DFT In Momentum Space

Intro

History

Some discussions

Goal

  • Self-consistent determination of momental density

    • Step 1: Find energy functionals of momental density

Action

  • II: Introduce effective kinetic energy

  • III: Momental TF approximation for atoms

  • IV: Scaling, Multiple counting of e-pairs

  • Appendix: Is Lowdin’s natrual orbitals same as single-electron states of independent particle description? No

Effective energy \(V\)

\[\begin{split}\begin{align*} & E=E_{\text{kin}}+\int dr\; V_{\text{ext}} n + E_{\text{int}}[n]+\mu(N-\int dr\; n)\\ &\Rightarrow {\delta E\over \delta n} = {\delta E_{\text{kin}}\over \delta n}+ \underbrace{V_{\text{ext}}+{\delta E_{\text{int}}\over \delta n}}_{V}+\mu(-1)=0 \\ & \Rightarrow E = \underbrace{E_{\text{kin}}+\int dr\; (V-\mu)n}_{E_1}+\mu N\\ & \begin{cases} & {\delta E_1\over \delta n}={\delta E\over \delta n}-0=0\\ & {\delta E_1\over \delta V}=n\\ & {\delta E_1\over \delta \mu} ={\delta E\over \delta \mu}-N=-N \end{cases} \end{align*}\end{split}\]

What is the problem?

  • We don’t know \(E_{\text{kin}}[n]\) and \(E_{\text{int}}[n]\)

Approximations

\[\begin{split}\begin{align*} & E_1=\langle \psi_0| \sum\limits_{i=1}^N [H(r_i,p_i)-\mu] |\psi_0 \rangle \\ & \boxed{= \text{Tr}[f(H-\mu)]\approx \text{Tr}[(H-\mu)\eta(\mu-H)] }\\ & \Rightarrow \begin{cases} &\boxed{ N\approx \text{Tr}[\eta(\mu-H)] }\\ &\boxed{ n(r')\approx 2\langle r'|\eta(\mu-H)|r' \rangle } \end{cases} \end{align*}\end{split}\]

Effective energy \(T\)

\[\begin{split}\begin{align*}\boxed{ r\to p\\ V_{\text{ext}} \to T_{\text{kin}}\\ E_{\text{kin}} \to E_{\text{ext}} }\end{align*}\end{split}\]
\[\begin{split}\begin{align*} & E=E_{\text{ext}}+\int dp \; T_{\text{kin}} n + E_{\text{int}}[n]+\mu(N-\int dp\; n)\\ &\Rightarrow {\delta E\over \delta n} = {\delta E_{\text{ext}}\over \delta n}+ \underbrace{T_{\text{kin}}+{\delta E_{\text{int}}\over \delta n}}_{T}+\mu(-1)=0 \\ & \Rightarrow E = \underbrace{E_{\text{ext}}+\int dp\; (T-\mu)n}_{E_1}+\mu N\\ & \begin{cases} & {\delta E_1\over \delta n}={\delta E\over \delta n}-0=0\\ & {\delta E_1\over \delta T}=n\\ & {\delta E_1\over \delta \mu} ={\delta E\over \delta \mu}-N=-N \end{cases} \end{align*}\end{split}\]

Approximations

\[\begin{align*} \boxed{ n(p')\approx 2\langle p'|\eta(\mu-H)|p' \rangle } \end{align*}\]

Thomas Fermi Approximation

\[\begin{split}\begin{align*} & E_1^{TF}[T-\mu]=g\int {drdp\over (2\pi)^3}[ T(p)+V_{\text{ext}}(r)-\mu ]\eta(\mu -T-V_{\text{ext}}) \\ &\Rightarrow n={\delta E_1^{TF} \over \delta T}= g\int {dr \over (2\pi)^3} 1\cdot \eta(\mu -T-V_{\text{ext}})\\ &= g{1 \over (2\pi)^3} {4\over 3}\pi R^3={g \over 6\pi^3}R^3 \\ &\text{Where }R \text{ is the radius of the sphere such that } \mu -T-V_{\text{ext}}>0 \end{align*}\end{split}\]

Task

\[\begin{split}\begin{align*} & V_{\text{ext}} = -{Z\over r}\Rightarrow r<{Z\over T-\mu}=R\\ & \Rightarrow E_1^{TF}=g\int {dp\over (2\pi)^3} (4\pi)\int_0^R dr \; r^2[T(p)-{Z\over r}-\mu] \\ & =g\int {dp\over (2\pi)^3} (4\pi) \left[ (T-\mu){R^3\over 3} -Z{R^2\over 2} \right]\\ & =g\int {dp\over (2\pi)^3} (4\pi) \left[-Z{R^2\over 6} \right]\\ & E_{\text{ext}}=g\int {dp\over (2\pi)^3} (4\pi) \left[-Z{R^2\over 2} \right] \end{align*}\end{split}\]
\[\begin{split}\begin{align*} & n^{(2)} (r,r';r,r')= \underbrace{n^{(1)}(r,r)n^{(1)}(r',r')}_{\text{Hatree}}\underbrace{ -{1\over 2}n^{(1)}(r,r')n^{(1)}(r',r)}_{\text{Exchange}} \quad\boxed{{1\over 2}???} \\ & n^{(1)}(r,r)=n(r) \\ & E_{\text{int}}= \int dr dr' \delta(r-r')n^{(2)} (r,r';r,r') \\ & ={1\over 2} \int dr n^2(r) \\ & ={1\over 2} \int dr 2\int {dp\over (2\pi)^3} \eta(R(p)-r) 2\int {dp'\over (2\pi)^3} \eta(R(p')-r) \\ & ={1\over 2} \int dr 2\int {dp\over (2\pi)^3}2\int {dp'\over (2\pi)^3}\; \eta(R(p)-r)\; \eta(R(p')-r) \\ & ={1\over 2} 2\int {dp\over (2\pi)^3}2\int {dp'\over (2\pi)^3} 4\pi{R_<^3\over 3} \\ & ={1\over 2} 2\int {dp\over (2\pi)^3}2\int {dp'\over (2\pi)^3} 4\pi{1\over 3}{6\pi^3\over g}n_< \\ \end{align*}\end{split}\]