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 $vtnad[\rho A,\rho B](r)$

$vtnad[\rho A,\rho B](r)$ defined as

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, $vtnad[\rho A,\rho B]$ is a component of the effective potential, for which a quantum-mechanical problem of *N*_{A} = ∫*ρ*_{A}(**r**)*d***r** electrons embedded in a field generated by the *N*_{B} = ∫*ρ*_{B}(**r**)*d***r** electrons is solved. The corresponding energy functional,

where the density functional of the kinetic energy (*T*_{s}[*ρ*]), is defined via the constrained search^{5} as

For the sake of subsequent considerations, we introduce the functional $Tsnad(subtr)[\rho tot,\rho B]$, which admits a more restricted pair of densities than does $Tsnad[\rho tot,\rho B]$,

and the corresponding potential

Any pair of densities *ρ*_{A} and *ρ*_{B} is admissible in $Tsnad[\rho A,\rho B]$, provided that each of them is *N*-representable (*N* being either *N*_{A} or *N*_{B}). Admissible pairs of densities in the functional $Tsnad(subtr)[\rho tot,\rho B]$ satisfy a different admissibility condition,

where Δ*V* 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,

*Condition***A** might be violated if *ρ*_{tot} and *ρ*_{B} are independent variables.

The admissibility condition for densities in the case of $vtnad(subtr)[\rho tot,\rho B]$ 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, $vs$-representability, and inversion of the Kohn–Sham equation

For a system of *N* = ∫*ρ*(**r**)*d***r** electrons in an external potential *v*_{ext}(**r**), the Kohn–Sham energy functional reads

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,

where *λ* is the Lagrange multiplier corresponding to the constraint due to normalization *N* = ∫*ρ*(**r**)*d***r**. 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 $(\rho =\u2211iNA|\varphi i|2)$. 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. $EvextKS[\rho o]$ lies above the ground-state energy. The latter minimizes the Hohenberg–Kohn density functional $EvextHK[\rho ]$, which is defined for a larger class of densities (*v*-representable densities). The present work concerns minima and infima of $EvextKS[\rho ]$.

Inserting the explicit form of $EvextKS[\rho ]$, 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,

where$vvext,\rho KS(r)$ denotes *a function*,

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 $Evext[\rho ]KS$ 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,

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,

that relates the functional derivative $\delta Ts[\rho ]\delta \rho (r)$ with the effective potential in the Kohn–Sham equation. In this work, if $vvext,\rho KS(r)$ 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 density^{5} is known as *inverting the Kohn–Sham equation* (see the pioneering work by Zhao, Morrison, and Parr^{7} 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 $Exc[\rho ]\u2248E\u0303xc[\rho ]$. The relevant expression of the energy functional and the corresponding effective potential are, thus,

and

For such a case, *ρ*_{tot} is *v*_{s}-representable by construction. Moreover, the functional *v*_{s}[*ρ*] and the function $v\u0303vext,\rho KS$ become fundamentally different quantities. The function $v\u0303vext,\rho KS$ can be evaluated at a density *ρ* that is not *v*_{s}-representable.

### C. Relation between $vtnad(subtr)[\rho tot,\rho B]$ 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,

and $vtnad(subtr)[\rho tot,\rho B](r)=vtnad[\rho A,\rho B](r)$.

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 $vtnad(subtr)[\rho tot,\rho B]$ numerically using Eq. (16).

## II. EULER–LAGRANGE EQUATION FOR $EvextKS[\rho ]$ 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, $Exc[\rho ]\u2248E\u0303xc[\rho ]$).The difference

*ρ*_{tot}−*ρ*_{B}vanishes on the Lebesgue measurable volume element (*V*_{zero}) on which*ρ*_{tot}> 0,

For such pairs, (*i*) $vs[\rho tot]=v\u0303vext,\rho totKS$ 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*) $E\u0303vextKS[\rho tot\u2212\rho B]=E\u0303vext\u2032KS[\rho tot\u2212\rho B]$ for any two external potentials *v*_{ext} and $vext\u2032$ 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 $E\u0303vextKS[\rho ]$ 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 $E\u0303vextKS[\rho ]$. 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 $E\u0303vextKS[\rho ]$. Some other functionals must be constructed. The functional $\Omega vext[\rho ]$ is defined as

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

for instance.

The lowest value of $\Omega vext[\rho ]$ corresponds to *ρ*_{min} = *ρ*_{tot} − *ρ*_{B} and equals $E\u0303vextKS[\rho min+\rho B]=E\u0303vextKS[\rho tot]$. Owing to the constraint *C*[*ρ*] = 0, *ρ*_{tot} − *ρ*_{B} is a minimum of $\Omega vext[\rho ]$ regardless of whether *ρ*_{tot} − *ρ*_{B} is a minimum or infimum of $E\u0303vextKS[\rho ]$. The density *ρ* = *ρ*_{tot} − *ρ*_{B} satisfies, thus, the following Euler–Lagrange equation:

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 $E\u0303vextKS[\rho ]$. Since *ρ*_{tot} − *ρ*_{B} is, however, a minimum of $\Omega vext[\rho ]$, 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 $E\u0303vextKS[\rho ]$ into Eq. (20) leads to

where

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 $\delta Ts[\rho ]\delta \rho (r)\rho =\rho tot$ and $\delta Ts[\rho ]\delta \rho (r)\rho =\rho tot\u2212\rho B$ used in Eq. (5) are, therefore, also not independent. As a result,

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

If $vextvs$ is used as *v*_{ext} in *v*_{Ω}[*ρ*], the term due to the constraint disappears. The potential $vt,\Omega nad(subtr)[\rho tot,\rho B]$ is uniquely (up to a constant) defined, and it is equal to $vtnad(subtr)[\rho tot,\rho B]$. 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 $E\u0303vextKS[\rho tot\u2212\rho B]$. Such a change of the potential does not affect $\Omega vextB[\rho ]$ 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 $C\u03f5,k[\rho ]=\u222bVzero\rho (r)\u2212\rho tot(r)2kdr\u2212\u03f5$ 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,

If the total density obtained from some Kohn–Sham calculations is partitioned in such a way that *Condition***A′** 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 *Condition***A′** 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 $vtnad(subtr)[\rho tot,\rho B]$ 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).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.