Nodes pseudopotential

Figure Al.3.13. All-electron and pseudopotential wavefiinction for the 3s state in silicon. The all-electron 3s state has nodes which arise because of an orthogonality requirement to tlie Is and 2s core states.

The EPM required some measured data to determine the Fourier coefficients of the pseudopotential. However, the most modem approaches follow the Fermi [5] concept of developing a pseudopotential to yield a wave function without nodes that coincides with the all-electron atomic wave function outside the core and is still normalized. Several methods were developed [16-19] in the 1970s and 1980s, and new methods for constructing useful pseudopotentials continue to appear in the literature. The applications discussed here are mostly based on the pseudopotentials developed using the approach described in Ref. [19]. The important point to empha-... 

Fig. 5.12 The Heine-Abarenkov (1964) pseudopotential for aluminium which has been normalized by the Fermi energy. The term q0 gives the position of the first node. The two large dots mark the zone boundary Fourier components, vDS(111) and (200).

As is seen from the behaviour of the more sophisticated Heine-Abarenkov pseudopotential in Fig. 5.12, the first node q0 in aluminium lies just to the left of (2 / ) / and g = (2n/a)2, the magnitude of the reciprocal lattice vectors that determine the band gaps at L and X respectively. This explains both the positive value and the smallness of the Fourier component of the potential, which we deduced from the observed band gap in eqn (5.45). Taking the equilibrium lattice constant of aluminium to be a = 7.7 au and reading off from Fig. 5.12 that q0 at 0.8(4 / ), we find from eqn (5.57) that the Ashcroft empty core radius for aluminium is Re = 1.2 au. Thus, the ion core occupies only 6% of the bulk atomic volume. Nevertheless, we will find that its strong repulsive influence has a marked effect not only on the equilibrium bond length but also on the crystal structure adopted. 

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 calculate the binding energy and bulk modulus of the MgO crystal, we have used the smallest cluster, namely, the diatomic MgO unit, embedded into two coordination spheres, whose nodes have effective potentials described by Eq. 20. Table 7 compares our results with the experimental data and the results of other calculations. One could see that even the smallest cluster embedded in the pseudopotential environment allows a decent reproduction of the bulk properties. Let us note that we have used also the smallest basis consisting of the numerical HFS orbitals of the 0 and Mg ions. 

We discuss now the choice of the spin orbitals. The spin-orbitals are conceptually more important than the pseudopotential because they provide the nodal structure of the trial function. With the fixed node approximation in RQMC, the projected ground state has the same nodal surfaces of the trial function, while the other details of the trial function are automatically optimized for increasing projection time. It is thus important that the nodes provided by given spin-orbitals be accurate. Moreover, the optimization of nodal parameters (see below) is, in general, more difficult and unstable than for the pseudopotential parameters [6]. 

The simplest form of spin-orbitals for a system with translational invariance are plane waves (PW) 9k r,a) = exp[ fe r]. This form was used in the first QMC study of metallic hydrogen [33]. It is particularly appealing for its simplicity and still qualitatively correct since electron-electron and electron-proton correlations are considered through the pseudopotential . The plane waves orbitals are expected to reasonably describe the nodal structure for metallic atomic hydrogen, but no information about the presence of protons appears in the nodes with PW orbitals. 

Figure 10. Radial part of orbital products 2s 2p for gC entering in the 2s-2p exchange integral. Whereas the product formed with nodeless pseudo-valence orbitals generated by a 4-valence electron ([jHe] core) pseudopotential (PP) is always positive, the one formed with valence orbitals from all-electron (AE) calculations has a negative contribution in the core region due to the 2s radial node.

The origin of shape-consistent pseudopotentials [131,160] lies in the insight that the admixture of only core orbitals to valence orbitals in order to remove the radial nodes leads to too contracted pseudo valence orbitals and finally as a consequence to poor molecular results, e.g., to too short bond distances. It has been recognized about 20 years ago that it is indispensable to have the same shape of the pseudo valence orbital and the original valence orbital in the spatial valence region, where chemical bonding occurs. Formally this requires also an admixture of virtual orbitals in Eq. 37. Since these are usually not obtained in finite difference atomic calculations, another approach was developed. Starting point... 

The electron affinity, which is very small for the Fe atom (0.15 eV), has so far not been reliably calculated. However, even the essentially zero affinity obtained is a tremendous improvement from the uncorrelated value of -2.36 eV. One of the reasons for the small remaining errors is that only simple trial functions were used. In particular, the determinants were constructed from Hartree-Fock orbitals. It is known that the Hartree-Fock wavefunction is usually more accurate for the neutral atom than for negative ion, and we conjecture that the unequal quality of the nodes could have created a bias on the order of the electron affinity, especially when the valence correlation energy is more than 20 eV. One can expect more accurate calculations with improved trial functions, algorithms, and pseudopotentials. 

Ensuring that the local form of the pseudopotential is well-behaved it contains no division by zero at nodes in X-... 

In order to maintain the wave function antisymmetry, the diffusion QMC is normally used within the fixed node approximation, i.e. the nodes are fixed by the initial trial wave function. Unfortunately, the location of nodes for the exact wave function is far from trivial to determine, although simple approximations such as HF can give quite reasonable estimates. The fixed node diffusion QMC thus determines the best wave function with the nodal structure of the initial trial wave function. If the trial wave function has the correct nodal structure, the QMC will provide the exact solution to the Schrodinger equation, including the electron correlation energy. It should be noted that the region near the nuclei contributes most to the statistical error in QMC methods, and in many apphcations the core electrons are therefore replaced by a pseudopotential. 

The method provides valence orbitals in (168) with internal nodes, which closely resemble the original valence Hartree-Fock orbitals of (166). This method has been developed mainly by Huzinaga and colleagues, who determine atomic pseudopotentials (model potentials in their terminology) of the form ... 

The normconserving pseudopotentials have proven to work well for many systems, notably semiconductors (see, e.g., Kune, this volume) and their surfaces (see, e.g., Morthrup and Cohen, 1982), ionic compounds (Froyen and Cohen, 1984), and simple metals (see, e.g., Lam and Cohen, 1981). Applications to transition metals also exist (see, e.g., Greenside and Schluter, 1983). The pseudopotential approximation becomes less satisfactory when valence and core electrons begin to have large overlap, both because of the pseudo-wavefunctions lacking nodes, and because the x-c potential ih the core region should also account for the presence of the core electrons. The latter problem can in many cases be treated well by "nonlinear" pseudopotentials (Louie et al., 1982). 

The KB separable form has, however, some disadvantages, leading sometimes to solutions with nodal surfaces that are lower in energy than solutions with no nodes [75,76]. These (ghost) states are an artifact of the KB procedure. To eliminate them one can use a different component of the pseudopotential as the local part of the KB form or choose a different set of core radii for the pseudo-potential generation. As a rule of thumb, the local component of the KB form should be the most repulsive pseudo-potential component. For example, for the Cu potential of Fig. 6.4, the choice of / = 2 as the local component leads to a ghost state, but choosing instead I = 0 remedies the problem. 

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