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.     

Optimized Gaussian basis sets for use with relativistic effective (core) potentials: K,Ca, Ga—Kr

Correlation-consistent valence basis sets were developed for the third-row main block elements (K, Ca, Ga—Kr) for use with relativistic effective core potentials. These basis sets are somewhat larger than double-zeta in size, with polarization functions, and are balanced for use in both Hartree–Fock and correlation calculations. Spin–orbit splittings for atoms and molecules are calculated and compared to experiment. These calculations use the approximate spin–orbit operator from the relativistic effective core potentials. The use of these results in the calculation of accurate thermochemical data is discussed. © 1997 John Wiley & Sons, Inc.

1 This article is a US Government work and, as such, is in the public domain in the United States of America.

     

Optimized accurate auxiliary basis sets for RI-MP2 and RI-CC2 calculations for the atoms Rb to Rn

The introduction of the resolution-of-the-identity (RI) approximation for electron repulsion integrals in quantum chemical calculations requires in addition to the orbital basis so-called auxiliary or fitting basis sets. We report here such auxiliary basis sets optimized for second-order Møller–Plesset perturbation theory for the recently published (Weigend and Ahlrichs Phys Chem Chem Phys, 2005, 7, 3297–3305) segmented contracted Gaussian basis sets of split, triple-ζ and quadruple-ζ valence quality for the atoms Rb–Rn (except lanthanides). These basis sets are designed for use in connection with small-core effective core potentials including scalar relativistic corrections. Hereby accurate resolution-of-the-identity calculations with second-order Møller–Plesset perturbation theory (MP2) and related methods can now be performed for molecules containing elements from H to Rn. The error of the RI approximation has been evaluated for a test set of 385 small and medium sized molecules, which represent the common oxidation states of each element, and is compared with the one-electron basis set error, estimated based on highly accurate explicitly correlated MP2–R12 calculations. With the reported auxiliary basis sets the RI error for MP2 correlation energies is typically two orders of magnitude smaller than the one-electron basis set error, independent on the position of the atoms in the periodic table.     

A two-step method of calculation of the electronic structure of molecules with heavy atoms: Theoretical aspect

An approach for a space-separated calculation of the wave function in the valence and core regions of a molecule is proposed. As the first step, the calculation of the orbitals (or two-component spinors in the relativistic case) in the valence region by the effective core potential (ECP ) method is performed. Then, it is followed by a restoration of orbitals (four-component spinors) expanded on spherical harmonics in the core regions of heavy atoms. Theoretical questions of the variational calculation of the molecular orbitals are considered in some core region limited by a sphere. Inclusion from the electronic cloud outside this region is reduced by the necessity of taking into account the orthonormality and boundary conditions together with an effective external field in respect to the selected core region. This method may be used for calculation of matrix elements of operators that are singular near nuclei (P, T-odd interactions, hyperfine structure, etc.). A substantial computational saving can be reached because the method enables, by the most optimal way, to combine the advantages of two well-developed approaches: molecular ECP calculations in the Gaussian basis set and one-center numerical atomic calculations with an external field. It is especially important when the relativistic effects are taken into account. © 1996 John Wiley & Sons, Inc.     

Generalized relativistic effective core potential: Theoretical grounds

The generalized relativistic effective core potential (GRECP) method is analyzed from theoretical and computational points of view. The Hamiltonian in the frozen‐core approximation is compared with the Hamiltonian containing the GRECP operator. It is demonstrated that the GRECP operator can be derived from rather natural physical grounds and the procedure of the GRECP generation can be justified theoretically. The accuracy of the RECP approximations in the simulation of the interactions and densities in the valence and outer‐core regions is analyzed. The reliability of the simulation of the interaction with the inner‐core electrons removed from the calculations with the GRECP is also studied. The importance of additional nonlocal terms both with the potentials for the outer‐core pseudospinors and with the potentials depending on the occupation numbers of the outermost core shells in the expression for the GRECP operator is demonstrated in calculations on the Ag, Ba, Hg, Tl, and U atoms. The difference between the outer core and valence potentials was investigated. It is shown that in the valence region the two‐component pseudospinors coincide with the large components of four‐component spinors in calculations for the same configuration states with a very high accuracy. Problems of Gaussian approximation caused by rather singular shapes of the potentials are considered. To attain a required high accuracy of approximation of the numerical potentials by Gaussians, serious additional efforts were undertaken. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 359–401, 1999     

Optimized Slater-type basis sets for the elements 1-118

Seven different types of Slater type basis sets for the elements H (Z = 1) up to E118 (Z = 118), ranging from a double zeta valence quality up to a quadruple zeta valence quality, are tested in their performance in neutral atomic and diatomic oxide calculations. The exponents of the Slater type functions are optimized for the use in (scalar relativistic) zeroth-order regular approximated (ZORA) equations. Atomic tests reveal that, on average, the absolute basis set error of 0.03 kcal/mol in the density functional calculation of the valence spinor energies of the neutral atoms with the largest all electron basis set of quadruple zeta quality is lower than the average absolute difference of 0.16 kcal/mol in these valence spinor energies if one compares the results of ZORA equation with those of the fully relativistic Dirac equation. This average absolute basis set error increases to about 1 kcal/mol for the all electron basis sets of triple zeta valence quality, and to approximately 4 kcal/mol for the all electron basis sets of double zeta quality. The molecular tests reveal that, on average, the calculated atomization energies of 118 neutral diatomic oxides MO, where the nuclear charge Z of M ranges from Z = 1-118, with the all electron basis sets of triple zeta quality with two polarization functions added are within 1-2 kcal/mol of the benchmark results with the much larger all electron basis sets, which are of quadruple zeta valence quality with four polarization functions added. The accuracy is reduced to about 4-5 kcal/mol if only one polarization function is used in the triple zeta basis sets, and further reduced to approximately 20 kcal/mol if the all electron basis sets of double zeta quality are used. The inclusion of g-type STOs to the large benchmark basis sets had an effect of less than 1 kcal/mol in the calculation of the atomization energies of the group 2 and group 14 diatomic oxides. The basis sets that are optimized for calculations using the frozen core approximation (frozen core basis sets) have a restricted basis set in the core region compared to the all electron basis sets. On average, the use of these frozen core basis sets give atomic basis set errors that are approximately twice as large as the corresponding all electron basis set errors and molecular atomization energies that are close to the corresponding all electron results. Only if spin-orbit coupling is included in the frozen core calculations larger errors are found, especially for the heavier elements, due to the additional approximation that is made that the basis functions are orthogonalized on scalar relativistic core orbitals.     

The error due to neglecting the core spin–orbit splitting in valence ZORA calculations with frozen core approximation and its elimination

In valence zeroth-order regular approximation (ZORA) calculations with frozen core approximation, when the basis set optimized to the related scalar relativistic ZORA calculations is used, neglecting the core spin–orbit splitting may result in additional basis set truncation errors. It is found that the error is negligible for most elements except the 6p-block elements. When the basis set is extended by a p-type STO function put on the 6p element atoms with the ζ value proper to 5p_1/2 orbitals, the error can be reduced to be negligible. The calculated atomic properties related to valence orbitals can be improved greatly by use of this extended basis set. The frozen core approximation calculations of some molecules containing Tl, Pb and Bi with closed shells show that neglecting the core spin–orbit splitting only slightly affects the calculated bond lengths and bond energies, and the calculated molecular property can also be improved slightly by use of the extended basis sets.     

Relativistic contributions to single and double core electron ionization energies of noble gases

We have performed relativistic calculations of single and double core 1s hole states of the noble gas atoms in order to explore the relativistic corrections and their additivity to the ionization potentials. Our study unravels the interplay of progression of relaxation, dominating in the single and double ionization potentials of the light elements, versus relativistic one-electron effects and quantum electrodynamic effects, which dominate toward the heavy end. The degree of direct relative additivity of the relativistic corrections for the single electron ionization potentials to the double electron ionization potentials is found to gradually improve toward the heavy elements. The Dirac-Coulomb Hamiltonian is found to predict a scaling ratio of ～4 for the relaxation induced relativistic energies between double and single ionization. Z-scaling of the computed quantities were obtained by fitting to power law. The effects of nuclear size and form were also investigated and found to be small. The results indicate that accurate predictions of double core hole ionization potentials can now be made for elements across the full periodic table.     

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 .     

Nodeless valence (pseudo)spinors

Atomic calculations using small-core relativistic effective core potentials (RECPs) explicitly treating outer core electrons are used to define two-component nodeless valence spinors (NVSs) and nodeless valence pseudospinors (NVPSs). Errors attributable to nonlocal electron repulsion interactions that arise from large-core RECPs are shown to result from the inherent arbitrariness in the choice of match points and number of derivatives that define shape-consistent pseudospinors, as well as the positions of radial nodes that reside in the outer core regions of atoms. Self-consistent field calculations in omegaomega-coupling for InH and InCl using RECPs derived from NVSs and NVPSs are reported. Increased bond distances relative to those calculated using very-large-core RECPs for In agree with those due to frozen 4d(3/2) and 4d(5/2) spinors and a small-core RECP. Results for AmCl+2 also reveal that the shortening in the bond length is recovered when the very-large-core RECP is derived using nodeless valence (pseudo)spinors.     

Spin-orbit configuration interaction study of the electronic structure of the 5f (2) manifold of U(4+) and the 5f manifold of U(5+)

The energy levels of the 5f configuration of U(5+) and 5f(2) configuration of U(4+) have been calculated in a dressed effective Hamiltonian relativistic spin-orbit configuration interaction framework. Electron correlation is treated in the scalar relativistic scheme with either the multistate multireference second-order multiconfigurational perturbation theory (MS-CASPT2) or with the multireference single and double configuration interaction (MRCI) and its size-extensive Davidson corrected variant. The CASPT2 method yields relative energies which are lower than those obtained with the MRCI method, the differences being the largest for the highest state (1)S(0) of the 5f(2) manifold. Both valence correlation effects and spin-orbit polarization of the outer-core orbitals are shown to be important. The satisfactory agreement of the results with experiments and four-component correlated calculations illustrates the relevance of dressed spin-orbit configuration interaction methods for spectroscopy studies of heavy elements.     

Ab initio relativistic effective potentials with spin-orbit operators. VI. Fr through Pu

Ab initio averaged relativistic effective core potentials (AREP ), spin-orbit (SO ) operators, and valence basis sets are reported for the elements Fr through Pu in the form of expansions in Gaussian-type functions. Gaussian basis sets with expansion coefficients for the low-energy states of each atom are given. Atomic orbital energies calculated under the j–j coupling scheme within the self-consistent field approximation and employing the AREP 'S in their unaveraged form (REP 'S) agree to within 10% of orbital energies due to numerical all-electron Dirac–Fock calculations. The accuracy of the AREP 'S and so operators is also shown to be good through comparisons of calculated so splitting energies with all-electron Dirac–Fock results.     

Density functional theory optimized basis sets for gradient corrected functionals: 3d transition metal systems

Density functional theory optimized basis sets for gradient corrected functionals for 3d transition metal atoms are presented. Double zeta valence polarization and triple zeta valence polarization basis sets are optimized with the PW86 functional. The performance of the newly optimized basis sets is tested in atomic and molecular calculations. Excitation energies of 3d transition metal atoms, as well as electronic configurations, structural parameters, dissociation energies, and harmonic vibrational frequencies of a large number of molecules containing 3d transition metal elements, are presented. The obtained results are compared with available experimental data as well as with other theoretical data from the literature.     

Core/valence partition and relativistic effects in effective potentials for transition metals

Two different versions, relativistic and nonrelativistic, of the “shape-consistent” effective potential (EP ) formalism are analyzed in ab initio calculations for transition metals. The influence of alternative core/valence partitions on the reliability of the procedure is discussed. The accuracy of EP results for transition metals is shown to depend on adequate choices of the valence subspace and proper inclusion of relativistic effects.     

Quasi-relativistic approximation in the SCF-MO-LCAO method

A quasi-relativistic approach to the MO-LCAO method is formulated taking into account the relativistic effects with an accuracy up to (v/c)² terms, the relativistic part of the electronic interaction in the Hamiltonian being neglected. In the framework of this approximation a set of SCF equations of the Roothaan form is derived; here only the relativistic analogue to the closed shell systems with one-determinant wave functions is considered. In so doing three types of relativistic corrections arise which are quite similar to those of the Pauli equation for one-electron atoms. The new matrix elements appearing due to these corrections can be reduced to some common integrals, which have to be calculated with relativistic radial atomic functions. The method allows a semi-empirical approach to the problem and does not require the Dirac four-component atomic functions (unknown in the most cases), thus making possible approximate quasi-relativistic electronic structure calculations of heavy-atom compounds.     

Frozen core and effective core potentials in symmetry-adapted perturbation theory

The application of the frozen-core approximation (FCA) and effective core potentials (ECPs) within symmetry-adapted perturbation theory (SAPT) has been investigated and implemented. Unlike in the case of conventional electronic-structure theories, the development of a frozen-core version of SAPT is not straightforward. In particular, the FCA realizations neglecting excitations from core orbitals and restricting all summation indices to valence orbitals only are no longer equivalent. It is shown that it is necessary in SAPT to keep some terms containing products of the valence orbitals of one monomer and the core orbitals of the other one in the exchange-energy components. When these terms are included or, equivalently, the "infinite-excitation-energy" approximation omitting only the excitations from the core orbitals is used, the accuracy of the frozen-core approximation in SAPT matches that obtained in supermolecular perturbational and coupled-cluster methods. If these terms are neglected, i.e., within the "index-range-restriction" approximation, several exchange corrections are significantly underestimated. When ECPs are used in SAPT, the accuracy of the interaction energies is as good as in conventional supermolecular methods, provided that the residual supermolecular Hartree-Fock term is included. We have found that only some types of ECPs can be reliably used for calculations of interaction energies both in SAPT and in supermolecular approaches. For systems containing heavy atoms, both FCA and the use of ECPs lead to very significant savings of computer time.