Intro to ab initio methods - University of Michiganerjank/Lectures/557/557-Lecture2a-QM.pdf · –Molecular orbital theory •Came from chemistry, since primarily developed for individual

ChE/MSE 557 Lecture 2a Fall 2006 1Computational Nanoscience of Soft [email protected]

Intro to ab initio methods

Lecture 2 Part A

Recommended reading:Leach, Chapters 2 & 3 for QM methods

For more QM methods: “Essentials of Computational Chemistry”by C.J. Cramer, Wiley (2002)


ab initio

• ab initio - from the beginningThe concise Oxford Dictionary, Oxford University Press, 2001

• ab initio calculation - A method of calculating atomic andmolecular structure directly from the first principles ofquantum mechanics, without using quantities derived fromexperiment (such as ionization energies found byspectroscopy) as parameters. A Dictionary of Chemistry, Oxford University Press, 2000– The most chemically accurate, physically precise computation

possible.– The holy grail of computational chemists.


Fundamentals

• The postulates and theorems of quantum mechanics formthe rigorous foundation for the prediction of observablechemical and physical properties of matter from firstprinciples.

• Any model of a material and its behavior, regardless of itssource, must ultimately find its basis in quantummechanics.

• All models of materials “include” QM either explicitly orimplicitly.


Fundamentals

• Matter is made of atoms, which are composed ofelectrons, protons, and neutrons.

• Over 75 years ago the laws of QM as formulated bySchrodinger, Dirac and others made it theoreticallypossible to understand and calculate how electrons andatomic nuclei interact to form matter.

• However, the solution of the governing equations of QM istoo difficult to solve exactly for anything but the simplestof systems (like the hydrogen atom).


QM Methods

• Over the past 10 years, ground-breaking advances in thedevelopment of QM techniques now allow QM calculationson molecular systems of real, practical interest.– Strength of carbon nanotubes– Optical spectra of quantum dots– Interaction of biological molecules (docking) with surfaces,

other bio-molecules, etc.– Energy of nanostructures on surfaces– Molecular structure and reactivity of complex molecules (e.g.

buckyballs and related carbonaceous molecules)– Bond strengths, angles for macromolecules.– Fracture of inorganic matter.


QM Methods

• Doing a QM simulation or calculation means including theelectrons explicitly.

• With QM methods, we can calculate properties thatdepend upon the electronic distribution, and to studyprocesses like chemical reactions in which bonds areformed and broken.

• The explicit consideration of electrons distinguishes QMmodels and methods from classical force field models andmethods.


Different QM Methods

• Several approaches exist. The two main ones are:– Molecular orbital theory

• Came from chemistry, since primarily developed forindividual molecules, gases and now liquids.

• Two “flavors”– Ab initio: all electrons included (considered exact)– Semi-empirical: only valence electrons included

– Density functional theory• Came from physics and materials science community,

since originally conceived for solids.• All electrons included via electronic density (considered

exact).


Fundamentals

• The fundamental postulates of QM assert that microscopicsystems are describable by ‘wave functions’ thatcompletely characterize all of the physical properties ofthe system.

• A wavefunction squared is a probability density.

• There are QM ‘operators’ corresponding to every physicalobservable that, when applied to a wave function, allowthe prediction of the probability of finding the system toexhibit a particular value or range of values for thatobservable.

x Ψ = xΨ Eigenvalue eqnfor position.


Fundamentals

• The operator that returns the system energy iscalled the Hamiltonian operator H.

HΨ = EΨ Time-independentSchrodinger Equation

!

H = "h2

2me

# i

2 "i

$h2

2mk

# k

2 "e2Zk

rik+

e2

rij+

e2ZkZl

rklk< l

$i< j

$k

$i

$k

$

Kineticenergy ofelectrons

Kineticenergy ofnuclei

Potentialenergy ofelectrons& nuclei

Potentialenergy ofelectrons

Potentialenergy ofnuclei

Eigenvalue equationfor system energy


Born-0ppenheimer Approximation

• Neutrons & protons are >1800 times more massive thanelectrons, and therefore move much more slowly.

• Thus, electronic “relaxation” is for all practical purposesinstantaneous with respect to nuclear motion.

• We can decouple the motion, and consider the electron-electron interactions independently of the nuclearinteractions. This is the Born-Oppenheimer approximation.

• For nearly all situations relevant to soft matter, thisassumption is entirely justified.

(Hel+Vn)Ψel(qi;qk) = EelΨel(qi;qk)

The Electronic Schrodinger Equation

Need to simplify!


QM Methods

• Goal of all QM methods in use today:

Solve the electronic Schrodinger equation forthe ground state energy of a system and thewavefunction that describes the positions of allthe electrons.

– The energy is calculated for a given trial wavefunction,and the “best” wavefunction is found as thatwavefunction that minimizes the energy.


QM Methods

• Solving SE is not so easy! Anything containing more thantwo elementary particles (i.e. one e- and one nucleon)can’t be solved exactly: the “many-body problem”.

• Even after invoking Born-Oppenheimer, still can’t solveexactly for anything containing more than two electrons.

• So -- all QM methods used today are APPROXIMATE afterall, even if considered “exact”! That is, they provideapproximate solutions to the Schrodinger equation.– Some are more approximate than others.


Molecular Orbital Theory

• MOT is expressed in terms of molecular wave functionscalled molecular orbitals.

• Most popular implementation: write molecular orbital as alinear combination of atomic orbitals φ (LCAO):

• Many different ways of writing “basis set”, which leads tomany different methods and implementations of MOT.

!

"i= aµi#µ

µ=1

K

$Eq. 2.68 in Leach K = # atomic orbitals


Molecular Orbital Theory

• Dozens of approaches for writing basis sets (e.g. in termsof Gaussian wavefunctions, or as linear combos ofGaussians).

• Different implementations retain different numbers ofterms.

• Semi-empirical MOT methods consider only valenceelectrons.

• Some methods include electron exchange.

• Some methods include electron correlation.


Density Functional Theory

• A different approach for solving Schrodinger’s equation forthe ground state energies of matter.

• Based on theory of Hohenberg and Kohn (1964) whichstates that it is not necessary to consider the motion ofeach individual electron in the system. Instead, it sufficesto know the average number of electrons at any one pointin space.

• The HK theorem enables us to write Eel as a functional ofthe electron density ρ.

• To perform a DFT calculation, one optimizes the energywith respect to the electron probability density, ratherthan with respect to the electronic wave function.

For a given density, the lowest energy is the best one.



• In the commonly used Kohn-Sham implementation, thedensity is written in terms of one-electron molecularorbitals called “Kohn-Sham orbitals”.

• This allows the energy to be optimized by solving a set ofone-electron Schrodinger equations (the KS equations),but with electron correlation included. This is a keyadvantage of the DFT method - it’s easier to includeelectron correlation.



• In DFT calculations, the MO’s are written as linearcombinations of atomic orbitals (LCAO) or basisfunctions which can be represented using Gaussianfunctions, plane waves, etc.

!

" = ai#i

i=1

N

$



• Different choices of basis sets, how many terms to use, ofwhat type, contribute to difficulty of calculation. More thana few hundred light atoms is still too time-consuming, evenon big computers.

• For molecules or systems with large numbers of electrons,pseudopotentials are used to represent the wavefunctions ofvalence electrons, and the core is treated in a simplified way.

This is a basic introductory summary of the DFT method.


Implementing MOT & DFT in Computer Code

• John Pople at Northwestern is a pioneer in computationalquantum chemistry.

• In 1970 Pople developed Gaussian, a quantum chemistrycode that solves approximations of the SE for molecules.

• In 1990, he included DFT in Gaussian.

This brought state-of-the-art QM computational methods to the masses.

Ab initio and semi-empirical methods (especially MOT methods)have revolutionized the Pharmaceuticals Industry, and are

now playing a major role in materials R&D.


1998 Nobel Prize in Chemistry

Walter Kohn"for his development

of the density-functional

theory"

John Pople"for his development of computational methods

in quantum chemistry"


QM Codes for Materials Research

• Commonly used codes:

– Gaussian: ab initio MOT and DFT• Available in Cerius2 from Accelrys

– NWChem: ab initio MOT and DFT• www.emsl.pnl.gov/pub/docs/nwchem


DFT Codes for Materials Research

• DFT codes used in materials research:– VASP - Vienna ab initio simulation package– Siesta - from Spain– Abinit– Gaussian– Castep - Cambridge sequential total energy

package– DMol3


Applications of ab initio computationsusing DMol3


Electronic band structure of POSS cubes functionalized withn benzene molecules (n = 0-8)

…

-10 -8 -6 -4 -2 0 2 4 6 8 10

Energy (eV)


Electronic band structure of POSS cubes functionalized with acenemolecules (benzene, naphtalene, anthracene, tetracene, and

pentacene)

-10 -8 -6 -4 -2 0 2 4 6 8 10Energy (eV)

Pure poss


Electron densities of acene-functionalized POSS

HOMO LUMO

8.4

4.842

3.325

2.295

1.486

0.999

Band Gap(eV)

13.776B-POSS

1.335P-POSS

3.348N-POSS

3.268A-POSS

1.597T-POSS

-POSS

Pure Acene(eV)Molecule


Electron densities of silica nanotubes

HOMO LUMO


Future developments of ab initio methods

• Used for calculating quantities like reaction rates, bondstrengths and angles, heats of formation, solubility, etc. ….Any properties that depend critically on electrondistribution.

• Lots of activity in MOT and DFT methods.

• Order N methods: The Holy Grail– Ab initio - computational effort now scales with the number of

electrons to a power n<4.– Semi-empirical - with only valence electrons, can get order N

scaling?


Too slow!

• Even with– all of the advances of the past 10 years in ab initio methods– ignoring some of the electrons (pseudopotentials)– implementation of ab initio codes on parallel machines

it is still not possible to use solely QM methods to simulatesystems that contain more than a few thousand atoms (achunk of matter containing less than 1 nm on a side), or formore than a picosecond for a very small number of atoms.


Still too slow!

• Ten years from now we may gain an order of magnitude inwhat can be simulated, but this is still not sufficient formany problems in soft matter.

– Assembly (especially if hierarchical)– Mechanical properties of composites– Rheology– Development of structure on length scales of 10 to 100’s of

nanometers.

Enter classical force fields.


Classical vs. ab initio methods

• No electronic properties

• Phenomenologicalpotential energy surface(typically 2-bodycontributions)

• Difficult to describe bondbreaking/formation

• Can do up to a billionparticles

• Electronic details included

• Potential energy surfacecalculated directly fromSchrodinger equation(many body termsincluded automatically)

• Describes bondbreaking/formation

• Limited to severalhundred atoms withsignificant dynamics

Classical ab initio

Documents

Intro to ab initio methods - University of Michiganerjank/Lectures/557/557-Lecture2a-QM.pdf · –Molecular orbital theory •Came from chemistry, since primarily developed for individual