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Pseudopotentials Pseudopotentials ”Replacement of one problem with another…” Basic idea: Replace the strong Coulomb potential of the nucleus and the effects of the tightly bound (chemically inert) core electrons by an effective potential acting on (chemically active) valence electrons. Physical reasoning: Core electron wave functions of an atom remain essentially unchanged when placed into different chemical environment and the core wave function’s only major contrigution to chemical bonding is to enforce the valence functions orthogonality to the core states. ”A pseudopotential can be generated in an atomic calculation and then used to compute properties of valence electron in molecules or solids, since the core states remain almost unchanged. Furthermore, the fact that pseudopotentials are not unique allows the freedom to choose forms that simplify the calculations and the interpretation of the resulting electronic structure.” - R.M. Martin ( Electronic structure: basic theory and practical method ) The first applications in scattering theory (atoms, nuclei) go back to the work of Fermi and Co. during the 1930’s. Hellmann used for solids in 1935 (scattering of electrons in metals). Revived interest in 1950’s (Phillips, Kleiman, Antoncik), invent of the orthogonalized plane wave (OPW) method, band structure of sp bonded metals and semiconductors. 1. Most modern pseudopotential calculations are based upon ”ab initio norm- conserving” potentials (e.g. Troullier-Martins 1991). Accurate and transferable , pseudopotential constructed in one environment (atom) describes valence properties in different environments including atoms, ions, molecules, and condensed matter 2. This approach has been extended by Blöchl and Vanderbilt (1990) who defined the ”ultrasoft pseudopotentials ”. Auxiliary localized functions, expressing pseudofunction as a sum of of a smooth part and a more rapidly varying function localized around each ion core Improved accuracy (?), less costly 3. Most recently projector augmented wave (PAW) formalism by Blöchl (1994) Reformulation of the OPW method, all-electron, no pseudopotentials Particularly suitable for DFT calculations (e.g. GPAW) Pseudopotential transformation: The pseudopotential can be chosen to be both smoother and weaker than the original potential V by taking advantage of the nonuniqueness of the pseudopotentials. Even thought the potential is a more complex object than a simple local potential, the fact that it is weaker and smoother (i.e., it can be expanded in a small number of Fourier components) has great advantages, conceptually and computationally . In particular, it immediately resolves the apparent contradiction that the valence bands in many materials are nearly-free-electron-like , although the wavefunctions must be orthogonal to the cores (non- free-electron-like). The resolution is that the bands are determined by the secular equation for the smooth wavefunction that involves the weak pseudopotential VPKA . ψ v →ψ~v nk nk Model ion potentials: Model potential replaces the potential of a nucleus and the core electrons. They are spherically symmetric and each angular momentum l,m can be treated separately non-local l-dependent model pseudopotential Vl(r) In general, pseudopotential is a non-local operator that can be written in ”semilocal” (SL) form = VSL ∑ Ylm Vl (r) Ylm lm All information is in the radial functions Vl(r) or their Fourier transforms Modern approach to the definition of potentials: ab initio potentials constructed to fit the valence properties calculated for the atom. Norm-conserving pseudopotentials Special role in the development of ab initio pseudopotentials Simplifies the application of pseudopotentials Norm-conserving pseudofunctions are normalized and are solutions of a model potential. The valence pseudofunctions satisfy the usual orthonormality conditions standard Kohn-Sham equations for the valence states (insert pseudopotential, pseudowavefunctions) More accurate and transferable Norm-conservation condition* 1. All-electron and pseudo valence eigenvalues agree for the chosen atomic reference configuration. 2. All-electron and pseudo valence wavefunctions agree beyond a chosen core radius Rc. 3. The logarithmic derivatives of the all-electron and pseudo wavefunctions agree at Rc. 4. The integrated charge inside Rc for each wavefunction agrees (norm-conservation). 5. The first energy derivative of the logarithmic derivatives of the all- electron and pseudowavefunctions agrees at Rc, and therefore for all r ≥ Rc. *According to Hamann, Schluter, and Chiang (1979) From Points 1 and 2 it follows that the NCPP equals the atomic potential outside the core region of radius Rc. Point 3 results in that the wavefunction and its derivative are continuous at Rc smooth potential Point 4 ensures that: (a) the total charge in the core region is correct and, (b) the normalized pseudo-orbital is equal to the true orbital outside Rc. R R = c 2 ψ 2 = c 2 φ 2 Ql drr l (r) drr l (r) ∫0 ∫0 Point 5 will reproduce the changes in the eigenvalues to linear order in the change in the self-consistent potential (e.g. molecule, solid). This is actually implied by Point 4 (not obvious). Generation of l-dependent norm-conserving pseudopotentials: Usual all-electron atomic calculation, radial Kohn-Sham equations Once the pseudo wavefunction is obtained (Points 1-5 above), the screened pseudopotential is recovered by inversion of the Schrödinger equation Next, remove the screening effects and generate an ionic pseudopotential Each angular-momentum component of the wavefunction will see a different potential. It is useful to separate the ionic pseudopotential into a local part (l-independent) of the potential plus non-local terms where Pl projects out the lth angular momentum component (”semilocal” operator). The local part includes all the long-range effects of the Coulomb potential and the non-local term vanishes at r > Rc (potential equals to all- electron potential). The semilocal potential can be transformed into a nonlocal form by using a procedure suggested by Kleinman and Bylander (KB, 1982), which includes the atomic reference pseudo wavefunction. Substantial savings in the computer time and storage can be achieved using this separable nonlocal expression. Even if one requires norm-conservation, there is still freedom of choice in the form of the pseudopotential. There is no one ”best pseudopotential” for any given element – there may be many ”best choices”, each optimized for some particular use of the pseudopotential. In general, there are two overall competing factors: Accuracy and transferability generally lead to the choice of small cutoff radius Rc and ”hard” potentials, since one wants to describe the wavefunctions as well as possible in the region near the atom Smoothness of the resulting pseudofunctions usually leads to choice of large cutoff radius Rc and ”soft” potentials, since one wants to describe the wavefunction with as few basis functions as possible (e.g. plane waves) Additional points: 1. Relativistic corrections Effects of special relativity can be incorporated into pseudopotentials (e.g. Au), since they originate deep in the interior of the atom near the nucleus, and the consequences can be easily carried into molecular or solid state calculations. This includes shifts due to scalar relativistic effects and spin- orbit interactions . 2. Non-linear core correction (NLCCs) The exchange-correlation potential is a non-linear functional of the density, and the core contribution has to be taken into account by applying a special correction term that involves the charge density of the core orbitals. The core density must be stored along with the pseudodensity. This increases significantly the transferability, but may be expensive computationally. The effect is particularly large for cases in which the core is extended (e.g. 3d- transition metals, 3p ”core” vs. 3d ”valence”). Norm-conserving pseudopotentials: examples This weeks exercise: Read the original article (course webpage) and write a two page essay of the Troullier- Martins prescription for pseudopotentials. Comparison of PPs: Plane-wave basis set TM91 form of PPs (”Present Method”) is a popular choice. Global convergence greatly improved (60 Ry is already good). The PP shape does matter. Convergence can be deceiving (see the shoulder). Ionic pseudopotentials for carbon: comparison Decay rate of the Fourier transform (Vanderbilt) or the shallowness of the potential (HSC) are not good convergence indicators (previous slide). More examples: oxides and copper Deep p potential of oxygen controls As a 3 d-metal, copper is very the convergence, 70 Ry cutoff energy challenging (deep d potential) is a good choice.
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