Core-Polarization Pseudopotentials

What remedies do we have The brute-force device tried in pioneer days, of incorporating core- and core-valence correlation effects into pseudopotentials just by fitting to experimental reference data containing these effects, does not work since the one-electron/one-center PP ansatz is insufficient for this purpose, cf. below. Certainly more reliable is a DFT description of core contributions to correlation effects which is possible with (and actually implied in) the non-linear core corrections discussed in Section 1.4. Another device, which has shown excellent performance in the context of quantum-chemical ab initio calculations180 and has later been adapted to PP work cf. e.g. refs. 139, 181-184), is that of core-polarization potentials (CPP) 

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Core polarization pseudopotentials, by Stoll and coworkers (4-valence electrons) from Reference 106. mDirac-Hartree-Fock calculations by Visser and coworkers From Reference 82. 

The ECP basis sets include basis functions only for the outermost one or two shells, whereas the remaining inner core electrons are replaced by an effective core or pseudopotential. The ECP basis keyword consists of a source identifier (such as LANL for Los Alamos National Laboratory ), the number of outer shells retained (1 or 2), and a conventional label for the number of sets for each shell (MB, DZ, TZ,...). For example, LANL1MB denotes the minimal LANL basis with minimal basis functions for the outermost shell only, whereas LANL2DZ is the set with double-zeta functions for each of the two outermost shells. The ECP basis set employed throughout Chapter 4 (denoted LACV3P in Jaguar terminology) is also of Los Alamos type, but with full triple-zeta valence flexibility and polarization and diffuse functions on all atoms (comparable to the 6-311+- -G++ all-electron basis used elsewhere in this book). 

The assumption that an electron can be described as a quasi-free particle implies that the interaction between the electron and any atom in the liquid is weak. It is then necessary that the attractive potential of the nucleus experienced by the electron penetrating the atomic core and the long range core polarization potential will be balanced by the electron s increased kinetic energy in the nuclear region. This restriction implies that the pseudopotential of the atom should be small. 

Other complications are associated with the partitioning of the core and valence space, which is a fundamental assumption of effective potential approximations. For instance, for the transition elements, in addition to the outermost s and d subshells, the next inner s and p subshells must also be included in the valence space in order to accurately compute certain properties (54). A related problem occurs in the alkali and alkaline earth elements, involving the outer s and next inner s and p subshells. In this case, however, the difficulties are related to core-valence correlation. Muller et al. (55) have developed semiempirical core polarization treatments for dealing with intershell correlation. Similar techniques have been used in pseudopotential calculations (56). These approaches assume that intershell correlation can be represented by a simple polarization of one shell (core) relative to the electrons in another (valence) and, therefore, the correlation energy adjustment will be... 

In order to treat polar semiconductors, we must make some assumption as to how the odd part of the pseudopotential, K, varies with distortion. The assumption that it is independent of shear, which was used in the LCAO theory, gives a polar value different from the homopolar value by a factor 2/( 2 + V Y = ae, or values for the isoelectronic series of Ge, GaAs, and ZnSe of 0.87,0.81, and 0.74, respectively, compared to the experimental values of 0.80, 0.65, and 0.32. The trend is right, though it is not quantitatively very accurate. To estimate a, we used the empty-core polarities from Table 18-2. The agreement is better if LCAO values are used but not significantly so. 

At present, the low-lying states of Na2 are better characterized computationally than experimentally, although multiphoton ionization experiments may change that picture eventually. We find reasonably close agreement between the results of our all-electron computations, pseudopotential (10), and model potential (11) computations. The latter two kinds of computations may give more accurate results than our ab initio computations since they may account for at least certain core polarization effects. 

A further reduction of the computational effort in investigations of electronic structure can be achieved by the restriction of the actual quantum chemical calculations to the valence electron system and the implicit inclusion of the influence of the chemically inert atomic cores by means of suitable parametrized effective (core) potentials (ECPs) and, if necessary, effective core polarization potentials (CPPs). Initiated by the pioneering work of Hellmann and Gombas around 1935, the ECP approach developed into two successful branches, i.e. the model potential (MP) and the pseudopotential (PP) techniques. Whereas the former method attempts to maintain the correct radial nodal structure of the atomic valence orbitals, the latter is formally based on the so-called pseudo-orbital transformation and uses valence orbitals with a simplified radial nodal structure, i.e. pseudovalence orbitals. Besides the computational savings due to the elimination of the core electrons, the main interest in standard ECP techniques results from the fact that they offer an efficient and accurate, albeit approximate, way of including implicitly, i.e. via parametrization of the ECPs, the major relativistic effects in formally nonrelativistic valence-only calculations. A number of reviews on ECPs has been published and the reader is referred to them for details (Bala-subramanian 1998 Bardsley 1974 Chelikowsky and Cohen 1992 Christiansen et... 

Figure 15. First (IPj) and second (IP2) ionization potentials of the lanthanide elements j La -2jLu. Experimental values are compared to results from 4f-in-core pseudopotential (PP) calculations with and without account of core-valence correlation effects by means of a core polarization potential (CPP) [95].

Molecular constants of selected Ge diatomics obtained with energy-consistent four-valence electron pseudopotential (PP) [197] and a core-polarization potential (CPP) [188] in connection with the optimized cc-pVnZ (n=T,Q) valence basis sets of Martin and Sundermann [241]. The label > denotes the result of an extrapolation to the basis set limit. 

Bond lengths R (A), binding energies D. (eV) and vibrational constants a>e (cm ) of the homonuclear halogen dimers from dl-electron (AE) Douglas-Kroll-HeB (DKH) and valence-only energy-consistent pseudopotential (EC-PP) Hartree-Fock self-consistent field (SCF) calculations. The effects of static and dynamic core-polarization at the valence-only level are modelled by a core-polarization potential (CPP). 

The static dipole polarizabilities of alkali dimers have been calculated as a function of the internuclear distance and of the vibrational index for both their electronic ground state and lowest triplet state. The method is based on /-dependent pseudopotentials for atomic core representation, Gaussian basis sets, effective core potentials to account for core polarization, the evaluation of molecular orbitals by the restricted HF method, and then a full valence Cl treatment. For all alkali pairs, the parallel and perpendicular components of the ground state a at equilibrium distance Rg scale as the cube of Re, which can be related to a simple electrostatic model of an ellipsoidal charge distribution. So, for the ground state, the longitudinal polarizability exhibits a maximum at a distance corresponding to 1.3-1.5 times the equilibrium distance. 

Another approach which has proven effective for including inner-shell effects is the core polarization potential (CPP) method introduced by Muller, Flesh, and Meyer. The method can be employed in both all-electron and pseudopotential calculation. It is based on the classical description of the interaction of a polarizable core with the field generated by the valence electrons and other cores ... 

A. Weigand, X. Cao, J. Yang, and M. Dolg, Quasirelativistic f-in-core pseudopotentials and core-polarization potentials for trivalent actinides and lanthanides molecular test for trifluorides, Theor. Chem. Acc., 126, 117-127 (2010). 

In the present work, correlation consistent basis sets have been developed for the transition metal atoms Y and Hg using small-core quasirelativistic PPs, i.e., the ns and (nA)d valence electrons as well as the outer-core (nA)sp electrons are explicitly included in the calculations. This can greatly reduce the errors due to the PP approximation, and in particular the pseudo-orbitals in the valence region retain some nodal structure. Series of basis sets from double-through quintuple-zeta have been developed and are denoted as cc-pVwZ-PP (correlation consistent polarized valence with pseudopotentials). The methodology used in this work is described in Sec. II, while molecular benchmark calculations on YC, HgH, and Hg2 are given in Sec. III. Lastly, the results are summarized in Sec. IV. 

Pseudopotential. A Basis Set which treats only Valence electrons in an explicit manner, all other electrons being considered as a part of a Core . LAVCP (and extensions including Polarization and/or Diffuse Functions) are pseudopotentials. 

The relaxation of the structure in the KMC-DR method was done using an approach based on the density functional theory and linear combination of atomic orbitals implemented in the Siesta code [97]. The minimum basis set of localized numerical orbitals of Sankey type [98] was used for all atoms except silicon atoms near the interface, for which polarization functions were added to improve the description of the SiOx layer. The core electrons were replaced with norm-conserving Troullier-Martins pseudopotentials [99] (Zr atoms also include 4p electrons in the valence shell). Calculations were done in the local density approximation (LDA) of DFT. The grid in the real space for the calculation of matrix elements has an equivalent cutoff energy of 60 Ry. The standard diagonalization scheme with Pulay mixing was used to get a self-consistent solution. In the framework of the KMC-DR method, it is not necessary to perform an accurate optimization of the structure, since structure relaxation is performed many times. 

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