Computational physicsOsamu Sugino and Ryotaro Arita

Computational physics

Sugino

1. Density functional theory (DFT)

2. Relation with many-body theory

3. Tutorial (Quantum espresso)

Prof. Arita

4. Introduction to strong correlation

5. Dynamical mean field theory (DMFT)

6. DFT+DMFT

Why density functional theory?

Many-body theory

1. Wave function theoryΨ(𝑥1, 𝑥2, ⋯ , 𝑥𝑁)

2. Density matrix theory𝑛 𝑥, 𝑥′ = 𝑁𝑑𝑥2⋯𝑑𝑥𝑁Ψ 𝑥, 𝑥2, ⋯ , 𝑥𝑁 Ψ∗(𝑥′, 𝑥2, ⋯ , 𝑥𝑁)

𝑛(2) 𝑥, 𝑥′ =𝑁 𝑁−1

2𝑑𝑥3⋯𝑑𝑥𝑁Ψ 𝑥, 𝑥′, 𝑥3, ⋯ , 𝑥𝑁 Ψ∗(𝑥, 𝑥′, 𝑥3, ⋯ , 𝑥𝑁)

3. Green’s function theory

𝐺 𝑥𝑡, 𝑥′𝑡′ =1

𝑖ℏΨ 𝑇 𝜓 𝑥𝑡 𝜓+ 𝑥′𝑡′ Ψ / Ψ Ψ

Quantum espresso

• Program package

Schrödinger equation

• Non-relativistic

• Born-Oppenheimer approximation

𝐻el = 𝑇el + 𝑊ee + 𝑉ne

𝐻elΨ = 𝐸Ψ

First principles calculation

Total energy of a matter

• Phase diagram • Chemical reaction

From Schrödinger eq. to Kohn-Sham eq.

• Solving time-independent Schrödinger eq. corresponds to solving full configuration interaction (CI)

• One can handle typically up to 10-20 electrons

• Kohn-Sham DFT provides a way to handle thousands of electrons at least approximately

Density functional theory

1. Hohenberg Kohn theorem

2. Method of Kohn and Sham

Hohenberg Kohn theorem

𝑣 𝑟 → 𝐻 = 𝑇 + 𝑊𝑒𝑒 + 𝑉 → Ψ 𝑥1, 𝑥2, ⋯ , 𝑥𝑁 → 𝑛 𝑟

All are functional of 𝑣(𝑟), i.e., function of a function

𝐸 = 𝛹 𝑇 + 𝑊ee 𝛹 +න𝑣 𝒓 𝑛 𝒓 𝑑𝒓

Hohenberg Kohn theorem

𝑣 𝑟 → 𝐻 = 𝑇 + 𝑊𝑒𝑒 + 𝑉 → Ψ 𝑥1, 𝑥2, ⋯ , 𝑥𝑁 → 𝑛 𝑟

𝑛 𝑟 → Ψ Reverse mapping exists.

All are functional of 𝑛 𝑟 .

This is not trivial at all because, roughly speaking, we are

going admit that the external potential 𝑣(𝑟) is uniquely

determined when the density distribution 𝑛 𝑟 is known.

Hohenberg Kohn theorem

𝑣 𝑟 → 𝐻 = 𝑇 + 𝑊𝑒𝑒 + 𝑉 → Ψ 𝑥1, 𝑥2, ⋯ , 𝑥𝑁 → 𝑛 𝑟

𝑛 𝑟 → Ψ, 𝐻, 𝑣(𝑟) Reverse mapping

Functional of 𝑛 𝑟

Suppose that two local potentials, 𝑣1(𝒓) and 𝑣2(𝒓) which differ more than an additive

constant, share the same density 𝑛(𝒓). Since the states are different and therefore ۧ|𝛹2

cannot be a ground-state wave function of 𝐻1 when non-degenerated,

𝐸1 = 𝛹1 𝐻1 𝛹1 < 𝛹2𝐻1 𝛹2 = 𝛹2

𝐻2 + 𝑉1 − 𝑉2 𝛹2 = 𝐸2 + 𝑣1 𝒓 − 𝑣2 𝒓 𝑛 𝒓 𝑑𝒓

holds. In parallel, we can show the following:

𝐸2 < 𝐸1 + 𝑣2 𝒓 − 𝑣1 𝒓 𝑛 𝒓 𝑑𝒓Then, we arrive at an inconsistent inequality 𝐸1 + 𝐸2 < 𝐸1 + 𝐸2.

Hohenberg Kohn theorem

𝑣 𝑛 𝑟 , 𝐸 𝑛 = 𝑇 𝑛 + 𝑉𝑒𝑒 𝑛 + න𝑣 𝑟 𝑛 𝑟 𝑑𝑟 ,

Ψ[𝑛] 𝑥1, 𝑥2, ⋯ , 𝑥𝑁

𝑛(𝑟) can be determined once the energy density functional

𝐸[𝑛] is known. The determination is possible if 𝐸[𝑛]behaves well.

𝛿𝐸 𝑛

𝛿𝑛 𝑟− 𝜆 = 0

𝑇 𝑛 + 𝑉𝑒𝑒 𝑛 ≡ 𝐹[𝑛] is called as the Hohenberg-Kohn functional

v-representability vs n-representability

• The electron density 𝑛 𝑟 is v-representable when the corresponding external potential 𝑣 𝑟 exists

• The optimal 𝑛 𝑟 is not necessarily v-representable even when n-representable

𝑛 𝑟 ≥ 0

න𝑛 𝑟 = 𝑁

න ∇ 𝑛 𝑟2< ∞

Thomas Fermi theory

Theory of an almost uniform electron system

4

3𝜋𝑘𝐹

3 =𝑁

2

2𝜋 3

Ω𝑇 =

𝑘≤𝑘𝐹

2 ×ℏ𝑘2

2𝑚

𝑘𝐹 𝑛 𝑟 = 3𝜋2𝑛 𝑟1/3

𝑇 = Ω3

5

ℏ2𝑘𝐹2

2𝑚𝑛 → න𝑑𝑟

3

5

ℏ2𝑘𝐹2 𝑛 𝑟

2𝑚𝑛(𝑟)

Thomas Fermi theory

𝑘𝐹 𝑛 𝑟 = 3𝜋2𝑛 𝑟1/3

𝑇 = Ω3

5

ℏ2𝑘𝐹2

2𝑚𝑛 → 𝑇[𝑛] ≃ න𝑑𝑟

3

5

ℏ2𝑘𝐹2 𝑛 𝑟

2𝑚𝑛(𝑟)

𝑉H 𝑛 =1

2න

)𝑛 𝒓 𝑛(𝒓′

𝒓 − 𝒓′𝑑𝒓𝑑𝒓′ , Vx 𝑛 ≃ −

3

4

3

𝜋

13

න𝑛43 𝑟 𝑑𝑟 , 𝑉c 𝑛 = 0

𝑉𝑒𝑒 𝑛 = 𝑉H 𝑛 + 𝑉xc 𝑛

The simplest local density approximation

Thomas Fermi theory

𝑇 𝑛 = න𝑑𝑟3

103𝜋2

23𝑛

53(𝑟)

𝑉𝑒𝑒 𝑛 =1

2

൯𝑛 𝒓 𝑛(𝒓′

𝒓−𝒓′𝑑𝒓𝑑𝒓′ −

3

4

3

𝜋

1

3𝑛

4

3 𝑟 𝑑𝑟

The 1st generation of DFT

𝛿𝑇 𝑛

𝛿𝑛 𝑟+𝛿𝑉𝑒𝑒 𝑛

𝛿𝑛 𝑟− 𝜆 = 0

ExerciseTo understand the difficulty of constructing the kinetic energy functional, let us test theone constructed according to the Thomas Fermi (TF) model. Note that TF model isappropriate for systems where the electron density has a small special dependence, i.e., fora nearly uniform system. According this model, the kinetic functional is given as

𝑇 =35/3𝜋4/3

10න𝑛 𝒓 5/3𝑑𝒓

for a three-dimensional system and as

𝑇 =𝜋2

6න0

𝐿

𝑑𝑥 𝑛 𝑥 3𝑑𝑥

for a one-dimensional system. From this, one can naively define the kinetic energy densityfunctional as

𝑡 𝑛 𝑥 =𝜋2

6𝑛 𝑥 3.

This is a local density approximation to the kinetic energy.

The integration can be done analytically for a 1D potential well of infinite wall height,where the wave function has a form

𝜑𝑛 𝑥 = 2/𝐿 sin𝑛𝜋𝑥

𝐿,

and the kinetic energy is given as

𝑇 =

𝑛

𝑁2

𝐿න

0

𝐿1

2

𝑛𝜋

𝐿

2

sin2𝑛𝜋𝑥

𝐿𝑑𝑥 =

𝜋2

6𝐿2𝑁 𝑁 + 1 𝑁 +

1

2.

When using the Thomas Fermi model, the kinetic energy is approximated as

𝑇TF =𝜋2

60𝐿 2

𝐿σ𝑛𝑁 sin2

𝑛𝜋𝑥

𝐿

3𝑑𝑥 =

𝜋2

6𝐿2𝑁 𝑁2 +

9

8𝑁 +

3

8.

TTF/T

10 20 30 40 50

0.005

0.010

0.015

0.020

0.025

Thomas Fermi theory for DFT (summary)

• Fully local density functional approximation

• The kinetic energy 𝑇[𝑛] appears the most problematic

• Gradient formula, given be Weizacker, improves it slightly, but the shell structure of an atom is not reproduced by the theory.

• Construction of 𝑇[𝑛] has attracted renewed attention recently.

Going beyond Thomas Fermi

• Kohn and Sham proposed a scheme for the energy density functional

• The KS-DFT is accepted as the standard and practical theory for modern electronic structure calculation

KS-DFT

𝐹 𝑛 = 𝑇𝑠 𝑛 + 𝑉H 𝑛 + 𝑉xc 𝑛 ,where

𝑇𝑠 𝑛 = minΦ→𝑛

Φ 𝑇 Φ = Φ[𝑛] 𝑇 Φ[𝑛]

Kinetic energy functional 𝑇[𝑛] can be approximated by that of a

non-interacting system which shares the same 𝑛(𝑟)

Non-interacting

Φ

Interacting

Ψ𝑛(𝑟)

KS-DFT

𝐹 𝑛 = 𝑇𝑠 𝑛 + 𝑉H 𝑛 + 𝑉xc 𝑛 ,where

𝑇𝑠 𝑛 = minΦ→𝑛

Φ 𝑇 Φ = Φ[𝑛] 𝑇 Φ[𝑛]

𝑇𝑠 𝑛 = න𝑑𝑟

1≤𝑖≤𝑁

𝜑𝑖∗ 𝑟 −

∇2

2𝜑𝑖(𝑟)

𝑛 𝑟 =

1≤𝑖≤𝑁

𝜑𝑖 𝑟2

Note that 𝜑𝑖 is also a

functional of 𝑛

See Eqs. (1.22-1.27) for more detail

KS-equation

𝛿𝐸 𝑛

𝛿𝜑𝑖∗ 𝑟

= −∇2

2𝜑𝑖 𝑟 +

𝛿𝑉Hxc 𝑛

𝛿𝑛 𝑟𝜑𝑖(𝑟)

𝑛 𝑟 =

1≤𝑖≤𝑁

𝜑𝑖 𝑟2−

∇2

2+ 𝑉Hxc[𝑛](𝑟) 𝜑𝑖 𝑟 =

𝑗

휀𝑖𝑗𝜑𝑗(𝑟)

A set of nonlinear differential equations

How to make the exchange-correlation?

1. Learn from the Thomas Fermi theory

2. Extrapolate/interpolate from systems of known xc potential

3. Use perturbation theory

4. Make the exact one (practical only for a few electron systems)

𝐸Hxc 𝑛 = 𝐸H 𝑛 + 𝐸x 𝑛 + 𝐸c 𝑛where

𝐸x 𝑛 ≡ Φ 𝑛 𝑊ee Φ 𝑛 − 𝐸H 𝑛 , (1.26)

and

𝐸c 𝑛 ≡ Ψ 𝑛 𝑇 + 𝑊ee Ψ 𝑛 − Φ 𝑛 𝑇 + 𝑊ee Φ 𝑛 (1.27)

𝑉x 𝑛 = −3

4

3

𝜋

1

3𝑛

4

3 𝒓 𝑑𝒓

𝑉H 𝑛 =1

2න

)𝑛 𝒓 𝑛(𝒓′

𝒓 − 𝒓′𝑑𝒓𝑑𝒓′

1. KS-DFT with TF exchange

2. Use the exchange-correlation of a uniform electron system

𝑉x𝑐 𝑛 = න𝑑𝒓 𝜖xc 𝑛 𝒓 𝑛(𝒓) 𝜖xc 𝑛 from QMC simulation

QMC simulation of uniform electron gas

Ceperley and Alder (1980)

3. Use perturbation theory(a) exchange-correlation hole

(b) Many-body perturbation theory (later)

Density matrix formalism

𝑛2 𝒓1, 𝒓2 = 𝑁 𝑁 − 1 ම Ψ 𝑛 𝒙1, ⋯ , 𝒙𝑁2𝑑𝜎1𝑑𝜎2𝑑𝒙3⋯𝑑𝒙𝑁

]Ψ[𝑛 𝑊ee ]Ψ[𝑛 =1

2ඵ

𝑛2 𝒓1, 𝒓2𝒓1 − 𝒓2

𝑑𝒓1𝑑𝒓2

We need the pair correlation only since we are dealing with a system

interacting via two-body force.

(So, for classical systems and nuclear systems, things are more

complicated!)

]Ψ[𝑛 𝑊ee ]Ψ[𝑛 =1

2ඵ

𝑛 𝒓1 𝑛 𝒓2𝒓1 − 𝒓2

𝑑𝒓1𝑑𝒓2 +1

2ඵ

𝑛 𝒓1 𝑛xc 𝒓1, 𝒓2𝒓1 − 𝒓2

𝑑𝒓1𝑑𝒓2

To make a practical DFT, one needs to model the exchange-correlation

hole.

𝑛2,xc 𝒓1, 𝒓2 = 𝑛 𝒓1 𝑛xc 𝒓1, 𝒓2 Exchange-correlation hole

න𝑛xc 𝒓1, 𝒓2 𝑑𝒓2 = −1

𝑛2 𝒓1, 𝒓2 = 𝑛 𝒓1 𝑛 𝒓2 + 𝑛2,xc 𝒓1, 𝒓2 Exchange-correlation

contribution

Exchange-correlation hole

Exact xc hole of a hydrogen moleculeOne electron is located at a point close to the right hand side of the nucleus

Approximate xc hole

• The xc hole can be approximately obtained using a perturbation theory (linear response theory)

• For that purpose, let us reformulate the Kohn-Sham theory using the adiabatic connection formula (AC)

• Then, let us transform the correlation function into the response function using the fluctuation dissipation theory (FDT)

• The latter treatment requires development of time-dependent density functional theory (TDDFT)

Adiabatic connection

• The Coulombic interaction is adiabatically introduced as𝐻𝜆 = 𝑇 + 𝜆 𝑊ee + V𝜆 (0 ≤ 𝜆 ≤ 1)

• Decompose the HK functional𝐹𝜆 𝑛 = min

Ψ→𝑛Ψ 𝑇 + 𝜆 𝑊ee Ψ = Ψλ[𝑛] 𝑇 + 𝜆 𝑊ee Ψ

λ[𝑛]

as𝐹𝜆 𝑛 = 𝑇𝑠 𝑛 + 𝐸H

𝜆 𝑛 + 𝐸xc𝜆 [𝑛]

𝐸H𝜆 𝑛 =

1

2ඵ

𝑛 𝒓1 𝑛 𝒓2|𝒓1 − 𝒓2|

𝜆𝑑𝒓1𝑑𝒓2 = 𝜆𝐸H 𝑛

𝐸x𝜆 𝑛 = Φ[𝑛] 𝜆 𝑊ee Φ[𝑛] − 𝐸H

𝜆 𝑛 = 𝜆𝐸x 𝑛

𝐸c𝜆 𝑛 = Ψλ[𝑛] 𝑇 + 𝜆 𝑊ee Ψ

λ[𝑛] − Φ 𝑛 𝑇 + 𝜆 𝑊ee Φ 𝑛

keeping the density 𝑛(𝑟) constant

Hellman Feynman theorem applied to the correlation functional

𝜕𝐸c𝜆 𝑛

𝜕𝜆= ൧Ψλ[𝑛 𝑊ee ൧Ψλ[𝑛 − Φ 𝑛 𝑊ee Φ 𝑛

𝐸𝑐 𝑛 = න

0

1

൧Ψλ[𝑛 𝑊ee ൧Ψλ[𝑛 𝑑𝜆 − Φ 𝑛 𝑊ee Φ 𝑛

𝐸𝑐 𝑛 =1

2න

0

1

𝑑𝜆ඵ𝑛 𝒓1 𝑛𝑐

𝜆 𝒓1, 𝒓2𝒓1 − 𝒓2

𝑑𝒓1𝑑𝒓2

𝐸𝑐 𝑛 =1

2ඵ

𝑛 𝒓1 ത𝑛𝑐 𝒓1, 𝒓2𝒓1 − 𝒓2

𝑑𝒓1𝑑𝒓2 Averaged xc hole

𝜆

𝑛 𝒓

𝑛𝑐𝜆 𝒓1, 𝒓2 ത𝑛𝑐 𝒓1, 𝒓2

𝐸𝑐[𝑛]

Exact 𝐸𝑐 for KS-DFT

𝐹 𝑛 = 𝑇𝑠 𝑛 + 𝐸H 𝑛 + 𝐸x 𝑛 +𝐸c 𝑛

𝐹 𝑛 = 𝑇 𝑛 + 𝐸H 𝑛 + 𝐸x 𝑛 +𝐸c 𝑛

For the original DFT

4. Calculate the exact xc potential

Calculated for Li-Be with elongated

interatomic distance

Actually this is more difficult than solving Schrödinger eq.

How to obtain the correlation hole?

• Normally, the correlation function is difficult to calculate

• The correlation function can be related to the response function via the fluctuation dissipation theorem

• Now we discuss how to obtain the response function

Density operator

• Ψ ො𝑛2(𝒓1, 𝒓2) Ψ is defined as follows

• ො𝑛1 𝒓 ≡ σ𝑖𝑁 𝛿(𝒓 − 𝒓𝑖) then 𝑛 𝒓 = Ψ ො𝑛1 𝒓 Ψ

• ො𝑛2 𝒓1, 𝒓2 ≡ ො𝑛1 𝒓1 ො𝑛1 𝒓2 − ො𝑛1 𝒓1 𝛿 𝒓1 − 𝒓2

𝑇 = −1

2න ∇𝒓

2 ො𝑛1 𝒓, 𝒓′ 𝒓′=𝒓𝑑𝒓

𝑉ee =1

2ඵ𝑤ee 𝒓1, 𝒓2 ො𝑛2 𝒓1, 𝒓2 𝑑𝒓1𝑑𝒓2

𝑉ne = න𝑣ne 𝒓 ො𝑛 𝒓 𝑑𝒓 1

𝒓1 − 𝒓2

We need the quantity

𝑛2,c𝜆 𝒓1, 𝒓2 = 𝑛2

𝜆 𝒓1, 𝒓2 − 𝑛2,KS 𝒓1, 𝒓2 .

Using the fact that the density is independent of 𝜆, we have

𝑛2,c𝜆 𝒓1, 𝒓2 = Ψλ ො𝑛1 𝒓1 ො𝑛1 𝒓2 Ψλ − Φ ො𝑛1 𝒓1 ො𝑛1 𝒓2 Φ

𝑛2𝜆 𝒓1, 𝒓2 = Ψλ ො𝑛2 𝒓1, 𝒓2 Ψλ

= Ψλ ො𝑛1 𝒓1 ො𝑛1 𝒓2 Ψλ − 𝛿 𝒓1 − 𝒓2 Ψλ ො𝑛1 𝒓1 Ψλ

𝑛2,KS 𝒓1, 𝒓2= Φ ො𝑛1 𝒓1 ො𝑛1 𝒓2 Φ − 𝛿 𝒓1 − 𝒓2 Φ ො𝑛1 𝒓1 Φ

Now let us consider the response

Response of the non-interacting system

χ0 𝒓′, 𝒓 =𝛿𝑛 𝒓′

𝛿𝑣𝑠 𝒓A time-dependent version

χ0 𝒓′𝑡′, 𝒓𝑡 =𝛿𝑛 𝒓′𝑡′

𝛿𝑣𝑠 𝒓𝑡.

Response of the interacting system

𝜒 𝒓′𝑡′, 𝒓𝑡 =𝛿𝑛 𝒓′𝑡′

𝛿𝑣ne 𝒓𝑡

𝑣𝑠 = 𝑣ne + 𝑣Hxc

Effective KS potential

The response and the effective potential are actually dependent on the AC switching parameter 𝜆

According to Runge and Gross, there exists a one-to-one correspondence betweenthe time-dependent density 𝑛 𝒓, 𝑡 and the time-dependent effective potential 𝑣s(𝒓, 𝑡),so that 𝑛 𝒓, 𝑡 can be obtained by solving the time-dependent Kohn-Sham equation:

𝑖𝜕

𝜕𝑡𝜑𝑖 𝒓, 𝑡 = −

1

2∇2 + 𝑣s 𝒓, 𝑡 𝜑𝑖 𝒓, 𝑡 ,

and

𝑛 𝒓, 𝑡 =

𝑖

𝑁

𝜑𝑖 𝒓, 𝑡2

Time-dependent DFT

Density response

𝛿𝑛 𝒓, 𝑡 = න

0

∞

𝑑𝑡′න𝑑𝒓′ 𝜒𝜆 𝒓𝑡, 𝒓′𝑡′ 𝛿𝑣ext,𝜆 𝒓′𝑡′

𝛿𝑛 𝒓, 𝑡 = න

0

∞

𝑑𝑡′න𝑑𝒓′ 𝜒KS,𝜆 𝒓𝑡, 𝒓′𝑡′ 𝛿𝑣eff,𝜆 𝒓′𝑡′

From the exact theory

From the KS-DFT

𝜒𝜆,KS 𝒓, 𝒓′, 𝜔 =𝑘𝑗

𝜑𝑘𝜆∗ 𝒓 𝜑𝑗

𝜆 𝒓 𝜑𝑗𝜆∗ 𝒓′ 𝜑𝑘

𝜆 𝒓′

𝜔 − 𝜔𝑗𝑘 + 𝑖𝜂

Defining the exchange-correlation kernel 𝐾 as

𝐾xc𝜆 𝒓𝑡, 𝒓′𝑡′ =

𝛿𝑣xc𝜆 𝒓𝑡

𝛿𝑛 𝒓′𝑡′,

we get the expression

δ𝑣s𝜆 𝒓𝑡 = 𝛿𝑣ext

𝜆 𝒓𝑡 + d𝐫′𝛿𝑛 𝒓’𝑡

𝐫 − 𝐫′+ 𝑑𝒓′𝑑𝑡′Kxc

𝜆 (𝒓𝑡, 𝐫′t′)𝛿𝑛 𝒓′𝑡′

𝜒𝜆 𝒓𝑡, 𝒓′𝑡′ = 𝜒λ,KS 𝒓𝑡, 𝒓′𝑡′ +ඵ𝜒𝜆,KS 𝒓𝑡, 𝒓1𝑡1𝛿 𝑡1 − 𝑡2𝒓1 − 𝒓2

+ 𝐾xc𝜆 𝒓1𝑡1, 𝒓2𝑡2 𝜒𝜆 𝒓2𝑡2, 𝒓

′𝑡′

Having discussed the fluctuation and response, let us related themFluctuation dissipation theorem

iχλ 𝒓1𝑡1, 𝒓2𝑡2= Ψ𝜆 𝑇[ො𝑛H(𝒓1𝑡1) ො𝑛

H(𝒓2𝑡2)] Ψ𝜆 − Ψ𝜆 ො𝑛(𝒓1) Ψ

𝜆 Ψ𝜆 ො𝑛(𝒓2) Ψ𝜆 (4.19)

If we set 𝑡2 = 𝑡1 + 0+, the Heisenberg phase factor is canceled out and

iχλ 𝒓1, 𝒓2; 𝜏 = 0− = Ψ𝜆 ො𝑛(𝒓1) ො𝑛(𝒓2) Ψ𝜆 − Ψ𝜆 ො𝑛 𝒓1 Ψ𝜆 Ψ𝜆 ො𝑛 𝒓2 Ψ𝜆 . (4.20)

It is possible to derive similar equation for λ = 0, givingiχ0 𝒓1, 𝒓2; 𝜏 = 0− = Φ ො𝑛(𝒓1) ො𝑛(𝒓2) Φ − Φ ො𝑛 𝒓1 Φ Φ ො𝑛 𝒓2 Φ . (4.21)

This makes it possible to write the two-body density correlation in terms of theresponse function as

𝑛2,c𝜆 𝒓1, 𝒓2 = i 𝜒𝜆 𝒓1, 𝒓2; 𝜏 = 0− − 𝜒0 𝒓1, 𝒓2; 𝜏 = 0− , (4.22)

With the causality taken into account,

using the Fourier transformed form as

𝑛2,c𝜆 𝒓1, 𝒓2 = − න

−∞

∞𝑑𝜔

2𝜋i𝑒𝑖𝜔0

+𝜒𝜆 𝒓1, 𝒓2; 𝜔 − 𝜒0 𝒓1, 𝒓2; 𝜔 , (4.23)

which is known as the fluctuation dissipation theorem. When this is combined with theadiabatic connection formula, we get

𝐸c = −1

2න

0

1

𝑑𝜆 න

−∞

∞𝑑𝜔

2𝜋i𝑒𝑖𝜔0

+ඵ

𝜒𝜆 𝒓1, 𝒓2; 𝜔 − 𝜒0 𝒓1, 𝒓2; 𝜔

𝒓1 − 𝒓2𝑑𝒓1𝑑𝒓2

KS-DFT within AC-FDT

One can obtain the correlation functional based on the exchange-correlation kernel

𝐾(𝑟𝑡, 𝑟′𝑡′), which has been constructed only approximately. However, there is a way

to construct it using the many-body Green’s function theory

The dynamical correlation, such as that

caused by the van der Waals

interaction, can be described.

Fractional number of electron and integer discontinuity problemWave and particle in quantum physics

Let us denote the number as 𝒩 which is also written as 𝒩 = 𝑁 − 1 + 𝑓 with 𝑁 ∈ 𝕀 and0 ≤ 𝑓 ≤ 1. The total-energy is described as

𝐸0𝒩 = min

ΓTr Γ 𝑇 + 𝑊ee + 𝑉ne , (2.30)

using the ensemble density matrix defined asΓ = 1 − 𝑓 ۧ|Ψ𝑁−1 |Ψ𝑁−1ۦ + 𝑓 ۧ|Ψ𝑁 .|Ψ𝑁ۦ (2.31)

(One may use many states with different number of electrons as well.) The variation isdone for Ψ𝑁−1 and Ψ𝑁, by which the ensemble density matrix is obtained as

Γ0 = 1 − 𝑓 ൿ|Ψ0𝑁−1 ൻΨ0

𝑁−1| + 𝑓 ൿ|Ψ0𝑁 ൻΨ0

𝑁|. (2.32)Then the total-energy can be described as

𝐸0𝒩 = 1 − 𝑓 𝐸0

𝑁−1 + 𝑓𝐸0𝑁.

From this definition, it is clear that the total-energy has a derivative discontinuity as

𝑑𝐸0𝒩

𝑑𝒩= ൝

𝐸0𝑁 − 𝐸0

𝑁−1 ≡ −𝐼𝑁 when 𝑁 − 1 < 𝒩 < 𝑁

𝐸0𝑁+1 − 𝐸0

𝑁 ≡ −𝐴𝑁 when 𝑁 < 𝒩 < 𝑁 + 1

System with fractional particle number

𝐹 𝑛 = minΓ→𝑛

Tr Γ 𝑇 + 𝑊ee . (2.38)

This can be decomposed, in the KS theory, into𝐹 𝑛 = 𝑇𝑠 𝑛 + 𝐸Hxc[𝑛] (2.39)

where the Kohn-Sham non-interacting kinetic energy is given as

𝑇𝑠 𝑛 = minΓs→𝑛

Tr[Γ𝑠 𝑇]

and the ensemble non-interacting density matrix isΓ𝑠 = (1 − 𝑓) ൿ|Φ𝑁−1,𝑓 ൻΦ𝑁−1,𝑓| + 𝑓 ൿ|Φ𝑁,𝑓 ൻΦ𝑁,𝑓|. (2.41)

By introducing partial occupation number of each KS orbitals 𝑛𝑖, the total-energy can be described as

𝐸 =

𝑖

𝑁

𝑛𝑖න𝜙𝑖∗ 𝒓 −

1

2∇2 + 𝑣ne 𝒓 𝜙𝑖 𝒓 𝑑𝒓 +𝐸Hxc 𝑛 , (2.42)

with the density

𝑛 𝒓 =

𝑖

𝑁

𝑛𝑖 𝜙𝑖 𝒓2 . (2.43)

Note that the occupation number satisfy the condition that 𝑛𝑖 = 1 for 𝑖 ≤ 𝑁 − 1 and 𝑛𝑁 = 𝑓. The KSequation is thereby unchanged

−1

2∇2 + 𝑣s 𝒓 𝜙𝑖 𝒓 = 휀𝑖𝜙𝑖(𝒓) (2.44)

𝑣s 𝒓 = 𝑣ne 𝒓 +𝛿𝐸Hxc 𝑛

𝛿𝑛 𝒓. (2.45)

There is uncertainty due to the additive constant

𝛿𝐸Hxc 𝑛 = න𝛿𝐸Hxc 𝑛

𝛿𝑛 𝒓+ const 𝛿𝑛 𝒓 𝑑𝒓

“const” should be carefully treated when 𝑁 crosses an integer

“const” is the additive constant to be determined elsewhere

𝜕𝐸

𝜕𝑛𝑖= 휀𝑖 Janak theorem

The one-electron energy can jump when crossing the integer occupancy

The derivative at a number slightly larger than an integer is,𝜕𝐸0

𝒩

𝜕𝒩 𝑁+𝛿= 휀HOMO

𝑁+𝛿 , = −𝐴𝑁

or

휀HOMO𝑁+𝛿 = න𝜙HOMO

𝑁+𝛿 𝒓 ∗ −1

2∇2 + 𝑣𝑠

𝑁+𝛿 𝒓 𝜙HOMO𝑁+𝛿 𝒓 𝑑𝒓

휀LUMO𝑁−𝛿 = න𝜙LUMO

𝑁−𝛿 𝒓 ∗ −1

2∇2 + 𝑣𝑠

𝑁−𝛿 𝒓 𝜙LUMO𝑁−𝛿 𝒓 𝑑𝒓 .

With decreasing 𝛿 → 0+, 𝑣𝑠𝑁+𝛿 𝒓 and 𝑣𝑠

𝑁−𝛿 𝒓 approach the potentials that are equal to each other except for the

additive constant, 𝑣𝑠𝑁+𝛿 𝒓 − 𝑣𝑠

𝑁−𝛿 𝒓 ≡ Δxc𝑁 .

Therefore,

휀HOMO𝑁+𝛿 = 휀LUMO

𝑁−𝛿 + Δxc𝑁

holds. Namely, after taking the limit,

휀LUMO𝑁 = −𝐴𝑁 − Δxc

𝑁 .This allows to rewrite the fundamental gap in terms of the derivative discontinuity as

𝐸gap𝑁 = 휀LUMO

𝑁 − 휀HOMO𝑁 + Δxc

𝑁

The constant appeared in the derivation of HK theorem turns out to be the derivative discontinuity.

Energy gap and integer discontinuity

Band structure of Ge w/ and w/o Δxc𝑁

Existing approximate xc

𝐸xcLDA 𝑛 = න𝑛 𝑟 휀xc

uniform gas𝑛 𝒓 𝑑𝒓 .

휀 is obtained by doing a Monte Carlo simulation of uniform electron gas of a fixed electron density. The result ofthe exchange-correlation energy is fitted to a form

−3

4

3

𝜋

13

𝑛13 𝒓 + 휀𝑐(𝑟𝑠(𝑛(𝒓)))

휀𝑐 𝑟𝑠 = ൞

𝐴 ln 𝑟𝑠 + 𝐵 + 𝐶 𝑟𝑠 ln 𝑟𝑠 when 𝑟𝑠 ≤ 𝑟𝑠,0𝑎

𝑟𝑠+

𝑏

𝑟𝑠3/2

when 𝑟𝑠 > 𝑟𝑠,0

where 𝑟𝑠 =3

4𝜋𝑛

1

3.

Local density approximation (LDA)

Generalized gradient approximation (GGA)

𝐸xcGEA 𝑛 = 𝐸xc

LDA 𝑛 + න𝐶xc 𝑛 𝒓 𝑛 𝒓43

∇𝑛 𝒓

𝑛 𝒓43

2

𝑑𝒓, (3.10)

The expansion parameter, however, can be large in some part of the target system making theexpansion break down. In this context, people have tried to construct a functional so that

𝐸xcGGA 𝑛 = න𝑓 𝑛 𝒓 , ∇𝑛 𝒓 𝑑𝒓 (3.11)

GGA[#1] B88 exchange functional: Becke 88 was constructed to reproduce the nonlocal exchange functional,such as the asymptotic behavior of the exchange energy per particle and the value of the nonlocal exchangeenergy of rare-gas atoms.GGA[#2] LYP correlation functional: Lee-Yang-Parr (LYP) correlation functional was constructed based onan approximation of the Hartree-Fock pair density.GGA[#3] PW91 exchange-correlation functional: Perdew-Wang 91 exchange-correlation functional is basedon a model of the exchange hole and the coupling-constant-averaged correlation hole.GGA[#4] PBE exchange-correlation functional: Perdew-Burke-Ernzerhof exchange-correlation functional is asimplified PW91.

Meta-GGA

𝐸xcmGGA = න𝑓 𝑛 𝒓 , ∇𝑛 𝒓 , ∇2𝑛 𝒓 , 𝜏 𝒓 𝑑𝒓 (3.13)

𝜏 𝒓 =1

2

𝑖

𝑁

∇𝜙𝑖 𝒓2 (3.14)

𝜏 𝒓 is constructed by the square of the KS gradient. This quantity is distinct from the electron density and thusthe ground-state energy is given formally as

𝐸0 = minΦ

Φ 𝑇 + 𝑉ne Φ + 𝐸H 𝑛Φ + 𝐸xc[𝑛Φ, 𝜏Φ] .

Hybrid DFT

Becke proposed to mix the nonlocal exchange energy, or the Hartree-Fock exchange energy,

𝐸xHF = −

1

2

𝜎

𝑖𝑗

𝑁𝜎

ඵ𝜙𝑖𝜎∗ 𝒓1 𝜙𝑗𝜎 𝒓𝟏 𝜙𝑗𝜎

∗ 𝒓2 𝜙𝑖𝜎 𝒓2

|𝒓1 − 𝒓2|𝑑𝒓1𝑑𝒓2, (3.18)

with the existing forms for the exchange and correlation, such as𝐸xc3H = 𝑎𝐸x

HF + 𝑏𝐸xGGA + 1 − 𝑎 − 𝑏 𝐸x

LDA + 𝑐𝐸cGGA + 1 − 𝑐 𝐸c

LDA. (3.17)The most famous parameterization is known as B3LYP, which was constructed by using the B88 exchange for theGGA exchange and the LYP correlation for the GGA correlation. Another form is

𝐸xc1H = 𝑎𝐸x

𝐻𝐹 + 1 − 𝑎 𝐸xDFA + 𝐸c

DFA, (3.19)where DFA stands for any semilocal density functional approximation (DFA). The most famous one is called PBE0 where the fitting parameter was set to be 0.25. Hybrid DFT is rather free from the self-interaction error. The mixing parameters, 𝑎 to 𝑐, are determined

Double hybrid DFT

Grimme constructed𝐸xc2DH = 𝑎x𝐸x

𝐻𝐹 + 1 − 𝑎x 𝐸xDFA + 1 − 𝑎c 𝐸c

DFA + 𝑎c𝐸cMP2, (3.22)

where MP2 stands for the second order approximation to the correlation energy

𝐸cMP2 = −

1

4

𝑖𝑗=1

𝑁

𝑎𝑏=𝑁+1

2𝑀ൻ𝜓𝑖𝜓𝑗| ۧ𝜓𝑎𝜓𝑏 − ൻ𝜓𝑖𝜓𝑗| ۧ𝜓𝑏𝜓𝑎

2

휀𝑎 + 휀𝑏 − 휀𝑖 − 휀𝑗, (3.23)

where the bracket appearing in the numerator indicates the Coulomb integral

ൻ𝜓𝑖𝜓𝑗| ۧ𝜓𝑎𝜓𝑏 =ඵ𝜓𝑖∗(𝒓1)𝜓𝑗

∗(𝒓𝟐)𝜓𝑎(𝒓1)𝜓𝑏(𝒓𝟐)

|𝒓1 − 𝒓𝟐|𝑑𝒙1𝑑𝒙2. (3.24)

Note that 𝑎 and 𝑏 is large than 𝑁 indicating that unoccupied KS orbitals are included in the formalism contrary toothers introduced so far, where only occupied orbitals are included. This way, the computation of the functional arebecoming more and more demanding although the accuracy is generally improving by that.

Range-separated hybrid DFT

Iikura, Tsuneda, Yanai, and Hirao proposed a long-range correction (LC) scheme to assign different forms for theexchange depending on the distance between the electrons as

𝐸xcLC = 𝐸x

long range,HF+ 𝐸x

short range,DFA+ 𝐸c

DFA.The range-separation is done using the error function as

𝐸xlong range,HF

= −1

2

𝜎

𝑖𝑗

𝑁𝜎

ඵ𝜙𝑖𝜎∗ 𝒓1 𝜙𝑗𝜎 𝒓𝟏 erf(𝜇|𝒓1 − 𝒓2|)𝜙𝑗𝜎

∗ 𝒓2 𝜙𝑖𝜎 𝒓2

|𝒓1 − 𝒓2|𝑑𝒓1𝑑𝒓2, (3.27)

This is based on the idea that the error of the semilocal approximation to the exchange is serious when two electronsare apart and the error is effectively corrected by the HF exchange. The cutoff distance for the range-separationneeds to be introduced empirically.

Exact exchangeThe exchange potential is from definition obtained as

𝛿𝐸x𝛿𝑣s 𝒓

= න𝛿𝐸x

𝛿𝑛 𝒓′𝛿𝑛 𝒓′

𝛿𝑣𝑠 𝒓𝑑𝒓′ . (4.2)

The derivative of the electron density can be obtained by using the definition of the response function

χ0 𝒓′, 𝒓 =𝛿𝑛 𝒓′

𝛿𝑣𝑠 𝒓

resulting as

න𝑣x 𝒓′ 𝜒0 𝒓′, 𝒓 𝑑𝒓′ =𝛿𝐸x𝛿𝑣s 𝐫

. (4.3)

That is, using the response function, change in the electron density with respect to the effective potential is expressedusing the KS orbital as

χ0 𝒓′, 𝒓 = −

𝜎

𝑖=1

𝑁𝜎

𝑎=𝑁𝜎+1

𝑀

ඵ𝜙𝑖𝜎∗ 𝒓′ 𝜙𝑎𝜎 𝒓 𝜙𝑎𝜎

∗ 𝒓 𝜙𝑖𝜎 𝒓′

휀𝑎𝜎 − 휀𝑖𝜎𝑑𝒓1𝑑𝒓2 + c. c. (4.5)

When this is used, we can derive the exchange potential using the two-electron integral as

𝛿𝐸x𝛿𝑣s 𝒓

=

𝜎

𝑖𝑗=1

𝑁𝜎

𝑎=𝑁𝜎+1

𝑀

𝜙𝑎𝜎𝜙𝑗𝜎 𝜙𝑗𝜎𝜙𝑖𝜎𝜙𝑎𝜎 𝒓 𝜙𝑖𝜎

∗ 𝒓

휀𝑎𝜎 − 휀𝑖𝜎+ c. c. (4.6)

Second order Görling Levy

The Hamiltonian appearing in the adiabatic connection, 𝐻𝜆 = 𝑇 + 𝜆 𝑊ee + V𝜆, is decomposed using V𝜆 = 𝑉s −

𝜆 𝑉Hx − 𝑉c𝜆 as

𝐻𝜆 = 𝑇 + 𝑉s + 𝜆 𝑊ee − 𝑉Hx − 𝑉c𝜆,

following the second order expansion of 𝑉c𝜆 with respect to 𝜆, which starts from the second order as 𝑉c

𝜆 = 𝜆2 𝑉c(2)

.Then the perturbation theory is applied to the KS state Φ𝑛=0 as

| ൿΨ 1 = −

𝑛≠0

| ۧΦ𝑛

Φ𝑛𝑊ee − 𝑉Hx Φ𝑛

ℇ𝑛 − ℇ0,

and this expression is inserted into the defining equation (Eq. (2.23))

𝐸c𝜆 = Ψλ 𝑇 + 𝜆 𝑊ee Ψ

λ − Φ 𝑇 + 𝜆 𝑊ee Φ

to obtain the correlation energy by expanding it in powers of 𝜆. The result is

𝐸c(2)

= Φ 𝑊ee Ψ(1) = Φ 𝑊ee − 𝑉Hx Ψ

(1) = −

𝑛≠0

Φ 𝑊ee − 𝑉Hx Φ𝑛2

ℇ𝑛 − ℇ0.

It was found this does not improve the correlation satisfactorily, indicating that it is necessary to go beyond thesecond order approximation.

Direct RPA

𝐾Hxc𝜆 𝒓1, 𝒓2; 𝜔 ≃ 𝐾H

𝜆 𝒓1, 𝒓2 = 𝜆𝑣 𝒓1, 𝒓2 . (4.28)When this is used, the resulting equation for the response function is

𝜒𝜆 𝒓1, 𝒓2; 𝜔 = 𝜒0 𝒓1, 𝒓2; 𝜔 + 𝜆ම𝜒0 𝒓1, 𝒓3; 𝜔 𝑣 𝒓3, 𝒓4 𝜒𝜆(𝒓4, 𝒓2; 𝜔) . (4.29)

By this we can define the correlation functional within this approximation called direct RPA as

𝐸cdRPA

= −1

2න0

1

𝑑𝜆න∞

∞ 𝑑𝜔

2𝜋i𝑒−i𝜔0

+[𝜆ම

𝜒0 𝒓1, 𝒓3; 𝜔 𝜒0 𝒓4, 𝒓2; 𝜔

𝒓1 − 𝒓2 𝒓3 − 𝒓4𝑑𝒓1𝑑𝒓2𝑑𝒓3𝑑𝒓4

+ 𝜆2ම𝜒0 𝒓1, 𝒓3; 𝜔 𝜒0 𝒓4, 𝒓5; 𝜔 𝜒0 𝒓6, 𝒓2; 𝜔

𝒓1 − 𝒓2 𝒓3 − 𝒓4 𝒓5 − 𝒓6𝑑𝒓1𝑑𝒓2𝑑𝒓3𝑑𝒓4𝑑𝒓5𝑑𝒓6 +⋯] (4.30)

Do simulations for deeper understanding

Preparation of the environment• Use MateriApps

• Linux (debian)

• Preinstalled applications• gfortran, mpirun, lapack, blas

• Electronic structure calculations (quantum espresso (QE))

• Visualization (band structure, density distribution)

• Download and install• Virtual Box

• Debian

• MateriApps

• Wannier90

• Run Wannier90 to make a tight-binding model of a diamond

• Do them within April

Report1

Show the band structure of a crystalline copper using quantum espresso.

1. Goto /usr/share/espresso/examples/PW/examples/example01 and run the program “run_example” and make sure that you have done a calculation of Cu as well as calculations of other elements.

2. Download tutorial files for July 1 from the QE web site or http://www.iiserpune.ac.in/~smr2626/talks-presentations.htmland follow the instructions to plot the band of Al.

3. Calculate the band of Cu following the Day2_exercise3 written by Prof. Marathe.

The results will be compared with the tight-binding band later in this course

Deadline is temporarily set at the end of April.

Send me (sugino@issp.u-tokyo.ac.jp) a pdf file.

• You will get the solid black line.

• Detail the approximations and

parameters you took to get the result.

Report2

Show the band structure of a crystalline copper using quantum espresso and wannier90.

• Read the tutorial #6 given in http://www.wannier.org/support/ and draw the s-component of the band as shown in Fig. 13.

• In doing so, obtain the density 𝑛(𝑟)using copper.scf

&controlcalculation='scf'restart_mode='from_scratch',pseudo_dir = '../../pseudo/',outdir='./'prefix='cu'

/&system

ibrav = 2, celldm(1) =6.822, nat= 1, ntyp= 1,ecutwfc = 30.0occupations='smearing', smearing='cold', degauss=0.02

/&electrons

diagonalization='cg'conv_thr = 1.0e-9mixing_beta = 0.7

/ATOMIC_SPECIESCu 63.546 Cu.pz-n-van_ak.UPF

ATOMIC_POSITIONSCu 0.0 0.0 0.0

K_POINTS (automatic)16 16 16 0 0 0

• Compute the KS eigenvalues

and eigenvectors using

copper.nscf

&controlcalculation='nscf'restart_mode='from_scratch',pseudo_dir = '../../pseudo/',outdir='./'prefix='cu'

/&system

ibrav = 2, celldm(1) =6.822, nat= 1, ntyp= 1,ecutwfc = 30.0occupations='smearing', smearing='cold', degauss=0.02nosym=.true.,nbnd=12

/&electrons

conv_thr = 1.0e-9diagonalization = 'cg'

/ATOMIC_SPECIESCu 55.85 Cu.pz-n-van_ak.UPF

ATOMIC_POSITIONSCu 0.0 0.0 0.0

K_POINTS {crystal}64

0.0000 0.0000 0.0000 0.01562500.0000 0.2500 0.0000 0.0156250…………………………………………………..0.7500 0.7500 0.7500 0.0156250

• Compute the overlap (copper.nnkp) required to make the maximally

localized Wannier orbitals using Wannier 90.

• Run pw2wannier90 to compute the overlap between Bloch states and the

projections for the starting guess (written in the copper.mmn and copper.amn

files).

• Run wannier90 to compute the MLWFs.

&inputppoutdir = './'prefix = 'cu'seedname = 'copper'spin_component = 'none'

/

5d centered at Cu site

2s centered at interstitial sites

• Read the tutorial and modify the input file “copper.win” to make the “copper_band.dat”

• Use gnuplot to plot the “copper_band.dat”

• Compare the plot with the original band, shown with black in Fig. 12.