Lecture on First-principles Computation (2): The Hartree-Fock theory and the homogeneous electron gas
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, 2016.9.18 Recall: the Hartree approximation
푁 푁 2 2 2 훻 1 휑푛(풓) 휑푚(풓′) 퐸 = 휑 − 휑 + 푑풓푑풓′ + 푣 풓 푛 풓 푑풓 푛 2푚 푛 2 풓 − 풓′ 푒푥푡 푛=1 푚≠푛 푁 푁 2 ∗ 푛 풓 = |휑푛 (풓)| = 휑푛(풓)휑푛(풓) 푛=1 푛=1
∗ Minimizing 퐸 = 퐸 휑푛 with respect to 휑푛(풓) under the constrain that 휑푛 휑푛 = 1,
훿 ∗ 퐸 휑푛 − 휖푛 휑푛 휑푛 − 1 = 0 훿휑푛 one obtains the Hartree equation:
푁 훻2 휑 (풓′) 2 − + 푑풓′ 푚 + 푣 풓 휑 풓 = 휖 휑 풓 2푚 풓 − 풓′ 푒푥푡 푛 푛 푛 푚≠푛 The Hartree equation
푁 훻2 휑 (풓′) 2 − + 푑풓′ 푚 + 푣 풓 휑 풓 = 휖 휑 풓 2푚 풓 − 풓′ 푒푥푡 푛 푛 푛 푚≠푛 푁 2 푛 풓 = |휑푛 (풓)| 푛=1 Thus the Hartree equation can also be expressed as:
훻2 푛 풓′ − 휑 (풓′) 2 − + 푑풓′ 푛 + 푣 풓 휑 풓 = 휖 휑 풓 2푚 풓 − 풓′ 푒푥푡 푛 푛 푛
The original Hartree approximation excludes the self-interaction term Nowadays, the Hartree approximation is understood as
훻2 푛 풓′ − + 푑풓′ + 푣 풓 휑 풓 = 휖 휑 풓 2푚 풓 − 풓′ 푒푥푡 푛 푛 푛 The anti-symmetrized wave function
The Hartree approximation ignores the Fermi-Dirac statistics of the electrons, and hence is intrinsically problematic. The Hartree-Fock approximation, instead, assumes a single-Slater determinant form of the wave function,
휑1(푥1) 휑2(푥1) ⋯ 휑푁(푥1) 1 휑1(푥2) 휑2(푥2) ⋯ 휑푁(푥2) Φ 푥1, 푥2, ⋯ , 푥푁 = 푥푖 = (풓푖, 휎푖) 푁! … … … ⋯
휑1(푥푁) 휑2(푥푁) ⋯ 휑푁(푥푁)
Electrons are fermions with spin ½ . The full electron coordinate 푥 contains a spatial coordinate 풓 and a spin coordinate 휎. The complete specification of an single-electron state is called spin orbital: ↑ Spin up 1 0 휎 = 훼↑ = , 훼↓ = 휑푛 풓푖, 휎푖 = 휑푛,휎푖 풓푖 훼휎푖(i) 푖 0 1 ↓ Spin down The Hartree-Fock approximation
With the single Slater determinant trial wavefunction, the total energy 퐸 = Φ 퐻 푒 Φ is given by
표푐푐. 훻2 퐸 휑 (풓) = 휑 − 휑 + 푣 풓 푛 풓 푑풓 푙 푛,휎 2푚 푛,휎 푒푥푡 푛,휎 표푐푐. 2 2 1 휑 (풓) 휑 (풓′) + 푑풓푑풓′ 푛,휎 푚,휎′ The Hartree term 2 풓 − 풓′ 푚,푛,휎,휎′ 표푐푐. ∗ ∗ 1 휑푛,휎(풓)휑푛,휎(풓′)휑푚,휎(풓)휑푚,휎(풓′) − 푑풓푑풓′ The exchange term 2 풓 − 풓′ 푚,푛,휎
표푐푐. 2 푛 풓 = 푛휎 풓 = 휑푛,휎(풓) 휎 푛,휎 Please note that 푚 = 푛, 휎 = 휎′ term in included in both the Hartree and the exchange terms, and hence cancel each other out. The Hartree-Fock approximation
∗ Again minimizing 퐸 = 퐸 휑푛 with respect to 휑푛 under the constrain that 휑푛 휑푛 = 1,
훿 ∗ 퐸 휑푛 − 휖푛 휑푛 휑푛 − 1 = 0 훿휑푛 one obtains the Hartree-Fock equation:
표푐푐. 훻2 휑∗ (풓′)휑 (풓′) − + 푣 풓 + 푑풓′ 푚,휎′ 푚,휎′ 휑 풓 2푚 푒푥푡 풓 − 풓′ 푛,휎 푚,휎′ 표푐푐. 휑∗ (풓′)휑 (풓′) − 푑풓′ 푚,휎 푛,휎 휑 풓 = 휖 휑 풓 풓 − 풓′ 푚,휎 푛 푛,휎 푚 The Hartree-Fock approximation
∗ Again minimizing 퐸 = 퐸 휑푛 with respect to 휑푛 under the constrain that 휑푛 휑푛 = 1,
훿 ∗ 퐸 휑푛 − 휖푛 휑푛 휑푛 − 1 = 0 훿휑푛
one obtains the Hartree-Fock equation:
표푐푐. 훻2 휑∗ (풓′)휑 (풓′) − + 푣 풓 + 푑풓′ 푚,휎′ 푚,휎′ 휑 풓 2푚 푒푥푡 풓 − 풓′ 푛,휎 푚,휎′ 표푐푐. 휑∗ (풓′)휑 (풓′) − 푑풓′ 푚,휎 푛,휎 휑 풓 = 휖 휑 풓 풓 − 풓′ 푚,휎 푛 푛,휎 푚
The second term can be seen as obtained by exchanging the orbital index 푚and 푛 of the first term. The Hartree-Fock equation
The Hartree-Fock equation can also be written as
⋯ 훻2 푛(풓′) − + 푣 풓 + 푑풓′ 휑 풓 2푚 푒푥푡 풓 − 풓′ 푛,휎 표푐푐. 휑 (풓′)휑∗ (풓′) + 푑풓′ 푛,휎 푚,휎 휑 풓 = 휖 휑 풓 풓 − 풓′ 푚,휎 푛 푛,휎 푚
표푐푐. 2 푛 풓 = 휑푛,휎(풓) 푛,휎 The Hartree-Fock equations are a set of non-linear equations that need to be solved self-consistently. The resultant set of eigenstates are called Hartree-Fock orbitals. The Koopman’s theorem
The Hartree-Fock orbital energy 휖푛 can be interpreted as the energy cost to remove an electron from occupied orbitals, or the energy gain to add an electron to unoccupied orbitals.
Tjalling Koopman Nobel Laureate in economics (1975) • The relaxation effect of orbitals upon adding/removing electrons is neglected.
• The electron correlation effect is neglected. The Hartree-Fock ground-state energy
|ΦHF =det{휑푙,휎}
푒 퐸 휑푙(풓) = ΦHF 퐻 ΦHF = 푇[휑푙] + 푉푒푥푡[푛] + J[n] + K[휑푙]
표푐푐. 훻2 푇 = 휑 − 휑 The kinetic energy 푛,휎 2푚 푛,휎 푛,휎 The external potential energy 푉푒푥푡 = 푣푒푥푡 풓 푛 풓 푑풓
1 푛 풓 푛(풓′) J = 푑풓푑풓′ The Hartree energy 2 풓 − 풓′ 표푐푐. 1 휑∗ (풓)휑 (풓′)휑 (풓)휑∗ (풓′) 퐾 = − 푑풓푑풓′ 푛,휎 푛,휎 푚,휎 푚,휎 2 풓 − 풓′ 푚,푛,휎 The exchange energy The correlation energy
The correlation energy of an electronic system is defined as the difference between the exact ground-state energy and the Hartree- Fock ground-state energy.
푒 푒 퐸푐 = Ψ0 퐻 Ψ0 − ΦHF 퐻 ΦHF
|Ψ0 : the exact ground state
|ΦHF : The Hartree-Fock ground state
By definition, the correlation energy is negative 퐸푐 < 0, and the Hartree-Fock approximation does not include the correlation effect. Summary of the Hartree-Fock theory
● Variational: the Hartree-Fock total energy is above the exact ground-state energy.
● The Hartree-Fock approximtion neglects the electron correlation effect.
● No single-particle self-interaction error.
● The Hartree-Fock single-particle energies has physical meanings.
● The Hartree-Fock theory is still important today. It is an important ingredient of hybrid density functional approximations. The homogeneous electron gas model
The homogeneous electron gas (HES) model is also called the “jellium” model, where the contribution of nuclear ions is treated as an uniform positive charge background.
As a first approximation, the 푞 = 0 component of the electron- electron interaction cancels the attraction between electrons and the positive charge background, and the residual (푞 ≠ 0) components are neglected. This is called the non-interacting HES model.
The Hamiltonian of the non-interacting HES is given by
푁 푁 2 퐻퐻퐸퐺 = ℎ 𝑖 = − 훻푖 2 푖=1 푖=1 Particles moving in a cubic box
The eigenvalue problem: 1 ℎ 휑 (풓) = − 훻2휑 (풓) = 휖 휑 (풓) 푛 2m 푛 푛 푛 It is convenient to choose Born von-Karmen (BvK) periodic boundary condition: 휑 푥, 푦, 푧 = 휑 푥 + 퐿, 푦, 푧 Torus：periodic in φ 푥, 푦, 푧 = 휑 푥, 푦 + 퐿, 푧 two dimensions φ 푥, 푦, 푧 = 휑 푥, 푦, 푧 + 퐿
Volume of the box: 푉 = 퐿3 The plane wave vector k is the 1 푖퐤퐫 휑 퐫 = 푒 = 휑푘 퐫 푉 quantum number here 푘2 ℏ2푘2 휖 = = 휖 , here ℏ = 1 2푚 푘 2푚 Consequence of the BvK boundary condition
휑 푥, 푦, 푧 = 휑 푥 + 퐿, 푦, 푧 3 φ 푥, 푦, 푧 = 휑 푥, 푦 + 퐿, 푧 2 φ 푥, 푦, 푧 = 휑 푥, 푦, 푧 + 퐿 1 0 -1 The BvK boundary condition implies: -2 푒푖푘푥퐿 = 1, 푒푖푘푦퐿 = 1, 푒푖푘푧퐿 = 1 -3 -3 -2 -1 0 1 2 3 4 Therefore
2π푛 2π푛푦 2π푛 푘 = 푥 , 푘 = , 푘 = 푧 , 푥 퐿 푦 퐿 푧 퐿 where 푛푥, 푛푦, 푛푧 are integer numbers.
The wave vectors of plane waves are quantized, and the allowed wave vectors (states) are uniformly distributed in the reciprocal space. Each state occupies a volume of 2휋/퐿 3 = (2휋)3/푉. The ab-initio electronic many-body Hamiltonian
푁 푁 푁 푒 훻2 1 푒 푒2 푒 퐻 푒 = − 푖 + + 푉푒푥푡 (풓 ) 2푚 2 풓 − 풓 푹푖,푍푖 푖 푖=1 푖≠푗 푖 푗 푖
= 푇 + 푉 푒푒 + 푉 푒푥푡
The ground state and excited states of the electronic Hamiltonian determine most properties of materials. The solution of the electronic Hamiltonian is often called the “electronic structure”.
Despite the considerable simplification that the Born-Oppenheimer approximation brings, the computational challenge remains. The exact solution of the interacting many-body Hamiltonian is not accessible even numerically for O(N)>10. Further simplification/approximation is a must! The Fermi vector and Fermi energy
The electronic states are occupied according to the “aufbau principle”, i.e., the energy levels are filled 푘퐹 from low to high, until certain highest energy, called 3 the Fermi energy. 2 1 Suppose there are 푁 electrons in system, we have 0 푘퐹 -1 푉 푁 = 2 1 = 2 푑3 푘 1 -2 8π3 -3 ∣퐤∣<푘퐹 0 푉 4π 푘3 = 2 푘3 = 푉 퐹 -3 -2 -1 0 1 2 3 4 8π3 3 퐹 3π2
Thus we obtain an important relation: Taking 푁, 푉 → ∞, while keeping 1/3 푁 푁 the density 푛 = fixed. The Fermi vector: 푘 = 3휋2 = 3휋2푛 1/3 푉 퐹 푉
2 2 2/3 The Fermi energy: 퐸퐹 = 푘퐹/2m = 3휋 푛 /2m The Fermi temperature and Fermi surface
2 1 3 Fermi vector: 푘퐹 = 3π 푛 푘2 3π2푛 2 3 Fermi energy: ϵ = 퐹 = 퐹 2m 2m
Fermi temperature: 푇퐹 = ϵ퐹 κ퐵
The Fermi temperatures of alkali metals are of Enrico Fermi 푂 104 kelvin.
The solution of equation 휖 푘 = 퐸퐹 defines sections in the k space; this is called the Fermi surface. The Fermi surface of a 3D HES is a sphere surface. Basic properties of HES
● The characteristic energy scale of HES is its Fermi energy 퐸퐹 or Fermi temperature 푘퐹 ，and this is uniquely determined by its electron density (or valence electron density for real metals). 4 ● The Fermi temperature of normal metals is ~10 kelvin or higher; this is much higher than the room temperature (273 kelvin). Therefore, under the room temperature, only electrons in the neighborhood of the Fermi surface will become active, and contribute to the thermodynamics of the system.
−1 푓 ϵ = 1 + 푒 ϵ−μ κ푇
푓𝑖푛𝑖푡푒 푇, μ ≈ κ푇퐹 푇 = 0, μ = κ푇퐹
κ 푇 퐵 퐹 κ퐵푇퐹 Fermi-Dirac distribution function under different temperatures Density of states (DOS)
• Suppose we want to get the energy of HES： 2푉 ∞ 퐸 = 푓 ϵ 풌 ϵ 풌 = 푑3 푘푓 ϵ 풌 ϵ 풌 8π3 풌 0 ∞ ∞ 푉 = 푑푘 푘2 푓 ϵ 풌 ϵ 풌 = 푑 ϵ퐷 ϵ 푓 ϵ ϵ 휋2 0 0 Density of states: 퐷 ϵ 푑ϵ: the number of states between the energy interval ϵ and 푑ϵ. 푚푉 푘 푚 퐷 ϵ = 2mϵ，with ϵ ≥ 0 푑푘 = 푑ϵ = 푑ϵ π2 푚 2ϵ
(2휋)3 1 In general, 퐷 휖 = 푑풌 훿(휖 − 휖(풌)) 푉 훻풌휖 풌 DOS of 3-dimensional HES
푚푉 퐷 ϵ = 2mϵ π2
The DOS around the Fermi level determines many properties of the electronic system. Homework I
Please try to derive the Hartree-Fock equations from the single Slater determinant assumption and the variational principle.