Multiple impurities and combined local density approximations in site-occupation embedding theory

Abstract

Site-occupation embedding theory (SOET) is an in-principle-exact multi-determinantal extension of density-functional theory for model Hamiltonians. Various extensions of recent developments in SOET (Senjean et al. in Phys Rev B 97:235105, 2018) are explored in this work. An important step forward is the generalization of the theory to multiple-impurity sites. We also propose a new single-impurity density-functional approximation (DFA) where the density-functional impurity correlation energy of the two-level (2L) Hubbard system is combined with the Bethe ansatz local density approximation (BALDA) to the full correlation energy of the (infinite) Hubbard model. In order to test the new DFAs, the impurity-interacting wavefunction has been computed self-consistently with the density-matrix renormalization group method (DMRG). Double occupation and per-site energy expressions have been derived and implemented in the one-dimensional case. A detailed analysis of the results is presented, with a particular focus on the errors induced either by the energy functionals solely or by the self-consistently converged densities. Among all the DFAs (including those previously proposed), the combined 2L-BALDA is the one that performs the best in all correlation and density regimes. Finally, extensions in new directions, like a partition-DFT-type reformulation of SOET, a projection-based SOET approach, or the combination of SOET with Green functions, are briefly discussed as a perspective.

Appendices

Appendix 1: Exact embedding potential at half-filling for multiple impurities

Let us consider any density ${\mathbf {n}}\equiv \lbrace n_i\rbrace _i$ summing up to a number $N=\sum _in_i$ of electrons. Under hole-particle symmetry, this density becomes $(\underline{2} - {\mathbf {n}})\equiv \lbrace 2-n_i\rbrace _i$ and the number of electrons equals $2L-N$ where L is the number of sites. We will prove that these two densities give the same correlation energy for the M-impurity-interacting system. Since, for any local potential ${\mathbf {v}}$, the variational principle in Eq. (5) reads as follows for an impurity-interacting system,

$$\begin{aligned} {\mathcal {E}}^{{\mathrm{imp}}}_M({\mathbf {v}})= \underset{{\mathbf {n}}}{\mathrm{min}} \left\{ F^{\mathrm{imp}}_M({\mathbf {n}}) + ({\mathbf {v}}| {\mathbf {n}}) \right\} , \end{aligned}$$

(65)

which gives, for any density ${\mathbf {n}}$,

$$\begin{aligned} F^{\mathrm{imp}}_M({\mathbf {n}})\ge {\mathcal {E}}^{\mathrm{imp}}_M({\mathbf {v}})-({\mathbf {v}}| {\mathbf {n}}), \end{aligned}$$

(66)

thus leading to the Legendre–Fenchel transform expression,

$$\begin{aligned} F^{\mathrm{imp}}_M({\mathbf {n}}) = \underset{{\mathbf {v}}}{\mathrm{sup}} \left\{ {\mathcal {E}}^{{\mathrm{imp}}}_M({\mathbf {v}}) - ({\mathbf {v}}|{\mathbf {n}})\right\} . \end{aligned}$$

(67)

By applying a hole-particle symmetry transformation to Eq. (67) [we will now indicate the number of particles in the impurity-interacting energies for clarity], we obtain

$$\begin{aligned} F^{\mathrm{imp}}_M(\underline{2} - {\mathbf {n}}) = \underset{{\mathbf {v}}}{\mathrm{sup}} \left\{ {\mathcal {E}}^{{\mathrm{imp}},2L-N}_M({\mathbf {v}}) - 2\sum _i v_i + ({\mathbf {v}}|{\mathbf {n}})\right\} , \end{aligned}$$

(68)

where ${\mathcal {E}}^{{\mathrm{imp}},2L-N}_M({\mathbf {v}})$ is the ($2L-N$)-particle ground-state of the following M-impurity-interacting Hamiltonian:

$$\begin{aligned} {\hat{H}}^{\mathrm{imp}}_M({\mathbf {v}}) &=-t \sum _{i\sigma } \left( {\hat{c}}_{i \sigma }^\dagger {\hat{c}}_{i+1\sigma } + \mathrm {H.c.}\right) + \sum _{i\sigma }v_i {\hat{c}}_{i\sigma }^\dagger {\hat{c}}_{i\sigma } \\& \quad +\, U \sum _{i=0}^{M-1}{\hat{c}}_{i\uparrow }^\dagger {\hat{c}}_{i\uparrow } {\hat{c}}_{i\downarrow }^\dagger {\hat{c}}_{i\downarrow }. \end{aligned}$$

(69)

Applying the hole-particle transformation to the creation and annihilation operators,

$$\begin{aligned} {\hat{c}}_{i\sigma } &^\dagger\rightarrow {\hat{b}}_{i\sigma }^\dagger =(-1)^i{\hat{c}}_{i\sigma } , \\ {\hat{c}}_{i\sigma }&\rightarrow {\hat{b}}_{i\sigma }=(-1)^i{\hat{c}}_{i\sigma }^\dagger , \end{aligned}$$

(70)

to the M-impurity-interacting Hamiltonian in Eq. (69) leads to

$$\begin{aligned} {\hat{H}}^{\mathrm{imp}}_M({\mathbf {v}}) &= -t \sum _{i\sigma } \left( {\hat{b}}_{i \sigma }^\dagger {\hat{b}}_{i+1\sigma } + \mathrm {H.c.}\right) + \sum _{i\sigma } v_i {\hat{b}}_{i\sigma } {\hat{b}}_{i\sigma }^\dagger \\&+ U \sum _{i=0}^{M-1}{\hat{b}}_{i\uparrow } {\hat{b}}_{i\uparrow }^\dagger {\hat{b}}_{i\downarrow } {\hat{b}}_{i\downarrow }^\dagger , \end{aligned}$$

(71)

or, equivalently,

$$\begin{aligned} {\hat{H}}^{\mathrm{imp}}_M({\mathbf {v}}) &= -\,t \sum _{i\sigma } \left( {\hat{b}}_{i \sigma }^\dagger {\hat{b}}_{i+1\sigma } + \mathrm {H.c.}\right) \\& \quad +\,2\sum _i v_i - \sum _{i\sigma } v_i {\hat{b}}_{i\sigma }^\dagger {\hat{b}}_{i\sigma } \\& \quad+\, UM - U \sum _{i=0}^{M-1}\sum _\sigma {\hat{b}}_{i\sigma }^\dagger {\hat{b}}_{i\sigma } \\& \quad +\, U \sum _{i} {\hat{b}}_{i\uparrow }^\dagger {\hat{b}}_{i\uparrow } {\hat{b}}_{i\downarrow }^\dagger {\hat{b}}_{i\downarrow } . \end{aligned}$$

(72)

Then, by substituting and shifting the potential as follows,

$$\begin{aligned} \tilde{v}_i = - v_i - U \sum _{j=0}^{M-1} \delta _{ij} \end{aligned}$$

(73)

we finally obtain

$$\begin{aligned} {\hat{H}}^{\mathrm{imp}}_M(\mathbf {\tilde{v}}) &= -t \sum _{i\sigma } \left( {\hat{b}}_{i \sigma }^\dagger {\hat{b}}_{i+1\sigma } + \mathrm {H.c.}\right) \\& \quad+\,2\sum _i v_i + \sum _{i\sigma } \tilde{v}_i {\hat{b}}_{i\sigma }^\dagger {\hat{b}}_{i\sigma } \\& \quad+\, UM + U \sum _{i} {\hat{b}}_{i\uparrow }^\dagger {\hat{b}}_{i\uparrow } {\hat{b}}_{i\downarrow }^\dagger {\hat{b}}_{i\downarrow }. \end{aligned}$$

(74)

As readily seen from Eqs. (69) and (74), the $(2L-N)$-electron ground-state energy ${\mathcal {E}}^{{\mathrm{imp}},2L-N}_M({\mathbf {v}})$ of ${\hat{H}}^{\mathrm{imp}}_M({\mathbf {v}})$ is connected to the N-electron ground-state energy ${\mathcal {E}}^{\mathrm{imp},N}_M(\mathbf {\tilde{v}})$ of ${\hat{H}}^\mathrm{imp}_M(\mathbf {\tilde{v}})$ by

$$\begin{aligned} {\mathcal {E}}^{{\mathrm{imp}},2L-N}_M ({\mathbf {v}}) = {\mathcal {E}}^{{\mathrm{imp}},N}_M(\mathbf {\tilde{v}}) + 2 \sum _i v_i + MU. \end{aligned}$$

(75)

Introducing Eq. (75) into Eq. (68) leads to

$$\begin{aligned} F^{\mathrm{imp}}_M(\underline{2} - {\mathbf {n}}) &= \underset{{\mathbf {v}}}{\mathrm{sup}} \left\{ {\mathcal {E}}^{{\mathrm{imp}},N}_M(\tilde{{\mathbf {v}}}) +({\mathbf {v}}|{\mathbf {n}})\right\} +MU \\ &= \underset{\tilde{{\mathbf {v}}}}{\mathrm{sup}} \left\{ {\mathcal {E}}^{{\mathrm{imp}},N}_M(\tilde{{\mathbf {v}}}) -(\tilde{{\mathbf {v}}}|{\mathbf {n}})\right\} +U\left( M-\sum _{i=0}^{M-1}n_i\right) \\ &= F^{{\mathrm{imp}}}_M({\mathbf {n}})+U\left( M-\sum _{i=0}^{M-1}n_i\right) . \end{aligned}$$

(76)

Note that the maximizing potential in Eq. (76), denoted by $\tilde{v}^{\mathrm{emb}}_M({\mathbf {n}})$, is nothing but the exact embedding potential $v^{\mathrm{emb}}_M({\mathbf {n}})$ which restores the exact density profile ${\mathbf {n}}$, by definition:

$$\begin{aligned} \tilde{v}^{\mathrm{emb}}_{M,i}({\mathbf {n}}) = v^{\mathrm{emb}}_{M,i}({\mathbf {n}}). \end{aligned}$$

(77)

According to the shift in Eq. (73), this maximizing potential is related to the maximizing one in Eq. (68), denoted by ${\mathbf {v}}^\mathrm{emb}_M(\underline{2} - {\mathbf {n}})$, by

$$\begin{aligned} \tilde{v}_{M,i}^{\mathrm{emb}}({\mathbf {n}}) = - v^{\mathrm{emb}}_{M,i}(\underline{2} - {\mathbf {n}}) - U \sum _{j=0}^{M-1} \delta _{ij}. \end{aligned}$$

(78)

From equality (77), it comes

$$\begin{aligned} {v}_{M,i}^\mathrm{emb}(\underline{2}-\underline{n}) = -{v}^{\mathrm{emb}}_{M,i}(\underline{n}) -U \sum _{j=0}^{M-1} \delta _{ij} \end{aligned}$$

(79)

thus leading to, at half-filling,

$$\begin{aligned} {v}^{\mathrm{emb}}_{M,i}(\underline{1})= - \dfrac{U}{2} \sum _{j=0}^{M-1} \delta _{ij}. \end{aligned}$$

(80)

Appendix 2: Fundamental relation between derivatives in t and U of the complementary bath per-site correlation energy for multiple impurities

If we denote ${\mathbf {v}}_M^{\mathrm{emb}}({\mathbf {n}})$ the maximizing potential in the Legendre–Fenchel transform of Eq. (67), we deduce from the linearity in t and U of the impurity-interacting Hamiltonian that [the dependence in t and U is now introduced for clarity]

$$\begin{aligned} F^{\mathrm{imp}}_M(t,U,{\mathbf {n}}) &= \left[ t \dfrac{\partial {\mathcal {E}}^{\mathrm{imp}}_M(t,U,{\mathbf {v}})}{\partial t} \right. \\&\left. \quad +\, U \dfrac{\partial {\mathcal {E}}^{\mathrm{imp}}_M(t,U,{\mathbf {v}})}{\partial U} \right] \Bigg |_{{\mathbf {v}}={\mathbf {v}}_M^{\mathrm{emb}}({\mathbf {n}})}, \end{aligned}$$

(81)

thus leading to the fundamental relation

$$\begin{aligned} F^{\mathrm{imp}}_M(t,U,{\mathbf {n}}) &= t\dfrac{\partial F^{\mathrm{imp}}_M(t,U,{\mathbf {n}})}{\partial t} \\&+ U\dfrac{\partial F^{\mathrm{imp}}_M(t,U,{\mathbf {n}})}{\partial U}, \end{aligned}$$

(82)

as a consequence of the stationarity condition fulfilled by ${\mathbf {v}}_M^{\mathrm{emb}}({\mathbf {n}})$. Since both the non-interacting kinetic energy [which is obtained when $U=0$] and the impurity Hx functional [first term in the right-hand side of Eq. (19)] fulfill the same relation, we conclude from the decomposition in Eq. (15) that

$$\begin{aligned} E^{\mathrm{imp}}_{\mathrm{c},M}(t,U,{\mathbf {n}}) &= t\dfrac{\partial E^{\mathrm{imp}}_{\mathrm{c},M}(t,U,{\mathbf {n}})}{\partial t} \\&+ U\dfrac{\partial E^{\mathrm{imp}}_{\mathrm{c},M}(t,U,{\mathbf {n}})}{\partial U}. \end{aligned}$$

(83)

We finally obtain, by combining Eqs. (23), (83) and (101), the fundamental relation in Eq. (40).

Appendix 3: Lieb maximization and correlation energy derivatives for a single impurity

The impurity-interacting LL functional in Eq. (7) [we consider the particular case of a single impurity ($M=1$) in the following] can be rewritten as a Legendre–Fenchel transform  [39, 40],

$$\begin{aligned} F^{\mathrm{imp}}(t,U,{\mathbf {n}}) = \underset{{\mathbf {v}}}{\mathrm{sup}} \left\{ {\mathcal {E}}^{\mathrm{imp}}(t,U,{\mathbf {v}}) - ({\mathbf {v}} | {\mathbf {n}}) \right\} , \end{aligned}$$

(84)

where ${\mathcal {E}}^{\mathrm{imp}}(t,U,{\mathbf {v}})$ is the ground-state energy of ${\hat{T}}+U{\hat{n}}_{0\uparrow }{\hat{n}}_{0\downarrow }+\sum _i v_i {\hat{n}}_i$. Note that the dependence in both t and U of $F^{\mathrm{imp}}({\mathbf {n}})$ and ${\mathcal {E}}^\mathrm{imp}({\mathbf {v}})$ has been introduced for clarity. The so-called Lieb maximization [70] procedure described in Eq. (84) has been used in this work in order to compute accurate values of $F^{\mathrm{imp}}(t,U,{\mathbf {n}})$ and $T_\mathrm{s}(t,{\mathbf {n}})=F^{\mathrm{imp}}(t,U=0,{\mathbf {n}})$ for a 8-site ring. The impurity-interacting energy ${\mathcal {E}}^\mathrm{imp}(t,U,{\mathbf {v}})$ has been obtained by performing an exact diagonalization calculation based on the Lanczos algorithm [71]. The impurity correlation energy is then obtained as follows,

$$\begin{aligned} E_\mathrm{c}^{\mathrm{imp}}(t,U,{\mathbf {n}}) = F^{\mathrm{imp}}(t,U,{\mathbf {n}}) - T_\mathrm{s}(t,{\mathbf {n}}) - \dfrac{U}{4}n_0^2. \end{aligned}$$

(85)

Since $\partial F^{\mathrm{imp}}(t,U,{\mathbf {n}})/\partial U=d^\mathrm{imp}(t,U,{\mathbf {n}})$ is the impurity site double occupation obtained for the maximizing potential in Eq. (84) (see Eq. (30) and Eq. (A5) in Ref. [41]), it comes from Eq. (85),

$$\begin{aligned} \dfrac{\partial E_\mathrm{c}^{\mathrm{imp}}(t,U,{\mathbf {n}})}{\partial U}=d^\mathrm{imp}(t,U,{\mathbf {n}})-\dfrac{n^2_0}{4}. \end{aligned}$$

(86)

Moreover, since

$$\begin{aligned}&t\dfrac{\partial F^{\mathrm{imp}}(t,U,{\mathbf {n}})}{\partial t}=T^\mathrm{imp}(t,U,{\mathbf {n}}) \\&=F^{\mathrm{imp}}(t,U,{\mathbf {n}})-Ud^\mathrm{imp}(t,U,{\mathbf {n}}) \end{aligned}$$

(87)

is the impurity-interacting kinetic energy obtained for the maximizing potential in Eq. (84) (see Eq. (35) and Eq. (B6) in Ref. [41]), which gives in the non-interacting case $t\,\partial T_\mathrm{s}(t,{\mathbf {n}})/\partial t=T_\mathrm{s}(t,{\mathbf {n}})$, we recover from Eq. (85) the expression in Eq. (B8) of Ref. [41],

$$\begin{aligned} \dfrac{\partial E_\mathrm{c}^{\mathrm{imp}}(t,U,{\mathbf {n}})}{\partial t} =\dfrac{T^\mathrm{imp}(t,U,{\mathbf {n}})-T_\mathrm{s}(t,{\mathbf {n}}) }{t}, \end{aligned}$$

(88)

which can be further simplified as follows,

$$\begin{aligned} \dfrac{\partial E_\mathrm{c}^{\mathrm{imp}}(t,U,{\mathbf {n}})}{\partial t} &= \dfrac{ E_\mathrm{c}^{\mathrm{imp}}(t,U,{\mathbf {n}})}{t} \\& \quad+\,\dfrac{U}{t}\left[ \dfrac{n_0^2}{4}-d^\mathrm{imp}(t,U,{\mathbf {n}})\right] . \end{aligned}$$

(89)

Interestingly, the derivatives in t and U are connected as follows, according to Eq. (86),

$$\begin{aligned} \dfrac{\partial E_\mathrm{c}^{\mathrm{imp}}(t,U,{\mathbf {n}})}{\partial t} &= \dfrac{ E_\mathrm{c}^{\mathrm{imp}}(t,U,{\mathbf {n}})}{t} \\& \quad -\,\dfrac{U}{t} \dfrac{\partial E_\mathrm{c}^{\mathrm{imp}}(t,U,{\mathbf {n}})}{\partial U}. \end{aligned}$$

(90)

Thus we recover Eq. (83) in the particular single-impurity case.

Similarly, in the fully interacting case, the LL functional can be rewritten as follows, as a consequence of Eq. (5),

$$\begin{aligned} F(t,U,{\mathbf {n}}) = \underset{{\mathbf {v}}}{\mathrm{sup}} \left\{ {E}(t,U,{\mathbf {v}}) - ({\mathbf {v}} | {\mathbf {n}}) \right\} , \end{aligned}$$

(91)

where the t- and U-dependence in both $F({\mathbf {n}})$ and $E({\mathbf {v}})$ is now made explicit. From the correlation energy expression,

$$\begin{aligned} E_\mathrm{c}(t,U,{\mathbf {n}})=F(t,U,{\mathbf {n}})-T_\mathrm{s}(t,{\mathbf {n}})-\dfrac{U}{4}\sum _in_i^2, \end{aligned}$$

(92)

and the expressions for the LL functional derivatives in t and U [those and their above-mentioned impurity-interacting analogs are deduced from the Hellmann–Feynman theorem],

$$\begin{aligned} \dfrac{\partial F(t,U,{\mathbf {n}})}{\partial U}=\sum _id_i(t,U,{\mathbf {n}}), \end{aligned}$$

(93)

and

$$\begin{aligned} t\dfrac{\partial F(t,U,{\mathbf {n}})}{\partial t} &= T (t,U,{\mathbf {n}}) \\ &= F(t,U,{\mathbf {n}})-U \sum _id_i(t,U,{\mathbf {n}}), \end{aligned}$$

(94)

it comes

$$\begin{aligned} \dfrac{\partial E_\mathrm{c}(t,U,{\mathbf {n}})}{\partial U}=\sum _id_i(t,U,{\mathbf {n}})-\dfrac{1}{4}\sum _in_i^2, \end{aligned}$$

(95)

and

$$\begin{aligned} \dfrac{\partial E_\mathrm{c}(t,U,{\mathbf {n}})}{\partial t}=\dfrac{T (t,U,{\mathbf {n}})-T_\mathrm{s} (t,{\mathbf {n}})}{t}. \end{aligned}$$

(96)

Note that $d_i(t,U,{\mathbf {n}})$ and $T (t,U,{\mathbf {n}})$, which have been introduced in Eqs. (93) and (94), denote the site i double occupation and the total (fully interacting) kinetic energy, respectively. Both are calculated for the maximizing potential in Eq. (91). For a uniform density profile $\underline{n}$, the per-site correlation energy reads

$$\begin{aligned} e_\mathrm{c}(t,U,n) &= \dfrac{E_\mathrm{c}(t,U,\underline{n})}{L} \\ &= \dfrac{ 1}{L} \left( F(t,U,\underline{n})-T_\mathrm{s}(t,\underline{n})\right) -\dfrac{U}{4}n^2. \end{aligned}$$

(97)

Since, in this case, $d_i(t,U,\underline{n})=d(t,U,n)$ is site-independent, we finally obtain from Eqs. (94), (95), and (96),

$$\begin{aligned} \dfrac{\partial e_\mathrm{c}(t,U,{n})}{\partial U}=d(t,U,{n})-\dfrac{n^2}{4}, \end{aligned}$$

(98)

and

$$\begin{aligned} \dfrac{\partial e_\mathrm{c}(t,U,{n})}{\partial t} &= \dfrac{F(t,U,\underline{n}) -T_\mathrm{s} (t,\underline{n})}{tL} -\dfrac{U}{t}d(t,U,{n}). \end{aligned}$$

(99)

By analogy with Eq. (89), the latter expression can be simplified as follows,

$$\begin{aligned} \dfrac{\partial e_\mathrm{c}(t,U,{n})}{\partial t}&= \dfrac{ e_\mathrm{c}(t,U,{n})}{t} +\dfrac{U}{t}\left[ \dfrac{n^2}{4}-d(t,U,{n})\right] , \end{aligned}$$

(100)

or, equivalently (see Eq. (98)),

$$\begin{aligned} \dfrac{\partial e_\mathrm{c}(t,U,{n})}{\partial t} &= \dfrac{ e_\mathrm{c}(t,U,{n})}{t} -\dfrac{U}{t} \dfrac{\partial e_\mathrm{c}(t,U,{n})}{\partial U} . \end{aligned}$$

(101)

Appendix 4: Derivatives of BALDA

1.1 Derivative with respect to U and t

As readily seen in Eq. (31), the derivative of the complementary bath per-site correlation energy functional with respect to U is necessary to compute double occupation in SOET. According to Eq. (24), it implies the derivative of the conventional per-site correlation energy, modeled with BALDA, which reads

$$\begin{aligned}&\dfrac{\partial e_\mathrm{c}^\mathrm{BALDA}(n \leqslant 1, U/t)}{\partial U} = \dfrac{\partial \beta (U/t)}{\partial U} \\&\quad \left[ \dfrac{-2t}{\pi } \sin \left( \dfrac{\pi n}{\beta (U/t)} \right) \right. \\&\quad \left. + \dfrac{2tn}{\beta (U/t)} \cos \left( \dfrac{\pi n}{\beta (U/t)}\right) \right] - \dfrac{n^2}{4}, \end{aligned}$$

(102)

and then for $n > 1$:

$$\begin{aligned}&\dfrac{\partial e_\mathrm{c}^\mathrm{BALDA}(n > 1, U/t)}{\partial U} = \dfrac{\partial \beta (U/t)}{\partial U} \\&\quad \left[ \dfrac{-2t}{\pi } \sin \left( \dfrac{\pi (2-n)}{\beta (U/t)} \right) + \dfrac{2t(2-n)}{\beta (U/t)} \cos \left( \dfrac{\pi (2-n)}{\beta (U/t)}\right) \right] \\&\quad + (n-1) - \dfrac{n^2}{4} \end{aligned}$$

(103)

where $\partial \beta (U/t) / \partial U = ( \partial \beta (U/t) / \partial (U/t) )/t$, is computed with finite differences by solving Eq. (47) for $\beta (U/t)$.

The derivative with respect to t is calculated according to Eq. (101).

1.2 Derivative with respect to n

To get the correlation embedding potential, the derivatives of the correlation functionals with respect to n is necessary. The derivative of the convention per-site density-functional correlation energy reads

$$\begin{aligned} \dfrac{\partial e_\mathrm{c}^\mathrm{BA}(n \leqslant 1)}{\partial n} &= - 2t \cos \left( \dfrac{\pi n}{\beta (U/t)}\right) \\& \quad +\, 2t \cos \left( \dfrac{\pi n}{2} \right) - \dfrac{Un}{2}, \end{aligned}$$

(104)

and

$$\begin{aligned} \dfrac{\partial e_\mathrm{c}^\mathrm{BA}(n > 1)}{\partial n}& = 2t \cos \left( \dfrac{\pi (2-n)}{\beta (U/t)}\right) \\& \quad-\, 2t \cos \left( \dfrac{\pi (2-n)}{2} \right) + U - \dfrac{Un}{2}. \end{aligned}$$

(105)

Appendix 5: Derivatives of SIAM-BALDA

The derivatives of the SIAM-BALDA impurity correlation functional [Eq. (51)] are given with respect to U for $n \leqslant 1$ as follows,

$$\begin{aligned}&\dfrac{\partial E_{\mathrm{c}, U/\Gamma \rightarrow 0}^{\mathrm{SIAM}}(U,\Gamma (t,n)) }{\partial U} = - \dfrac{2 \times 0.0369}{\pi }\left( \dfrac{U}{\Gamma (t,n)}\right) \\&\quad + \dfrac{4\times 0.0008}{\pi ^3}\left( \dfrac{U}{\Gamma (t,n)}\right) ^3. \end{aligned}$$

(106)

The derivative with respect to t is given according to Eq. (90). Then, the impurity correlation potential is determined by the derivative of the functional with respect to the occupation number n:

$$\begin{aligned}&\dfrac{\partial E_{\mathrm{c}, U/\Gamma \rightarrow 0}^{\mathrm{SIAM}}(U,\Gamma (t,n))}{\partial n} \\&\quad = \dfrac{\partial \Gamma (t,n)}{\partial n} \dfrac{\partial E_{\mathrm{c}, U/\Gamma \rightarrow 0}^{\mathrm{SIAM}}(U,\Gamma )}{\partial \Gamma }\Bigg |_{\Gamma = \Gamma (t,n)}, \end{aligned}$$

(107)

where

$$\begin{aligned} \dfrac{\partial E_{\mathrm{c}, U/\Gamma \rightarrow 0}^{\mathrm{SIAM}}(U,\Gamma ) }{\partial \Gamma } = \dfrac{0.0369}{\pi } \left( \dfrac{U}{\Gamma }\right) ^2 - \dfrac{3\times 0.0008}{\pi ^3}\left( \dfrac{U}{\Gamma }\right) ^4, \end{aligned}$$

(108)

and

$$\begin{aligned}&\dfrac{\partial \Gamma (t,n)}{\partial n} \\&\quad = t \left( \dfrac{-\dfrac{\pi }{2}\sin ^2 (\pi n /2) - (1 + \cos (\pi n /2)) \dfrac{\pi }{2}\cos (\pi n /2)}{\sin ^2(\pi n /2)} \right) \\&\quad = - \dfrac{\pi t}{2} \left( \dfrac{1 + \cos (\pi n /2)}{\sin ^2(\pi n /2)} \right) = - \dfrac{\pi \Gamma (t,n)}{2\sin (\pi n /2)}. \end{aligned}$$

(109)

If $n > 1$, the particle-hole formalism imposes to use $\Gamma (t,2-n)$ instead of $\Gamma (t,n)$. The derivatives with respect to n should be changed accordingly.

Appendix 6: Derivatives of 2L-BALDA

1.1 Parametrization of the correlation energy of the dimer

In this section, we summarize the parametrization of the Hubbard dimer correlation energy by Carrascal and co-workers [51, 52], necessary to understand the following derivations. The equations coming from their paper are referred to as (& N), where N is the number of the equation. We start from the definition of the correlation energy, where n is the occupation of the site 0 and $u = U/2t$ is a dimensionless parameter,

$$\begin{aligned} E_\mathrm{c}^\mathrm{2L}(U,n) = f(g,\rho )\Bigg |_{ \begin{array}{c} g=g(\rho ,u)\\ {\rho } = | n-1 | \end{array} } - T_\mathrm{s}(n) - E_\mathrm{Hx}(U,n), \end{aligned}$$

(110)

where 2L refers to “two-level”, and

$$\begin{aligned} T_\mathrm{s}(n) = -2t \sqrt{n(2-n)}, ~~~~ E_\mathrm{Hx}(U,n) = U\left( 1 - n\left( 1 - \dfrac{n}{2} \right) \right) . \end{aligned}$$

(111)

To account for particle-hole symmetry of the functional, the variable $\rho = | n-1 |$ is used rather than n directly. We now simply follow the guidelines from Eq.(& 102) to (& 107), leading to

$$\begin{aligned} f(g, \rho ) = -2t g + U h(g,\rho ), \end{aligned}$$

(112)

and

$$\begin{aligned} h(g,\rho ) = \dfrac{g^2\left( 1 - \sqrt{1 - \rho ^2 - g^2}\right) + 2\rho ^2}{2(g^2 + \rho ^2)}. \end{aligned}$$

(113)

Then, they proposed a first approximation to $g(\rho ,u)$, denoted by the label 0:

$$\begin{aligned} g_0(\rho ,u) = \sqrt{\dfrac{(1 - \rho )(1 + \rho (1 + (1+\rho )^3 u a_1(\rho ,u)))}{1 + (1+\rho )^3 u a_2(\rho ,u)}}, \end{aligned}$$

(114)

where

$$\begin{aligned} a_i(\rho ,u) = a_{i1}(\rho ) + u a_{i2}(\rho ), ~~~ i = 1, 2 \end{aligned}$$

(115)

and

$$ \begin{aligned}a_{21}(\rho ) = \dfrac{1}{2} \sqrt{\dfrac{\rho (1-\rho)}{2}} , \quad a_{12}(\rho ) = \dfrac{1}{2}(1-\rho ), \\ \quad a_{11}(\rho ) = a_{12}\left( 1 + \dfrac{1}{\rho } \right), \quad a_{22}(\rho ) = \dfrac{a_{12}(\rho )}{2}.\end{aligned}$$

(116)

Plugging $g=g_0(\rho ,u)$ into $f(g,\rho )$ leads to the first parametrization of $E_\mathrm{c}^{2L}(n)$ in Eq. (110). In this work, we implemented the more accurate parametrization, given in Eq.(& 114) [52]:

$$\begin{aligned} g_1(\rho ,u) = g_0(\rho ,u) + \left( u \dfrac{\partial h(g,\rho )}{\partial \rho }\Bigg |_{g = g_0(\rho ,u)} -1 \right) q(\rho , u), \end{aligned}$$

(117)

and where $q(\rho ,u)$ is given in Eq. (& 115) by [52]:

$$\begin{aligned} q(\rho , u) = \dfrac{(1-\rho )(1+\rho )^3 u^2 [ (3\rho /2 - 1 + \rho (1+\rho )^3 u a_2(\rho ,u))a_{12}(\rho ) - \rho (1 + (1 + \rho )^3 u a_1(\rho ,u))a_{22}(\rho )]}{2 g_0(\rho , u) (1 + (1 + \rho )^3 u a_2(\rho ,u) )^2}. \end{aligned}$$

(118)

The accurate pametrization of $E_\mathrm{c}^{2L}(n)$ is obtained by plugging this $g_1(\rho ,u)$ into $f(g,\rho )$, instead of $g_0(\rho , u)$.

In order to obtain the impurity correlation energy, a simple scaling of the interaction parameter U has to be applied on the conventional correlation energy, as demonstrated in Ref. [40] and given in Eq. (55), leading to

$$\begin{aligned}&E_\mathrm{c}^\mathrm{imp, 2L}(U, n) = E_\mathrm{c}^{2L}(U/2,n) \\&\quad = f(g,\rho )\Bigg |_ {\begin{array}{c} g=g(\rho ,u/2)\\ {\rho } = | n-1 | \end{array} } - T_\mathrm{s}(n) - E_\mathrm{Hx}(U/2,n). \end{aligned}$$

(119)

1.2 Derivative with respect to U and t

We compute the derivative with respect to the dimensionless parameter $u = U/2t$. The $\rho$- and u- dependence of $g(\rho , u)$ will be omitted for readability. Besides, many functions will be introduced, aiming to make the implementation and its numerical verification easier. Starting with

$$\begin{aligned} \dfrac{\partial E_\mathrm{c}^\mathrm{2L}(n)}{\partial U}& = \dfrac{1}{2t} \dfrac{\partial f(g,\rho )}{\partial u}\Bigg |_ {\begin{array}{c} g=g(\rho ,u)\\ {\rho } = | n-1 | \end{array} } \\& \quad-\, \left( 1 + n\left( \dfrac{1}{2}n - 1\right) \right) , \end{aligned}$$

(120)

the impurity correlation functional reads, according to Eq. (119),

$$\begin{aligned} \dfrac{\partial E_\mathrm{c}^\mathrm{imp,2L}(n)}{\partial U}& = \dfrac{1}{4t} \dfrac{\partial f(g,\rho )}{\partial u}\Bigg |_ {\begin{array}{c} g=g(\rho ,u/2)\\ {\rho } = | n-1 | \end{array} } \\& \quad-\, \dfrac{1}{2}\left( 1 + n\left( \dfrac{1}{2}n - 1\right) \right) , \end{aligned}$$

(121)

with

$$\begin{aligned} \dfrac{\partial f(g,\rho )}{\partial u} = -2t\left( \dfrac{\partial g}{\partial u} - h(g,\rho ) - u \times \dfrac{\partial h(g,\rho )}{\partial u}\right) . \end{aligned}$$

(122)

The derivative of $h(g,\rho )$ is quite easy, as its only u-dependence is contained in g, so that:

$$\begin{aligned} \dfrac{\partial h(g,\rho )}{\partial u} = \dfrac{\partial g}{\partial u} \dfrac{\partial h(g, \rho )}{\partial g}, \end{aligned}$$

(123)

with

$$\begin{aligned} \dfrac{\partial h(g,\rho )}{\partial g} = g\dfrac{g^4 + 3g^2\rho ^2 + 2\rho ^2\left( \rho ^2 - 1 - Y(g,\rho )\right) }{2\left( g^2 + \rho ^2\right) ^2Y(g,\rho )}, \end{aligned}$$

(124)

where the function $Y(g,\rho ) = \sqrt{1 - g^2 - \rho ^2}$ has been introduced. For the first approximation, $g = g_0$ and

$$\begin{aligned} \dfrac{\partial g_0(\rho , u)}{\partial u} = \dfrac{\partial \sqrt{G(\rho , u)}}{\partial u} = \dfrac{\partial G(\rho , u)/\partial u}{2\sqrt{G(\rho , u)}}, \end{aligned}$$

(125)

where $G(\rho ,u) = N(\rho ,u)/D(\rho ,u)$ and

$$\begin{aligned} N(\rho ,u)& = (1 - \rho ) \left[ 1 + \rho \left( 1 + (1 + \rho )^3 u a_1(\rho , u)\right) \right] , \end{aligned}$$

(126)

and

$$\begin{aligned} D(\rho , u)& = 1 + (1 + \rho )^3 u a_2(\rho , u). \end{aligned}$$

(127)

Their respective derivative with respect to u reads

$$\begin{aligned} \dfrac{\partial N(\rho , u)}{\partial u}& = (1 - \rho ) \rho (1 + \rho )^3 \\&\left( a_1(\rho ,u) + u \dfrac{\partial a_1(\rho , u)}{\partial u} \right) \end{aligned}$$

(128)

and

$$\begin{aligned} \dfrac{\partial D(\rho ,u)}{\partial u}& = \left( 1 + \rho \right) ^3 \left( a_2(\rho ,u) + u \dfrac{\partial a_2(\rho , u)}{\partial u} \right) , \end{aligned}$$

(129)

with

$$\begin{aligned} \dfrac{\partial a_{2}(\rho , u)}{\partial u} = a_{22}(\rho ), ~~~~ \dfrac{\partial a_{1}(\rho , u)}{\partial u} = a_{12}(\rho ). \end{aligned}$$

(130)

Turning to the second approximation $g = g_1$ implemented in this work, one get from the derivative of Eq. (117),

$$\begin{aligned} \dfrac{\partial g_1}{\partial u}& = \dfrac{\partial g_0}{\partial u} + \left( \dfrac{\partial h(g,u)}{\partial g} \Bigg |_{g=g_0} + u\dfrac{\partial }{\partial u}\left( \dfrac{\partial h(g,u)}{\partial g}\Bigg |_{g=g_0}\right) \right) \\&q(\rho ,u)+ \left( u \dfrac{\partial h(g,u)}{\partial g}\Bigg |_{g=g_0} - 1 \right) \dfrac{\partial q(\rho ,u)}{\partial u}. \end{aligned}$$

(131)

For convenience, we introduce two functions w(g, u) and v(g, u) so that

$$\begin{aligned}&\dfrac{\partial }{\partial u} \left( \dfrac{\partial h(g,u)}{\partial g} \right) = \left( \dfrac{\partial w(g,u)}{\partial u}v(g,u)\right. \\&\quad \left. - w(g,u)\dfrac{\partial v(g,u)}{\partial u}\right) \Big /w(g,u)^2, \end{aligned}$$

(132)

with

$$\begin{aligned} w(g,u)& = g\left[ g^4 + 3g^2\rho ^2 + 2\rho ^2\left( \rho ^2 - 1 - Y(g,\rho )\right) \right] , \end{aligned}$$

(133)

$$\begin{aligned} v(g,u)& = 2Y(g,\rho ) \left( g^2 + \rho ^2\right) ^2, \end{aligned}$$

(134)

and

$$\begin{aligned} \dfrac{\partial w(g,u)}{\partial u}& = \dfrac{\partial g}{\partial u} \left[ g^4 + 3g^2\rho ^2 + 2\rho ^2 \left( \rho ^2 - 1 - Y(g,\rho )\right) \right. \\&\left. quad +\, g\left( 4g^3 + 6g\rho ^2+ \dfrac{2\rho ^2g}{Y(g,\rho )} \right) \right] , \end{aligned}$$

(135)

$$\begin{aligned} \dfrac{\partial v(g,u)}{\partial u}& = g\left( g^2 + \rho ^2\right) \dfrac{\partial g}{\partial u} \\& \quad \left[ \dfrac{-2\left( g^2 + \rho ^2\right) }{Y(g,\rho )} + 8Y(g,\rho ) \right] . \end{aligned}$$

(136)

Finally, the last term in Eq. (131) reads, for $q(\rho ,u) = j(\rho ,u)k(\rho ,u)/l(\rho ,u)$:

$$\begin{aligned} \dfrac{\partial q(\rho ,u)}{\partial u} = \dfrac{\left( \dfrac{\partial j(\rho ,u)}{\partial u}k(\rho ,u) + j(\rho ,u)\dfrac{\partial k(\rho ,u)}{\partial u}\right) l(\rho ,u) - j(\rho ,u)k(\rho ,u)\dfrac{\partial l(\rho ,u)}{\partial u}}{l(\rho ,u)^2} \end{aligned}$$

(137)

with

$$\begin{aligned} j(\rho ,u)& = (1-\rho )(1+\rho )^3 u^2, \end{aligned}$$

(138)

$$\begin{aligned} k(\rho , u)& = \left( 3\rho /2 - 1 + \rho (1 + \rho )^3 u a_2 (\rho , u)\right) a_{12}(\rho ) \\& \quad-\, \rho \left( 1 + (1+\rho )^3 \lambda u a_1 (\rho , u)\right) a_{22}(\rho ), \end{aligned}$$

(139)

and

$$\begin{aligned} l(u)& = 2g_0(u) \left[ 1 + (1+\rho )^3 \lambda u a_2 (\rho , u,\lambda ) \right] ^2, \end{aligned}$$

(140)

and their derivative with respect to u:

$$\begin{aligned}&\dfrac{\partial j(\rho ,u)}{\partial u} = 2(1-\rho )(1+\rho )^3\lambda u, \end{aligned}$$

(141)

$$\begin{aligned}&\dfrac{\partial k(u)}{\partial u} = a_{12}(\rho )\left[ \rho (1+\rho )^3 \left( a_2 (\rho ,u) + u \dfrac{\partial a_2 (\rho ,u)}{\partial u} \right) \right] \\&\quad - a_{22}(\rho ) \left[ \rho (1+\rho )^3 \left( a_1 (\rho ,u) + u\dfrac{\partial a_1(\rho ,u)}{\partial u}\right) \right], \end{aligned}$$

(142)

and

$$\begin{aligned}&\dfrac{\partial l(u)}{\partial u} = 4g_0(u) \left[ 1 + (1+\rho )^3 u a_2 (\rho ,u)\right] (1+\rho )^3 \\&\quad \left( a_2(\rho ,u) + u \dfrac{\partial a_2(\rho ,u)}{\partial u} \right) \\&\quad + 2\dfrac{\partial g_0}{\partial u} \left[ 1 + (1+\rho )^3 \lambda u a_2 (\rho ,u) \right] ^2. \end{aligned}$$

(143)

The derivative with respect to t is given according to Eq. (90).

1.3 Derivative with respect to n

Regarding the derivative with respect to n which is necessary to get the embedded correlation potential, it comes

$$\begin{aligned} \dfrac{\partial E_\mathrm{c}^\mathrm{imp,2L}(n)}{\partial n} = \dfrac{\partial \rho }{\partial n} \dfrac{f(g,\rho )}{\partial \rho }\Bigg |_ {\begin{array}{c} g=g(\rho ,u/2)\\ {\rho } = | n-1 | \end{array} } - \dfrac{\partial {\mathcal {T}}_s(n)}{\partial n} - \dfrac{U}{2} \end{aligned}$$

(144)

where $\partial \rho / \partial n = {\text {sign}}(n-1)$ and

$$\begin{aligned} \dfrac{\partial {\mathcal {T}}_s(n)}{\partial n} = - \dfrac{2t(1 - n)}{\sqrt{n(2-n)}}. \end{aligned}$$

(145)

We start with

$$\begin{aligned} \dfrac{\partial f(g,\rho )}{\partial \rho } = -2t\dfrac{\partial g}{\partial \rho } + U \dfrac{\partial h(g,\rho )}{\partial \rho }, \end{aligned}$$

(146)

where, for the first parametrization using $g = g_0(\rho ,u)$,

$$\begin{aligned} \dfrac{\partial g_0}{\partial \rho } = =\dfrac{1}{2g_0D(\rho ,u)} \left( \dfrac{\partial N(\rho ,u)}{\partial \rho } - g_0^2 \dfrac{\partial D(\rho ,u)}{\partial \rho } \right) , \end{aligned}$$

(147)

with

$$\begin{aligned}&\dfrac{\partial N(\rho , u)}{\partial \rho } = -1 + (1 - 2\rho )\left( 1+(1+\rho )^3ua_1(\rho , u)\right) \\&\quad +\, \rho u(1-\rho )(1+\rho )^2 \\&\quad \left( 3a_1(\rho , u) + (1+\rho )\dfrac{\partial a_1(\rho , u)}{\partial \rho }\right) , \end{aligned}$$

(148)

$$\begin{aligned}&\dfrac{\partial D(\rho , u)}{\partial \rho } = u (1+\rho )^2 \\&\quad \left( 3a_2(\rho , u) + (1-\rho )\dfrac{\partial a_2(\rho , u)}{\partial \rho }\right) , \end{aligned}$$

(149)

and

$$\begin{aligned}&\dfrac{\partial a_1(\rho , u)}{\partial \rho } = \dfrac{\partial a_{11}(\rho )}{\partial \rho } + u\dfrac{\partial a_{12}(\rho )}{\partial \rho }, \\&\quad \dfrac{\partial a_2(\rho , u)}{\partial \rho } = \dfrac{\partial a_{21}(\rho )}{\partial \rho } + u\dfrac{\partial a_{22}(\rho )}{\partial \rho } ,\end{aligned}$$

(150)

$$\begin{aligned}&\dfrac{\partial a_{12}(\rho )}{\partial \rho } = 2 \dfrac{\partial a_{22}(\rho )}{\partial \rho } = - \dfrac{1}{2}, \\&\quad \dfrac{\partial a_{21}(\rho )}{\partial \rho } = \dfrac{1 - 2\rho }{2\sqrt{(1-\rho )\rho /2}} , \\&\quad \dfrac{\partial a_{11}(\rho )}{\partial \rho } = \dfrac{\partial a_{21}(\rho )}{\partial \rho } \left( 1 + \dfrac{1}{\rho } \right) - \dfrac{1}{\rho ^2} a_{21}(\rho ). \end{aligned}$$

(151)

Then, the right term in the right-hand side of Eq. (146) is derived as:

$$\begin{aligned}&\dfrac{\partial h(g,\rho )}{\partial \rho } = \dfrac{1}{2(g^2 + \rho ^2)} \\&\quad \left( 4\rho + 2g\dfrac{\partial g}{\partial \rho } \left( 1 - Y(g,\rho ) \right) + g^2 \dfrac{g (\partial g/\partial \rho ) + \rho }{Y(g,\rho )} \right) \\&\quad - \dfrac{g(\partial g / \partial \rho ) + \rho }{(g^2 + \rho ^2)^2} \left( 2\rho ^2 + g^2 \left( 1 - Y(g,\rho ) \right) \right) . \end{aligned}$$

(152)

Turning to the second parametrization $g = g_1$, the derivative with respect to $\rho$ leads to

$$\begin{aligned} \dfrac{\partial g_1}{\partial \rho }& = \dfrac{\partial g_0}{\partial \rho } + \left( u\dfrac{\partial h(g,\rho )}{\partial g}\Bigg |_{g=g_0} - 1\right) \dfrac{\partial q(\rho ,u)}{\partial \rho } \\&+ u\dfrac{\partial }{\partial \rho } \left( \dfrac{\partial h(g,\rho )}{\partial g}\Bigg |_{g=g_0}\right) q(\rho ,u) \end{aligned}$$

(153)

with

$$\begin{aligned}&\dfrac{\partial }{\partial \rho } \left( \dfrac{\partial h(g,\rho )}{\partial g}\right) = \dfrac{- (\partial g / \partial \rho ) (g^2 + \rho ^2) + 4g(g ( \partial g / \partial \rho ) + \rho ) }{(g^2 + \rho ^2)^3} \\&\quad \left( 2\rho ^2 + g^2(1 - Y(g,\rho )) \right) \\&\quad - \dfrac{g}{(g^2 + \rho ^2)^2} \left( 4\rho + 2g\dfrac{\partial g}{\partial \rho } \left( 1 - Y(g,\rho ) \right) + g^2 \dfrac{g (\partial g / \partial \rho ) + \rho }{Y(g,\rho )} \right) \\&\quad - \dfrac{g(\partial g / \partial \rho ) + \rho }{(g^2 + \rho ^2)^2} \left( 2g(1 - Y(g,\rho )) + \dfrac{g^3}{Y(g,\rho )} \right) \\&\quad + \dfrac{1}{2(g^2 + \rho ^2)} \left( 2\dfrac{\partial g}{\partial \rho } (1 - Y(g,\rho )) + \dfrac{2g \rho }{Y(g,\rho )}\right. \\&\quad \left. + \dfrac{5g^2 (\partial g / \partial \rho ) }{Y(g,\rho )} + g^3\dfrac{g(\partial g / \partial \rho ) + \rho }{Y(g,\rho )^3} \right) . \end{aligned}$$

(154)

Finally,

$$\begin{aligned} \dfrac{\partial q(\rho ,u)}{\partial \rho }& = \left( \dfrac{\partial P(\rho ,u)}{\partial \rho }Q(\rho ,u)\right. \\&\left. - P(\rho ,u)\dfrac{\partial Q(\rho ,u)}{\partial \rho }\right) \Big / Q(\rho ,u)^2, \end{aligned}$$

(155)

with

$$\begin{aligned}&\dfrac{\partial P(\rho ,u)}{\partial \rho } = \left( 3(1-\rho )(1+\rho )^2 - (1+\rho )^3 \right) u^2 \\&\quad \left[ \left( \dfrac{3\rho }{2} - 1 + \rho (1 + \rho )^3 u a_2(\rho , u) \right) a_{12}(\rho )\right. \\&\quad \left. - \rho \left( 1 + (1 + \rho )^3 u a_1(\rho ,u)\right) a_{22}(\rho ) \right] \\&\quad + (1-\rho )(1+\rho )^3 u^2 \left[ \left( \dfrac{3}{2} + 3 u (1+\rho )^2 \rho a_2(\rho ,u)\right. \right. \\&\quad \left. \left. + u(1+\rho )^3 \left( a_2(\rho ,u) + \rho \dfrac{\partial a_2(\rho ,u)}{\partial \rho } \right) \right) a_{12}(\rho ) \right. \\&\quad + \left( \dfrac{3\rho }{2} - 1 + \rho (1+\rho )^3 u a_2(\rho ,u)\right) \\&\quad \dfrac{\partial a_{12}(\rho )}{\partial \rho } - \left( \rho \dfrac{\partial a_{22}(\rho )}{\partial \rho } + a_{22}(\rho )\right) \\&\quad \left( 1 + (1+\rho )^3 u a_1(\rho ,u)\right) \\&\left. - \rho a_{22}(\rho )\left( 3(1+\rho )^2 u a_1(\rho ,u) + (1+\rho )^3 u \dfrac{\partial a_1(\rho ,u)}{\partial \rho } \right) \right] \end{aligned}$$

(156)

and

$$\begin{aligned}&\dfrac{\partial Q(\rho ,u)}{\partial \rho } = 2\dfrac{\partial g_0}{\partial \rho } \left( 1+(1+\rho )^3 u a_2(\rho ,u)\right) ^2 \\&\quad + 4g_0\left( 1+(1+\rho )^3 u a_2(\rho ,u)\right) u \\&\quad \left( 3(1+\rho )^2a_2(\rho ,u) + (1+\rho )^3 \dfrac{\partial a_2(\rho ,u)}{\partial \rho }\right) . \end{aligned}$$

(157)