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On the Physical Origin of the 1 / 4 Exact Exchange in Density Functional Theory

We investigate the physical origin of the broadly used 1/4 fraction of exact exchange in density functional theory (DFT)’s hybrid functionals. We compare two models of the exchange energy of an electron pair in an unpolarized electron gas. One model treats the spin states of the electron pair as a statistical ensemble, while the other includes their quantum superposition. The quantum mechanical treatment is shown to possess an exchange energy equal to that of the statistical ensemble, plus an additional 1/4 exact exchange energy due to a spin-flip exchange interaction between electron pair states with antiparallel spin. This exchange interaction originates from the coupling between doubly-excited configurations, and is not explicitly included in semilocal DFT. By showing that the statistical ensemble contains 3/4 of the density-based exchange energy, we can attribute, for each pair of occupied orbitals, the 1/4 exact exchange used in hybrid DFT to the spin-flip exchange processes mentioned above. Practical and conceptual implications of our findings are outlined.


I. INTRODUCTION
In the framework of density functional theory (DFT), the total energy can be expressed as a functional of the electron density [1], and minimized to obtain the ground state energy [2]. In practice, since the electron-electron interactions cannot at present be expressed exactly in terms of the density, one needs to approximate the exchange and correlation energies, often using a local functional of the density as is done in the local density approximation (LDA) of DFT [3]. A recurring observation is that replacing roughly one quarter of the densitybased exchange energy with one quarter of the orbitalbased Hartree-Fock (HF) (so-called "exact") exchange improves the accuracy of several computed properties, including the ground state energy and the equilibrium atomic structure. This approach of mixing the densitybased and exact exchange energies is adopted in the socalled hybrid exchange-correlation functionals, a popular choice in modern DFT calculations.
In the simplest hybrid scheme [4][5][6], a single parameter α is employed for the exchange mixing, according to E XC = E DFT XC + α (E HF X − E DFT X ), where E X and E XC are, respectively, the exchange and exchangecorrelation energies, and the DFT and HF superscripts denote density-based and HF-based quantities. While it is common knowledge [7,8] that an exact-exchange fraction of roughly α = 1/4 gives optimal results, as Perdew put it [7], one should think of the 1/4 fraction as an empirical parameter, because a rigorous argument for it is still missing. There have been efforts to rationalize the origin of the 1/4 exact exchange [8], but a plausible physical explanation has not yet been proposed [7].
Let us examine more closely the exact exchange mixing problem. When α=0 and only the local density-based exchange is employed, as is done in the LDA, the results systematically overbind matter. This is surprising, since E XC employed in the LDA is derived from two variational calculations, HF for exchange and quantum Monte Carlo (QMC) for correlation interactions [3,9]. Since E XC is dominated by exchange, LDA's overbinding indicates that the density-based exchange binds matter too strongly. Equally surprising is the fact that using exact exchange alone together with density-based correlation, which corresponds to the α=1 limit, does not give accurate results either. This trend suggests that correlation effects are not fully described by local interactions. Why then should mixing the α=0 and α=1 limits, each with its own pitfalls, give better results at all? If that happens, as it does for a fraction of roughly α = 1/4 exact exchange, it must be that mixing the density-based and exact exchange corrects both exchange and correlation effects, since neither is accurate in the LDA. However, if mixing the density-based and exact exchange is an ad hoc attempt to correct exchange-correlation interactions, its physical justification remains unclear.
This work aims to advance our understanding of the 1/4 exact exchange mixing, and explain its physical origin. We focus on an electron pair in the interacting electron gas of a material, and show that its exchange energy equals that of an electron pair described as a statistical ensemble of four possible spin states, each with equal probability, plus a correction equal to 1/4 of the exact exchange energy, which stems from a spin-flip exchange interaction that is missing in the statistical ensemble. The exchange energy of the statistical ensemble, which represents a probability-based model of a pair, is shown to be equal to 3/4 of the density-based exchange energy. We conclude that the 3/4 density-based plus 1/4 exact exchange mimics the correct quantum mechanical (QM) treatment of a pair in the interacting electron gas. The 1/4 exact exchange is associated with an exchange interaction between electron pair states with antiparallel spin, which originates from the coupling between doublyexcited configurations and is not included in semilocal DFT. The factor of 4 in the 1/4 fraction reflects the number of possible spin states of an electron pair. Some implications of these results are discussed.

A. Two views of exchange interactions
In its spirit, DFT is a classical theory of the electron gas. As Hohenberg and Kohn put it in their seminal work [1], in DFT "the system of electrons is pictured like a classical liquid" with electron density n. This statement has rather profound implications. The electron density n(x)dv, where x = {r, σ} combines the position r and spin σ variables, is equal to the number of particles times the probability of finding a particle with spin σ within the volume dv around the point r, when all other particles have arbitrary positions and spins [10]. DFT is thus a QM theory based on a real-valued probability − the electron density − rather than a complex-valued probability amplitude, as is the wave function.
One main problem with an electronic structure theory based on probability is that exchange is an inherently QM interaction, since it is related to spin. Exchange interactions lower the electron repulsion energy by keeping a pair of electrons apart, which requires a spatially antisymmetric wave function and a symmetric (triplet) spin state for the electron pair. In HF, a pairwise treatment of exchange is adopted, in which only particles occupying spin-orbitals with parallel spin contribute to the exchange energy. The density-based picture of DFT is rather different − exchange is not treated as a sum of pairwise interactions, but rather, it depends only on the electron density at a point. This means that all particles in the density contribute the same to exchange in semilocal DFT, regardless of their spin state. Another key QM concept, quantum superposition, is also missing in DFT's probability-based description of the electron gas.
The main challenge in explaining the origin of the 1/4 exact exchange lies in marrying these two widely different approaches to treating exchange − one density-based, which amounts to a probability-based QM (and thus, classical) treatment, and another based on pairwise exchange interactions, as is HF. Our goal will be to write the exchange energy of an electron pair as the sum of the exchange energy in a statistical ensemble, which will be treated in the density-based picture, plus a correction equal to 1/4 of the HF exchange energy. We will then show that the statistical ensemble contains 3/4 of the density-based exchange energy, thus explaining, for each electron pair in the electron gas, the physical origin of the 3/4 density-based plus 1/4 exact exchange mixing.

B. Preliminaries on the exchange energy
For the N -particle interacting electron gas in a material, the Hamiltonian, |ri−rj | , is the sum of one-body operators, h i = −∇ 2 i /2m + v(r i ), and two-body Coulomb interactions, V ij = e 2 /|r i − r j |. We assume that the electron gas is unpolarized, and that its ground state [see Fig. 1(a)] consists of N /2 doublyoccupied orthonormal orbitals, ϕ i (i = 1, ... , N/2). Its HF exchange energy is: where J is the exchange energy between the orbitals ϕ i and ϕ j [11], and the factor of 2 multiplying J in Eq. 1 accounts for the sum over spin. According to Eq. 1, the total HF exchange energy is the sum of the exchange interaction between all orbital pairs, with each orbital pair contributing 2J to the total exchange energy. When the same system is treated in DFT, the exchange energy is a functional of the electron density, n(r) = 2 i |ϕ i (r)| 2 , but its expression is unknown and needs to be approximated. In the LDA [3], E DFT X is approximated as a sum of local contributions that depend only on the electron density at a given point; the exchange energy per particle as a function of the density is obtained from a HF calculation of the homogeneous electron gas (HEG) as ε HEG X (n) = (3 e 2 /4π)(3π 2 ) 1/3 n 1/3 [11], and the total LDA exchange energy is obtained with the integral E DFT X = dr n(r) ε HEG X [n(r)]. Hybrid functionals mix the semilocal DFT and HF pictures. In the simplest hybrid approximation [4][5][6], the exchange energy E hybrid X combines a fraction α of HF exchange energy, E HF X , with a fraction 1−α of density-based exchange, E DFT X . For a mixing fraction of α = 1/4, We focus on two orbitals ϕ i and ϕ j composing the ground state [see Fig. 1(a)], both of which are doubly occupied. As noted above, the exchange energy contribution from these two orbitals in HF (neglecting selfinteraction) is E HF X = 2J, so that one quarter of their exchange energy contribution is 1 4 Since the density-based exchange energy E DFT X [n] cannot be apportioned into pairwise terms, how much just two orbitals contribute to the total density-based exchange energy cannot be rigorously computed. Yet, let us take for the time being such a pairwise approach, and imagine that ϕ i and ϕ j are the only two orbitals contributing to the charge density. Then one can write their exchange energy in hybrid DFT with 1/4 exchange mixing as where n(r) = 2 ( |ϕ i (r)| 2 + |ϕ j (r)| 2 ) is the density from the four electrons occupying the two orbitals ϕ i and ϕ j .
In the next section, we introduce a model that attempts to explain the physical origin of this 1/4 hybrid mixing scheme, and rationalize why it works in practice.

A. Models of an electron pair
Since the Coulomb interaction is pairwise, an accurate treatment of exchange needs a theory which, unlike DFT, explicitly treats two-particle correlations. Therefore, we seek to formulate a two-particle model to understand exchange processes and extend their DFT treatment. The two-particle density matrix [10] is a natural extension of the electron density to treat two-body correlations. The electron repulsion energy E e−e of an electron pair can be written explicitly as a functional of its two-particle density matrix, ρ(x 1 , x 2 ), using [10,12] where V 12 = e 2 |r1−r2| is the two-body Coulomb interaction. We focus again on the two doubly-occupied orbitals ϕ i and ϕ j . Since we neglect self-interaction, the only relevant exchange processes are those between electrons occupying the two different orbitals. We thus model a pair made up by an electron in orbital ϕ i with spin σ and an electron in orbital ϕ j with spin σ , and investigate its two-particle density matrix and contribution to exchange. We imagine measuring the spin of the two electrons composing the pair. Since the electron gas is unpolarized, we expect to measure one of these four outcomes with equal probability: spin up for both electrons, spin down for both electrons, spin up for the electron in orbital ϕ i and spin down for the electron in orbital ϕ j , or viceversa. Given that electrons are indistinguishable, we express each of these four scenarios, respectively, with a two-particle Slater determinant formed with spin-orbitals ϕ iσ and ϕ jσ : The explicit expressions for these states are, for example, and similar ones for |Ψ 3 and |Ψ 4 . Here and below, we adopt a convention commonly used in quantum information [13], in which |0 and |1 represent spin up and spin down states, respectively, and |σσ is a collective spin state of the two electrons in the pair, formally corresponding to the tensor product |σσ = |σ ⊗ |σ . (c) Configurations employed to derive ρ (2) . Four configurations − one for each collective value of the spin variables σ and σ − are mixed with equal weight. The orbitals ϕi and ϕj are each occupied by one electron, with spin σ and σ , respectively. In (b) and (c), one electron has been promoted from each of the orbitals ϕi and ϕj to ϕa and ϕ b , in a way that conserves the spin unpolarized character of the electron gas. Other states that are not relevant to the exchange processes between ϕi and ϕj are shown as shaded.
Using the states in Eq. 6 as a basis set, we formulate two models of the electron pair, both consistent with the spin measurements discussed above. The first approach, called here model 1 , describes the pair with a wave function |Ψ p consisting of an equal quantum superposition of the four states: The corresponding two-particle density matrix, ρ (1) , is a so-called pure state [13], This description of the electron pair is the simplest ansatz that includes quantum superposition and twobody correlations. We contrast this description of the pair with one based on probability alone, called model 2, which does not use quantum superposition and is "classical" and similar in spirit to DFT. In model 2, the pair is described as a statistical ensemble composed in equal parts by the four pair basis states in Eq. 6. This model corresponds to a mixed state [13], generally described by the density matrix ρ = µ P µ |Ψ µ Ψ µ |. For our Model 1 -Pure state with quantum superposition Model 2 -Mixed state, no quantum superposition model 2, the two-particle density matrix is so that in our case, P µ = 1/4 is the probability of finding the electron pair in state |Ψ µ . Note that spin measurements on the electron pair in orbitals ϕ i and ϕ j will give, both in model 1 and 2 (but for different reasons), one of the four basis states in Eq. 6 with equal probability of 1/4, consistent with our assumption of an unpolarized electron gas. An important property of ρ (1) and ρ (2) is that they are N -representable [12], in the sense that they can both be derived from two-body reduced density matrices, each obtained from an N -body density matrix by tracing out N − 2 electrons [12]. In particular, one can derive ρ (1) and ρ (2) from doubly-excited configurations described by Slater determinants in which an electron has been removed from each of the orbitals ϕ i and ϕ j and placed into unoccupied states [see Fig. 1 As is well known in configuration interaction and coupled cluster theories [14], such doubly-excited configurations contribute substantially to the ground state energy and the Coulomb interactions.
Our definitions given above for ρ (1) and ρ (2) include only the terms in the reduced density matrices contributing to the exchange interaction between orbitals ϕ i and ϕ j . In detail, one can obtain ρ (1) from the 2-body reduced density matrix of a pure state |Ψ .. , ϕ i |s , ... , ϕ j |s , ... ], namely a Slater determinant in which each electron occupying ϕ i and ϕ j is in the spin superposition state |s = (|0 +|1 )/ √ 2 [see Fig. 1(b)] [15]. Similarly, ρ (2) can be obtained as the equal-probability ensemble of four density matrices, Γ σσ , each obtained from the 2-body reduced density matrix for a pure state |Ψ  Fig. 1(c)]. Detailed derivations of these results are given in Appendix A. One may wonder why in formulating ρ (1) the electrons occupying ϕ i and ϕ j have been placed in the spin quantum superposition |s . The reason is that since quantum fluctuations generate the doubly-excited configurations employed to obtain ρ (1) , one should not attribute a given spin to the electrons in orbitals ϕ i and ϕ j to fully capture the possible interactions between configurations.
The two models of the pair, and their associated density matrices ρ (1) and ρ (2) , are represented schematically in Fig. 2. We will employ these models to investigate the physical origin of the 1/4 exact exchange and justify Eq. 4 for the exchange energy contributed by two orbitals in a hybrid DFT functional. Below, we shall regard model 1 as a QM treatment of the pair and model 2 as a classical treatment based on probability, and thus a two-particle analogue of DFT.

B. Physical origin of the 1/4 exact exchange
To compute the exchange energy of the electron pair in the two models, we focus on the two-body Coulomb term V 12 = e 2 |r1−r2| of the Hamiltonian, keeping only the exchange integral J (and neglecting the direct Coulomb integral) resulting from this interaction [11]. Using a property of the trace, Tr |p p|Ô = p|Ô|p , with |p p| a projector andÔ an operator, we obtain the exchange energy E (i) X of the electron pair in model i = 1, 2 by computing the matrix elements of the Coulomb interaction between the two-particle determinant basis states, |Ψ µ in Eq. 6: where the sums run over the four basis states, and we defined the exchange part of the matrix elements of the Coulomb interaction, where the subscript X denotes that only the exchange integrals are kept. With these definitions, the exchange energy matrix M reads: Let us discuss briefly how this result is obtained. The non-zero diagonal matrix elements are M 11 = M 22 = J, while M 33 = M 44 = 0 since for these matrix elements the exchange integral vanishes. We take the orbitals ϕ i and ϕ j to be real, so that M is symmetric, and obtain the off-diagonal part of M using the properties of the matrix elements of the two-body operator V 12 between Slater determinants that differ by one or two spin-orbitals (see Ref. 11). For example, M 12 = ↑↑ |V 12 | ↓↓ X is a matrix element taken between determinants differing by two spin-orbitals; it vanishes since no term survives the integration over spin. All of the M 13 , M 14 , M 23 , M 24 also vanish since they are matrix elements between two determinants differing by one spin-orbital, which can be obtained from one another by flipping one spin.
The only nonzero off-diagonal matrix element is M 34 = ↑↓ |V 12 | ↓↑ X = J (and M 43 = M 34 ), which corresponds to a spin-flip exchange interaction (see Fig. 3) between pair states with antiparallel spins in the orbitals ϕ i and ϕ j . This interaction contributes to the exchange energy of the pair only in model 1 (see Eq. 11), and is associated with an exchange process between two doublyexcited configurations in which an electron in spin-orbital ϕ i↑ scatters into ϕ j↓ , and viceversa, due to the Coulomb interaction (see Fig. 3). The strength of this interaction is J, the exact exchange integral used in HF, even though in HF such interactions coupling different configurations do not contribute to exchange. As such, it should be regarded as a form of correlation − in the guise of an exchange process − arising from the coupling between doubly-excited electronic configurations that contribute to the ground state energy. Note that these spin-flip exchange interactions are not included in the HF calculation of the HEG employed to define the exchange energy in the LDA. They are also not explicitly included in the LDA correlation energy of the HEG [9]. We argue below that such spin-flip exchange interactions are the origin of the 1/4 exact exchange missing in semilocal DFT.
The exchange energy of the pair in the two models can be obtained using Eq. 11 with the matrix M given above: Note that the QM treatment of model 1 accounts for all of the four processes contributing to exchange, including the parallel-spin exchange processes M 11 and M 22 , and the two spin-flip exchange process M 34 and M 43 . By contrast, the ensemble description of the pair in model 2 misses the two spin-flip exchange process M 34 and M 43 . Since model 1 is a proper QM treatment of the pair in the interacting electron gas, we rename E (1) X to just E X , the exchange energy of the electron pair, and write: As we noted above, the last term in Eq. 15 equals one quarter of the HF exchange energy contribution from the interaction between the two orbitals ϕ i and ϕ j , so we can write: While E X equals J/2 in the pairwise exact exchange picture, our goal is to express it in the density-based picture. This is natural since the ensemble treatment of model 2 is a probability-based approach similar to that of DFT. We will prove in the next section that the exchange energy E (2) X of the pair described as a statistical ensemble in model 2 corresponds to 3/4 of the densitybased exchange energy, namely, that E . This equality will give us the result we want to prove, namely, that the exchange energy for a pair of occupied orbitals is Comparing this result with Eq. 4 tells us that a hybrid functional with 1/4 mixing contains, for each pair of orbitals, the correct exchange interactions, which account for both the same-spin exchange and the spin-flip exchange from interacting doubly-excited configurations.

C. Origin of the 3/4 density-based exchange
Expressing in terms of the density the exchange energy E (2) X of an electron pair in model 2 requires shifting from a pairwise to a density-based view of exchange interactions. Model 2 is a statistical ensemble of the four possible spin states of the pair; within each state, the exchange interaction is described by the matrix element M µµ , where µ = 1, ... , 4 labels the spin state of the electron pair in Eq. 6. As we noted earlier, all particles in the electron density contribute the same to exchange in semilocal DFT, regardless of their spin state. This means that one can identify the fraction of electrons contributing to exchange, and then assign to all such electrons the same contribution, which is a functional of the total electron density.
It is somewhat intuitive that, if the electron gas is described as an ensemble of pairs using a probabilitybased treatment, only the pairs in the electron gas in a triplet state should contribute to exchange. This is because, in essence, exchange interactions lower the Coulomb repulsive energy by keeping a pair of electrons apart due to their spatially antisymmetric wave function, which requires a symmetric (triplet) spin state. In our model 2 of the electron pair, both |Ψ 1 = A ij |00 and |Ψ 2 = A ij |11 are spatially antisymmetric (see Eq. 7) triplet states. However, the spatial parts of |Ψ 3 = |↑↓ and |Ψ 4 = |↓↑ lack a well-defined symmetry. The problem is clearer when shown in a pictorial view of the statistical ensemble ρ (2) Fig. 4 (a)], where the portions of the system in states |Ψ 1 and |Ψ 2 clearly contribute to exchange with their corresponding matrix elements M 11 = M 22 = J, but it is unclear how the other half of the system, in states |Ψ 3 and |Ψ 4 , contributes, since the spin-flip exchange process hides in the interaction M 34 between these two states.
This subtle issue can be resolved by diagonalizing M in the subspace of states |Ψ 3 and |Ψ 4 , which provides a singlet and a triplet state with well-defined spatial wave function symmetry. Such triplet state |Ψ T and singlet state |Ψ S can be written in terms of |Ψ 3 and |Ψ 4 as 2 is the antisymmetric spatial wave function defined above, and S ij (r 1 , r 2 ) = [ϕ i (r 1 )ϕ j (r 2 ) + ϕ j (r 1 )ϕ i (r 2 )]/ √ 2 is a symmetric spatial wave function. It is clear that only |Ψ T contributes to exchange due to its antisymmetric spatial wave function, but |Ψ S does not.
The electron pair ensemble ρ (2) can be rewritten as formed for 3/4 by triplet (|Ψ 1 , |Ψ 2 and |Ψ T ) and 1/4 by singlet (|Ψ S ) spin states: This different way of expressing ρ (2) , in terms of pair states with a well-defined total spin, is shown schematically in Fig. 4 (b). The part of the ensemble contributing to exchange, which is called below ρ (2) E , contains the terms in ρ (2) proportional to |A ij (r 1 , r 2 )| 2 : We can thus show that the fraction of electrons α E contributing to the density-based exchange equals 3/4: Above, the electron density of the pair, n(x), is obtained from the two-body density matrix as [10,12] n which correctly integrates to 2 electrons, while the electron density contributing to exchange is which integrates to 2 × 3 [n], concluding the proof of Eq. 17 and justifying Eq. 4 for the hybrid DFT exchange energy contributed by a pair of electron orbitals. Note that E DFT X depends non-linearly on the density, and it needs to be computed with the total electron density n, which includes also the density of electrons in singlet states.
There is an important subtlety in comparing Eq. 4 and Eq. 17. In the former, E hybrid X is computed using the electron density for doubly-occupied orbitals ϕ i and ϕ j , which is due to four electrons, while in Eq. 17 the electron density integrates to two electrons. This subtlety can be resolved by observing that for the spin-flip exchange interaction to occur, only two electrons (one per  (2) is an ensemble of states with well-defined single particle spin, but one is unsure of how to account for the spin-flip exchange interaction M34. (b) Our view of density-based exchange, in which ρ (2) is a mixture of 3 triplet and 1 singlet states. Each triplet spin state contributes 1/4 E DFT X to exchange, so that a total of 3/4 of the density-based exchange interaction needs to be included. (c) Schematic representation of exchange in hybrid DFT with 1/4 mixing. The portions of the system in red contribute to density-based exchange; the quarter in blue is the exact exchange contribution, which accounts, as we argue, for spin-flip exchange processes, thus completing the exchange hole. orbital) can occupy the orbitals ϕ i and ϕ j , since the initial and final states possess opposite spins in each orbital [see Fig. 3]. Therefore, to extend Eq. 17 to four electrons, one needs to use the density n(r) = 2 ( |ϕ i (r)| 2 +|ϕ j (r)| 2 ) for four electrons in the 3/4 E DFT X [n] term, as in Eq. 4, while keeping the spin-flip interaction as the one due to two electrons, which equals 1/4 E HF X , since this term derives from doubly-excited configurations. After this step, our result in Eq. 17 matches exactly the hybrid DFT result in Eq. 4 we aimed to explain. This reasoning also provides the basis for extending the result to the entire electron gas. To extend Eq. 17 to all the electrons, one should use the density due to all occupied orbitals, and sum over all orbital pairs the pairwise part accounting for the spin-flip (exact) exchange, thus obtaining the total exchange energy for all the electrons in a hybrid functional with 1/4 mixing, E X = 3/4 E DFT X [n] + 1/4 E HF X . Our results can thus rationalize, both for each pair of occupied orbitals and (somewhat less rigorously) for the entire electron gas, the 1/4 exact exchange employed in hybrid DFT.
Finally, we have argued that in the density-based picture only 3/4 of the electrons contributes to exchange. This result is at odds with the approach currently taken in the LDA, where all the electrons − rather than only 3/4 of them − contribute to exchange. Our result was derived by treating the spin states of an electron pair as an ensemble. On the other hand, there is no rigorous justification for the approach taken in the LDA, where through a HF calculation of the HEG, in which only half of the pairs contribute to exchange, one derives a per-particle exchange energy that is then applied to all particles, regardless of their spin. The important step of going from a pairwise to a density-based view of exchange is tacitly neglected in the LDA due to the assumed locality.

IV. DISCUSSION
A. Why does the LDA overbind?
A corollary of our results is that a maximum fraction of 3/4 of density-based exchange should be used. Using a larger fraction is incorrect because, by doing so, one fails to take into account the spin-dependent nature of exchange. Is it possible that including a unit fraction of density-based exchange is responsible for LDA's tendency to overbind matter and overestimate cohesive and atomization energies? As noted above, the LDA overbinds even though its nature is essentially variational since its exchange and correlation parts derive, respectively, from a HF and a QMC calculation of the HEG. Since E XC is dominated by exchange, it is intuitive that exchange should be responsible for the overbinding of the LDA.
If one were to include only 3/4 of the density-based exchange (without adding a corresponding fraction of 1/4 exact exchange) the LDA may well underbind and recover its variational character, but the system would not be described accurately due to the absence of a complete exchange hole [16]. To complete the exchange hole, in hybrid DFT one adds a 1/4 fraction of exact exchange, which mimics, as we argue, correlation processes that are not included in the density-based exchange and that originate from spin-flip exchange interactions between doubly-excited configurations. Besides revisiting our intuition about density-based exchange, improving the treatment of density-based correlation may also be needed to properly account for spin. One could, for example, combine the HEG correlation energies of the unpolarized and polarized electron gas [9] in a way that more accurately accounts for spin-dependent exchange and correlation processes in the unpolarized electron gas. We discuss numerical results that corroborate the interpretation given here on the origin of the 1/4 exact exchange. We focus on trends obtained with hybrid functionals in systems with less than four electrons. In particular, Perdew et al. [8] compiled a table of atomization energies of several molecules, and showed that a simple hybrid with 1/4 exact exchange dramatically improves the accuracy of the computed atomization energy compared to LDA calculations, in which atomization energies are significantly overestimated. Three molecules in the data set, H 2 , Li 2 , and LiH, are an exception to this trend [8]; their atomization energies computed with the hybrid functional exhibit nearly the same or a larger discrepancy with experiment than in the LDA. In Table I  Our treatment of the physical origin of the 1/4 exact exchange is valid only when four or more electrons are present, since otherwise 1/4 E HF X does not equal the contributions from the spin-flip exchange processes from doubly-excited configurations. Therefore, when less than four electrons (in two doubly-occupied orbitals) are present, the exact exchange added by hand cannot successfully mimic these exchange interactions. One could argue that the trends in the data of Ref. 8 for H 2 , Li 2 , and LiH may be a mere coincidence. Yet, it should be noted that these are the only three molecules in the data set with less than four valence electrons, and also the only molecules for which the hybrid result does not improve over the LDA. This observation does not constitute a complete characterization of the performance of hybrid functionals for systems with less than four valence electrons, and the topic admittedly deserves further investigation. It is nevertheless a useful indication that the interpretation put forward in this work on the physical origin of the 1/4 exact exchange is backed in some measure by hybrid DFT calculations.

V. CONCLUSIONS
Let us summarize and further discuss our main results. We have shown that the 1/4 exact exchange employed in hybrid DFT can be attributed to spin-flip exchange interactions between pair states with antiparallel spin. These interactions are not included in HF and semilocal DFT and can be seen as correlation processes in the guise of exchange. Such exchange processes arise from the interaction between doubly-excited configurations that contribute to the ground state energy. They need to be added by hand to the density-based exchange picture, using exact exchange because they are orbitaldependent. Including a 1/4 fraction of exact exchange can thus be seen as a rudimentary attempt to include explicitly pair correlation and multireference effects in the density functional picture. This interpretation may also explain why hybrid functionals can successfully describe the ground state of several strongly correlated materials [17], despite the challenges with the latter are due mainly to correlation rather than exchange.
We have also shown that, within a probability-based treatment in which the spin states of an electron pair are modeled as a statistical ensemble, the correct fraction of density-based exchange is 3/4, since only 3 out of 4 pairs in the electron gas are in a triplet state and therefore kept apart by the exchange interaction. Adding the 3/4 density-based and 1/4 exact exchange completes the exchange hole, thus satisfying the sum rule that one electron is missing around each electron due to exchange.
The picture that emerges from our model is rather different from the conventional view of hybrid DFT. We argue that a fraction of 3/4 is the correct amount of density-based exchange, and that using a larger fraction is incorrect because, by doing so, one fails to take into account the spin-dependent nature of exchange. This result may explain the well-known (yet unexplained) tendency of the LDA to overbind matter in spite of its variational nature. Within our model, the factor of 4 in the 1/4 exact exchange fraction reflects the number of possible spin states of an electron pair, rather than an order of perturbation theory as has been previously proposed. We also observed that hybrid functionals may only be effective for systems with more than four electrons (or, more precisely, with at least two doublyoccupied orbitals). The results obtained here have assumed an unpolarized electron gas, a density-based exchange that depends only on the density and not on its gradient, and a global hybrid functional. It should be possible, though nontrivial, to extend our treatment to a spin-polarized electron gas, and to gradient-corrected exchange and range-separated hybrid functionals.
M.B. thanks Nicola Marzari and Fernando Brandao for fruitful discussions. M.B. also thanks Megan Schill for discussion and for help with the initial stage of this project. This work was supported by the National Science Foundation under Grant CAREER no. CAREER-1750613.
Following Lowdin [10], we define the 2-body reduced density matrix (2-RMD) Γ(x 1 , x 2 ) for N electrons as is an antisymmetric wave function for N electrons, and x = {r, σ} denotes both the spatial coordinate r and the spin σ.
We specialize to the case in which the wave function Ψ N is a single Slater determinant, Ψ N = (1/ √ N !) det[{ϕ k }], with {ϕ k } a set of orthonormal spinorbitals labeled by the index k = 1, ... , N . In this case, the 2-RMD takes the simple form [16] Γ(x 1 , x 2 ) = 1 2 where γ(x, x ) = N k=1 ϕ * k (x)ϕ k (x ) is the 1-body density matrix. For a Slater determinant wave function, we can thus write the 2-RDM in Eq. A2 as In our work, we focus on the contribution to exchange from an electron pair occupying two given spin-orbitals, ϕ iσ and ϕ jσ . We thus keep in the 2-RDM only the terms that involve these orbitals, and obtain the density matrix: where h.c. denotes the Hermitian conjugate of the previous term in brackets, and we removed the tilde to indicate we are no longer working with the full 2-RDM for N electrons, but rather, with its two-electron part of relevance here. We write explicitly the spatial and spin parts, for each spin-orbital ϕ iσ (x) and its Hermitian conjugate ϕ * iσ (x), as ϕ iσ (x) = ϕ i (r) |σ ϕ * iσ (x) = ϕ * i (r) σ| .