554017 Advanced Quantum Chemistry

Pekka ManninenJussi Lehtola

University of Helsinki, 2012

Overview

• Lectures: Lectures (26 hours in total) on Mondays and Wednesdays from 2 pm to 4 pm. Room A128 on Mondays, A120 on Wednesdays.

• Demos & exercises: Supporting computer and paper-and-pencil demonstrations and exercise sessions (12 hours in total) on Fridays from 2 to 4 pm. Room A120.

• Credits: 4 ECTS. Successful examination or a project work and a couple of obligatory exercises is required for the credits.

• Literature: Selected parts of T. Helgaker, P. Jørgensen, and J. Olsen, Molecular Electronic-Structure Theory (Wiley, 2002) and R. McWeeny, Methods of Molecular Quantum Mechanics (Academic Press, 1992). Lecture notes and other material will be available online.

• Course page: http://www.chem.helsinki.fi/~manninen/aqc2012

Course ScheduleWeek Monday (PM) Wednesday (PM) Friday (JL)

11 Molecular QM No lecture (travelling) MQM demo calculations

12 MQM MQM & Secondquantization formalism

MQM demos, softwareintro

13 HF theory No lecture (travelling) HF calculations & further topics

14 HF theory & MPPT Correlated wave-function methods

No lecture (Easter)

15 No lecture (Easter) Correlated wave-function methods

Correlated wave-function calculations

16 Atomic basis functionsand Gaussian basis sets

Molecular integralevaluation

Further topics in basissets and integralevaluation

17 Relativistic QC Relativistic QC Relativistic QC demos

18 No lecture (May Day) Course wrap-up, project work presentations

Project workpresentationts

Exercises

• Each week a set of exercises is given• Preparatory

– Do these before the exercise session

• For discussion• For discussion– These will be demonstrated and discussed in the session

• For completion– You should solve these on your own or (preferably) as

teamwork– Return them (all) in writing by Friday April 27 to Jussi

Lecture 1: A Gentle Introductionto Many-Electron Quantum

MechanicsMechanicsMolecular Hamiltonians, wave

functions and all that

Introduction

• Quantum chemistry: application of quantum mechanics in physical models and experiments of chemical systems

• Calculation of the predictions of quantum theory as applied to polyatomic species (first principles applied to polyatomic species (first principles studies)

• Involves interplay of experimental and theoretical methods

• Many-body problem, these calculations are performed using the largest computers

Introduction

• Quantum chemistry: application of quantum mechanics in physical models and experiments of chemical systems– e.g. the ground state of individual atoms and molecules,

the excited states, and the transition states that occur the excited states, and the transition states that occur during chemical reactions

– Practically all experimental properties can be computedfrom first principles

• This course: determining the electronic structure of a rigid molecule by solving the molecular SchödingerEquation

Molecular Schrödiger Equation

• Molecule = a quantum system of N electrons and Mnuclei

• (Almost) fully described by the Schrödinger Equation

Ξ <K K� ( , , , ; , , , ; )H tx x x X X XΞ <¶ Ξ

K K

K K

1 2 1 2

1 2 1 2

� ( , , , ; , , , ; )

( , , , ; , , , ; )N M

t N M

H t

i t

x x x X X Xx x x X X X

Atomic units are being used; i.e. h=4οδ=e=me =1; c=1/»1/137

MolecularHamiltonian

Wave function

Molecular Hamiltonian

,

< < < ∗

< ∗ ∗

<, Ñ ∗ ,å åå1

2

1 1 1

� � � �

1�2

e n enN N N

e i i ji i j i

H H H H

H r r< < < ∗

< < < ∗

< <

<, Ñ ∗ ,

<, Ñ ∗,

<,,

å åå

å å å

åå

1 1 1

2

1 1 1

1 1

21 1�2

�

e i i ji i j iM M M

K Ln K

K K L KK K LN M

Ken

i K i K

H

Z ZH

m

ZH

r r

R R

r R

Born-Oppenheimer approximation

• Invoke an approximation– nuclei move slowly as compared to electrons– keep them fixed in the space

• The Hamiltonian simplifies into the molecular• The Hamiltonian simplifies into the molecularelectronic Hamiltonian

– The wave function will be dependent on the electroniccoordinates only

,

< < < < < ∗

<, Ñ , ∗ ,,å åå åå

12

1 1 1 1 1|1�2 |

N N M N NK

i i ji i K i j ii K

ZH r r

r R

Ξ < Ξ K1 2

( , , , ; )N

tx x x

Eigenvalue equation

• Restrict the consideration into the time-independentform of the molecular S.E.– The problem becomes an eigenvalue problem

Ξ < ΞK K� ( , , , ) ( , , , )H Ex x x x x x

– The eigenfunctions are orthogonal and square-integrable,

Ξ < ΞK K1 2 1 2

� ( , , , ) ( , , , )n N n n N

H Ex x x x x x

dΞ Ξ <n m nm

Limitations

• The Born-Opperheimer approximation itself• Other limitations in the Hamiltionian

– Non-relativistic kinematics, instant interactions– Point-like nuclei, electronic interaction only– Point-like nuclei, electronic interaction only

• Observation always involves interaction with the system; thus the time-independent form may not beadequate

• We have to invoke further approximations also in the solution of the equation

Spin-orbitals

• Electrons are described by their spatial location and an internal degree of freedom, spin– The one-electron wave functions must have a dependency

on them both, i.e. y j h s<( ) ( ) ( )

i i ix r

– These spin-orbitals are states where an electron hassimultaneously a definite energy and z-component of spin

– In case of electrons, the projected spin operator has onlytwo possible solutions, 1/2=ߣ or 1/2-=ߣ

y j h s<( ) ( ) ( )i i i

x r

y e y

y l y

<

<

�

�z

h

s

Antisymmetricity

• Electrons are indistinguishable, i.e.

• Pauli exclusion principle:

Ξ Ξ Ξ Ξ= Û = ±2 2

1 2 2 1 1 2 2 1( , ) ( , ) ( , ) ( , )x x x x x x x x

• Pauli exclusion principle:

for the interchange of any two electrons in the N-electron wave function

Ξ

Ξ

K K K

K K K

=

-1 2

1 2

ij i j N

j i N

P x x x x x

x x x x x

ˆ ( , , , , , , , )

( , , , , , , , )

Slater determinant

• How to construct a proper many-electron wave function on spin-orbitals?

• The requirement for antisymmetricity is straight-forwardly met when constructing the wave function forwardly met when constructing the wave function in the form of the Slater determinant

y y yy y y

y y y

Ε <

L

L

K

M M O M

L

1 1 2 1 1

1 2 2 2 21 2

1 2

( ) ( ) ( )

( ) ( ) ( )1( , , , )

!( ) ( ) ( )

n

nN

N N n N

N

x x xx x x

x x x

x x x

One- and N-electron expansions

• We can expand the exact wave function in a linearcombination of SD’s

• The (spatial part of) spin-orbitals is expanded in a linear combination of atom-centred functions

Ξ < Εå i ii

C

linear combination of atom-centred functions(atomic orbitals, AO)

• The AO’s are further expanded in some simplerfunctions (basis set functions, bf)

• This is referred to as the ”standard model” of quantum chemistry

m mm

j f<å ii

c

One- and N-electron expansions

expa

nsio

n(#

SD’s) Exact solutionexact solution in a given

one-electron basisN

-ele

ctro

nex

pans

ion

One-electron expansion (#bf’s)

the basis set limit fora given N-electron model

”Levels of theory”ex

pans

ion

(#SD

’s)

MN

æ öç ÷ç ÷è ø

Full configuration-interaction (CI) wave function

Multi-configurational methods

Ξ < Εå i ii

C

Sc. Post-HF methods

N-e

lect

ron

expa

nsio

n

1 The Hartree-Fock theory

Truncated CI methods

Coupled-cluster methods

Sc. Post-HF methodsDifference referred to as”correlation energy”

Things to think about

• Quantum chemistry: why bother?– Mere reproduction of experiments does not make too

much sense– What is the added value from the computational

approach?approach?

• Molecular wave functions: what are they?– Can some kind of meaning be given to

• The full N-electron wave function• One-electron wave functions (spin-orbitals)

– Or are they just mathematical constructs? What is ”real” then?

• Have a look at: Lundeen et al, Nature 474, 188–191 (2011)

Further homework

• Do the preparatory exercise from the first set of exercise assignments by Friday

• Study the Chapter 1 from the lecture notes• Skim through the Chapter 2• Skim through the Chapter 2

www.csc.fi/courses/archive/summerschool2012