Picture change error correction in the radial distributions of canonical orbital densities and total electron density of radon atom: the effect of the size of nucleus and the basis set limit

The 2nd order Douglas-Kroll-Hess (DKH2) and the Infinite Order Two Component (IOTC) radial distributions of electron density of canonical Hartree-Fock (HF) orbitals of radon atom are presented. Furthermore, the total electron density is revisited. The picture change error (PCE) correction is investigated by analytical means. The point charge model of nucleus and the Gaussian nucleus model are employed. The basis set is extrapolated by means of including tight s and also p Gaussians within the original triple zeta basis set. It is found that the DKH1 PCE corrected DKH2 total electron and s orbital contact densities are negative for the point charge model of nucleus if tight enough s Gaussians are included in the basis set. It is shown that this failure is caused due to the missing terms of the second order Douglas-Kroll transformation for the DKH2 electron density. PCE is found the most striking in the DKH2/IOTC electron density of s orbitals close to the nucleus. The radial distributions of the 2-component p _1/2 orbital densities are considerably affected by PCE at the nucleus as well. Furthermore, the PCE corrected DKH2/IOTC scalar p orbital densities have a non-zero value of electron density at nucleus and can be considered as an spin-orbit (SO) average of the p _1/2 and p _3/2 orbitals. The d and f orbitals are affected by PCE in the vicinity of the nucleus only little. The PCE corrected DKH2 and IOTC radial distributions of orbital densities are nodeless, which is completely in agreement with the radial distribution of the analytic or numeric DCH orbital densities.     

An infinite-order two-component relativistic Hamiltonian by a simple one-step transformation

The authors report the implementation of a simple one-step method for obtaining an infinite-order two-component (IOTC) relativistic Hamiltonian using matrix algebra. They apply the IOTC Hamiltonian to calculations of excitation and ionization energies as well as electric and magnetic properties of the radon atom. The results are compared to corresponding calculations using identical basis sets and based on the four-component Dirac-Coulomb Hamiltonian as well as Douglas-Kroll-Hess and zeroth-order regular approximation Hamiltonians, all implemented in the DIRAC program package, thus allowing a comprehensive comparison of relativistic Hamiltonians within the finite basis approximation.     

Relativistic effects in HgHe and HgXe CCSD(T) ground state potential curves. Low‐density viscosity simulations of Hg:Xe mixture

The comparison of coupled cluster with single and double excitations and with perturbative correction of triple excitations [CCSD(T)] ground state potential curves of mercury with rare gases (RG): HgHe and HgXe, at several levels of theory is presented. The scalar relativistic (REL) effects and spin‐orbit coupling effects in the ground state potential curves of these weakly bounded dimers are considered. The CCSD(T) ground state potential curves at the level of the Dirac‐Coulomb Hamiltonian (DCH) are compared with CCSD(T) curves at the level of 4‐component spin‐free modified DCH, the scalar 2nd order Douglas‐Kroll‐Hess (DKH2) and the nonrelativistic (NR‐LL) (Lévy‐Leblond) Hamiltonian. In addition, London‐Drude formula and SCF interaction energy curves are employed in the analysis of different contributions of REL effects in dissociation energies of HgRG and Hg₂ dimers. Moreover, the large anharmonicity of the HgHe ground state potential curve is highlighted. The computationally less demanding scalar DKH2 Hamiltonian is employed to calculate the HgXe, Hg₂, and Xe₂ all electron CCSD(T) ground state potential curves in highly augmented quadruple zeta basis sets. These potential curves are used to simulate the shear viscosity of mercury, xenon, and mercury‐xenon (Hg:Xe) mixture. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

Subshell fitting of relativistic atomic core electron densities for use in QTAIM analyses of ECP-based wave functions

Scalar-relativistic, all-electron density functional theory (DFT) calculations were done for free, neutral atoms of all elements of the periodic table using the universal Gaussian basis set. Each core, closed-subshell contribution to a total atomic electron density distribution was separately fitted to a spherical electron density function: a linear combination of s-type Gaussian functions. The resulting core subshell electron densities are useful for systematically and compactly approximating total core electron densities of atoms in molecules, for any atomic core defined in terms of closed subshells. When used to augment the electron density from a wave function based on a calculation using effective core potentials (ECPs) in the Hamiltonian, the atomic core electron densities are sufficient to restore the otherwise-absent electron density maxima at the nuclear positions and eliminate spurious critical points in the neighborhood of the atom, thus enabling quantum theory of atoms in molecules (QTAIM) analyses to be done in the neighborhoods of atoms for which ECPs were used. Comparison of results from QTAIM analyses with all-electron, relativistic and nonrelativistic molecular wave functions validates the use of the atomic core electron densities for augmenting electron densities from ECP-based wave functions. For an atom in a molecule for which a small-core or medium-core ECPs is used, simply representing the core using a simplistic, tightly localized electron density function is actually sufficient to obtain a correct electron density topology and perform QTAIM analyses to obtain at least semiquantitatively meaningful results, but this is often not true when a large-core ECP is used. Comparison of QTAIM results from augmenting ECP-based molecular wave functions with the realistic atomic core electron densities presented here versus augmenting with the limiting case of tight core densities may be useful for diagnosing the reliability of large-core ECP models in particular cases. For molecules containing atoms of any elements of the periodic table, the production of extended wave function files that include the appropriate atomic core densities for ECP-based calculations, and the use of these wave functions for QTAIM analyses, has been automated.     

Reductive dechlorination of hexachloroethane in the environment: mechanistic studies via computational electrochemistry.

Ab initio and density functional levels of electronic structure theory are applied to characterize alternative mechanisms for the reductive dechlorination of hexachloroethane (HCA) to perchloroethylene (PCE). Aqueous solvation effects are included using the SM5.42R continuum solvation model. After correction for a small systematic error in the electron affinity of the chlorine atom, theoretical predictions are accurate to within 23 mV for four aqueous reduction potentials relevant to HCA. A single pathway that proceeds via two successive single-electron transfer/barrierless chloride elimination steps, is predicted to be the dominant mechanism for reductive dechlorination. An alternative pathway predicted to be accessible involves trichloromethylchlorocarbene as a reactive intermediate. Bimolecular reactions of the carbene with other species at millimolar or higher concentrations are predicted to potentially be competitive with its unimolecular rearrangement to form PCE.     

Exact non-additive kinetic potentials in realistic chemical systems

In methods based on frozen-density embedding theory or subsystem formulation of density functional theory, the non-additive kinetic potential (v(t) (nad)(r)) needs to be approximated. Since v(t) (nad)(r) is defined as a bifunctional, the common strategies rely on approximating v(t) (nad)[ρ(A),ρ(B)](r). In this work, the exact potentials (not bifunctionals) are constructed for chemically relevant pairs of electron densities (ρ(A) and ρ(B)) representing: dissociating molecules, two parts of a molecule linked by a covalent bond, or valence and core electrons. The method used is applicable only for particular case, where ρ(A) is a one-electron or spin-compensated two-electron density, for which the analytic relation between the density and potential exists. The sum ρ(A) + ρ(B) is, however, not limited to such restrictions. Kohn-Sham molecular densities are used for this purpose. The constructed potentials are analyzed to identify the properties which must be taken into account when constructing approximations to the corresponding bifunctional. It is comprehensively shown that the full von Weizsa?cker component is indispensable in order to approximate adequately the non-additive kinetic potential for such pairs of densities.     

Relativistic and electron‐correlation effects on magnetizabilities investigated by the Douglas‐Kroll‐Hess method and the second‐order Møller‐Plesset perturbation theory

Isotropic and anisotropic magnetizabilities for noble gas atoms and a series of singlet and triplet molecules were calculated using the second‐order Douglas‐Kroll‐Hess (DKH2) Hamiltonian containing the vector potential A and in part using second‐order generalized unrestricted Møller‐Plesset (GUMP2) theory. The DKH2 Hamiltonian was resolved into three parts (spin‐free terms, spin‐dependent terms, and magnetic perturbation terms), and the magnetizabilities were decomposed into diamagnetic and paramagnetic terms to investigate the relativistic and electron‐correlation effects in detail. For Ne, Kr, and Xe, the calculated magnetizabilities approached the experimental values, once relativistic and electron‐correlation effects were included. For the IF molecule, the magnetizability was strongly affected by the spin‐orbit interaction, and the total relativistic contribution amounted to 22%. For group 17, 16, 15, and 14 hydrides, the calculated relativistic effects were small (less than 3%), and trends were observed in relativistic and electron‐correlation effects across groups and periods. The magnetizability anisotropies of triplet molecules were generally larger than those of similar singlet molecules. The so‐called relativistic‐correlation interference for the magnetizabilities computed using the relativistic GUMP2 method can be neglected for the molecules evaluated, with exception of triplet SbH. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

NMR shielding constants of CuX,AgX, and AuX (X = F,Cl, Br,and I) investigated by density functional theory based on the Douglas–Kroll–Hess Hamiltonian

Two‐component relativistic density functional theory (DFT) with the second‐order Douglas–Kroll–Hess (DKH2) one‐electron Hamiltonian was applied to the calculation of nuclear magnetic resonance (NMR) shielding constant. Large basis set dependence was observed in the shielding constant of Xe atom. The DKH2‐DFT‐calculated shielding constants of I and Xe in HI, I₂, CuI, AgI, and XeF₂ agree well with those obtained by the four‐component relativistic theory and experiments. The Au NMR shielding constant in AuF is extremely more positive than in AuCl, AuBr, and AuI, as reported recently. This extremely positive shielding constant arises from the much larger Fermi contact (FC) term of AuF than in others. Interestingly, the absolute values of the paramagnetic and the FC terms are considerably larger in CuF and AuF than in others. The large paramagnetic term of AuF arises from the large d‐components in the Au d_π –F p_π and Au sd_σ–F p_σ molecular orbitals (MOs). The large FC term in AuF arises from the small energy difference between the Au sd_σ + F p_σ and Au sd_σ–F p_σ MOs. The second‐order magnetically relativistic effect, which is the effect of DKH2 magnetic operator, is important even in CuF. This effect considerably improves the overestimation of the spin‐orbit effect calculated by the Breit–Pauli magnetic operator. © 2013 Wiley Periodicals, Inc.     

Thomas–Fermi–Dirac models of atoms constrained by nuclear cusp and long-range conditions

The traditional Thomas–Fermi–Dirac model of the electronic structure for a neutral atom is deficient in that it predicts an infinite electron density at the nucleus and a sharp cutoff of the electron density at a finite radius. This study was carried out to remedy these faults in the model. Extending an idea used earlier in Thomas–Fermi (TF ) theory [Proc. Natl. Acad. Sci. U.S.A. 83 , 3577 (1985)], the Thomas–Fermi–Dirac (TFD ) energy functional is minimized under constraints ∫ρ( r ) d r = N, ∫e^?2kr▽²ρ( r )d r < ∞ and ∫(1 ? e^?k′r)ρ^4/3( r )d r < ∞, with k and k′ determined by the nuclear cusp condition and the correct asymptotic behavior. Optimum coordinate scaling also is considered. It is found that the TFD model is substantially improved by constraining the minimization search domain of the energy functional in this way. Energies are given for five noble gas atoms, and Compton profiles for these atoms are calculated. The behavior of electrons in momentum space is improved in both this modified TFD model and in the corresponding modified TF model.     

Charge densities for singlet and triplet electron pairs

Analytic properties of charge densities associated with singlet and triplet electron pairs, ρ₀( r ) and ρ₁( r ), are presented. In an N‐electron system with total spin S, distributions ρ₀( r ) and ρ₁( r ) are independent of the spin projection quantum number M (spin rotation invariance), as opposed to the usual spin‐up and spin‐down electron densities, ρ_α( r ) and ρ_β( r ). We derive equations showing that in the case of a wave function which is a spin‐eigenfunction, ρ₀( r ) and ρ₁( r ) are linear combinations of the total charge density ρ( r ) and the uncompensated spin density ρ_s( r )=[ρ_α( r )−ρ_β( r )]/2M. For a wave function which is not an eigenfunction of $\mathcal{S}^{2}$, no such relationship exists. In a related discussion, a definition of the high‐spin solution corresponding to a given spin‐unrestricted Hartree–Fock wave function is proposed, and a notion of effectively unpaired electrons is introduced. The distributions ρ₀( r ) and ρ₁( r ) are shown not to be invariant under spin coupling between isolated systems. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 651–660, 2000     

One-electron contributions to the g-tensor for second-order Douglas-Kroll-Hess theory

The electric g-tensor is a central quantity for the interpretation of electron paramagnetic resonance spectra. In this paper, a detailed derivation of the 1-electron contributions to the g-tensor is presented in the framework of linear response theory and the second-order Douglas-Kroll-Hess (DKH) transformation. Importantly, the DKH transformation in the presence of a magnetic field is not unique. Whether or not the magnetic field is included in the required Foldy-Wouthuysen transformation, different transformation matrices and, consequently, Hamiltonians result. In this paper, a detailed comparison of both approaches is presented, paying particular attention to the mathematical properties of the resulting Hamiltonians. In contrast to previous studies that address the g-tensor in the framework of DKH theory, the resulting terms are compared to those of the conventional Pauli theory and are given a physical interpretation. Based on these mathematical and physical arguments, we establish that the proper DKH transformation for systems with constant magnetic fields is based on a gauge-invariant Foldy-Wouthuysen transformation, i.e., a Foldy-Wouthuysen transformation including the magnetic field. Calculations using density functional theory (DFT) are carried out on a set of heavy, diatomic molecules, and a set of transition-metal complexes. Based on these calculations, the performance of the relativistic calculation with and without inclusion of picture-change effects is compared. Additionally, the g-tensor is calculated for the Lanthanide dihydrides. Together with the results from the other two molecular test sets, these calculations serve to quantify the magnitude of picture-change effects and elucidate trends across the periodic table.     

On the unusual weak intramolecular C...C interactions in Ru3(CO)12: a case of bond path artifacts introduced by the multipole model?

In a recent publication in this journal, an experimental charge density analysis on the triruthenium cluster Ru(3)(CO)(12) showed unusual C...C bond paths linking the axial carbonyl ligands [Gervasio, G.; Marabello, D.; Bianchi, R.; Forni, A. J. Phys. Chem. A 2010, 114, 9368, hereafter GMBF]. These were also observed in one theoretical DFT calculation, and are associated with very low values of ρ(r(b)) and ?(2)ρ(r(b)). Our independent experimental charge density analysis on Ru(3)(CO)(12) is entirely consistent with GMBF and confirms the presence of these apparent weak interactions in the multipole model density. However, we conclusively demonstrate that these unusual C...C bond paths between the axial carbonyl ligands are in fact artifacts arising from the Hansen-Coppens multipole model, which is used to analyze the experimental data. Numerous relativistic and nonrelativistic gas-phase DFT calculations, using very extensive basis sets and with corrections for dispersion effects, uniformly fail to reproduce these intramolecular features in the QTAIM topology of the electron density. Moreover, multipole fitting of theoretical static structure factors computed from these quantum electron densities results in the reappearance of the C...C bond paths between the axial carbonyl ligands in the derived molecular graphs. On the other hand, using the experimental structure factors to generate "experimental" X-ray constrained DFT wave functions once again yields molecular graphs which do not show these secondary C...C bond paths. The evidence therefore strongly implicates the multipole model as the source of these spurious features and in turn suggests that great caution should be applied in the interpretation of bond paths where the values of ρ(r(b)) and ?(2)ρ(r(b)) are very low.     

Two-component relativistic methods for the heaviest elements

Different generalized Douglas-Kroll transformed Hamiltonians (DKn, n=1, 2,...,5) proposed recently by Hess et al. are investigated with respect to their performance in calculations of the spin-orbit splittings. The results are compared with those obtained in the exact infinite-order two-component (IOTC) formalism which is fully equivalent to the four-component Dirac approach. This is a comprehensive investigation of the ability of approximate DKn methods to correctly predict the spin-orbit splittings. On comparing the DKn results with the IOTC (Dirac) data one finds that the calculated spin-orbit splittings are systematically improved with the increasing order of the DK approximation. However, even the highest-order approximate two-component DK5 scheme shows certain deficiencies with respect to the treatment of the spin-orbit coupling terms in very heavy systems. The meaning of the removal of the spin-dependent terms in the so-called spin-free (scalar) relativistic methods for many-electron systems is discussed and a computational investigation of the performance of the spin-free DKn and IOTC methods for many-electron Hamiltonians is carried out. It is argued that the spin-free IOTC rather than the Dirac-Coulomb results give the appropriate reference for other spin-free schemes which are based on approximate two-component Hamiltonians. This is illustrated by calculations of spin-free DKn and IOTC total energies, r(-1) expectation values, ionization potentials, and electron affinities of heavy atomic systems.     

Mechanisms of reductive eliminations in square planar Pd(II) complexes: nature of eliminated bonds and role of trans influence

The trans influence of various phosphine ligands (L) in direct as well as dissociative reductive elimination pathways yielding CH(3)CH(3) from Pd(CH(3))(2)L(2) and CH(3)Cl from Pd(CH(3))(Cl)L(2) has been quantified in terms of isodesmic reaction energy, E(trans), using the MPWB1K level of density functional theory. In the absence of a large steric effect, E(trans) correlated linearly with the activation barrier (E(act)) of both direct and dissociation pathways. The minimum of molecular electrostatic potential (V(min)) at the lone pair region of phosphine ligands has been used to assess their electron donating power. E(trans) increased linearly with an increase in the negative V(min) values. Further, the nature of bonds that are eliminated during reductive elimination have been analyzed in terms of AIM parameters, viz. electron density (ρ(r)), Laplacian of the electron density (?(2)ρ(r)), total electron energy density (H(r)), and ratio of potential and kinetic electron energy densities (k(r)). Interestingly, E(act) correlated inversely with the strength of the eliminated metal-ligand bonds measured in terms of the bond length or the ρ(r). Analysis of H(r) showed that elimination of the C-C/C-Cl bond becomes more facile when the covalent character of the Pd-C/Pd-Cl bond increases. Thus, AIM details clearly showed that the strength of the eliminated bond is not the deciding factor for the reductive elimination but the nature of the bond, covalent or ionic. Further, a unified picture showing the relationship between the nature of the eliminated chemical bond and the tendency of reductive elimination is obtained from the k(r) values: the E(act) of both direct and dissociative mechanisms for the elimination of CH(3)CH(3) and CH(3)Cl decreased linearly when the sum of k(r) at the cleaved bonds showed a more negative character. It means that the potential electron energy density dominates over the kinetic electron energy density when the bonds (Pd-C/Pd-Cl) become more covalent and the eliminated fragments attain more radical character leading to the easy formation of C-C/C-Cl bond.     

The effect of electron correlation on the topological and atomic properties of the electron density distributions of molecules

The topological properties of the electron density and the properties of an atom in a molecule are calculated by means of second-order Møller-Plesset perturbation theory (MP2) and compared with the results of configuration interaction calculations (C12) which include all single and double substitutions from the Hartree-Fock reference configuration. A software package for analyzing the effects of electron correlation on the topological properties of the electron density of molecules is described. H₂CO is used to provide a numerical example and to indicate that the number of bond critical points is unaffected by the inclusion of electron correlation. Correlation leads to only a small shift in the positions of bond critical points and a small change in the electron density at bond critical points. It is further shown that the energy of an atom in a molecule can be calculated to an accuracy of 1 kcal/mol and the electron population of an atom to about 0.001e. A statistical method is used to show that the deviation of the MP2 correlation correction relative to the CI2 correlation correction for a variety of atomic properties is about 25%.     

Douglas–Kroll–Hess Theory: a relativistic electrons-only theory for chemistry

A unitary transformation allows to separate (block-diagonalize) the Dirac Hamiltonian into two parts one part: solely describes electrons, while the other gives rise to negative-energy states, which are the so-called positronic states. The block-diagonal form of the Hamiltonian no longer accounts for the coupling of both kinds of states. The positive-energy (‘electrons-only’) part can serve as a ‘fully’ relativistic electrons-only theory, which can be understood as a rigorous basis for chemistry. Recent developments of the Douglas–Kroll–Hess (DKH) method allowed to derive a sequence of expressions, which approximate this electrons-only Hamiltonian up to arbitrary-order. While all previous work focused on the numerical stability and accuracy of these arbitrary-order DKH Hamiltonians, conceptual issues and paradoxa of the method were mostly left aside. In this work, the conceptual side of DKH theory is revisited in order to identify essential aspects of the theory to be distinguished from purely computational consideration.