Pseudopotentials frozen-core

Flere we distinguish between nuclear coordinates R and electronic coordinates r is the single-particle kinetic energy operator, and Vp is the total pseudopotential operator for the interaction between the valence electrons and the combined nucleus + frozen core electrons. The electron-electron and micleus-micleus Coulomb interactions are easily recognized, and the remaining tenu electronic exchange and correlation... 

The most important approach to reducing the computational burden due to core electrons is to use pseudopotentials. Conceptually, a pseudopotential replaces the electron density from a chosen set of core electrons with a smoothed density chosen to match various important physical and mathematical properties of the true ion core. The properties of the core electrons are then fixed in this approximate fashion in all subsequent calculations this is the frozen core approximation. Calculations that do not include a frozen core are called all-electron calculations, and they are used much less widely than frozen core methods. Ideally, a pseudopotential is developed by considering an isolated atom of one element, but the resulting pseudopotential can then be used reliably for calculations that place this atom in any chemical environment without further adjustment of the pseudopotential. This desirable property is referred to as the transferability of the pseudopotential. Current DFT codes typically provide a library of pseudopotentials that includes an entry for each (or at least most) elements in the periodic table. 

Except for occasional discussions of the basis set dependence of the results, the numerical implementation issues such as grid integration techniques, electron-density fitting, frozen-cores, pseudopotentials, and linear-scaling techniques, are omitted. 

Besides the reduction of frozen-core errors when going from large-core to medium-core or small-core potentials also the valence correlation energies obtained in pseudopotential calculations become more accurate since the radial nodal structure is partially restored [97,98]. Clearly the accuracy of small-core potentials is traded against the low computational cost of the large-core po-... 

The results of this procedure for alkaline and alkaline-earth systems were quite good [186,187], at least for atoms, and pseudopotentials of this type were generated [188] and applied [189] for most of the main group elements. However, due to the limited validity of the frozen-core approximation when going from a medium or highly charged one-valence electron ion to a neutral atom or nearly neutral ion, the approach is bound to fail for most other elements. This is especially the case for transition metals, lanthanides and actinides, where small cores are indispensable for accurate pseudopotentials. More recent calibration studies of alkaline and alkaline earth elements exhibited however, that for accu-... 

Modern relativistic effective core potentials provide a useful tool for accurate quantum chemical investigations of heavy atom systems. If sufficiently small cores are used to minimize frozen-core and other errors, they are able to compete in accuracy with the more rigorous all-electron approaches and still are, at the same time, economically more attractive. Successful developments in the field of valence-only Hamiltonians turned relativistic effects into a smaller problem than electron correlation in practical calculations. Both the model potential and the pseudopotential variant have advantages and disadvantages, and the answer to the question which approach to follow may be a matter of personal taste. Highly accurate correlated all-electron calculations are becoming... 

Shape-Consistent Pseudopotentials. - While with model potentials the wavefunction is (ideally) not changed with respect to the valence part of an AE frozen-core wavefunction, such a change is desirable for computational reasons. The nodal structure of the valence orbitals in the core region requires highly localized basis functions these are not really needed for the description of bonding properties in molecules but rather for the purpose of core-valence orthogonalization. The idea to incorporate this Pauli repulsion of the core into the pseudopotential is as old as pseudopotential theory itself.62,63 Modem ab initio pseudopotentials of this type have been developed since the end of the seventies, cf. e.g. refs. 64-68. 

Effective potentials also depend on the type of basis set used, hi atomic orbital calculations, they are sometimes referred to as frozen-core potentials. In most cases, only the highest-energy s, p and d electrons are included in the calculation. In plane-wave calculations, effective potentials are known as pseudopotentials They come in different varieties soft or ultrasoft pseudopotentials need only a relatively low energy cut-off as they involve a larger atomic core. ... 

The choice of the basis set is strictly related to another ingredient of several implementations of the DFT scheme, i.e. pseudopotentials. As is well known, the chemistry of the elements depends predominantly on the valence electrons, i.e. on those in the highest-energy incomplete atomic shell. It is natural, therefore, to include only the chemically active electrons explicitly in the computation. Two different but closely related approaches have been introduced to exploit this basic simplification (1) the frozen-core approximation, which assumes that the core is not modified by the formation of chemical bonds, and (2) the pseudopotential formalism, which replaces the interaction between valence and core electrons by an external potential acting on the former and does not explicitly include the latter. 

One way to reduce the computational cost of DFT (or WFT) calculations is to recognize that the core electrons of an atom have only an indirect influence on the atom chemistry. It thus makes sense to look for ways to precompute the atomic cores, essentially factoring them out of the larger electronic structure problem. The simplest way to do this is to freeze the core electrons, or to not allow their density to vary from that of a reference atom. This frozen core approach is generally more computationally efficient. One class of frozen core methods is the pseudopotential (PP) approach. The pseudopotential replaces the core electrons with an effective atom-centered potential that represents their influence on valence electrons and allows relativistic effects important to the core electrons to be incorporated. The advent of ultrasoft pseudopotentials (US-PPs) [18] enabled the explosion in supercell DFT calculations we have seen over the last 15 years. The projector-augmented wave (PAW) [19] is a less empirical and more accurate and transferable approach to partitioning the relativistic core and valence electrons and is also widely used today. Both the PP and PAW approaches require careful parameterizations of each atom type. 

Full ab initio treatments for complex transition metal systems are difficult owing to the expense of accurately simulating all of the electronic states of the metal. Much of the chemistry that we are interested in, however, is localized around the valence band. The basis functions used to describe the core electronic states can thus be reduced in order to save on CPU time. The two approximations that are typically used to simplify the basis functions are the frozen core and the pseudopotential approximations. In the frozen core approximations, the electrons which reside in the core states are combined with the nuclei and frozen in the SCF. Only the valence states are optimized. The assumption here is that the chemistry predominantly takes place through interactions with the valence states. The pseudopotential approach is similar. 

The valence electrons oscillate in the core region as is shown in Fig. A5, which is difficult to treat using plane wave basis functions. Since the core electrons are typically insensitive to the environment, they are replaced by a simpler smooth analytical function inside the core region. This core can also now include possible scalar relativistic effects. Both the frozen core and pseudopotential approximations can lead to significant reductions in the CPU requirements but one should always test the accuracy of such approximations. 

Calculations using pseudopotentials instead of dealing with all-electron problem rely on the validity of what is called "Frozen Core Approximation" as for the behavior of the charge densities of the valence electrons, it is expected to resemble the "real" (all-electron) charge distribution as closely as possible outside the core-region. 

The initio pseudopotentials together with the local density approximation have proven to yield excellent results as will be illustrated by the examples in this review. The basic assumptions in their construction are the frozen-core approximation and a decoupling of the core charge in the determination of the exchange-correlation potential seen by the valence electrons. 

When combined with a frozen core, this approximation is equivalent to pseudopotential or ab initio model potential methods, which are developed in chapter 20. These methods incorporate relativistic effects into a one-electron operator for heavy atoms, and the rest of the molecule is treated nonrelativistically. 

To formalize these concepts, we first develop the frozen-core approximation, and then examine pseudoorbitals or pseudospinors and pseudopotentials. From this foundation we develop the theory for effective core potentials and ab initio model potentials. The fundamental theory is the same whether the Hamiltonian is relativistic or nonrela-tivistic the form of the one- and two-electron operators is not critical. Likewise with the orbitals the development here will be given in terms of spinors, which may be either nonrelativistic spin-orbitals or relativistic 2-spinors derived from a Foldy-Wouthuysen transformation. We will use the terminology pseudospinors because we wish to be as general as possible, but wherever this term is used, the term pseudoorbitals can be substituted. As in previous chapters, the indices p, q, r, and s will be used for any spinor, t, u, v, and w will be used for valence spinors, and a, b, c, and d will be used for virtual spinors. For core spinors we will use k, I, m, and n, and reserve i and j for electron indices and other summation indices. 

The sums are over the valence electrons only, and the frozen-core pseudopotential is... 

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