Recent progress in atomic and chemical group effective potentials

Recent progress on atomic and chemical group effective potentials is presented. The reviewed effective potentials follow a shape-consistent extraction technique from ab initio data, within a scalar relativistic approximation. Two types of averaged relativistic effective core potentials are considered: the correlated ones where a part of the correlation energy is included in the effective potential, and the polarized ones for which only the core polarization effects are taken into account. In addition spin-orbit polarized pseudopotentials have been extracted, and the effects of the core polarization are tested on the atomic spectroscopy of iodine. Finally a very recent chemical group effective methodology is presented, reducing the number of both electrons and nuclei explicitly treated. Chemical transferability is investigated, and test calculations on a cyclopentadienyl effective group potential are presented.     

A proper account of core-polarization with pseudopotentials: single valence-electron alkali compounds

A polarization potential is incorporated into semi-empirical and Hartree-Fock adjusted pseudopotentials. For molecules, the pseudopotentials become geometry-dependent. The omission of this effect has been the reason for too smallR_e values in earlier semi-empirical work. Very accurate results are obtained for the dimer and hydride ions of alkali elements up to K.     

Norm-conserving Hartree-Fock pseudopotentials and their asymptotic behavior

We investigate the properties of norm-conserving pseudopotentials (effective core potentials) generated by inversion of the Hartree-Fock equations. In particular, we investigate the asymptotic behavior as r-->infinity and find that such pseudopotentials are nonlocal over all space, apart from a few special cases such as H and He. Such extreme nonlocality leads to a lack of transferability and, within periodic boundary conditions, an undefined total energy. The extreme nonlocality must therefore be removed, and we argue that the best way to accomplish this is a minor relaxation of the norm-conservation condition. This is implemented, and pseudopotentials for the atoms H-Ar are constructed and tested.     

Calculation of accurate permanent dipole moments of the lowest 1,3Sigma+ states of heteronuclear alkali dimers using extended basis sets

Obtaining ultracold samples of dipolar molecules is a current challenge which requires an accurate knowledge of their electronic properties to guide the ongoing experiments. In this paper, we systematically investigate the ground state and the lowest triplet state of mixed alkali dimers (involving Li, Na, K, Rb, Cs) using a standard quantum chemistry approach based on pseudopotentials for atomic core representation, Gaussian basis sets, and effective terms for core polarization effects. We emphasize on the convergence of the results for permanent dipole moments regarding the size of the Gaussian basis set, and we discuss their predicted accuracy by comparing to other theoretical calculations or available experimental values. We also revisit the difficulty to compare computed potential curves among published papers, due to the differences in the modelization of core-core interaction.     

On the accuracy of one-component pseudopotential spin-orbit calculations

Improvements on current one-component extraction procedures of spin-orbit pseudopotentials are investigated for high accuracy computation of spin-orbit coupling energies. By means of the perturbation-theory formalism we first show that spin-orbit pseudopotentials, extracted at the one-component self-consistent-field level from a reference all-electron Dirac-Coulomb or Dirac-Coulomb-Breit calculation, include valence spin-orbit polarization and relaxation effects. As a consequence the use of these pseudopotentials in uncontracted spin-orbit configuration interaction (CI) with singles from the reference ground-state configuration gives rise to double counting of these spin-orbit effects. Two new methods that avoid such double counting have been investigated. The first, so-called "explicit" method, calculates explicitly, by means of a four-component spin-orbit CI, the double-counted spin-orbit effects and removes them from the pseudopotentials. Due to the nonadditivity of the core and valence spin-orbit effects as well as the so-called "pseudovariational collapse," this method is shown to be cumbersome. In the second "implicit" method the spin-orbit pseudopotential is extracted at the spin-orbit polarized and relaxed level by means of a single-excitation spin-orbit CI calculation. Atomic tests on iodine demonstrate the ability of the latter method to solve the double-counting problem.     

Development of new pseudopotential methods: improved model core potentials for the first-row transition metals

We have recently developed new nonrelativistic and scalar-relativistic pseudopotentials for the first-row transition metal and several main-group elements. These improved Model Core Potentials were tested on a variety of transition metal complexes to determine their accuracy in reproducing electronic structures, bond lengths, and harmonic vibrational frequencies with respect to both all-electron reference data as well as experimental data. The new potentials are also compared with the previous model core potentials available for the first-row transition metals. The new potentials do a superior job at reproducing atomic data, reproduce molecular data as well as the previous version, and in conjunction with new main-group pseudopotentials that have L-shell structure of the valence basis set, they are slightly faster.     

A combination of pseudopotentials and density functionals: Results for Cun, Cun, Agn, and Agn clusters (n ≤ 4)

Structures and binding energies have been calculated for neutral and singly ionized Cu and Ag clusters with up to four atoms. The valence-electron system is treated at the ab initio SCF level; core-valence interaction is accounted for by semi-empirical pseudopotentials, corrected for core polarization; valence correlation is included using a local spin-density-functional approximation, corrected for self-interaction. The results are compared to previous calculations for Li, Na and K clusters, and it is shown that, in spite of the different bulk crystal structure, small group 1a and 1b clusters have many properties in common.     

Ultrasoft pseudopotentials for lanthanide solvation complexes: core or valence character of the 4f electrons

The 4f electrons of lanthanides, because of their strong localization in the region around the nucleus, are traditionally included in a pseudopotential core. This approximation is scrutinized by optimizing the structures and calculating the interaction energies of Gd(3+)(H(2)O) and Gd(3+)(NH(3)) microsolvation complexes within plane wave Perdew-Burke-Ernzerhof calculations using ultrasoft pseudopotentials where the 4f electrons are included either in the core or in the valence space. Upon comparison to quantum chemical MP2 and CCSD(T) reference calculations it is found that the explicit treatment of the 4f electrons in the valence shell yields quite accurate results including the required small spin polarization due to ligand charge transfer with only modest computational overhead.     

A combination of quasirelativistic pseudopotential and ligand field calculations for lanthanoid compounds

Summary Improved energy-adjusted quasirelativistic pseudopotentials for lanthanoid atoms with fixed valency are presented and tested in molecular calculations for CeO, CeF, EuO, GdO, YbO, and YbF. The pseudopotential calculations treat the lanthanoid 4f shell as part of the core and yield accurate estimates for average bond lengths, vibrational frequencies and dissociation energies of all states belonging to a superconfiguration. Information for each individual state of the considered superconfiguration may be obtained from subsequent ligand field model calculations. The results of this combined pseudo-potential and ligand field approach (PPLFT) are compared to more accurate calculations with ab initio pseudopotentials that include the lanthanoid 4f orbitals explicitly in the valence shell and to available experimental data.     

Spectroscopic properties of alkali atoms embedded in Ar matrix

We present a theoretical investigation of visible absorption and related luminescence of alkali atoms (Li, Na, and K) embedded in Ar matrix. We used a model based on core polarization pseudopotentials, which allows us to determine accurately the gas-to-matrix shifts of various trapping sites. The remarkable agreement between our calculated results and the experimental spectra recorded by several authors allows us to establish a clear assignment of the observed spectra, which are made of contributions from crystalline sites on the one hand, and of grain boundary sites on the other hand. Our study reveals remarkably large Stokes shifts, up to 9000 cm(-1), which could be observed experimentally to identify definitely the trapping sites.     

Pseudopotential study of the ground and excited states of Yb2

The ground state of the van der Waals-type lanthanide dimer Yb₂ has been studied by means of relativistic energy-consistent ab initio pseudopotentials using three different core definitions. Electron correlation was treated by coupled-cluster theory, whereby core-valence correlation effects have been accounted for either explicitly by correlating the energetically highest coreorbitals or implicitly by means of an effective core-polarization potential. Results for the first and second atomic ionization potentials, the atomic dipole polarizability, and the spectroscopic constants of the molecular ground state are reported. Low-lying excited states have been investigated with spin-orbit configuration interaction calculations. It is also demonstrated for the whole lanthanide series that correlation effects due to the atomic-like, possibly open 4f-shell in lanthanides can be modeled effectively by adding a core-polarization potential to pseudopotentials attributing the 4f-shell to the core. Received: 3 April 1998 / Accepted: 27 July 1998 / Published online: 9 October 1998     

A systematic ab initio study of the equilibrium geometry and vibrational wave numbers of bismuthine

The equilibrium structure and the harmonic and anharmonic force fields of BiH(3) are determined by high-level ab initio calculations using a variety of correlation treatments, basis sets, and pseudopotentials, partly in combination with core polarization potentials. Spin-orbit effects are included by a configuration interaction treatment. This systematic study serves to establish a reliable computational protocol for such calculations and, in particular, to minimize basis set superposition errors through an improved new basis set and/or counterpoise corrections. Using the recommended procedures, the best ab initio results for the equilibrium geometry and the fundamental vibrational wave numbers are in good agreement with the available experimental data, which further supports the recent spectroscopic identification of BiH(3). The ground-state total atomization energy of BiH(3) is predicted to be 153.1 kcal/mol.     

How deeply are core electrons perturbed when valence electrons are spin polarized? The case study of transition metal compounds

The ferromagnetic and antiferromagnetic wave functions of the KMnF₃ perovskite have been evaluated quantum-mechanically by using an all electron approach and, for comparison, pseudopotentials on the transition metal and the fluorine ions. It is shown that the different number of α and β electrons in the d shell of Mn perturbs the inner shells, with shifts between the α and β eigenvalues that can be as large as 6 eV for the 3s level, and is far from negligible also for the 2s and 2p states. The valence electrons of F are polarized by the majority spin electrons of Mn, and in turn, spin polarize their 1s electrons. When a pseudopotential is used, such a spin polarization of the core functions of Mn and F can obviously not take place. The importance of such a spin polarization can be appreciated by comparing (i) the spin density at the Mn and F nuclear position, and then the Fermi contact constant, a crucial quantity for the hyperfine coupling, and (ii) the ferromagnetic–antiferromagnetic energy difference, when obtained with an all electron or a pseudopotential scheme, and exploring how the latter varies with pressure. This difference is as large as 50% of the all electron datum, and is mainly due to the rigid treatment of the F ion core. The effect of five different functionals on the core spin polarization is documented.     

Quasi-relativistic density functional study of aurophilic interactions

Fifteen molecules containing the Au(I) species have been calculated by ab initio HF and MP2 methods and by five different density functional approaches. The aurophilic Au(d10)-Au(d10) bonding mechanism has been investigated. Both, one-electron interactions (i.e., electrostatic, polarization, charge transfer, and orbital interference) and two-electron effects (i.e., correlation, dispersion) contribute significantly to the so-called 'secondary' or metallophilic bonds representing the Au-Au interaction. Second, the applicability of density functional approaches to this type of bonding has been tested. It is well-known that present day density functionals are not yet designed to simulate the long-range London dispersion forces between nonoverlapping systems, whereas they approximately reproduce the short range dynamical electron correlations of strongly overlapping chemically bonded nondegenerate species. It is found here empirically for the investigated groups of gold(I) cluster compounds that simple local density functionals (LDF) of the Slater (or Slater plus Vosko) type yield rather reasonable estimates for the equilibrium distances, and (on the average) also for the aurophilic interaction energies, though with rather large standard deviations. Still LDF are useful for survey investigations of Au cluster compounds. Common gradient corrected DF are not recommended here, nor are the large core pseudopotentials for Au.     

Pseudopotential method for the direct construction of localized orbitals by multiconfiguration self-consistent field theory

The orbital equations for the direct construction of localized fixed orbitals by multiconfiguration self-consistent field theory (MCSCF-FXO) are transformed without approximation into pseudopotential form by a two-step process. First the utilization of a particular family of localization is shown to separate the set of orbital equations into two sets of coupled equations, one describing “valence” orbitals and one describing “core” orbitals. In addition we obtain by appropriate choice of localization potential three different sets of MCSCF-FXO orbitals, namely: maximally screened, “one-center” and “intermediate” orbitals. In the second step the orbital equations are transformed into pseudopotential form and explicit non-local pseudopotentials yielding and core orbitals are obtained. Finally, several different physically motivated approximations to the exact pseudopotentials, and the frozen-core approximation are discussed.     

Core polarization and relativistic effects competition in the first ionization potentials for some systems in the Cu,Ag, and Au isoelectronic sequences

Single-configuration relativistic Hartree–Fock values of the first ionization potentials for Cu through Kr⁷⁺, Ag through I⁶⁺, and Au through Pb³⁺ are computed in “frozen” and “relaxed core” approximations with and without allowance for core polarization. Effects of polarization of the atomic core by the valence electron are included by introducing a polarization potential in the one-electron Hamiltonian of the valence electron. The core polarization potential depends on two parameters, the static dipole polarizability of the core α and the cut-off radius r₀, which are chosen independently of the ionization potential data. It is demonstrated that by including the core polarization potential with α and r₀ parameters, which are simply chosen instead of being empirically fitted, it is still possible to account, on the average, for at least 70% of the discrepancy between the single-configuration relativistic Hartree–Fock ionization potentials and the experiment, a discrepancy usually ascribed to the contribution of valence-core electron correlations, and to bring the theoretical ionization potentials to an average agreement with experiment of around 1%. It can be concluded from this study that for low and medium Z elements the core polarization dominates for neutral systems or systems in low ionization stages, whereas for highly ionized systems the relativistic effects prevail. For heavy elements, however, the core polarization influence is comparable to the relativistic one only for neutral systems, whereas for ions the relativistic effects are overwhelmingly predominant.     

Ionic core effects on the mie resonance in lithium clusters

We investigate the effects of the atomic cores on the Mie resonance in lithium metal clusters, perturbing a jellium Hamiltonian with zero-range pseudopotentials. The resonance is red-shifted with respect to the classical formula by core effects, most important of which is the increased effective mass due to the core potentials. Much of the large shift seen in lithium clusters is thereby explained if the strength of the pseudopotentials is taken from band structure calculations. However, such pseudopotentials cause the resonance to be greatly broadened, contrary to observation.     

Structural and spectroscopic study of the LiRb molecule beyond the Born-Oppenheimer approximation

Adiabatic and diabatic potential energy curves and the permanent and transition dipole moments of the low-lying electronic states of the LiRb molecule dissociating into Rb(5s, 5p, 4d, 6s, 6p, 5d, 7s, 6d) + Li(2s, 2p) have been investigated. The molecular calculations are performed with an ab initio approach based on nonempirical pseudopotentials for Rb(+) and Li(+) cores, parametrized l-dependent core polarization potentials and full configuration interaction calculations. The derived spectroscopic constants (R(e), D(e), T(e), ω(e), ω(e)x(e), and B(e)) of the ground state and lower excited states are in good agreement with the available theoretical works. However, the 8-10(1)Σ(+), 8-10(3)Σ(+), 6(1,3)Π, and 3(1,3)Δ excited states are studied for the first time. In addition, to the potential energy, accurate permanent and transition dipole moments have been determined for a wide interval of internuclear distances. The permanent dipole moment of LiRb has revealed ionic characters both relating to electron transfer and yielding Li(-)Rb(+) and Li(+)Rb(-) arrangements. The diabatic potential energy for the (1,3)Σ(+), (1,3)Π, and (1,3)Δ symmetries has been performed for this molecule for the first time. The diabatization method is based on variational effective Hamiltonian theory and effective metric, where the adiabatic and diabatic states are connected by an appropriate unitary transformation.     

Breakdown of the pseudopotential approximation for magnetizabilities and electric multipole moments: test calculations for Au, AuF, and Sn(n) cluster (n ≤ 20)

The response of the electronic wavefunction to an external electric or magnetic field is widely considered to be a typical valence property and should, therefore, be adequately described by accurately adjusted pseudopotentials, especially if a small-core definition is used within this approximation. In this paper we show for atomic Au and Au(+), as well as for the molecule AuF and tin clusters, that in contrast to the case of the static electric dipole polarizability or the electric dipole moment, core contributions to the static magnetizability are non-negligible, and can therefore lead to erroneous results within the pseudopotential approximation. This error increases with increasing size of the core chosen. For tin clusters, which are of interest in ongoing molecular beam experiments currently carried out by the Darmstadt group, the diamagnetic and paramagnetic isotropic components of the magnetizability tensor almost cancel out and large-core pseudopotentials do not even predict the correct sign for this property due to erroneous results in both the diamagnetic and (more importantly) the paramagnetic terms. Hence, all-electron calculations or pseudopotentials with very small cores are required to adequately predict magnetizabilities for atoms, molecules and the solid state, making it computationally more difficult to obtain this quantity for future investigations in heavy atom containing molecules or clusters. We also demonstrate for this property that all-electron density functional calculations are quite robust and give results close to wavefunction based methods for the atoms and molecules studied here.     

Molecular structure,vibrational frequencies and ionization potential of tin dihalides: An ab initio SCF and CI study

The geometrical parameters for SnX₂ (X = Cl, Br, I) and SnX₂⁺ have been optimized at the SCF and Cl levels using non-empirical pseudopotentials and a basis set of double-zeta plus polarization quality. For the neutral molecules the geometrical parameters are in agreement with the experimental values, the difference being of only 2%. Vibrational frequencies and ionization potentials are also in agreement with experiment.