Troullier-Martins Potential

A different method to construct the pseudo wave-functions was proposed by Troullier and Martins [58,62], based on earlier work by Kerker [63]. This method is much simpler than Hamaim s and emphasizes the desired smoothness of the pseudo-potential (although it introduces additional constraints to obtain it). It achieves softer p seudo-potentials for the 2p valence states of the first row elements and for the d valence states of the transition metals. For other elements both methods produce equivalent potentials. 

Fernando Nogueira, Alberto Castro, and Miguel A.L. Marques 

The coefEcients of p r) are adjusted by imposing norm-conservation, the continuity of the pseudo wave-functions and their first four derivatives at r = rj, and that the screened pseudo-potential has zero curvature at the origin. This 

There are many other not so widely used norm-conserving pseudo-potentials [64,65,66,67,68]. Note that, in some cases, norm-conservation was abandoned in favor of increased pseudo-potential smoothness [69]. 

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In this work we recalculate the structures of Au clusters with 6scalar relativistic Troullier-Martins pseudo-potentials , respectively, and within the SIESTA code" . In Fig 2 we present our results for the structures and relative binding energies. We see that GGA leads to planar structures whereas LDA favors 3D structures for n>7 clusters. Thus, in addition to relativistic effects, the observed planarity of Au clusters is accounted for using only the GGA level of theory. 

Figure 4-2. Energy conservation in CP-MD the potential energy (Ee, main axis), temperature (kinetic energy, T, auxiliary, right-hand side axis), physical energy (T + Ee, auxiliary axis), and conserved energy (Econs). The difference between Ec0 s and T + Ee is the fictitious kinetic energy of the wavefunction. The data from the simulation for the ethylene molecule with the CPMD program13 (Troullier-Martins pseudopotentials1415, time step of 4 a.u., fictitious mass 400 a.u., cut-off energy 70 Ry, unit cell 12 Ax 12 A xl2 A)...

Structures of a large amount of different molecules without a chemical Leitmotiv were studied using Troullier-Martins-type pseudopotentials and DFT by Chen et a/.228 For a study using model potentials, see ref. 229. Molecular shapes of metal alkyls and hydrides, ML , were discussed at Valence Bond level by Landis et al.230... 

Fig. 6.3. Troullier-Martins pseudo-potential for Al, with ro = n = r2 = 2.60bohr pseudo wave-functions vs. true wave-functions (left) and pseudo-potentials (right)...

Fig. 6.4. Troullier-Martins pseudo-potential for Cu, with ro = V2 = 2.2 and ri = 2.4bohr. Notice that the LDA and GGA pseudo-potential are essentially identical, the main difference being the GGA potential oscillations near the origin...

To illustrate the use of real-space methods, we again chose to study methane (CH4). For all calculations, we used the program octopus [90] (see also http //www.tddft.org/programs/octopus), which was written by some of the authors, and is freely available under an open source license. Furthermore, we employed the Troullier-Martins pseudo-potentials which are distributed with the code, and the GGA in the parameterization of Perdew, Burke and Ernzerhof. 

The way i>f p is generated from the atomic calculation is not unique. Common pseudopotentials are generated following the prescription of, e.g., Bachelet, Hamann and Schlriter [82], Kleinman and Bylander [83], Vanderbilt [84] or Troullier and Martins [85]. Useful reviews are Refs. [86, 87, 88]. The pseudopotential approach is very convenient because it reduces the number of electrons treated explicitly, making it possible to perform density-functional calculations on systems with tens of thousands of electrons. Moreover, the pseudopotentials upp are much smoother than the bare nuclear potentials vext. The remaining valence electrons are thus well described by plane-wave basis sets. 

Figure 6.3 shows the Troullier and Martins pseudo-potential for Al, with ro = ri = r 2 = 2.60 bohr. The 3d wave-functions are not shown since the state is unbound for this potential. 

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