Is the non-additive kinetic potential always equal to the difference of effective potentials from inverting the Kohn–Sham equation?

Wesolowski, Tomasz Adam

doi:10.1063/5.0101791

The relation used frequently in the literature according to which the non-additive kinetic potential which is a functional depending on a pair of electron densities is equal (up to a constant) to the difference of two potentials obtained from inverting two Kohn–Sham equations, is examined. The relation is based on a silent assumption that the two densities can be obtained from two independent Kohn–Sham equations, i.e., are v_s-representable. It is shown that this assumption does not hold for pairs of densities: ρ_tot being the Kohn–Sham density in some system and ρ_B obtained from such partitioning of ρ_tot that the difference ρ_tot − ρ_B vanishes on a Lebesgue measurable volume element. The inversion procedure is still applicable for ρ_tot − ρ_B but cannot be interpreted as the inversion of the Kohn–Sham equation. It is rather the inversion of a Kohn–Sham-like equation. The effective potential in the latter equation comprises a “contaminant” that might even not be unique. It is shown that the construction of the non-additive kinetic potential based on the examined relation is not applicable for such pairs.

I. PRELIMINARY CONSIDERATIONS AND NOTATION

A. The bi-functional of the non-additive kinetic potential $v_{t}^{nad} [ρ_{A}, ρ_{B}] (r)$

$v_{t}^{nad} [ρ_{A}, ρ_{B}] (r)$ defined as

v_{t}^{nad} [ρ_{A}, ρ_{B}] (r) \equiv {(\frac{δ T_{s}^{nad} [ρ, ρ_{B}]}{δ ρ (r)}|}_{ρ = ρ_{A}}

(1)

is a key element in subsystem formulation of density functional theory (DFT)¹ and in Frozen-Density Embedding Theory (FDET).^2–4 In both formalisms found based on the Hohenberg–Kohn theorems, $v_{t}^{nad} [ρ_{A}, ρ_{B}]$ is a component of the effective potential, for which a quantum-mechanical problem of N_A = ∫ρ_A(r)dr electrons embedded in a field generated by the N_B = ∫ρ_B(r)dr electrons is solved. The corresponding energy functional,

T_{s}^{nad} [ρ_{A}, ρ_{B}] \equiv T_{s} [ρ_{A} + ρ_{B}] - T_{s} [ρ_{A}] - T_{s} [ρ_{B}],

(2)

where the density functional of the kinetic energy (T_s[ρ]), is defined via the constrained search⁵ as

T_{s} [ρ] = \min_{Φ \to ρ} ⟨Φ | \hat{T} | Φ⟩ .

(3)

For the sake of subsequent considerations, we introduce the functional $T_{s}^{nad(subtr)} [ρ_{tot}, ρ_{B}]$ ⁠, which admits a more restricted pair of densities than does $T_{s}^{nad} [ρ_{tot}, ρ_{B}]$ ⁠,

T_{s}^{nad(subtr)} [ρ_{tot}, ρ_{B}] \equiv T_{s} [ρ_{tot}] - T_{s} [ρ_{B}] - T_{s} [ρ_{tot} - ρ_{B}],

(4)

and the corresponding potential

v_{t}^{nad(subtr)} [ρ_{tot}, ρ_{B}] \equiv {(\frac{δ T_{s} [ρ]}{δ ρ (r)}|}_{ρ = ρ_{tot}} - {(\frac{δ T_{s} [ρ]}{δ ρ (r)}|}_{ρ = ρ_{tot} - ρ_{B}} .

(5)

Any pair of densities ρ_A and ρ_B is admissible in $T_{s}^{nad} [ρ_{A}, ρ_{B}]$ ⁠, provided that each of them is N-representable (N being either N_A or N_B). Admissible pairs of densities in the functional $T_{s}^{nad(subtr)} [ρ_{tot}, ρ_{B}]$ satisfy a different admissibility condition,

C o n d i t i o n A : \forall_{Δ V} \forall_{r \in Δ V} ρ_{tot} (r) \geq ρ_{B} (r),

(6)

where ΔV is a Lebesgue measurable volume element.

Condition A is satisfied automatically if ρ_tot is constructed as a sum of two N-representable components. In such a case,

T_{s}^{nad(subtr)} [ρ_{A} + ρ_{B}, ρ_{B}] = T_{s}^{nad} [ρ_{A}, ρ_{B}] .

(7)

ConditionA might be violated if ρ_tot and ρ_B are independent variables.

The admissibility condition for densities in the case of $v_{t}^{nad(subtr)} [ρ_{tot}, ρ_{B}]$ is stronger than Condition A. Formulating it involves elements of the Kohn–Sham formalism given below.

B. Elements of the Kohn–Sham formalism: Energy functional, Euler–Lagrange equation, $v_{s}$ -representability, and inversion of the Kohn–Sham equation

For a system of N = ∫ρ(r)dr electrons in an external potential v_ext(r), the Kohn–Sham energy functional reads

E_{v_{ext}}^{K S} [ρ] \equiv T_{s} [ρ] + \int v_{ext} (r) ρ (r) d r + \frac{1}{2} \iint \frac{ρ (r) ρ (r^{'})}{| r - r^{'} |} d r^{'} d r + E_{x c} [ρ],

(8)

with E_xc[ρ] being the exchange–correlation functional in the Kohn–Sham formulation of DFT.⁶

The density ρ is pure-state non-interacting v-representable (v_s-representable in short), if there exists an external potential v_ext such that ρ_o is the lowest energy solution of the Euler–Lagrange equation,

{(\frac{δ E_{v_{ext}}^{K S} [ρ]}{δ ρ (r)}|}_{ρ = ρ_{o}} = λ,

(9)

where λ is the Lagrange multiplier corresponding to the constraint due to normalization N = ∫ρ(r)dr. Equation (9) takes the form of the Kohn–Sham equation if a Slater determinant built up from N_A orbitals (ϕ_i) is used to represent the density $(ρ = \sum_{i}^{N_{A}} | ϕ_{i} |^{2})$ ⁠. For systems for which the ground-state densities are not v_s-representable, the density ρ_o that minimizes the Kohn–Sham functional is not the ground-state density. $E_{v_{ext}}^{K S} [ρ_{o}]$ lies above the ground-state energy. The latter minimizes the Hohenberg–Kohn density functional $E_{v_{ext}}^{H K} [ρ]$ ⁠, which is defined for a larger class of densities (v-representable densities). The present work concerns minima and infima of $E_{v_{ext}}^{K S} [ρ]$ ⁠.

Inserting the explicit form of $E_{v_{ext}}^{K S} [ρ]$ ⁠, given in Eq. (8), to the Euler–Lagrange equation leads to the relation between the functional derivative of T_s[ρ] and a local potential,

{(\frac{δ T_{s} [ρ]}{δ ρ (r)}|}_{ρ = ρ_{o}} + \overset{v_{v_{ext}, ρ_{o}}^{K S} (r)}{\overset{︷}{v_{ext} (r) + \int \frac{ρ_{o} (r^{'})}{| r - r^{'} |} d r^{'} {(+ \frac{δ E_{x c} [ρ]}{δ ρ (r)}|}_{ρ = ρ_{o}}}} - λ = 0,

(10)

where $v_{v_{ext}, ρ}^{K S} (r)$ denotes a function,

v_{v_{ext}, ρ}^{K S} (r) \equiv v_{ext} (r) + \int \frac{ρ (r^{'})}{| r - r^{'} |} d r^{'} + \frac{δ E_{x c} [ρ]}{δ ρ (r)},

(11)

that can be evaluated for any v_ext and admissible ρ. In Eq. (10), however, v_ext and ρ are not independent. This equation holds only if the density ρ_o minimizes $E_{v_{ext} [ρ]}^{K S}$ for v_ext specified in the subscript.

All terms besides v_ext in Eq. (10) are explicit density functionals. The external potential v_ext is, thus, determined uniquely (up to a constant) by the ground-state density ρ_o. The Kohn–Sham potential is, therefore, an implicit functional of density. It will be denoted with v_s[ρ]. Both potentials are equal for v_s-representable densities,

\forall_{v_{s} - r e p r e s e n t a b l e ρ} v_{s} [ρ] (r) = v_{v_{ext}, ρ}^{K S} (r) + c o n s t .,

(12)

where v_ext is the external potential for which ρ is the ground-state density.

For v_s-representable densities, Eq. (10) provides the desired external-potential-free relation,

\forall_{v_{s} - r e p r e s e n t a b l e ρ} \frac{δ T_{s} [ρ]}{δ ρ (r)} + v_{s} [ρ] (r) - λ = 0,

(13)

that relates the functional derivative $\frac{δ T_{s} [ρ]}{δ ρ (r)}$ with the effective potential in the Kohn–Sham equation. In this work, if $v_{v_{ext}, ρ}^{K S} (r)$ is used, it indicates that this potential is available numerically, whereas v_s[ρ] indicates that it is either a mathematical construction or the result of the inversion procedure.

Obtaining the effective potential for a given density⁵ is known as inverting the Kohn–Sham equation (see the pioneering work by Zhao, Morrison, and Parr⁷ or a recent review by Jensen and Wasserman,⁸ for instance). The procedures to invert the Kohn–Sham equation are numerically tedious. Additional assumptions are made in order to obtain the inverted potential uniquely if finite basis sets are used.^7–14 In this work, we understand that the inverted potential is the potential that corresponds exactly to the target density. For the sake of subsequent consideration, we also note that the inversion might be applied for a more general type of equations–Kohn–Sham equations in which additional constraints are imposed on density, for instance. In such a case, the target density does not have to be v_s-representable.

The subsequent considerations concern the case where the exchange–correlation functional is approximated by means of an explicit expression $E_{x c} [ρ] \approx {\tilde{E}}_{x c} [ρ]$ ⁠. The relevant expression of the energy functional and the corresponding effective potential are, thus,

\begin{align} {\tilde{E}}_{v_{ext}}^{K S} [ρ] & \equiv T_{s} [ρ] + \int v_{ext} (r) ρ (r) d r \\ + \frac{1}{2} \iint \frac{ρ (r) ρ (r^{'})}{| r - r^{'} |} d r^{'} d r + {\tilde{E}}_{x c} [ρ] \end{align}

(14)

and

{\tilde{v}}_{v_{ext}, ρ}^{K S} (r) \equiv v_{ext} (r) + \int \frac{ρ (r^{'})}{| r - r^{'} |} d r^{'} + \frac{δ {\tilde{E}}_{x c} [ρ]}{δ ρ (r)} .

(15)

For such a case, ρ_tot is v_s-representable by construction. Moreover, the functional v_s[ρ] and the function ${\tilde{v}}_{v_{ext}, ρ}^{K S}$ become fundamentally different quantities. The function ${\tilde{v}}_{v_{ext}, ρ}^{K S}$ can be evaluated at a density ρ that is not v_s-representable.

C. Relation between $v_{t}^{nad(subtr)} [ρ_{tot}, ρ_{B}]$ and Kohn–Sham potentials in the case of partitioning ρ_tot

Equation (10) makes it possible to express (up to a constant) each functional that is independent of external potential in the right-hand-side of Eq. (5) by means of the potentials that depend explicitly on the external potential, provided that both ρ_tot − ρ_B and ρ_tot are v_s-representable. The present work concerns such cases that ρ_tot is given and it is v_s-representable, whereas ρ_B satisfies Condition A, i.e., it is obtained from some partitioning of ρ_tot. Such pairs of densities are relevant for DFT-based partition formalisms^13,15 in which ρ_tot is the ground-state density corresponding to a given external potential and some approximation for the exchange–correlation potential, whereas ρ_tot − ρ_B is the density obtained from its partitioning. In FDET, ρ_tot is not known a priori, and such pairs are considered only in specially designed methodology oriented studies.^16–19 The requirement that ρ_tot − ρ_B is v_s-representable is stronger than Condition A. It will be referred to as Condition B. If it is satisfied, Eq. (13) applied for the two densities yields the desired relation,

v_{t}^{nad(subtr)} [ρ_{tot}, ρ_{B}] = v_{s} [ρ_{tot} - ρ_{B}] - v_{s} [ρ_{tot}] + c o n s t

(16)

and $v_{t}^{nad(subtr)} [ρ_{tot}, ρ_{B}] (r) = v_{t}^{nad} [ρ_{A}, ρ_{B}] (r)$ ⁠.

If ρ_tot is obtained from some Kohn–Sham calculations, the demanding tasks–the inversion of the Kohn–Sham equation–is needed only for ρ = ρ_tot − ρ_B in order to evaluate $v_{t}^{nad(subtr)} [ρ_{tot}, ρ_{B}]$ numerically using Eq. (16).

In Sec. II, the applicability of Eq. (16), which hinges on Condition B, will be examined for ρ_B, which results from partitioning of some v_s-representable ρ_tot.

II. EULER–LAGRANGE EQUATION FOR $E_{v_{ext}}^{K S} [ρ]$ WITH ADDITIONAL CONSTRAINT

The following pair of densities is considered:

The partitioned density ρ_tot is obtained from the Kohn–Sham equation (from here on, we use tildes to indicate that in practical applications, $E_{x c} [ρ] \approx {\tilde{E}}_{x c} [ρ]$ ⁠).
The difference ρ_tot − ρ_B vanishes on the Lebesgue measurable volume element (V_zero) on which ρ_tot > 0,

\forall_{r \in V_{zero}} ρ_{tot} (r) - ρ_{B} (r) = 0 .

(17)

For such pairs, (i) $v_{s} [ρ_{tot}] = {\tilde{v}}_{v_{ext}, ρ_{tot}}^{K S}$ is finite on V_zero with a possible exception at the positions of the nuclei, where v_ext is singular; (ii) the potential v_s[ρ_tot − ρ_B]–if it exists–is expected to diverge on V_zero because ρ_tot − ρ_B vanishes there; and (iii) ${\tilde{E}}_{v_{ext}}^{K S} [ρ_{tot} - ρ_{B}] = {\tilde{E}}_{v_{ext}^{'}}^{K S} [ρ_{tot} - ρ_{B}]$ for any two external potentials v_ext and $v_{ext}^{'}$ that differ only on V_zero. These observations indicate that the density ρ_tot − ρ_B is not v_s-representable. The potential for which ${\tilde{E}}_{v_{ext}}^{K S} [ρ]$ reaches the lowest value at ρ_tot − ρ_B diverges on V_zero. The addition of any constant or even a non-constant but finite potential on V_zero does not affect ${\tilde{E}}_{v_{ext}}^{K S} [ρ]$ ⁠. If ρ_tot − ρ_B is obtained from the Euler–Lagrange equation (not that of the Kohn–Sham formalism), the effective potential in this equation is, therefore, not defined uniquely on V_zero. As a result, the difference between the two effective potentials corresponding to ρ_tot and ρ_tot − ρ_B might be ambiguous in V_zero.

To further investigate this indication, we construct a generalized Euler–Lagrange equation that guarantees that ρ_tot − ρ_B is stationary. The prerequisite for the existence of an effective potential obtainable from the inversion is that ρ_tot − ρ_B is a minimum of some density functionals that can be used in the Euler–Lagrange equation. If ρ_tot − ρ_B is not v_s-representable, it cannot be ${\tilde{E}}_{v_{ext}}^{K S} [ρ]$ ⁠. Some other functionals must be constructed. The functional $Ω_{v_{ext}} [ρ]$ is defined as

Ω_{v_{ext}} [ρ] = {\tilde{E}}_{v_{ext}}^{K S} [ρ + ρ_{B}] + λ^{C} \cdot C [ρ],

(18)

where C[ρ] = 0 represents the constrained due to Eq. (17) such as

C [ρ] = \int_{V_{zero}} {(ρ (r) - ρ_{tot} (r))}^{2} d r = 0,

(19)

for instance.

The lowest value of $Ω_{v_{ext}} [ρ]$ corresponds to ρ_min = ρ_tot − ρ_B and equals ${\tilde{E}}_{v_{ext}}^{K S} [ρ_{\min} + ρ_{B}] = {\tilde{E}}_{v_{ext}}^{K S} [ρ_{tot}]$ ⁠. Owing to the constraint C[ρ] = 0, ρ_tot − ρ_B is a minimum of $Ω_{v_{ext}} [ρ]$ regardless of whether ρ_tot − ρ_B is a minimum or infimum of ${\tilde{E}}_{v_{ext}}^{K S} [ρ]$ ⁠. The density ρ = ρ_tot − ρ_B satisfies, thus, the following Euler–Lagrange equation:

\begin{align} {(\frac{δ Ω_{v_{ext}} [ρ]}{δ ρ (r)}|}_{ρ = ρ_{tot} - ρ_{B}} & = {(\frac{δ {\tilde{E}}_{v_{ext}}^{K S} [ρ]}{δ ρ (r)}|}_{ρ = ρ_{tot} - ρ_{B}} {(+ λ^{C} \cdot \frac{δ C [ρ]}{δ ρ (r)}|}_{ρ = ρ_{tot} - ρ_{B}} = λ^{″}, \end{align}

(20)

where λ″ is the Lagrange multiplier associated with the normalization condition.

The fact that v_ext is some fixed external potential [the same as the one in Eq. (10), for instance] is crucial in this construction. Instead of searching for the potential, for which ρ_tot − ρ_B is the minimizer of the corresponding Kohn–Sham equation, we introduce a constraint. Owing to the constraint, the requirement of v_s-representability of ρ_tot − ρ_B is not needed and ρ_tot − ρ_B can be just an infimum of ${\tilde{E}}_{v_{ext}}^{K S} [ρ]$ ⁠. Since ρ_tot − ρ_B is, however, a minimum of $Ω_{v_{ext}} [ρ]$ ⁠, the same inversion techniques can be applied as the ones used for the Kohn–Sham equation. The obtained effective potential has, however, another interpretation. If λ^C ≠ 0, it is no longer the Kohn–Sham potential corresponding to any v_ext.

Inserting the definition of ${\tilde{E}}_{v_{ext}}^{K S} [ρ]$ into Eq. (20) leads to

{(\frac{δ T_{s} [ρ]}{δ ρ (r)}|}_{ρ = ρ_{tot} - ρ_{B}} + v_{Ω} [ρ_{tot} - ρ_{B}] - λ^{″} = 0,

(21)

where

\begin{align} v_{Ω} [ρ_{tot} - ρ_{B}] & = λ^{C} \cdot {(\frac{δ C [ρ]}{δ ρ (r)}|}_{ρ = ρ_{tot} - ρ_{B}} + v_{ext} (r) \\ + \int \frac{ρ_{tot} (r^{'}) - ρ_{B} (r^{'})}{| r - r^{'} |} d r^{'} {(+ \frac{δ {\tilde{E}}_{x c} [ρ]}{δ ρ (r)}|}_{ρ = ρ_{tot} - ρ_{B}} . \end{align}

(22)

The above construction of the effective potential v_Ω[ρ_tot − ρ_B] is our principal result, which we will discuss below. If ρ_tot − ρ_B is not v_s-representable, the external potential for which the component of v_Ω[ρ_tot − ρ_B] due to the constraint vanishes does not exist. The component of v_Ω[ρ_tot − ρ_B] due to the constraint does not vanish. Since it depends on ρ_tot, the effective potentials in two Euler–Lagrange equations, for ρ_tot − ρ_B and for ρ_tot, are not independent. The derivatives ${(\frac{δ T_{s} [ρ]}{δ ρ (r)}|}_{ρ = ρ_{tot}}$ and ${(\frac{δ T_{s} [ρ]}{δ ρ (r)}|}_{ρ = ρ_{tot} - ρ_{B}}$ used in Eq. (5) are, therefore, also not independent. As a result,

\begin{align} v_{t, Ω}^{nad(subtr)} [ρ_{tot}, ρ_{B}] & = v_{Ω} [ρ_{tot} - ρ_{B}] - {\tilde{v}}_{v_{ext}, ρ_{tot}}^{K S} \\ \neq v_{t}^{nad(subtr)} [ρ_{tot}, ρ_{B}] + c o n s t ., \end{align}

(23)

if ρ_tot − ρ_B is not v_s-representable.

If ρ_tot − ρ_B is v_s-representable, the external potential for which ρ_tot − ρ_B is the solution of the Kohn–Sham equation exists and is given as

\begin{align} v_{ext}^{v_{s}} (r) & = v_{s} [ρ_{tot} - ρ_{B}] - \int \frac{ρ_{tot} (r^{'}) - ρ_{B} (r^{'})}{| r - r^{'} |} d r^{'} {(- \frac{δ {\tilde{E}}_{x c} [ρ]}{δ ρ (r)}|}_{ρ = ρ_{tot} - ρ_{B}} . \end{align}

(24)

If $v_{ext}^{v_{s}}$ is used as v_ext in v_Ω[ρ], the term due to the constraint disappears. The potential $v_{t, Ω}^{nad(subtr)} [ρ_{tot}, ρ_{B}]$ is uniquely (up to a constant) defined, and it is equal to $v_{t}^{nad(subtr)} [ρ_{tot}, ρ_{B}]$ ⁠. Equation (16), thus, holds without being affected by the present construction.

Turning back to the case of not v_s-representable ρ_tot − ρ_B, we note that any change of the external potential (v_ext) localized only in V_zero does not affect ${\tilde{E}}_{v_{ext}}^{K S} [ρ_{tot} - ρ_{B}]$ ⁠. Such a change of the potential does not affect $Ω_{v_{ext}}^{B} [ρ]$ either. The potential v_Ω[ρ_tot − ρ_B] might depend on the form of the constraint. The constraint C[ρ], which assures that ρ = ρ_tot − ρ_B obtained as a solution of the corresponding Euler–Lagrange equation is a minimum, might have different forms than the example given in Eq. (19). To avoid infinite Lagrange multiplier, one might consider $C_{ϵ, k} [ρ] = \int_{V_{zero}} {(ρ (r) - ρ_{tot} (r))}^{2 k} d r - ϵ$ with an integer k and look for the solution of the Euler–Lagrange equation at the ϵ → 0 limit, for instance. The pathway to reach the limit depends obviously on k. The way the constraint is enforced in V_zero might lead, thus, to different potentials v_Ω[ρ_tot − ρ_B].

III. DISCUSSION

In some multi-level molecular simulations (see Refs. 11–13 and 20–22, for instance), the inversion of the Kohn–Sham equation is made to overcome the difficulties in approximating the non-additive kinetic potential component of the embedding potential by means of an explicit bi-functional. The inversion is made for pairs of densities such as the ones considered in this work (i.e., satisfying Condition A and for v_s-representable ρ_tot). In practice, additional approximations combine with the silent assumption examined in more detail in this work that ρ_tot − ρ_B is v_s-representable. The present analysis provides rather new interpretation of the potential obtained in such methods. It is not the approximation for the non-additive kinetic potential, but it is rather the potential enforcing desired properties of the target density such as localizing it in the pre-defined domain.

The present work also provides a clear hint of how to avoid obtaining meaningless non-additive kinetic potentials from differences of effective potentials in the case of partitioning. A practically impossible to verify condition of the v_s-representability of ρ_tot − ρ_B, which is needed for Eq. (16) to hold, can be replaced by a weaker one,

C o n d i t i o n A^{'} : \forall_{Δ V} \forall_{r \in Δ V} ρ_{tot} (r) > ρ_{B} (r) .

(25)

If the total density obtained from some Kohn–Sham calculations is partitioned in such a way that ConditionA′ is satisfied, then the constraint imposing Eq. (17) in the Euler–Lagrange equation is not needed. Equation (16) can be expected to hold. The verification whether the pair of densities ρ_tot and ρ_B satisfies ConditionA′ is straightforward if ρ_tot is given.

Finally, the present work, might seem related to the treatment of the kinetic energy by Hadjisavvas and Theophilou,²³ in their derivation of the Kohn–Sham equation, which bypasses the v-representability requirement. In our work, we do not bypass the requirement of the v_s-representability of ρ_tot − ρ_B. We rather demonstrate that the potential $v_{t}^{nad(subtr)} [ρ_{tot}, ρ_{B}]$ cannot be obtained from the effective potentials in the corresponding Euler–Lagrange equations if ρ_tot − ρ_B is not v_s-representable. Our approach to deal with densities that are non-v_s-representable in variational calculations is similar in spirit to the one by Gould and Pittalis.²⁴

ACKNOWLEDGMENTS

The author would like to thank Professor Tim Gould and Professor Nikitas Gidopoulos for helpful comments concerning the presented derivations and an anonymous reviewer for bringing to the author’s attention the work of Hadjisavvas and Theophilou and for sharing unpublished notes.

AUTHOR DECLARATIONS

Conflict of Interest

The author has no conflicts to disclose.

Author Contributions

Tomasz Adam Wesolowski: Conceptualization (lead); Formal analysis (lead); Methodology (lead); Writing – original draft (lead); Writing – review & editing (lead).