QM -- the basicsBrazilian Summerschool of Molecular Modeling
Sao Paulo, 2013
O. Anatole von LilienfeldArgonne Leadership Computing Facility
Argonne National Laboratory, USAChemistry Department
University of Basel, Switzerland
First principles --- whyyyy???
1. Chemical reactions: Break/create bonds2. Variable coordination3. Changing interactions/regimes4. Difficult elements (metals)5. Unknown force-fields6. QM properties (spectroscopy, tunneling,
band-structure)
First Principles
Schrödinger
Born Oppenheimer
Benzene: 12 nuclei + 42 electrons = 54 particles -> 3x54 = 162 dimensions
Clamp nucleiSolve electronic problems
why awful??
Schrödinger
Just imagine storing it* ... e.g.● 10 orbitals, e.g. 20 electrons, e.g. water
dimer● 100 points/orbital (not much)● 4 bytes/point
*not mentioning optimizing it
why awful??
Schrödinger
Just imagine storing it* ... e.g.● 10 orbitals, e.g. 20 electrons, e.g. water dimer● 100 points/orbital (not much)● 4 bytes/point-> 10010 points ~ 4 x 1020 bytes = 0.4 EB = 0.4B TB
*not mentioning optimizing it... pretty awful object
ApproximationsQuantum chemistry● Hueckel ● (ZI)NDO ● Hartree-Fock (Single Slater determinant)● MP2/MP3/MP4 ... (Perturbation theory)● Configuration Interaction (CI) ...● Coupled Cluster (CCS, CCSD, CCSDT ... )● MR-SCF ... ● CAS ...● ``model chemistry''
ApproximationsSolid state physics● Muffin-Tin (Slater)● Bond-order potentials● tight-binding DFT● orbital free DFT (Thomas-Fermi)● DFT (Hohenberg-Kohn/Kohn-Sham)● Jacob's ladder of DFT by Perdew● Many-body perturbation
1. GW2. DMFT
● Quantum Monte Carlo
Thomas Fermi
Kohn Hohenberg
Sham
Perdew
Ceperley
Density-functional theory (LDA/GGA, meta, hybrids, ...)
Semi-empirical methods (AM1, PM6, ZINDO, TB-DFT)
Interatomic potentials (“force fields”)(Tersoff, Brenner, Foiles, Pettifor, Karplus etc)
Correlated wavefunctions, MBPT(MP2, RPA, CCSD(T), GW, ...)
Full CI, Quantum Monte Carlo
Accuracy/Transferability
Why DFT?
Computational Cost
Why DFT?
Interatomic potentials
Semi-empirical methods
KS-DFTCorrelated Ψ
Exact
K's Nobel '98~10k paper/year
Accuracy/Transferability
Take home message --or-- reasons to fail a PhD ... Explain1. Hohenberg-Kohn theorem 1 & 22. Kohn-Sham potential3. Exchange-correlation potentials4. Hellmann-Feynman theorem5. Verlet algorithm6. Car-Parrinello vs. Born-Oppenheimer7. Basis-sets: Atomic vs. planewave8. Pseudopotentials
Density Functional TheoryCould we use electron density instead of wavefunction?● measured experimentally● only 3D
1. Hohenberg-Kohn theorem: Unless identical, two systems can not have same density (up to a constant)
2. Hohenberg-Kohn theorem: The ground state energy is
minimal in density
Take home message --or-- reasons to fail a PhD ... Explain1. Hohenberg-Kohn theorem 1 & 22. Kohn-Sham potential3. Exchange-correlation potentials4. Hellmann-Feynman theorem5. Verlet algorithm6. Car-Parrinello vs. Born-Oppenheimer7. Basis-sets: Atomic vs. planewave8. Pseudopotentials
Take home message --or-- reasons to fail a PhD ... Explain1. Hohenberg-Kohn theorem 1 & 22. Kohn-Sham potential3. Exchange-correlation potentials4. Hellmann-Feynman theorem5. Verlet algorithm6. Car-Parrinello vs. Born-Oppenheimer7. Basis-sets: Atomic vs. planewave8. Pseudopotentials
So what about xc-potential?
Perdew: Jacob's ladder1. rung: LDA2. rung: GGA3. rung: meta-GGA4. rung: orbitals (hybrid)5. rung: unoccupied orbitals
Perdew
Take home message --or-- reasons to fail a PhD ... Explain1. Hohenberg-Kohn theorem 1 & 22. Kohn-Sham potential3. Exchange-correlation potentials4. Hellmann-Feynman theorem5. Verlet algorithm6. Car-Parrinello vs. Born-Oppenheimer7. Basis-sets: Atomic vs. planewave8. Pseudopotentials
Take home message --or-- reasons to fail a PhD ... Explain1. Hohenberg-Kohn theorem 1 & 22. Kohn-Sham potential3. Exchange-correlation potentials4. Hellmann-Feynman theorem5. Verlet algorithm6. Car-Parrinello vs. Born-Oppenheimer7. Basis-sets: Atomic vs. planewave8. Pseudopotentials
AIMD: Verlet algorithm (rather than Euler)
Reasons1. simple2. needs only forces (no higher derivatives)3. exact up to 3rd order4. time reversible5. symplectic (conserves space volume)6. stable for long time (conserves energy)
choose dt << period of highest frequency mode in system
-> Blackboard!!!
Take home message --or-- reasons to fail a PhD ... Explain1. Hohenberg-Kohn theorem 1 & 22. Kohn-Sham potential3. Exchange-correlation potentials4. Hellmann-Feynman theorem5. Verlet algorithm6. Car-Parrinello vs. Born-Oppenheimer7. Basis-sets: Atomic vs. planewave8. Pseudopotentials
AIMD: BOMD vs CPMD
Note1. in BOMD, move nuclei first, quench electrons, calculate
forces2. in CPMD, electrons and nuclei evolve simultaneously3. in CPMD, thus no need to optimize wavefunction @
every time step4. doesn't the non-optimal wavefunction introduce errors?
a. yes, fictitious electron dynamics affects forces on ions, buti. averages outii. can be kept small (``adiabatic separation'')
5. careful: Use small time steps, small fictitious mass & thermostats (electronic & nuclear) to control adiabicity
6. works best for large gap systems
Take home message --or-- reasons to fail a PhD ... Explain1. Hohenberg-Kohn theorem 1 & 22. Kohn-Sham potential3. Exchange-correlation potentials4. Hellmann-Feynman theorem5. Verlet algorithm6. Car-Parrinello vs. Born-Oppenheimer7. Basis-sets: Atomic vs. planewave8. Pseudopotentials
Atomic orbital basis
Philosophy: Molecules consist of distorted atoms
○ chemical insights○ small basis often yields good results○ non-orthogonal○ depend on nuclear position (Pulay-forces)○ basis set superposition errors (BSSE)○ multiple convergence cutoffs
Planewave basis
Philosophy: Molecules are assemblies of atoms that distort free electrons
○ orthogonal (linearly independent/superposition principle)○ independent of nuclear position (no Pulay forces)○ exploit Fourier transforms - some integrals way easier to evaluate○ no BSSE○ naturally periodic○ single convergence cutoff○ many functions needed
Take home message --or-- reasons to fail a PhD ... Explain1. Hohenberg-Kohn theorem 1 & 22. Kohn-Sham potential3. Exchange-correlation potentials4. Hellmann-Feynman theorem5. Verlet algorithm6. Car-Parrinello vs. Born-Oppenheimer7. Basis-sets: Atomic vs. planewave8. Pseudopotentials
Pseudopotentials - why?
● Old idea: Effective core potential
● Reduce basis set size (especially for plane waves)
● Reduce number of electrons
● Include effects○ relativistic○ self-interaction correction○ widened band-gap○ London dispersion effects
Hellmann
Pseudopotentials - how?
Replace core electrons by a potential, and solve only valence electron problem
● core/valence-> atomic ref.
● no core overlap-> small rc
● non-linear xc -> NLCC
● test test test
Take home message --or-- reasons to fail a PhD ... Explain1. Hohenberg-Kohn theorem 1 & 22. Kohn-Sham potential3. Exchange-correlation potentials4. Hellmann-Feynman theorem5. Verlet algorithm6. Car-Parrinello vs. Born-Oppenheimer7. Basis-sets: Atomic vs. planewave8. Pseudopotentials
Exercise Run CPMD program to calculate hydrogen molecule (http://cpmd.org/documentation/cpmd-html-manual)
Download input and pseudopotential file fromhttp://www.alcf.anl.gov/~anatole/Brazil2013/Exercise.1.tar.gz
Calculate1. wavefunction (play with different atomic positions/cutoffs
& optimization convergence criteria)2. geometry (adapt input file to RESTART)3. electrostatic potential (adapt input & use cpmd2cube.x
to generate cube-file and vmd to visualize)
Execute using: ./cpmd.x H2.inp