lecture 4 2/7/18 - materials intelligence research

Quantum mechanicsLecture 42/7/18

1Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky

References for electronic structure

Richard M. Martin, Electronic Structure: Basic Theory and Practical Methods, Cambridge University Press (2004).

Mike Finnis, Interatomic Forces in Condensed Matter, Oxford University Press (2003).

Efthimios Kaxiras, Atomic and Electronic Structure of Solids, Cambridge University Press (2003).


Why solve Quantum Mechanics?

Potential models have limited transferability (universality)Need to describe bond breaking Electrons needed for electronic, optical, magnetic properties


CH3Br: Br displacement by NH3

Bonding and structure

Paraelectric (high T) and ferroelectric (low T) phases of PbTiO3


Quantum electrons hold matter together

• Atoms are made by massive, point‐like nuclei • Surrounded by tightly bound, rigid shells of core electrons that

do not evolve much• Bound together by a glue of valence electrons• Why don’t electrons fall on the nucleus?

5

Figure from https://suncat.stanford.edu

Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky

Material properties from first principles

Energy scale kBT at our living conditions (~300 K): 0.025 eV

Chemistry: differences in bonding energies are within one order of magnitude of 0.29 eV (hydrogen bond).

Binding energy of an electron to a proton (hydrogen):13.6058 eV = 1 Rydberg (Ry) = 0.5 Hartree (Ha) = 0.5 a.u


Uncertainty principle

The position and momentum of a particle cannot be simultaneously measured with arbitrarily high precision. There is a minimum for the product of the uncertainties of these two measurements.


When is particle like a wave?


De Broglie hypothesis

Einstein (1905): light is emitted and absorbed discretely De Broglie (1924): all matter is a waveThe smaller the dimension, the more wave‐like the physics

9

40

2220

2 cmcpcmKEmcE

hchE

pcE

ph

Bose‐Einstein condensate


Light: a wave and a particle

First simultaneous measurement of light as a particle and a wave3.4um x 45nm silver wire with a standing light wave interacting with a beam of single electrons

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L.Piazza et al, Nature Com. 6, 407 (2015)


Typical length scales

Atomic diameter is 0.1 nmElectron accelerated through 100V : λ = 0.12 nmNitrogen molecule at 300K: λ = 10‐14 nmBaseball at 90mph: λ = 10‐25 nm

The wavelength of heavy objects is not relevant for bonding


Atomic units convenient for simulations

• 1 unit of charge = absolute charge on an electron = 1.60219 x 10‐19 C

• 1 mass unit = mass of an electron = 9.10593 x 10‐31 kg

• 1 unit of length (1 Bohr) = 5.29177 x 10‐11 m

• 1 unit of energy (1 Hartree) = 4.35981 x 10‐18 J

A couple of observations:1 Bohr = the radius of the first orbit in Bohr’s treatment of the H atom1 Hartree = the interaction between two electrons separated by 1 BohrThe energy of the 1s electron in the H atom is ‐0.5 Hartree


A particle is like a localized wave

Adding several waves of different wavelength will produce an interference pattern that can localize the “wave packet”

The wavefunction fully describes the state of the systemA differentiable complex function of position and time


Probabilistic interpretation of QM

Instead of specific point positions we talk about probability densities

Wavefunction is “normalized” if

Need to specify amplitude and phase of a continuous function of position!

Much more difficult to describe numerically than a classical point particle

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is the probability of finding an electron in r and t


Schrodinger equation

Wave equation of motion: linear partial differential equationPredicts the evolution of the wavefunction in time

15

Hamiltonian


Evolution of a wavefunction

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Initially a very localized gaussian wave function of a free particle in 2D

figure from Wikipedia


Time‐independent Schrodinger equation

Consider a time‐independent potential V(r,t)=V(r)

e.g. when electrons are much faster than nuclei (adiabatic approximation)

We expect there will be standing wave solutions

After some algebra we get

two eigenvalue equations


Time‐independent Schrodinger equation

The time‐dependent part is trivial to solve

The position part depends on the potential and is difficult


Free particle in one dimension

The Hamiltonian is the kinetic energy operator:

The Schrodinger equation:

Solutions:


Plane wave

Completely delocalized in position spaceFully localized in momentum space (p is known)


Eigenstates: stationary waves

21

Eigenstate solutions are called “stationary” because position probability density does not depend on time

Only need to solve the time‐independent (position) equationThe complete time dependent solution is constructed from eigenvalues and eigenfunctions


Quantum tunneling

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The energy of the tunneled particle is the same but the probability amplitude is decreased.

STM image of a graphite surface



Infinite square well

Boundary condition: wavefunction vanishes at the walls

Find all plane‐wave like solutions subject to boundary conditions

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0 x a


Infinite square well

Only a discrete set of standing wavesEven the ground state has nonzero kinetic energyHigher states have more energy (curvature) and more nodes


a

Quantum dots

Optical spectrum of nanoparticles depends on sizeQuantum confinement of electrons in the nanoparticle

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Nature 523, 39–40 (2015)


Quantum harmonic oscillator

26

Equally spaced energy levels


Harmonic oscillator wavefunctions

27



Quantum atom

Coulomb interaction between the point nucleus and electron

The Laplacian operator term in spherical coordinates looks like

Solution for a 1‐electron atom can be found analytically


Quantum atom solutions

Make an solution ansatz by separation of variables

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),()()( , lmlnnlm YrR r

The wavefunctions are called “orbitals” and are characterized by three quantum numbers n, l, and m.

n is the principal quantum number: 0, 1, 2, …l is the azimuthal quantum number: 0, 1, … (n‐1)m is the magnetic quantum number: ‐l, ‐(l‐1), …0…,(l‐1), l

and there is spin s


Hydrogen atom orbitals

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radial probability densities

angular part of the orbital wavefunctions


Energy is a function of the principal quantum number n onlyOrbitals with the same n but different l and m are “degenerate.”

The entire chemical alphabet is derived from the stationary solutions of the Coulomb potential Schrodinger equationFor higher‐Z atoms this solution is approximate and the degeneracy is not exactA good approximation assuming the core electrons give rise to a spherically symmetric potential

Quantum atom solutions


Atomic orbitals


Operators

For each observable, there exists an Hermitean operator. Measurement of the observable yields an eigenvalue of that operator

Examples of operators for single particle systems:

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rrA 'ˆ ˆ A a r aa r eigenfunction

eigenvalue

ˆ momentumx rx

irpx

rm

rmpprT 2

2

22ˆˆˆ energy kinetic

operator


<Bra|Ket> notation

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)(r

ijjiji rdrr )()(*

iiiii EHrdrrVm

r

ˆ)()(

2)( 2

2*


Expectation values (averages)

The expectation value of a quantity such as energy, position, momentum, etc., is determined by the corresponding operator

Expectation values of Hermitean operators are real numbers


Let operator A have two eigenstates with different eigenvalues

We can write

Eigenfunctions are orthogonal


Matrix formulation

How to discretize the problem for a numerical solution?

Expand a wavefunction as a linear combination of basis vectors

37

EHrErH ˆ)()(ˆ

functions orthogonalk ..1

nkn

nnc

mm EH ˆmn

knmn EcHc

ˆ..1


Linear algebra: diagonalization

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mnmkn

n EcHc

ˆ,1

mkn

nmn EccH ,1

kkkkk

k

c

c

E

c

c

HH

HH

.

.

.

.

.

.

............

...... 11

1

111

HH T*


Variational principle

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Expectation value of the energy in the ground state is the lowest possible

Equality holds only if we guessed the exact ground state wavefunction


Functional

A functional takes a function as input and gives a number as output

An example is an integral or an expectation value

A functional derivative of an integral is

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yxfF )]([

dxxVxnxnF )()()]([

)(][ xVnnF

H

Eˆ

][


Variational principle is equivalent to Schrodinger equation

Consider the average quantity

Take functional derivative with respect to the wavefunction

This is “stationary” only if Schrodinger equation is satisfied

Functional derivatives


Energy of a Hydrogen atom

First: guess a trial wavefunction

This happens to be the exact ground state of the H atom

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HE

expC r

2 2 22

3 2

1 1,2 2

C C Cr

normalization constant


lecture 4 2/7/18 - materials intelligence research

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