汎函数微分
電位の古典的な部分に対して、トマスとフェルミはクーロン位置エネルギー汎函数
J [ ρ ] = 1 2 ∫∫ ρ ( r ) ρ ( r ′ ) | r − r ′ | d r d r ′ = ∫ ( 1 2 ∫ ρ ( r ) ρ ( r ′ ) | r − r ′ | d r ′ ) d r = ∫ j [ r , ρ ( r ) ] d r {\displaystyle J[\rho ]={\frac {1}{2}}\iint {\frac {\rho ({\boldsymbol {r}})\rho ({\boldsymbol {r}}')}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}\,d{\boldsymbol {r}}d{\boldsymbol {r}}'=\int \left({\frac {1}{2}}\int {\frac {\rho ({\boldsymbol {r}})\rho ({\boldsymbol {r}}')}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}d{\boldsymbol {r}}'\right)d{\boldsymbol {r}}=\int j[{\boldsymbol {r}},\rho ({\boldsymbol {r}})]\,d{\boldsymbol {r}}}
δ T W [ ρ ] δ ρ = ∂ t ∂ ρ − ∇ ⋅ ∂ t ∂ ( ∇ ρ ) = − 1 8 ( ∇ ρ ( r ) ) 2 ρ ( r ) 2 − ∇ ⋅ ( 1 4 ∇ ρ ( r ) ρ ( r ) ) = 1 8 ( ∇ ρ ( r ) ) 2 ρ ( r ) 2 − 1 4 ∇ 2 ρ ( r ) ρ ( r ) {\displaystyle {\begin{aligned}{\frac {\delta T_{\mathrm {W} }[\rho ]}{\delta \rho }}&={\frac {\partial t}{\partial \rho }}-\nabla \cdot {\frac {\partial t}{\partial (\nabla \rho )}}\\&=-{\frac {1}{8}}{\frac {(\nabla \rho ({\boldsymbol {r}}))^{2}}{\rho ({\boldsymbol {r}})^{2}}}-\nabla \cdot \left({\frac {1}{4}}{\frac {\nabla \rho ({\boldsymbol {r}})}{\rho ({\boldsymbol {r}})}}\right)\\&={\frac {1}{8}}{\frac {(\nabla \rho ({\boldsymbol {r}}))^{2}}{\rho ({\boldsymbol {r}})^{2}}}-{\frac {1}{4}}{\frac {\nabla ^{2}\rho ({\boldsymbol {r}})}{\rho ({\boldsymbol {r}})}}\end{aligned}}}
となる。