Exposing minimal composition of Kohn-Sham theory and its extendability

In the KS theory Kohn and Sham (1965), which realizes the HK theorem, the energy variation is reduced to equations defining a set of s.p. orbitals, called KS orbitals. This is a striking feature of the KS theory. To disclose what ingredients at minimum enable to solve the many-fermion problem by s.p. equations, the equations are rederived by means of the 1-body density matrix $\varrho$.

Impose $Q^{A}=Q^{A}[\varrho]$, i.e., all $Q^{A}$’s depend solely on the 1-body density matrix $\varrho$, not on $\mathcal{C}^{(\nu)}$ ($\nu\geq 2$). Moreover, $Q^{A}$’s have to be independent of one another as a function of $\varrho$. The energy $E[\Psi]=E[\varrho,\mathcal{C}^{(2)},\mathcal{C}^{(3)},\cdots,\mathcal{C}^{(N)}]$ is partitioned into two parts, regular and irregular parts with respect to $\mathbf{Q}$. The irregular part is assumed to be represented only with $\varrho$, without $\mathcal{C}^{(\nu)}$ ($\nu\geq 2$), and to be differentiable with respect to $\varrho$ though not with respect to $\mathbf{Q}$. The irregular part may explicitly contain $Q^{A}$ without contradiction, because $Q^{A}=Q^{A}[\varrho]$. The partitioning is then expressed as

This partitioning may correspond to that of the underlying Hamiltonian,

although it is not requisite.

The individual theory is not defined until fixing $\mathbf{Q}$ and the partitioning (viz., $E^{\mathrm{KS}}_{\mathrm{irr}}$). In the standard KS equation, $\mathbf{Q}=n(\mathbf{r})$ and $E^{\mathrm{KS}}_{\mathrm{irr}}$ corresponds to the kinetic energy term so that the shell effects could be tractable. This also holds in the example of Appendix C. However, the source of the irregularity is not necessarily restricted to the kinetic energy in general cases. The choice of $\mathbf{Q}$ and $E^{\mathrm{KS}}_{\mathrm{irr}}$ is not always obvious; it may count on physical insight. Still, the structure of the KS theory can be discussed without specifying $\mathbf{Q}$ and $E^{\mathrm{KS}}_{\mathrm{irr}}$, as long as the influence of $\mathcal{C}^{(\nu)}$ ($\nu\geq 2$) on them is masked.

With the partitioning of Eq. (19), the result of the constrained search of Eq. (16) is written as

Whereas $E^{\mathrm{KS}}_{\mathrm{irr}}$ in Eq. (21) is not determined until knowing $\varrho[\Psi^{\mathrm{HK}}_{\mathbf{Q}}]$, it is a difficult task to obtain $\varrho[\Psi^{\mathrm{HK}}_{\mathbf{Q}}]$, not more accessible than $|\Psi^{\mathrm{HK}}_{\mathbf{Q}}\rangle$. A practical way is to minimize $E^{\mathrm{KS}}_{\mathrm{irr}}$ for $\varrho$ under the constraint that $\varrho$ should provide a given $\mathbf{Q}$, $\min_{\varrho\to\mathbf{Q}}E^{\mathrm{KS}}_{\mathrm{irr}}[\varrho;\mathbf{Q}]$. The resultant $\varrho$ is not equal to $\varrho[\Psi^{\mathrm{HK}}_{\mathbf{Q}}]$, and thereby $\min_{\varrho\to\mathbf{Q}}E^{\mathrm{KS}}_{\mathrm{irr}}[\varrho;\mathbf{Q}]\neq E^{\mathrm{KS}}_{\mathrm{irr}}\big{[}\varrho[\Psi^{\mathrm{HK}}_{\mathbf{Q}}];\mathbf{Q}\big{]}$. However, this deviation is assumed to be a regular function of $\mathbf{Q}$. Then, the deviation can be incorporated into the regular part of the EDF, deriving the modified partitioning as

Although the principal variables $\mathbf{Q}$ should be a function of the density matrix $\varrho$, it is convenient to handle them as if they were independent in $E^{\mathrm{KS}}_{\mathrm{irr}}$. To do it, $Q^{A}$ is replaced by an auxiliary variable $q^{A}$, with setting $q^{A}=Q^{A}$ as a constraint. The EDF within the KS scheme is now defined as follows, with the Lagrange multiplier $\Lambda^{\mathrm{KS}}_{A}$,

This procedure is analogous to the Legendre transformation used in Refs. Fukuda et al. (1994); Valiev and Fernando (1997). Then, after the treatment analogous to Eq. (9),

the condition $\delta\tilde{E}^{\mathrm{KS}}[\varrho;\mathbf{q}]\geq 0$ should be fulfilled at the energy minimum, where the variation is performed for $\varrho$, $\mathbf{q}$ and $\mu$. The variation of $E^{\mathrm{KS}}$ for $q^{A}$ yields

Here the differentiation with respect to $q^{A}$ is expressed by the partial derivative, but it should read as a functional derivative when $A$ is continuous. The variation provides s.p. Hamiltonian,

$\Gamma^{\mathrm{KS}}$ is the KS potential. It is reasonable to assume that $(Q^{A})^{\ast}$ is also contained in $\mathbf{Q}$ for a complex $Q^{A}$. Then $h^{\mathrm{KS}}$ is hermitian and therefore diagonalizable, having real eigenvalues,

similarly to the HF equation (7). This is the KS equation, and the s.p. state diagonalizing $h^{\mathrm{KS}}$, $\varphi^{\mathrm{KS}}_{i}=\sum_{k}\mathcal{U}^{\mathrm{KS}}_{ki}\,\phi_{k}$, is the KS orbital. Because of the identical mathematical structure, all the arguments on the solutions of the HF equation in Sec. II hold for those of the KS equation (27) as they are. The resultant 1-body density matrix $\varrho^{\mathrm{KS}}$ satisfies Eq. (15), corresponding to the state $|\Phi^{\mathrm{KS}}\rangle$ in which the lowest $N$ KS orbitals are occupied. However, while all $Q^{A}$’s are physical, individual $\varrho^{\mathrm{KS}}_{k\ell}$ (equivalently, $\mathcal{U}^{\mathrm{KS}}_{ki}$ and $\varphi^{\mathrm{KS}}_{i}$) is artificial, losing clear physical meaning, as is recognized from the derivation.

In the conventional formulation of the KS theory, the subspace $\mathcal{V}_{\mathrm{idem}}$ is considered from the beginning by introducing the non-interacting reference system and the KS orbitals. On the other hand, the functional $E^{\mathrm{KS}}[\varrho;\mathbf{q}]$ is here defined in the full Hilbert space $\mathcal{V}_{\mathrm{full}}$. Hence it can be identified as an effective Hamiltonian, although it depends on $\mathbf{q}$ and is not necessarily representable in the second-quantized form. Solutions of the KS equation fall on $\mathcal{V}_{\mathrm{idem}}$ as a result. Still, it is legitimate to state that the idempotent $\varrho$ ($\varrho^{\mathrm{KS}}$, to be precise) is implicitly assumed when postulating that $E^{\mathrm{KS}}_{\mathrm{irr}}$ does not depend on $\mathcal{C}^{(\nu)}$ ($\nu\geq 2$).

As already mentioned, the HK theorem was extended to the case $\mathbf{Q}=\{\varrho_{k\ell}\}$ Gilbert (1975), above which the density-matrix functional theory (DMFT) was exploited Donnely and Parr (1978); Zumbach and Maschke (1985); Schindlmayr and Godby (1995); López-Sandoval and Pastor (2000); Requistl and Pankratov (2008); Pernal and Giesbertz (2016). The present reformulation should be distinguished from the DMFT. It is noticed that $E^{\mathrm{HK}}[\varrho_{k\ell}]$ (i.e., $\mathbf{Q}=\{\varrho_{k\ell}\}$) itself should not be handled in the KS theory because the 1-body density matrix of the KS theory $\varrho\,(=\varrho^{\mathrm{KS}})$ cannot be physical, as will be discussed in Sec. V. The ensemble representability Valone (1980) is irrelevant to $\varrho$ here.

If $E^{\mathrm{KS}}_{\mathrm{irr}}[\varrho;\mathbf{q}]$ consists of two parts, one which linearly depends on $\varrho$ but does not depend explicitly on $\mathbf{q}$ and the other expressed only with $\mathbf{q}$,

Eq. (26) becomes

This is the case of the standard KS equation (see Sec. VI.1).

Irregularity of $E^{\mathrm{HK}}[\mathbf{Q}]$ may arise via correlations linked to a pair condensate, as in superconductivity. It is then reasonable to include the pairing tensor $\kappa$ and $\kappa^{\ast}$ in the irregular part of the EDF, and to extend Eq. (19) as

Here $Q^{A}$’s are postulated to depend on $(\varrho,\kappa,\kappa^{\ast})$. Arguments analogous to Eqs. (21) and (22) yield

As the KS functional in Eq. (23), the Kohn-Sham–Bogolyubov-de Gennes (KSBdG) functional is obtained by regarding $q^{A}=Q^{A}$ as a constraint,

In addition, the particle-number condition (8) is considered,

By defining

where

the KSBdG equation is derived,

in analogy to the HFB equation (63). Above, the relation obtained by the variation with respect to $q^{A}$,

has been inserted. The identical mathematical structure guarantees that the arguments on the solutions of the HFB equation in Appendix A also apply to those of the KSBdG equation (36).

It is noted that $\kappa$ and $\kappa^{\ast}$ do not need to be physical in the KSBdG scheme. The KSBdG solution can be exact for systems where the particle number is fixed, provided that $E^{\mathrm{KSBdG}}$ is appropriately determined.

Whereas the HK theorem in Sec. III does not demand it, all $Q^{A}$’s have been assumed to depend solely on $\varrho$ in Sec. IV.1, not on $\mathcal{C}^{(\nu)}$ ($\nu\geq 2$). Comments on this point are given.

The KS equation closes within the s.p. degrees of freedom, not leading to hierarchical equations. This remarkable property is actualized because everything in $E^{\mathrm{KS}}$ does not depend on the correlation functions $\mathcal{C}^{(\nu)}$ ($\nu\geq 2$). The principal variables $\mathbf{Q}$ should not depend on $\mathcal{C}^{(\nu)}$ for the KS equation to keep mathematical rigor.

One might consider that the KS equation can be derived even when $Q^{A}$ depends on $\mathcal{C}^{(\nu)}$ via a constrained search similar to Eq. (16) Donnely and Parr (1978). For instance, suppose that $Q^{A}$ depends on $\mathcal{C}^{(2)}$ as well as on $\varrho$. Then $E^{\mathrm{KS}}$ of Eq. (23) also depends on $\mathcal{C}^{(2)}$. Not to lead to hierarchical equations, a new EDF should be constructed via the constrained search $\min_{(\varrho,\mathcal{C}^{(2)})\to\varrho}E^{\mathrm{KS}}$. However, this introduces another step in the minimization process, which likely damages the differentiability of $Q^{A}$ for $\varrho$, as illustrated in Appendix B. Because the KS theory relies on this differentiability, it is desired that the principal variables do not depend on $\mathcal{C}^{(\nu)}$. Conversely, if $\mathcal{C}^{(\nu)}$ is initially contained in $Q^{A}$ but is finally eliminated in $E^{\mathrm{KS}}$, one should pay special attention to the differentiability.

It is worth recalling that the KS equation (27) is derived primarily from the variation of $E^{\mathrm{KS}}[\varrho;\mathbf{q}]$ with respect to $\varrho$. Apart from the physical importance, the variation with respect to $q^{A}$ is subsidiary in deriving the equational form, only determining the KS potential $\Gamma^{\mathrm{KS}}$ through the Lagrange multiplier $\Lambda^{\mathrm{KS}}_{A}$ as in Eqs. (25) and (26). Indeed, variation of $E^{\mathrm{KS}}$ for $\varrho$ after inserting $q^{A}=Q^{A}[\varrho]$leads to the same s.p. Hamiltonian as in Eq. (26),

The g.s. energy is given at $\varrho=\varrho^{\mathrm{KS}}$,

This feature implies that equations equivalent to (27) are obtained by removing or adding some variables to $\mathbf{Q}$, as long as the total $E^{\mathrm{KS}}$ is an identical functional of $\varrho$. Consider the case that some of the principal variables are taken out. The new set is denoted by $\mathbf{Q}^{\prime}=\{Q^{B^{\prime}}\}$ and the removed set by $\bar{\mathbf{Q}}=\{Q^{\bar{B}}\}$; $\mathbf{Q}=\mathbf{Q}^{\prime}\oplus\bar{\mathbf{Q}}$. $E^{\mathrm{KS}}_{\mathrm{reg}}$ is separated into two parts, $E^{\mathrm{KS}}_{\mathrm{reg}}=\bar{E}^{\mathrm{KS}}_{\mathrm{reg}}+E^{\prime\,\mathrm{KS}}_{\mathrm{reg}}$, where $\bar{E}^{\mathrm{KS}}_{\mathrm{reg}}$ includes $Q^{\bar{B}}$ and $E^{\prime\,\mathrm{KS}}_{\mathrm{reg}}$ does not. By newly defining $E^{\prime\,\mathrm{KS}}_{\mathrm{irr}}:=E^{\mathrm{KS}}_{\mathrm{irr}}+\bar{E}^{\mathrm{KS}}_{\mathrm{reg}}$, the following equality,

is fulfilled up to the functional form of $\varrho$. This relation guarantees the equivalence at the level of the equations as well as the resultant energy. The opposite is true: principal variables may be added without influencing the s.p. equations. We shall call this feature equational equivalence.

Despite the equational equivalence, the choice of $\mathbf{Q}$ is physically significant. As already discussed, while the energy $E$ and the principal variables $\mathbf{Q}$ obtained from the KS equation are physical, other quantities do not have strict physical meaning. The equational equivalence indicates that the equational form does not tell what variables are physical. It entirely depends on how the EDF has been constructed. This point matters when the EDF is phenomenologically fixed as in Sec. VI.2.

Several limiting cases are instructive. First, if all $Q^{A}$’s are taken out by setting $\bar{\mathbf{Q}}=\mathbf{Q}$, $\mathbf{Q}^{\prime}$ becomes an empty set $\emptyset$, leading to $E^{\prime\,\mathrm{KS}}_{\mathrm{reg}}=0$ and $E^{\mathrm{KS}}[\varrho;\emptyset]=E^{\prime\,\mathrm{KS}}_{\mathrm{irr}}[\varrho;\emptyset]$. In this case, no quantity other than the total energy is considered physical. Second, it is possible to set $E^{\prime\,\mathrm{KS}}_{\mathrm{irr}}=0$, opposite to the first limiting case. There are practical cases in which $E^{\mathrm{KS}}_{\mathrm{irr}}$ can be represented in terms of certain currents. For instance, in the standard KS theory for electronic systems, the kinetic density of the non-interacting system is expressed as $\xi(\mathbf{r})/2m$, where

If all those currents are added to $\mathbf{Q}$, we have $E^{\prime\,\mathrm{KS}}_{\mathrm{irr}}=0$. Still, we should consider $E^{\prime\,\mathrm{KS}}_{\mathrm{irr}}$ to depend on $\varrho$, since it is essential to derive the KS equation. An extreme case along this line is that of $\mathbf{Q}=\{\varrho_{k\ell}\}$, the whole 1-body density matrix Donnely and Parr (1978); Zumbach and Maschke (1985); Requistl and Pankratov (2008); Pernal and Giesbertz (2016). Then all components of $\varrho$ would be considered physical. However, an idempotent $\varrho$, which necessarily results from the KS equation, cannot be exact in general, even though there remains room to interpret the s.p. states as those of dressed particles. These limiting cases help to understand issues of representability, as discussed in the subsequent subsection.

Significance of representability has been perceived for the HK theorem. The HK theorem in Sec. III was initially asserted by showing the uniqueness of the mapping from the electron density $n(\mathbf{r})$ to the external electron-ion potential $U_{ei}(\mathbf{r})$ Hohenberg and Kohn (1964), which are connected via the Legendre transformation. This mapping should exist for any $n(\mathbf{r})$ with $n(\mathbf{r})\geq 0$ and $\int d^{3}r\,n(\mathbf{r})=N$ (the $v$-representability). While this $v$-representability issue can be circumvented by the constrained search of Eq. (16), it is transferred to the differentiability of the EDF as mentioned in Sec. III, and this problem could arise even at the KS level. Once $\mathbf{\Lambda}^{\mathrm{KS}}=\{\Lambda^{\mathrm{KS}}_{A}\}$ are fixed, the KS equation (27) with the KS Hamiltonian (26) determines $\varrho^{\mathrm{KS}}$ and thereby the principal variables $\mathbf{Q}[\varrho^{\mathrm{KS}}]$, defining the mapping from $\mathbf{\Lambda}^{\mathrm{KS}}$ to $\mathbf{Q}$. The question is in the inverse mapping from $\mathbf{Q}$ to $\mathbf{\Lambda}^{\mathrm{KS}}$, i.e., whether $\mathbf{\Lambda}^{\mathrm{KS}}$ exists for any $\mathbf{Q}$ in its physical domain. The existence of this inverse mapping may be regarded as $v$-representability at the KS level, which has been called ‘non-interacting $v$-representability’ in some literature. Equation (25) tells us that the $v$-representability at the KS level arrives at the differentiability of $E^{\mathrm{KS}}_{\mathrm{irr}}[\varrho;\mathbf{Q}]$ and $E^{\mathrm{KS}}_{\mathrm{reg}}[\mathbf{Q}]$ with respect to $Q^{A}$.

As already mentioned, any $\mathbf{Q}$ in the physical domain should be represented by the 1-body density matrix $\varrho$ that satisfies Eqs. (8) and (11). This condition is called $N$-representability and has been proven for the original case $\mathbf{Q}=n(\mathbf{r})$ Harriman (1981). For the KS theory to be applicable, any $\mathbf{Q}$ in its physical domain should be obtained by an idempotent $\varrho$. This condition is stronger than the $N$-representability for the HK theorem and the ‘pure-state representability’ in Ref. Valone (1980). We shall call it $N$-representability at the KS level. For $\mathbf{Q}=n(\mathbf{r})$, the $N$-representability is also satisfied at the KS level Harriman (1981).

Two representabilities have been distinguished in the fermionic DFT, the $v$- and the $N$-representabilities. Both are further distinguished between those at the HK and KS levels. The $v$-representability is linked to the differentiability of the EDF, and it is usually weaker at the KS level than at the HK level. On the other hand, the $N$-representability at the KS level, which requires the idempotency of the 1-body density matrix, is stronger than that at the HK level.

The reformulation of the KS theory with the least postulates addresses the criteria for its extendability. The following conditions should be satisfied for the KS equation to be rigorous and practical, and they give the criteria.

• (a)

  Principal variables $\mathbf{Q}$ should be appropriately chosen. They should help to minimize energy and contain all quantities under physical interest other than energy. At the same time, all $Q^{A}$’s should depend only on $\varrho$ and not on the correlation functions $\mathcal{C}^{(\nu)}$ ($\nu\geq 2$), or well approximated by functionals only of $\varrho$. Moreover, $\mathbf{Q}$ should be $N$-representable at the KS level in the domain under search, i.e., representable by an idempotent $\varrho$ with $\mathrm{tr}(\varrho)=N$.

• (b)

  Irregularity in $E^{\mathrm{HK}}[\mathbf{Q}]$ can be removed via $E^{\mathrm{KS}}_{\mathrm{irr}}[\varrho;\mathbf{Q}]$, which is a differentiable functional of $\varrho$. Furthermore, $E^{\mathrm{KS}}_{\mathrm{irr}}$ should properly be chosen so that $E^{\mathrm{KS}}_{\mathrm{reg}}[\mathbf{Q}]$ should be regular for $\mathbf{Q}$. Because $E^{\mathrm{KS}}_{\mathrm{reg}}$ is defined by (22), this condition usually requires that both $E^{\mathrm{HK}}_{\mathrm{reg}}[\mathbf{Q}]$ and $\Big{(}E^{\mathrm{KS}}_{\mathrm{irr}}\big{[}\varrho[\Psi^{\mathrm{HK}}_{\mathbf{Q}}];\mathbf{Q}\big{]}-\min_{\varrho\to\mathbf{Q}}E^{\mathrm{KS}}_{\mathrm{irr}}[\varrho;\mathbf{Q}]\Big{)}$ are regular for $\mathbf{Q}$.

• (c)

  The entire $E^{\mathrm{KS}}[\varrho;\mathbf{Q}]$ should be fixed before starting computation, which requires that $E^{\mathrm{KS}}_{\mathrm{reg}}[\mathbf{Q}]$ is constructed appropriately.

Condition (a) is the first step for applying the KS approach. One might think that the greater number of principal variables provide the more accurate description of the system. However, the $N$-representability becomes the more difficult to be satisfied. Therefore the set $\mathbf{Q}$ should be neither too big nor too small. Significance of (b) is exemplified by the normal-to-superconductor transition. The pair correlation carried by $\kappa$ and $\kappa^{\ast}$ plays an essential role in this phenomenon, connected to the order parameter. It is difficult to remove the irregularity at the transition only by $\varrho$, indicating that the KS theory itself cannot be applied, though possible via the extension to the KSBdG equation⁷⁷7 For the KSBdG scheme, $\varrho$ in Conditions (a–c) should be read as $(\varrho,\kappa,\kappa^{\ast})$. . Another example could be the confinement of constituent quarks, in which the 3-body correlation forming the color singlet is essential. It does not seem possible to ascribe irregularity in $E^{\mathrm{HK}}$ to $E^{\mathrm{KS}}_{\mathrm{irr}}$ that is a functional only of $\varrho$, without using $\mathcal{C}^{(3)}$.

If the underlying Hamiltonian $H$ is known and is partitioned as in Eq. (20), it may be possible to consider $H_{\mathrm{irr}}$ and $H_{\mathrm{reg}}$ unperturbed Hamiltonian and perturbation, respectively. Then, when setting on $H_{\mathrm{reg}}$ adiabatically, the wave-function $|\Psi\rangle$ changes gradually, as the adiabatic theorem Messiah (1961) tells us. At a glance, it looks reasonable to expect that $E^{\mathrm{KS}}_{\mathrm{reg}}\big{[}\mathbf{Q}]$ is regular in this case. However, this is not always true because it breaks down for the $\mathbf{Q}=\{\varrho_{k\ell}\}$ case discussed in Sec. V.2, in which the $N$-representability at the KS level is lost.

Unless all conditions (a–c) are fulfilled, approaches using the KS equation lose firmness or practicality. Conversely, as long as an EDF meets these conditions, the many-body system can be solved via s.p. equations, following the procedure in Sec. IV. Extensions may be adding principal variables, modifying the EDF, or applying to other many-fermion systems than those composed of electrons.

The KS theory has been successful for electronic systems, from atoms and molecules to solids. Under $\mathbf{Q}=n(\mathbf{r})$, it has been established as a standard ab initio method for many-electron systems. The Hamiltonian of the system is comprised of the kinetic energy $K$, the external potential provided by ions $U_{ei}$, and the electron-electron interaction $V_{ee}$. In the context of Eq. (20), these terms are partitioned as⁸⁸8 Although the contribution of $U_{ei}$ is regular for $n(\mathbf{r})$, we put it into $H_{\mathrm{irr}}$ so that $H_{\mathrm{reg}}$ could be universal. Shifting it to $H_{\mathrm{reg}}$ does not influence the resultant equations.,

Correspondingly, the EDF follows Eq. (20). As in Eq. (28), $E^{\mathrm{KS}}_{\mathrm{irr}}$ is separated into individual terms of Eq. (42) as

where $E_{K}[\varrho]$ corresponds to the first term of Eq. (28) and $E_{ei}[n(\mathbf{r})]=\mathit{\Delta}E^{\mathrm{KS}}_{\mathrm{irr}}$. Then $E^{\mathrm{KS}}_{\mathrm{reg}}$ in Eq. (22) becomes

having no contribution from $E_{ei}$. Since $U_{ei}$ in Eq. (42) carries characteristics of individual systems, $E^{\mathrm{KS}}_{\mathrm{reg}}$ is expected to be universal. As long as $H_{\mathrm{reg}}$ or its exchange-correlation part shown below is perturbative, the regularity of $E^{\mathrm{KS}}_{\mathrm{reg}}$ is likely satisfied. There have been attempts to extend the theory to include other variables in $\mathbf{Q}$ as mentioned in Sec. I, although the representability issues have not been explored sufficiently.

$E^{\mathrm{KS}}_{\mathrm{reg}}$ is taken as a sum of the direct term of $V_{ee}$ and the rest:

All of the complicated many-body effects are attributed to the exchange-correlation term $E^{\mathrm{KS}}_{\mathrm{xc}}$. It should be noted that the general theory does not demand $E^{\mathrm{KS}}_{\mathrm{reg}}$ to be represented by an integral of a local function. Indeed, $E_{\mathrm{dir}}$ is an integral containing the densities at distant positions $\mathbf{r}^{\prime}$ and $\mathbf{r}^{\prime\prime}$, multiplied by the interaction having the non-local form $1/|\mathbf{r}^{\prime}-\mathbf{r}^{\prime\prime}|$. In contrast, $E^{\mathrm{KS}}_{\mathrm{xc}}$ is customarily assumed to be the integral of a local function $\mathcal{H}_{\mathrm{xc}}$, a function depending on the density at a single position $\mathbf{r}^{\prime}$. The kernel $\mathcal{H}_{\mathrm{xc}}$ has been determined by the local-density approximation (LDA) Hohenberg and Kohn (1964); Sahni et al. (1988) or the generalized gradient approximation (GGA) Perdew et al. (1996). A method for obtaining $\mathcal{H}_{\mathrm{xc}}$ called ‘meta-GGA’ has been developed Tao et al. (2003). A machine-learning-based method has been proposed recently Nagai et al. (2022). While the locality assumption on $E^{\mathrm{KS}}_{\mathrm{xc}}$ has helped the KS theory to be practical in electronic systems, it is an approximation whose appropriateness should be scrutinized.