Diata Traore

Emmanuel Giner

Julien Toulouse

Basis-set correction based on density-functional theory: Rigorous framework for a one-dimensional model

We reexamine the recently introduced basis-set correction theory based on density-functional theory consisting in correcting the basis-set incompleteness error of wave-function methods using a density functional. We use a onedimensional model Hamiltonian with delta-potential interactions which has the advantage of making easier to perform a more systematic analysis than for three-dimensional Coulombic systems while keeping the essence of the slow basis convergence problem of wave-function methods. We provide some mathematical details about the theory and propose a new variant of basis-set correction which has the advantage of being suited to the development of an adapted localdensity approximation. We show indeed how to develop a local-density approximation for the basis-set correction functional which is automatically adapted to the basis set employed, without resorting to range-separated density-functional theory as in previous works, but using instead a finite uniform electron gas whose electron-electron interaction is projected on the basis set. The work puts the basis-set correction theory on firmer grounds and provides an interesting strategy for the improvement of this approach.

I. INTRODUCTION

In electronic-structure theory of atoms, molecules or solids, one of the main limitations of standard correlated wavefunction computational methods for solving the Schrödinger equation is the slow convergence of the energy and other properties with respect to the size of the one-electron basis set employed (see, e.g., Refs. 1-4). This slow basis convergence originates from the singular behavior of the repulsive Coulomb electron-electron interaction at small interelectronic distances, which creates a depletion in the wave function with a characteristic derivative discontinuity at electron-electron coalescence -the infamous electron-electron cusp 5,6 .

The two main approaches for dealing with this problem are (i) extrapolation to the complete-basis-set limit by using increasingly large basis sets 1,2 , and (ii) explicitly correlated methods which incorporate in the wave function a correlation factor reproducing the electron-electron cusp (see, e.g., Ref . 7). Recently, some of the present authors introduced an alternative basis-set correction scheme based on densityfunctional theory (DFT) 8 . This latter scheme consists in correcting the energy calculated by a wave-function method with a finite basis set by a density functional incorporating the short-range electron correlation effects missing in the basis set. This basis-set correction scheme was further developed and tested in Refs. 9-15, demonstrating that it successfully accelerates the basis convergence of wave-function methods for various properties and systems.

The advantages of the basis-set correction scheme is its conceptual simplicity and computational efficiency. In practice, however, the limpidity of this approach is somewhat diminished by the fact that in all previously cited works the a) Electronic mail: dtraore@lct.jussieu.fr b) Electronic mail: emmanuel.giner@lct.jussieu.fr c) Electronic mail: toulouse@lct.jussieu.fr basis-set correction functional was approximated by shortrange correlation functionals borrowed from range-separated DFT [START_REF] Savin | Recent Developments of Modern Density Functional Theory[END_REF][START_REF] Toulouse | [END_REF][18][19][20] ), relying on an approximate mapping between the basis-set correction theory and range-separated DFT.

In the present work, we reexamine more closely the basisset correction theory. For this, we use a one-dimensional (1D) model Hamiltonian with delta-potential interactions [21][22][23][24] which has the advantage of making easier to perform a more systematic analysis than for three-dimensional (3D) Coulombic systems, while keeping the essence of the slow basis convergence problem of wave-function methods. After introducing the 1D model and discussing its relevance in Section II, we present the basis-set correction theory in some mathematical details in Section III. In particular, we introduce a new variant of basis-set correction which has the advantage of being suited for development of an adapted local-density approximation (LDA). In Section IV, we show indeed how to develop a LDA for the basis-set correction functional which is automatically adapted to the basis set employed without resorting to range-separated DFT but using instead a finite uniform electron gas (UEG) whose electron-electron interaction is projected on the basis set. Section IV C contains our conclusion and outlook. Hartree atomic units (a.u.) are used throughout this work.

II. ONE-DIMENSIONAL MODEL SYSTEM

A. Description of the model

We consider N = 2 electrons in a 1D He-like atom with delta-potential interactions described by the Hamiltonian [21][22][23][24]

H = T + W ee + V ne , (1)
where

T = - 1 2 N i=1 ∂ 2 ∂x 2 i , W ee = δ(x 1 -x 2 ), V ne = -Z N i=1 δ(x i ) (2)
are the kinetic-energy operator, the Dirac-delta electronelectron interaction, and the Dirac-delta nucleus-electron potential with nuclear charge Z = 2, respectively. Since we will be only interested in the spin-singlet ground state, we can ignore spin and antisymmetry, and thus work on the oneelectron Hilbert space h = L 2 (R, C) and the two-electron (non-antisymmetrized) tensor-product Hilbert space H = h ⊗ h. The ground-state energy can be expressed as

E 0 = min Ψ∈W Ψ, HΨ , (3)
where W is the set of all admissible wave functions

W = Ψ ∈ H | Ψ ∈ H 1 (R 2 , C), Ψ, Ψ = 1 , (4)
where

H 1 (R 2 , C) = { f ∈ L 2 (R 2 , C) | ∂ i f ∈ L 2 (R 2 , C), i = 1, 2}
is the first-order Sobolev space and , designates the L 2 inner product.

The ground state of this 1D He-like atom with deltapotential interactions can be considered as a model for the ground state of the real 3D He atom. Indeed, it can be shown that the ground state of a generalization to arbitrary dimension D of the electronic Hamiltonian of the He atom with Coulomb-potential interactions exactly reduces for D = 1, after appropriate scaling of the energies and distances, to the ground state of the Hamiltonian in Eq. ( 1) 23,[25][26][27][START_REF]Dimensional Scaling in Chemical Physics[END_REF] . Our main interest in this model is that it gives an electron-electron cusp (or derivative discontinuity) condition identical to the familiar 3D one 5,6 , i.e. for small interelectronic distances x 12 = x 1x 2 the exact ground-state wave function behaves as

Ψ 0 (x 1 , x 2 ) = Ψ 0 (x 1 , x 1 ) 1 + 1 2 |x 12 | + O(x 2 12 ) . (5)
When using a finite one-electron basis set, we thus expect a slow convergence with the basis size very similar to the slow convergence observed in 3D quantum systems with the Coulomb electron-electron interaction. This is why we prefer this model to other 1D quantum systems (see, e.g., Refs. 29 and 30).

Another neat fact about the present model is that it can be solved analytically at the Hartree-Fock (HF) level 31 . The total HF ground-state energy is

E HF = -Z 2 + Z 2 - 1 12 = -3.083333... a.u., (6)
and the HF (doubly) occupied spatial orbital is

∀x ∈ R, φ 1 (x) = 2β √ γ e -β|x| 1 -γ e -2β|x| , (7)
with

β = Z -1/2 = 3/2 and γ = 1/(4Z -1) = 1/7.
The exact ground-state energy cannot be calculated analytically but has been accurately estimated numerically 21,24 to be E 0 = -3.155390 a.u..

B. Full-configuration interaction in a basis set

We now consider full-configuration-interaction (FCI) calculations in a finite one-electron basis set B ⊂ H 1 (R, C). To have a systematically improvable basis set, we use Hermite (or Hermite-Gaussian) basis functions

∀x ∈ R, f α n (x) = N α n H n ( √ 2αx) e -αx 2 , ( 8
)
where n is a natural number, H n are the Hermite polynomials, N α n = (2 n n!) -1/2 (2α/π) 1/4 is the normalization factor, and α > 0 is a real constant. It is well known that the set { f α n } n=0,...,n max converges to a complete orthonormal basis of L 2 (R, C) in the limit n max → ∞ for any fixed exponent α. We deliberately use the same exponent in all basis functions, namely α = 11.5, instead of multiple exponents in order to avoid optimizing them. Except for that, this basis set is quite similar to the Gaussian-type-orbital basis sets widely used in quantum chemistry. Since we are not interested in the convergence of the HF energy with this basis set (which is slow, see Appendix A 1) but only in the convergence of the FCI correlation energy, we add the exact occupied HF orbital φ 1 given in Eq. (7) to the basis set. Our final basis set is thus

B = φ 1 ∪ f α n n=0,...,n max ≡ χ µ µ=1,...,M , (9)
and contains M = n max + 2 basis functions:

χ 1 = φ 1 , χ 2 = f α 0 , ..., χ M = f α n max .
We introduce now h B = span(B) as the M-dimensional one-electron Hilbert space spanned by this basis set B, and

H B = h B ⊗ h B the corresponding two-electron Hilbert space of dimension M 2 .
The FCI ground-state energy for this basis set B is

E B FCI = min Ψ∈W B Ψ, HΨ , (10)
where W B is the set of normalized wave functions restricted to

H B W B = Ψ ∈ H B | Ψ, Ψ = 1 . (11)
In practice, we proceed as follows. For each basis size n max , we first perform a HF calculation [START_REF] Helgaker | Molecular Electronic-Structure Theory[END_REF][START_REF] Szabo | Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory[END_REF] . The nucleus-electron integrals are just χ µ , v ne χ ν = -Zχ µ (0)χ ν (0), and the kinetic integrals χ µ , tχ ν = (1/2) R χ µ (x)χ ν (x)dx and the two-electron integrals χ µ χ ν , W ee χ λ χ σ = R χ µ (x)χ ν (x)χ λ (x)χ σ (x)dx are calculated by Romberg numerical integration [START_REF] Press | Numerical Recipes[END_REF] . We then use the obtained HF orbitals {φ i } i=1,...,M to expand the FCI groundstate wave function as

Ψ B FCI (x 1 , x 2 ) = M i=1 M i=1 c i, j φ i (x 1 )φ j (x 2 ). ( 12
)
The FCI coefficients c i, j and the associated FCI ground-state energy E B FCI are found by diagonalization of the Hamiltonian. Parity inversion symmetry is exploited in all our calculations.

In Fig. 1, we report the FCI ground-state energy E B FCI as a function of the basis size n max . We observe a quite slow convergence of E B FCI with n max toward the exact ground-state energy E 0 . A numerical fit from n max = 50 and 70 gives the following power-law convergence gives A ≈ 0.077 a.u. and b ≈ 0.68. Equivalently, in terms of the number of basis functions M, this convergence law can be stated as 1/M b . Note that since our basis set includes the exact HF occupied orbital this slow convergence is entirely due to the correlation energy. According to the analysis given in Appendix A 2, we theoretically expect b = 0.5. The difference most likely means that larger values of n max are needed to reach the asymptotic regime. Note that for the 3D Coulomb case it is well known that the correlation energy exhibits a cubic-law convergence with respect to either the maximal angular momentum or the maximal principal quantum number of the basis set, or equivalently a 1/M convergence law in terms of the number of basis functions [START_REF] Helgaker | Molecular Electronic-Structure Theory[END_REF][START_REF] Klopper | [END_REF] . In the present work, the use of a basis of Hermite functions with a single exponent thus leads to an even slower convergence rate.

E B FCI ∼ n max →∞ E 0 + A n b max , (13)
To illustrate further the slow basis convergence, we also calculate the FCI correlation pair density

ρ B 2,c (x 1 , x 2 ) = 2 |Ψ B FCI (x 1 , x 2 )| 2 -|Φ HF (x 1 , x 2 )| 2 , ( 14
)
where Φ HF (x 1 , x 2 ) = φ 1 (x 1 )φ 1 (x 2 ) is the HF wave function. In Fig. 2, this quantity is plotted with respect to the second electron coordinate x 2 for a fixed value of the first electron coordinate x 1 = 0. The convergence of ρ B 2,c (x 1 , x 2 ) with n max is again slow and reminiscent of the well-known slow basis convergence of the correlation pair density for the 3D Coulomb electron-electron interaction (see, e.g., Refs. 3, 35, and 36). Note that the small derivative discontinuity seen on all the curves of Fig. 2 at x 2 = 0 is due to the fact that we include in our basis set the exact HF orbital in Eq. ( 7) which has itself an electron-nucleus cusp, namely φ 1 (x) = φ 1 (0)(1 -Z|x| + O(x 2 )). The electron-electron cusp condition in Eq. ( 5) is only recovered for large n max .

In conclusion, the present 1D model adequately captures the main characteristics of the slow basis convergence problem of standard quantum-chemistry wave-function methods, and it is thus appropriate for applying our basis-set correction approach.

III. BASIS-SET CORRECTION THEORY

We now develop the basis-set correction theory based on DFT for the present 1D model which aims at removing the basis-set incompleteness error in the FCI ground-state energy. This requires to develop an extension of standard DFT from the usual complete-basis-set setting to the case of the incomplete finite one-electron basis set B. In such a finite basis set, it is known that the original Hohenberg-Kohn theorem 37 does not hold anymore [38][39][40] , in the sense that there is an infinite number of local potentials which give, after projection in a finite basis set, the same ground-state density 41 . However, we will show that we can still define density functionals associated with a finite basis set.

A. Density-functional theory for the one-dimensional model

We start by reviewing some useful definitions of standard DFT specialized to the 1D model. For mathematically oriented reviews of DFT, see for instance Refs. 42-49, and in particular Ref. 50 which encompasses the 1D case.

Working on the same Hilbert space H as before, we now consider the following 1D Hamiltonian for N = 2 electrons with a general external potential v

H[v] = T + W ee + V, (15)
where again

T = -(1/2) N i=1 ∂ 2 /∂x 2 i and W ee = δ(x 1 -x 2 )
, and

V = N i=1 v(x i
) is now a general external potential operator. We will still take admissible wave functions to be in the space W given in Eq. ( 4). The convex set of densities representable by a wave function Ψ ∈ W, the so-called N-representable densities, is then 42,[START_REF] Lewin | [END_REF]

R = {ρ | ∃ Ψ ∈ W, ρ Ψ = ρ} = ρ ∈ L 1 (R) | ρ ≥ 0, R ρ(x)dx = N, √ ρ ∈ H 1 (R) , (16)
where

ρ Ψ (x 1 ) = N R |Ψ(x 1 , x 2 )| 2 dx 2
is the density of the wave function Ψ. We have R ⊂ X where X is the Banach space X = C 0 (R) ∩ L 1 (R) with C 0 (R) the space of continuous functions vanishing at infinity. Therefore, the space of external potentials v that we can consider is the continuous dual space of X, i.e. V = X = M(R)+ L ∞ (R) where M(R) is the space of bounded Radon measures. Note that the set V includes the external potential considered in Section II, i.e. v ne (x) = -Zδ(x). For v ∈ V, we then define the ground-state energy as

E 0 [v] = inf Ψ∈W Ψ, H[v]Ψ . ( 17
)
The Levy-Lieb density functional 42,[START_REF] Levy | [END_REF] is defined as a constrained-search over wave functions yielding the density

ρ ∀ρ ∈ R, F[ρ] = min Ψ∈W ρ Ψ, (T + W ee )Ψ , (18)
where

W ρ = {Ψ ∈ W | ρ Ψ = ρ}.
It gives the ground-state energy as

E 0 [v] = inf ρ∈R (F[ρ] + (v, ρ)) , (19)
where we have introduced the notation (v, ρ) = R v(x)ρ(x)dx. If a minimizing density ρ 0 exists in Eq. ( 19) then it is an exact ground-state density for the potential v.

One can also define the Lieb density functional 42 , which is the Legendre

-Fenchel convex-conjugate of E 0 [v] ∀ρ ∈ R, F L [ρ] = sup v∈V (E 0 [v] -(v, ρ)) . (20)
Just like the Levy-Lieb functional F, the Lieb functional F L gives the exact ground-state energy as

E 0 [v] = inf ρ∈R (F L [ρ] + (v, ρ)).
In general, the functionals F and F L are different, the Lieb functional being in fact the lower semicontinuous convex envelope (lscv) of the Levy-Lieb functional, i.e.

F L = lscv(F) ≤ F. (21)
It turns out that the Lieb functional can also be expressed as a generalization of the Levy-Lieb functional in which the constrained search is extended from pure-state wave functions to ensemble density matrices 42,52 . This implies that

F[ρ] = F L [ρ]
for densities ρ which are densities of a nondegenerate ground state of the Hamiltonian H[v] for some potential v. In the present case of two spin-singlet electrons, the ground state is always non-degenerate and thus the Levy-Lieb and Lieb functionals are identical, i.e. F = F L .

B. First variant of basis-set correction

The first variant of basis-set correction corresponds to the one introduced for the 3D Coulombic case in Ref. 8 and further developed in Refs. 9-12, and 14. We consider the Hamiltonian H[v] in Eq. ( 15) on the two-electron Hilbert space H B associated with the basis set B. For v ∈ V, the FCI groundstate energy is

E B FCI [v] = min Ψ∈W B Ψ, H[v]Ψ , (22)
where W B , given in Eq. ( 11), is the set of normalized wave functions restricted to H B . We define the corresponding Levy-Lieb density functional for the basis set B as

∀ρ ∈ R B , F B [ρ] = min Ψ∈W B ρ Ψ, (T + W ee )Ψ , (23)
where

W B ρ = Ψ ∈ W B | ρ Ψ = ρ and R B is the set of densi- ties representable by a wave function Ψ ∈ W B R B = ρ | ∃ Ψ ∈ W B , ρ Ψ = ρ . (24)
A priori, this set is not convex and not easily characterized.

The FCI ground-state energy can be expressed as

E B FCI [v] = min ρ∈R B F B [ρ] + (v, ρ) . ( 25
)
We now decompose the exact Levy-Lieb density functional F[ρ] in Eq. ( 18) as

∀ρ ∈ R B , F[ρ] = F B [ρ] + ĒB [ρ], (26)
where ĒB [ρ] is the complementary basis-set correction density functional

ĒB [ρ] = Ψ[ρ], (T + W ee )Ψ[ρ] -Ψ B [ρ], (T + W ee )Ψ B [ρ] , (27)
and Ψ[ρ] and Ψ B [ρ] are minimizing wave functions in Eqs. ( 18) and ( 23), respectively. Clearly, since

W B ρ ⊂ W ρ , we have ∀ρ ∈ R B , F B [ρ] ≥ F[ρ]
, and thus ĒB [ρ] ≤ 0. Since the decomposition in Eq. ( 26) is defined only for ρ ∈ R B , we cannot recover the exact ground-state energy E 0 [v]. Instead, we can obtain the following approximate energy obtained by restricting the minimization in Eq. (19)

to densities ρ in R B E B 0 [v] = min ρ∈R B (F[ρ] + (v, ρ)) = min ρ∈R B       min Ψ∈W B ρ Ψ, (T + W ee )Ψ + ĒB [ρ] + (v, ρ)       = min Ψ∈W B Ψ, (T + W ee + V)Ψ + ĒB [ρ Ψ ] , (28)
or, designating by Ψ B 0 ∈ W B a minimizing wave function in Eq. ( 28),

E B 0 [v] = Ψ B 0 , (T + W ee + V)Ψ B 0 + ĒB [ρ Ψ B 0 ]. ( 29
)
It is easy to see that

E 0 [v] ≤ E B 0 [v] ≤ E B FCI [v]
. For a given basis set B, the functional ĒB [ρ] provides a (self-consistent) basis-set correction to the FCI energy so that

E B 0 [v] is a better approximation to E 0 [v] than E B FCI [v] is. Moreover, as the basis set is increased toward completeness, E B 0 [v] should converge much faster to E 0 [v] than E B FCI [v]
does, since, roughly speaking, densities ρ typically converges faster than wave functions Ψ with respect to the basis set.

For simplicity, instead of performing the minimization in Eq. ( 28), one may use a non-self-consistent approximation consisting in using the FCI ground-state wave function Ψ B FCI in place of Ψ B 0 , giving what we will call a "FCI+DFT" energy

E B FCI+DFT [v] = Ψ B FCI , (T + W ee + V)Ψ B FCI + ĒB [ρ Ψ B FCI ], (30)
which is an upper bound of

E B 0 [v], i.e. E B FCI+DFT [v] ≥ E B 0 [v]. Again, as the basis set is increased toward completeness, E B FCI+DFT [v] should converge much faster to E 0 [v] than E B FCI [v] does.
One inconvenient of this basis-set correction scheme is that, for a given finite basis set B, it does not give the exact groundstate energy, even in principle if we knew the exact complementary basis-set correction functional ĒB [ρ]. This is due to the fact that F B [ρ] is defined only on the restricted set of densities R B . Another related inconvenient is that since ĒB [ρ] is defined only on this restricted set of densities, it is not clear how to define the LDA for it. Defining the LDA would indeed require to consider uniform densities, but uniform densities are not in R B . Even though uniform densities are not in R either, they can be approached with densities from R 53,54 , so it would be preferable to have a complementary basis-set correction functional defined on the entire set R. This would also permit to connect in principle the basis-set correction scheme to the exact ground-state energy. This is what is achieved by the second variant of basis-set correction.

C. Second variant of basis-set correction

For the second variant of basis-set correction, we work on the full Hilbert space H (not restricted to the basis set B), and define the following Hamiltonian

H wB [v] = T + W B ee + V, (31)
where the kinetic-energy operator T and the external potential operator V are still defined as before, but the electron-electron interaction operator is now projected in the basis set B

W B ee = P B W ee P B , (32)
where P B is the orthogonal projector onto the basis-setrestricted Hilbert space H B . The notation "wB" is to remind us that only W ee is projected. Of course, W B ee is a complicated non-local two-electron operator. Using an orthonormal orbital basis {φ i } i=1,...,M spanning the same space as B, its integral kernel can be written as

W B ee (x 1 , x 2 ; x 1 , x 2 ) = M i=1 M j=1 M k=1 M l=1 φ i (x 1 )φ j (x 2 ) φ i φ j , W ee φ k φ l φ * k (x 1 )φ * l (x 2 ), (33)
where

φ i φ j , W ee φ k φ l = R φ * i (x)φ * j (x)φ k (x)φ l (x)
dx are the two-electron integrals in the orbital basis {φ i }. For v ∈ V, the associated ground-state energy is

E wB 0 [v] = inf Ψ∈W Ψ, H wB [v]Ψ . (34)
Clearly, if we were to restrict the minimization in Eq. (34) to the set W B , E wB 0 [v] would reduce to E B FCI [v] [Eq. (22)]. Therefore, we have

E wB 0 [v] ≤ E B FCI [v]
. Moreover, because W ee is a positive operator, one would expect that projecting it in a finite basis should typically decrease the ground-state energy, i.e. E wB 0 [v] ≤ E 0 [v], but this may not be generally true. We then define the corresponding Levy-Lieb functional for all densities ρ ∈ R as

∀ρ ∈ R, F wB [ρ] = min Ψ∈W ρ Ψ, (T + W B ee )Ψ . ( 35
)
For the same reasons as before, comparison with Eq. (23) shows that, for

ρ ∈ R B , F wB [ρ] ≤ F B [ρ].
The ground-state energy E wB 0 [v] can be written as

E wB 0 [v] = inf ρ∈R F wB [ρ] + (v, ρ) . ( 36
)
We now decompose the exact Levy-Lieb density functional

F[ρ] as ∀ρ ∈ R, F[ρ] = F wB [ρ] + ĒwB Hxc [ρ], (37)
which defines the complementary Hartree-exchangecorrelation (Hxc) basis-set correction functional ĒwB Hxc [ρ].

Analogously to what is done in multideterminant rangeseparated DFT 18,20,[55][56][57][58] , the functional ĒwB Hxc [ρ] can be decomposed as

ĒwB Hxc [ρ] = ĒwB Hx,md [ρ] + ĒwB c,md [ρ], (38)
where ĒwB Hx,md [ρ] is the Hartree-exchange (Hx) contribution defined as the expectation value of the complementary interaction WB ee = W ee -W B ee over the minimizing multideterminant (md) wave function Ψ wB [ρ] (that we will assume to be unique up to a global phase factor) in Eq. ( 35)

ĒwB Hx,md [ρ] = Ψ wB [ρ], WB ee Ψ wB [ρ] , (39)
and ĒwB

c,md [ρ] is the remaining correlation (c) contribution ĒwB c,md [ρ] = Ψ[ρ], (T + W ee )Ψ[ρ] -Ψ wB [ρ], (T + W ee )Ψ wB [ρ] . (40)
Clearly, since Ψ[ρ] minimizes Ψ, (T + W ee )Ψ , we have ĒwB c,md [ρ] ≤ 0. Since the decomposition in Eq. ( 37) is defined for all densities ρ ∈ R, we can obtain the exact ground-state energy as

E 0 [v] = inf ρ∈R F wB [ρ] + ĒwB Hxc [ρ] + (v, ρ) = inf ρ∈R min Ψ∈W ρ Ψ, (T + W B ee )Ψ + ĒwB Hxc [ρ] + (v, ρ) = inf Ψ∈W Ψ, (T + W B ee + V)Ψ + ĒwB Hxc [ρ Ψ ] . (41)
For potentials v for which there exists a minimizing wave function Ψ wB 0 ∈ W in Eq. ( 41), this wave function yields a exact ground-state density ρ 0 , i.e. ρ Ψ wB 0 = ρ 0 . It is in fact the minimizing wave function in Eq. (35) giving the density ρ 0 , i.e. Ψ wB 0 = Ψ wB [ρ 0 ]. Using this fact and combining Eqs. ( 41), (38), and (39), we can express the exact groundstate energy as

E 0 [v] = Ψ wB 0 , (T + W ee + V)Ψ wB 0 + ĒwB c,md [ρ Ψ wB 0 ]. ( 42
) Thus, this second variant of basis-set correction leads to an energy expression similar to Eq. ( 29) obtained for the first variant of basis-set correction, with the functional ĒwB c,md [ρ] replacing the functional ĒB [ρ] and the wave function Ψ wB 0 replacing the wave function Ψ B 0 . One advantage of this second variant of basis-set correction is that it gives the exact groundstate energy E 0 [v]. The price to pay is that the minimization in Eq. ( 41) is more complicated than in Eq. ( 28) since it is over general wave functions Ψ ∈ W and not simply wave functions restricted to W B . Moreover, the minimization involves not only the functional ĒwB c,md [ρ] but also the functional ĒwB Hx,md [ρ] in Eq. ( 39) which did not appear in the first variant of basis-set correction.

Similarly to the first variant of basis-set correction, we can define a non-self-consistent approximation consisting in using the FCI ground-state wave function Ψ B FCI in place of Ψ wB 0 in Eq. (42), giving an alternative "FCI+DFT" energy

E wB FCI+DFT [v] = Ψ B FCI , (T + W ee + V)Ψ B FCI + ĒwB c,md [ρ Ψ B FCI ], (43)
which is quite similar but not equivalent to Eq. ( 30) since the complementary basis-set correction functional is different. Like for the first variant of basis-set correction, when the basis set is increased toward completeness, E wB FCI+DFT [v] should converge much faster to E 0 [v] than E B FCI [v] does. Finally, we define the Lieb density functional for this second variant of basis-set correction

∀ρ ∈ R, F wB L [ρ] = sup v∈V E wB 0 [v] -(v, ρ) . (44)
Like in standard DFT, by the theory of Legendre-Fenchel transformations, this Lieb functional F wB L must be the lower semi-continuous convex envelope of the Levy-Lieb functional F wB , i.e.

F wB L = lscv(F wB ) ≤ F wB . ( 45
)
One could also write down this Lieb functional as a constrained-search over ensemble density matrices, and again we should have F wB [ρ] = F wB L [ρ] for densities ρ which are densities of a non-degenerate ground state of the Hamiltonian H wB [v] for some potential v. As already mentioned, in the present case of two spin-singlet electrons, the ground state is always non-degenerate and thus the Levy-Lieb and Lieb functionals are identical, i.e. F wB = F wB L . As we will see in Section IV B, the definition in Eq. ( 44) is useful to calculate the functional in practice since it involves a unconstrained maximization over potentials v whereas the definition in Eq. ( 35) involves a potentially more complicated constrained minimization over wave functions yielding a fixed density ρ.

In summary, the advantage of the second variant of basis-set correction over the first variant is that it is connected with the exact ground-state energy [via Eq. ( 41) or (42)] and that it involves a complementary basis-set correction functional which is defined for all densities in R. In the next section, we exploit this in order to construct a LDA for the functional ĒwB c,md [ρ].

IV. LOCAL-DENSITY APPROXIMATION FROM FINITE UNIFORM-ELECTRON GAS

In standard DFT, the LDA is based on the infinite UEG. Essentially, for the 1D case, calculating the energy per particle of the infinite UEG amounts to plugging an uniform density ρ unif : x → ρ 0 ∈ (0, +∞) in the density functional F[ρ] and taking the thermodynamic limit, i.e. lim N→∞ F[ρ unif ]/N. One difficulty is that a non-zero uniform density function ρ unif defined on the entire real line R is obviously not Nrepresentable. For the 3D case, the infinite UEG was rigorously mathematically defined in Refs. 53 and 54 by first convoluting the uniform density with a function of compact support (so that the convoluted density is N-representable) and then taking the thermodynamic limit N → ∞ (after removing the Hartree energy which is divergent for the Coulombic 3D case). Here, in the spirit of Ref. 59, we will instead consider a finite UEG, i.e., for a finite electron number N.

A. Finite uniform-electron gas for the complete-basis-set case

To define a 1D finite UEG, we generalize the standard DFT of Section III A from the real line R to a finite interval Ω a = (-a/2, a/2) of length a. Hence, the one-electron Hilbert space is h a = L 2 (Ω a , C) and the two-electron Hilbert space is H a = h a ⊗ h a . For external local potentials v ∈ V a (where the space V a will be specified below) and N = 2 electrons, we define the ground-state energy of the Hamiltonian H[v] = T +W ee +V [Eq. (15)] restricted to the Hilbert space H a as

E 0,a [v] = inf Ψ∈W a Ψ, H[v]Ψ a , (46)
with the set of admissible wave functions

W a = Ψ ∈ H a | Ψ ∈ H 1 per (Ω 2 a , C), Ψ, Ψ a = 1 , (47)
where

Ψ 1 , Ψ 2 a = Ω 2 a Ψ * 1 (x 1 , x 2 )Ψ 2 (x 1 , x 2 )
∀ρ ∈ R a , F a [ρ] = min Ψ∈W a,ρ Ψ, (T + W ee )Ψ a , (48)
where

W a,ρ = {Ψ ∈ W a , ρ Ψ = ρ} and R a is the set of N- representable densities on Ω a R a = {ρ | ∃ Ψ ∈ W a , ρ Ψ = ρ} = ρ ∈ L 1 (Ω a ) | ρ ≥ 0, Ω a ρ(x)dx = N, √ ρ ∈ H 1 per (Ω a ) , (49)
and

H 1 per (Ω a ) = { f ∈ H 1 (Ω a ) | lim x→-a/2 f (x) = lim x→a/2 f (x)}.
We have R a ⊂ X a where X a is the Banach space X a = C per (Ω a ) ∩ L 1 (Ω a ) with C per (Ω a ) the space of continuous functions on Ω a with periodic boundary conditions. Therefore, the space of external potentials to consider is the continuous dual space of X a , i.e. V a = X a = M per (Ω a ) + L ∞ (Ω a ) where M per (Ω a ) is the space of bounded Radon measures on Ω a with periodic boundary conditions. In the limit of an infinite interval (a → ∞), we recover the standard theory on the real line R, i.e. lim a→∞ F a [ρ] = F[ρ] where F[ρ] is the standard Levy-Lieb functional defined in Eq. ( 18). Similarly, we could generalize the Lieb density functionals in Eq. ( 20) to the finite interval Ω a .

We now define a finite UEG (fUEG) by considering the uniform density ρ unif : x → ρ 0 = N/a on the interval Ω a for the fixed electron number N = 2. Note that ρ unif is in fact the unique uniform density belonging to R a . The energy per particle of this finite UEG is

ε fUEG (ρ 0 ) = F a [ρ unif ] N , (50)
and is a function of the only variable ρ 0 since N is fixed and a = N/ρ 0 . The value of F a [ρ unif ] corresponds to the groundstate energy of the two-electron Hamiltonian with zero external potential

H fUEG = H[0] = T + W ee , (51)
with periodic boundary conditions on Ω a , provided that the ground-state density is the uniform density ρ unif (i.e., no translational symmetry breaking). We note that in Refs. 61-64 1D finite UEGs mapped to a ring were introduced using the Coulomb electron-electron interaction. Here, instead we use the Dirac-delta electron-electron interaction and we do not work on a ring. For a given density ρ 0 in the range [0, 10] a.u. and for the fixed electron number N = 2 and interval length a = N/ρ 0 , we calculate the ground-state energy by performing a FCI calculation using a one-electron plane-wave (pw) orthonormal basis {p n } n∈Z, |n|≤n pw max where p n (x) = (1/ √ a)e ik n x and k n = 2πn/a.

The one-electron kinetic integrals p n 1 , tp n 2 a = (2π 2 n 2 1 /a 2 )δ n 1 ,n 2 and the two-electron integrals p n 1 p n 2 , W ee p n 3 p n 4 a = (1/a)δ n 1 +n 2 ,n 3 +n 4 are trivial. We use a plane-wave cutoff n pw max = 60 which leads to FCI energies converged to at least 1 mhartree (and in fact generally better than that). As usual, the finite UEG energy per particle can be decomposed as ε fUEG (ρ) = t s,fUEG (ρ) + ε H,fUEG (ρ) + ε x,fUEG (ρ) + ε c,fUEG (ρ), (52) with the non-interacting kinetic energy per particle t s,fUEG (ρ) = 0 (since N = 2 the only occupied orbital is the 50) and ( 52)] of the 1D finite UEG as a function of the density ρ for N = 2 electrons and a plane-wave cutoff n pw max = 60. The exact PT2 correlation energy per particle, which is independent of N and ρ, is indicated as a horizontal line. The correlation energy per particle of the infinite UEG (N → ∞) as parametrized by Magyar and Burke 24 from essentially numerically exact Bethe-ansatz results is also plotted for comparison. constant plane wave p 0 (x) = 1/ √ a which has a zero kinetic energy), the Hartree energy per particle ε H,fUEG (ρ) = ρ/2, the exchange energy per particle ε x,fUEG (ρ) = -ρ/4, and the correlation energy per particle ε c,fUEG (ρ).

The correlation energy per particle ε c,fUEG (ρ) is plotted in Fig. 3. In the high-density limit, ε c,fUEG (ρ) tends to the correlation energy per particle in second-order perturbation theory (PT2) with respect to the electron-electron interaction W ee lim

ρ→∞ ε c,fUEG (ρ) = ε PT2 c,fUEG = - 1 24 , (53)
which is a constant independent of ρ. It turns out that this constant is the same for N = 2 and N → ∞ 24 . In the low-density limit, ε c,fUEG (ρ) goes to zero linearly with ρ (see Ref. 65)

ε c,fUEG (ρ) ∼ ρ→0 - ρ 4 , (54)
so as to exactly cancel out the Hartree and exchange energies per particle. This is due to the fact that, in this limit, the probability density of finding the electrons at the same point of space is zero and thus the Dirac-delta electron-electron interaction has not effect. This is the 1D version of the stronginteraction limit of DFT [66][67][68][69] . Equation ( 54) is also true for N → ∞ 24,70 , and is in fact true independently of N 65 . In Fig. 3, we also show the correlation energy per particle of the infinite UEG (N → ∞) as parametrized by Magyar and Burke 24 from essentially numerically exact Bethe-ansatz results. Not only the correlation energies per particle for N = 2 and N → ∞ agree well for small and large densities, as they should since they have the same N-independent asymptotic behaviors [Eqs. (53) and ( 54)], but they also agree very well for intermediate densities (the maximal deviation between the two curves being about 0.4 mhartree), showing that the thermodynamic limit N → ∞ is essentially already reached at N = 2 for this 1D UEG. Hence, there is no need considering 1D UEGs with N > 2 electrons. This must be due to the very short-range nature of the Dirac-delta electron-electron interaction. For the 1D UEG with the Coulomb interaction, the correlation energy per particle depends much more strongly on the electron number 62 .

B. Finite uniform-electron gas for the incomplete-basis-set case

We now generalize the second variant of basis-set correction of Section III C from the real line R to a finite interval Ω a = (-a/2, a/2) of length a. For v ∈ V a and N = 2 electrons, we define the ground-state energy of the restriction to the Hilbert space H a of the Hamiltonian H wB [v] = T +W B ee +V [Eq. ( 31)], featuring the electron-electron interaction projected in the basis set B used for the 1D He-like atom, as

E wB 0,a [v] = inf Ψ∈W a Ψ, H wB [v]Ψ a . (55)
The corresponding Levy-Lieb density functional is

∀ρ ∈ R a , F wB a [ρ] = min Ψ∈W a,ρ Ψ, (T + W B ee )Ψ a , (56)
and the corresponding Lieb density functional is

∀ρ ∈ R a , F wB L,a [ρ] = sup v∈V a E wB 0,a [v] -(v, ρ) a , (57)
where (v, ρ) a = Ω a v(x)ρ(x)dx. Again, in the limit of an infinite interval (a → ∞), we recover the theory of Section III C.

For a given basis set B, we now define an associated finite UEG by considering the uniform density ρ unif : x → ρ 0 = N/a on the interval Ω a for the fixed electron number N = 2.

The kinetic + electron-electron energy per particle of this Bdependent finite UEG is

f wB fUEG (ρ 0 ) = F wB a [ρ unif ] N , (58)
where

F wB a [ρ unif ] = Ψ wB [ρ unif ], (T + W B ee )Ψ wB [ρ unif ] a , (59)
and Ψ wB [ρ unif ] is the ground-state wave function (assumed to be unique up to a global phase factor) of the two-electron Hamiltonian

H wB fUEG = T + W B ee + V wB , (60)
with periodic boundary conditions on Ω a and with V wB = N i=1 v wB (x i ) where v wB (x) is the local potential (that we assume to exist and which is defined up to an additive constant) which enforces the constraint that the ground-state wave function Ψ wB [ρ unif ] yields indeed the uniform density ρ unif . Since the projected electron-electron interaction W B ee breaks translation invariance, the addition of the potential v wB is necessary to restore a uniform density. This is in contrast with the UEG for the complete-basis-set case for which no external potential was necessary to obtain a uniform density [Eq. ( 51)]. To conveniently obtain the potential v wB , we use the fact that, since the two-electron finite UEG has a nondegenerate ground state, the Levy-Lieb functional F wB a and the Lieb functional F wB L,a are identical. The potential v wB then just corresponds to the maximizing potential in Eq. ( 57) for ρ = ρ unif (see Refs. 71-73)

v wB = argmax v∈V a E wB 0,a [v] -(v, ρ unif ) a . (61)
For a given basis set B, for a given density ρ 0 in the range [0, 10] a.u., and for the fixed electron number N = 2 and interval length a = N/ρ 0 , we calculate the energy E wB 0,a [v] by performing a FCI calculation using a plane-wave orthonormal basis {p n } n∈Z, |n|≤n pw max . The one-electron kinetic integrals are still p n 1 , tp n 2 a = (2π 2 n 2 1 /a 2 )δ n 1 ,n 2 . The integrals of the electron-electron interaction projected in the basis set B [see Eq. (33)] can be calculated as

p n 1 p n 2 , W B ee p n 3 p n 4 a = M i=1 M j=1 M k=1 M l=1 S * i,n 1 S * j,n 2 φ i φ j , W ee φ k φ l S k,n 3 S l,n 4 , (62)
where φ i φ j , W ee φ k φ l are the two-electron integrals in terms of the HF orbitals {φ i } i=1,...,M of the 1D He-like atom expanded in the basis set B (see Section II B) and S i,n = φ i , p n a = Ω a φ * i (x)p n (x)dx are the overlap integrals between the HF orbitals and the plane-wave basis functions. The potential v to optimize in Eq. ( 61) is also expanded on the same plane-wave basis set

v(x) = n∈Z,|n|≤n pw max c n p n (x), (63)
with coefficients c n ∈ R and we impose c -n = c n in order to have a real-valued and parity-even potential. The one-electron potential integrals, needed to calculate E wB 0,a [v], are

p n 1 , vp n 2 a = Ω a p * n 1 (x)v(x)p n 2 (x)dx = c n 1 -n 2 √ a , (64)
and the second term in Eq. ( 61) is simply (v, ρ unif ) a = √ ac 0 ρ 0 . Finally, for the optimization of the potential, it is useful to have the derivative of F[v] = E wB 0,a [v] -(v, ρ unif ) a with respect to the coefficient c n . Using the Hellmann-Feynman theorem, we find [v]. In practice, we use a plane-wave cutoff of n pw max = 30. We can use a smaller cutoff than the cutoff used for the complete-basis-set UEG in Section IV A since the FCI energy E wB 0,a [v] has a fast convergence with n pw max due to the presence of the projected electron-electron interaction W B ee . To optimize the coefficients {c n } of the potential, we use the conjugate gradient method [START_REF] Press | Numerical Recipes[END_REF] . Since the term n = 0 in Eq. ( 63) is just an arbitrary constant, we keep the coefficient c 0 fixed to 0. With a zero potential v, the FCI density ρ Ψ v=0 (x) can deviate from the target density ρ 0 by about as much as 0.2 a.u. for the basis set B of smallest size (i.e., M = 1). With our optimized potentials v wB , the density ρ Ψ v wB (x) deviates from the target density ρ 0 to at most about 10 -4 a.u..

∂F[v] ∂c n = (p n , ρ Ψ v ) a -(p n , ρ unif ) a = 1 √ a n 1 ∈Z,|n 1 |≤n pw max γ n 1 ,n 1 +n - √ aρ 0 δ n,0 , (65)
As an aside, it might be worthwhile to stress here that in the Hamiltonian in Eq. ( 60), the kinetic-energy operator T is not projected in the basis set B. We observed that if T is also projected in the basis set B, then the high-lying states of T collapse to the lower part of the spectrum, which inevitably pollutes the nature of the ground state of the finite UEG Hamiltonian and renders numerically impossible to find a potential restoring a uniform density. This is why in the second-variant of basis-set correction in Section III C we have decided to project only the electron-electron interaction W ee in the basis set B.

The optimized potentials v wB (x) obtained from Eq. ( 61) are plotted in Fig. 4 for the example of the target density ρ 0 = 2 a.u. and for different sizes n max of the basis set B of the 1D Helike atom introduced in Section II B. To compensate for the breaking of translation invariance of the projected electronelectron interaction, all potentials show oscillations with maximum amplitude on the edges of the interval. As expected, when n max increases, the amplitude of the potential decreases, as it must eventually go to zero in the complete-basis-set limit n max → ∞.

Once the FCI ground-state wave function Ψ wB [ρ unif ] = Ψ v wB for the optimal potential v wB has been obtained, we calculate the following energy per particle using this wave func- tion

ε wB fUEG (ρ 0 ) = Ψ wB [ρ unif ], (T + W ee )Ψ wB [ρ unif ] a N , (66)
which we can decompose in the same way as in Eq. ( 52)

ε wB fUEG (ρ) = t s,fUEG (ρ) + ε H,fUEG (ρ) + ε x,fUEG (ρ) + ε wB c,fUEG (ρ), (67)
with the same kinetic, Hartree, and exchange contributions as in Eq. ( 52), and a new correlation energy per particle ε wB c,fUEG (ρ). This latter quantity is plotted in Fig. 5 for different sizes n max of the basis set B of the 1D He-like atom. As expected, when n max increases, ε wB c,fUEG (ρ) becomes more negative and gets closer to the correlation energy per particle ε c,fUEG (ρ) of the complete-basis-set limit n max → ∞. For finite n max , it can be observed that, in the high-density limit, the correlation energy per particle ε wB c,fUEG (ρ) goes to zero, unlike in Eq. ( 53). This is due to the fact that, as the density increases, the relevant electron-electron distances contributing to the correlation energy become smaller and the basis set B is unable to resolve the Dirac-delta electron-electron interaction at a fine enough distance scale.

Finally, we calculate the complementary multideterminant correlation energy per particle of the finite UEG for the basis set B [see Eq. ( 40

)] εwB c,md,fUEG (ρ) = ε fUEG (ρ) -ε wB fUEG (ρ), (68)
which is plotted in Fig. 6 for different basis sizes n max . As n max increases, the magnitude of εwB c,md,fUEG (ρ) decreases and must eventually go to zero in the complete-basis-set limit n max → ∞. The magnitude is largest for high densities since εwB c,md,fUEG (ρ) must compensate for the inability of the basis set B to represent the Dirac-delta electron-electron interaction at a small distance scale. Perhaps surprisingly, there is also a local maximum of the magnitude of εwB c,md,fUEG (ρ) at small densities. This is due to the fact that, at small densities, εwB c,md,fUEG (ρ) does not exactly cancel out the Hartree and 66) and ( 67)] of the 1D finite UEG as a function of the density ρ for N = 2 electrons and for different sizes n max of the basis set B of the 1D He-like atom. The curve labelled by "HF" corresponds to the limiting case where the basis set B contains only the exact HF occupied orbital. exchange energies per particle, in contrast to the completebasis-set case [Eq. ( 54)]. Again, this must come from the inability of the basis set B to represent the Dirac-delta electronelectron interaction sufficiently precisely to give a zero probability density of finding the electrons as the same point of space in the low-density limit. Interestingly, in between the small and the large-density regimes, for n max ≥ 5, the magnitude of εwB c,md,fUEG (ρ) passes through a minimum. In particular, for n max = 70, εwB c,md,fUEG (ρ) is almost zero at around ρ ≈ 0.5 a.u., which means that the basis set B accurately captures the effect of the Dirac-delta electron-electron interaction at this density.

C. Finite local-density approximation

We can now define the finite LDA (fLDA) for the complementary multideterminant correlation density functional ĒwB c,md [ρ] [Eq. (40)] involved in the second-variant of basisset correction using the previously determined complementary correlation energy per particle εwB c,md,fUEG (ρ) of the 1D finite UEG

ĒwB c,md,fLDA [ρ] = R ρ(x) εwB c,md,fUEG (ρ(x))dx. ( 69
)
We recall that in standard LDA the functional of an inhomogeneous system for a finite electron number N is approximated using the UEG for infinite electron number N → ∞. Here, instead, in the present finite LDA the functional of the inhomogeneous system is approximated using the UEG of the same electron number N. The use of this finite LDA in lieu of the standard LDA should not be seen as a crucial point for the basis-set correction theory but more like a convenient alternative. For a sufficiently short-range complementary interaction FCI complementary correlation energy per particle εwB c,md,fUEG (ρ) [Eq. ( 68)] of the 1D finite UEG as a function of the density ρ for N = 2 electrons and for different sizes n max of the basis set B of the 1D He-like atom. The curve labelled by "HF" corresponds to the limiting case where the basis set B contains only the exact HF occupied orbital. (10)] and FCI+fLDA ground-state energy E wB FCI+fLDA [Eq. ( 70)] of the 1D He-like atom as a function of the basis size n max . The first point labelled by "HF" corresponds to the limiting case where the basis set B contains only the exact HF occupied orbital (in this case, FCI simply reduces to HF). The exact energy is taken from Ref. 24.

WB ee = W ee -W B ee , the LDA should not depend much on the electron number used in the definition of the underlying UEG.

We then correct the FCI energy of the 1D He-like atom using this fLDA functional in the non-self-consistent approximation introduced in Eq. (43), obtaining what we will call the "FCI+fLDA" energy

E wB FCI+fLDA = Ψ B FCI , (T + W ee + V ne )Ψ B FCI + ĒwB c,md,fLDA [ρ Ψ B FCI ]. (70)
In practice, we calculate ĒwB c,md,fLDA [ρ Ψ B FCI ] by numerical integration using cubic interpolation between calculated values of εwB c,md,fUEG (ρ). The FCI densities of the 1D He-like atom take values from 0 to about 3.5 a.u.. In Fig. 7, the FCI+fLDA energy is plotted as a function of the basis size n max . It is clear that the basis-set correction provides a spectacular improvement of the FCI energy. For example, for n max = 0, the FCI energy is about 55 mhartree above the exact energy while the FCI+fLDA energy is only 1.5 mhartree below the exact energy. For n max ≥ 20, the FCI+fLDA energy is within 1 mhartree of the exact energy. We note that the residual error must come from the fact that in Eq. ( 42) the functional ĒwB c,md [ρ] is approximated with the fLDA functional and also that the wave function Ψ wB 0 is approximated by the FCI wave function Ψ B FCI .

CONCLUSION

In this work, we have reexamined the recently introduced DFT-based basis-set correction theory on a 1D model with delta-potential interactions, which is a convenient setting to carefully study the slow basis convergence problem of quantum-chemistry wave-function methods. We provided mathematical details about the formulation of the theory, as well as a new variant of basis-set correction which has the advantage that the basis-set correction functional is defined for all N-representable densities. This allowed us to define a LDA for the basis-set correction functional, not based on range-separated DFT as in all previous works, but directly on a 1D finite UEG adapted to the basis set employed. We showed that this approach is very effective to correct for the basis-set incompleteness error in the FCI ground-state energy. We believe that the present work puts the basis-set correction theory on firmer grounds.

Future efforts will focus on the extension of the present work to 3D Coulombic systems. The extension of the theory is straightforward. What remains to be seen is whether the present work adapts well to the standard Gaussian-typeorbital basis sets used in quantum chemistry and whether we can still construct an accurate LDA for the basis-set correction functional based on a 3D UEG with the Coulomb electronelectron interaction projected in the basis set.

We thus see that the two-electron energy converges as 1/n 1/2 max in a basis of Hermite functions.

FIG. 1 .FIG. 2 .

 12 FIG. 1. FCI ground-state energy E BFCI [Eq. (10)] of the 1D He-like atom as a function of the basis size n max . The exact energy is taken from Ref.24.

FIG. 3 .

 3 FIG.3. FCI correlation energy per particle ε c,fUEG (ρ) [Eqs.(50) and (52)] of the 1D finite UEG as a function of the density ρ for N = 2 electrons and a plane-wave cutoff n pw max = 60. The exact PT2 correlation energy per particle, which is independent of N and ρ, is indicated as a horizontal line. The correlation energy per particle of the infinite UEG (N → ∞) as parametrized by Magyar and Burke 24 from essentially numerically exact Bethe-ansatz results is also plotted for comparison.

FIG. 4 .

 4 FIG.4. External optimized potential v wB (x) [Eq. (61)] keeping a uniform density ρ 0 = 2 a.u. for the 1D finite UEG for N = 2 electrons and for different sizes n max of the basis set B of the 1D He-like atom.

FIG. 5 .

 5 FIG.5. FCI correlation energy per particle ε wB c,fUEG (ρ) [Eqs. (66) and (67)] of the 1D finite UEG as a function of the density ρ for N = 2 electrons and for different sizes n max of the basis set B of the 1D He-like atom. The curve labelled by "HF" corresponds to the limiting case where the basis set B contains only the exact HF occupied orbital.

  FIG. 6.FCI complementary correlation energy per particle εwB c,md,fUEG (ρ) [Eq. (68)] of the 1D finite UEG as a function of the density ρ for N = 2 electrons and for different sizes n max of the basis set B of the 1D He-like atom. The curve labelled by "HF" corresponds to the limiting case where the basis set B contains only the exact HF occupied orbital.

FIG. 7 .

 7 FIG. 7. FCI ground-state energy E BFCI [Eq.(10)] and FCI+fLDA ground-state energy E wB FCI+fLDA [Eq. (70)] of the 1D He-like atom as a function of the basis size n max . The first point labelled by "HF" corresponds to the limiting case where the basis set B contains only the exact HF occupied orbital (in this case, FCI simply reduces to HF). The exact energy is taken from Ref.24.

  ∈Z,|n 2 |≤n pw max γ n 1 ,n 2 p n 1 (x)p * n 2 (x) with the oneparticle reduced density matrix γ of the ground-state wave function Ψ v of H wB

	where	we	have	used	ρ Ψ v (x)	=
	n 1 ∈Z,|n 1 |≤n	pw max	n 2

ACKNOWLEDGEMENTS

We thank Éric Cancès, Louis Garrigue, Paola Gori-Giorgi, Muhammad Hassan, Antoine Levitt, Pierre-Francois Loos, and Mi-Song Dupuy for useful comments on the manuscript. We particularly thank Éric Cancès for pointing out to us the appropriate space V of potentials to consider and Antoine Levitt for giving us the correct definition of the space H 1 per (Ω 2 a , C). This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme Grant agreement No. 810367 (EMC2).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

AUTHOR DECLARATIONS

The authors have no conflicts to disclose.

Appendix A: Convergence rate of the expectation value of a Dirac-delta potential in a basis of Hermite functions

One-electron Dirac-delta potential

Let us consider the 1D hydrogen-like Hamiltonian

with nuclear charge Z ∈ (0, +∞). The ground-state wave function is (see, e.g., Refs. 74 and 75)

which exhibits a cusp identical to the 3D Coulombic case. We expand ϕ in the orthonormal basis of Hermite functions

with coefficients c n = R f α n (x)ϕ(x)dx, which are non-zero only for even integers n. Using the following asymptotic equivalent of the unnormalized Hermite functions, for fixed x, [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] H n (

and the well-known asymptotic equivalent of the gamma function

we obtain the following asymptotic equivalent of the normalized Hermite functions

Writing n = 2p with p ∈ N, we then find the leading term of the asymptotic expansion of the coefficients c 2p by integrating over x

This is perfectly consistent with the analysis of Refs. 77 and 78 which shows that an exponentially decaying function ϕ having a square-integrable first weak derivative (i.e., ϕ ∈ H 1 (R)) but a non-square-integrable second weak derivative (i.e., ϕ H 2 (R)) must have Hermite expansion coefficients c n going to zero as n -k with k ∈ (1, 3/2]. The leading term of the asymptotic expansion of c 2p f α 2p (x) is thus

and, in particular at x = 0,

Calling φ the best approximation (in the sense of the L 2 norm) to ϕ obtained with a maximal quantum number n max , i.e.

we find that φ(0) converges slowly to the exact value ϕ(0

and the expectation value of the Dirac-delta potential

We also expect the total energy to converge as 1/n 1/2 max in a basis of Hermite functions.

Two-electron Dirac-delta interaction

Let us consider the Hamiltonian of the 1D two-electron Hooke's atom 24

where ω is the angular frequency parameter of the external harmonic potential. In contrast to the 1D He-like atom [Eq. ( 1)], the 1D two-electron Hooke's atom has the advantage to be solvable in terms of special functions. Indeed, changing the variables to the center-of-mass (cm) coordinate X = (x 1 + x 2 )/2 and the relative (rel) coordinate x 12 = (x 1x 2 ) makes the Hamiltonian separable

where

and

The total ground-state energy is then

where E 0 = ω/2 is the ground-state energy of h cm and ε 0 is the ground-state energy of h rel which is found from the equation 24,79 2

where ν 0 = ε 0 /ω -1/2. For example, for ω = 1 a.u., we have ε 0 = 0.806746 a.u. 24 . The ground-state wave function is

where the center-of-mass wave function is just given by the first Hermite function Φ(X) = f 2ω 0 (X) and the relative wave function is given by 79

where c is a real-valued normalization constant and x → D ν (x) with ν ∈ R is the parabolic cylinder function [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] . The relative wave function has the same cusp as in Eq. ( 5), i.e.

]. Let us consider now the expansion of the wave function Ψ in the tensor-product orthonormal basis of Hermite functions

Due to invariance of the harmonic-oscillator Hamiltonian to a rotation of coordinates, the same space is spanned by the rotated orthonormal basis {(X, x 12 ) → f 2α n 1 (X) f α/2 n 2 (x 12 )} (n 1 ,n 2 )∈N 2 (see Refs. 77 and 78). This means that the relative wave function ψ is independently expanded as

with coefficients d n = R f α/2 n (x 12 ) ψ(x 12 )dx 12 , which are nonzero only for even integers n. Using Eq. (A6) and with the help of Mathematica 80 , we find the leading term of the asymptotic expansion of the coefficients d 2p

where µ 0 = -ν 0 /2 + 1/2. Introducing the best approximation to ψ obtained with a maximal quantum number n max


Celia Kessassi

Modeling the Induction of Psychosocial Stress in Virtual Reality Simulations

Keywords: Psychosocial stress, Trier Social Stress Test TSST, Threat and challenge theory, computational model

During the last few years, a wide number of virtual reality applications dealing with psychosocial stress have emerged. However, our current understanding of stress and psychosocial stress in virtual reality hinders our ability to finely control stress induction. In my PhD project I plan to develop a computational model which will describe the respective impact of each factor inducing psychosocial stress, including virtual reality factors, personal factors and other situational factors.

I. MOTIVATION AND RESEARCH QUESTIONS

The human-computer interaction field is gaining increasing interest and much effort is invested to optimize interfaces and build more human-centred applications. For instance, O. Carsten and M. H. Martens [START_REF] Carsten | How can humans understand their automated cars? hmi principles, problems and solutions[END_REF] state that the design and function of the Human Machine Interfaces is crucial for ensuring human safety, this is particularly true in car design. Affective computing, in the other hand, is a special sub-field where the emphasis is put on the detection, understanding and generation of human affect, and where the aim is to develop technologies fulfilling human needs, such as applications providing mental health services in virtual reality [START_REF] Park | A literature overview of virtual reality (vr) in treatment of psychiatric disorders: recent advances and limitations[END_REF].

In fact, during the last few years, an increasing number of studies and applications in virtual reality have emerged, some of which, such as social skills training, function by inducing psychosocial stress in virtual environment [START_REF] Parsons | Affective outcomes of virtual reality exposure therapy for anxiety and specific phobias: A meta-analysis[END_REF] [START_REF] Pertaub | An experiment on public speaking anxiety in response to three different types of virtual audience[END_REF]. Although they were proven to be effective for triggering stress, the mechanism of stress induction is still loosely understood, that is to say, a virtual system that is proven to trigger stress in certain individuals may not induce stress for others. In order to better understand this mechanism, we aim at developing a computational model describing the various parameters in virtual reality affecting psychosocial stress. We believe the work will subsequently contribute to other research domains such as human-machine interactions and stress research in general. Moreover, it will inspire the design of various applications including: social skills training in virtual reality, stress detection, stress prevention, personalized gamification among others.

In order to achieve our goal, we started by defining the problem, that is to say defining what we consider to be stress and psychosocial stress. Among the things we need to take note of are the various parameters impacting stress in virtual interactions, including the external (ex. room size, virtual agent's behaviour..) and internal (ex. personality traits, experience..) parameters. Moreover, virtual reality is a singular environment, different from the real world in term of cognitive impact. Immersion, for instance, plays an important role in the user's behaviour and perceptions [START_REF] Alshaer | Immersion factors affecting perception and behaviour in a virtual reality power wheelchair simulator[END_REF]. In our research we consider and explore all of these aspects to build our model describing psychosocial stress induction in virtual reality.

II. BACKGROUND AND RELATED WORK

A. Definition of stress

Hans Selye, known as the "father of stress" is considered to be the first to incorporate the term "stress" into the medical lexicon [START_REF] Tan | Hans selye (1907-1982): Founder of the stress theory[END_REF] [7] [START_REF] Lazarus | Stress, appraisal, and coping[END_REF] [9] [START_REF] Selye | Forty years of stress research: principal remaining problems and misconceptions[END_REF]. During his medical school, he observed that patients with a variety of illnesses had many of the same "non-specific" symptoms. To learn more about these symptoms, he conducted a series of experiments where rats were placed under the exposure of various noxious agents, such as cold, surgical injury, production of spinal shock, excessive muscular exercise, or intoxication with sublethal doses of diverse drugs. Consequently, he observed the same symptoms regardless of the nature of the damaging agents.

Stress was described by Selye as "the non-specific response of the body to any demand". In fact, stress is regarded as the universal set of reactions and processes created by an environmental demand. It is designated as non-specific because the body shows the same effect regardless of the source of stress.

B. Psychosocial stress

Among the many sources of stress, we are interested in the psychosocial stress. Psychosocial stress refers to the stress induced in social relationships or arrangement, and affecting the organism through the mediation of higher nervous processes [START_REF] Levi | Stress and Distress in Response to Psychosocial Stimuli: Laboratory and Real-Life Studies on Sympatho-Adrenomedullary and Related Reactions[END_REF]. According to Blascovich's model of psychosocial stress, also recognized as the BPS model [START_REF] Blascovich | The biopsychosocial model of arousal regulation[END_REF], socially stressful situations can fall into two categories, namely threat or challenge. To determine whether the situation constitutes a threat or a challenge, one makes a cognitive appraisal of the situation, if the comparative appraisals results in the perception of situation's demands that can not be overcome with the perceived resources at one's disposal, then threat becomes the overall resulting appraisal. Conversely, if the comparative appraisals results in a perception of resources that can overcome situation's demands, then the challenge is the overall resulting appraisal. Demands, as described in the theory, refers to any demand, uncertainty and/or danger. This threat and challenge theory was influenced by prior work confirming the potential of interpersonal and intrapersonal factors to moderate autonomic responses [START_REF] Blascovich | The biopsychosocial model of arousal regulation[END_REF]. In one of Blascovich early experiments, he studied the impact of varied presence of others during a mental arithmetic task. In the experiment, autonomic responses were recorded while participants (middle-age women) performed the task with either a human friend, their pet dog or alone, however the experimenter was present in all three conditions. Results indicated that participants who performed the mental arithmetic task in the presence of their human friend resulted in significantly larger increases in autonomic responses compared to those performing the task with their pet dog. Conversely, the presence of the pet dog induced smaller autonomic responses compared to the presence of the experimenter alone. Blascovich interpreted the results by suggesting that the human friend was more likely to be perceived as evaluative than the canine friend. Thus, interpersonal and intrapersonal factors plays a key role in the level of stress.

Following the development of his theory, he conducted a series of studies that corroborated with the threat and challenge model [START_REF] Tomaka | Cognitive and physiological antecedents of threat and challenge appraisal[END_REF]. For instance, he measured physiological, behavioural and subjective responses of participants interacting with stigmatized partners during a collaborative task. The stigma included facial-birthmarks, race and socioeconomic status. Measures indicated that participants interacting with stigmatized partners exhibited consistent threat responses compared to participants interacting with non-stigmatized partners, who exhibited challenge responses. In addition to the to stereotypes, a similar effect was observed with varied instructions during experiments . The aim of varying instructions was to influence perception of demands by either emphasizing accuracy of task performance and potential evaluation, or doing one's best. As hypothesized threat and challenge responses were consistent with the nature of instructions.

Through those experiments, Blasovich and his colleagues try to provide evidences supporting the BPS model. Moreover, they indicate physiological responses to threat distinct from the physiological responses of challenge, hence, both induce different levels of stress. For instance, results suggests that cardiac output levels increase during challenge compared to threat. In addition to that, self-reported stress levels were observed to be positively related to cognitive appraisal, such that the more threatened one is the more he reports experiencing stress during a task. Although the model has been criticized [START_REF] Wright | Cardiovascular correlates of challenge and threat appraisals: A critical examination of the biopsychosocial analysis[END_REF], these findings provide essential information to understand how stress affect the user in general and provide necessary grounds to study psychosocial stress in virtual reality. It is valuable for building the hypotheses and the experimental protocols and it is relevant for establishing measures that indicate stress levels, including physiological measures and self-reported measures.

C. Stress induction tools

One of the tools used to trigger stress in laboratory setting is the Trier Social Stress Test (TSST). It's mostly used for its reliability to trigger physiological responses of stress. It consists of an anticipation phase and a speech delivery phase that each lasts 10 minutes. Participants are asked to introduce themselves to a panel of three persons, presented as body language experts, and are told that they would be recorded with a video camera. Participants prepare the speech during the anticipation phase in one room, and they deliver the speech during the speech delivery phase in an other room where the committee and video camera are placed. Following the public speaking phase, the participants are asked to perform a mental arithmetic task in front of the same audience, counting backward from 1022 with steps of 13. This protocol is recognized for successfully triggering stress responses [START_REF] Kirschbaum | The 'trier social stress test'-a tool for investigating psychobiological stress responses in a laboratory setting[END_REF] [16], including, saliva cortisol increase (70% of the population respond with an increase of 2.5nmol/l above baseline) and heart rate increase.

While the TSST is a very effective tool for inducing psychosocial stress and is valuable as an experimental tool in virtual reality, as it has already been studied [START_REF] Montero-López | A virtual reality approach to the Trier Social Stress Test: Contrasting two distinct protocols[END_REF] [16] [START_REF] Zimmer | Virtually stressed? A refined virtual reality adaptation of the Trier Social Stress Test (TSST) induces robust endocrine responses[END_REF] [19] [START_REF] Liu | Sex differences in stress reactivity to the trier social stress test in virtual reality[END_REF], our ambition, however, is to identify the characteristics of stressful social situations, in virtual interactions, that can account for different situational contexts, and not only in the context of the TSST protocol.

D. Stressful situation's features

In order to explore the different features that makes a situation stressful, L. Lebois et al. [START_REF] Lebois | Establishing the situated features associated with perceived stress[END_REF] carried out an experiment where 12 participants had to evaluate 572 sentences describing life events situation. Participants rated how much those sentences possessed 19 features potentially associated with perceived stressful situations, such as experience or familiarity with the situation. The findings indicates high correlation between perceived stress and some hypothesized features including expectation violation, outcome certainty, negative valence, arousal or coping certainty. These findings are very beneficial since it helps understand the factors correlating with stress, it will be leveraged to study these parameters in virtual reality and help build hypothesis and experimental protocols using varied situations in virtual simulations.

While Lebois and her colleagues studied the features of stressful situations in general, L. B. Fich et al. [START_REF] Fich | Can architectural design alter the physiological reaction to psychosocial stress? a virtual tsst experiment[END_REF] investigated one particular feature in virtual reality. The study compares the impact of stress between two groups, one group that went through the TSST protocol in a virtual room with openings (three windows in front of the participant) and the other group that participated to the TSST in a virtual room without openings. The hypothesized argument was that in a stressful situation such as public speaking, the physiological responses of stress would be dampened through the presence of windows in the virtual room. This hypothesis stems from the studies suggesting that view through a window influence recovery from surgery. However, apart from saliva cortisol no other biomarker of stress (ex. heart rate, T-wave amplitude) was significantly influenced by the presence nor the absence of windows. These suggestions provide an insight on how virtual parameters might induce particular stress patterns.

III. HYPOTHESIS AND THE SCOPE OF THE TECHNICAL

PROBLEM

At the present moment, a wide range of hypothesis are being considered for the project. Main ones are described in the following.

A. Situational factors and stress responses

One of the directions we consider is leveraging Blascovich's BPS theory to define the factors that explains stress in virtual reality. As it was described above in section II-B, the effect of particular parameters was investigated in vivo, including varied stereotypes and instructions. Going further, these parameters can also be studied in virtual reality simulations.

Some studies on social exclusion [START_REF] Zöller | No effect of social exclusion on salivary cortisol secretion in women in a randomized controlled study[END_REF] [23] explored the impact of social exclusion on psychological state and stress. One protocol that is frequently used is a virtual ball-tossing game called Cyberball, it is a standardized tool for social exclusion simulation. The virtual game counts 3 players, one that represents and can be controlled by the participant and the others represent two other players. The players has to toss the ball to one of the other players, and at a particular moment, the two other player stop tossing the ball to the participant. Results shows significant effect on mood, and even neurological evidences of social pain, however, Weik and his colleagues [START_REF] Weik | Pre-experience of social exclusion suppresses cortisol response to psychosocial stress in women but not in men[END_REF] reported no effect on saliva cortisol, which is one of stress biomarkers. These findings raises questions on the nature and type of demands that triggers stress. One may speculate that this social exclusion experiment create situational demands that exceeds appraised resources, consequently, according to the BPS model, stress and in particular threat should be experienced. Nevertheless, this account contradicts with experimental results. Thus, in order to predict how a virtual social interaction affects stress, a full and comprehensive understanding of the appraisal of demand and resources is required. For this reason, we envisage the possibility to explore more about cues impacting the perception of demands and perception of the resources at one's disposal.

Despite all, the endeavour of studying the nature of demands and resources appraisal and how they combine seem challenging. According to Blaschovich the "appraisal" process is influenced by multivariate and multi-process systems including conscious and unconscious processes, and the complex influence of conscious and unconscious on the relationship between demand and resources can not be obtained with self-reported data [START_REF] Blascovich | The Robust Nature of the Biopsychosocial Model Challenge and Threat: A Reply to Wright and Kirby[END_REF].

B. Perspective as a factor

According to previous research related to perspective (ex. first-person perspective) in virtual reality, the perspective from which one observes a situation determines the way he experiences it [START_REF] Salamin | The benefits of third-person perspective in virtual and augmented reality?[END_REF] [START_REF] Monteiro | Evaluating engagement of virtual reality games based on first and third person perspective using eeg and subjective metrics[END_REF]. For instance, Bergström and his colleagues suggest that observing a situation from the first-person rather than the third-person perspective leads to high subjective body ownership illusion, which consequently leads to higher stress responses in uncomfortable virtual body posture situation [START_REF] Bergström | First-person perspective virtual body posture influences stress: a virtual reality body ownership study[END_REF]. In contrast, another study suggests that higher degrees of anxiety appear to be linked to increased third-person perspective in social phobia [START_REF] Wallace-Hadrill | The Impact of Perspective Change As a Cognitive Reappraisal Strategy on Affect: A Systematic Review[END_REF], moreover, Clark and Wells propose that individuals with social phobia view themselves in a negative impression of how they appear to others, as the observer perspective [START_REF] Spurr | The observer perspective: effects on social anxiety and performance[END_REF]. These suggestions point out the probable relevance of perspective to influence social stress. However, whether psychosocial stress level is more important in first-person or third-person perspective is not certain yet, more extensive research should be conducted to answer these questions.

Another hypothesis related to the suggestions above involves the potential of body ownership illusion to influence stress. For instance, we consider maximizing the body ownership illusion using other techniques, such as using a virtual 3D scanned version of the person and analyze the impact on social stress.

C. Mutiplicy of stressors in the TSST

Common findings demonstrated the reliably of the TSST to induce physiological responses of stress, however, the protocol is not limited to a public speaking task, it includes an apprehension phase, that can in itself be a stressor because it creates demands for anticipation and preparation, more importantly, it creates uncertainty that may lead to experiencing threat. In addition, the mental arithmetic phase is a mental task that can introduce heart rate increase which is not related to psychosocial stress, because mental demanding tasks are known to alter cardiovascular responses [START_REF] Sloan | Components of heart rate reactivity during mental arithmetic with and without speaking[END_REF]. In the other hand, the unexpected nature of the mental task can be the source of the stress response. Studying the respective effect of these phases might provide a better insight about which situation will be perceived as socially stressful or non-stressful in the context of the TSST protocol.

IV. OBJECTIVES AND METHODOLOGY

To summarize, our goal is to understand the various factors in a given virtual social interaction situation impacting stress responses and determine, according to these factors, how stress bio-markers are impacted. Based on the literature review, there is 3 main components that are essential for the model describing psychosocial stress in virtual reality (see fig. 1). First, one main component is the virtual reality parameters. As mentioned above, many studies report significant effect on psychosocial stress while virtual system parameters were varied. These parameters may include the virtual architectural design, perspective, the audience size or audience behaviour [START_REF] Mostajeran | The effects of virtual audience size on social anxiety during public speaking[END_REF] [4] [START_REF] Fich | Can architectural design alter the physiological reaction to psychosocial stress? a virtual tsst experiment[END_REF]. The second component of the model is the situational parameters. We believe that the situational settings in which someone experiences a stressful situation plays a key role, for instance, maybe if someone experiences a stressful situation in laboratory settings rather than at home, maybe stress wouldn't be experienced the same way. Finally, there is the component encompassing personal factors. As it has been supported from a wide number of theories and research studies, this component seems to be the key factor that dictates, depending on the individual, if a particular situation is perceived as stressful or non-stressful . It may include personal factors such as past experience, personal state or personal traits. The model we aim to develop is also a model that is able to predict stress responses according to various parameters, doing so requires understanding the respective impact of each parameter. In addition, we have seen in a number of studies where varying a parameter such as the presence of windows in virtual environment has an impact on some but not all biomarkers of stress. It suggests that different stressors will induce different stress patterns.

Accordingly, we will continue reviewing the literature related to factors of stressful situation, with the purpose of collecting information and determining which parameter is relevant to be incorporated in the model and which pattern of stress is induced.

If one parameter is not sufficiently explored, we shall endeavour to study it further. This should be completed by means of experimental studies where we will vary virtual and situational parameters and analyze the stress response.

Main stress responses may be investigated through perceptual, behavioural, and physiological responses. The evaluation of perceptual responses to a stressor involves subjective estimations and perceptions. Indeed, to measure self-reported stress level, numerous questionnaires have been used in research, including the Cohen's Perceived Stress Scale (PSS) that measures the degree to which situations are considered stressful, doing so by addressing events experienced in the preceding month [START_REF] Cohen | A global measure of perceived stress[END_REF]. Another one is the Strain Questionnaire (SQ) a multi-model perception of stress that measures behavioral, cognitive, and physical stress perception of stress [START_REF] Lefebvre | A multi-modal questionnaire for stress[END_REF].

Regarding the physiological responses of stress, saliva cortisol along with heart rate are two of stress biomarkers frequently investigated. The more stressed someone is, the more cortisol and heart rate increase. Prior studies demonstrated that stress has also electroencephalogram correlates, thus, it can indicate stress level [START_REF] Ehrhardt | Separating eeg correlates of stress: Cognitive effort, time pressure, and social-evaluative threat[END_REF]. Moreover Elevated skin conductance is found to be correlated with stress elevation [START_REF] Jeunet | Design and validation of a mental and social stress induction protocol -towards load-invariant physiologybased stress detection[END_REF]. The above described biomarkers will be important biomarkers to assess stress responses during experiments.

Finally, the goal is to build a reliable model. Its reliability will have to be validated using experimental data, doing so by comparing the model with the experimental data.

V. A DESCRIPTION OF THE WORK DONE SO FAR AND A TENTATIVE PLAN FOR FUTURE WORK

As the PhD project started only recently, much attention is currently being paid to literature and establishing the list of parameters relevant to the model. Once the literature has been sufficiently explored the subject should be narrowed down and clear hypothesis should be formulated. From there, we will develop an experimental framework for virtual interactions where the parameters under study will be varied to analyze their impacts. These parameters might include the virtual room design, or the instructions given to the participants, it may include a variety of parameters, however, at the present moment there is no definite parameter, as our hypotheses may evolve. After the development of the virtual simulation framework, we plan to carry out experiments and collect data that will serve to study our hypothesis and to develop the computational model.

VI. EXPECTED CONTRIBUTIONS

We hope that our work will contribute to provide a thorough understanding of the mechanisms underlying psychosocial stress induction in virtual reality simulations. Such understanding will have impacts in the field of virtual reality in general, and in human-agent interactions in particular, as this will further our knowledge about the factors shaping user's experience during virtual social interactions. Furthermore, the model might serve as support to build, using the appropriate characteristic, applications that need full control of stress induction.

The aimed computation model should be able to detect, based on stress responses of the user, if the corresponding virtual-social interaction is appraised as stressful or nonstressful. Moreover, it will be able to predict, based on the user's personal factors, if a particular virtual-social interaction will be perceived as stressful. The methods, tools and results produced in the project will enable the creation of experiments to test hypotheses and make predictions which will in turn contribute to progress in social and cognitive psychology.

Finally, the experiments we intend to conduct will lead to the creation of a multimodal corpora that will be made available to the scientific community, after a strict audit of the data, ensuring that the anonymisation procedure was carried out properly. These corpora will be invaluable tools for further research on virtual social interactions, social stress induction mechanisms, and computational models applied to social behaviour and physiological signals.

Fig. 1 .

 1 Fig. 1. An overview of the components under study for stress induction