Pseudoorbitals

So, for some match point / to infinity, the atomic pseudo-orbital is identical to the valence HF atomic orbital. For radial distances less than / the pseudoorbital is defined by a polynomial expansion that goes to zero. The values of the polynomial are found by matching the value and first three derivatives of the HF orbital at / . 

The shape-consistent (or norm-conserving ) RECP approaches are most widely employed in calculations of heavy-atom molecules though ener-gy-adjusted/consistent pseudopotentials [58] by Stuttgart team are also actively used as well as the Huzinaga-type ab initio model potentials [66]. In plane wave calculations of many-atom systems and in molecular dynamics, the separable pseudopotentials [61, 62, 63] are more popular now because they provide linear scaling of computational effort with the basis set size in contrast to the radially-local RECPs. The nonrelativistic shape-consistent effective core potential was first proposed by Durand Barthelat [71] and then a modified scheme of the pseudoorbital construction was suggested by Christiansen et al. [72] and by Hamann et al. [73]. 

All the above restoration schemes are called nonvariational as compared to the variational one-center restoration (VOCR, see below) procedure proposed in [79, 80]. Proper behavior of the molecular orbitals (four-component spinors) in atomic cores of molecules can be restored in the scope of a variational procedure if the molecular pseudoorbitals (two-component pseudospinors) match correctly the original orbitals (large components of bispinors) in the valence region after the molecular RECP calculation. As is demonstrated in [69, 44], this condition is rather correct when the shape-consistent RECP is involved to the molecular calculation with explicitly... 

One not so obvious problem with the shape-consistent REP formalism (or any nodeless pseudoorbital approach) is that some molecular properties are determined primarily by the electron density in the core region (some molecular moments, Breit corrections, etc.) and cannot be computed directly from the valence-only wave function. For Phillips-Kleinman (21) types of wave functions, Daasch et al. (52) have shown that the core electron density can be approximated quite accurately by adding in the atomic core orbitals and then Schmidt orthogonalizing the valence orbitals to the core. This new set of orbitals (core plus orthogonalized valence) is a reasonable approximation to the all-electron set and can be used to compute the desired properties. This will not work for the shape-consistent case because / from Eq. (18) cannot be accurately described in terms of the core orbitals alone. On the other hand, it is clear from that equation that the corelike portion of the valence orbitals could be reintroduced by adding in fy (53),... 

Semiempirical spin-orbit operators play an important role in all-electron and in REP calculations based on Co wen- Griffin pseudoorbitals. These operators are based on rather severe approximations, but have been shown to give good results in many cases. An alternative is to employ the complete microscopic Breit-Pauli spin-orbit operator, which adds considerably to the complexity of the problem because of the necessity to include two-electron terms. However, it is also inappropriate in heavy-element molecules unless used in the presence of mass-velocity and Darwin terms. 

Within the density functional theory (DFT), several schemes for generation of pseudopotentials were developed. Some of them construct pseudopotentials for pseudoorbitals derived from atomic calculations [29] - [31], while the others make use [32] - [36] of parameterized analytical pseudopotentials. In a specific implementation of the numerical integration for solving the DFT one-electron equations, named Discrete-Variational Method (DVM) [37]- [41], one does not need to fit pseudoorbitals or pseudopotentials by any analytical functions, because the matrix elements of an effective Hamiltonian can be computed directly with either analytical or numerical basis set (or a mixed one). 

Two pseudopotential schemes were developed in the framework of the DV-Xq method. One of them [42], [43] is based on the explicit inclusion of core atomic orbitals in the valence pseudoorbitals (the Phillips-Kleinman ansatz)... 

These pseudopotentials are inconvenient because the resulting pseudoorbitals are eigenfunctions to different Hamiltonians. We developed a more suitable scheme [44], [45] based on the Christiansen-Lee-Pitzer approach [9]. Let us consider first the construction of pseudopotentials, then their use for molecular systems, and, finally, for simulating surfaces and bulk of solids. 

In order to construct atomic pseudopotentials, one needs to formulate some requirements on a pseudoorbital which is to be an eigenfunction to the atomic pseudopotential equation with the same eigenvalue as in the all-electron case. We assume that outermost part of a pseudoorbital coincides with the numerical Hartree-Fock-Slater (HFS) function as close to the nucleus as possible. The pseudoorbital should be nodeless and normalized. 

According to these requirements, the pseudoorbital of each outer atomic shell is constructed from the corresponding numerical HFS radial function Rni r) by substituting its portion for r < Tc, where Tc is some core radius to be determined, by the five-term polynomial. The polynomial coefficients are chosen so as to match the amplitude and the first three derivatives of the inner portion and the remaining part of R i r). The matching point is chosen to be the innermost point at which the matching results in a nodeless function with no more than two inflections in the entire region. This can be attained when the smoothness index... 

Typical pseudoorbitals constructed from the HFS 3s and 3p orbitals of the chlorine atom are presented in Fig. 1 and 2, respectively. Since a pseudoorbital constructed in such a way has no nodes, Eq. 3 can be inverted to represent explicitly the corresponding pseudopotential... 

In order to estimate the reliability of the pseudopotential variant of our DVM package [40], [54], [55], the molecules O2, O3, CI2 as well as the complex anions PdCl and PdCl are considered. The following valence configurations of the neutral atoms were assumed O (2s 2p ), Cl (3s 3p ), and Pd (4p 4d 5s ) when constructing the pseudopotentials, and the a parameters are the Schwartz s [56] ones. We made use of the numerical pseudoorbitals as the basis functions. Several calculations were performed with adding some diffuse STOs from Roetti-Clementi s double- basis [57] to our numerical bases. 

Extended stands for the numerical pseudoorbitals augmented by some diffuse STOs of Roetti-Clementi s double- basis set. 

As is seen, the difference in the eigenvalues calculated with both approaches does not exceed 0.1 eT, as a rule. We did not find any essential dependence on the form of the core density added to the valence pseudodensity when calculating the exchange (i. e., as a superposition of the numerical atomic densities or simulated by a spherical Bessel function pc = Asin Br)fr [58]). The orbital energy of the 2cr LUMO calculated in both approaches differs by 1 eV, and the difference depends on the shape of the innermost section of the pseudoorbitals. Such a behavior is rather common [59] - [61]. 

Table 4. DV-X orbital energies (in eV) of the PdCl dianion calculated within the all-electron and pseudopotential approaches. The latter includes the atomic pseudopotentials and pseudoorbitals on the chlorine atoms only. R Pd — O) = 2.30 A.

In Fig. 2 we compare the 6s orbitals obtained for the two different couplings of the ion core. The difference in the calculations for these two orbitals is that the A a-coefficient for the exchange-correlation term in the Hartree-Slater Hamiltonian is varied to shift the calculated orbital energy to agree with the respective binding energy. The Hartree-Slater orbital for the 6s [ Fi/2 core] is also shown in Fig. 2. The inner nodes in this orbital are removed to obtain the 6s pseudoorbital. 

Fig. 2. Orbitals for the [5p (3/2)]6s [light solid line) and the [5p (l/2)]6s [heavy solid line) pseudoorbitals from which the j-wave effective potentials were determined The 6s Hartree-Slater orbital is also shown [heavy dashed line)...

The most straightforward and formally sound route to account for non-electrostatic effects is provided by the use of the pseudopotential concept. As shown by Phillips and Kleinman63, the orbitals in a selected subsystem (pseudoorbitals - 4>iK ) can be obtained from the following equation involving the pseudo-Hamiltonian (HPK) ... 

Spin-orbit (SO) coupling corrections were calculated for the Pt atom since the relativistic effects are essential for species containing heavy elements. Other scalar relativistic corrections like the Darwin and mass-velocity terms are supposed to be implicitly included in (quasi)relativistic pseudopotentials because they mostly affect the core region of the considered heavy element. Their secondary influence can be seen in the contraction of the outer s-orbitals and the expansion of the d-orbitals. This is considered in the construction of the pseudoorbitals. The effective SO operator can be written within pseudopotential (PS) treatment in the form71 75... 

Using pseudopotentials has several major beneficial consequences (i) Only the valence electrons must be treated explicitly, thus the number of equations to be solved [Eqs. (13)] can be reduced drastically (ii) the pseudoorbitals are very smooth near the atomic core, and thus Tout can be reduced drastically and (iii) important relativistic effects of the core electrons of heavy elements such as the 5d elements can be included in nonrelativistic calculations. The major downsides are that the potential v(r) in Eq. (3) must be replaced with a more complicated and computationally expensive nonlocal pseudopotential and, more importantly, that the transferability of the pseudopotential, i.e., its accuracy in different bonding environments, may not be perfect. Developing highly transferable pseudopotentials that can be used at as low an cut as possible is a major current topic of research. 

Here a pseudo-potential is used to represent the core-valence interaction. Thereby the SCF procedure is reduced in scale to encompass only the valence electrons. The development of the method is described in an application to the uranium atom [60]. The procedure is as follows. First a set of valence pseudoorbitals is formed from a linear combination of atomic orbitals, with coefficients... 

Figure 9. Contributions (%) of the all-electron Is- and 2s-orbitals to the ten energetically lowest ns-pseudoorbitals of gC in the 2s 2p ground state calculated with an uncontracted 15s9p all-electron basis set [98]. The vertical dashed line denotes the approximate position of the Is core orbital in the spectrum for model potential calculations after the typical energy shift of-2 ,.

In 1992 Dmitriev, Khait, Kozlov, Labzowsky, Mitrushenkov, Shtoff and Titov [151] used shape consistent relativistic effective core potentials (RECP) to compute the spin-dependent parity violating contribution to the effective spin-rotation Hamiltonian of the diatomic molecules PbF and HgF. Their procedure involved five steps (see also [32]) i) an atomic Dirac-Hartree-Fock calculation for the metal cation in order to obtain the valence orbitals of Pb and Hg, ii) a construction of the shape consistent RECP, which is divided in a electron spin-independent part (ARECP) and an effective spin-orbit potential (ESOP), iii) a molecular SCF calculation with the ARECP in the minimal basis set consisting of the valence pseudoorbitals of the metal atom as well as the core and valence orbitals of the fluorine atom in order to obtain the lowest and the lowest H molecular state, iv) a diagonalisation of the total molecular Hamiltonian, which... 

Effective Core Potential methods are classified in two families, according to their basic grounds. On the one hand, the Pseudopotential methods (PP) rely on an orbital transformation called the pseudoorbital transformation and they are ultimately related to the Phillips-Kleinman equation [2]. On the other hand, the Model Potential methods (MP) do not rely on any pseudoorbital transformation and they are ultimately related to the Huzinaga-Cantu equation [3,4]. The Ab Initio Model Potential method (AIMP) belongs to the latter family and it has as a... 

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