Lecture on First-principles Computation (5): General Introduction to Density Functional Theory

任新国 (Xinguo Ren)

中国科学技术大学 量子信息重点实验室 Key Laboratory of Quantum Information, USTC

Hefei, 2016.9.28 Basic idea behind density functional theory

The many-body Hamiltonian: 푁 푁 푁 훻2 1 푒2 퐻 = 푇 + 푉 + 푉 = − 푖 + + 푉 (풓 ) 푒푒 푒푥푡 2푚 2 풓 − 풓 푒푥푡 푖 푖 푖≠푗 푖 푗 푖=1 The ground-state energy:

퐸0 Ψ = Ψ ∣ 푇 ∣ Ψ + Ψ ∣ 푉 푒푒 ∣ Ψ + Ψ ∣ 푉 푒푥푡 ∣ Ψ The ground-state density 푁

푛 = Ψ ∣ 푛 ∣ Ψ = Ψ ∣ δ 퐫 − 퐫퐢 ∣ Ψ 푖=1 ? 퐸0 = 퐸0 Ψ 퐸0 = 퐸0 푛

If one can avoid complex many-body wave function, and express the ground- state energy as a function of the electron density, then the problem will be significantly simplified. Early-day DFT: Thomas-Fermi approximation

퐸 Ψ = Ψ ∣ 푇 ∣ Ψ + Ψ ∣ 푉 푒푒 ∣ Ψ + Ψ ∣ 푉 푒푥푡 ∣ Ψ

3 3 0 Kinetic energy: 푇 = Ψ ∣ 푇 ∣ Ψ = 푑 푟푛 퐫 ϵ푘 푛 퐫 ≈ 푑 푟푛 퐫 ϵ푘 푛 퐫

ϵ푘 푛 풓 : The kinetic energy density at point r, which depends on the electron density in the whole space

0 ϵ푘 푛 풓 : kinetic energy density of non-interacting HES, which only depends on the electron density at r: 푛 풓

0 ϵ푘 푛 퐫 ϵ푘 푛 퐫 ϵ푘 푛 퐫 Local approximation Non-interacting approximation The Thomas-Fermi approximation (1927,1928)

퐸 Ψ = Ψ ∣ 푇 ∣ Ψ + Ψ ∣ 푉 푒푒 ∣ Ψ + Ψ ∣ 푉 e푥푡 ∣ Ψ

Electron-electron and electron-nuclear interaction energy:

푉ee = Ψ ∣ 푉 ee ∣ Ψ = 퐸Hartree 푛 퐫 + 푈XC 푛 퐫

3 퐸푒푥푡 = Ψ ∣ 푉 푒푥푡 ∣ Ψ = 푑 푟푛 퐫 푣푒푥푡 퐫

In summary:

1 푛 풓 푛 풓′ 퐸 푛 퐫 = 푑3 푟푛 풓 ϵ0 푛 풓 + 푑3 푟푑3푟′ TF 푘 2 ∣ 풓 − 풓′ ∣

3 + 푑 푟푛 풓 푣푒푥푡 풓 The Thomas-Fermi equation

Under the Thomas-Fermi approximation, the ground-state energy of an electronic system is an explicit functional of the electron density!

3 3 3 ϵ0 푛 = ϵ (푛) = 3π 2 3푛2 3 = 퐶 푛2 3, 퐶 = 3π 2 3 푘 5 퐹 10푚 푘 푘 10푚

1 푛 풓 푛 풓′ 퐸 푛 풓 = 퐶 푑3 푟푛5 3 풓 + 푑3 푟푑3푟′ + 푑3 푟푛 풓 푣 풓 TF 푘 2 ∣ 풓 − 풓′ ∣ 푒푥푡

Minimizing the ground-state energy with respect to the density, under the constraint 푑3 푟푛 퐫 = 푁

3 δ 퐸TF 푛 퐫 − μ 푑 푟푛 퐫 − 푁 = 0 The Thomas-Fermi equation

1 푛 풓 푛 풓′ 퐸 푛 풓 = 퐶 푑3 푟푛5 3 풓 + 푑3 푟푑3푟′ + 푑3 푟푛 풓 푣 풓 TF 푘 2 ∣ 풓 − 풓′ ∣ 푒푥푡

3 δ 퐸TF 푛 퐫 − μ 푑 푟푛 퐫 − 푁 = 0

5 3C 푛2 3 퐫 + 푣 퐫 − μ = 0 푘 푒푓푓 Solving this equation, one obtains the ground-state electron density 푛 퐫′ 푣 퐫 = 푑3 푟′ + 푣 퐫 and the ground-state energy 푒푓푓 ∣ 퐫 − 퐫′ ∣ 푒푥푡

푉퐻푎푟푡푟푒푒 퐫 Dirac's exchange correction (1930)

Further accounting for the exchange contribution:

0 퐸TFD 푛 풓 = 퐸TF 푛 풓 + 퐸푥 푛 풓 The exact exchange energy of an inhomogeneous systems cannot be expressed as an explicit functional of the electron, and here we again adopt the local approximation.

0 3 0 3 4 3 퐸푥 푛 풓 = 푑 푟푛 풓 ϵ푥 푛 풓 = 퐶푥 푑 푟푛 풓

1 3 3 퐶 = − 3 4 ≈ −0.739 푥 π

3 δ 퐸TFD 푛 풓 − μ 푑 푟푛 풓 − 푁 = 0 − 3 π 1 3푛1 3 퐫 2 3 5 3C푘 푛 풓 + 푣푒푓푓 풓 − μ = 0

푣푒푓푓 풓 = 푣퐻푎푟푡푟푒푒 풓 + 푣푒푥푡 풓 + 푣푥 풓 Deficiencies of the Thomas-Fermi approximation

● The charge density at the nuclear position is infinite

● The charge density away from the nucleus shows a 1/r6 decay, instead of the correct exponential decay.

● Atoms have no shell structure, and hence the periodic table of elements cannot be obtained.

● Atoms cannot bind and form molecules and solids. Slater's Xα method (1951)

• Hartree-Fock-Slater equation: （Initially as an approximation to the Hartree-Fock equation）

훻2 − + 푣 퐫 + 푣 퐫 + 푣 퐫 ϕ 퐫 = ϵ ϕ 퐫 2m 푒푥푡 퐻푎푟푡푟푒푒 푋α 푙 푙 푙

3 푣푋ϕ푙 퐫 = 푑 푟푣푋 퐫, 퐫′ ϕ푙 퐫′ ≈ 푣푋α 퐫 ϕ푙 퐫

1 3 3 푣 퐫 = −α 3 2 푛1 3 퐫 푋α π John C. Slater α=1  Slater local exchange α=2/3 Kohn-Sham local exchange Slater's Xα method (1951)

• Hartree-Fock-Slater equation:

훻2 − + 푣 퐫 + 푣 퐫 + 푣 퐫 ϕ 퐫 = ϵ ϕ 퐫 2m 푒푥푡 퐻푎푟푡푟푒푒 푋α 푙 푙 푙 3 푣 퐫 = −α 3 2 푛1 3 퐫 푋α π

Slater Xα (α=2/3) method: the ground-state energy

푁 훻2 퐸 = ϕ ∣ − ∣ ϕ + 퐸 푛 + 퐸 푛 + 퐸0 푛 HFS 푙 2m 푙 퐻푎푟푡푟푒푒 푒푥푡 푥 푖=1

Apart from the kinetic energy, the total energy form of Slater Xα (α=2/3) is the same as the Thomas-Fermi-Dirac method. Slater's Xα method (1950's-1970's)

Slater Xα (2/3<=α<=1 as an adjustable parameter) method has been widely used for computations of atoms, molecules, and solids, and was de facto the main method for calculating realistic systems during 1950's – 1970's. Historically Slater's Xα method was viewed as an approximation to the Hartree-Fock method. As a matter of fact, Slater Xα can be better seen as an approximate density functional theory. Kohn's thought on the Thomas-Fermi theory

Walter Kohn, Nobel lecture (1999) Kohn's thought on Thomas-Fermi (TH) theory

3 5 3C 푛2 3 풓 + 푉 풓 − μ = 0, 퐶 = 3π 2 3 푘 푒푓푓 푘 10푚

1 μ − 푉 풓 = 3π푛 풓 2 3 푒푓푓 2m

푛 풓′ 푉 풓 = 푉 풓 − 푑3 푟′ 푒푥푡 푒푓푓 ∣ 풓 − 풓′ ∣

Walter Kohn

Therefore, in Thomas-Fermi theory, (apart from a constant,) the external potential is uniquely determined by the electron density. Furthermore, because 푁 = 푑 3 푟 ′ 푛 퐫 ′ , thus for given ground-state electron density, the Hamiltonian of an electronic system is completely determined. Kohn's thought on Thomas-Fermi (TH) theory

This raised a general question in my mind: Is a complete, exact description of ground-state electronic structure in terms of n(r) possible in principle? A key question was whether the density n(r) completely characterized the system. ---- Walter Kohn, before 1963 Hohenberg-Kohn theorem I

푁 푁 푁 훻2 1 퐻 = − 푖 + 푉 풓 , 풓 + 푣 풓 2m 2 푖 풋 푒푥푡 푖 푖 푖≠푗 푖 Theorem I:

For any system of interacting particles in an external potential 푣푒푥푡(풓), the potential is determined uniquely, except for a constant, by the ground-state particle density 푛0 풓 .

P. Hohenberg & W. Kohn, (1964)

Phys. Rev. 136, B864 (1964). Proof of the Hohenberg-Kohn Theorem I

“Reductio ad absurdum” (reduction to absurd,归谬法） 퐴 퐵 Assume there exists two external potential 푣푒푥푡 푟 , 푣푒푥푡 푟 , which differ by more than a constant, but yield the same ground-state electron density 푛0 풓 , 퐴 퐴 퐴 Non-degenerate 퐸 = Ψ퐴 ∣ 퐻 ∣ Ψ퐴 < Ψ퐵 ∣ 퐻 ∣ Ψ퐵 ground state

퐴 퐵 3 퐴 퐵 Ψ퐵 ∣ 퐻 ∣ Ψ퐵 = 퐸 + 푑 푛 퐫 푣푒푥푡 퐫 − 푣푒푥푡 퐫

퐴 퐵 3 퐴 퐵 퐸 < 퐸 + 푑 푛 퐫 푣푒푥푡 퐫 − 푣푒푥푡 퐫

퐵 퐴 3 퐵 퐴 퐸 < 퐸 + 푑 푛 퐫 푣푒푥푡 퐫 − 푣푒푥푡 퐫 Summing up the two equations This is not possible, and the 퐸퐴 + 퐸퐵 < 퐸퐴 + 퐸퐵 original assumption is wrong. Deduction of the Hohenberg-Kohn theorem I

The Hamiltonian of an interacting electron systems is fully determined by its ground-state density, apart from a constant. Therefore, both the ground state and excited states, and hence all properties of the system are completely determined by its ground- state density. Hohenberg-Kohn (HK) theorem II

A universal functional for the energy E in terms of the density 푛(풓), 퐸 = 퐸[푛], can be defined, valid for any external potential 푣푒푥푡 풓 . The exact ground-state energy 퐸0 of the system is the global minimum value of this functional, and the density 푛(풓) that minimizes the functional is the exact ground state density 푛0 풓 . Proof of Hohenberg-Kohn Theorem II

Assume 푛(풓) is a “V-representable” particle density, i.e., 푛(풓) is a ground-state particle density associated with a physically realizable external potential 푣푒푥푡(풓).

퐸HK 푛 풓 = Ψ0 푛 퐫 ∣ 퐻 ∣ Ψ0 푛 풓 3 = Ψ0 푛 풓 ∣ 푇 + 푉 푒푒 ∣ Ψ0 푛 풓 + 푑 푟푣푒푥푡 풓 푛 풓

3 = 퐹HK 푛 풓 + 푑 푟푣푒푥푡 풓 푛 풓

퐹HK 푛 퐫 : does not depend the external potential and is hence called “universal functional”. Proof of the Hohenberg-Kohn Theorem II

퐸HK 푛 퐫 = Ψ0 푛 퐫 ∣ 퐻 ∣ Ψ0 푛 퐫 3 = 퐹HK 푛 퐫 + 푑 푟푣푒푥푡 퐫 푛 퐫

Assume the external potential of current system is 푣푒푥푡(풓), and ′ its ground-state density is 푛0(풓). For any other density 푛 풓 ≠ 푛0 풓 its ground-state wave function is

퐸HK 푛0 풓 = Ψ0 푛0 풓 ∣ 퐻 ∣ Ψ0 푛0 풓 < Ψ′ 푛′ 풓 ∣ 퐻 ∣ Ψ′ 푛′ 풓 = 퐸HK 푛′ 풓