Electronic structure of molecules using one-component wave functions and relativistic effective core potentials

A one-component approach to molecular electronic structure is discussed that includes the dominant relativistic effects on valence electrons and yet allows the use of the traditional quantum-chemistry techniques. The approach starts with one-component Cowan–Griffin relativistic orbitals that successfully incorporate the effects of the mass-velocity and Darwin terms present in more complicated wave functions such as the Dirac–Hartree–Fock. The approach then constructs “relativistic” effective core potentials (RECPS ) from these orbitals, and uses these to bring the relativistic effects into the molecular electronic calculations. The use of effective one-electron spin-orbit operators in conjunction with these one-component wave functions to include the effects of spin-orbit coupling is discussed. Applications to molecular systems involving heavy atoms and comparisons with available spectroscopic data on molecular geometries and excitation energies are presented. Finally, a new approach to the construction of RECPS encompassing the Hamiltonian and shapeconsistent approach is presented together with a novel analysis of the long-range behavior of the RECPS .     

Electronic spectrum of UO2(2+) and [UO2Cl4]2- calculated with time-dependent density functional theory

The electronic spectra of UO(2) (2+) and [UO(2)Cl(4)](2-) are calculated with a recently proposed relativistic time-dependent density functional theory method based on the two-component zeroth-order regular approximation for the inclusion of spin-orbit coupling and a noncollinear exchange-correlation functional. All excitations out of the bonding sigma(u) (+) orbital into the nonbonding delta(u) or phi(u) orbitals for UO(2) (2+) and the corresponding excitations for [UO(2)Cl(4)](2-) are considered. Scalar relativistic vertical excitation energies are compared to values from previous calculations with the CASPT2 method. Two-component adiabatic excitation energies, U-O equilibrium distances, and symmetric stretching frequencies are compared to CASPT2 and combined configuration-interaction and spin-orbit coupling results, as well as to experimental data. The composition of the excited states in terms of the spin-orbit free states is analyzed. The results point to a significant effect of the chlorine ligands on the electronic spectrum, thereby confirming the CASPT2 results: The excitation energies are shifted and a different luminescent state is found.     

New relativistic ANO basis sets for transition metal atoms

New basis sets of the atomic natural orbital (ANO) type have been developed for the first, second, and third row transition metal atoms. The ANOs have been obtained from the average density matrix of the ground and lowest excited states of the atom, the positive and negative ions, and the atom in an electric field. Scalar relativistic effects are included through the use of a Douglas-Kroll-Hess Hamiltonian. Multiconfigurational wave functions have been used with dynamic correlation included using second order perturbation theory (CASSCF/CASPT2). The basis sets are applied in calculations of ionization energies, electron affinities, and excitation energies for all atoms and polarizabilities for spherically symmetric atoms. These calculations include spin-orbit coupling using a variation-perturbation approach. Computed ionization energies have an accuracy better than 0.2 eV in most cases. The accuracy of computed electron affinities is the same except in cases where the experimental values are smaller than 0.5 eV. Accurate results are obtained for the polarizabilities of atoms with spherical symmetry. Multiplet levels are presented for some of the third row transition metals.     

Theoretical investigation of electronic structures and excitation energies of doubly N-confused porphyrin and its group 11 transition metal (III) complexes

Density functional theory is carried out to study cis-doubly N-confused porphyrin and its metal (Cu3+, Ag3+, and Au3+) complexes. The electronic structures and bonding situations of these molecules have been investigated by using the natural bond orbital analysis and the topological analysis of the electron localization function. We have studied the electronic spectra of cis-doubly N-confused porphyrin and its metal complexes with time-dependent density functional theory. The introduction of group 11 transition metals leads to blueshifts of their electronic spectra with respect to that of cis-doubly N-confused porphyrin. In particular, the absorption spectra of the copper complex show some weak Q bands that mainly arise from a combination of ligand-to-metal charge transfer and ligand-to-ligand charge transfer transitions. The relativistic time-dependent density functional theory with spin-orbit coupling calculations indicates that the effects of spin-orbit coupling on the excitation energies of the copper and silver complexes are so small that it is safe enough to neglect spin-orbit interactions for these two complexes. However, it has a significant effect on the absorption spectra of the gold complex.     

Theoretical investigation of electronic structures and excitation energies of hexaphyrin and its group 11 transition metal (III) complexes

Density functional theory is carried out to study hexaphyrin and its bis-metal and mixed bis-metal (M = Cu³⁺, Ag³⁺, and Au³⁺) complexes. The electronic structures and bonding situations of them are studied by using natural bond orbital approach and the topological analysis of the electron localization function. Electronic spectra are investigated by using time-dependent density functional theory. The introduction of group 11 transition metals leads to red shifts in the spectra of these metal complexes with respect to that of hexaphyrin. Moreover, it is noteworthy that the spectra of copper contained complexes are mainly derived from combination of ligand-to-metal charge transfer and ligand-to-ligand charge transfer transitions. In addition, the relativistic time-dependent density functional theory with spin-orbit coupling calculations indicate that the effects of spin-orbit coupling on the excitation energies are so small that it is safe enough to neglect spin-orbit coupling for these systems.     

On Koopmans' theorem in density functional theory

This paper clarifies why long-range corrected (LC) density functional theory gives orbital energies quantitatively. First, the highest occupied molecular orbital and the lowest unoccupied molecular orbital energies of typical molecules are compared with the minus vertical ionization potentials (IPs) and electron affinities (EAs), respectively. Consequently, only LC exchange functionals are found to give the orbital energies close to the minus IPs and EAs, while other functionals considerably underestimate them. The reproducibility of orbital energies is hardly affected by the difference in the short-range part of LC functionals. Fractional occupation calculations are then carried out to clarify the reason for the accurate orbital energies of LC functionals. As a result, only LC functionals are found to keep the orbital energies almost constant for fractional occupied orbitals. The direct orbital energy dependence on the fractional occupation is expressed by the exchange self-interaction (SI) energy through the potential derivative of the exchange functional plus the Coulomb SI energy. On the basis of this, the exchange SI energies through the potential derivatives are compared with the minus Coulomb SI energy. Consequently, these are revealed to be cancelled out only by LC functionals except for H, He, and Ne atoms.     

水溶液中碳酸铀酰化合物的电子结构

应用相对论密度泛函理论系统研究了水溶液中非水合化和水合化碳酸铀酰化合物Cn/m(其中n和m分别为结构中碳酸配体和水配体的个数)的结构.溶剂效应采用类导体屏蔽模型(COSMO),并采用零级规整近似(ZORA)方法考虑标量相对论效应和旋-轨耦合相对论效应.电子跃迁采用包含旋-轨耦合相对论效应的含时密度泛函理论并在相关交换势中采用轨道势能统计平均(SAOP)做近似计算.结果表明碳酸配体对配合物结构和电子跃迁有很大的影响.C3/0配合物的稳定性可归于5f轨道参与了高占据轨道的成键作用.增加碳酸盐配体导致最大波长的蓝移,并在近可见光区域出现高强度的吸收.     

PtF6(2-) dianion and its detachment spectrum: a fully relativistic study

In this work we calculate the photoelectron spectrum of the PtF(6)2- dianion by application of the third-order Dirac-Hartree-Fock one-particle propagator technique. Relativistic effects and electron correlation are hereby treated on a consistent theoretical basis which is mandatory for systems containing heavy elements. A PtF6(2-) gas phase photoelectron spectrum is not yet available and our calculations therefore have predictive character. As it is characteristic for dianionic systems a strong dependence on basis set size and molecular geometry is observed. In contrast to the already calculated PtCl(6)2- photoelectron spectrum no valence orbital inversion due to strong interplay of spin-orbit coupling and electron correlation is observed. Furthermore an unusually strong spin-orbit splitting was found for the sigma-type subvalence 1t1u molecular spinor despite its very small platinum p population. The double ionization threshold is strongly lowered by relativistic effects now enabling an interatomic Coulombic decay process after ionization from the sigma-bonding orbitals. The results stress the importance of spin-orbit coupling for the understanding of the spectral structure which cannot be reproduced by a scalar-relativistic treatment only.     

Analysis of the effect of spin-orbit coupling on the electronic structure and excitation spectrum of the Bi2(2-) anion in (K-crypt)2Bi2 on the basis of relativistic electronic structure calculations

The Bi2(2-) anions that have been characterized in (K-crypt)2Bi2 are isoelectronic with O2 but are diamagnetic and EPR-silent, unlike O2. The UV-vis spectrum measured for (K-crypt)2Bi2 shows two broad absorption peaks located at 2.05 and 2.85 eV, but no absorption at lower energies down to 0.62 eV. To account for these observations, the electronic structures of the isoelectronic diatomic dianions Q2(2-) (Q = N, P, As, Sb, Bi) were compared on the basis of relativistic density functional theory calculations, and the electronic excitations of Bi2(2-) were analyzed on the basis of relativistic configuration interaction calculations. The extent of spin-orbit coupling, brought about by the relativistic effect, increases steadily in the order N < P < As < Sb < Bi such that the "closed-shell" state is more stable than the "open-shell" state for Bi2(2-), while the opposite is the case for N2(2-), P2(2-), As2(2-), and Sb2(2-). The nature of the electronic excitations of Bi2(2-) was assigned and discussed from the viewpoint of molecular orbitals in the absence of spin-orbit coupling.     

A theoretical study of the excited states of AmO2n+, n=1,2,3

The ground and excited states of the AmO(2) (+), AmO(2) (2+), and AmO(2) (3+) ions have been studied using the four-component configuration interaction singles doubles, spin-orbit complete active space self-consistent field, and spin-orbit complete active space-order perturbation theory methods. The roles of scalar relativistic effects and spin-orbit coupling are analyzed; results with different methods are carefully compared by a precise analysis of the wave functions. A molecular spinor diagram is used in relation to the four-component calculations while a ligand field model is used for the two-step method. States with the same number of electrons in the four nonbonding orbitals are in very good agreement with the two methods while ligand field and charge transfer states do not have the same excitation energies.     

Theoretical study of the electronic spectroscopy of CO adsorbed on Pt(111)

The excited states of CO adsorbed on the Pt(111) surface are studied using a time-dependent density functional theory formalism. To reduce the computational cost, electronic excitations are computed within a reduced single excitation space. Using cluster models of the surface, excitation energies are computed for CO in the on-top, threefold, and bridge binding sites. On adsorption, there is a lowering of the 5sigma orbital energy. This leads to a large blueshift in the 5sigma- -> pi(CO*) excitation energy for all adsorption sites. The 1pi and 4sigma orbital energies are lowered to a lesser extent, and smaller shifts in the corresponding excitation energies are predicted. For the larger clusters, pi* excitations at lower energies are observed. These transitions correspond to excitations to virtual orbitals of pi* character which lie below the pi* orbitals of gas phase CO. These orbitals are associated predominantly with the metal atoms of the cluster. The excitation energies are also found to be sensitive to changes in the adsorption geometry. The electronic spectrum of CO on Pt(111) is simulated and the assignment of the bands observed in experimental electron energy loss spectroscopy discussed.     

Time-dependent four-component relativistic density functional theory for excitation energies

Time-dependent four-component relativistic density functional theory within the linear response regime is developed for calculating excitation energies of heavy element containing systems. Since spin is no longer a good quantum number in this context, we resort to time-reversal adapted Kramers basis when deriving the coupled Dirac-Kohn-Sham equation. The particular implementation of the formalism into the Beijing density functional program package utilizes the multipolar expansion of the induced density to facilitate the construction of the induced Coulomb potential. As the first application, pilot calculations on the valence excitation energies and fine structures of the rare gas (Ne to Rn) and Group 12 (Zn to Hg) atoms are reported. To the best of our knowledge, it is the first time to be able to account for spin-orbit coupling within time-dependent density functional theory for excitation energies.     

A simplified relativistic time-dependent density-functional theory formalism for the calculations of excitation energies including spin-orbit coupling effect

In the present work we have proposed an approximate time-dependent density-functional theory (TDDFT) formalism to deal with the influence of spin-orbit coupling effect on the excitation energies for closed-shell systems. In this formalism scalar relativistic TDDFT calculations are first performed to determine the lowest single-group excited states and the spin-orbit coupling operator is applied to these single-group excited states to obtain the excitation energies with spin-orbit coupling effects included. The computational effort of the present method is much smaller than that of the two-component TDDFT formalism and this method can be applied to medium-size systems containing heavy elements. The compositions of the double-group excited states in terms of single-group singlet and triplet excited states are obtained automatically from the calculations. The calculated excitation energies based on the present formalism show that this formalism affords reasonable excitation energies for transitions not involving 5p and 6p orbitals. For transitions involving 5p orbitals, one can still obtain acceptable results for excitations with a small truncation error, while the formalism will fail for transitions involving 6p orbitals, especially 6p1/2 spinors.     

SCMEH-MO calculations on lanthanide systems. III. Ln(CO)6, Ln(OC)6 (Ln = Nd,Sm)

The SCMEH-MO method with average relativistic and spin-orbit effects has recently been applied to study the electronic structure and bonding in samarium pentamethylcyclopentadienyls. In this report the same approach has been utilized in studying the electronic structures of Nd and Sm hexacarbonyls. In contrast to the stable transition metal d-block carbonyls, these lanthanide carbonyls are found to be quite unstable. These findings are based on calculated electronic structures and bond energies. © 1996 John Wiley & Sons, Inc.     

Spin-orbit effects on a gold-based superatom: a relativistic Jellium model

The inclusion of relativistic effects always brings to the scientific community great and stimulating surprises. To consider the spin-orbit term, which accounts for the interaction between the spatial and spin coordinates, requires the use of double point groups of symmetry in order to solve the Dirac equation or the two component approximation to it, leading to total angular momenta (j) functions, atomic or molecular spinors, instead of pure orbital angular momenta (l), atomic or molecular orbitals. Large and small components, derived from the Dirac treatment, depict wavefunctions corresponding to fermions, electrons, which are described for the first time for a superatom case. In addition, their behavior is revisited in order to clarify the effects of the inclusion of the spin-orbit coupling into the electronic structure calculations, which can be extended to other superatoms, clusters, molecules and atoms.     

Excitation energies expressed as orbital energies of Kohn–Sham density functional theory with long-range corrected functionals

A new simple and conceptual theoretical scheme is proposed for estimating one-electron excitation energies using Kohn–Sham (KS) solutions. One-electron transitions that are dominated by the promotion from one initially occupied orbital to one unoccupied orbital of a molecular system can be expressed in a two-step process, ionization, and electron attachment. KS with long-range corrected (LC) functionals satisfies Janak's theorem and LC total energy varies almost linearly as a function of its fractional occupation number between the integer electron points. Thus, LC reproduces ionization energies (IPs) and electron affinities (EAs) with high accuracy and one-electron excitation energies are expressed as the difference between the occupied orbital energy of a neutral molecule and the corresponding unoccupied orbital energy of its cation. Two such expressions can be used, with one employing the orbital energies for the neutral and cationic systems, while the other utilizes orbital energies of just the cation. Because the EA of a molecule is the IP of its anion, if we utilize this identity, the two expressions coincide and give the same excitation energies. Reasonable results are obtained for valence and core excitations using only orbital energies.     

Ionization potentials and electron affinities in the Perdew-Zunger self-interaction corrected density-functional theory

Using a self-consistent implementation of the Perdew-Zunger self-interaction corrected (PZ-SIC) density-functional theory, we have calculated ionization potentials (IP) and electron affinities (EA) of first- and second-row atoms and a set of small molecules. Several exchange-correlation functionals were tested. IPs and EAs were obtained by two methods: as the difference in self-consistent field (SCF) energies of neutrals and ions (deltaSCF) and as negatives of highest-occupied orbital energies. We found that, except for local spin-density approximation, PZ-SIC worsens DeltaSCF IPs and EAs. On the other hand, PZ-SIC brings orbital eigenvalues into much better agreement with electron removal energies. The Perdew-Zunger SIC seems to over-correct many-electron systems; for molecules it performs worse than for atoms. We also discuss several common approximations to PZ-SIC such as spherical averaging of orbital densities in atoms.     

Density functional perturbational orbital theory of spin polarization in electronic systems. II. Transition metal dimer complexes

We present a theoretical scheme for a semiquantitative analysis of electronic structures of magnetic transition metal dimer complexes within spin density functional theory (DFT). Based on the spin polarization perturbational orbital theory [D.-K. Seo, J. Chem. Phys. 125, 154105 (2006)], explicit spin-dependent expressions of the spin orbital energies and coefficients are derived, which allows to understand how spin orbitals form and change their energies and shapes when two magnetic sites are coupled either ferromagnetically or antiferromagnetically. Upon employment of the concept of magnetic orbitals in the active-electron approximation, a general mathematical formula is obtained for the magnetic coupling constant J from the analytical expression for the electronic energy difference between low-spin broken-symmetry and high-spin states. The origin of the potential exchange and kinetic exchange terms based on the one-electron picture is also elucidated. In addition, we provide a general account of the DFT analysis of the magnetic exchange interactions in compounds for which the active-electron approximation is not appropriate.     

Spin-orbit effects on structures of closed-shell polyatomic molecules containing heavy atoms calculated by two-component Hartree–Fock method

We have implemented geometry optimization using an analytic gradient to a two-component Kramers' restricted Hartree–Fock (KRHF) method for polyatomic molecules with closed-shell configurations. The KRHF method is a Hartree–Fock method based on relativistic effective core potentials with effective spin-orbit operators. The derivatives of spin-orbit integrals are obtained by numerical differentiation. Geometries for the various forms of polyatomic hydrides containing row 6 p-block elements are optimized with and without spin-orbit interactions. The structural changes due to spin-orbit interactions are small, but show definite trends, which correlate well with the p_1/2 spinor population. Atomization energies are reduced significantly by incorporating spin-orbit interactions for all molecules considered. The KRHF calculations of several methylhalides demonstrate that the spinor energies from the KRHF method can be useful for the interpretation of experimental photoelectron spectra of molecules exhibiting spin-orbit splittings. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1526–1533, 1998