First Principles CalculationsA brief Introduction & Density Functional Theory

Ling-Ti Kong (�-N)

EMail: konglt@sjtu.edu.cn

School of Materials Science and Engineering, Shanghai Jiao Tong University

Nov 27, 2019

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Schedules

Topic Date Venue

Fundamentals of Quantum Mechanics Nov 27, 2019 DX 213Electronic structure for crystals Dec 04, 2019 DX 213DFT calculations: practical concerns Dec 11, 2019 DX 213

Hand-on session I Dec 18, 2019 DX 213Hand-on session II Dec 25, 2019 DX 213

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Outline for today

1 Fundamentals of Quantum Mechanics

2 Born-Oppenheimer Approximation

3 Density Functional Theory

4 Further readings

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Fundamentals of Quantum Mechanics

Classic Mechanics

Newton’s Laws

mr = −5r U

r =d2r

dt2

5r =∂

∂rr(t): atomic trajectory;

U : potential (interaction) energy;

Quantum Mechanics

Schrodinger Equations

i~∂

∂tΨ = HΨ

Ψ: the wave function;

the probability amplitude for different

configurations of the system at

different times;

i~ ∂∂t : energy operator;

H: Hamiltonian operator;

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Quantum Mechanics

Time dependent Schrodinger equation

i~∂

∂tΨ = HΨ

H = T + V

T = Te + Tn, V = Ve−e + Ve−n + Vn−n+Vext

Time independent or stationary Schrodinger equation

HΨ = EΨ

H = −∑

i

~2

2me52

i −1

4πε0

∑i ,I

ZI e2

|ri − RI |+

1

8πε0

∑i 6=j

e2

|rj − ri |

−∑

I

~2

2MI52

I +1

8πε0

∑I 6=J

ZIZJe2

|RJ − RI |

Ψ = Ψ(r1, r2, · · · , ri , · · · ,R1,R2, · · · ,RI , · · · )5 / 25

Solvable Quantum Systems: two examples

Free particle

The Schrodinger equation

− ~2

2m52 ψ(r, t) = i~

∂

∂tψ(r, t)

Solution for a particular momentum

ψ(r, t) = e i(kr−ωt)

with the constraint

~2k2

2m= ~ω.

〈p〉 = 〈ψ| − i~5 |ψ〉 = ~k〈E 〉 = 〈ψ|i~ ∂

∂t |ψ〉 = ~ω

1D infinite potential well

Potential:

Schrodinger equation:

Solution:

Constraint:

Energy levels: En = ~2k2n

2m = n2h2

8mL2

For more: http://en.wikipedia.org/wiki/List of quantum-mechanical systems with analytical solutions 6 / 25

Born-Oppenheimer/Adiabatic Approximation

Wave function for general system:

Ψtotal = Ψ(r1, · · · , ri , · · · ,R1, · · · ,RI , · · · )

BO Approximation:Ψtotal = ψ(r;R)·φ(R)

First step of BO Approximation:

Heψ(r,R) = Eeψ(r,R)

where He = Te + Ve−e + Ve−n

Second step of BO Approximation:

[Tn + Ee(R) + Vn−n]φ(R) = Eφ(R)

M Born and J R Oppenheimer, Ann. Physik 84, 457 (1927)

http://en.wikipedia.org/wiki/Born%E2%80%93Oppenheimer approximation

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Born-Oppenheimer/Adiabatic Approximation

Neglect quantum effect in ionic system → replacing Schrodinger Eq. byNewton Eq.

MI∂2~RI (t)

∂t2= −5I E0(~R)

E0(~R) = Ee(~R) + Vnn(~R)

Force −5I E0(~R) contains contributions from the direct ion-ioninteraction and a term from the gradient of the electronic total energy.

Hellmann-Feynman theorem

5IEe(~R) = 〈ψ(~R)| 5I He(~R)|ψ(~R)〉

Couples the electronic Schrodinger Eq. with the ionic Newtonian Eq..Car-Parrinellohttp://en.wikipedia.org/wiki/Hellmann%E2%80%93Feynman theorem

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Schrodinger for electrons

Hψ(r) = Eψ(r)

H = Te + Ve−e + v(r)

= −∑

i

~2

2me52

i +1

8πε0

∑i 6=j

e2

|rj − ri |− 1

4πε0

∑i ,I

ZI e2

|ri − RI |

Usual procedure to solve SE:

v(r) SE−→ ψ(r) 〈ψ| · · · |ψ〉−−−−−−→

observables.

Define the system by choosing v(r);

Plug into SE and solves ψ;

Get observables by operations over ψ.

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Chemical approaches: Hartree approximation

Postulation

ψ can be written as a simple product of one-electron wave functions

ψ(r) =∏

i

φi (ri )

[− ~2

2m52 +V

(i)eff (R, r)

]φi (ri ) = εiφi (ri )

V(i)eff (R, r) =

∫ ∑j 6=i ρj (r

′)

|r − r′|dr′ + V (R, r)

ρj (r) = |φj (r)|2

Problem

Electrons are Fermions and indistinguishable, follow Pauli exclusionprinciples.

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Chemical approaches: Hartree-Fock approximation (SCF)

Introduce Pauli exclusion principle

ψ(r) = SD{φi (ri , σi )}

[− ~2

2m52 +

∫ ∑j 6=i ρj (r

′)

|r − r′|dr′ + V (R, r)

]φi

−∑

j

[∑σ′

∫φ∗j φi

|r − r′|dr′

]φj =

∑j

λijφj

Problem

The implied mean field approximation neglects electron correlation for theelectrons of opposite spin.

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Density Functional Theory: Preliminaries

Function

Function is a relation that uniquely associates members of one set withmembers of another set.f : x −→ f (x)

Functional

Functional is a real-valued function on a vector space V , usually offunctions.V = {f : [0, 1] −→ < : f is contineous}

A functional is a function of functions. It maps a function to a number.

Extremum

x0 minimizes/maximizes f (x) if df (x)dx |x=x0 = 0.

f0(x) minimizes/maximizes F [f (x)] if δF [f (x)]δf (x) |f =f0 = 0.

Ref: http://mathworld.wolfram.com

K. Kang, C. Weinberger, and W. Cai, A Short Essay on Variational Calculus, Stanford University, 2006.

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Density Functional Theory (DFT)

n(r) = N

∫d3r2

∫d3r3 · · ·

∫d3rNψ

∗(r, r2, · · · , rN)ψ(r, r2, · · · , rN)

DFT procedure to solve SE:

n(r) =⇒ ψ(r) =⇒ v(r).

Vext(r)HK⇐= n(r)

⇓ ⇑Ψi ({r}) =⇒ Ψ0({r})

Pierre C. Hohenberg

(1934.10-2017.12)

Walter Kohn

(1923.3-2016.4)

Lujeu Sham (!½Ê)

(1938.4- )

P. Hohenberg and W. Kohn, Phys. Rev. 136:B864, 1964. W. Kohn and L. J. Sham, Phys. Rev. 140:A1133, 1965.

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Hohenberg-Kohn theorem

First (existence) theorem

For any system of interacting particles in vext(r), n0(r) is uniquely deter-mined, i.e., there is a one-to-one correspondence between vext(r) and n0(r).

Second (variational) theorem

A universal functional E [n] can be defined. The exact ground state is theglobal minimum value of this functional, i.e, Ev [n0] ≤ Ev [n].

Interpretation

Unique mapping of v(r)⇔ n0;

All properties are given by n0(r);

Each property is a functional ofn0(r), f [n0];

A functional f [n0] maps a functionto a result: n0(r)→ f .

Advantages

reduces problem dim.from 3N to 3;

n(r) is an experimentalobservable;

Easy to visualize.

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Hohenberg-Kohn theorem

Hamiltonian of electrons

H = −∑

i

~2

2me52

i +1

8πε0

∑i 6=j

e2

|rj − ri |− 1

4πε0

∑i ,I

ZI e2

|ri − RI |

Specially for energy

ψ0 produces n0, minimizes E ⇒ n0 reproduces ψ0, minimizes E ;

Total energy functional:

E [n] = F [n] + V [n] = T [n] + U[n] + V [n].

T [n] and U[n]: universal (system independent);V [n]: system dependent, but once v(r) is known, V [n] =

∫n(r)v(r)d3r .

For a given system (v(r)), all one has to do is to minimize E [n] withrespect to n.

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Hohenberg-Kohn theorem

pros

simple; exact.

cons

T [n] and U[n] are unknown.

Borrow ideas from Thomas-Fermi approximation ?

U[n] ≈ UH [n] =e2

2

∫d3r

∫d3r ′

n(r)n(r′)

|r − r′|,

T [n] ≈ TLDAsingle particle[n] =

∫thoms (n(r))d3r .

thoms (n) =

3~2(3π)2/3n5/3

10m

Unfortunately: unstable molecules.Reasons: 1) neglect of correlations; 2) LDA in T [n].

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Kohn-Sham equations

Decompose T [n] and U[n]:

T [n] = Ts [n] + Tc [n]

U[n] = UH [n] + Ux [n]

Rewrite the exact energy functional:

E [n] = Ts [n] + UH [n] + Exc [n] + V [n]

Energy minimization:

δE [n]

δn(r)=

δTs [n]

δn(r)+δV [n]

δn(r)+δUH [n]

δn(r)+δExc [n]

δn(r)

=δTs [n]

δn(r)+ v(r) + vH(r) + vxc (r)

= 0.

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Kohn-Sham equations

Noninteracting particles moving in the potential field of vs(r):

0 =δE [n]

δn(r)=δTs [n]

δn(r)+δVs [n]

δn(r)=δTs [n]

δn(r)+ vs(r)

Effective external field:

vs(r) ≡ veff (r) = v(r) + vH(r) + vxc (r)

Auxiliary Schrodinger Equation:[−~252

2m+ vs(r)

]φi (r) = εiφi (r)

n(r) ≡ ns(r) =N∑i

fi · |φi (r)|2.

Ts [n] = − ~2

2m

∑i

∫φ∗i (r)52 φi (r)d

3r , Ts [n] + Vs [n] =∑

i

fi · εi .

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Kohn-Sham equations

Total energy:

E0

= Ts [n] + UH [n] + Exc [n] + V [n]

= Ts [n] + UH [n] + Exc [n] +

∫v(r)n(r)dr

= Ts [n] + UH [n] + Exc [n] +

∫[vs(r)− vH(r)− vxc(r)]n(r)dr

= Ts [n] + UH [n] + Exc [n] + Vs [n]−∫

vH(r)n(r)dr −∫

vxc (r)n(r)dr

= Ts [n] + Vs [n] + UH [n]−∫

vH(r)n(r)dr + Exc [n]−∫

vxc (r)n(r)dr

=∑

i

fiεi −e2

8πε0

∫ ∫n0(r)n0(r′)

|r − r′|drdr′ + Exc [n]−

∫δExc [n]

δnn(r)dr

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Kohn-Sham equations

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Kohn-Sham equations

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DFT in practice: approximations

Local Density Approximation (LDA)

ELDAx = −3e2

4

(3

π

)1/3 ∫n(r)4/3d3r

Exc [n] ' ELDAxc [n] =

∫ehom

xc (n)d3r |n→n(r) =

∫ehom

xc [n(r)]d3r .

Generalized Gradient Approximation (GGA)

EGGAxc [n] =

∫f (n(r),5n(r))d3r .

Chemistry: BLYP Physics: PBE

Spin-polarized

ELSDAxc [nα, nβ] =

∫f (nα, nβ)n(r)d3r .

n = nα + nβ. 22 / 25

Homework

1 Go through all the slides and deduce all equations by yourself.

2 Learn bash by yourself.

3 Learn some basics on Linux by yourself.

4 Think: If you evaluate the kinetic energy of a Kohn-Sham system, isit equal to the physical kinetic energy? why?

5 Hand in: Solve the free particle and 1-D infinite potential wellproblems illustrated in page 6.Due: Dec. 4, 2019. Send electronic version (word or pdf) toModelMaterials@ALiYun.com.

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References and further readings

1 K. Capelle, A Bird’s-Eye View of DFT,http://arxiv.org/abs/cond-mat/0211443.

2 Fabio Finocchi, Density Functional Theory for Beginners: BasicPrinciples and Practical Approaches.

3 R. O. Jones and O. Gunnarsson, Review of Modern Physics61(3):689, 1989.

4 J. Hafner, Atomic-scale computational materials science, ActaMaterialia 48:71-92, 2000.

5 R.M. Dreizler and E.K.U. Gross, Density Functional Theory ,Springer-Verlag (1990), ISBN 3-540-51993-9.

Lecture downloads:http://cms.sjtu.edu.cn/gs/download.html

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