Treating Core Electrons

Unlike semiempirical methods that are formulated to completely neglect the core electrons, ah initio methods must represent all the electrons in some manner. However, for heavy atoms it is desirable to reduce the amount of computation necessary. This is done by replacing the core electrons and their basis functions in the wave function by a potential term in the Hamiltonian. These are called core potentials, elfective core potentials (ECP), or relativistic effective core potentials (RECP). Core potentials must be used along with a valence basis set that was created to accompany them. As well as reducing the computation time, core potentials can include the effects of the relativistic mass defect and spin coupling terms that are significant near the nuclei of heavy atoms. This is often the method of choice for heavy atoms, Rb and up. 

The energy obtained from a calculation using ECP basis sets is termed valence energy. Also, the virial theorem no longer applies to the calculation. Some molecular properties may no longer be computed accurately if they are dependent on the electron density near the nucleus. 

There are several issues to consider when using ECP basis sets. The core potential may represent all but the outermost electrons. In other ECP sets, the outermost electrons and the last filled shell will be in the valence orbital space. Having more electrons in the core will speed the calculation, but results are more accurate if the —1 shell is outside of the core potential. Some ECP sets are designated as shape-consistent sets, which means that the shape of the atomic orbitals in the valence region matches that for all electron basis sets. ECP sets are usually named with an acronym that stands for the authors names or the location where it was developed. Some common core potential basis sets are listed below. The number of primitives given are those describing the valence region. 

CREN Available for SC(4.v) through Hs(0.v6/)6d), this is a shape-consistent basis set developed by Ermler and coworkers that has a large core region and small valence. This is also called the CEP—4G basis set. The CEP—31G and CEP—121G sets are related split valence sets. 

SBKJC VDZ Available for Li(4.v4/ ) through Hg(7.v7/ 5d), this is a relativistic basis set created by Stevens and coworkers to replace all but the outermost electrons. The double-zeta valence contraction is designed to have an accuracy comparable to that of the 3—21G all-electron basis set. Hay-Wadt MB Available for K(5.v5/ ) through Au(5.v6/ 5r/), this basis set contains the valence region with the outermost electrons and the previous shell of electrons. Elements beyond Kr are relativistic core potentials. This basis set uses a minimal valence contraction scheme. These sets are also given names starting with LA for Los Alamos, where they were developed. 

Hay Wadt VDZ Available for K(555p) through Au(556p5J), this basis 

---

There has also been a recent extension of the APW method to include arbitrary variations in the potential. In this "fuli-potential linear" (FLAPW) method, the sphere in the APW method becomes merely a convenient surface for matching wave functions—both inside and outside the sphere, the potential is allowed to have ali non-spherical components, and the equations is solved with essentially arbitrary accuracy. This can provide a major avenue to be able to treat core electrons, d and f states, and low-symmetry situations simultaneously. It has been applied only in limited cases thus far. ... 

The orbitals from which electrons are removed and those into which electrons are excited can be restricted to focus attention on correlations among certain orbitals. For example, if excitations out of core electrons are excluded, one computes a total energy that contains no correlation corrections for these core orbitals. Often it is possible to so limit the nature of the orbital excitations to focus on the energetic quantities of interest (e.g., the CC bond breaking in ethane requires correlation of the acc orbital but the 1 s Carbon core orbitals and the CH bond orbitals may be treated in a non-correlated manner). 

By convention, n must be greater than nn for a system with an od number of electrons. Also, this counting should ignore the core electrons i the molecule (these are treated in step 6). Gaussian will indicate the numbt of electrons of each type. Look for the line containing NOB in the outpt from the single point energy calculation in step 3 ... 

We begin with a presentation of the ideas of the electronic structure of metals. A liquid or solid metal of course consists of positively charged nuclei and electrons. However, since most of the electrons are tightly bound to individual nuclei, one can treat a system of positive ions or ion cores (nuclei plus core electrons) and free electrons, bound to the metal as a whole. In a simple metal, the electrons of the latter type, which are treated explicitly, are the conduction electrons, whose parentage is the valence electrons of the metal atoms all others are considered as part of the cores. In some metals, such as the transition elements, the distinction between core and conduction electrons is not as sharp. 

Most of the calculations have been done for Cu since it has the least number of electrons of the metals of interest. The clusters represent the Cu(100) surface and the positions of the metal atoms are fixed by bulk fee geometry. The adsorption site metal atom is usually treated with all its electrons while the rest are treated with one 4s electron and a pseudopotential for the core electrons. Higher z metals can be studied by using pseudopotentials for all the metals in the cluster. The adsorbed molecule is treated with all its electrons and the equilibrium positions are determined by minimizing the SCF energy. The positions of the adsorbate atoms are varied around the equilibrium position and SCF energies at several points are fitted to a potential surface to obtain the interatomic force constants and the vibrational frequency. 

Slovenia), using the DFT implementation in the Gaussian03 code. Revision C.02 (8). The orbitals were described by a mixed basis set. A fully uncontracted basis set from LANL2DZ was used for the valence electrons of Re (9), augmented by two / functions Q =1.14 and 0.40) in the full optimization. Re core electrons were treated by the Hay-Wadt relativistic effective core potential (ECP) given by the standard LANL2 parameter set (electron-electron and nucleus-electron). The 6-3IG basis set was used to describe the rest of the system. The B3PW91 density functional was used in all calculations. 

Let us consider approximations in accounting for the Breit interaction, that we made when outer core and valence electrons are included in GRECP calculations with Coulomb two-electron interactions, but inner core electrons are absorbed into the GRECP. When both electrons belong to the inner core shells, the Breit effect is of the same order as the Coulomb interaction between them. Though Bff does not contribute to differential (valence) properties directly, it can lead to essential relaxation of both core and valence shells. This relaxation is taken into account when the Breit interaction is treated by self-consistent way in the framework of the HEDB method [33, 34]. 

It was Hellmann (1935) who first proposed a rather radical solution to this problem -replace the electrons with analytical functions that would reasonably accurately, and much more efficiently, represent the combined nuclear-electronic core to the remaining electrons. Such functions are referred to as effective core potentials (ECPs). In a sense, we have already seen ECPs in a very crude form in semiempirical MO theory, where, since only valence electrons are treated, the ECP is a nuclear point charge reduced in magnitude by the number of core electrons. 

An even simpler approach to relativity, for heavy elements, is to use effective core potentials (ECPs) to represent the core electrons, taking the potentials from various compilations in the literature that explicitly include relativistic effects in the generation of the ECPs. References to such ECPs are given by Dyall et al. [103]. These relativistic ECPs (RECPs) allow the inclusion of some relativistic effects into a nonrelativistic calculation. Since ECPs will be treated in detail elsewhere, we will not pursue this approach further here. We may note, however, that recent comparisons with Dirac-Fock calculations suggest that the main weakness in the RECPs is not the treatment of relativity but the quality of the ECPs themselves [103]. Different RECPs gave spectroscopic constants with a noticeable scatter, compared to Dirac-Fock, but the relativistic corrections (difference between an RECP and the corresponding ECP value) were fairly consistent with one another. 

Treating the core electrons in effect as part of the atomic nuclei means that we need basis functions only for the valence electrons. With a minimal basis set... 

Atoms beyond the first row of the periodic table and molecules containing such atoms are commonly treated in QMC by an approach that avoids explicit consideration of core electrons see, for example [3,27]. The reason is the very considerable increase in computer time imposed by core electrons. For many properties of chemical interest, core electrons play a relatively minor role. The origin of the increase of computer time has been analyzed in various ways, but a major factor is differing time scales for valence and core electrons [28-30]. Just as pseudopotentials can serve to reduce the computational effort in basis set methods, they play an equivalent, if not more after important, role in QMC because of the heavy computational demands of sampling core electrons relative to valence electrons. 

---