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 ~~~~~~~~~~~~~~~ .. math:: \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*} Interacting ~~~~~~~~~~~ .. math:: \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*} Hartree Fock Equation --------------------- - only consider **exchange** - interacting electron gas can be solved analytically at HF level - hamiltonian .. math:: \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*} - wavefunction approximated by slater determinant .. math:: \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*} - total energy .. math:: \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*} - variational principle .. math:: \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*} - total energy is **NOT** sum of eigenvalues .. math:: \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*} Exchange Energy --------------- .. math:: \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*} - claim: planewave :math:`e^{ikr}` are still eigenstates .. math:: \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*} - finding eigenvalues .. math:: \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*} - interpretation: exchange term reduces :math:`E`, by making :math:`e^-` with same spin apart from each other - total energy .. math:: \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*} Task: contact interaction ------------------------- - contact interaction: :math:`{1\over |r-r'|}\to \delta(r-r')` .. math:: \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*} - finding eigenvalues .. math:: \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*} - total energy (the previous case did not multiply spin factor :math:`2`) .. math:: \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*} correlation ----------- .. math:: \begin{align*} & E_c:=E_{\text{real}}-T-E_x \end{align*} - collective behavior of :math:`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 :math:`r_s` by assuming :math:`\epsilon_c\to \text{const}` for large :math:`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 :math:`r_s\Rightarrow \epsilon_c=0.311\ln(r_s)...`, agrees with wigner - **Quantum Monte Carlo** Helper: Delta Function ---------------------- .. math:: \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*}