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arXiv:0705.0721v2 [physics.chem-ph] 29 Jun 2007
Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and
system averaged pair density
Julien Toulouse∗
Cornell Theory Center, Cornell University, Ithaca, New York 14853, USA.
Roland Assaraf†
Laboratoire de Chimie Th´ eorique, Universit´ e Pierre et Marie Curie and
Centre National de la Recherche Scientifique, 75005 Paris, France.
C. J. Umrigar‡
Cornell Theory Center and Laboratory of Atomic and Solid State Physics,
Cornell University, Ithaca, New York 14853, USA.
(Dated: February 1, 2008)
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 delta-
function 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 impor-
tant place in electronic structure theory between the sim-
plicity of one-electron densities and the complexity of the
many-electron wave function. In particular, the system-
averaged 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. 1–31). Density func-
tional theory-like approaches have been proposed based
on the position intracule density [32–36] or on the closely-
related Wigner intracule density [37–39].
Position intracule densities have been extracted from
experimental X-ray scattering intensities for small atoms
and molecules [40–42].They have been calculated in
the Hartree-Fock (HF) approximation for systems rang-
ing from small atoms to large molecules [7, 12, 16,
19, 21, 27, 43–46]. Calculations using common quan-
tum chemistry correlated methods such as second-order
Møller-Plesset perturbation theory, multi-configurational
self-consistent-field (MCSCF) and configuration interac-
∗Electronic address: toulouse@tc.cornell.edu
†Electronic address: assaraf@lct.jussieu.fr
‡Electronic address: cyrus@tc.cornell.edu
tion approaches have been limited to atoms and small
molecules [6, 11, 14, 18, 24, 28, 47, 48]. Very accurate cal-
culations using Hylleraas-type explicitly-correlated wave
functions have been done only for the helium and lithium
isoelectronic series [1, 5, 8, 15, 17, 49–53]. Variational
Monte Carlo (VMC) calculations using Jastrow-Slater
wave functions have been used to compute correlated po-
sition 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 calcula-
tion 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 =
(r1,r2,...,rN) is the 3N−dimensional vector of electron
coordinates (ignoring spin for now), is defined as the
quantum-mechanical average of the delta-function oper-
ator δ(rij− u)
I(u) =1
2
?
i?=j
?
dRΨ(R)2δ(rij− u), (1)
where rij= rj−riand the sum is over all electron pairs,
and its spherical average is
I(u) =
?
dΩu
4π
I(u). (2)
Among the most important properties of I(u) are the
normalization sum rule (giving the total number of elec-
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2
tron pairs)
?∞
0
du4πu2I(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–
59). The moments of the radial intracule density
Mk=?∞
Coulomb interaction energy
0du4πu2+kI(u) are related to physical observ-
ables [60], in particular M−1is just the electron-electron
Wee= M−1=
?∞
0
du4πuI(u). (6)
In standard correlated methods based on an expansion
of the wave function in Slater determinants, the impor-
tant short-range part of the position intracule density
converges very slowly with the one-electron and many-
electron basis. The advantage of employing QMC meth-
ods [61] lies in the possibility of using compact, explicitly-
correlated wave functions which are able to describe
properly the short-range part of I(u). However, the prob-
lem 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 num-
ber 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 sta-
tistical average
Ihisto(u) =?Ihisto
L
(u,R)?
(u,R)
Ψ2, (7)
over the trial wave function density Ψ(R)2, of the local
“histogram” estimator Ihisto
L
Ihisto
L
(u,R) =1
2
?
i?=j
1[u−∆u/2,u+∆u/2[(rij)
4πu2∆u
, (8)
where 1[a,b[(r) = 1 if r ∈ [a,b[ and 0 otherwise. Here
and in the following, ?f(R)?Ψ2 = (1/M)?M
f(R) over M configurations Rk 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 vari-
ance [22, 62, 63], the calculation of position intracule
k=1f(Rk)
designates the statistical average of the local quantity
densities in QMC remains very inefficient: very long runs
have to be performed to reach an acceptably small sta-
tistical uncertainty. Moreover, the histogram estimator
has a discretization error: even in the limit of an infinite
sample M → ∞, Ihisto(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 pos-
sible to greatly reduce the discretization error by choos-
ing 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 [64]
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 [65–67], which has been
recently applied to calculations of one-electron densi-
ties [68]. These new ZVZB estimators of I(u) have vari-
ances 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 dis-
cretization 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 meth-
ods [52, 69–83], 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 estima-
tors previously proposed for computations of averages of
probability densities at particle coalescences [62, 84–86].
Moreover, we make use in this work of the recently-
developed linear energy minimization method [87, 88] to
finely optimize the Jastrow parameters, the configuration
state functions (CSFs) coefficients and the orbital coef-
ficients of our trial Jastrow-Slater wave functions. Op-
timization of the determinantal part of the wave func-
tion with an increasing number of CSFs allows us to ob-
tain a systematic improvement of wave functions. This
provides a practical route for calculating intracule densi-
ties 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 estima-
tors for an arbitrary observable in QMC. In Sec. III, we
give improved estimators for the case of the position in-
tracule density. Sec. IV contains computational details
of the calculations, and Sec. V discusses results for the
He and C atoms and for the C2and N2molecules to il-
lustrate the technique. Sec. VI contains our conclusions.
Hartree atomic units are used throughout this work.
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II. ZERO-VARIANCE ZERO-BIAS IMPROVED
ESTIMATORS
We review here the principle of the zero-variance zero-
bias improved QMC estimators for an arbitrary observ-
able that does not commute with the Hamiltonian, de-
veloped in Refs. 65–67. Throughout this section, all the
averages ?···?Ψ2 are considered in the limit of an infinite
Monte Carlo (MC) sample M → ∞.
A.Estimators in variational Monte Carlo
In VMC, the exact energy E0 = ?Ψ0|ˆH|Ψ0?/?Ψ0|Ψ0?
of some exact eigenfunction Ψ0of the electronic Hamil-
tonianˆH is estimated by the statistical average of the
local energy EL(R) = ?R|ˆH|Ψ?/Ψ(R)
E = ?EL(R)?Ψ2,
using an approximate trial wave function Ψ(R).
systematic error (or bias) of this estimator δE = E −
E0and its variance σ2(EL) = ?(EL(R) − E)2?Ψ2, 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 − Ψ0| (where |···|
designates the Hilbert space norm)
(9)
The
δE = O(|δΨ|2), (10a)
σ2(EL) = O(|δΨ|2), (10b)
which is referred to as the quadratic zero-variance zero-
bias property of the local energy. This is easily shown by
writing Ψ = Ψ0+ 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 O0= ?Ψ0|ˆO|Ψ0?/?Ψ0|Ψ0? can be estimated by
the statistical average of the local observable OL(R) =
?R|ˆO|Ψ?/Ψ(R)
O = ?OL(R)?Ψ2,
but, as Ψ0 is generally not an eigenstate ofˆO, the sys-
tematic error of this estimator δO = O − O0 vanishes
only linearly with |δΨ|, while its variance σ2(OL) =
?(OL(R) − O)2?Ψ2 generally does not even vanish with
|δΨ|
δO = O(|δΨ|),
(11)
(12a)
σ2(OL) = O(1), (12b)
which often makes the calculations of observables inac-
curate 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
ˆHλ=ˆH + λˆO,(13)
with some associated exact λ-dependent eigenfunction
Ψλ
0= Ψ0+ λΨ′
0+ ··· , (14)
where Ψ′
dependent energy Eλ
exact value of the observable is given by the deriva-
tive of the energy with respect to λ at λ = 0: O0 =
?dEλ
Ψλ= Ψ + λΨ′+ ··· ,
where Ψ′=
?dΨλ/dλ?
local energy Eλ
0=
?dΨλ
0/dλ?
λ=0and corresponding exact λ-
0= ?Ψλ
0|ˆHλ|Ψλ
0?/?Ψλ
0|Ψλ
0?, then the
0/dλ?
λ=0. Introducing an approximate λ-dependent
trial wave function
(15)
λ=0, the λ-dependent energy can
be estimated as the statistical average of the λ-dependent
L(R) = ?R|ˆHλ|Ψλ?/Ψλ(R)
Eλ=?Eλ
over the probability density Ψλ(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 Eλwith respect to λ at λ = 0
L(R)?
Ψλ2,(16)
OZVZB=
?OZVZB
?dEλ
= ?OL(R)?Ψ2 +?∆OZV
+?∆OZB
L
(R)?
?
Ψ2
=
dλ
λ=0
L(R)?
Ψ2
L(R)?
Ψ2,(17)
where the zero-variance (ZV) contribution
∆OZV
L(R) =
?
?R|ˆH|Ψ′?
Ψ′(R)
− EL(R)
?
Ψ′(R)
Ψ(R), (18)
does not contribute to the average,?∆OZV
the Hermiticity of the Hamiltonian) but can decrease the
variance, and the zero-bias (ZB) contribution
L(R)?
Ψ2= 0
(in the limit of an infinite MC sample M → ∞, from
∆OZB
L(R) = 2[EL(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 van-
ishes 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 + ǫΨ′. For the special
case of the calculation of the derivatives of the energy
with respect to the nuclear coordinates, the average of
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the ZB term is known in the electronic structure litera-
ture 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 OZV
tematic 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′
δOZV= O(|δΨ|),
L(R) = OL(R) + ∆OZV
L(R) has a sys-
0|
(20a)
σ2?OZV
L
?= O(|δΨ|2+ |δΨ′|2+ |δΨ||δΨ′|).
If one also includes the ZB correction, then both the sys-
tematic error and the variance of the ZVZB estimator
OZVZB
L
(R) vanish quadratically with |δΨ| and |δΨ′|
δOZVZB= O(|δΨ|2+ |δΨ||δΨ′|),
(20b)
(21a)
σ2?OZVZB
This is easily shown by writing Ψ = Ψ0 + δΨ and
Ψ′= Ψ′
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 ex-
act wave function Ψ0and its derivative Ψ′
perturbation theory: (ˆH − E0)|Ψ′
Thus, the ZVZB estimators permit in principle accu-
rate and efficient calculations for observables, provided
that a good approximation for the derivative Ψ′
able. 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 system-
atic error to the quality of the approximate derivative Ψ′,
since in this case Ψ′only needs to be a good approxima-
tion of Ψ′
spanned by the wave function derivatives with respect
to the parameters at the optimal parameter values popt,
i.e. Ψi= (∂Ψ(p)/∂pi)p=popt. Indeed, the contribution to
the average of the ZB term of any component of Ψ′along
Ψiis proportional to the energy gradient with respect to
parameter piand thus vanishes at the stationary point:
?2[EL(R) − E]Ψi(R)/Ψ(R)?Ψ2 = (∂E/∂pi)p=popt= 0.
This point is at the origin of the improved accuracy in
the calculations of forces in QMC in Refs. 90, 91.
L
?= O(|δΨ|2+ |δΨ′|2+ |δΨ||δΨ′|). (21b)
0+ δΨ′in the expressions of the average and the
0in first-order
0? = −(ˆO − O0)|Ψ0?.
0is avail-
0in the orthogonal complement of the space
B.Estimators in diffusion Monte Carlo
DMC calculations within the FN approximation go
beyond VMC calculations by replacing the VMC dis-
tribution Ψ(R)2by the more accurate mixed FN-DMC
distribution ΨFN(R)Ψ(R) in statistical averages, i.e.
?···?Ψ2 → ?···?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 |δΨFN| = |ΨFN− Ψ0|
δE = O(|δΨ||δΨFN|).
For an arbitrary observable, the systematic error still
vanishes only linearly with |δΨ| or |δΨFN|
δO = O(|δΨ| + |δΨFN|),
but the use of the hybrid estimator “2DMC − VMC”,
Ohybrid = 2?OL(R)?ΨFNΨ− ?OL(R)?Ψ2, allows one to
remove the dominant linear term |δΨ|
δOhybrid= O(|δΨFN| + |δΨ|2+ |δΨFN|2+ |δΨ||δΨFN|).
(22)
(23)
(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
δOZVZB= O(|δΨ|2+ |δΨ||δΨ′| + |δΨ||δΨFN|
+|δΨ′||δΨFN|),
with no linear term |δΨFN| 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,
OZVZB
L
(R)?
δOZVZB
(25)
hybrid= 2?OZVZB
hybrid= O(|δΨ|2+ |δΨ||δΨFN| + |δΨ′||δΨFN|),
ΨFNΨ−?OZVZB
L
(R)?
Ψ2, whose
systematic error also vanishes quadratically as
(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|ˆH|R′?
H(R)δ(R−R′) where H(R) = −(1/2)?N
=
k=1∇2
rk+V (R)
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contains kinetic and potential energy contributions, the
ZV term takes the form
∆OZV
L(R) = −1
2
N
?
k=1
?
∇2
rkQ(R) + 2vk(R) · ∇rkQ(R)
?
,
(27)
where vk(R) = ∇rkΨ(R)/Ψ(R) is the drift velocity of
the trial wave function. The ZB term is simply
∆OZB
L(R) = 2∆EL(R)Q(R), (28)
where ∆EL(R) = EL(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 Hamil-
tonian, 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 ad-
ditional term is required for the variance to rigorously
vanish quadratically as in Eq. (20b), in practice the es-
timator 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 intrac-
ule density, the local observable IL(u,R) is
IL(u,R) =1
2
?
i?=j
?
dΩu
4π
δ(rij− 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 IL(u,R) is
Q1(u,R) = −1
8π
?
i?=j
?
dΩu
4π
1
|rij− u|, (30)
which gives the following ZV contribution
∆IZV1
L
(u,R) =
1
8π
?
i?=j
?
dΩu
4π
?
∇2
rij
1
|rij− u|
?
+2vi(R) · ∇rij
1
|rij− u|
. (31)
Using the identity ∇2
is seen that the first term in ∆IZV1
cels the delta function in IL(u,R), and the ZV estimator
IZV1
L
(u,R) = IL(u,R) + ∆IZV1
L
rij1/|rij− u| = −4πδ(rij− u), it
L
(u,R) exactly can-
(u,R) is simply
IZV1
L
(u,R) =
1
4π
?
i?=j
vi(R) ·
?
dΩu
4π
∇rij
1
|rij− u|
= −1
4π
?
i?=j
vi(R) ·rij
r3
ij
1[u,+∞[(rij). (32)
The form of Q1(u,R) of Eq. (30) could also have been
guessed from the behavior at small rij of the first-order
solution Ψ′(R) of the perturbation theory with respect to
δ(rij−u) (see Refs. 72, 92). In contrast to the histogram
estimator of Eq. (8) where only electron pairs with rela-
tive distance rijin 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 rij ≥ u contribute
to I(u). For u ?= 0, the variance of this ZV estimator
is finite and much smaller than the variance of the his-
togram 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 ap-
plied in Ref. 22 for calculations of intracule densities at
the coalescence point.
The ZB correction corresponding to Q1(u,R) of
Eq. (30) is
∆IZB1
L
(u,R) = −∆EL(R)
4π
?
i?=j
?
+1[u,+∞[(rij)
dΩu
4π
1
|rij− u|
= −∆EL(R)
4π
?
i?=j
?1[0,u[(rij)
urij
?
. (33)
The
IZV1
L
tematic error due to the trial wave function as shown
in Sec. V. Note that, in contrast to the histogram esti-
mator, 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 rij encountered in the
Monte Carlo run. Also, because the form of Q1(u,R)
of Eq. (30) has been chosen only to cure the deficiency
of the histogram estimator at small u, the ZB correc-
tion 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 in-
tracule density using the ZVZB estimator IZV1ZB1
The second choice for Q(R) presented next remedies this
limitation.
resulting
(u,R) + ∆IZB1
ZVZB
(u,R) allows one to reduce the sys-
estimatorIZV1ZB1
L
(u,R)=
L
L
(u,R).
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B. Second improved estimator
A more general choice for Q(R) that cancels the infi-
nite variance of IL(u,R) is
Q2(u,R) = −1
8π
?
i?=j
?
dΩu
4π
g(rij,u)
|rij− u|, (34)
where g(rij,u) is some function satisfying g(rij,u) = 1
for rij= u. For example, using
g(rij,u) = e−ζ|rij−u|, (35)
with ζ > 0, has the advantage of ensuring a correct ex-
ponential decay of the estimator at infinity for a finite
system.
The corresponding ZV estimator is
IZV2
L
(u,R) =
1
8π
?
i?=j
?
dΩu
4π
?
2vi(R) · ∇rij
e−ζ|rij−u|
|rij− u|
+2∇rij
1
|rij− u|· ∇rije−ζ|rij−u|
|rij− u|∇2
= −1
4π
i?=j
+
1
rije−ζ|rij−u|?
vi(R) ·rij
?
?
r3
ij
?
rijA(rij,u)
+rij
uB(rij,u)
?
−ζ2
2A(rij,u)
?
, (36)
and its ZB correction is
∆IZB2
L
(u,R) = −∆EL(R)
4π
?
?
i?=j
?
dΩu
4π
e−ζ|rij−u|
|rij− u|
= −∆EL(R)
4π
i?=j
A(rij,u), (37)
where
A(rij,u) =e−ζ|rij−u|− e−ζ(rij+u)
2ζriju
, (38)
B(rij,u) =1
2
?
sgn(rij− u)e−ζ|rij−u|− e−ζ(rij+u)?
and where sgn is the sign function. These refined ZV esti-
mator IZV2
L
(u,R) and ZVZB estimator IZV2ZB2
IZV2
L
(u,R)+∆IZB2
L
(u,R) now both decay correctly expo-
nentially 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 com-
putational cost is larger per Monte Carlo step than that
of the simpler estimators of the previous subsection.
,(39)
L
(u,R) =
IV.COMPUTATIONAL DETAILS
We have chosen to illustrate the efficiency and ac-
curacy of the new improved QMC estimators by cal-
culating the intracule density of the He and C atoms
and the C2 and N2 molecules in their ground states.
The molecules are considered at their experimental ge-
ometries (dC−C = 2.3481 Bohr [93] and dN−N = 2.075
Bohr [94]).
We startbygenerating
wave function using the quantum chemistry program
GAMESS [95], 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 va-
lence 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 reop-
timized 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
C2and N2molecules, we use the CVB2 basis of the same
authors. In GAMESS, each Slater function is actually
approximated by a fit to 14 Gaussian functions [99–101]
This standard ab initio wave function is then multi-
plied by a Jastrow factor, imposing the electron-electron
cusp condition, but with essentially all other free pa-
rameters chosen to be zero to form our starting trial
Jastrow-Slater wave function, and QMC calculations are
performed with the program CHAMP [102] or QMC-
MOL [103] using the true Slater basis set rather than
its Gaussian expansion. The Jastrow parameters, the
orbital coefficients and the configuration state func-
tions (CSFs) coefficients of this trial wave function are
optimized in VMC using the very efficient, recently-
developed linear energy minimization method [87, 88]
and an accelerated Metropolis algorithm [104, 105]. Once
the trial wave function has been optimized, we com-
pute 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−1in an efficient DMC algorithm featur-
ing very small time-step errors [110].
a standard
ab initio
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 den-
sity I(u) as a function of the electron-electron distance
u, calculated in VMC using the standard histogram esti-
mator of Eq. (8), the ZV1 improved estimator of Eq. (32)
and the ZV1ZB1 improved estimator of Eq. (33), employ-
ing only 100 000 MC configurations sampled from a HF
trial wave function (without a Jastrow factor). In this
Page 7
7
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.5 1 1.5 2
I(u) (a.u.)
u (a.u.)
histogram estimator with HF wave function
ZV1 estimator with HF wave function
ZV1ZB1 estimator with HF wave function
accurate intracule
He atom
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 his-
togram, ZV1 and ZV1ZB1 estimators using only 100 000 MC
configurations sampled from a HF trial wave function (with-
out a Jastrow factor) are compared. For the histogram esti-
mator, a grid step of ∆u = 0.005 has been used. The intrac-
ule density obtained with an accurate, explicitly-correlated
Hylleraas-type wave function is also shown.
subsection, we intentionally use a poor trial wave func-
tion to demonstrate the large reduction in the bias re-
sulting from the ZVZB estimator. The calculations take
a few seconds on a present-day single-processor personal
computer. The intracule density obtained with an accu-
rate, 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 mag-
nitude 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 intrac-
ule 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 vari-
ance 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. IIIB, the relative statis-
tical 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, spec-
tacularly reduces the statistical uncertainty, its variance
being smaller by about 3 orders of magnitude at u = 3.5.
-0.0002
-0.0001
0
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
3 3.2 3.4 3.6 3.8 4
I(u) (a.u.)
u (a.u.)
ZV1ZB1 estimator with HF wave function
ZV2ZB2 estimator with HF wave function
accurate intracule
He atom
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.
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 intrac-
ule densities of the elements of the He isoelectronic series
from Z = 2 (He) to Z = 20 (Ar18+) using Jastrow-Slater
wave functions. We found that, in the relevant range of
u (u ? 5), the intracule density roughly obeys the scal-
ing law I(u/Z) ∼ Znwith 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 rela-
tive 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 an-
other advantage of our improved estimators: because the
estimator of I(u) is strongly correlated with the estima-
tor 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 uncer-
tainty. This makes the calculated intracule densities di-
rectly 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 in-
tegrand of the electron-electron Coulomb interaction en-
ergy over the electron-electron distance u, i.e. 4πuI(u)
[see Eq. (6)]. The ZVZB2 estimator is used with an ac-
Page 8
8
0
2
4
6
8
10
0 1 2 3 4 5
4 π u I (u) (a.u.)
u (a.u.)
total intracule
same spin contribution
opposite spin contribution
C atom
FIG. 3: Integrand of the electron-electron Coulomb inter-
action energy 4πuI(u) over the electron-electron distance u
for the C atom, calculated in VMC using the ZVZB2 (with
ζ = 2√2I ≈ 1.82 [114]) improved estimator and a fully opti-
mized Jastrow-Slater CAS(4,4) trial wave function. The same
spin contribution 4πuISS(u) and the opposite spin contribu-
tion 4πuIOS(u) are also shown.
curate Jastrow-Slater CAS(4,4) wave function in which
the Jastrow, CSF and orbital parameters have been si-
multaneously optimized. Note however that, at the scale
of the plot, the HF intracule 4πuIHF(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 dis-
tance (u ≈ 1.0) corresponds essentially to the K-L and
L-L electron pairs. Fig. 3 also shows the same spin (SS)
contribution
ISS(u) = I↑↑(u) + I↓↓(u), (40)
and opposite spin (OS) contribution
IOS(u) = I↑↓(u) + I↓↑(u), (41)
where Iσσ′(u) is the spherical average of the spin-resolved
intracule densities
Iσσ′(u) =1
2
?
i?=j
?
dRΨ(R)2δ(rij− u)δsi,σδsj,σ′, (42)
where the spin coordinates of the N = N↑+ N↓ elec-
trons in the spin-assigned wave function Ψ(R) are fixed
at si =↑ for i = 1,...,N↑ and si =↓ for i = N↑+
1,...,N↑+ N↓.
at short distance is dominated by the OS contribution,
while at long distance the SS and OS contributions be-
come nearly identical.
Figure 4 shows the correlation part of the radial in-
tracule density 4πu2[I(u) − IHF(u)], also called corre-
lation hole, using the ZVZB2 improved estimator and
the same Jastrow-Slater CAS(4,4) trial wave function.
Correlation decreases the radial probability density at
Not surprisingly, the intracule density
-0.15
-0.1
-0.05
0
0.05
0.1
0 1 2 3 4 5
4 π u2 [ I (u) - IHF (u) ] (a.u.)
u (a.u.)
total correlation intracule
same spin contribution
opposite spin contribution
C atom
FIG. 4: Correlation part of the radial position intracule den-
sity 4πu2[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
4πu2ˆISS(u) − ISS
4πu2ˆIOS(u) − IOS
HF(u)˜
HF(u)˜are also shown.
and the opposite spin contribution
-0.15
-0.1
-0.05
0
0.05
0.1
0 1 2 3 4 5
4 π u2 [ I (u) - IHF (u) ] (a.u.)
u (a.u.)
HF
Jastrow × HF
Jastrow × SD
Jastrow × CAS(4,4)
C atom
VMC with ZVZB2 estimator
FIG. 5: Correlation part of the radial position intracule den-
sity 4πu2[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]) im-
proved estimator for a series of trial wave functions of increas-
ing 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
4πu2?ISS(u) − ISS
The OS component constitutes the main contribution to
the correlation hole.
The effect of the chosen trial wave function on the ac-
curacy on the correlation hole is examined in Fig. 5. Four
HF(u)?and the opposite spin contribu-
tion 4πu2?IOS(u) − IOS
HF(u)?
are also shown in Fig. 4.
Page 9
9
0
5
10
15
20
0 1 2 3 4 5
4 π u I (u) (a.u.)
u (a.u.)
total intracule
same spin contribution
opposite spin contribution
C2 molecule
FIG. 6: Integrand of the electron-electron Coulomb interac-
tion energy 4πuI(u) versus the electron-electron distance u
for the C2 molecule, calculated in VMC using the ZVZB2
(with ζ = 2√2I ≈ 1.83 [114]) improved estimator and a fully
optimized Jastrow-Slater CAS(8,8) trial wave function. The
same spin contribution 4πuISS(u) and the opposite spin con-
tribution 4πuIOS(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 quantita-
tive 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 in-
tracules agree closely.
C.C2 molecule
We now discuss the more difficult case of the C2
molecule. The ground-state wave function of this system
has a strong multi-configurational character due to ener-
getic near-degeneraciesamong the valence orbitals (“non-
dynamical correlation” in chemists’ jargon, or “strong
correlation” in physicists’ jargon), making it a challeng-
ing system despite its small size.
Figure 6 plots 4πuI(u) versus u using the ZVZB2 esti-
mator with an accurate Jastrow-Slater CAS(8,8) wave
function (Jastrow, CSF and orbital parameters opti-
mized). 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-
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 1 2 3 4 5
4 π u2 [ I (u) - IHF (u) ] (a.u.)
u (a.u.)
CAS(8,8)
Jastrow × CAS(8,8) [Jastrow optimized]
Jastrow × CAS(8,8) [Jastrow + CSFs optimized]
Jastrow × CAS(8,8) [Jastrow + CSFs + orbitals optimized]
C2 molecule
VMC with ZV2 estimator
FIG. 7: Correlation part of the radial position intracule den-
sity 4πu2[I(u) − IHF(u)] as a function of the electron-electron
distance u for the C2 molecule, where I(u) has been cal-
culated in VMC using the ZV2 improved estimator (with
ζ = 2√2I ≈ 1.83 [114]) with a MCSCF CAS(8,8) wave func-
tion 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 in-
teratomic 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 C2molecule
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 intra-
atomic 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 electron-
electron distances, but in fact importantly modifies the
correlation hole up to distances u ≈ 4. The Jastrow fac-
tor has also an indirect action via reoptimization of the
CSF and orbital coefficients in its presence. The reop-
timization 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 in-
tracule 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
Page 10
10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 1 2 3 4 5
4 π u2 [ I (u) - IHF (u) ] (a.u.)
u (a.u.)
Jastrow × HF
Jastrow × SD
Jastrow × CAS(8,5)
Jastrow × CAS(8,7)
Jastrow × CAS(8,8)
C2 molecule
a) VMC with ZVZB2 estimator
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 1 2 3 4 5
4 π u2 [ I (u) - IHF (u) ] (a.u.)
u (a.u.)
Jastrow × HF
Jastrow × SD
Jastrow × CAS(8,5)
Jastrow × CAS(8,7)
Jastrow × CAS(8,8)
C2 molecule
b) DMC with ZVZB2 estimator
FIG. 8: Correlation part of the radial position intracule density 4πu2[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 [114]) 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 in-
creasing accuracy: Jastrow × HF (with only the Jas-
trow parameters optimized), Jastrow × SD (with the
Jastrow and orbital parameters optimized), and multi-
determinant Jastrow × CAS(8,5), Jastrow × CAS(8,7)
and Jastrow × CAS(8,8) (with the Jastrow, CSF and or-
bital parameters optimized). At core distances (u ≈ 0.2),
the correlation hole is essentially converged with only
a single-determinant Jastrow-Slater wave function, pro-
vided that the orbitals are reoptimized together with the
Jastrow factor. In contrast, at longer valence distances,
the correlation hole depends very strongly on the deter-
minantal 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 pseudopo-
tential 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 rea-
sonable convergence. We have checked that further exci-
tations 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 correspond-
ing VMC intracules, the accuracy of the intracule densi-
ties in the valence region being still essentially controlled
by the trial wave function.
-0.4
-0.2
0
0.2
0.4
0.6
0 1 2 3 4 5
4 π u2 [ I (u) - IHF (u) ] (a.u.)
u (a.u.)
Jastrow × HF
Jastrow × SD
Jastrow × CAS(10,8)
N2 molecule
VMC with ZVZB2 estimator
FIG. 9: Correlation part of the radial position intracule den-
sity 4πu2[I(u) − IHF(u)] as a function of the electron-electron
distance u for the N2 molecule, where I(u) has been calcu-
lated in VMC using the ZVZB2 (with ζ = 2√2I ≈ 2.14 [114])
improved estimator for a series of trial wave functions of in-
creasing 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 intrac-
ule densities with the N2 molecule, which has recently
been the object of experimental and theoretical investi-
gations [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 C2 molecule, no depletion of intracule
Page 11
11
density is observed at the bond length. Also, the correla-
tion hole is here essentially converged with only a single-
determinant Jastrow-Slater wave function provided that
the orbitals are optimized with the Jastrowfactor. Unlike
the C2molecule, going to a multi-determinant wave func-
tion 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 config-
uration 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), con-
structed using the general zero-variance zero-bias princi-
ple for observables that do not commute with the Hamil-
tonian.By replacing the average of the local delta-
function operator by the average of a smooth non-local
operator, these estimators decrease the variance of the
standard histogram estimator by several orders of mag-
nitude, and thus make the calculation of this quantity
in QMC vastly more efficient. Interestingly, they per-
mit 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 er-
ror with respect to u and the possibility to obtain very
smooth curves for I(u). These improved estimators, to-
gether 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 triv-
ial adaptations for QMC calculations of the companion
entity of I(u), namely the spherically averaged extracule
density E(r), representing the probability density of find-
ing 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 n2(r1,r2). More generally, the variance
reduction technique presented here can be applied to cal-
culations of any pair-correlation function in classical and
quantum Monte Carlo calculations.
Acknowledgments
We would like to thank Andreas Savin and Paola Gori-
Giorgi for stimulating discussions. We also thank Alexan-
der Kollias for providing us the Gaussian fits of Slater
basis functions of Ref. 101. J. T. acknowledges finan-
cial support from a Marie Curie Outgoing International
Fellowship (039750-QMC-DFT). This work was also sup-
ported by the Centre National de la Recherche Scien-
tifique 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).
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