© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Periodic Boundary Methods and Applications: Ab-initio Quantum Mechanics for Band Structures CH121b Jamil Tahir-Kheli MSC, Caltech May 4, 2011

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Outlines (1)What is different about crystalline solids? (2)Bloch theorem (3)First Brillouin zone (4)Reciprocal space sampling (5)Plane wave, APW, Gaussian basis sets (6)SeqQuest (7)Crystal06

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 What is different about solids? HHHHHHH a Infinite repeating pattern of atoms with translational symmetry Even if you have 1 basis function per atom, there is still an infinite number of atoms leading to diagonalization of an infinite matrix! This implies we can never solve crystals By exploiting the translational symmetry of the crystal, we can find a way to break the problem into finite pieces that approximate the solution

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Bloch Theorem (simplification due to translation symmetry) K-Vectors

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Bloch Theorem (example: one dimensional hydrogen chain) HHHHHHH a

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Bloch Theorem (example: one dimensional hydrogen chain) band structure k = 0 k =  /a

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Density of States

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Bloch Theorem (example: two dimensional hydrogen surface)

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 The First Brillouin zone The first Brillouin zone contains all possible interactions between two adjacent unit cells.

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Hartree-Fock-Roothaan Equation in periodic systems Finite diagonalizations

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 We can solve for each k-point, but there are an infinite number of them By evaluating each k-point at the first Brillouin zone and summing them together, we can obtain the properties such as total energy or electron density of the system In practice, the only computationally feasible approach is to approximate the full BZ integral with summation over a finite set of k-points. Impossible !!!

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Reciprocal Space Sampling (Monkhost-Pack grids)

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Differences between Molecular and Periodic Codes There is an infinity far away from the molecule where the density decays to zero as an exponential. The exponent is the ionization potential (up to a factor) and can be shown to equal the HOMO eigenvalue. DFT obtains exact density and thus IP.

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 There is no vacuum away from infinite crystal where we can define the zero of the electrostatic potential. No physical significance can be attached to the Kohn-Sham eigenvalues for solid calculations. Empirically, we do it anyway.

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Orbital energies are arbitrary up to a constant. To obtain the work functions, you need to know the surface charge distribution of a finite sample. Ionization potential Fermi level

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Plane Wave Augmented Plane Waves Gaussian Orbitals Ewald (CRYSTAL) Reference Density (SeqQuest) Ab-Initio Methods FLAPW, Wien2k VASP “Exact”  GW

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Numerical Basis Sets DMOL3 SIESTA Green’s Function (GW)

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Plane Waves Basis functions for each k in Brillouin Zone, where G is a reciprocal lattice vector. Solve for wavefunctions and energies,

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Practically, to obtain a finite set of states, the basis functions are cutoff, The cutoff is quoted as an energy, or as a cutoff wavelength,

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Assembling the Fock matrix to diagonalize is easy with Plane waves. Kinetic Nuclear Coulomb + Exchange

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Problem: cutoff G must be chosen extremely large to capture variation of wavefunction near nuclei. Fock matrix to diagonalize cheap to assemble, but large. Diagonalization becomes time consuming. CASSTEP is a pure plane wave code.

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Augmented Plane Wave codes try to reduce the number of basis functions of pure plane wave by using atomic orbitals in the vicinity of nuclei that are smoothly joined to plane waves in the interstitial region. Self-Consistent spherical potential inside spheres Constant potential in interstitial regions Wavefunctions in two regions are smoothly joined

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 APW works well for computing band structures, but has three drawbacks: 1.) There are no standard basis functions. This makes it difficult to visualize the wavefunction in terms of atomic orbitals. Mulliken populations are hard to quantify. 2.) Exact exchange is hard to compute. Thus, modern hybrid functionals that include Hartree-Fock exchange are not presently available with this approach. 3.) There is a certain arbitrariness to the choice of sphere radii. Wien2k and FLAPW

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 GW Method Feynman diagram method =+ = ++ + ….. Gives good bandgaps and excitations, but computationally very very expensive. Not competitive with DFT. Poles of propagator are physical excitation energies.

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Gaussian Orbitals Trial wavefunctions for crystal momentum k are built up from linear combinations of localized atomic Gaussian orbitals. Atomic Gaussian localized at R

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Advantages: 1.) Fewer basis functions needed to solve problem. 2.) Intuitive wavefunctions that are easily visulalized. 3.) Mulliken populations 4.) Can do surface problems Disadvantage: 1.) Much harder to calculate elements in Fock matrix.

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 SeqQuest (Sequential QUantum Electronic STructure) Worked out onceVaries slowly so solve in Fourier space using Poisson equation, Can obtain linear scaling!!

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 The linear scaling method does not lend itself to an easy way to compute exact Hartree-Fock exchange. HF exchange requires brute force calculation taking the scaling back to O(N^3). In fact, no one has found a fast way to compute exact exchange for periodic systems. If you can, PUBLISH!

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 GaN Quest Input Deck Bohr

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Online manual for Quest

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 CRYSTAL: A Gaussian Code Input Structure of CRYSTAL Structure Basis set (atomic orbital) Method (HF or DFT) SCF control

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Input Structure of CRYSTAL (example) Your personal note about this calculation “crystal” “slab” “polymer” “molecule” Space group sequence number Cell parameters Number of non-equivent atoms Atomic coordiantes Basis set

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Input Structure of CRYSTAL (Basis set) atomic number For example: C: 6 O: 8 Ni: 28 Ni: 228 number of shells all electron basis set effective core potential 1 st shell 2 nd shell 3 th shell 4 th shell End of basis set section basis set type 0: input by hand 1: STO-nG 2: 3(or 6)-21G shell (orbital) type 0: s orbital 1: s+p orbital 2: p orbital 3: d orbital 4: f orbital number of Gaussian functions number of electrons at this shell scale factor Si (1s 2 2s 2 2p 6 3s 2 3p 2 ) 14 electrons Si ((function)3s 2 3p 2 ) 4 electrons

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Crystal06 Input (basis set)

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L14 Crystal06 Input (SCF control) k-point net for insulator: n n for metal n 2n maximum SCF iterations mixing control 30% P % P 1 for second step