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 vs-representable. It is shown that this assumption does not hold for pairs of densities: ρtot being the Kohn–Sham density in some system and ρB obtained from such partitioning of ρtot that the difference ρtotρB vanishes on a Lebesgue measurable volume element. The inversion procedure is still applicable for ρtotρB 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.

vtnad[ρA,ρB](r) defined as

vtnad[ρA,ρB](r)δTsnad[ρ,ρB]δρ(r)ρ=ρA
(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, vtnad[ρA,ρB] is a component of the effective potential, for which a quantum-mechanical problem of NA = ∫ρA(r)dr electrons embedded in a field generated by the NB = ∫ρB(r)dr electrons is solved. The corresponding energy functional,

Tsnad[ρA,ρB]Ts[ρA+ρB]Ts[ρA]Ts[ρB],
(2)

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

Ts[ρ]=minΦρΦ|T̂|Φ.
(3)

For the sake of subsequent considerations, we introduce the functional Tsnad(subtr)[ρtot,ρB], which admits a more restricted pair of densities than does Tsnad[ρtot,ρB],

Tsnad(subtr)[ρtot,ρB]Ts[ρtot]Ts[ρB]Ts[ρtotρB],
(4)

and the corresponding potential

vtnad(subtr)[ρtot,ρB]δTs[ρ]δρ(r)ρ=ρtotδTs[ρ]δρ(r)ρ=ρtotρB.
(5)

Any pair of densities ρA and ρB is admissible in Tsnad[ρA,ρB], provided that each of them is N-representable (N being either NA or NB). Admissible pairs of densities in the functional Tsnad(subtr)[ρtot,ρB] satisfy a different admissibility condition,

ConditionA:ΔVrΔVρtot(r)ρB(r),
(6)

where ΔV is a Lebesgue measurable volume element.

ConditionA is satisfied automatically if ρtot is constructed as a sum of two N-representable components. In such a case,

Tsnad(subtr)[ρA+ρB,ρB]=Tsnad[ρA,ρB].
(7)

ConditionA might be violated if ρtot and ρB are independent variables.

The admissibility condition for densities in the case of vtnad(subtr)[ρtot,ρB] is stronger than ConditionA. Formulating it involves elements of the Kohn–Sham formalism given below.

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

EvextKS[ρ]Ts[ρ]+vext(r)ρ(r)dr+12ρ(r)ρ(r)|rr|drdr+Exc[ρ],
(8)

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

The density ρ is pure-state non-interactingv-representable (vs-representable in short), if there exists an external potential vext such that ρo is the lowest energy solution of the Euler–Lagrange equation,

δEvextKS[ρ]δρ(r)ρ=ρo=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 NA orbitals (ϕi) is used to represent the density (ρ=iNA|ϕi|2). For systems for which the ground-state densities are not vs-representable, the density ρo that minimizes the Kohn–Sham functional is not the ground-state density. EvextKS[ρo] lies above the ground-state energy. The latter minimizes the Hohenberg–Kohn density functional EvextHK[ρ], which is defined for a larger class of densities (v-representable densities). The present work concerns minima and infima of EvextKS[ρ].

Inserting the explicit form of EvextKS[ρ], given in Eq. (8), to the Euler–Lagrange equation leads to the relation between the functional derivative of Ts[ρ] and a local potential,

δTs[ρ]δρ(r)ρ=ρo+vext(r)+ρo(r)|rr|dr+δExc[ρ]δρ(r)ρ=ρovvext,ρoKS(r)λ=0,
(10)

wherevvext,ρKS(r) denotes a function,

vvext,ρKS(r)vext(r)+ρ(r)|rr|dr+δExc[ρ]δρ(r),
(11)

that can be evaluated for any vext and admissible ρ. In Eq. (10), however, vext and ρ are not independent. This equation holds only if the density ρo minimizes Evext[ρ]KS for vext specified in the subscript.

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

vsrepresentableρvs[ρ](r)=vvext,ρKS(r)+const.,
(12)

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

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

vsrepresentableρδTs[ρ]δρ(r)+vs[ρ](r)λ=0,
(13)

that relates the functional derivative δTs[ρ]δρ(r) with the effective potential in the Kohn–Sham equation. In this work, if vvext,ρKS(r) is used, it indicates that this potential is available numerically, whereas vs[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 vs-representable.

The subsequent considerations concern the case where the exchange–correlation functional is approximated by means of an explicit expression Exc[ρ]Ẽxc[ρ]. The relevant expression of the energy functional and the corresponding effective potential are, thus,

ẼvextKS[ρ]Ts[ρ]+vext(r)ρ(r)dr+12ρ(r)ρ(r)|rr|drdr+Ẽxc[ρ]
(14)

and

ṽvext,ρKS(r)vext(r)+ρ(r)|rr|dr+δẼxc[ρ]δρ(r).
(15)

For such a case, ρtot is vs-representable by construction. Moreover, the functional vs[ρ] and the function ṽvext,ρKS become fundamentally different quantities. The function ṽvext,ρKS can be evaluated at a density ρ that is not vs-representable.

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 ρtotρB and ρtot are vs-representable. The present work concerns such cases that ρtot is given and it is vs-representable, whereas ρB satisfies ConditionA, i.e., it is obtained from some partitioning of ρtot. Such pairs of densities are relevant for DFT-based partition formalisms13,15 in which ρtot is the ground-state density corresponding to a given external potential and some approximation for the exchange–correlation potential, whereas ρtotρB is the density obtained from its partitioning. In FDET, ρtot is not known a priori, and such pairs are considered only in specially designed methodology oriented studies.16–19 The requirement that ρtotρB is vs-representable is stronger than ConditionA. It will be referred to as ConditionB. If it is satisfied, Eq. (13) applied for the two densities yields the desired relation,

vtnad(subtr)[ρtot,ρB]=vs[ρtotρB]vs[ρtot]+const
(16)

and vtnad(subtr)[ρtot,ρB](r)=vtnad[ρA,ρB](r).

If ρtot is obtained from some Kohn–Sham calculations, the demanding tasks–the inversion of the Kohn–Sham equation–is needed only for ρ = ρtotρB in order to evaluate vtnad(subtr)[ρtot,ρB] numerically using Eq. (16).

In Sec. II, the applicability of Eq. (16), which hinges on ConditionB, will be examined for ρB, which results from partitioning of some vs-representable ρtot.

The following pair of densities is considered:

  • The partitioned density ρtot is obtained from the Kohn–Sham equation (from here on, we use tildes to indicate that in practical applications, Exc[ρ]Ẽxc[ρ]).

  • The difference ρtotρB vanishes on the Lebesgue measurable volume element (Vzero) on which ρtot > 0,

rVzeroρtot(r)ρB(r)=0.
(17)

For such pairs, (i) vs[ρtot]=ṽvext,ρtotKS is finite on Vzero with a possible exception at the positions of the nuclei, where vext is singular; (ii) the potential vs[ρtotρB]–if it exists–is expected to diverge on Vzero because ρtotρB vanishes there; and (iii) ẼvextKS[ρtotρB]=ẼvextKS[ρtotρB] for any two external potentials vext and vext that differ only on Vzero. These observations indicate that the density ρtotρB is not vs-representable. The potential for which ẼvextKS[ρ] reaches the lowest value at ρtotρB diverges on Vzero. The addition of any constant or even a non-constant but finite potential on Vzero does not affect ẼvextKS[ρ]. If ρtotρB 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 Vzero. As a result, the difference between the two effective potentials corresponding to ρtot and ρtotρB might be ambiguous in Vzero.

To further investigate this indication, we construct a generalized Euler–Lagrange equation that guarantees that ρtotρB is stationary. The prerequisite for the existence of an effective potential obtainable from the inversion is that ρtotρB is a minimum of some density functionals that can be used in the Euler–Lagrange equation. If ρtotρB is not vs-representable, it cannot be ẼvextKS[ρ]. Some other functionals must be constructed. The functional Ωvext[ρ] is defined as

Ωvext[ρ]=ẼvextKS[ρ+ρB]+λCC[ρ],
(18)

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

C[ρ]=Vzeroρ(r)ρtot(r)2dr=0,
(19)

for instance.

The lowest value of Ωvext[ρ] corresponds to ρmin = ρtotρB and equals ẼvextKS[ρmin+ρB]=ẼvextKS[ρtot]. Owing to the constraint C[ρ] = 0, ρtotρB is a minimum of Ωvext[ρ] regardless of whether ρtotρB is a minimum or infimum of ẼvextKS[ρ]. The density ρ = ρtotρB satisfies, thus, the following Euler–Lagrange equation:

δΩvext[ρ]δρ(r)ρ=ρtotρB=δẼvextKS[ρ]δρ(r)ρ=ρtotρB+λCδC[ρ]δρ(r)ρ=ρtotρB=λ,
(20)

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

The fact that vext 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 ρtotρB is the minimizer of the corresponding Kohn–Sham equation, we introduce a constraint. Owing to the constraint, the requirement of vs-representability of ρtotρB is not needed and ρtotρB can be just an infimum of ẼvextKS[ρ]. Since ρtotρB is, however, a minimum of Ωvext[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 λC ≠ 0, it is no longer the Kohn–Sham potential corresponding to any vext.

Inserting the definition of ẼvextKS[ρ] into Eq. (20) leads to

δTs[ρ]δρ(r)ρ=ρtotρB+vΩ[ρtotρB]λ=0,
(21)

where

vΩ[ρtotρB]=λCδC[ρ]δρ(r)ρ=ρtotρB+vext(r)+ρtot(r)ρB(r)|rr|dr+δẼxc[ρ]δρ(r)ρ=ρtotρB.
(22)

The above construction of the effective potential vΩ[ρtotρB] is our principal result, which we will discuss below. If ρtotρB is not vs-representable, the external potential for which the component of vΩ[ρtotρB] due to the constraint vanishes does not exist. The component of vΩ[ρtotρB] due to the constraint does not vanish. Since it depends on ρtot, the effective potentials in two Euler–Lagrange equations, for ρtotρB and for ρtot, are not independent. The derivatives δTs[ρ]δρ(r)ρ=ρtot and δTs[ρ]δρ(r)ρ=ρtotρB used in Eq. (5) are, therefore, also not independent. As a result,

vt,Ωnad(subtr)[ρtot,ρB]=vΩ[ρtotρB]ṽvext,ρtotKSvtnad(subtr)[ρtot,ρB]+const.,
(23)

if ρtotρB is not vs-representable.

If ρtotρB is vs-representable, the external potential for which ρtotρB is the solution of the Kohn–Sham equation exists and is given as

vextvs(r)=vs[ρtotρB]ρtot(r)ρB(r)|rr|drδẼxc[ρ]δρ(r)ρ=ρtotρB.
(24)

If vextvs is used as vext in vΩ[ρ], the term due to the constraint disappears. The potential vt,Ωnad(subtr)[ρtot,ρB] is uniquely (up to a constant) defined, and it is equal to vtnad(subtr)[ρtot,ρB]. Equation (16), thus, holds without being affected by the present construction.

Turning back to the case of not vs-representable ρtotρB, we note that any change of the external potential (vext) localized only in Vzero does not affect ẼvextKS[ρtotρB]. Such a change of the potential does not affect ΩvextB[ρ] either. The potential vΩ[ρtotρB] might depend on the form of the constraint. The constraint C[ρ], which assures that ρ = ρtotρB 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 Cϵ,k[ρ]=Vzeroρ(r)ρtot(r)2kdrϵ 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 Vzero might lead, thus, to different potentials vΩ[ρtotρB].

In some multi-level molecular simulations (see Refs. 1113 and 2022, 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 ConditionA and for vs-representable ρtot). In practice, additional approximations combine with the silent assumption examined in more detail in this work that ρtotρB is vs-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 vs-representability of ρtotρB, which is needed for Eq. (16) to hold, can be replaced by a weaker one,

ConditionA:ΔVrΔVρtot(r)>ρB(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 ρtot and ρB satisfies ConditionA′ is straightforward if ρtot 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 vs-representability of ρtotρB. We rather demonstrate that the potential vtnad(subtr)[ρtot,ρB] cannot be obtained from the effective potentials in the corresponding Euler–Lagrange equations if ρtotρB is not vs-representable. Our approach to deal with densities that are non-vs-representable in variational calculations is similar in spirit to the one by Gould and Pittalis.24 

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.

The author has no conflicts to disclose.

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

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

1.
2.
T. A.
Wesołowski
,
Phys. Rev. A
77
,
012504
(
2008
).
3.
K.
Pernal
and
T. A.
Wesolowski
,
Int. J. Quantum Chem.
109
,
2520
(
2009
).
4.
T. A.
Wesolowski
,
J. Chem. Theory Comput.
16
,
6880
(
2020
).
6.
W.
Kohn
and
L. J.
Sham
,
Phys. Rev.
140
,
A1133
(
1965
).
7.
Q.
Zhao
,
R. C.
Morrison
, and
R. G.
Parr
,
Phys. Rev. A
50
,
2138
(
1994
).
8.
D. S.
Jensen
and
A.
Wasserman
,
Int. J. Quantum Chem.
118
,
e25425
(
2018
).
9.
R.
van Leeuven
and
E. J.
Baerends
,
Phys. Rev. A
49
,
2421
(
1994
).
10.
Q.
Wu
and
W.
Yang
,
J. Chem. Phys.
118
,
2498
(
2003
).
11.
B.
Zhou
,
Y.
Alexander Wang
, and
E. A.
Carter
,
Phys. Rev. B
69
,
125109
(
2004
).
12.
C. R.
Jacob
,
J. Chem. Phys.
135
,
244102
(
2011
).
13.
C.
Huang
,
M.
Pavone
, and
E. A.
Carter
,
J. Chem. Phys.
134
,
154110
(
2011
).
14.
P.
de Silva
and
T. A.
Wesolowski
,
Phys. Rev. A
85
,
032518
(
2012
).
15.
P.
Elliott
,
K.
Burke
,
M. H.
Cohen
, and
A.
Wasserman
,
Phys. Rev. A
82
,
024501
(
2010
).
16.
A.
Savin
and
T. A.
Wesolowski
,
Prog. Theor. Chem. Phys.
19
,
311
326
(
2009
).
17.
P.
de Silva
and
T. A.
Wesolowski
,
J. Chem. Phys.
137
,
094110
(
2012
).
18.
T. A.
Wesolowski
and
A.
Savin
, in
Recent Progress in Orbital-Free Density Functional Theory
, Recent Advances in Computational Chemistry Vol. 6, edited by
T. A.
Wesolowski
and
Y. A.
Wang
(
World Scientific
,
Singapore
,
2013
), pp.
275
295
.
19.
M.
Banafsheh
and
T.
Adam Wesolowski
,
Int. J. Quantum Chem.
118
,
e25410
(
2018
).
20.
O.
Roncero
,
M. P.
de Lara-Castells
,
P.
Villarreal
,
F.
Flores
,
J.
Ortega
,
M.
Paniagua
, and
A.
Aguado
,
J. Chem. Phys.
129
,
184104
(
2008
).
21.
J. D.
Goodpaster
,
N.
Ananth
,
F. R.
Manby
, and
T. F.
Miller
,
J. Chem. Phys.
133
,
084103
(
2010
).
22.
S.
Fux
,
C. R.
Jacob
,
J.
Neugebauer
,
L.
Visscher
, and
M.
Reiher
,
J. Chem. Phys.
132
,
164101
(
2010
).
23.
N.
Hadjisavvas
and
A.
Theophilou
,
Phys. Rev. A
30
,
2183
(
1984
).
24.
T.
Gould
and
S.
Pittalis
,
Phys. Rev. Lett.
123
,
016401
(
2019
).