Exchange and Correlation
- date:
3 Nov 2019
What is the idea?
The task is to find the exchange term
In local density approximation (LDA), the exchange term is from Hatree Fock Theory, and the correlation term is from Quantum Monte Carlo
Thus first review Hatree Fock theory
Electron gas
Non-Interacting
\[\begin{split}\begin{align*}
& r_s:\text{radius of sphere with one }e^- \quad 4\pi {r_s^3\over 3}={1\over n}\\
& \sigma:\text{spin}\\
& \text{planewave eigenstates}:e^{ikr}\\
& \text{Fermi wavevector}:k_F^\sigma =\max k^\sigma\\
& k^\sigma \text{ means dependence on }\sigma \\
& \text{Average Energy}: {T_\text{tot}^\sigma\over N}={\int dk\; k^2/2\over \int dk }={3\over 10}(k_F^\sigma)^2\\
\end{align*}\end{split}\]
Interacting
\[\begin{split}\begin{align*}
& H= \sum_i {-\nabla^2\over2}
\underbrace{+{1\over 2} \sum_{i\ne j}{1\over |r_i-r_j|}}_{V_{\text{e-e}}}
\underbrace{-{1\over2}\int drdr'{n(r)n(r')\over |r-r'|}}_{V_{\text{e-ion}}} \\
& \text{with assumption }n_{\text{ion}}(r)=n(r) \\
\end{align*}\end{split}\]
Hartree Fock Equation
only consider exchange
interacting electron gas can be solved analytically at HF level
hamiltonian
\[\begin{split}\begin{align*}
& H=\sum_i^{N_e}{-\nabla_i^2\over 2}-\sum_i^{N_e}\sum_I^{N_n}{Z_I\over |r_i-R_I|}
+{1\over 2}\sum_i^{N_e}\sum_{j\ne i}^{N_e} {1\over |r_i-r_j|}+{1\over 2}\sum_I^{N_n}\sum_{J\ne I}^{N_n}{Z_I Z_J\over |R_I-R_J|}\\
& = H_e+H_n \\
& H_e\equiv \sum_i h_1(r_i)+{1\over 2}\sum_{i\ne j}h_2(r_i,r_j) \\
\end{align*}\end{split}\]
wavefunction approximated by slater determinant
\[\begin{split}\begin{align*}
& \Phi = {1\over \sqrt{N!} }
\begin{vmatrix}
\phi_1(x_1) & \phi_2(x_1) & ... & \phi_N(x_1)\\
\phi_1(x_2) & \phi_2(x_2) & ... & \phi_N(x_2)\\
...\\
\phi_1(x_N) & \phi_2(x_N) & ... & \phi_N(x_N)
\end{vmatrix} \\
\end{align*}\end{split}\]
total energy
\[\begin{split}\begin{align*}
& E_{HF}=\langle \Phi| H_e|\Phi \rangle=\sum_i \langle \phi_i| h_1|\phi_i \rangle +\boxed{{1\over 2}\times}\sum_{i, j} [ \langle \phi_i \phi_j| h_2|\phi_i\phi_j \rangle - \langle \phi_j\phi_i| h_2|\phi_i \phi_j \rangle ] \\
\end{align*}\end{split}\]
variational principle
\[\begin{split}\begin{align*}
& \phi_k\to\phi_k+\delta\phi_k\Rightarrow \delta\left[ \langle \Phi| H_e|\Phi \rangle - \sum_{i,j} \lambda_{ij}(\langle \phi_i|\ \phi_j \rangle -\delta_{ij} )\right]=0 \\
& h_1\phi_k(x_1)+\sum_i\left[ \int dx_2\; \phi_i^*(x_2) h_2 \phi_i(x_2)\phi_k(x_1) -\int dx_2\; \phi_i^*(x_2) h_2 \phi_i(x_1)\phi_k(x_2) \right] =\sum_i \lambda_{ki}\phi_i(x_1)\\
& \left[ h_1+\sum_i(J_i-K_i) \right]\phi_k=\sum_i \lambda_{ki}\phi_i \\
& \text{diagnalize } \lambda_{ki}=\delta_{ki}\epsilon_k \\
\end{align*}\end{split}\]
total energy is NOT sum of eigenvalues
\[\begin{split}\begin{align*}
& \sum_k \epsilon_k = \sum_i \langle \phi_i| h_1|\phi_i \rangle +\boxed{1\times}\sum_{i, j} [ \langle \phi_i \phi_j| h_2|\phi_i\phi_j \rangle - \langle \phi_j\phi_i| h_2|\phi_i \phi_j \rangle ] \\
\end{align*}\end{split}\]
Exchange Energy
\[\begin{split}\begin{align*}
& E_i\phi_i(r)=\left[{-\nabla^2\over 2}+V_{\text{e-e}} +V_{\text{e-ion}}\right]\phi_i(r) \\
& \qquad \qquad\boxed{- \sum_j \delta_{\sigma_i\sigma_j} \int dr' {\phi_j^*(r')\phi_i(r')\phi_j(r)\over|r-r'|} \quad\text{exchange}}\\
& i,j\text{ are combined orbit/spin indices} \\
&V_{\text{e-e}} = \sum_j \int dr' {|\phi_j(r')|^2\over |r-r'|}= \int dr' {\sum_j|\phi_j(r')|^2\over |r-r'|}=\int dr' {n(r')\over |r-r'|}\\
&V_{\text{e-ion}} = \sum_k -{Z_k\over |r-R_k|}\to\int -{n(R)dR\over |r-R|} \\
& \text{with assumption }n_{\text{ion}}(r)=n(r) \\
& \Rightarrow V_{\text{e-e}}+V_{\text{e-ion}}=0 \\
\end{align*}\end{split}\]
claim: planewave \(e^{ikr}\) are still eigenstates
\[\begin{split}\begin{align*}
& \text{Proof by substitution}\\
& E_ke^{ikr}={1\over2}k^2 e^{ikr} -\sum_{k'} \underbrace{\int dr'{ e^{-ik'r'}e^{ikr'}e^{ik'r} \over|r-r'|} }_I\\
& I=e^{ik'r} \int dr'{ e^{i(k-k')r'} \over|r-r'|}=e^{ik'r}e^{i(k-k')r} \int_{|\vec{u}|>0} d\vec{u}{ e^{i(k-k')\vec{u}} \over |\vec{u}|} \\
&\qquad = e^{ikr} 4\pi \int_{u=0}^{\infty} du \; ue^{i(k-k')u}\\
& \qquad = 2\times \text{integration by parts}\\
&\qquad = e^{ikr} {4\pi\over (k-k')^2} \qquad \boxed{}\\
\end{align*}\end{split}\]
finding eigenvalues
\[\begin{split}\begin{align*}
& \sum_{k'}{4\pi\over (k-k')^2}\to \int {d\vec{k}'\over (2\pi)^3}{4\pi\over (\vec{k}-\vec{k}')^2}\\
&={4\pi\over(2\pi)^2}\int \sin\theta \;d\theta\int_0^{k_F} dk'{k'^2\over k^2-2kk'\cos\theta +k'^2} \\
& = {4\pi\over(2\pi)^2}\int_0^{k_F}{k'^2\over 2kk'} dk'\ln( k^2+2kk's +k'^2 )_{s=-1}^1 \\
& = {1\over 2\pi}\int_0^{k_F}{k'\over k} dk' \left| {k^2+2kk' +k'^2 \over k^2-2kk's +k'^2} \right| \\
& = {1\over \pi}\int_0^{k_F}{k'\over k} dk' \left| {k+k' \over k-k'} \right| =...\\
& \Rightarrow \boxed{E_k={1\over 2}k^2-{k_F\over \pi }f({k\over k_F})\\
f(x)=1+{1-x^2\over 2x}\ln|{1+x\over 1-x}| } \\
\end{align*}\end{split}\]
interpretation: exchange term reduces \(E\), by making \(e^-\) with same spin apart from each other
total energy
\[\begin{split}\begin{align*}
& E_{HF}={1\over (2\pi)^3}4\pi\int_0^{k_F} k^2\left[{k^2\over 2}+\boxed{{1\over 2}\times}\left(-{k_F\over \pi}f({k\over k_F})\right)\right] \\
& {E_{HF}\over N}=...={2.21\over r_s^2}\underbrace{-{0.916\over r_s}}_{\epsilon_x}
\end{align*}\end{split}\]
Task: contact interaction
contact interaction: \({1\over |r-r'|}\to \delta(r-r')\)
\[\begin{split}\begin{align*}
& \sum_j \int dr' \phi_j^*(r')\phi_j(r)\phi_i(r')\delta(r-r') \quad \text{exchange} \\
& \Rightarrow I=e^{ik'r}\int dr' e^{i(k-k')r'}
\delta(r-r')=e^{ikr}\qquad \boxed \\
\end{align*}\end{split}\]
finding eigenvalues
\[\begin{split}\begin{align*}
& \sum_{k'} 1\to \int{d\vec{k}' \over (2\pi)^3}={1\over(2\pi)^3}4\pi {k_F^3\over 3}\\
& \Rightarrow E_k={1\over 2}k^2-{k_F^3\over 6 \pi^2} \\
\end{align*}\end{split}\]
total energy (the previous case did not multiply spin factor \(2\))
\[\begin{split}\begin{align*}
& E_{HF}={2\over (2\pi)^3}4\pi\int_0^{k_F} k^2\left[{k^2\over 2}+\boxed{{1\over 2}\times}\left(-{k_F^3\over 6 \pi^2}\right)\right] \\
&\quad = {1\over 10\pi^2}k_F^5 -{1\over 36\pi^4}k_F^6 \\
& n={2\over(2\pi)^3}4\pi {k_F^3\over 3}={1\over 3\pi^2}k_F^3 \\
\end{align*}\end{split}\]
correlation
\[\begin{align*}
& E_c:=E_{\text{real}}-T-E_x
\end{align*}\]
collective behavior of \(e^-\) to screen and decrease Coulomb interaction
Unlike Exchange, correlation becomes more pronounced for opposite spins since they are more close
attempts
Wigner crystal solved exactly for small \(r_s\) by assuming \(\epsilon_c\to \text{const}\) for large \(r_s\Rightarrow \epsilon_c=-{0.44\over r_s+7.8}\)
Gellman and Breuckner summed an infinite series of Feynman diagrams and calculated exactly for large \(r_s\Rightarrow \epsilon_c=0.311\ln(r_s)...\), agrees with wigner
Quantum Monte Carlo
Helper: Delta Function
\[\begin{split}\begin{align*}
& f(x)={1\over 2\pi} \int dk\; e^{ikx}\left[\int dx' \;e^{-ikx'}f(x') \right] \\
& =\int dx' \; f(x')\underbrace{{1\over 2\pi}\int dk\; e^{ik(x-x')}}_{\delta(x-x')} \\
\end{align*}\end{split}\]