Bases

Javier JunqueraAtomic orbitals of finite rangeas basis setsMost important reference followed in this lecture(basis)Different methods proposedifferent basis functionsEach method has its own advantages:- most appropriate for a range of problems- provide insightful information in its realm of applicationEach method has its own pitfalls:- importance to understand the method, the pros and the cons.- what can be computed and what can not be computedThree main families of methods depending on the basis setsAtomic sphere methodsLocalized basis setsPlane wave and gridsThree main families of methods depending on the basis setsAtomic sphere methodsLocalized basis setsPlane wave and gridsyNumerical atomic orbitalsNumerical solution of the Kohn-Sham Hamiltonian for the isolated pseudoatom with the same approximations(xc,pseudos) as for the condensed systemThis equation is solved in a logarithmic grid using the Numerov methoddegree of freedom to play withSize:Number of atomic orbitals per atomfH HBasis set generated for isolated atoms… …but used in molecules or condensed systemsStarting from the function we want to suplementThe second-ζ function reproduces the tail of the of the first-ζ outside a radiusAnd continuous smoothly towards the origin as(two parameters: the second-ζ and its first derivative continuous at rThe same Hilbert space can be expanded if we use the difference, with theadvantage that now the second-ζ vanishes at rm (more efficient)Finally, the second-ζ is normalizedPolarize: add l= -1 m = 0 m = +2New orbitals directed indifferent directions withrespect the original basisImproving the quality of the basis ⇒more atomic orbitals per atomCohesion curves PW and NAO convergenceAtomic orbitals show nice convergence with respect the sizePolarization orbitals very important for convergence (more than multiple-ζ)= 0Particle in a confinement potential: Imposing a finite+has a node∞ whenThe range and shape might be also controlled by an extra charge and/or by a confinement potentialExtra charge δQOrbitals in anions tend to be more delocalizedOrbitals in cations tend to be more localized(For instance, this parameter might be important in some oxides)Confinement potentialsSolve the Schrödinger equation for the isolated atom insidean confinement potentialaDetermined by the energy shiftproduces orbitals with first derivative discontinuous at rcSoft-confinement potentialPitfall:two new parameters to play with, more exploratory calculationsAdvantages:orbital continuous with all the derivatives continuos diverges at r c (orbital exactly vanishes there)zero at the core regionAvailable in SIESTAJ. Junquera et al., Phys. Rev. B 64, 235111 (2001)How to introduce the basis set in SIESTA Effort on defining a systematic with minimum parameters If nothing is specified: defaultDefault valueBasis size: PAO.BasisSize DZPRange of first-zeta: PAO.EnergyShift 0.02 RySecond-zeta: PAO.BasisType SplitRange of second-zeta: PAO.SplitNorm 0.15Confinement: Hard wellGood basis set in terms of accuracy versus efficiencyMore global control on the basis with a few input variables: size and rangeSize:Basis size:PAO.BasisSize SZDZSZPDZP Range:Range of first-zeta: PAO.EnergyShift 0.02 RyRange of second-zeta: PAO.SplitNorm 0.15The larger both values, the more confined the basis functionsMore specific control on the basis: the PAO.Basis blockThese variables calculated from PAO.EnergyShift and PAO.SplitNorm valuesV0 in Ry r i in bohrsRecapNumerical Atomic OrbitalsA very efficient basis setEspecially suitable for Order-N methodsSmooth transition from quick exploratory calculations tohighly convergedLack of systematic convergenceSimple handles for tuning the basis setsGenerate multiple-ζ: Split ValenceGenerate polarization orbitals: Perturbative polarizationControl the range of the orbitals in a balanced way: Energy ShiftConfine the orbitals: Soft-confinement potentialA DZP basis set, the same deviations as DFT functional or PseudoSuplementary informationaBoundary conditions:k must satisfy。