Julien Toulouse
email: toulouse@tc.cornell.edu

Roland Assaraf
email: assaraf@lct.jussieu.fr

C J Umrigar

Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system averaged pair density

We construct improved quantum Monte Carlo estimators for the spherically-and system-averaged electron pair density (i.e. the probability density of finding two electrons separated by a relative distance u), also known as the spherically-averaged electron position intracule density I(u), using the general zero-variance zero-bias principle for observables, introduced by Assaraf and Caffarel. The calculation of I(u) is made vastly more efficient by replacing the average of the local deltafunction operator by the average of a smooth non-local operator that has several orders of magnitude smaller variance. These new estimators also reduce the systematic error (or bias) of the intracule density due to the approximate trial wave function. Used in combination with the optimization of an increasing number of parameters in trial Jastrow-Slater wave functions, they allow one to obtain well converged correlated intracule densities for atoms and molecules. These ideas can be applied to calculating any pair-correlation function in classical or quantum Monte Carlo calculations.

I. INTRODUCTION

Two-electron distribution functions occupy an important place in electronic structure theory between the simplicity of one-electron densities and the complexity of the many-electron wave function. In particular, the systemaveraged electron pair density, the probability density of finding two electrons separated by the relative position vector u, also known as the electron position intracule density I(u), plays an important role in qualitative and quantitative descriptions of electronic systems. Position intracule densities have been extensively used to analyze shell structure, electron correlation, Hund's rules and chemical bonding (see, e.g., Refs. . Density functional theory-like approaches have been proposed based on the position intracule density [32][33][34][35][36] or on the closelyrelated Wigner intracule density [37][38][39].

Position intracule densities have been extracted from experimental X-ray scattering intensities for small atoms and molecules [40][41][42]. They have been calculated in the Hartree-Fock (HF) approximation for systems ranging from small atoms to large molecules [7,12,16,19,21,27,[43][44][45][46]. Calculations using common quantum chemistry correlated methods such as second-order Møller-Plesset perturbation theory, multi-configurational self-consistent-field (MCSCF) and configuration interac-tion approaches have been limited to atoms and small molecules [6,11,14,18,24,[START_REF] Mercero | NATO-ASI Series in Metal-Ligand Interaction in Molecular-, Nano-, Micro, and Macro-systems in Complex Environments[END_REF]47,48]. Very accurate calculations using Hylleraas-type explicitly-correlated wave functions have been done only for the helium and lithium isoelectronic series [1,5,8,15,17,[49][START_REF] Thakkar | Density Matrices and Density Functionals[END_REF][START_REF] Dressel | [END_REF][52][53]. Variational Monte Carlo (VMC) calculations using Jastrow-Slater wave functions have been used to compute correlated position intracule densities for atoms from helium to neon and some of their isoelectronic series [22, 23, 25, 26, 29-31, 52, 54, 55]. In this paper, we show that the calculation of position intracule densities using quantum Monte Carlo (QMC) methods can be made much more accurate and efficient, opening new possibilities of investigation.

The position intracule density associated with an N -electron (real) wave function Ψ(R), where R = (r 1 , r 2 , ..., r N ) is the 3N -dimensional vector of electron coordinates (ignoring spin for now), is defined as the quantum-mechanical average of the delta-function operator δ(r iju)

I(u) = 1 2 i =j dR Ψ(R) 2 δ(r ij -u), (1)
where r ij = r jr i and the sum is over all electron pairs, and its spherical average is

I(u) = dΩ u 4π I(u). (2)
Among the most important properties of I(u) are the normalization sum rule (giving the total number of elec-tron pairs)

∞ 0 du 4πu 2 I(u) = N (N -1) 2 , (3)
the electron-electron cusp condition [56] dI(u) du

u=0 = I(0), (4)
and, for finite systems, the exponential decay at large u determined by the spherically averaged one-electron density n(u) evaluated at the distance u from the chosen origin

I(u) ∼ u→∞ (N -1) 2 n(u) ∝ u→∞ e -2 √ 2I u , ( 5
)
where I is the vertical ionization energy (see Refs. [57][58][59].

The moments of the radial intracule density M k = ∞ 0 du 4πu 2+k I(u) are related to physical observables [60], in particular M -1 is just the electron-electron Coulomb interaction energy

W ee = M -1 = ∞ 0 du 4πu I(u). (6)
In standard correlated methods based on an expansion of the wave function in Slater determinants, the important short-range part of the position intracule density converges very slowly with the one-electron and manyelectron basis. The advantage of employing QMC methods [61] lies in the possibility of using compact, explicitlycorrelated wave functions which are able to describe properly the short-range part of I(u). However, the problem in this approach is that one calculates the average of a delta-function operator which has an infinite variance. As for other probability densities, the standard procedure in QMC approaches simply consists of counting the number of electron pairs separated by the distance u within ∆u encountered in the Monte Carlo run. More precisely, e.g. in VMC calculations, I(u) is estimated as the statistical average

I histo (u) = I histo L (u, R) Ψ 2 , (7)
over the trial wave function density Ψ(R) 2 , of the local "histogram" estimator I histo L (u, R)

I histo L (u, R) = 1 2 i =j 1 [u-∆u/2 , u+∆u/2[ (r ij ) 4πu 2 ∆u , (8)
where

1 [a,b[ (r) = 1 if r ∈ [a, b[ and 0 otherwise.
Here and in the following, f (R)

Ψ 2 = (1/M ) M k=1 f (R k ) designates the statistical average of the local quantity f (R) over M configurations R k sampled from Ψ(R) 2 .
For u = 0, this histogram estimator has a finite but large variance for small u or small ∆u. Although the use of importance sampling can help decrease the variance [22,62,63], the calculation of position intracule densities in QMC remains very inefficient: very long runs have to be performed to reach an acceptably small statistical uncertainty. Moreover, the histogram estimator has a discretization error : even in the limit of an infinite sample M → ∞, I histo (u) remains only an approximation of first order in ∆u to the position intracule density I(u) of the wave function Ψ(R). Note however that it is possible to greatly reduce the discretization error by choosing a flexible analytic form for I(u) that obeys known conditions (such as the cusp condition of Eq. ( 4)) and fitting in each interval ∆u the integral of I(u) (instead of I(u)) to the computed data. This approach has been used to calculate radial electron densities for atoms [START_REF] Filippi | Recent Developments of Modern Density Functional Theory[END_REF] and spherically-averaged intracule densities [52] but the method is not as effective for multidimensional densities.

In this work, we develop improved QMC estimators for the spherically averaged position intracule density based on the zero-variance zero-bias (ZVZB) principle for observables introduced by Assaraf and Caffarel for calculations of electronic forces [START_REF] Assaraf | [END_REF][66][67], which has been recently applied to calculations of one-electron densities [68]. These new ZVZB estimators of I(u) have variances several orders of magnitude smaller than the ones obtained with the standard estimator of Eq. ( 8) and thus dramatically increase the efficiency of these calculations. Moreover, these estimators do not suffer from any discretization error and can even reduce the systematic error due to the approximate trial wave function. Like related techniques proposed for calculations of one-electron or two-electron densities using deterministic ab initio methods [52,[69][70][71][72][73][74][75][76][77][78][79][80][81][82][83], these estimators replace the average of the local delta-function operator by the average of an operator which is nonlocal in real space. They can be viewed as a generalization of the improved QMC estimators previously proposed for computations of averages of probability densities at particle coalescences [62,[84][85][86].

Moreover, we make use in this work of the recentlydeveloped linear energy minimization method [87,88] to finely optimize the Jastrow parameters, the configuration state functions (CSFs) coefficients and the orbital coefficients of our trial Jastrow-Slater wave functions. Optimization of the determinantal part of the wave function with an increasing number of CSFs allows us to obtain a systematic improvement of wave functions. This provides a practical route for calculating intracule densities in VMC and fixed-node (FN) diffusion Monte Carlo (DMC) with progressively smaller systematic errors.

The paper is organized as follows. In Sec. II, we review the principle of zero-variance zero-bias improved estimators for an arbitrary observable in QMC. In Sec. III, we give improved estimators for the case of the position intracule density. Sec. IV contains computational details of the calculations, and Sec. V discusses results for the He and C atoms and for the C 2 and N 2 molecules to illustrate the technique. Sec. VI contains our conclusions.

Hartree atomic units are used throughout this work.

II. ZERO-VARIANCE ZERO-BIAS IMPROVED ESTIMATORS

We review here the principle of the zero-variance zerobias improved QMC estimators for an arbitrary observable that does not commute with the Hamiltonian, developed in Refs. 65-67. Throughout this section, all the averages • • • Ψ 2 are considered in the limit of an infinite Monte Carlo (MC) sample M → ∞.

A. Estimators in variational Monte Carlo

In VMC, the exact energy E 0 = Ψ 0 | Ĥ|Ψ 0 / Ψ 0 |Ψ 0 of some exact eigenfunction Ψ 0 of the electronic Hamiltonian Ĥ is estimated by the statistical average of the local energy

E L (R) = R| Ĥ|Ψ /Ψ(R) E = E L (R) Ψ 2 , (9)
using an approximate trial wave function Ψ(R). The systematic error (or bias) of this estimator δE = E -E 0 and its variance

σ 2 (E L ) = (E L (R) -E) 2 Ψ 2 ,
δE = O(|δΨ| 2 ), (10a)
σ 2 (E L ) = O(|δΨ| 2 ), (10b)
which is referred to as the quadratic zero-variance zerobias property of the local energy. This is easily shown by writing Ψ = Ψ 0 + δΨ in the expressions of the average and the variance, and expanding to second-order in δΨ.

Similarly, the exact expectation value of an arbitrary observable O 0 = Ψ 0 | Ô|Ψ 0 / Ψ 0 |Ψ 0 can be estimated by the statistical average of the local observable

O L (R) = R| Ô|Ψ /Ψ(R) O = O L (R) Ψ 2 , (11)
but, as Ψ 0 is generally not an eigenstate of Ô, the systematic error of this estimator δO = O -O 0 vanishes only linearly with |δΨ|, while its variance

σ 2 (O L ) = (O L (R) -O) 2 Ψ 2 generally does not even vanish with |δΨ| δO = O(|δΨ|), (12a)
σ 2 (O L ) = O(1), (12b)
which often makes the calculations of observables inaccurate and inefficient. However, Assaraf and Caffarel [67] have pointed out that the quadratic zero-variance zero-bias property of the energy can be extended to an arbitrary observable by expressing it as an energy derivative. This is based on the Hellmann-Feynman theorem which states that, if one defines the λ-dependent Hamiltonian Ĥλ = Ĥ + λ Ô, (13) with some associated exact λ-dependent eigenfunction

Ψ λ 0 = Ψ 0 + λ Ψ ′ 0 + • • • , (14)
where Ψ ′ 0 = dΨ λ 0 /dλ λ=0 and corresponding exact λdependent energy

E λ 0 = Ψ λ 0 | Ĥλ |Ψ λ 0 / Ψ λ 0 |Ψ λ 0 ,
then the exact value of the observable is given by the derivative of the energy with respect to λ at λ = 0: O 0 = dE λ 0 /dλ λ=0 . Introducing an approximate λ-dependent trial wave function

Ψ λ = Ψ + λ Ψ ′ + • • • , (15)
where Ψ ′ = dΨ λ /dλ λ=0 , the λ-dependent energy can be estimated as the statistical average of the λ-dependent

local energy E λ L (R) = R| Ĥλ |Ψ λ /Ψ λ (R) E λ = E λ L (R) Ψ λ 2 , (16)
over the probability density Ψ λ (R) 2 , and the Hellmann-Feynman theorem suggests now to define a zero-variance zero-bias (ZVZB) estimator for the observable as the derivative of E λ with respect to λ at λ = 0

O ZVZB = O ZVZB L (R) Ψ 2 = dE λ dλ λ=0 = O L (R) Ψ 2 + ∆O ZV L (R) Ψ 2 + ∆O ZB L (R) Ψ 2 , (17)
where the zero-variance (ZV) contribution

∆O ZV L (R) = R| Ĥ|Ψ ′ Ψ ′ (R) -E L (R) Ψ ′ (R) Ψ(R) , (18)
does not contribute to the average, ∆O ZV L (R) Ψ 2 = 0 (in the limit of an infinite MC sample M → ∞, from the Hermiticity of the Hamiltonian) but can decrease the variance, and the zero-bias (ZB) contribution

∆O ZB L (R) = 2 [E L (R) -E] Ψ ′ (R) Ψ(R) , (19)
usually has little effect on the variance but can decrease the systematic error. The average of this ZB term vanishes if the trial wave function Ψ is an exact eigenstate or, more generally, if the energy E is stationary with respect to an infinitesimal variation of the trial wave function along the derivative Ψ ′ : Ψ → Ψ + ǫ Ψ ′ . For the special case of the calculation of the derivatives of the energy with respect to the nuclear coordinates, the average of the ZB term is known in the electronic structure literature as the Pulay contribution [89].

If one wants simply to compute an observable for a given trial wave function without changing the average, the ZB correction has to be omitted. In this case, the ZV estimator

O ZV L (R) = O L (R) + ∆O ZV L (R)
has a systematic error that still vanishes linearly with |δΨ| but a variance that now vanishes quadratically with |δΨ| and the error in its derivative

|δΨ ′ | = |Ψ ′ -Ψ ′ 0 | δO ZV = O(|δΨ|), (20a)
σ 2 O ZV L = O(|δΨ| 2 + |δΨ ′ | 2 + |δΨ| |δΨ ′ |). (20b)
If one also includes the ZB correction, then both the systematic error and the variance of the ZVZB estimator

O ZVZB L (R) vanish quadratically with |δΨ| and |δΨ ′ | δO ZVZB = O(|δΨ| 2 + |δΨ| |δΨ ′ |), (21a)
σ 2 O ZVZB L = O(|δΨ| 2 + |δΨ ′ | 2 + |δΨ| |δΨ ′ |). (21b)
This is easily shown by writing Ψ = Ψ 0 + δΨ and Ψ ′ = Ψ ′ 0 + δΨ ′ in the expressions of the average and the variance, and expanding with respect to δΨ and δΨ ′ . One can verify that the exact cancellation of the dominant contributions to the systematic error and the variance of Eqs. (12a) and (12b) by the ZV and ZB corrections is a direct consequence of the relationship between the exact wave function Ψ 0 and its derivative Ψ ′ 0 in first-order perturbation theory:

( Ĥ -E 0 )|Ψ ′ 0 = -( Ô -O 0 )|Ψ 0 .
Thus, the ZVZB estimators permit in principle accurate and efficient calculations for observables, provided that a good approximation for the derivative Ψ ′ 0 is available. We note that optimizing the parameters p in the trial wave function Ψ(p) by energy minimization presents the advantage of decreasing the sensitivity of the systematic error to the quality of the approximate derivative Ψ ′ , since in this case Ψ ′ only needs to be a good approximation of Ψ ′ 0 in the orthogonal complement of the space spanned by the wave function derivatives with respect to the parameters at the optimal parameter values p opt , i.e. Ψ i = (∂Ψ(p)/∂p i ) p=popt . Indeed, the contribution to the average of the ZB term of any component of Ψ ′ along Ψ i is proportional to the energy gradient with respect to parameter p i and thus vanishes at the stationary point:

2[E L (R) -E]Ψ i (R)/Ψ(R) Ψ 2 = (∂E/∂p i ) p=popt = 0.
This point is at the origin of the improved accuracy in the calculations of forces in QMC in Refs. 90, 91.

B. Estimators in diffusion Monte Carlo

DMC calculations within the FN approximation go beyond VMC calculations by replacing the VMC distribution Ψ(R) 2 by the more accurate mixed FN-DMC distribution Ψ FN (R)Ψ(R) in statistical averages, i.e.

• • • Ψ 2 → • • • ΨFNΨ .
All the DMC variances have the same lowest order behaviors with respect to the error in the trial wave function as in VMC. The systematic error of the energy now vanishes as the product of the error in the trial wave function |δΨ| and the error in the FN wave function

|δΨ FN | = |Ψ FN -Ψ 0 | δE = O(|δΨ| |δΨ FN |). (22)
For an arbitrary observable, the systematic error still vanishes only linearly with |δΨ| or

|δΨ FN | δO = O(|δΨ| + |δΨ FN |), (23)
but the use of the hybrid estimator "2DMC -VMC",

O hybrid = 2 O L (R) ΨFNΨ -O L (R) Ψ 2
, allows one to remove the dominant linear term |δΨ|

δO hybrid = O(|δΨ FN | + |δΨ| 2 + |δΨ FN | 2 + |δΨ| |δΨ FN |). (24)
The same improved estimators defined in the previous section can also be used straightforwardly in FN-DMC calculations. Note that, in this case, the average of the ZV term of Eq. ( 18) no longer vanishes on an infinite MC sample. Nevertheless, this term gives the same order of reduction of the variance as in VMC. The systematic error and the variance are still given by Eqs. (20a) and (20b), though the prefactors can be different in VMC and DMC. We note that it is possible to define a new ZV correction of vanishing average in FN-DMC [66] but this does not change the leading order of either the systematic error or the variance.

Adding the ZB term of Eq. ( 19), the systematic error of the ZVZB estimator vanishes quadratically

δO ZVZB = O(|δΨ| 2 + |δΨ| |δΨ ′ | + |δΨ| |δΨ FN | +|δΨ ′ | |δΨ FN |), (25)
with no linear term |δΨ FN | in contrast to the hybrid estimator of Eq. ( 24). The behavior of the variance of the ZVZB estimator in FN-DMC is still given by Eq. (21b). One can also define a hybrid ZVZB estimator,

O ZVZB hybrid = 2 O ZVZB L (R) ΨFNΨ -O ZVZB L (R) Ψ 2
, whose systematic error also vanishes quadratically as

δO ZVZB hybrid = O(|δΨ| 2 + |δΨ| |δΨ FN | + |δΨ ′ | |δΨ FN |), (26)
with no term in |δΨ| |δΨ ′ | in contrast to Eq. ( 25).

C. Expressions of the estimators for local Hamiltonians

We give now more explicit expressions of the estimators in terms of the convenient logarithmic derivative

Q(R) = Ψ ′ (R)/Ψ(R).
In the case of local Hamiltonians

R| Ĥ|R ′ = H(R)δ(R-R ′ ) where H(R) = -(1/2) N k=1 ∇ 2 r k +V (R)
contains kinetic and potential energy contributions, the ZV term takes the form

∆O ZV L (R) = - 1 2 N k=1 ∇ 2 r k Q(R) + 2 v k (R) • ∇ r k Q(R) , (27)
where

v k (R) = ∇ r k Ψ(R)/Ψ(R)
is the drift velocity of the trial wave function. The ZB term is simply

∆O ZB L (R) = 2∆E L (R)Q(R), (28)
where ∆E L (R) = E L (R) -E. For each observable, a specific form for Q(R) has to be chosen. In principle, one could optimize Q(R) for each system by minimizing the variance of the ZVZB estimator. Besides Q(R), the ZV and ZB terms require only the evaluation of the drift velocity and the local energy of the trial wave function that are already available in QMC programs.

Clearly, a nonlocal operator V (R, R ′ ) in the Hamiltonian, e.g. a nonlocal pseudopotential, would yield an additional term in the simplified expression of the ZV correction of Eq. ( 27). Although the inclusion of this additional term is required for the variance to rigorously vanish quadratically as in Eq. (20b), in practice the estimator using the simple form of Eq. ( 27) can already achieve a considerable variance reduction.

III. ZERO-VARIANCE ZERO-BIAS IMPROVED ESTIMATORS FOR THE SPHERICALLY AVERAGED POSITION INTRACULE DENSITY

In the case of the spherically averaged position intracule density, the local observable I L (u, R) is

I L (u, R) = 1 2 i =j dΩ u 4π δ(r ij -u), (29)
which has an infinite variance. We will give two possible choices for Q(R) which define good improved estimators.

A. First improved estimator

The minimal choice for Q(R) that cancels the infinite variance of

I L (u, R) is Q 1 (u, R) = - 1 8π i =j dΩ u 4π 1 |r ij -u| , (30)
which gives the following ZV contribution

∆I ZV1 L (u, R) = 1 8π i =j dΩ u 4π ∇ 2 rij 1 |r ij -u| +2 v i (R) • ∇ rij 1 |r ij -u| . ( 31
)
Using the identity ∇ 2 rij 1/|r ij -u| = -4πδ(r iju), it is seen that the first term in ∆I ZV1 L (u, R) exactly cancels the delta function in I L (u, R), and the ZV estimator

I ZV1 L (u, R) = I L (u, R) + ∆I ZV1 L (u, R) is simply I ZV1 L (u, R) = 1 4π i =j v i (R) • dΩ u 4π ∇ rij 1 |r ij -u| = - 1 4π i =j v i (R) • r ij r 3 ij 1 [u,+∞[ (r ij ). ( 32
)
The form of Q 1 (u, R) of Eq. ( 30) could also have been guessed from the behavior at small r ij of the first-order solution Ψ ′ (R) of the perturbation theory with respect to δ(r iju) (see Refs. 72,92). In contrast to the histogram estimator of Eq. ( 8) where only electron pairs with relative distance r ij in the small interval [u-∆u/2, u+∆u/2[ contribute to I(u), for the ZV estimator of Eq. ( 32) all the electron pairs with relative distance r ij ≥ u contribute to I(u). For u = 0, the variance of this ZV estimator is finite and much smaller than the variance of the histogram estimator as shown in Sec. V. For u = 0, this ZV estimator reduces to the estimator proposed in Ref. 62 to compute one-electron densities at the nucleus and applied in Ref. 22 for calculations of intracule densities at the coalescence point.

The ZB correction corresponding to Q 1 (u, R) of Eq. ( 30) is

∆I ZB1 L (u, R) = - ∆E L (R) 4π i =j dΩ u 4π 1 |r ij -u| = - ∆E L (R) 4π i =j 1 [0,u[ (r ij ) u + 1 [u,+∞[ (r ij ) r ij . (33)
The resulting ZVZB estimator

I ZV1ZB1 L (u, R) = I ZV1 L (u, R) + ∆I ZB1
L (u, R) allows one to reduce the systematic error due to the trial wave function as shown in Sec. V. Note that, in contrast to the histogram estimator, these ZV and ZVZB improved estimators do not suffer from any discretization error: they are estimators of I(u) for an infinitely precise value of u. Both the ZV estimator of Eq. ( 32) and its ZB correction of Eq. ( 33) are very simple to program and very fast to compute on a grid over u.

Like the histogram estimator, the ZV estimator of Eq. (32) gives simply zero for distances u beyond the largest electron-electron distance r ij encountered in the Monte Carlo run. Also, because the form of Q 1 (u, R) of Eq. ( 30) has been chosen only to cure the deficiency of the histogram estimator at small u, the ZB correction of Eq. ( 33) actually makes the histogram estimator worse at large u. The ZB correction decays only as 1/u as u → ∞ instead of the correct exponential decay and consequently gives large variances at large u. Thus, it is inaccurate to extract the long-range behavior of the intracule density using the ZVZB estimator I ZV1ZB1 L (u, R). The second choice for Q(R) presented next remedies this limitation.

B. Second improved estimator

A more general choice for Q(R) that cancels the infinite variance of

I L (u, R) is Q 2 (u, R) = - 1 8π i =j dΩ u 4π g(r ij , u) |r ij -u| , (34)
where g(r ij , u) is some function satisfying g(r ij , u) = 1 for r ij = u. For example, using

g(r ij , u) = e -ζ|rij -u| , (35)
with ζ > 0, has the advantage of ensuring a correct exponential decay of the estimator at infinity for a finite system.

The corresponding ZV estimator is

I ZV2 L (u, R) = 1 8π i =j dΩ u 4π 2 v i (R) • ∇ rij e -ζ|rij-u| |r ij -u| +2 ∇ rij 1 |r ij -u| • ∇ rij e -ζ|rij -u| + 1 |r ij -u| ∇ 2 rij e -ζ|rij -u| = - 1 4π i =j v i (R) • r ij r 3 ij r ij A(r ij , u) + r ij u B(r ij , u) - ζ 2 2 A(r ij , u) , (36)
and its ZB correction is

∆I ZB2 L (u, R) = - ∆E L (R) 4π i =j dΩ u 4π e -ζ|rij-u| |r ij -u| = - ∆E L (R) 4π i =j A(r ij , u), (37)
where

A(r ij , u) = e -ζ|rij-u| -e -ζ(rij+u) 2ζr ij u , (38)
B(r ij , u) = 1 2 sgn(r ij -u)e -ζ|rij-u| -e -ζ(rij+u) ,( 39
)
and where sgn is the sign function. These refined ZV estimator

I ZV2 L (u, R) and ZVZB estimator I ZV2ZB2 L (u, R) = I ZV2 L (u, R)+∆I ZB2
L (u, R) now both decay correctly exponentially as u → ∞. The constant ζ is chosen according to Eq. ( 5), i.e. ζ = 2 √ 2I where I is an estimate of the vertical ionization energy. On the other hand, the computational cost is larger per Monte Carlo step than that of the simpler estimators of the previous subsection.

IV. COMPUTATIONAL DETAILS

We have chosen to illustrate the efficiency and accuracy of the new improved QMC estimators by calculating the intracule density of the He and C atoms and the C 2 and N 2 molecules in their ground states. The molecules are considered at their experimental geometries (d C-C = 2.3481 Bohr [93] and d N-N = 2.075 Bohr [START_REF] Huber | Molecular Spectra and Molecular Structure -IV. Constants of Diatomic Molecules[END_REF]).

We start by generating a standard ab initio wave function using the quantum chemistry program GAMESS [START_REF] Schmidt | [END_REF], typically a HF wave function or a MC-SCF wave function with a complete active space (CAS) generated by distributing n valence electrons in m valence orbitals [CAS(n,m)]. For each system considered, we choose the one-electron Slater basis so as to ensure that the HF intracule density is reasonably converged with respect to the basis. For the He atom, we use the basis of Clementi and Roetti [96], with exponents reoptimized at the HF level by Koga et al. [97]. For the C atom, we use the CVB1 basis of Ema et al. [98]. For the C 2 and N 2 molecules, we use the CVB2 basis of the same authors. In GAMESS, each Slater function is actually approximated by a fit to 14 Gaussian functions [99][100][101] This standard ab initio wave function is then multiplied by a Jastrow factor, imposing the electron-electron cusp condition, but with essentially all other free parameters chosen to be zero to form our starting trial Jastrow-Slater wave function, and QMC calculations are performed with the program CHAMP [START_REF] Champ | a quantum Monte Carlo program written[END_REF] or QMC-MOL [START_REF] Qmcmol | a quantum Monte Carlo program written[END_REF] using the true Slater basis set rather than its Gaussian expansion. The Jastrow parameters, the orbital coefficients and the configuration state functions (CSFs) coefficients of this trial wave function are optimized in VMC using the very efficient, recentlydeveloped linear energy minimization method [87,88] and an accelerated Metropolis algorithm [START_REF] Umrigar | [END_REF][START_REF] Umrigar | Quantum Monte Carlo Methods in Physics and Chemistry[END_REF]. Once the trial wave function has been optimized, we compute the intracule density in VMC, and in DMC using the fixed-node and the short-time approximations (see, e.g., Refs. 106-109). We use an imaginary time-step of τ = 0.01 hartree -1 in an efficient DMC algorithm featuring very small time-step errors [110].

V. RESULTS AND DISCUSSION

A. He atom

The improvement due to the ZV and ZVZB estimators is illustrated for the simple case of He atom.

Figure 1 shows the spherically averaged intracule density I(u) as a function of the electron-electron distance u, calculated in VMC using the standard histogram estimator of Eq. ( 8), the ZV1 improved estimator of Eq. ( 32) and the ZV1ZB1 improved estimator of Eq. ( 33), employing only 100 000 MC configurations sampled from a HF trial wave function (without a Jastrow factor). In this subsection, we intentionally use a poor trial wave function to demonstrate the large reduction in the bias resulting from the ZVZB estimator. The calculations take a few seconds on a present-day single-processor personal computer. The intracule density obtained with an accurate, explicitly-correlated Hylleraas-type wave function with 491 terms [52,111,112] is also shown as a reference.

The statistical uncertainty obtained with the histogram estimator is very large, especially at small u, making it impossible to extract the on-top intracule density I(0). The ZV1 estimator spectacularly reduces the statistical uncertainty which becomes invisible at the scale of the plot, its variance being smaller by nearly 4 orders of magnitude for small and large u and by about 2 orders of magnitude for intermediate u. The ZV1ZB1 estimator in turn spectacularly reduces the systematic error due to the use of a HF trial wave function, the obtained intracule density agreeing well with the accurate reference. In particular, the maximum at small u caused by the cusp at u = 0 is accurately reproduced. On the other hand, the use of the ZV1ZB1 estimator increases slightly the variance at small u and more importantly at large u where the ZB1 correction of Eq. ( 33) decays too slowly as 1/u. The behavior of the estimators in the long-range tail of I(u) is explored in detail in Fig. 2 which compares the ZV1ZB1 and ZV2ZB2 estimators of Eqs. ( 36) and (37) for u > 3. As mentioned in Sec. III B, the relative statistical uncertainty obtained with the ZV1ZB1 is very large (about 200% at u = 3.5), whereas the ZV2ZB2 estimator, which has the correct exponential decay at large u, spectacularly reduces the statistical uncertainty, its variance being smaller by about 3 orders of magnitude at u = 3.5. The improvement is even more dramatic for larger u.

We now discuss the scaling of the intracule density and its statistical uncertainty with respect to the nuclear charge Z. For this purpose, we have calculated the intracule densities of the elements of the He isoelectronic series from Z = 2 (He) to Z = 20 (Ar 18+ ) using Jastrow-Slater wave functions. We found that, in the relevant range of u (u 5), the intracule density roughly obeys the scaling law I(u/Z) ∼ Z n with n ≈ 3, in agreement with the value predicted by a simple hydrogenic model [113]. The statistical uncertainty of I(u/Z) computed with the ZV1 estimator roughly obeys the same law, so that the relative statistical accuracy on I(u/Z) does not deteriorate with Z in the series (at least up to Z = 20).

Finally, we note in passing that Fig. 2 also reveals another advantage of our improved estimators: because the estimator of I(u) is strongly correlated with the estimator of I(u + ∆u), as obvious for instance in Eq. ( 32), the obtained intracule density curves are always very smooth, even at a scale much smaller than the statistical uncertainty. This makes the calculated intracule densities directly suitable for subsequent manipulations.

B. C atom

We continue the discussion of the improved estimators for the more interesting case of the C atom.

The repercussion of the shell structure of the system on the intracule is apparent in Fig. 3 which shows the integrand of the electron-electron Coulomb interaction energy over the electron-electron distance u, i.e. 4πu I(u) [see Eq. ( 6)]. The ZVZB2 estimator is used with an ac- √ 2I ≈ 1.82 [START_REF]Nist chemistry webbook[END_REF]) improved estimator and a fully optimized Jastrow-Slater CAS(4,4) trial wave function. The same spin contribution 4πu I SS (u) and the opposite spin contribution 4πu I OS (u) are also shown.

curate Jastrow-Slater CAS (4,4) wave function in which the Jastrow, CSF and orbital parameters have been simultaneously optimized. Note however that, at the scale of the plot, the HF intracule 4πu I HF (u) would look the same. The curve displays two maxima: the maximum at short distance (u ≈ 0.2) corresponds essentially to the K-K electron pair and the maximum at longer distance (u ≈ 1.0) corresponds essentially to the K-L and L-L electron pairs. Fig. 3 also shows the same spin (SS) contribution

I SS (u) = I ↑↑ (u) + I ↓↓ (u), (40)
and opposite spin (OS) contribution

I OS (u) = I ↑↓ (u) + I ↓↑ (u), (41)
where I σσ ′ (u) is the spherical average of the spin-resolved intracule densities

I σσ ′ (u) = 1 2 i =j dR Ψ(R) 2 δ(r ij -u)δ si,σ δ sj ,σ ′ , (42)
where the spin coordinates of the N = N ↑ + N ↓ electrons in the spin-assigned wave function Ψ(R) are fixed at s i =↑ for i = 1, ..., N ↑ and s i =↓ for i = N ↑ + 1, ..., N ↑ + N ↓ . Not surprisingly, the intracule density at short distance is dominated by the OS contribution, while at long distance the SS and OS contributions become nearly identical. Figure 4 shows the correlation part of the radial intracule density 4πu 2 [I(u) -I HF (u)], also called correlation hole, using the ZVZB2 improved estimator and the same Jastrow-Slater CAS(4,4) trial wave function. Correlation decreases the radial probability density at √ 2I ≈ 1.82 [START_REF]Nist chemistry webbook[END_REF]) improved estimator for a series of trial wave functions of increasing accuracy: HF, Jastrow × HF, fully-optimized Jastrow × SD and Jastrow × CAS(4,4) wave functions.

the two previously-mentioned maxima and increases it at longer distances. The same spin contribution 4πu 2 I SS (u) -I SS HF (u) and the opposite spin contribution 4πu 2 I OS (u) -I OS HF (u) are also shown in Fig. 4. The OS component constitutes the main contribution to the correlation hole.

The effect of the chosen trial wave function on the accuracy on the correlation hole is examined in Fig. 5. Four √ 2I ≈ 1.83 [START_REF]Nist chemistry webbook[END_REF]) improved estimator and a fully optimized Jastrow-Slater CAS (8,8) trial wave function. The same spin contribution 4πu I SS (u) and the opposite spin contribution 4πu I OS (u) are also shown.

trial wave functions of increasing accuracy have been tested: HF, Jastrow × HF (with only Jastrow parameters optimized), Jastrow × single-determinant (SD) (with Jastrow and orbital parameters optimized) and Jastrow × CAS(4,4) (with Jastrow, CSF and orbital parameters optimized). Remarkably, the ZVZB2 estimator gives a correlation hole with a correct overall structure even with the uncorrelated HF wave function. For more quantitative predictions, the use of a Jastrow factor is however necessary. Optimization of the orbitals in a Jastrow-Slater single-determinant wave function brings a further significant improvement in the interesting short-range region (u 2). The Jastrow-Slater multi-determinant CAS (4,4) and the Jastrow-Slater single-determinant intracules agree closely.

C. C2 molecule

We now discuss the more difficult case of the C 2 molecule. The ground-state wave function of this system has a strong multi-configurational character due to energetic near-degeneracies among the valence orbitals ("nondynamical correlation" in chemists' jargon, or "strong correlation" in physicists' jargon), making it a challenging system despite its small size.

Figure 6 plots 4πu I(u) versus u using the ZVZB2 estimator with an accurate Jastrow-Slater CAS (8,8) wave function (Jastrow, CSF and orbital parameters optimized). The curve displays three maxima: the maximum at short distance (u ≈ 0.2) corresponds essentially to the intra-atomic K-K electron pairs; the maximum at long distance (u ≈ 2.3) is located at about the interatomic distance and corresponds mainly to the interatomic K- √ 2I ≈ 1.83 [START_REF]Nist chemistry webbook[END_REF]) with a MCSCF CAS (8,8) wave function and a series of Jastrow × CAS (8,8) wave functions with different levels of optimization.

K electron pairs; the maximum at intermediate distance (u ≈ 1.2) must correspond mainly to intra-atomic and interatomic K-L and L-L electron pairs. As for the C atom, the intracule density at short distance is dominated by the OS contribution, while at long distance the SS and OS contributions become nearly identical.

Figure 7 shows the correlation hole of the C 2 molecule using the ZV2 improved estimator calculated in VMC with a MCSCF CAS (8,8) wave function and a series of Jastrow × CAS (8,8) wave functions for different levels of optimization. The MCSCF CAS (8,8) wave function does not correlate the core electrons and thus gives essentially no correlation hole at distances corresponding to intraatomic core electron pairs (u ≈ 0.2). At longer valence distances, the MCSCF CAS (8,8) wave function gives only a gross overall shape of the correlation hole. Introduction of a Jastrow factor yields a correct correlation hole at core distances, which as expected is about twice as deep as the core correlation hole of the C atom, and reduces the correlation hole at valence distances. The action of the Jastrow factor is thus not limited to very short electronelectron distances, but in fact importantly modifies the correlation hole up to distances u ≈ 4. The Jastrow factor has also an indirect action via reoptimization of the CSF and orbital coefficients in its presence. The reoptimization of the CSF coefficients changes slightly the correlation hole at valence distances. The reoptimization of the CSF and orbital coefficients has a more important impact of the correlation hole at valence distances and also at core distances.

We next scrutinize in detail the convergence of the intracule density with respect to the determinantal part of the trial wave function. Figure 8a shows the correlation hole using the ZVZB2 improved estimator calculated in 2I ≈ 1.83 [START_REF]Nist chemistry webbook[END_REF]) improved estimator for a series of trial wave functions of increasing accuracy: Jastrow × HF, fully-optimized Jastrow × SD, Jastrow × CAS (8,5), Jastrow × CAS(8,7) and Jastrow × CAS (8,8) wave functions.

VMC with five trial Jastrow-Slater wave functions of increasing accuracy: Jastrow × HF (with only the Jastrow parameters optimized), Jastrow × SD (with the Jastrow and orbital parameters optimized), and multideterminant Jastrow × CAS(8,5), Jastrow × CAS(8,7) and Jastrow × CAS(8,8) (with the Jastrow, CSF and orbital parameters optimized). At core distances (u ≈ 0.2), the correlation hole is essentially converged with only a single-determinant Jastrow-Slater wave function, provided that the orbitals are reoptimized together with the Jastrow factor. In contrast, at longer valence distances, the correlation hole depends very strongly on the determinantal part of the wave function. In particular, the multi-determinant wave functions yield a depletion of the intracule density at about the bond length, a feature not present at the single-determinant level. We have verified that this minimum disappears when using a pseudopotential removing the 1s electrons, and we thus interpret it as a decrease of probability of finding two interatomic core electrons separated by the bond distance. Overall, it appears necessary to use at least a multi-determinant CAS (8,7) wave function which includes configurations constructed from the antibonding π orbitals to reach reasonable convergence. We have checked that further excitations beyond the CAS (8,8) wave function barely change the correlation hole. The corresponding correlation holes calculated in FN-DMC with the same ZVZB2 estimator and trial wave functions are reported in Fig. 8b. The DMC intracules do not differ much from the corresponding VMC intracules, the accuracy of the intracule densities in the valence region being still essentially controlled by the trial wave function. √ 2I ≈ 2.14 [START_REF]Nist chemistry webbook[END_REF]) improved estimator for a series of trial wave functions of increasing accuracy: Jastrow × HF, fully-optimized Jastrow × SD and Jastrow × CAS (10,8) wave functions.

D. N2 molecule

We finish our illustration of the calculation of intracule densities with the N 2 molecule, which has recently been the object of experimental and theoretical investigations [42].

The correlation holes of this molecule calculated in VMC with the ZVZB2 improved estimator for a Jastrow × HF, and fully optimized Jastrow × SD and Jastrow × CAS (10,8) wave functions are plotted in Fig. 9. In sharp contrast to the C 2 molecule, no depletion of intracule density is observed at the bond length. Also, the correlation hole is here essentially converged with only a singledeterminant Jastrow-Slater wave function provided that the orbitals are optimized with the Jastrow factor. Unlike the C 2 molecule, going to a multi-determinant wave function does not have a large effect on the correlation hole at valence distances. The correlation hole calculated here agrees reasonably well with recent multi-reference configuration interaction and coupled-cluster calculations, and with experiment [42].

VI. CONCLUSIONS

We have presented improved QMC estimators for the spherically averaged position intracule density I(u), constructed using the general zero-variance zero-bias principle for observables that do not commute with the Hamiltonian. By replacing the average of the local deltafunction operator by the average of a smooth non-local operator, these estimators decrease the variance of the standard histogram estimator by several orders of magnitude, and thus make the calculation of this quantity in QMC vastly more efficient. Interestingly, they permit calculations of I(u) for very short and very large interelectronic distances u that are never realized in the Monte Carlo run. These new estimators can also decrease the systematic error of the intracule density due to the approximate trial wave function. Other advantages of these estimators are the absence of any discretization error with respect to u and the possibility to obtain very smooth curves for I(u). These improved estimators, together with the achievement of systematically reducing the systematic error in both VMC and DMC calculations by optimization of trial wave functions with an increasing number of parameters, have allowed us to obtain accurate correlated intracule densities for atoms and molecules.

The estimators presented here can be used with trivial adaptations for QMC calculations of the companion entity of I(u), namely the spherically averaged extracule density E(r), representing the probability density of finding two electrons with center of mass at a radial distance r with respect to the chosen origin. Similar improved estimators can be constructed for the three-dimensional intracule density I(u), extracule density E(r) and the full pair density n 2 (r 1 , r 2 ). More generally, the variance reduction technique presented here can be applied to calculations of any pair-correlation function in classical and quantum Monte Carlo calculations.

  whose square root is proportional to the statistical uncertainty, both vanish quadratically as a function of the error in the trial wave function |δΨ| = |Ψ -Ψ 0 | (where | • • • | designates the Hilbert space norm)

FIG. 1 :

 1 FIG.1: Spherically averaged position intracule density I(u) as a function of the electron-electron distance u for the He atom. The intracule densities calculated in VMC with the histogram, ZV1 and ZV1ZB1 estimators using only 100 000 MC configurations sampled from a HF trial wave function (without a Jastrow factor) are compared. For the histogram estimator, a grid step of ∆u = 0.005 has been used. The intracule density obtained with an accurate, explicitly-correlated Hylleraas-type wave function is also shown.

FIG. 2 :

 2 FIG. 2: Long-range tail of the spherically averaged position intracule density I(u) of the He atom. The intracule densities calculated in VMC with the ZV1ZB1 and ZV2ZB2 (with ζ = 2 √ 2I ≈ 2.7) estimators using only 100 000 MC configurations sampled from a HF trial wave function (without a Jastrow factor) are compared. The intracule density obtained with an accurate, explicitly-correlated Hylleraas-like wave function is also shown.

FIG. 3 :

 3 FIG.3: Integrand of the electron-electron Coulomb interaction energy 4πu I(u) over the electron-electron distance u for the C atom, calculated in VMC using the ZVZB2 (with ζ = 2 √ 2I ≈ 1.82[START_REF]Nist chemistry webbook[END_REF]) improved estimator and a fully optimized Jastrow-Slater CAS(4,4) trial wave function. The same spin contribution 4πu I SS (u) and the opposite spin contribution 4πu I OS (u) are also shown.

FIG. 4 :FIG. 5 :

 45 FIG. 4: Correlation part of the radial position intracule density 4πu 2 [I(u) -IHF(u)] as a function of the electron-electron distance u for the C atom, where I(u) has been calculated in VMC using the ZVZB2 (with ζ = 2 √ 2I ≈ 1.82 [114]) improved estimator and a fully optimized Jastrow-Slater CAS(4,4) trial wave function. The same spin contribution 4πu 2 I SS (u) -I SS HF (u) and the opposite spin contribution 4πu 2 I OS (u) -I OS HF (u) are also shown.

FIG. 6 :

 6 FIG.6: Integrand of the electron-electron Coulomb interaction energy 4πu I(u) versus the electron-electron distance u for the C2 molecule, calculated in VMC using the ZVZB2 (with ζ = 2 √ 2I ≈ 1.83[START_REF]Nist chemistry webbook[END_REF]) improved estimator and a fully optimized Jastrow-Slater CAS(8,8) trial wave function. The same spin contribution 4πu I SS (u) and the opposite spin contribution 4πu I OS (u) are also shown.

FIG. 7 :

 7 FIG. 7: Correlation part of the radial position intracule density 4πu 2 [I(u) -IHF(u)] as a function of the electron-electron distance u for the C2 molecule, where I(u) has been calculated in VMC using the ZV2 improved estimator (with ζ = 2√ 2I ≈ 1.83[START_REF]Nist chemistry webbook[END_REF]) with a MCSCF CAS(8,8) wave function and a series of Jastrow × CAS(8,8) wave functions with different levels of optimization.

FIG. 8 :

 8 FIG. 8: Correlation part of the radial position intracule density 4πu 2 [I(u) -IHF(u)] as a function of the electron-electron distance u for the C2 molecule, where I(u) has been calculated in VMC (a) and FN-DMC (b) using the ZVZB2 (with ζ = 2 √2I ≈ 1.83[START_REF]Nist chemistry webbook[END_REF]) improved estimator for a series of trial wave functions of increasing accuracy: Jastrow × HF, fully-optimized Jastrow × SD, Jastrow × CAS(8,5), Jastrow × CAS(8,7) and Jastrow × CAS(8,8) wave functions.

FIG. 9 :

 9 FIG. 9: Correlation part of the radial position intracule density 4πu 2 [I(u) -IHF(u)] as a function of the electron-electron distance u for the N2 molecule, where I(u) has been calculated in VMC using the ZVZB2 (with ζ = 2 √ 2I ≈ 2.14[START_REF]Nist chemistry webbook[END_REF]) improved estimator for a series of trial wave functions of increasing accuracy: Jastrow × HF, fully-optimized Jastrow × SD and Jastrow × CAS(10,8) wave functions.

Acknowledgments

We would like to thank Andreas Savin and Paola Gori-Giorgi for stimulating discussions. We also thank Alexander Kollias for providing us the Gaussian fits of Slater basis functions of Ref. 101. J. T. acknowledges financial support from a Marie Curie Outgoing International Fellowship (039750-QMC-DFT). This work was also supported by the Centre National de la Recherche Scientifique and by the National Science Foundation (DMR-0205328, EAR-0530301). Most of the calculations were been performed at the Cornell Theory Center and on the Intel cluster at the Cornell Nanoscale Facility (a member of the National Nanotechnology Infrastructure Network supported by the National Science Foundation).


Olivier Aubert
email: olivier.aubert@univ-nantes.fr

Yannick Prié
email: yannick.prie@univ-nantes.fr

Camila Canellas
email: camila.canellas@univ-nantes.fr

Leveraging video annotations in video-based e-learning

Keywords: E-learning, MOOCs, Video Annotation, Pedagogical Processes

The e-learning community has been producing and using video content for a long time, and in the last years, the advent of MOOCs greatly relied on video recordings of teacher courses. Video annotations are information pieces that can be anchored in the temporality of the video so as to sustain various processes ranging from active reading to rich media editing. In this position paper we study how video annotations can be used in an e-learning context -especially MOOCs -from the triple point of view of pedagogical processes, current technical platforms functionalities, and current challenges. Our analysis is that there is still plenty of room for leveraging video annotations in MOOCs beyond simple active reading, namely live annotation, performance annotation and annotation for assignment; and that new developments are needed to accompany this evolution.

INTRODUCTION

While video material had been used for several decades as a learning support, the development of web-based e-learning first caused a setback in the usage of pedagogical videos, due to lack of network bandwidth or standardized formats and software. However, video streaming, video hosting and the dissemination of capture and editing tools have come along and supported the exponential growth of video usage on the Web. Again video became an important component of e-learning setups, through the OpenCourseWare movement and the recent advent of Massive Online Open Courses (MOOCs).

Video annotations (section 2) are information pieces that can be anchored in the temporality of the video so as to sustain various processes ranging from active reading to rich media editing (section 3). Our main interest in this position paper is related to how video annotations are and can be used in e-learning context -especially MOOCs -from the triple point of view of pedagogical processes (section 4), current technical platforms functionalities (section 5), and current challenges (section 6). One of the most important results of our analysis1 is that there is still plenty of room for leveraging video annotations in MOOCs beyond simple active reading, namely live annota-tion, performance annotation and annotation for assignment; and that technological improvements are needed to accompany this evolution.

VIDEO ANNOTATIONS

Active reading is a process where a reader assimilates and re-uses the object of his reading, as part of his knowledge work [START_REF] Waller | Functionality in digital annotation: Imitating and supporting real-world annotation[END_REF]. The goals may be the exploration of a document, its enrichment or its analysis, for oneself or within a collaborative activity. Active reading usually relies on annotations, that add some information to a specific section or fragment of the target document, and can thereafter be reused along it for searching, navigating, repurposing, etc.

The link between the annotation and the original document may be more or less explicit, from the handwritten note in the margin of a book to the note taken on a notebook while watching a movie (which will involve more work from the annotator to specify the targeted information). Two main components of an annotation are usually considered: its content and its anchor. The content may take any form, as long as the underlying support allows it, and can be more or less structured. Anchoring will depend on the nature of the annotated document, and will be more or less explicit and easy to navigate.

We specifically position ourselves in this article in the audiovisual context. Video documents present a number of specific or more stringent issues. First, contrary to text content, they do not have any implicit semantics: any active reading must thus be mediated through annotations that provide an explicit information layer along the document. Second, contrary to text reading, the reading speed is imposed by the player. The annotation process then requires some kind of interruption or interference from the viewing process, and this conflict between the temporality of the video and that of the annotation process must be addressed somehow. A video annotation is composed of data explicitly associated to a spatiotemporal fragment of a video. As illustrated in Figure 1, the spatiotemporal anchors define at least a timestamp (in the case of durationless fragments, specifying a single time in the document, e.g. 354), but more generally a begin and an end timecodes (e.g. 1321 and 1521). They may additionally address a specific static or dynamic zone of the displayed video (e.g. a rectangle shape that would follow a player in football video).

Audiovisual document

Video annotation content data can be of any type. Textual data is most often used, since it is the easiest to produce and to consume, but any content (audio, images, video, key/values...) can as well be associated. For instance, in a language learning context, the tutor can take textual notes about a video recording of a session, and also annotate by recording some spoken words to indicate the correct pronunciation. Annotations can also be articulated through some structure, such as a type classification: a feature movie analysis could for instance define different annotation types like Shot, Sequence or Character appearance (see figure 1).

USING VIDEO ANNOTATIONS

Annotations can be created manually or automatically. Manual creation involves various user interfaces, depending on the nature of the task and on the information that has to be captured (see VideoAnt and Advene in figure 2). Annotations may also automatically be created by extracting features from an actual video document (through speech recognition, or automatic shot detection for example) [START_REF] Nixon | TV's future is linked: Web and television across screens 4th international workshop on future television at EuroITV 2013[END_REF], or by capturing synchronized information during the very recording of the document. This last case is used for instance when recording information about the activity of a user: an ergonomics researcher studying the use of a software can capture a recording of the user screen while using the software, along with more discrete information capture from the software (button clicks, file openings, etc.).

Beyond information retrieval and search, which is routinely carried out in active reading activities or in video monitoring systems, video annotations can be used in a variety of ways, such as enrichment and document creation.

Enrichment of the rendering of the video document is not new: subtitles can indeed be considered as video annotations, that are displayed as a caption over the video. But such overlays can also be graphical, to attract the attention of the viewer on a specific visual element. Video enrichments produced from the annotations can also be placed along the (original) video player, to produce a navigable table of contents for instance. Video annotations can also be used to create other documents, be it other videos as it is the case in video summarization (automatic or guided); or more rich-media documents as an article illustrated with some fragments (through the annotations) of the video; or even an annotation-based hypervideo that permits the navigation from one video to the other. Eventually, collaborative activities can greatly benefit from annotations, that here serve as an interpretative layer between participants.

These different types of uses can be put to use in different application domains. Video archives (e.g. TV, surveillance) can propose an enhanced access to their collections through video annotations, allowing to find specific video fragments. The Yovisto platform2 (Waitelonis and Sack, 2012) offers for example access to video through semantic annotations, allowing to look for specific location, people, events... Sports analysis also greatly relies on video material, which can be used in a reflective way by offering the sportsman to view his own performance, or to analyse the behaviour of adversaries on recordings of previous competitions. Many applications such as EliteSportsAnalysis or MotionView Video Analysis Software offer tools to annotate and analyse sport performances. Research on activity in domains such as ergonomics, animal behaviour, linguistics, etc. also uses annotation software, since researchers need to perform a precise analysis of video recordings. There exist a number of research tools such as Advene, Anvil or Transana, as well as commercial offers like Noldus. They all offer annotation capabilities accompanied by various visualisation and analysis capabilities. Pedagogy is obviously an important domain for video annotation practices. First, any matter dealing with videos content as learning material, such as language or movie courses [START_REF] Canellas | Advene: active reading through hypervideo[END_REF][START_REF] Puig | Lignes de temps: Involving cinema exhibition visitors in mobile and on-line film annotation[END_REF], can benefit from the usage of annotations on a course support, as in VideoNot.es. Second, discussion about self-reflective activities can be enhanced by annotation-based tools. For instance, VideoTraces has been used for a long time in dance courses [START_REF] Cherry | Using a digital video annotation tool to teach dance composition[END_REF][START_REF] Bailey | Dancing on the grid: using e-science tools to extend choreographic research[END_REF] for annotating dance sessions. More generally, video recordings of learners presentations or interaction are used to implement self-reflection activities in classrooms [START_REF] Rich | Video annotation tools technologies to scaffold, structure, and transform teacher reflection[END_REF], supported by a number of tools such as VideoTraces, CLAS or MediaNotes.

It appears that a great number of practices have been experimented in different contexts and application domains, from the 1990s VHS based reflective activities to the more recent collaborative and online video analyses. The experience accumulated on these tools and practices could fruitfully be incorporated in e-learning.

E-LEARNING PROCESSES BASED ON VIDEO ANNOTATIONS

In order to assess in what measure we can leverage the existing experience in video annotation systems in an e-learning context, we organize these processes along four classes of scenarios.

For active reading with annotations, video is a learning material whose content has to be assimilated or evaluated. This task is carried out through an it-erative process, dedicated to the analysis of the audiovisual source through its enrichment with annotations and the definition of appropriate visualisations. In a learning context, students may annotate the video material by taking notes, by themselves or collaboratively. Conversely, teachers may use the same techniques for evaluating and grading videos produced by students. And both learners and teachers can engage into a discussion about a video through annotations.

Live annotation occurs during a live lecture, which is recorded and annotated at the same time. Students take notes during the lecture, and reuse these notes as a basic indexing system when replaying the recording. Teachers may also let students ask questions through annotations, and answer them at the end of the lecture [START_REF] Bétrancourt | Assessing the use of a trace-based synchronous tool for distant language tutoring[END_REF].

Performance annotation also implies video as a trace of a performance, be it recorded in a conventional classroom or during a synchronous online session. The recording may already be augmented automatically by the capture of annotations containing information about the activity (sent documents, chat messages, etc). Based on this recorded video and activity trace, students may annotate their own performance, for self-reflection or for sharing an analysis with their teacher [START_REF] Rich | Video annotation tools technologies to scaffold, structure, and transform teacher reflection[END_REF]. Teachers may as well annotate their own performance in a self-reflective way, to improve their practice (ibid). Eventually, students may annotate a recorded course for suggesting improvements or pointing out difficult sections. The teacher can use that feedback when preparing the next course or the next version of the same course [START_REF] Sadallah | A framework for usage-based document reengineering[END_REF].

Eventually, in annotation for assignment, the video is a material that has to be used to prepare an assignment (a feature movie, a recording of new for media analysis, etc). The work may require stu-dents to analyse some aspects of the video, and produce annotations reflecting their analysis. The annotations are then later assessed by the teacher [START_REF] Wong | Teaching developmental psychology using an interactive online video platform[END_REF]. Further, the annotations resulting from the analysis may be reused to produce a new document, like an abstract or a video collage. At Columbia University, students use the MediaThread platform [START_REF] Bossewitch | Teaching and learning with video annotations[END_REF] to produce critical video composition or critical multimedia essays, by combining annotations. The teachers then assess their productions.

We can identify distinguishing features between these different classes of scenarios, considering on the one hand the status of the video, and on the other hand the actors producing and using the annotations. The annotated video can be a base learning material, such as a movie or a documentary to study, or can be a recording a lecture (which the students may or may not have seen live). It can also be the recording of student contributions. The actors producing and/or using the annotations can be the students, the teachers, student colleagues or teacher colleagues, or even the general public.

ANNOTATION USES IN E-LEARNING SYSTEMS

Table 1 provides an overview of the various functionalities related to video annotation offered by mainstream MOOC platforms 3 or more specifically dedicated tools.

From our analysis, it appears first that features that facilitate the comprehension of the discourse -such as the possibility to adjust the video speed or activate subtitles and transcriptions -are largely present in MOOCs. These tools seem to be important in a multicultural context where subtitles or transcription are frequently produced in collaboration with student in the case of translations to other languages. Second, there is the use of interactive enrichments in MOOC platforms, usually to have the video stop so that students answer a question in order to verify the understanding of what had just been explained. Third, if many MOOC platforms allow adding comments on video lectures, annotations refering to a part of the video are only possible via external tools. Fourth, the 3 Based on courses available in late December 2013. Some platforms such as Udemy were not considered due to the fact that they do not have an open access. This table will be actualized for the final version of the paper. A more detailed version is available on the web and constantly actualized on http://comin-ocw.org/video-annotations/platformfeatures/. dedicated tools we analyzed offer more features regarding the annotation process -such as interactive timelines, export of annotated data, sharing of annotation, etc. -than MOOC platforms.

Thus, most of the solutions presented in Table 1 provide the possibility to develop pedagogical activities aiming to achieve active reading with annotation from students. Indeed, this is the main use of such tools by MOOC students, who are seeking a better understanding of the proposed video material. Annotation-based active reading can be carried out individually as well as collaboratively in the majority of the tools. This last possibility is even more significant: as students cannot always rely on having their doubts/difficulties solved by the teacher, collaboration with peers via annotation tools can increase their understanding along with secondary benefits of developing cognitive capacities on learning from video, observational skills and increasing focus and attention.

As far as our other classes of scenarios are concerned, performance annotation was not observed in the MOOC context although it could lead to an improvement of courses as it would be based on facilitated self-reflection for the teacher or, even better, if made in collaboration with students. Tasks involving annotation as assignment were not observed either, though it is clear that the possibility of critical exploration where students must find evidences to support their thinking could be used within the MOOC context, even so with the use of external annotation tools and peer evaluation (a well developed practice already used regarding text assignments).

SOME CHALLENGES

It appears that video material and its use in MOOCs are massive nowadays; nevertheless a more advanced use of video enrichments and more specifically of video annotations is still not a reality. From the classes of scenarios we described earlier and the overview of the current situation of e-learning and MOOC platforms, we would like to put forward a number of challenges that we think should be addressed in future versions of e-learning systems, linked with annotation issues.

Manual production of annotations. Manual annotation processes raise specific ergonomic and usability issues, all the more in the video domain where playing the document may interfere with annotation entry. The variety of targeted devices like mobile phones exacerbates these issues. Moreover, as mentioned above, annotations are also an ideal vehicle for collaboration activities around videos, in a syn-

Visualization Multilanguage subtitles X 1,2 X X 1 X X X X Transcription X X X X X X Other synchronized enrichments (e.g. slides)
X 3 List of annotations (navigable) X X X X Timeline with annotations X X Interactive enrichments on the video or aside (e.g. embedded questions, alternative endings, hypertext links to external content) X 4

X 4 X 5

Editing/sharing features Comment (about the whole video) X X X X X X Video markers (single timecode + comment) X 6 X 6 X 6 X X X X X Internal annotation tools (natively on the platform) X X X X X X External annotation tools 7 (third-party tools) X X 8 X X Exportation of temporalized data X X Internal annotations sharing X X External annotations sharing 9 X X Notes: 1. Usually generated automatically with the possibility of correction by the students. 2. Translations into other languages are carried out in collaboration with students (crowd sourced translation). 3. Synchronized slides. 4. Use of multiple choice embedded question. 5. Video Questions is a new feature available in beta version. There is also a possibility to choose the ending of a video. 6. Navigation of the video through transcriptions. 7. Usually VideoNot.es, those are the platforms featured on its website. 8. The use of the tool is promoted on the Coursera wiki page. 9. In most cases, videos on YouTube can be used by the external tools.

chronous [START_REF] Nathan | CollaboraTV: making television viewing social again[END_REF] or asynchronous way. This brings some specifics issues, notably around the ergonomics of video co-annotation; as well as privacy (e.g. the level of shared information must be clearly displayed and tunable by users).

Semi-automatic generation of annotations. Most video-based e-learning systems use only plain videos, sometimes fragmented into small independent videos, providing only basic features. In order to make these videos more accessible, e-learning platforms should commonly provide features such as transcription or chaptering. Some projects such as TransLectures aim at providing automatic or semiautomatic transcription of video, so that users may use the transcription as entry points into the video, either for querying and finding specific fragments, or as a simple navigation means.

Rich media and hypervideos. Beyond the basic video layout (side to side, overlay) that can be used to display the video material, annotations can be used to enrich the video or to produce whole new documents, such as hypervideos [START_REF] Chambel | Hypervideo and cognition: Designing video-based hypermedia for individual learning and collaborative knowledge building[END_REF] that are documents combining video and assets originating from annotations. Challenges here pertain to ergonomics, document modelling and (semi-)automatic production, for instance through an annotation-guided summarization.

Video annotations related learning analytics. With MOOCs, learning analytics have become a major concern for all organisations, by necessity -on this kind of scale, it is important to take informed decisions -and by opportunity -we now have the technological and processing capacity to capture and analyse the huge amount of information generated by thousands of learners. The annotation process and the annotations themselves offer an additional source of information for learning analytics at a finer scale, that could qualify as micro-analytics. Given the importance of video resources, it is undoubtedly important to have precise feedback on its reception. This new source of information could be used for example in course re-engineering [START_REF] Sadallah | A framework for usage-based document reengineering[END_REF].

Annotation model and sharing. Numerous tools provide video annotation features, and many use custom data models for storing annotation information (Cinelab, Exmeralda). However, standardization efforts are underway to define more interoperable and generic annotation models, able to encompass various annotation practices on different source documents and to integrate well with the current semantic web efforts (OpenAnnotation). Let us remark that some universities, mainly in the US, are strongly committed to pushing forward and generalizing annotation practices among students and faculty members, building annotation ecosystems: Columbia (Bossewitch and Preston, 2011), Stanford [START_REF] Pea | The diver project: interactive digital video repurposing[END_REF] and Harvard.

All these challenges share common concerns. First, mobile phones and tablets have become important platforms for consulting various resources, and among them, pedagogical resources. It is important to propose the most complete experience on annotationenhanced e-learning platforms on all devices, and especially on mobile ones, which have important constraints in terms of display size and general capacity. Second, copyright and licensing issues are even more stringent, since they concern not only the video document (which has to be shareable to allow collaborative work), but also the produced annotations. Clear licenses for this additional data should be specified, hopefully with a bias towards openness and reuse. Eventually, the question of accessibility -mainly for sensory deficiencies -has to be considered as video annotations are clearly a means to provide a better level of accessibility to video content [START_REF] Encelle | Annotation-based video enrichment for blind people: A pilot study on the use of earcons and speech synthesis[END_REF].

We have proposed four classes of scenarios illustrating how video annotations can be used in elearning contexts. To evaluate in what measure these scenarios are feasible or already present, we have reviewed a number of e-learning platforms (focusing on MOOCs) and tools, in order to identify existing annotation features. It appears that if some support already exists, there is still plenty of room to efficiently implement the scenarios that go beyond simple active reading, and a number of challenges related to video annotation still remain. These challenges should be addressed in future versions of e-learning systems, and we will tackle some of them in our future work on the COCo platform4 .

Figure 1 :

 1 Figure 1: Video annotations anchors and content

Figure 2 :

 2 Figure2: On the left, the VideoAnt video online annotation system displays the video with some annotations organised in a list. On the right, the Advene video annotation tool features several annotation display and creation interfaces: here a timeline at the bottom and a temporalized note-taking view on the right of the video.

Table 1 :

 1 Annotation-related functionalities offered by mainstream MOOC platforms or dedicated tools

	EdX	Coursera	Canvas Network	Khan Academy	Iversity	Open2Study	VideoANT	VideoNot.es	Annotated HTML	Mediathread	YouTube	Matterhorn Player

This work has received a French government support granted to the COMIN Labs excellence laboratory and managed by the National Research Agency in the "Investing for the Futures" program ANR-JO-LABX-07-0J.

Underlined terms have an associated URL given in the Webography annex at the end of the article.

The authors of the paper are involved in the COCo project (Cominlabs Open Courseware) based in University of Nantes, which is a recent initiative of the Cominlabs laboratory. The project goals are to build and animate a research platform for both disseminating and promoting rich media open courseware content.