Quantum Monte Carlo with reoptimized perturbatively selected configuration-interaction wave functions

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by  Emmanuel Giner

Quantum Monte Carlo with reoptimized perturbatively selected
configuration-interaction wave functions
Emmanuel Giner1, Roland Assaraf2, and Julien Toulouse2
1Dipartimento di Scienze Chimiche e Farmaceutiche,
Universita di Ferrara, Via Fossato di Mortara 17, I-44121 Ferrara, Italy
2Sorbonne Universit´es, UPMC Univ Paris 06, CNRS,
Laboratoire de Chimie Th´eorique, 4 place Jussieu, F-75005, Paris, France
(Dated: January 14, 2016)
6
1 WeexploretheuseinquantumMonteCarlo(QMC)oftrialwavefunctionsconsistingofaJastrow
0 factor multiplied by a truncated configuration-interaction (CI) expansion in Slater determinants
2 obtainedfromaCIperturbativelyselected iteratively(CIPSI)calculation. IntheCIPSIalgorithm,
theCIexpansionisiterativelyenlargedbyselectingthebestdeterminantsusingperturbationtheory,
n
whichprovidesanoptimalandautomaticwayofconstructingtruncatedCIexpansionsapproaching
a
the full CI limit. We perform a systematic study of variational Monte Carlo (VMC) and fixed-
J
node diffusion Monte Carlo (DMC) total energies of first-row atoms from B to Ne with different
2
levels of optimization of the parameters (Jastrow parameters, coefficients of the determinants, and
2
orbital parameters) in these trial wave functions. The results show that the reoptimization of the

coefficients of the determinants in VMC (together with the Jastrow factor) leads to an important
]
h loweringofbothVMCandDMCtotalenergies,andtotheirmonotonicconvergencewiththenumber
p of determinants. In addition, we show that the reoptimization of the orbitals is also important in
- both VMC and DMC for the Be atom when using a large basis set. These reoptimized Jastrow-
m
CIPSIwavefunctionsappearaspromising,systematicallyimprovabletrialwavefunctionsforQMC
e calculations.
h
c
. I. INTRODUCTION In recent years, a lot of effort has been devoted to
s
c developing efficient methods for optimizing a large num-
i berofparametersinQMCtrialwavefunctions (see,e.g.,
s
Thedevelopmentofaccurateelectronic-structurecom-
y Refs. 6–14). One of the most effective approaches is the
putationalmethods remainsa topicalresearchsubjectof
h linearoptimizationmethodofRefs.11–13. Thisisanex-
p high interest. Currently, the two most popular families tension of the zero-variancegeneralizedeigenvalue equa-
[ of electronic-structure computational methods in quan-
tion approach of Nightingale and Melik-Alaverdian [15]
tum chemistry are density-functional theory (see, e.g.,
1 to arbitrary nonlinear parameters, and it permits a
Ref. 1) and post-Hartree-Fock wave-functionapproaches
v very efficient and robust energy minimization in a VMC
5 (see, e.g., Ref. 2). Quantum Monte Carlo (QMC) meth- framework. The availability of such optimization meth-
1 ods(see,e.g,Refs.3–5)arealternativeapproaches,which
ods have recently lead to the exploration of various
9 become more and more attractive thanks to recent the-
forms of trial wave functions: Jastrow-antisymmetrized-
5 oretical developments and to the important growing of
geminal-power wave functions [16–18], Jastrow-pfaffians
0
available computational resources. Indeed, QMC meth-
. wave functions [19, 20], Jastrow-backflow wave func-
1 ods need little memory and are embarrassingly parallel
tions [21], orbital-attached multi-Jastrow wave func-
0 whichmakethemideallysuitedtomodernmassivelypar-
tions [22], and various types of Jastrow-valence-bond
6 allel supercomputers.
wave functions [23–25].
1
: The two most commonly used variants of QMC meth- For atoms and molecules, the most used and sys-
v
ods are variational Monte Carlo (VMC) and fixed-node tematically improvable form of QMC trial wave func-
i
X diffusion Monte Carlo (DMC). The VMC method uses tions remains a Jastrowfactormultiplied by a truncated
r a flexible trial wave function, usually including an ex- configuration-interaction(CI) expansion in Slater deter-
a plicit correlation factor called the Jastrow factor, and minants(see,e.g.,Refs.11–14,26–32). However,amajor
appliesMonteCarlonumericalintegrationtechniquesfor problemishowtosystematicallyselectthebestdetermi-
calculating the multidimensional integrals of quantum nants entering the truncated CI expansion. StandardCI
mechanics. The DMC methods goes beyond VMC by calculationsareusuallytruncatedbasedonorbitalactive
extracting the projection of the trial wave function on spaceand/orexcitationcriteria,whicharefarfrombeing
the exact ground-state wave function of the system. In optimal for rapidly approaching the full CI (FCI) limit.
practice, except for a few very simple systems, it is nec- Very recently, Giner et al. [33–36] have proposed to
essary to impose the fixed-node approximation in DMC use in QMC truncated CI expansions constructed from
and one only obtains the energy of the best variational the CI perturbatively selected iteratively (CIPSI) algo-
wave function having the same nodes as the trial wave rithm[37]. Inthis approach,the truncatedCIexpansion
function. The construction of optimal trial wave func- is iteratively enlarged by selecting the most important
tions is thus crucial for accurate results in both VMC missing determinants using perturbation theory. The
and DMC. advantage of such an approach resides in the fact that
2
the selecteddeterminants are by constructionthose hav- and the coefficients c are obtained by minimization of
I
ing the largest impact on electronic correlation, both of the variational energy E(0),
static and dynamic nature, in the specific system under
consideration. Thisalgorithmthusallowsonetosystem- E(0) =minhΨCIPSI|Hˆ|ΨCISPIi, (2)
atically and rapidly approach the FCI wave function in {cI} hΨCIPSI|ΨCIPSIi
an automatic way, and obtain truncated CI expansions
with limited numbers of determinants that are quanti- where Hˆ is the many-body Hamiltonian operator. One
tative approximations to the FCI wave function. Giner then considers the determinants that do not belong to
et al. used these CIPSI wave functions in DMC calcu- the S space. For each such determinant |µi, its first-
lations, without any Jastrow factor and without reopti- order coefficient using the Epstein-Nesbet zeroth-order
mizing any parameters in QMC, and showed that, with Hamiltonian [47, 48] is given by
enough determinants, accurate results can be obtained
hµ|Hˆ|Ψ i
with these trialwave functions for atomic and molecular c(1) = CIPSI
systems. µ ECIPSI−hµ|Hˆ|µi
(3)
The present work continues this line of research by hµ|Hˆ|Ii
studyingtheeffectofaddingasophisticatedJastrowfac- =XcI E −hµ|Hˆ|µi,
tortotheseCIPSIwavefunctionsandoptimizingthepa- I∈S CIPSI
rameters (Jastrow parameters, coefficients of the deter-
and then the contribution of |µi at second order to the
minants, and orbital parameters) in VMC using the lin-
energy is
earmethod. In particular,we wantto know whether the
reoptimization in VMC of the coefficients of the deter- e(2) =c(1)hΨ |Hˆ|µi
µ µ CIPSI
minants and/orthe orbitalsin these trialwavefunctions
|hµ|Hˆ|Ψ i|2 (4)
in the presence of the Jastrow factor brings important = CIPSI .
improvements in VMC and DMC. The paper is orga- E −hµ|Hˆ|µi
CIPSI
nizedasfollows. SectionII describesthe methodologyof
At each CIPSI iteration, the total second-order correc-
theCIPSIalgorithm,theparametrizationoftheJastrow-
tion to the energy E(2) is defined as the sum of all the
CIPSI wave functions used in QMC, and gives compu-
energycontributionse(2) overall|µinotbelongingtothe
tational details on the calculations performed. Section µ
S space,
III gives and discusses the numerical results obtained on
first-row atoms from Be to Ne. Finally, we give our con-
clusions in Section IV. E(2) = Xe(µ2), (5)
µ∈/S
andtheCIPSIenergyis definedasE =E(0)+E(2),
CIPSI
II. METHODOLOGY
which is an estimation of the FCI energy. In the ver-
sion of CIPSI used here, the determinant |µi is added to
A. CIPSI wave functions theS spaceifitssecond-ordercontributiontotheenergy
e(2) is larger than a certain threshold η set at the begin-
µ
The CIPSI method is a selected CI algorithm where ning of the iteration. When the selection procedure has
the determinants are chosen according to a perturbative been done, one obtains a new CIPSI wave function con-
estimationoftheirimportance,whichallowsonetobuild structed with a new set determinants belonging to the
CI expansionskeepingonlythe mostimportantdetermi- new S space (i.e., the determinants present at the be-
nants. As this idea is somewhat intuitive, various se- ginning of the iterationtogether with the ones that have
lected CI schemes guided by perturbation theory have just been added). This new CIPSI wave function serves
been proposed [37–42], but the CIPSI version is one of asthereferencewavefunctionforthenextiteration. The
the best algorithms in this field, as it introduces an iter- selectionthresholdηisloweredateachiteration,andthis
ative procedure and various stopping criteria to control process is repeated until one reaches a given criterion of
the quality of the wave functions and energies. The con- convergence. The criterion used here is a given thresh-
vergence of the CIPSI wave functions and energies has old on the remaining second-order contribution to the
been intensively studied in the past decades [37, 43–46] energy E(2) [36], but one can find in the literature other
and a revival of such ideas has appeared in the DMC stopping criteria based on an estimation of the norm of
framework in the last few years [33, 34, 36]. We now the first-order wave function [45] or simply based on the
summarize the CIPSI algorithm used in this work. number of determinants belonging to the S space. It
At the beginning of a givenCIPSI iteration, one has a should be noted that thanks to perturbation theory and
reference CIPSI wavefunction |Ψ i built with Slater to its iterative nature, the CIPSI algorithm allows one
CIPSI
determinants |Ii that span a space S, to select determinants in the whole FCI space, whereas
standardtruncatedCImethodsrestrictthedeterminants
|ΨCIPSIi=XcI|Ii, (1) to a given pattern of excitations with respect to a refer-
ence space and/or an orbital space.
I∈S
3
B. Jastrow-CIPSI wave functions the Slater determinants are obtained froma large CIPSI
calculationwheretheremainingsecond-orderenergycor-
After the CIPSI calculation is done, we construct a rection |E(2)| [see Eq. (5)] is smaller than 10−4 hartree.
Jastrow-CIPSI wave function parametrized as In practice, for the VB1 basis set, the number of de-
terminants in the large CIPSI calculations ranges from
|ΨJCIPSI(p)i=Jˆ(α)eκˆ(κ) XcI|Ii, (6) 19 for the Be atom to approximatively 104 for the Ne
atom. Then, the determinants are sorted according to
I∈S
the absolute values of their coefficients, and the wave
where Jˆ(α) is a Jastrow-factor operator depending on functionistruncatedatvariousnumbersofdeterminants
some parameters α, and eκˆ(κ) is an orbital rotation op- keeping the coefficients as they are in the large CIPSI
erator. wave function (i.e., at the near FCI level). We prefer
The orbital excitation operator is defined as κˆ(κ) = to use this procedure instead of using the coefficients
κ (Eˆ −Eˆ ) where κ are the orbital rotation from CIPSI wave functions with small numbers of de-
Pk<l kl kl lk kl
parameters and Eˆ =aˆ† aˆ +aˆ† aˆ is the spin-singlet terminants. Indeed, the selected determinants and their
kl k↑ l↑ k↓ l↓ coefficients from a near FCI calculation are expected to
excitation operator from orbital l to orbital k. The
use of the unitary operator eκˆ(κ) allows one to have a be moreoptimalforVMC inthe presenceofthe Jastrow
factororforDMC.Nevertheless,forthebestpossiblere-
non-redundantparametrizationoftheorbitalcoefficients,
sults in VMC, the selection of the determinants should
automatically preserving the orthonormality of the or-
be done in VMC in the presence of the Jastrow factor,
bitals (see Refs. 11 and13). The orbitalsare partitioned
which is not currently implemented.
into three sets: inactive (doubly occupied in all determi-
These CIPSI wave functions are then multiplied by
nants),active(occupiedinsomedeterminantsandunoc-
ourJastrowfactor,andQMCcalculationsareperformed
cupiedinothers),andvirtual(unoccupiedinalldetermi-
withtheprogramCHAMP[57]usingthetrueSlaterbasis
nants). Thenon-redundantexcitationstoconsidera pri-
set rather than its Gaussian expansion. The parameters
ori are: inactive → active, inactive → virtual, active →
are optimized by minimizing the energy with the linear
virtual, andactive → active. Some redundancies canac-
optimization method [11–13] in VMC, using an acceler-
tually occur with the active-active excitations, and they
ated Metropolis algorithm [58, 59]. We test three levels
must be detected and eliminated. Also, only excitations
of optimization: optimization of the Jastrow parameters
between orbitals of the same irreducible representation
only, simultaneous optimization of the Jastrow param-
need to be considered.
eters and the coefficients of the determinants, and si-
WeuseaflexibleJastrowfactorconsistingoftheexpo-
multaneous optimization of the Jastrow parameters, the
nential of the sum of electron-nucleus, electron-electron,
coefficients of the determinants, and the orbital parame-
and electron-electron-nucleus terms, written as system-
ters. Once the trialwavefunctions have been optimized,
atic polynomial and Pad´e expansions [49] (see, also,
we perform DMC calculations within the short-time and
Refs. 26 and 50) with 24 free parameters. The total
parameters p = (α,c,κ) to be optimized in the wave fixed-node approximations (see, e.g., Refs. 60–64). We
function are the Jastrow parameters α, the coefficients use an imaginary time step of τ = 0.0025 hartree−1 in
an efficient DMC algorithm with very small time-step
of the determinants c, and the orbital rotation parame-
ters κ. errors [65].
III. RESULTS AND DISCUSSION
C. Computational details
SectionIIIAreportsthenumericalresultsobtainedfor
In the present work, the starting orbitals, both
theseriesofatomsrangingfromBtoNe. Then,thecase
occupied and unoccupied, are obtained from re-
of the Be atom is investigated in more details in Section
stricted Hartree-Fock (RHF) or restricted open-shell
IIIBasaprototypeofasystemshowingimportantstatic
Hartree-Fock (ROHF) calculations performed using the
correlationeffects.
GAMESS(US) package [51]. The basis set used here for
the CIPSIcalculationsisa fitofthe polarizedtriple-zeta
VB1 Slater basis set of Ref. 52. For the case of the Be
atom, we also perform calculations with the polarized A. Results for atoms from B to Ne
quadruple-zeta VB2 Slater basis set. Each Slater basis
function is fitted to a linear combination of 10 Gaussian InFigure1wereporttheconvergenceofthetotalVMC
basis functions [53–55]. The CIPSI calculations are per- energies of the atoms with respect to the number of de-
formedusingtheQuantumPackage[56]. The1selectrons terminants at various optimization levels. As expected
are kept frozen in all calculations, consistently with the atthe VMC level,for a givennumber of determinants in
fact that the VB1 or VB2 basis set does not provideany the CI expansion, the energy lowers (within the statisti-
basisfunctions adaptedforcore-coreorcore-valencecor- caluncertainty)asoneincreasesthenumberofoptimized
relation. For all the systems, the starting coefficients of variational parameters in the wave function. Consider-
4
-24.625
opt j opt j
B atom VMC opt j+c -37.815 C atom VMC opt j+c
opt j+c+o opt j+c+o
-24.630 Exact Exact
-37.820
Total energy (hartree) ---222444...666443505 Total energy (hartree) ---333777...888332505
-37.840
-24.650
-37.845
-24.655 1 10 50 100 250 1 50 100 250 500 1000
Number of determinants Number of determinants
-75.025
opt j opt j
-54.560 N atom VMC opt j+c -75.030 O atom VMC opt j+c
opt j+c+o opt j+c+o
Exact Exact
e) -54.565 e) -75.035
e e
artr -54.570 artr -75.040
h h
y ( y ( -75.045
g g
ner -54.575 ner -75.050
e e
otal -54.580 otal -75.055
T T
-75.060
-54.585
-75.065
-54.590
1 50 100 250 500 1000 1 50 100 250 500 1000
Number of determinants Number of determinants
-99.685 -128.885
opt j opt j
-99.690 F atom VMC opt j+c -128.890 Ne atom VMC opt j+c
opt j+c+o opt j+c+o
-99.695 Exact -128.895 Exact
ee) -99.700 ee) -128.900
artr -99.705 artr -128.905
h h
gy ( -99.710 gy ( -128.910
er er -128.915
n -99.715 n
otal e -99.720 otal e --112288..992250
T T
-99.725
-128.930
-99.730
-128.935
-99.735 -128.940
1 50 100 250 500 1000 1 50 100 250 500 1000
Number of determinants Number of determinants
FIG.1. VMCtotalenergiesofatomswithJastrow-CIPSIwavefunctionswithincreasingnumbersofdeterminantsanddifferent
levelsofoptimizationinVMC:optimizationoftheJastrowfactor(optj),optimizationoftheJastrowfactorandthecoefficients
of the determinants (opt j+c), and optimization of the Jastrow factor, the coefficients of the determinants, and the orbitals
(opt j+c+o). The basis set used is VB1.
ing the convergence of the VMC energy as a function as it should. Also, the gain in the total VMC energy is
of the number of determinants, several observations can important in all the calculations reported here, includ-
be made. First, optimizing only the Jastrow factor, and ing the ones with the largest numbers of determinants.
thuskeepingtheCIcoefficientsoptimizedatthenearFCI Third, the gain in the total VMC energy brought by the
level, does not lead to a systematically monotonic low- optimization of the orbitals (together with the Jastrow
ering of the VMC energy upon increasing the number of factor andthe CI coefficients) is much smallerand tends
determinants (cases of the O, F, and Ne atoms). Never- to reduce when increasing the number of determinants.
theless, after reachinga certainnumber of determinants, Thisis expectedsince inthe FCI limitthe wavefunction
the VMC energy tends to lower and converge for a large becomes invariant with respect to orbital rotations.
numberofdeterminants. Second,theoptimizationofthe
Figure 2 shows the convergence of the DMC total en-
CI coefficients in the presence of the Jastrow factor re-
ergies as a function of the number of determinants using
moves this non-monotonic behavior of the VMC energy,
the Jastrow-CIPSI wave functions previously optimized
5
opt j opt j
-24.640 B atom DMC opot pj+tc j++oc -37.830 C atom DMC opot pj+tc j++oc
Exact Exact
Total energy (hartree) --2244..665405 Total energy (hartree) --3377..884305
-37.845
-24.655 1 10 50 100 250 1 50 100 250 500 1000
Number of determinants Number of determinants
opt j -75.050 opt j
-54.575 N atom DMC opt j+c O atom DMC opt j+c
opt j+c+o opt j+c+o
Exact Exact
ee) ee) -75.055
hartr -54.580 hartr
y ( y (
g g
er er -75.060
n n
e e
otal -54.585 otal
T T
-75.065
-54.590
1 50 100 250 500 1000 1 50 100 250 500 1000
Number of determinants Number of determinants
-99.710 -128.915
opt j opt j
F atom DMC opt j+c Ne atom DMC opt j+c
opt j+c+o opt j+c+o
-99.715 Exact -128.920 Exact
e) e)
e e
hartr -99.720 hartr -128.925
y ( y (
g g
ener -99.725 ener -128.930
al al
ot ot
T -99.730 T
-128.935
-99.735
-128.940
1 50 100 250 500 1000 1 50 100 250 500 1000
Number of determinants Number of determinants
FIG.2. DMCtotalenergiesofatomswithJastrow-CIPSIwavefunctionswithincreasingnumbersofdeterminantsanddifferent
levelsofoptimizationinVMC:optimizationoftheJastrowfactor(optj),optimizationoftheJastrowfactorandthecoefficients
of the determinants (opt j+c), and optimization of the Jastrow factor, the coefficients of the determinants, and the orbitals
(opt j+c+o). The basis set used is VB1.
at the VMC level. Similarly to what was observed for crease monotonically with the number of determinants.
the VMC total energies, when only the Jastrow factor MovingnowtotheDMCresultsusingthewavefunctions
has been optimized, the DMC total energies do not sys- where the Jastrow factor and the CI coefficients have
tematically converge monotonically with the number of been simultaneously optimized, two observations can be
determinants. A transient region clearly occurs for the made. First, at a given number of determinants, a sub-
F and Ne atoms where the DMC energy increases with stantially lower DMC energy is obtained in comparison
respecttotheoneobtainedusingtheRHF/ROHFdeter- tothe oneobtainedwithoutthe reoptimizationofthe CI
minant. Fortheseatoms,obtainingaDMC energylower coefficients (except for the B atom, for which the gain
thantheoneobtainedwiththeRHF/ROHFdeterminant is comparable to the statistical uncertainty). Thus, the
requires at least 100 and 250 determinants, respectively. CI coefficients obtainedfrom the largeCIPSI wavefunc-
For larger numbers of determinants, the DMC energy tion clearly do not provide the best nodal structure,and
withthenon-reoptimizedCIexpansionsdoestendtode- the dynamical correlation brought by the Jastrow fac-
6
TABLEI.TotalVMCandDMCenergiesofatomswithJastrow-CIPSIwavefunctionswiththelargestnumbersofdeterminants
used (Ndmeatx) and different levels of optimization in VMC: optimization of the Jastrow factor (opt j), optimization of the
Jastrowfactorandthecoefficientsofthedeterminants(optj+c),andoptimization oftheJastrowfactor,thecoefficientsofthe
determinants, and the orbitals (opt j+c+o). Thebasis set used is VB1.
E E Ea
VMC DMC exact
Ndmeatx opt j opt j+c opt j+c+o opt j opt j+c opt j+c+o
B 250 -24.6486(5) -24.6514(5) -24.6523(4) -24.6532(3) -24.6534(2) -24.6536(2) -24.65390
C 1000 -37.8367(5) -37.8423(5) -37.8431(5) -37.8429(4) -37.8436(2) -37.8442(1) -37.8450
N 1000 -54.5769(5) -54.5839(5) -54.5856(5) -54.5869(5) -54.5879(3) -54.5885(3) -54.5893
O 1000 -75.0488(5) -75.0584(5) -75.0588(5) -75.0621(6) -75.0645(4) -75.0651(1) -75.0674
F 1000 -99.7107(5) -99.7235(5) -99.7248(5) -99.7266(7) -99.7301(4) -99.7309(4) -99.7341
Ne 1000 -128.9106(5) -128.9284(8) -128.9280(5) -128.9313(6) -128.9341(4) -128.9354(5) -128.9383
a
Estimated non-relativistic total energies from Ref. 66.
torsignificantlychangestheCIcoefficientsandimproves coreexcitationsintheCIexpansions. Theimpactonthe
the nodes of the wave function. Second, the previously DMC energy of core excitations together with the effect
observednon-monotonicbehavior of the DMC energy at of using an appropriatebasis setfor corecorrelationwas
smallnumbers ofdeterminants is avoided,andthe DMC illustratedinapreviousworkbyGineret al.[33]andwill
energynowconvergesmonotonically(withinthe statisti- not be repeated here.
cal uncertainties) and more rapidly with the number of
determinants. Even though having a monotonic conver-
gence seems reasonable, note that there is in principle B. The case of the Be atom
no guarantee that optimizing more parameters in VMC
always improves the nodes of the trial wave functions
Now we investigate in more details the effect of the
and therefore lowers the DMC energy. The fact that it
basis set and of the level of optimization in the case of
is in practice the case must mean that our trial wave
the Be atom which is known to present important static
functions are reasonably accurate. Considering now also
correlation effects due to the near degeneracy of the 2s
the optimization of the orbitals, it seems that the gain
and2pshells. For this system, we use the VB1 and VB2
in the DMC total energy with respect to the situation
basissets[52]andthelargestCIPSIwavefunctionscorre-
when only the Jastrow factor and the CI coefficients are
spondtotheFCIlimitwithinthesebasissetswiththe1s
optimized is quite small if there is any. More precisely,
frozen-core approximation. The minimal multidetermi-
the DMC energies obtained using the two sets of varia-
nantwavefunctioncontainsfourdeterminants,the RHF
tional parameters are almost always compatible within
determinant and the three double excitations 2s → 2p
one, two, or three standard deviations, even for small i
(with i = x,y,z). These three excited determinants are
numbers of determinants. Reoptimizing the orbitals has
responsible for the strong multideterminant character of
thus only a small effect on the nodes of these trial wave
the ground-state wave function. We report in Figures 3
functions, at least for the systems considered here.
and 4 the convergence of the VMC and DMC total en-
The VMC and DMC total energies obtained at the ergies with the number of determinants for the VB1 and
three levels of optimization using the largest CI expan- VB2 basis sets, respectively. Some corresponding ener-
sions are reported in Table I. This table shows that the gies are also given in Table II.
VMC and DMC errors with respect to the estimated WiththeVB1basisset,theresultsaregloballysimilar
exact energies [66] are reduced by about a factor of 2 to the results obtained for the atoms from B to Ne. The
when going from the optimization of the Jastrow fac- VMC and DMC energies are almost convergedusing the
tor only to the optimization of the Jastrow factor and four-determinant wave function. In comparison to the
the CI coefficients. As already noticed on Figures 1 situation where only the Jastrow factor and the coeffi-
and 2, the effect of reoptimizing the orbitals is very cientsofthedeterminantsareoptimized,theconvergence
small. The best DMC total energies obtained in the oftheVMCenergywiththenumberofdeterminantsisa
present work are much lower than the DMC total ener- bit faster when the orbitals are reoptimized in VMC. As
giesobtainedwithfullyreoptimizedJastrow-full-valence- regardstheDMCcalculations,weobservethattheDMC
complete-active-space wave functions [13]. This shows energy slightly increases when going from 4 to 19 deter-
theimportanceofincludingexciteddeterminantsbeyond minantswhenoptimizingonlytheJastrowfactorandthe
the valence orbital space for improving the nodes of the coefficients of the determinants. This effect can be seen
wave function. We note that lower DMC energies have thanks to the smallness of the errorbars,and we believe
beenreportedwithtruncatedCIwavefunctionsforsome thatitisreal,i.e. notduetoafailureoftheoptimization
atomsintheseriesintheliterature[32,33]. Theremain- butsimply due tothe factthatminimizing the VMC en-
ing errors in the present DMC energies are essentially ergy does not necessarily lead to a lower DMC energy.
due to the limited basis set used and to the exclusion of This slight non-monotonic behavior of the DMC energy
7
-14.645 -14.655
opt j opt j
Be atom VMC VB1 opt j+c Be atom DMC VB1 opt j+c
opt j+c+o opt j+c+o
-14.650 Exact Exact
e) e)
y (hartre -14.655 -14.664 y (hartre -14.660 -14.6670
g -14.666 g
er er
en -14.660 en
al -14.668 al -14.665 -14.6675
Tot 4 10 19 Tot 4 10 19
-14.665
-14.670 -14.670
1 4 10 19 1 4 10 19
Number of determinants Number of determinants
FIG.3. VMCandDMCtotalenergiesoftheBeatomwithJastrow-CIPSI wavefunctionswithincreasingnumbersofdetermi-
nantsanddifferentlevelsofoptimizationinVMC:optimizationoftheJastrowfactor(optj),optimizationoftheJastrowfactor
andthecoefficientsofthedeterminants(optj+c),andoptimization oftheJastrow factor, thecoefficients ofthedeterminants,
and the orbitals (opt j+c+o). The error bars are smaller than the point symbols. The basis set used is VB1.
-14.645 -14.655
opt j opt j
Be atom VMC VB2 opt j+c Be atom DMC VB2 opt j+c
opt j+c+o opt j+c+o
-14.650 Exact Exact
e) e)
y (hartre -14.655 -14.6640 y (hartre -14.660 -14.6670
g g
er er
en -14.660 en
al al -14.665
Tot 4 10 20 34 Tot 4 10 20 34
-14.665
-14.670 -14.670
1 4 10 20 34 1 4 10 20 34
Number of determinants Number of determinants
FIG.4. VMCandDMCtotalenergiesoftheBeatomwithJastrow-CIPSI wavefunctionswithincreasingnumbersofdetermi-
nantsanddifferentlevelsofoptimizationinVMC:optimizationoftheJastrowfactor(optj),optimizationoftheJastrowfactor
andthecoefficientsofthedeterminants(optj+c),andoptimization oftheJastrow factor, thecoefficients ofthedeterminants,
and the orbitals (opt j+c+o). The error bars are smaller than the point symbols. The basis set used is VB2.
is eliminated when the orbitals are also optimized. to the VB2 basis set, since one would naively expect an
With the VB2 basis set, when the orbitals are not re- improvement when increasing the number of basis func-
optimized, the convergence of the VMC and DMC en- tions. This effect is related to the much more diffuse
ergies with the number of determinants is much slower characterof the RHF 2porbitals obtained with the VB2
than with the VB1 basis set. Also, the reoptimization basis set compared to those obtained with the VB1 ba-
of the coefficients of determinants in the presence of the sis set. Indeed, in accordance to Koopmans’ theorem, a
Jastrow factor does not lead to any significant lowering RHFvirtual2porbitaloftheneutralBeatomrepresents
of the VMC and DMC energies. Focussing on the four- an approximation of an occupied 2p orbital of the Be−
determinant wave function, the DMC energy obtained anion, which is very diffuse. The VB1 basis set contains
when only the Jastrow factor and the coefficients of the only one compact 2p basis function and thus does not
determinants are optimized is 7.8 mhartree higher with have any flexibility to create such a diffuse 2p orbital.
the VB2 basis set with respect to the value obtained at By contrast, the VB2 basis set also contains a diffuse 2p
the same level of optimization with the VB1 basis set basis function, and therefore the RHF optimization pro-
(see Table II). The reoptimization of the orbitals in the cedurehastheflexibilitytogenerateamuchmorediffuse
four-determinant wave function has a very large impact 2p orbital. Even though such diffuse 2p orbitals are bet-
on both the VMC and DMC energies, and allows one to terapproximationstothe exactvirtualRHF2porbitals,
avoidthis deteriorationeffect uponincreasingthe size of they are much worse to describe ground-state electronic
the basis set. correlation when used in multideterminant expansions.
Onemaywonderaboutthe originofthis deterioration The reoptimization in VMC of the orbitals in the four-
observedwiththeRHForbitalswhengoingfromtheVB1 determinant wave function with the VB2 basis set leads
8
TABLE II. Total VMC and DMC energies of the Be atom with Jastrow-CIPSI wave functions with different numbers of
determinants (Ndet) and different levels of optimization in VMC: optimization of the Jastrow factor (opt j), optimization of
the Jastrow factor and the coefficients of the determinants (opt j+c), and optimization of the Jastrow factor, the coefficients
of the determinants, and theorbitals (opt j+c+o). The basis sets used are VB1 and VB2.
E E Ea
VMC DMC exact
Basis set Ndet opt j opt j+c opt j+c+o opt j opt j+c opt j+c+o
VB1 4 -14.66488(1) -14.66517(1) -14.666379(5) -14.66707(1) -14.66715(1) -14.66713(1)
19 -14.665190(5) -14.666186(5) -14.666402(5) -14.66707(1) -14.66698(1) -14.66712(1)
VB2 4 -14.65111(3) -14.65116(3) -14.66639(3) -14.65954(4) -14.65939(4) -14.66712(1)
34 -14.66522(3) -14.66569(3) -14.66643(3) -14.66704(2) -14.66713(2) -14.66711(1) -14.66739
a
Estimated non-relativistic total energies from Ref. 66.
to muchmore compact 2porbitals which better describe wayssystematicallydecreasemonotonicallywith respect
ground-state correlation. This explanation is confirmed to the number of determinants.
by the calculation of the expectation value of r2 over (2)Ifthecoefficientsofthedeterminantsarereoptimized
one 2p orbital. With the VB2 basis set, this quantity in VMC simultaneously with the Jastrow factor, an im-
is equal to 28.3 bohr2 at the RHF level, and decreases portantenergeticgainisobtainedinVMCandDMC(the
to 7.7 bohr2 after reoptimizationof the orbitals in VMC errorsinthe totalenergiesarereducedby abouta factor
using a four-determinant wave function. With the VB1 of 2), even for large numbers of determinants, and both
basisset,the samequantityis alwaysequalto 7.7bohr2. VMC andDMC totalenergiesconvergenearlymonoton-
Thishighlightsthe importanceoforbitaloptimizationin ically with respect to the number of determinants.
cases with important static correlation effects. (3) The reoptimization in VMC of the orbitals, together
Finally,we notethat, withthe VB2 basissetandwith with the Jastrow factor and the coefficients of the de-
thelargestnumberofdeterminants,theVMCenergyob- terminants, has a much smaller effect on the VMC and
tainedwhenoptimizingalltheparametersissignificantly DMC total energies.
lower than the one obtained when optimizing only the Inaddition,themoredetailedstudyoftheBeatom,rep-
Jastrow factor and the coefficients of the determinants. resenting a prototype of a system with important static
This results shows the impact of the reoptimization of correlationeffects,showsthatinthiscasethereoptimiza-
the 1sorbitalin the presence ofthe Jastrowfactor. This tion in VMC of the orbitals can have a large impact on
effect is notaccountedfor in the FCI wavefunction with both the VMC and DMC energies when using the larger
thefrozen-coreapproximation. However,thereoptimiza- Slater VB2 basis set.
tion of the 1s orbital has no effect on the DMC energy. Thus,eventhoughtheseconclusionsshouldbechecked
onmore systems,Jastrow-CIPSIwavefunctions withre-
optimized coefficients of the determinants (and, in some
IV. CONCLUSIONS cases, with reoptimized orbitals) appear as promising,
systematically improvable trial wave functions for QMC
In this work, we have explored the use in VMC and calculations. Future possible works on this topic include
DMC of trial wave functions consisting of a Jastrow fac- checkingtheaccuracyofenergydifferencesobtainedfrom
tor multiplied by a truncated CI expansion in Slater de- reoptimizedJastrow-CIPSIwavefunctions,andperform-
terminants obtained from a prior CIPSI calculation. In ing the selection of the best determinants in the trun-
the CIPSI algorithm, the CI expansion is iteratively en- cated CI expansions directly in QMC. The exploration
larged by selecting the best determinants using pertur- of this last strategy would be indeed interesting since
bation theory, which provides an optimal and automatic one can expect that the dynamic correlation brought by
way of constructing truncated CI expansions approach- the Jastrow factor impacts the selection.
ing the FCI limit and not based on a priori criteria on
orbital active spaces and/or excitation classes.
All-electron QMC calculations on a series of atoms
ACKNOWLEDGEMENTS
ranging from B to Ne with the Slater VB1 basis set and
different levels of optimization of the parameters (Jas-
trow parameters, coefficients of the determinants, and It is a great pleasure to dedicate this paper to An-
orbital parameters) show that: dreas Savin who is among the early contributors to the
(1) If the coefficients of the determinants are not reop- developmentofquantumMonteCarloforquantumchem-
timized in VMC in the presence of the Jastrow factor istry [67–70] and who has always encouraged this line of
(i.e., kept as they have been obtained in a large CIPSI research. We thank Michel Caffarel, Anthony Scemama,
calculation),theVMCandDMCtotalenergiesdonotal- and Cyrus Umrigar for discussions.
9
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