Ensemble Density Functional Theory of Neutral and Charged Excitations

Abstract

Recent progress in the field of (time-independent) ensemble density-functional theory (DFT) for excited states are reviewed. Both Gross–Oliveira–Kohn (GOK) and N-centered ensemble formalisms, which are mathematically very similar and allow for an in-principle-exact description of neutral and charged electronic excitations, respectively, are discussed. Key exact results, for example, the equivalence between the infamous derivative discontinuity problem and the description of weight dependencies in the ensemble exchange-correlation density functional, are highlighted. The variational evaluation of orbital-dependent ensemble Hartree-exchange (Hx) energies is discussed in detail. We show in passing that state-averaging individual exact Hx energies can lead to severe (although solvable) v-representability issues. Finally, we explore the possibility of using the concept of density-driven correlation, which has been introduced recently and does not exist in regular ground-state DFT, for improving state-of-the-art correlation density-functional approximations for ensembles. The present review reflects the efforts of a growing community to turn ensemble DFT into a rigorous and reliable low-cost computational method for excited states. We hope that, in the near future, this contribution will stimulate new formal and practical developments in the field.

Appendices

Asymptotic Behavior of the xc Potential

Let us consider the simpler one-dimensional (1D) case in which the KS-PPLB equations read as

$$\begin{aligned} -\dfrac{1}{2}\dfrac{d^2 \varphi ^\alpha _i(x)}{dx^2}+\left( v_{\mathrm{ext}}(x)+v^\alpha _{\mathrm{Hxc}}(x)\right) \varphi ^\alpha _i(x)=\varepsilon ^\alpha _i\varphi ^\alpha _i(x), \end{aligned}$$

(290)

thus leading to

$$\begin{aligned} \dfrac{d^2 \varphi ^\alpha _i(x)}{d x^2}\underset{{|{x}|}\rightarrow +\infty }{=}-2\left( \varepsilon ^\alpha _i-v^\alpha _{\mathrm{xc}}(\infty )\right) \varphi ^\alpha _i(x), \end{aligned}$$

(291)

where we used the limits $v_{\mathrm{ext}}(\infty )=v^\alpha _{\mathrm{H}}(\infty )=0$. Note that ${|{\varphi ^\alpha _i(x)}|}$ is expected to decay as ${|{x}|}\rightarrow +\infty$, which implies $-2\left( \varepsilon ^\alpha _i-v^\alpha _{\mathrm{xc}}(\infty )\right) >0$. Therefore,

$$\begin{aligned} \varphi ^\alpha _i(x)\underset{{|{x}|}\rightarrow +\infty }{\sim }e^{-\sqrt{-2(\varepsilon ^\alpha _i-v^\alpha _{\mathrm{xc}}(\infty ))}{|{x}|}}, \end{aligned}$$

(292)

and

$$\begin{aligned} n_{\hat{\gamma }_{\mathrm{KS}}^\alpha }(x)\underset{{|{x}|}\rightarrow +\infty }{\sim } \alpha {|{\varphi ^\alpha _N(x)}|}^2 \sim \alpha \,e^{-2\sqrt{-2(\varepsilon ^\alpha _N-v^\alpha _{\mathrm{xc}}(\infty ))}{|{x}|}}. \end{aligned}$$

(293)

In the true interacting system, the N-electron ground-state wave function $\varPsi ^N_0$ fulfills

$$\begin{aligned} \begin{aligned}&\left[ \sum ^N_{i=1}\left( -\dfrac{1}{2}\dfrac{\partial ^2}{\partial x_i^2}+v_{\mathrm{ext}}(x_i)\right) +\sum ^N_{1\le i<j}w_{\mathrm{ee}}({|{x_i-x_j}|})\right] \varPsi ^N_0(x_1,\ldots ,x_N) \\&\quad =E^N_0\varPsi ^N_0(x_1,\ldots ,x_N), \end{aligned} \end{aligned}$$

(294)

where $w_{\mathrm{ee}}({|{x_i-x_j}|})$ is a well-behaved two-electron repulsion energy in 1D. Let us consider the situation where ${|{x_1}|}\rightarrow +\infty$ while $x_2,\ldots ,x_N$ remain in the region of the system, which corresponds to an ionization process in the ground state. Since $w_{\mathrm{ee}}({|{x_1-x_j}|})\rightarrow 0$, the (to-be-antisymmetrized) wave function and its density can be rewritten as

$$\begin{aligned} \varPsi ^N_0(x_1,\ldots ,x_N)\underset{{|{x_1}|}\rightarrow +\infty }{\sim }\varphi ^{[N]}(x_1)\,\varPsi ^{N-1}_0(x_2,\ldots ,x_N) \end{aligned}$$

(295)

and

$$\begin{aligned} n_{\varPsi ^N_0}(x_1)\underset{{|{x_1}|}\rightarrow +\infty }{\sim }{|{\varphi ^{[N]}(x_1)}|}^2, \end{aligned}$$

(296)

respectively, where

$$\begin{aligned} \dfrac{{\mathrm{d}}^2\varphi ^{[N]}(x_1)}{{\mathrm{d}} x_1^2}\underset{{|{x_1}|}\rightarrow +\infty }{\sim }-2\left( E^N_0-E^{N-1}_0\right) \varphi ^{[N]}(x_1)=2I_0^N\varphi ^{[N]}(x_1), \end{aligned}$$

(297)

thus leading to the explicit expression

$$\begin{aligned} \varphi ^{[N]}(x)\underset{{|{x}|}\rightarrow +\infty }{\sim }e^{-\sqrt{2I_0^N}{|{x}|}}. \end{aligned}$$

(298)

From the exact mapping of the ensemble PPLB density onto the KS system, we deduce from Eqs. (296) and (298) that

$$\begin{aligned} \begin{aligned} n_{\hat{\gamma }_{\mathrm{KS}}^\alpha }(x)&\underset{{|{x}|}\rightarrow +\infty }{\sim }(1-\alpha )e^{-2\sqrt{2I_0^{N-1}}{|{x}|}}+\alpha \,e^{-2\sqrt{2I_0^N}{|{x}|}}\sim \alpha \,e^{-2\sqrt{2I_0^N}{|{x}|}}, \end{aligned} \end{aligned}$$

(299)

where we assumed that $E^{N-1}_g=I_0^{N-1}-I_0^{N}>0$. Thus, we conclude from Eq. (293) that

$$\begin{aligned} I_0^N=-(\varepsilon ^\alpha _N-v^\alpha _{\mathrm{xc}}(\infty )). \end{aligned}$$

(300)

Any constant shift in the xc potential $v^\alpha _{\mathrm{xc}}({\mathbf {r}})$ does not affect the above expression. Since, according to Janak’s theorem, $I_0^N=-\varepsilon ^\alpha _N$, the constant is imposed in PPLB and

$$\begin{aligned} v^\alpha _{\mathrm{xc}}(\infty )=0. \end{aligned}$$

(301)

We now turn to the left and right formulations of N-centered eDFT. We recall the shorthand notations $(\xi _-,0)\overset{notation}{\equiv } \xi _-$ and $(0,\xi _+)\overset{notation}{\equiv } \xi _+$. When $\xi _+>0$, the right N-centered ensemble density, which is mapped onto a non-interacting KS ensemble, has the following asymptotic behavior [we just need to substitute $N+1$ for N in Eqs. (293), (296), and (298)],

$$\begin{aligned} n^{\xi _+}(x)&\underset{{|{x}|}\rightarrow +\infty }{\sim }&\xi _+\,e^{-2\sqrt{2I_0^{N+1}}{|{x}|}} \end{aligned}$$

(302)

$$\begin{aligned}\sim & {} \xi _+\,e^{-2\sqrt{-2(\varepsilon ^{\xi _+}_{N+1}-v^{\xi _+}_{\mathrm{xc}}(\infty ))}{|{x}|}}. \end{aligned}$$

(303)

Similarly, for $\xi _-\ge 0$, we have

$$\begin{aligned} n^{\xi _-}(x)&\underset{{|{x}|}\rightarrow +\infty }{\sim }&\left( 1-\dfrac{(N-1)\xi _-}{N}\right) \,e^{-2\sqrt{2I_0^{N}}{|{x}|}} \end{aligned}$$

(304)

$$\begin{aligned}\sim & {} \left( 1-\dfrac{(N-1)\xi _-}{N}\right) \,e^{-2\sqrt{-2(\varepsilon ^{\xi _-}_{N}-v^{\xi _-}_{\mathrm{xc}}(\infty ))}{|{x}|}}. \end{aligned}$$

(305)

Thus, we conclude that

$$\begin{aligned} A_0^N=I_0^{N+1}\overset{\xi _+>0}{=}-\varepsilon ^{\xi _+}_{N+1}+v^{\xi _+}_{\mathrm{xc}}(\infty ) \end{aligned}$$

(306)

and

$$\begin{aligned} I_0^{N}\overset{\xi _-\ge 0}{=}-\varepsilon ^{\xi _-}_{N}+v^{\xi _-}_{\mathrm{xc}}(\infty ). \end{aligned}$$

(307)

Derivation of the eDMHF Equations

For convenience, we use the following exponential parameterization of the single-configuration wave functions [126],

$$\begin{aligned} {|{\varPhi _I}\rangle }\equiv {|{\varPhi _I(\varvec{\kappa })}\rangle }=e^{-\hat{\kappa }}{|{\overline{\varPhi }^{\mathbf {w}}_I}\rangle }, \end{aligned}$$

(308)

where $\varvec{\kappa }\equiv \left\{ \kappa _{pq}\right\} _{p<q}$ are the variational orbital rotation parameters and $\hat{\kappa }$ is the corresponding real singlet rotation quantum operator. The latter reads as follows in second quantization,

$$\begin{aligned} \hat{\kappa }=\sum _{p<q}\kappa _{pq}(\hat{E}_{pq}-\hat{E}_{qp})=-\hat{\kappa }^\dagger , \end{aligned}$$

(309)

where the index p refers to the orbital $\overline{\varphi }^{\mathbf {w}}_p$ and $\hat{E}_{pq}=\sum _{\tau =\uparrow ,\downarrow }\hat{a}_{p\tau }^\dagger \hat{a}_{q\tau }$. Therefore, the eDMHF energy becomes a function of $\varvec{\kappa }$,

$$\begin{aligned} E^{\mathbf {w}}_\mathrm{eDMHF}(\varvec{\kappa })=E_\mathrm{HF}\left( \mathbf{{D}}^{\mathbf {w}}(\varvec{\kappa })\right) , \end{aligned}$$

(310)

where $\mathbf{{D}}^{\mathbf {w}}(\varvec{\kappa })=\sum _I{\mathtt{{w}}}_I\mathbf{{D}}^{\varPhi _I(\varvec{\kappa })}$ is a trial ensemble density matrix, and $E_\mathrm{HF}(\mathbf{{D}})$ is the conventional ground-state HF density matrix functional energy:

$$\begin{aligned} E_\mathrm{HF}(\mathbf{{D}})=\sum _{mk}h_{mk}D_{mk}+\dfrac{1}{2}\sum _{klmn}\left( {\langle {mn}}{|{kl}\rangle }-\dfrac{1}{2}{\langle {mn}}{|{lk}\rangle }\right) D_{mk}D_{nl}. \end{aligned}$$

(311)

By construction, the minimum is reached when $\varvec{\kappa }=0$, and we denote $\mathbf{{D}}^{\mathbf {w}}=\mathbf{{D}}^{\mathbf {w}}(\varvec{\kappa }=0)$. Note that

$$\begin{aligned} D^{\overline{\varPhi }^{\mathbf {w}}_I}_{pq}={\left\langle {\overline{\varPhi }^{\mathbf {w}}_I}\left| \hat{E}_{pq} \right| {\overline{\varPhi }^{\mathbf {w}}_I}\right\rangle }=\delta _{pq}n^I_{p}, \end{aligned}$$

(312)

where the occupation number $n^I_{p}$ is an integer, and

$$\begin{aligned} D^{\mathbf {w}}_{pq}=\sum _I{\mathtt{{w}}}_ID^{\overline{\varPhi }^{\mathbf {w}}_I}_{pq}=\delta _{pq}\sum _I{\mathtt{{w}}}_In^I_{p}=\delta _{pq}\theta ^{\mathbf {w}}_p, \end{aligned}$$

(313)

where $\theta ^{\mathbf {w}}_p$ can be fractional. The stationarity condition that is fulfilled by the minimizing eDMHF orbitals can now be written explicitly as follows,

$$\begin{aligned} \left. \dfrac{\partial E^{\mathbf {w}}_\mathrm{eDMHF}(\varvec{\kappa })}{\partial \kappa _{pq}} \right| _{\varvec{\kappa }=0} =\sum _{rs} \left. \dfrac{\partial D^{\mathbf {w}}_{rs}(\varvec{\kappa })}{\partial \kappa _{pq}}\right| _{\varvec{\kappa }=0} \left. \dfrac{\partial E_\mathrm{HF}\left( \mathbf{{D}}\right) }{\partial D_{rs}}\right| _{\mathbf{{D}}=\mathbf{{D}}^{\mathbf {w}}} =0, \end{aligned}$$

(314)

where

$$\begin{aligned} \begin{aligned} \dfrac{\partial E_\mathrm{HF}\left( \mathbf{{D}}\right) }{\partial D_{rs}}&=h_{rs}+ \dfrac{1}{2}\sum _{nl}\left( {\langle {rn}}{|{sl}\rangle }-\dfrac{1}{2}{\langle {rn}}{|{ls}\rangle }\right) D_{nl} \\&\quad +\dfrac{1}{2}\sum _{mk}\left( {\langle {mr}}{|{ks}\rangle }-\dfrac{1}{2}{\langle {mr}}{|{sk}\rangle }\right) D_{mk} \\&=h_{rs}+\sum _{nl}\left( {\langle {rn}}{|{sl}\rangle }-\dfrac{1}{2}{\langle {rn}}{|{ls}\rangle }\right) D_{nl} \\&\equiv f_{rs}(\mathbf{{D}}) \end{aligned} \end{aligned}$$

(315)

is the conventional density matrix functional Fock operator matrix element, and

$$\begin{aligned} \begin{aligned} \left. \dfrac{\partial D^{\mathbf {w}}_{rs}(\varvec{\kappa })}{\partial \kappa _{pq}} \right| _{\varvec{\kappa }=0}&=\sum _I{\mathtt{{w}}}_I {\left\langle \left[ \hat{E}_{pq}-\hat{E}_{qp},\hat{E}_{rs}\right] \right\rangle }_{\varPhi ^{\mathbf {w}}_I} \\&=\sum _I{\mathtt{{w}}}_I\left( \delta _{qr}\delta _{ps}n^I_{p}-\delta _{ps}\delta _{qr}n^I_{q}-\delta _{pr}\delta _{qs}n^I_{q}+\delta _{qs}\delta _{pr}n^I_{p}\right) \\&=\left( \delta _{qr}\delta _{ps}+\delta _{pr}\delta _{qs}\right) \sum _I{\mathtt{{w}}}_I\left( n^I_{p}-n^I_{q}\right) \\&=\left( \delta _{qr}\delta _{ps}+\delta _{pr}\delta _{qs}\right) (\theta ^{\mathbf {w}}_p-\theta ^{\mathbf {w}}_q), \end{aligned} \end{aligned}$$

(316)

where we used the relation $[\hat{E}_{pq},\hat{E}_{rs}]=\delta _{qr}\hat{E}_{ps}-\delta _{ps}\hat{E}_{rq}$ (see Ref. [126]) with Eqs. (312) and (313). If we denote $f^{\mathbf {w}}_{rs}=f_{rs}(\mathbf{{D}}^{\mathbf {w}})$, Eq. (314) can be written in a compact form as follows,

$$\begin{aligned} \begin{aligned} (\theta ^{\mathbf {w}}_p-\theta ^{\mathbf {w}}_q)\sum _{rs}\left( \delta _{qr}\delta _{ps}+\delta _{pr}\delta _{qs}\right) f^{\mathbf {w}}_{rs}=0, \end{aligned} \end{aligned}$$

(317)

thus leading to the final result:

$$\begin{aligned} \left( \theta _p^{\mathbf {w}}-\theta _q^{\mathbf {w}}\right) f^{\mathbf {w}}_{qp}=0. \end{aligned}$$

(318)

Derivation of the SAHF Equations

We use the same parameterization as in Appendix 2, i.e.,

$$\begin{aligned} {|{\varPhi _I}\rangle }\equiv {|{\varPhi _I(\varvec{\kappa })}\rangle }=e^{-\hat{\kappa }}{|{\tilde{\varPhi }^{\mathbf {w}}_I}\rangle }, \end{aligned}$$

(319)

where the indices $\{p\}$ in creation/annihilation operators (as well as in one- and two-electron integrals) now refer to the minimizing SAHF orbitals $\left\{ \tilde{\varphi }^{\mathbf {w}}_p\right\}$. The to-be-minimized SAHF energy can be expressed as follows,

$$\begin{aligned} E^{\mathbf {w}}_\mathrm{SAHF}(\varvec{\kappa })=\sum _I{\mathtt{{w}}}_I\left( \sum _{rs}h_{rs}D^{\varPhi _I(\varvec{\kappa })}_{rs} +E_{\mathrm{H}}[n_{\varPhi _I(\varvec{\kappa })}]+\mathcal {E}^I_{\mathrm{x}}\left[ \mathbf{D}^{\varPhi _I(\varvec{\kappa })}\right] \right) , \end{aligned}$$

(320)

so that the stationarity condition reads as

$$\begin{aligned} \begin{aligned}&\left. \dfrac{\partial E^{\mathbf {w}}_\mathrm{SAHF}(\varvec{\kappa })}{\partial \kappa _{pq}}\right| _{\varvec{\kappa }=0} =0 \\&= \sum _I{\mathtt{{w}}}_I\left[ \sum _{rs}\left( h_{rs}+\left[ v^{{\mathbf {w}}}_{{\mathrm{x}},I}\right] _{rs}\right) \dfrac{\partial D_{rs}^{{\varPhi }_I(\varvec{\kappa })}}{\partial \kappa _{pq}} +\int {\mathrm{d}}{\mathbf {r}}\, v_{\mathrm{H}}[n_{\tilde{\varPhi }^{\mathbf {w}}_I}]({\mathbf {r}})\dfrac{\partial n_{{\varPhi }_I(\varvec{\kappa })}({\mathbf {r}})}{\partial \kappa _{pq}} \right] _{\varvec{\kappa }=0}, \end{aligned} \end{aligned}$$

(321)

where $\left[ v^{{\mathbf {w}}}_{{\mathrm{x}},I}\right] _{rs}\equiv \left. \partial \mathcal {E}^I_{\mathrm{x}}\left[ \mathbf{D}\right] /\partial D_{rs}\right| _{\mathbf{{D}}=\mathbf{{D}}^{\tilde{\varPhi }^{\mathbf {w}}_I}}$ and $v_{\mathrm{H}}[n]({\mathbf {r}})=\delta E_{\mathrm{H}}[n]/\delta n({\mathbf {r}})$. Note that the individual densities are recovered from the density matrices as follows,

$$\begin{aligned} n_{\varPhi _I(\varvec{\kappa })}({\mathbf {r}})=\gamma ^{\varPhi _I(\varvec{\kappa })}({\mathbf {r}},{\mathbf {r}})=\sum _{rs}\tilde{\varphi }^{\mathbf {w}}_r({\mathbf {r}})\tilde{\varphi }^{\mathbf {w}}_s({\mathbf {r}})D^{\varPhi _I(\varvec{\kappa })}_{rs}. \end{aligned}$$

(322)

Therefore, if we use the notation

$$\begin{aligned} {\left\langle {\tilde{\varphi }^{\mathbf {w}}_r}\left| \hat{h}+\hat{v}^{{\mathbf {w}}}_{{\mathrm{Hx}},I} \right| {\tilde{\varphi }^{\mathbf {w}}_s}\right\rangle }=h_{rs} +\int {\mathrm{d}}{\mathbf {r}}\,\tilde{\varphi }^{\mathbf {w}}_r({\mathbf {r}}) v_{\mathrm{H}}[n_{\tilde{\varPhi }^{\mathbf {w}}_I}]({\mathbf {r}})\tilde{\varphi }^{\mathbf {w}}_s({\mathbf {r}}) +\left[ v^{{\mathbf {w}}}_{{\mathrm{x}},I}\right] _{rs} , \end{aligned}$$

(323)

Eq. (321) can be rewritten in a compact form as follows,

$$\begin{aligned} \sum _I{\mathtt{{w}}}_I\sum _{rs}{\left\langle {\tilde{\varphi }^{\mathbf {w}}_r}\left| \hat{h}+\hat{v}^{{\mathbf {w}}}_{{\mathrm{Hx}},I} \right| {\tilde{\varphi }^{\mathbf {w}}_s}\right\rangle }\left. \dfrac{\partial D_{rs}^{{\varPhi }_I(\varvec{\kappa })}}{\partial \kappa _{pq}}\right| _{\varvec{\kappa }=0}=0. \end{aligned}$$

(324)

We conclude from Eq. (316) that

$$\begin{aligned} \begin{aligned} 0&=\sum _I{\mathtt{{w}}}_I\left( n^I_{p}-n^I_{q}\right) \sum _{rs}\left( \delta _{qr}\delta _{ps}+\delta _{pr}\delta _{qs}\right) {\left\langle {\tilde{\varphi }^{\mathbf {w}}_r}\left| \hat{h}+\hat{v}^{{\mathbf {w}}}_{{\mathrm{Hx}},I} \right| {\tilde{\varphi }^{\mathbf {w}}_s}\right\rangle } \\&=2\sum _I{\mathtt{{w}}}_I\left( n^I_{p}-n^I_{q}\right) {\left\langle {\tilde{\varphi }^{\mathbf {w}}_p}\left| \hat{h}+\hat{v}^{{\mathbf {w}}}_{{\mathrm{Hx}},I} \right| {\tilde{\varphi }^{\mathbf {w}}_q}\right\rangle }, \end{aligned} \end{aligned}$$

(325)

thus leading to the final result:

$$\begin{aligned} (\theta ^{\mathbf {w}}_p-\theta ^{\mathbf {w}}_q){\left\langle {\tilde{\varphi }^{\mathbf {w}}_p}\left| \hat{h} \right| {\tilde{\varphi }^{\mathbf {w}}_q}\right\rangle }+\sum _I{\mathtt{{w}}}_I\left( n^I_{p}-n^I_{q}\right) {\left\langle {\tilde{\varphi }^{\mathbf {w}}_p}\left| \hat{v}^{{\mathbf {w}}}_{{\mathrm{Hx}},I} \right| {\tilde{\varphi }^{\mathbf {w}}_q}\right\rangle }=0. \end{aligned}$$

(326)

Exact DD Ensemble Correlation Energy in the Hubbard Dimer

For convenience, we will use the following exact expression for the ensemble DD correlation energy:

$$\begin{aligned} E_{\mathrm{c}}^{{\mathtt{{w}}},\mathrm{DD}}(n^{\mathtt{{w}}})=-(1-{\mathtt{{w}}})^2{\mathtt{{w}}}\dfrac{\partial f_0^{\mathtt{{w}}}(n^{\mathtt{{w}}})}{\partial {\mathtt{{w}}}}+{\mathtt{{w}}}^2(1-{\mathtt{{w}}})\dfrac{\partial f_1^{\mathtt{{w}}}(n^{\mathtt{{w}}})}{\partial {\mathtt{{w}}}}. \end{aligned}$$

(327)

The individual Hx-only GOK energies are extracted from the ensemble energy,

$$\begin{aligned} f^\xi (n)= -2t\sqrt{(1-\xi )^2-(1-n)^2} +\dfrac{U}{2}\left[ 1+\xi -\dfrac{(3\xi -1)(1-n)^2}{(1-\xi )^2}\right] , \end{aligned}$$

(328)

as follows,

$$\begin{aligned} f^{\mathtt{{w}}}_{0}\left( n^{\mathtt{{w}}}\right)= {} f^{\mathtt{{w}}}\left( n^{\mathtt{{w}}}\right) -{\mathtt{{w}}}\left. \dfrac{\partial f^\xi \left( n^{\xi ,{\mathtt{{w}}}}\right) }{\partial \xi }\right| _{\xi ={\mathtt{{w}}}}, \end{aligned}$$

(329)

and

$$\begin{aligned} f^{\mathtt{{w}}}_{1}\left( n^{\mathtt{{w}}}\right)= {} f^{\mathtt{{w}}}\left( n^{\mathtt{{w}}}\right) +(1-{\mathtt{{w}}})\left. \dfrac{\partial f^\xi \left( n^{\xi ,{\mathtt{{w}}}}\right) }{\partial \xi }\right| _{\xi ={\mathtt{{w}}}}, \end{aligned}$$

(330)

where

$$\begin{aligned} n^{\xi ,{\mathtt{{w}}}}=(1-\xi ) n_{\varPhi _{0}^{{\mathtt{{w}}}}}+ \xi n_{\varPhi _{1}^{{\mathtt{{w}}}}}. \end{aligned}$$

(331)

Since $n_{\varPhi _{1}^{{\mathtt{{w}}}}}=1$ and

$$\begin{aligned} (1-{\mathtt{{w}}})n_{\varPhi _{0}^{{\mathtt{{w}}}}}+{\mathtt{{w}}}n_{\varPhi _{1}^{{\mathtt{{w}}}}}=n^{\mathtt{{w}}}, \end{aligned}$$

(332)

or, equivalently,

$$\begin{aligned} n_{\varPhi _{0}^{{\mathtt{{w}}}}}=\dfrac{n^{\mathtt{{w}}}-{\mathtt{{w}}}}{(1-{\mathtt{{w}}})}, \end{aligned}$$

(333)

it comes

$$\begin{aligned} n^{\xi ,{\mathtt{{w}}}}=(1-\xi )\frac{(n^{{\mathtt{{w}}}}-{\mathtt{{w}}})}{1-{\mathtt{{w}}}}+ \xi \end{aligned}$$

(334)

and

$$\begin{aligned} \left. \dfrac{\partial n^{\xi ,{\mathtt{{w}}}}}{\partial \xi }\right| _{\xi ={\mathtt{{w}}}}=1-\frac{(n^{{\mathtt{{w}}}}-{\mathtt{{w}}})}{1-{\mathtt{{w}}}}=\dfrac{1-n^{\mathtt{{w}}}}{1-{\mathtt{{w}}}}. \end{aligned}$$

(335)

From the weight derivative expression

$$\begin{aligned} \begin{aligned} \left. \dfrac{\partial f^\xi \left( n^{\xi ,{\mathtt{{w}}}}\right) }{\partial \xi }\right| _{\xi ={\mathtt{{w}}}}=\left. \dfrac{\partial f^\xi (n^{\mathtt{{w}}})}{\partial \xi }\right| _{\xi ={\mathtt{{w}}}}+\left. \dfrac{\partial n^{\xi ,{\mathtt{{w}}}}}{\partial \xi }\right| _{\xi ={\mathtt{{w}}}}\times \left. \dfrac{\partial f^{\mathtt{{w}}}(n)}{\partial n}\right| _{n=n^{\mathtt{{w}}}}, \end{aligned} \end{aligned}$$

(336)

where

$$\begin{aligned} \dfrac{\partial f^\xi (n)}{\partial \xi }=\dfrac{2t(1-\xi )}{\sqrt{(1-\xi )^2-(1-n)^2}}+\dfrac{U}{2}\left[ 1-\dfrac{(n-1)^2(1+3\xi )}{(1-\xi )^3}\right] \end{aligned}$$

(337)

and

$$\begin{aligned} \dfrac{\partial f^{\mathtt{{w}}}(n)}{\partial n}=\dfrac{2t(n-1)}{\sqrt{(1-{\mathtt{{w}}})^2-(1-n)^2}}+U\dfrac{(3{\mathtt{{w}}}-1)(1-n)}{(1-{\mathtt{{w}}})^2}, \end{aligned}$$

(338)

thus leading to

$$\begin{aligned} \begin{aligned} \left. \dfrac{\partial f^\xi \left( n^{\xi ,{\mathtt{{w}}}}\right) }{\partial \xi }\right| _{\xi ={\mathtt{{w}}}}&=\dfrac{2t(1-{\mathtt{{w}}})}{\sqrt{(1-{\mathtt{{w}}})^2-(1-n^{\mathtt{{w}}})^2}} \\&\quad -\dfrac{2t(1-n^{\mathtt{{w}}})^2}{(1-{\mathtt{{w}}})\sqrt{(1-{\mathtt{{w}}})^2-(1-n^{\mathtt{{w}}})^2}} \\&\quad + \dfrac{U}{2}\left[ 1-\dfrac{(n^{\mathtt{{w}}}-1)^2(1+3{\mathtt{{w}}})}{(1-{\mathtt{{w}}})^3}\right] \\&\quad +U\dfrac{(3{\mathtt{{w}}}-1)(1-n^{\mathtt{{w}}})^2}{(1-{\mathtt{{w}}})^3}, \end{aligned} \end{aligned}$$

(339)

or, equivalently,

$$\begin{aligned} \left. \dfrac{\partial f^\xi \left( n^{\xi ,{\mathtt{{w}}}}\right) }{\partial \xi }\right| _{\xi ={\mathtt{{w}}}}=\dfrac{2t\sqrt{(1-{\mathtt{{w}}})^2-(1-n^{\mathtt{{w}}})^2}}{(1-{\mathtt{{w}}})} +\dfrac{U}{2}\left[ 1-\dfrac{3(1-n^{\mathtt{{w}}})^2}{(1-{\mathtt{{w}}})^2}\right] , \end{aligned}$$

(340)

it comes

$$\begin{aligned} f^{\mathtt{{w}}}_{0}\left( n^{\mathtt{{w}}}\right) =-\dfrac{2t\sqrt{(1-{\mathtt{{w}}})^2-(1-n^{\mathtt{{w}}})^2}}{(1-{\mathtt{{w}}})}+\dfrac{U}{2}\left[ 1+\dfrac{(1-n^{\mathtt{{w}}})^2}{(1-{\mathtt{{w}}})^2}\right] \end{aligned}$$

(341)

and

$$\begin{aligned} f^{\mathtt{{w}}}_{1}\left( n^{\mathtt{{w}}}\right) =U\left[ 1-\dfrac{(1-n^{\mathtt{{w}}})^2}{(1-{\mathtt{{w}}})^2}\right] . \end{aligned}$$

(342)

As a result,

$$\begin{aligned} \dfrac{\partial f_0^{\mathtt{{w}}}(n^{\mathtt{{w}}})}{\partial {\mathtt{{w}}}}=\dfrac{2t(n^{\mathtt{{w}}}-1)(n_{\varPsi _1}-1)}{(1-{\mathtt{{w}}})^2\sqrt{(1-{\mathtt{{w}}})^2-(1-n^{\mathtt{{w}}})^2}}+U\dfrac{(n^{\mathtt{{w}}}-1)(n_{\varPsi _1}-1)}{(1-{\mathtt{{w}}})^3} \end{aligned}$$

(343)

and

$$\begin{aligned} \dfrac{\partial f_1^{\mathtt{{w}}}(n^{\mathtt{{w}}})}{\partial {\mathtt{{w}}}}=-\dfrac{2U(n^{\mathtt{{w}}}-1)(n_{\varPsi _1}-1)}{(1-{\mathtt{{w}}})^3}, \end{aligned}$$

(344)

which leads, according to Eq. (327), to the final compact expression

$$\begin{aligned} \begin{aligned} E_{\mathrm{c}}^{{\mathtt{{w}}},\mathrm{DD}}(n^{\mathtt{{w}}})&=-{\mathtt{{w}}}(n^{\mathtt{{w}}}-1)(n_{\varPsi _1}-1) \\&\quad \times \left[ \dfrac{2t}{\sqrt{(1-{\mathtt{{w}}})^2-(1-n^{\mathtt{{w}}})^2}}+\dfrac{U(1+{\mathtt{{w}}})}{(1-{\mathtt{{w}}})^2} \right] . \end{aligned} \end{aligned}$$

(345)