Gaussian Basis Sets for Solid-State Calculations

Transcription

1 Gaussian Basis Sets for Solid-State Calculations K. Doll Molpro Quantum Chemistry Software Institute of Theoretical Chemistry, D Stuttgart, Germany MW-MSSC 2017, Minneapolis, July 10, 2017

2 Introduction about basis sets Outline How to make up your own basis set - all-electron example: carbon (for diamond) - ECP example: Sr (for SrO) - Miscellaneous hints: reoptimising exponents, exponents for free atom, brute force approach (even tempered), how to deal with error messages Note: Many, many numbers in this talk catch main ideas

3 Introduction

4 Basis set types Hartree-Fock/Kohn-Sham equations can in general not be solved analytically => expand molecular/crystalline orbitals in basis functions plane waves: ħ 2 ( k + K ) 2 E m cutoff el Φ k ( r )= c k, K exp( i( k+ K ) r) K with: ħ 2 2m el ( k+ K ) 2 E cutoff local basis sets (linear combination of atomic orbitals, LCAO): - Slater type orbitals (STO): ϕ STO ( r)=n x a y b z c exp( ζ r) solve H-atom exactly, core region properly described - Gaussian type orbitals (GTO): integrals easier than for STOs ϕ GTO ( r )= N x a y b z c exp( ζ r 2 )

5 Gaussian basis sets ϕ primitive ( r )=N (x A x ) a ( y A y ) b ( z A z ) c exp ( ζ ( r A ) 2 ) primitive basis function, atom-centered, atom at A Exponent: ζ: unit: 1/bohr 2 1 bohr = Å large ζ: tight important for core region Small ζ ( < 1 ) : diffuse, important for valence region, chemical bond

6 Contraction Idea: the core region does not change very much when a chemical bond is formed => make linear combination which describe core region well: contraction : n ϕ( r )= N i=1 c i x a y b z c exp( ζ i r 2 ) primitive Gaussian c i : contraction coefficients Total basis set: set of various ϕ μ ( r ) with different exponents, contractions coefficients

7 Orbitals Molecular orbital: Ψ i ( r )= μ a i μ ϕ μ ( r ) Crystalline orbital: Ψ i ( r, k )= μ a iμ Φ μ ( r, k) with: Φ μ ( r, k)= g ϕ μ ( r A μ g)exp(i k g) The orbitals are obtained when the Hartree-Fock/Kohn-Sham equations are solved. All basis functions contribute to each orbital.

8 Angular part, 1 Y lm (θ,φ) : spherical harmonics r l Y lm (θ,φ) : solid harmonics L z eigenstates only for atom or linear molecule => real combinations are preferably used (Condon-Shortley convention, normalisation ignored in the following!): l=0 : Y 00 => 1 l=1 : Y 11 - Y 1-1 => x Y 11 + Y 1-1 => y Y 10 => z

9 Angular part, 2 l=2 : Y 20 => 3z 2 -r 2 =2z 2 -x 2 -y 2 Y 21 Y 2-1 => xz Y 21 + Y 2-1 => yz Y 22 + Y 2-2 => x 2 -y 2 Y 22 Y 2-2 => xy The order is in the manual, page 22

10 Angular part, 3 Why does the order matter? Population analysis (keyword PPAN), projected density of states (keyword DOSS): this is how the orbitals are ordered Typical application of population analysis: Ion embedded in crystal field of other ions (octahedral, tetrahedral ) what is the d-occupancy? Use ROTREF if surrounding ions are not textbook-like arranged

11 Exponents: where from? Easiest: even tempered: exponents separated by common factor Example: factor 4, start from 0.25, stop at 4096: 0.25, 1, 4, 16, 64, 256, 1024, 4096 general: αβ k here: α=0.25 β=4 k=0...7 must be done for the different angular momenta: α s, β s, α p, β p, α d, β d Optimise α, β (lowest total energy of the respective atom)

12 Even tempered basis sets: optimise parameters α, β αβ k Hartree-Fock limit can be reached with more and more basis functions For practical purposes: use exponents separated by factors of 2-4

13 Basis set choice: databases Molecular case: databases available EMSL basis set exchange: MOLPRO code: Huge number of basis sets available Periodic systems: CRYSTAL website Basis sets for most atoms available, see also CRYSTAL manual chapter 12, page

14 Basis set choice: general recommendations Type of calculation is important: here Hartree-Fock or DFT, but no wavefunction based correlation (CCSD, CI, MP2 ) => relatively small basis set is sufficient, basis set does not have to be correlation-consistent Basis set depends on charge state (0, -1, +1?): negative ions have larger radius than positive ions => extra diffuse exponents required for negatively charged atoms (and metals in the periodic case), may be removed for positively charged atoms usually good to include polarisation function: e.g. if s and p orbitals are occupied, add d-exponents effective core potentials may be advantageous from 1st row transition metals on (to reduce calculational effort + include scalar-relativistic effects) but you can still do all-electron calculations even for heavy atoms (e.g. to look at core-levels)

15 Basis set: general recommendations adding more basis functions lowers the energy because of the variational principle! diffuse functions are important for the chemical bond very large exponents may improve the core region lead to lower energy, but only little improvement of the region where the chemical bond takes place => usually the part of the basis set which describes the region close to the nucleus can be kept as is diffuse functions may have to be altered (e.g. use educated guess such as even tempered basis functions, or reoptimise)

16 Basis set: personal favorites for light elements: a basis set similar to a molecular 6-31G basis set, usually already on CRYSTAL basis set website for heavy elements: Stuttgart/Cologne effective core potentials, diffuse exponents must be reoptimised

17 Basis set names huge confusion due to enormous number of basis sets, and enormous number of names most general: (number of s-, p-, d- exponents) / [number of s-, p-, d- contractions] e.g. (10s4p) / [3s2p]

18 Is my basis set good enough? Try it! If possible, compare with enhanced basis set (e.g. add diffuse function, polarisation function...): total energy, band structure, population analysis How large is the Mulliken population of the diffuse functions? Compute the band structure for a simple system where the element has a similar charge state - when it works for a simple system, it should work for a bigger system with similar charge state

19 Possible problems with Gaussian basis sets Very diffuse basis functions can cause numerical instabilities: => avoid exponents lower than for periodic systems (molecular codes use test on eigenvalues of overlap matrix and reduce basis set) But neither use a basis set which was calibrated for a positively charged ion for the neutral atom or negatively charged ion it needs extra diffuse functions e.g. a basis set calibrated for Li + in LiF may not work for Li metal without extra diffuse functions Exponents should be separated by factor > 2 remember even tempered basis sets: factor 2-4 between exponents

20 In the following: How to make up your own basis set

21 Example: Diamond

22 Example: Diamond, choose from EMSL web site

23 Example: Diamond Neutral => try standard basis set for molecules, e.g. 6-31G* * : polarisation function (here: d-shell) From EMSL website: 6-31G* basis set BASIS "ao basis" PRINT #BASIS SET: (10s,4p,1d) -> [3s,2p,1d] C S C SP C SP C D END s-shell sp-shell sp-shell d-shell exponent s-contraction p-contraction

24 Diamond quick analysis of the basis set From EMSL website: BASIS "ao basis" PRINT #BASIS SET: (10s,4p,1d) -> [3s,2p,1d] C S C SP C SP C D END Quick analysis: most diffuse exponent is => not too diffuse, try unchanged 6-31G* should be able to describe neutral carbon atom in molecular calculation => should be alright for diamond where carbon is neutral => try unchanged! exponent s-contraction p-contraction

25 Example: Diamond from EMSL to CRYSTAL format From EMSL website: BASIS "ao basis" PRINT #BASIS SET: (10s,4p,1d) -> [3s,2p,1d] C S C SP C SP C D END CRYSTAL format: This basis set had been previously used in CRYSTAL calculations and is on the web at

26 Diamond band structure: crystal input diamond.d12 part 1 part 2 Diamond 6-31G* basis set CRYSTAL END END DFT EXCHANGE LDA CORRELAT PZ END NODIRECT SHRINK 8 16 MAXCYCLE 60 FMIXING 70 NOSHIFT PPAN END Full input1 = put part1 + part2 together in 1 file: diamond.d12

27 Diamond band structure: properties input diamond.d3 BAND diamond band structure G-X X-W W-L L-G G-X END agrees well with H. Bross, R. Bader, phys. stat. sol. (b) 191, 369 (1995) Deviations possible for very high bands (note that we have only as many bands as basis functions!) => basis set reasonable for carbon in neutral, positive, or slightly negative state with a good band structure, the other properties are usually also reasonable, e.g. equilibrium geometry

28 ECP-Input and example: SrO

29 Effective core potentials Idea: reduce computational effort with a potential which describes nucleus and inner electrons + include a part of the relativistic effects Shape-consistent: fit shape of valence orbitals to all-electron shape Energy-consistent: fit to a set of energies of electronic states of an atom/ion In the following: Stuttgart/Cologne energy-consistent (ab initio) pseudopotentials

30 General recommendation Labeled as: ECPnXY ECP : effective core potential n: number of core electrons X: S or M S: single-valence-electron ion used in fit (use this only for one- or two valence electron atoms) M: multi-valence-electron ion used in fit (use M if available) Y: HF Hartree-Fock (only for light elements) WB Wood-Boring (simplification of DF) DF Dirac-Fock (fully relativistic, 4 components as in Dirac equation) use WB or DF if available e.g. ECP10MWB : nucleus + 1s 2 2s 2 2p 6 replaced by pseudopotential ( Ne core, 10 electrons) Multi-valence-electron atom used in fit, relativistic

31 Example: SrO O: use from CRYSTAL web site, here: O_8-411_towler_1994 Sr (38 electrons): select at Click on Sr, new window opens Options: ECP36SHF ECP36SDF ECP28MHF ECP28MWB ECP28MDF } } large core (36 electrons in core) small core (28 electrons in core) Let us choose ECP28MWB for the effective core potential and for the basis set also the one labeled ECP28MWB

32 Example: SrO ECP main web site

33 ECP for Sr Pseudopotential ECP28MWB for Sr # COMMENT # Q=10., MEFIT, WB, Ref 13. # COMMENT # Sr 238 <here, the number of valence-basis shells is needed> INPUT <please add the valence-basis input, without blank line> # COMMENT # References: # COMMENT # [13] M. Kaupp, P. v. R. Schleyer, H. Stoll, H. Preuss, J. Chem. Phys. 94, 1360 (1991). Available basis sets for Sr, ECP28MWB: ECP28MWB ECP28MDF_GUESS (ECP28MHF_GUESS ECP28MWB_GUESS) Core: 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 : 28 electrons in core Use ECP data as is no change when used with CRYSTAL

34 Basis set for Sr, compatible with ECP Basis set ECP28MWB for Sr, ECP28MWB # COMMENT # (6s6p5d)/[4s4p2d)-Basissatz fuer PP. von Ref X X X X X X X X X X X: here, the number of electrons per valence shell is needed CRYSTAL format: replace X with number X=2 electrons X=2 electrons X=0 electrons X=0 electrons X=6 electrons X=0 electrons X=0 electrons X=0 electrons X=0 electrons X=0 electrons 4s 2 4p 6 5s 2 (neutral)

35 Basis set for Sr, compatible with ECP Quick analysis: Many diffuse functions < 0.1 Will make problems => remove But then the outermost diffuse is for s for p for d not diffuse enough? => use educated guess, between Factor 2-4 smaller than smallest exponent e.g.: add extra diffuse s 0.1 p 0.1 d 0.1

36 Basis set: summary of the strategy for Sr Start from the initial basis set Remove diffuse exponents, e.g. exponents < 0.1 Fill up with new diffuse exponents, but only if by at least factor 2 smaller than smallest exponent Test!

37 Basis set for Sr, compatible with ECP Full basis set Remove diffuse Exponents, e.g. < 0.1 too diffuse too diffuse too diffuse too diffuse too diffuse Educated guess: add diffuse s,p,d with exponent 0.1 each (by at least factor 2 smaller than smallest exponent) Final basis set

38 Input for Sr and O INPUT :Sr, ECP 8 shells Effective core potential 10 valence electrons Sr: O: Basis set: s-shell, 2 electrons s-shell, 2 electrons s-shell, 0 electrons p-shell, 6 electrons p-shell, 0 electrons p-shell, 0 electrons d-shell, 0 electrons d-shell, 0 electrons 4s 2 4p 6 5s 2 : 10 electrons from Turin web page Starting configuration: 1s 2 2s 2 2p 4 : 8 electrons

39 SrO CRYSTAL END INPUT SrO: input for crystal sro.d12 Part 1 part 2 part 3 part 4 geometry, ECP Sr basis set O basis set SCF part END Put all 4 parts together in 1 file: sro.d12 DFT EXCHANGE LDA CORRELAT PZ END NODIRECT SHRINK 8 16 MAXCYCLE 60 FMIXING 70 NOSHIFT PPAN END

40 Test: SrO band structure, input sro.d3 BAND Input for fcc structure L-G G-X X-W W-K K-G END Good agreement with B. Pan, N.-P. Wang, M. Rohlfing, Appl. Phys. A (2015)

41 SrO: possible Sr basis set deficiencies full Sr basis set diffuse s omitted diffuse p omitted diffuse d omitted

42 SrO: possible Sr basis set deficiencies neutral Sr has configuration 5s 2 lowest unoccupied orbitals are 4d 5p => diffuse functions important for s, d ; less for p

43 diffuse p omitted diffuse d omitted SrO: possible Sr basis set deficiencies note: configuration of Sr 2+ is: 4s 2 4p 6 5s 0 4d 0 5p 0 full Sr basis set: ECP+[3s3p2d] diffuse s omitted

44 Miscellaneous hints

45 Improve an existing basis set: reoptimise exponents Assume we want to do calculation for Cu bulk Existing copper basis set, calibrated for KCuF 3, Cu 2+ state M.D. Towler, R. Dovesi and V.R. Saunders Phys. Rev. B 52, (1995) Quick analysis: sp is not diffuse enough for Cu metal typical values are sp d exponent? => educated guess: sp as initial guess then optimise: sp d criterion: lowest energy

46 Improve an existing basis set: reoptimise exponents Optimise sp d 0.430: Perform total energy calculation for Cu bulk, at experimental lattice constant, optimise iteratively: Initial energy (PWGGA) I 1) optimise sp 0.6 => 0.55 (keep other exponents fixed) E E+03 2) optimise sp 0.2 => E+03 3) optimise d 0.43 => E+03 II 1) optimise sp 0.55 => E+03 2) optimise sp 0.14 => 0.14 same energy 3) optimise d 0.40 => 0.40 same energy Done! Finally we have optimised: sp and d 0.40 Energy gain: about hartree band structure Thursday

47 Basis set for a single free atom Idea: linear dependence problem much less severe for a single free atom (keyword MOLECULE) => use basis set with many diffuse functions to describe the neutral atom (necessary for e.g. cohesive energy) but do not change the tight exponents, otherwise inconsistent with bulk!

48 Cu bulk Basis set for a free Cu atom Cu atom: extra diffuse functions

49 A basis set from scratch: even tempered for diamond If anything else fails, we can make up an even tempered basis set: Example for carbon, similar for other elements: factor ~3 between exponents LDA band structure with this basis set

50 Error message: number of electrons and nuclear charge do not match Necessary for periodic systems: number of electrons = nuclear charge Otherwise Coulomb energy diverges Error message is: ERROR **** INPBAS **** UNIT CELL NOT NEUTRAL remedy: properly count all the electrons in the basis set input simplest: make each atom neutral SrO example: Sr is neutral, O is neutral

51 Error message: linear dependence If this error message shows up: ERROR **** CHOLSK **** BASIS SET LINEARLY DEPENDENT Then most likely one or more exponents are too diffuse. Remedy: check all the diffuse exponents, remove small ones, possibly reoptimise remaining exponents

52 Summary Many good basis sets for periodic systems are already available on the web Charge state important, check diffuse exponents, use educated guess : - even tempered rule: factor 2-4 between exponents - avoid exponents below ~ compute band structure as test if basis set is large enough Optimise exponent: use total energy