Microcanonical Ensemble
P = 1 / W P=1 / W
P = 1 / W
Isolated systems with constant N, V, E
Canonical Ensemble
P ( E ) = e ( F − E ) / K T P(E)=e^{(F-E) / K T}
P ( E ) = e ( F − E ) / K T
Systems in contact with a heat bath (constant N , V , T \mathrm{N}, \mathrm{V}, \mathrm{T} N , V , T
Grand Canonical Ensemble
P ( E , N ) = e ( Ω + μ N − E ) / K T P(E, N)=e^{(\Omega+\mu N-E) / K T}
P ( E , N ) = e ( Ω + μ N − E ) / K T
Systems that have exchange of particles and energies with environment (constant V , T \mathrm{V}, \mathrm{T} V , T μ \mu μ
Equilibrium
Average values of physical properties do not vary with time and independent from momentum of particles.
Nonequilibrium processes
Time-dependent processes
Steady-state Transport processes
Equilibrium State
Associated with a particular temperature T
The system has a universal chemical potential
Average values of physical measurables do not vary with time
Steady State
Average values of physical measurables do not vary with time
Can be in equilibrium or nonequilibrium
Materials: Ions + Electrons
Time-dependent (TD) processes
TD Ion motions: Structure phase changes; Chemicalreactions.
TD Electron properties: Electronic phase changes; Electronic excitations.
Steady-state Transport processes
DC electric/spin current: Transport of electrons or ions.
Thermoelectric effects: Electron and heat transport.
The central issue for understanding materials
H ^ ψ = E ψ ⟹ A ˉ = ⟨ ψ ∣ A ^ ∣ ψ ⟩ \hat{H} \psi=E \psi \quad \Longrightarrow \bar{A}=\langle\psi| \hat{A}|\psi\rangle
H ^ ψ = E ψ ⟹ A ˉ = ⟨ ψ ∣ A ^ ∣ ψ ⟩
H ^ = − h 2 2 m ∑ i ∇ i 2 − 1 κ ∑ m ; i = 1 , N e q m r i m + 1 κ ∑ i ; j > i e 2 r i j \hat{H}=-\frac{\mathrm{h}^2}{2 m} \sum_i \nabla_i^2-\frac{1}{\kappa} \sum_{m ; i=1, N} \frac{e q_m}{r_{i m}}+\frac{1}{\kappa} \sum_{i ; j>i} \frac{e^2}{r_{i j}}
H ^ = − 2 m h 2 i ∑ ∇ i 2 − κ 1 m ; i = 1 , N ∑ r i m e q m + κ 1 i ; j > i ∑ r i j e 2
1st Hohenberg-Kohn (HK) theorem (1965) ρ ( r ) \rho(r) ρ ( r )
⟹ A ˉ = A [ ρ ( r ) ] \Longrightarrow \bar{A}=A[\rho(r)]
⟹ A ˉ = A [ ρ ( r ) ]
2nd Hohenberg-Kohn (HK) theorem ρ ~ ( r ) \widetilde{\rho}(r) ρ ( r ) ρ ~ ( r ) ≥ 0 , ∫ ρ ~ ( r ) d r = N \widetilde{\rho}(r) \geq 0, \int \widetilde{\rho}(r) d r=N ρ ( r ) ≥ 0 , ∫ ρ ( r ) d r = N E 0 E_0 E 0 E ~ [ ρ ~ ] \widetilde{E}[\widetilde{\rho}] E [ ρ ]
DFT Variational Principle
E [ ρ ] = T ^ [ ρ ] + V ^ e x t [ ρ ] + V ^ e e [ ρ ] = V ^ e x t [ ρ ] + F H K [ ρ ] E [ ρ ] → Min { ρ V ^ exp [ ρ ] + F H K [ ρ ] } \begin{gathered}
E[\rho]=\hat{T}[\rho]+\hat{V}_{e x t}[\rho]+\hat{V}_{e e}[\rho]=\hat{V}_{e x t}[\rho]+F_{H K}[\rho] \\
E[\rho] \rightarrow \operatorname{Min} \{ _\rho \hat{V}_{\exp }[\rho]+F_{H K}[\rho] \}
\end{gathered}
E [ ρ ] = T ^ [ ρ ] + V ^ e x t [ ρ ] + V ^ e e [ ρ ] = V ^ e x t [ ρ ] + F H K [ ρ ] E [ ρ ] → M i n { ρ V ^ e x p [ ρ ] + F H K [ ρ ] }
Assumption: A non-interacting reference system
ψ ~ G S = 1 N ! det ( φ 1 φ 2 φ 3 … φ N ) ρ ~ reference = ∣ ψ ~ G S ( r ) ∣ 2 ≡ ρ true G S \widetilde{\psi}_{G S}=\frac{1}{\sqrt{N!}} \operatorname{det}\left(\varphi_1 \varphi_2 \varphi_3 \ldots \varphi_N\right) \quad \quad \widetilde{\rho}_{\text {reference }}=\left|\widetilde{\psi}_{G S}(r)\right|^2 \equiv \rho_{\text {true } G S}
ψ G S = N ! 1 d e t ( φ 1 φ 2 φ 3 … φ N ) ρ reference = ∣ ∣ ∣ ∣ ψ G S ( r ) ∣ ∣ ∣ ∣ 2 ≡ ρ true G S
Approximate the kinetic energy of the true system by the kinetic energy of the reference system.
T [ ρ ] ≈ T ~ [ ρ ] ≡ ∑ i ⟨ φ i ∣ − 1 2 ∇ i 2 ∣ φ i ⟩ T[\rho] \approx \widetilde{T}[\rho] \equiv \sum_i\left\langle\varphi_i\right|-\frac{1}{2} \nabla_i^2\left|\varphi_i\right\rangle
T [ ρ ] ≈ T [ ρ ] ≡ i ∑ ⟨ φ i ∣ − 2 1 ∇ i 2 ∣ φ i ⟩
Search the non-interacting reference system
E [ ρ ] = ∑ i ⟨ φ i ∣ − 1 2 ∇ i 2 ∣ φ i ⟩ + ∫ ρ ( r ) V e x t ( r ) d r + J [ ρ ] + E x c [ ρ ] ∫ φ i φ j d r = δ i j ρ ( r ) = ∑ i ∣ φ i ( r ) ∣ 2 Ω ( { φ i } ) ≡ E [ ρ ] − ∑ i , j ε i j ∫ φ i φ j d r \begin{aligned}
& E[\rho]=\sum_i\left\langle\varphi_i\right|-\frac{1}{2} \nabla_i^2\left|\varphi_i\right\rangle+\int \rho(r) V_{e x t}(r) d r+J[\rho]+E_{x c}[\rho] \\
& \int \varphi_i \varphi_j d r=\delta_{i j} \quad \rho(r)=\sum_i\left|\varphi_i(r)\right|^2 \\
& \Omega\left(\left\{\varphi_i\right\}\right) \equiv E[\rho]-\sum_{i, j} \varepsilon_{i j} \int \varphi_i \varphi_j d r
\end{aligned}
E [ ρ ] = i ∑ ⟨ φ i ∣ − 2 1 ∇ i 2 ∣ φ i ⟩ + ∫ ρ ( r ) V e x t ( r ) d r + J [ ρ ] + E x c [ ρ ] ∫ φ i φ j d r = δ i j ρ ( r ) = i ∑ ∣ φ i ( r ) ∣ 2 Ω ( { φ i } ) ≡ E [ ρ ] − i , j ∑ ε i j ∫ φ i φ j d r
δ Ω ( { φ i } ) δ { φ i } = 0 \frac{\delta \Omega\left(\left\{\varphi_i\right\}\right)}{\delta\left\{\varphi_i\right\}}=0
δ { φ i } δ Ω ( { φ i } ) = 0
Kohn-Sham equation
( − 1 2 ∇ 2 + V eff ) φ i = ε i φ i \left(-\frac{1}{2} \nabla^2+V_{\text {eff }}\right) \varphi_i=\varepsilon_i \varphi_i
( − 2 1 ∇ 2 + V eff ) φ i = ε i φ i
V e f f ( r ) = V e x t ( r ) + ∫ ρ ( r ′ ) ∣ r − r ′ ∣ d r ′ + V x c ( r ) V_{e f f}(r)=V_{e x t}(r)+\int \frac{\rho\left(r^{\prime}\right)}{\left|r-r^{\prime}\right|} d r^{\prime}+V_{x c}(r)
V e f f ( r ) = V e x t ( r ) + ∫ ∣ r − r ′ ∣ ρ ( r ′ ) d r ′ + V x c ( r )
V x c ( r ) = δ E x c [ ρ ] δ ρ V_{x c}(r)=\frac{\delta E_{x c}[\rho]}{\delta \rho} V x c ( r ) = δ ρ δ E x c [ ρ ]
Understanding periodic systems is the basis of modelling materials in equilibrium.
For a periodic system, H ^ ( x + n a ) = H ^ ( x ) \widehat{H}(x+n a)=\widehat{H}(x) H ( x + n a ) = H ( x ) T ^ ( n a ) φ ( x ) = φ ( x + n a ) \hat{T}(n a) \varphi(x)=\varphi(x+n a) T ^ ( n a ) φ ( x ) = φ ( x + n a ) T ^ \widehat{T} T H ^ \widehat{H} H [ T ^ , H ^ ] = 0 [\widehat{T}, \widehat{H}]=0 [ T , H ] = 0 T ^ \widehat{T} T H ^ \widehat{H} H
H ^ ψ ( x ) = E ψ ( x ) T ^ ( n a ) ψ ( x ) = T ψ ( x ) T ^ ( n a ) ψ ( x ) = ψ ( x + n a ) T ψ ( x ) = ψ ( x + n a ) ∣ T ψ ( x ) ∣ 2 = ∣ ψ ( x + n a ) ∣ 2 ⟹ T = e i α ( n a ) \begin{gathered}
\widehat{H} \psi(x)=E \psi(x) \\
\widehat{T}(n a) \psi(x)=T \psi(x) \quad \widehat{T}(n a) \psi(x)=\psi(x+n a) \\
\mathrm{T} \psi(x)=\psi(x+n a) \\
|\mathrm{T} \psi(x)|^2=|\psi(x+n a)|^2 \Longrightarrow T=e^{i \alpha(n a)}
\end{gathered}
H ψ ( x ) = E ψ ( x ) T ( n a ) ψ ( x ) = T ψ ( x ) T ( n a ) ψ ( x ) = ψ ( x + n a ) T ψ ( x ) = ψ ( x + n a ) ∣ T ψ ( x ) ∣ 2 = ∣ ψ ( x + n a ) ∣ 2 ⟹ T = e i α ( n a )
Bloch Theorem :
ψ ( x ) = e i k x ϕ ( x ) \quad \boldsymbol{\psi}(\boldsymbol{x})=\boldsymbol{e}^{\boldsymbol{i} \boldsymbol{k} \boldsymbol{x}} \boldsymbol{\phi}(\boldsymbol{x})
ψ ( x ) = e i k x ϕ ( x )
ϕ ( x + n a ) = ϕ ( x ) \phi(x+n a)=\phi(x)
ϕ ( x + n a ) = ϕ ( x )
