QM -- the basics - unibas.chanatole/teaching/Notes.1.pdf · QM -- the basics Brazilian ......

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

First Principles

Schrödinger

First Principles

Feynman

Schrödinger

variations

Hellmann

-> Blackboard!!!

Electronic wavefunction

... pretty awful object

Schrödinger

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

Proof of first HK theorem

Kohn-Sham ANSATZ (!)

Unknown for now

Kohn-Sham

put all the dirt under carpet ...

Kohn-Sham

Can you find this? If yes -> free trip to Stockholm

Kohn-Sham

So what about xc-potential?

Perdew: Jacob's ladder1. rung: LDA2. rung: GGA3. rung: meta-GGA4. rung: orbitals (hybrid)5. rung: unoccupied orbitals

Perdew

Perdew: Jacob's ladder

So what about xc-potential? Perdew



Mattsson, J Chem Phys (2008)

Forces

Feynman

Schrödinger

variations

Hellmann

-> Blackboard!!!

Forces

Follow forces for geometry relaxation ... or ab initio molecular dynamics (AIMD)

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!!!

AIMD: BOMD

Born Oppenheimer

AIMD: BOMD

AIMD: CPMD

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

AIMD: BOMD vs CPMD

Basis sets

Linear combination

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

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

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

QM -- the basics - unibas.chanatole/teaching/Notes.1.pdf · QM -- the basics Brazilian ......

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