a stable cluster to redistribute its interatomic bonds and thus to remain attached to the surface. An unstable cluster does not have this characteristic and as a result this must leave the surface.

Theoretical Study of the Electronic States of the Rb2 Molecule

We have calculated the electronic states of Rb2 by multireference configuration interactions using the averaged relativistic effective small-core potential and the core-polarization potential. The potential energy curves for a large number of states dissociating into from 5s+5s up to 7s+5s asymptotic limits are calculated and the spectroscopic constants are reported. The spin–orbit effects for the states dissociating into 5p+5s and 4d+5s are calculated using the effective spin–orbit potential. The results are compared with available experimental data and other theoretical works.

Molecular potential energy function and reaction dynamics for LaH2 (C2V, X̃ 2A1)

The present work has derived an analytical potential energy function for the ground state (C2V, X̃ 2A1) of LaH2. The electronic state and reasonable dissociation limits are correctly determined based on Atomic and Molecular Reaction Statics (AMRS), and then, using a relativistic compact effective potential (RCEP) for La. The equilibrium geometry, dissociation energy and harmonic frequencies for LaH2 have been calculated by ab initio methods. The results show that R(LaH)=2.1945Å, ∠HLaH=124.4° and De(LaH2)=5.599eV, and ν1, ν2 and ν3 are 1216.521, 1087.417 and 2156.957cm−1, respectively. Molecular reaction dynamics for the collision La(2Dg)+H2(X1Σg+,v=j=0) has been studied based on the analytical potential energy function of LaH2(X̃ 2A1) by using the Monte Carlo quasi-classical trajectory approach. The results for the collision process indicate that the main channel is the exchange reaction La(2Dg)+H2(X1Σg+,v=j=0)→LaH(X1Σ+,v′,j′)+H(2Sg) with the product LaH, and without the formation of the complex compound LaH2. The relationship of the reactive cross-section σr with the relative translational energy Et shows that there is a threshold energy of 40kcal/mol. Because of the tremendous difference in the masses of La and H2, these is a direct collision, and the distributions of the products LaH and H2 are along the direction of forward scattering.

Ab initio structure of the active site of phosphotriesterase.

This research exploits two recent developments to obtain a fundamental understanding of the metalloenzyme active site using the bimetallic enzyme phosphotriesterase as an example of this class. First is the theoretical prediction that the structure and spectroscopy of a native metalloenzyme active site is qualitatively determined by the supermolecule complex of the metal(s) and the first shell of ligands with proper charge states including waters directly bonded to ionic ligands. The second is the development of an effective potential for representing the molecular environment interacting with an all-electron active site in the quantum Hamiltonian. The GAMESS suite of electronic structure codes has implemented this new methodology, effective fragment potentials (EFP), to make theoretical calculations on structure, spectroscopy, and reactivity tractable for systems involving hundreds of atoms. Since there are transition metal cations at the active site of these enzymes, the all-electron part of the complex is calculated with relativistic compact effective potentials (CEP) and their concomitant basis sets. A realistic representation of the active site with its protein environment can be obtained using a combination of the CEP and EFP. This presentation will determine the inherent electronic and structural characteristics of phosphotriesterase using ab initio quantum mechanical methods. A single X-ray structure for the Zn-Zn enzyme is leveraged to obtain the structure of the Cd-Cd enzyme and to examine the consequences of protonating the active site.

AB initio Calculations of the structure and spectra of chromium, molybdenum, and tungsten tetrafluorides. nontetrahedral molecular structure of WF4

The properties of MF4 molecules (M = Cr, Mo, W) are investigated by the restricted Hartree-Fock method using Moller-Plesset second-order perturbation theory and by the second-order configuration interaction method using the multiconfigumtion wave function derived in a complete active space approximation, in wide bases complemented with polarization d and f functions. Relativistic effective potentials are used to describe the core electrons. For CrF4 and MoF4, the tetrahedral configuration of nuclei in the electronic state of3k2 symmetry is energetically most favorable. In the WF4 molecule, the least-energy structure is a D2hstructure in the singlet state 1A1. The D4h,(1 A1g) and Td ( 3A2) configurations in the WF4 molecule are higher on the energy scale than the ground state by 4 and 6305 cm’1 and are saddle points. For all of the analyzed configurations of MF4 molecules, the geometrical parameters, the vibrational spectra, and the energies of vertical electronic transitions are found. The chemical bonding is analyzed and a simple model is proposed to explain the variation of the relative energies of states in the series CrF4→MoF4→WF4.