World Journal of Condensed Matter Physics, 2013, 3, 203-206
Published Online November 2013 (
Open Access WJCMP
A Quantum Monte Carlo Study of Lanthanum
Nagat Elkahwagy1, Atif Ismail1, Sana Maize2, Kamal Reyad Mahmoud1
1Physics Department, Faculty of Science, Kafrelsheikh University, Kafr El-Sheikh, Egypt; 2Physics Department, Faculty of Science,
Menoufia University, Shebin El-Kom, Egypt.
Received August 25th, 2013; revised October 6th, 2013; accepted October 21st, 2013
Copyright © 2013 Nagat Elkahwagy et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Pseudopotential calculations of the ground state energies of the Lanthanum neutral atom, first and second corresponding
cations by means of the variational Monte Carlo (VMC) and the diffusion Monte Carlo (DMC) methods are performed.
The first and the second ionization potentials have been calculated for Lanthanum. The obtained results are satisfactory
and comparable with the available experimental data. Studying the DMC energy of the La atom at different time steps,
gave us a time step error of the order 0.0019 Hartree for the smallest time step, τ = 0.0001 Hartree1, and 0.0104 Har-
tree for the largest time step, τ = 0.01 Hartree1. This paper demonstrates the ability of extending the QMC method for
lanthanides and obtaining highly accurate results.
Keywords: Variational Monte Carlo; Diffusion Monte Carlo; Lanthanum; Hartree; Time Step
1. Introduction
Quantum Monte Carlo (QMC) is a powerful technique by
which one can perform computational electronic struc-
ture calculations with high accuracy. One of the advan-
tages of the QMC technique is that its computational
efforts scales with N3 where N is the number of electrons
in the system. This technique is favorable over other
computational many-body methods. The most common
QMC techniques, for atoms and molecules, are the varia-
tional Monte Carlo (VMC) and the diffusion Monte
Carlo (DMC). Accurate calculations for extremely light
atoms using QMC methods are performed by a large
number of researchers [1-4]. For the atoms heavier than
Ne, the situation is more difficult. However, there have
been some studies which gave satisfactory results [5-8].
For our knowledge, it is the first time Qwalk code is to
be used in dealing with lanthanides.
The study of chemical systems that contain f-elements
is still a particularly challenging branch of computational
chemistry. The difficulties presented by f-elements in
quantum mechanical calculations arise from the large
magnitude of the relativistic effect and the limitation in
the electron correlation treatment.
In the present work, by means of VMC and DMC
methods, we have done calculations for the ground state
energies of La atom and its charged cations with the hope
“achieving high accuracy”. In addition, we study the
DMC energies at different time steps and the accurate
extrapolated value of the ground state energy of La atom
is derived. To allow the QMC calculations of this heavy
atom, pseudopotential valence-only calculations have been
performed, since the presence of the inert core electrons
introduces a large fluctuation in the energies and this
reduces the computational efficiency. In our study, the
basic form of the wave function is the Slater-Jastrow
wave function which is considered the most common and
simplest one.
In the next section, we outline a brief description of
the QMC methods. The results are then presented and
discussed. Atomic units are used throughout this work
unless otherwise indicated.
2. Computational Methods
Quantum Monte Carlo methods have been extensively
described in the literatures [9-11], so we give here a brief
description of the two methods, the variational and diffu-
sion Monte Carlo methods.
The variational Monte Carlo (VMC) technique de-
pends on the familiar variational principle for finding the
ground energies of quantum mechanical systems. By the
variational principle, the expectation value of the ground
state energy of a many body system of N particles evalu-
A Quantum Monte Carlo Study of Lanthanum
ated with a trial wavefunction ψT is given by
 
 
which provide an upper bound to the exact ground state
energy E0.
The VMC method rewrites the last integral in the fol-
lowing form:
 
is the local energy EL of an electronic
configuration, and
is the probability density for
the configuration R.
The Metropolis algorithm is used to sample a series of
points, Ri, from the probability density in the configura-
tion space. At each of these points the local energy EL is
evaluated [12]. After a sufficient number of evaluations
of the local energy have been made, the average is taken
So the VMC is a simple technique in which the statistical
efficiency of the results depends on the whole on the trial
wavefunction. The better the wavefunction guess, the
more efficient the VMC result.
The more accurate diffusion Monte Carlo (DMC)
method is a stochastic projector method for solving the
imaginary time many-body Schrödinger equation:
 
where τ is the imaginary time, τ = it and ET is the energy
Importance sampling with a trial wavefunction
is used to improve the statistical accuracy of the
simulation and this is can be achieved by multiplying
Equation (4) by and rearranging
 
fR fR
fRv REE fR
 
 
interpreted as a prob-
ability density and
 
is the local en-
This equation can be simulated with a random walk
having diffusion, a draft, and a branching step and may
be written in the integral form:
 
 
 
where the Green’s function
is a solution
of the same initial Equation (5) and can be interpreted as
a probability of transition from a state R to R'. It is pos-
sible to use QMC method to solve the integral in Equa-
tion (6) but the difficulty is that the precise form of
is not known. Fortunately the comparison
of the Schrodinger equation with the diffusion equation
gives us a clue about how one might approximate the
unknown Green’s function.
The evolution during the long time interval τ can be
generated repeating a large number of short time steps τ.
In the limit 0
, one can make use of the short time
approximation for Green’s function [13]:
 
,; 2π
exp 2
exp2 2
 
 
But due to the fermionic nature of electrons, the
wavefunction must have positive and negative parts and
this is opposite to the assumed nature of ψ which is a
probability distribution. So the fixed-node approximation
[14] had been used to treat the fermionic antisymmetry
which constrains the nodal surface of ψ to equal that of
the antisymmetric trial wavefunction ψT.
So far, the main difference between VMC and DMC is:
In VMC, a Monte Carlo method evaluates the many-
dimensional integral to calculate quantum mechanical
expectation values. Accuracy of the results depends cru-
cially on the quality of the trial wavefunction, which is
controlled by the functional form of the wavefunction
and the optimization of the wavefunctions parameters.
DMC removes most of the error in the trial wavefunc-
tion by stochastically projecting out the ground state us-
ing an integral form of the imaginary-time Schrödinger
equation [15].
The form of the trial wavefunction is therefore very
important; it must be both accurate and easy to evaluate.
The simplest and most common wavefunction used in
QMC is the Slater Jastrow wavefunction which consists
of a Slater determinant multiplied by the exponential Jas-
trow correlation factor which includes the dynamic cor-
relation among the electrons so it plays a crucial role in
treating many-body systems. The basic functional form
of the Slater-Jastrow wavefunction is
sjn n
Re cDR
Open Access WJCMP
A Quantum Monte Carlo Study of Lanthanum 205
Rrr r denote the space coordinate of
N electrons, J(R) is the Jastrow factor, cn are coefficient,
and Dn(R) is a Slater determinant of single particle orbi-
tals which usually obtained from self-consistent DFT or
Hartree-Fock calculations.
3. Results and Discussion
All our QMC calculations were performed by means of
Qwalk code [16]. The basic form of the wavefunction is
a product of Slater determinants for spin-up and spin-
down electrons multiplied by a Jastrow correlation factor.
The initial orbitals of the trial wavefunction are gener-
ated in GAMESS package [17] via spin-restricted open
Hartree-Fock calculations. In the present work we have
performed pseudopotential calculations by using the
CRENBS ECP basis set [18] which eliminates 54 elec-
trons (Xe-core) so three electrons only are treated as va-
lence electrons in the Lanthanum atom. The usage of
large pseudopotential introduces additional errors but the
Monte Carlo errors are much decreased. We used a mean
population of 2000 configurations. At first, all our calcu-
lations in Table 1 were performed with a time step of τ =
0.0001 H1.
In Table 1, we present pseudopotential calculations of
the Hartree Fock, EHF, the variational Monte Carlo, EVMC,
and the diffusion Monte Carlo, EDMC, ground state ener-
gies for the Lanthanum atom, first and second charged
cations, alongside with the fluctuations of the local en-
ergy; σ, for each method. In the last column, we measure
the accuracy of the Jastrow factor by estimating
which is the percentage of the DMC correlation energy
retrieved within VMC
It is worth mentioning here that because of the lack of
experimental data we didn’t able to make a comparison
for the estimated values of the pseudopotential ground
state energies for La. To our knowledge, these are the
first values for La published to date by the QMC method.
The tabulated results in Table 1 tell us that we have
performed high accuracy QMC calculations. As expected
the best values for fluctuations is the diffusion Monte
Carlo one, σDMC. Indeed, we obtained the results with
high efficiency, η, which is greater than 90% for the La
atom and its doubly charged cation, La+2, and greater
than 80% for its singly charged cation La+1.
From the estimated DMC energies for the La atom and
its cations, we calculated the first and second ionization
potentials for La and compared it with the experimental
values [19]. These values are shown in Table 2. The first
ionization potential is in a good agreement with the ex-
perimental value while the error is larger for the second
ionization potential, we attributed this to the large dif-
ference in the fluctuations of the estimated diffusion
Monte Carlo energies, σDMC, between the La atom and its
double charged cation, La+2.
Practical calculations of the DMC energy suffer from
the time step error that originates from the use of finite
time step in the short time approximation. To investigate
the time step error in our diffusion Monte Carlo calcula-
tions for La atom, we made calculations for several val-
ues of τ to get an accurate extrapolation to zero time step.
Table 3 presents a number of calculations of the DMC
energies at different steps. The values of the DMC en-
ergy of La atom as a function of the time step are plotted
in Figure 1.
From Figure 1, it can be seen that the relation between
Table 1. Ground state total energies computed within Har-
tree Fock, EHF, variational Monte Carlo, EVMC, and diffu-
sion Monte Carlo, EDMC, for La, first and second charged
cations. All energies are in Hartrees.
La 1.21380.31 1.2511 0.20 1.2535 0.1493.95
La+1 1.01060.35 1.0374 0.19 1.0431 0.1782.46
La+2 0.66430.23 0.6724 0.04 0.6727 0.0396.42
Table 2. First and second ionization potentials for the La
atom computed within diffusion Monte Carlo, IPDMC, com-
pared to the available experimental values, IPEXP. The ioni-
zation potentials are in (eV).
First 5.72 5.58
Second 15.80 11.06
Table 3. Time step depende nce of the diffusion Monte Carlo
Energy, EDMC, for the La atom. The last column indicates
the values of the time step errors.
Time step (Hartree)1EDMC (Hartree) Time step error (Hartree)
0.00010 1.2535 0.0019
0.00025 1.2557 0.0003
0.00050 1.2567 0.0013
0.00100 1.2594 0.0040
0.00150 1.2610 0.0056
0.00200 1.2582 0.0028
0.00400 1.2624 0.0070
0.00600 1.2639 0.0085
0.00800 1.2654 0.0100
0.01000 1.2658 0.0104
Open Access WJCMP
A Quantum Monte Carlo Study of Lanthanum
Open Access WJCMP
Figure 1. Time step dependence of the diffusion Monte
Carlo (DMC) energies for La atom.
the DMC energies and the time steps follows a polyno-
mial relationship, this is because the presence of Jastrow
factor that introduces polynomial behavior in the energy
as a function of time step. We performed a polynomial
extrapolation of the energies to zero time step. The ex-
trapolated value at τ = 0 for La is 1.25659 ± 0.000839 H.
For the smallest time step, τ = 0.0001 H1, we found a
time step error of 0.0019 H and for the largest time step,
τ = 0.01 H1, a value of 0.0104 H has been found.
In conclusion, the small statistical errors which have
been reported for La and its charged cations in this paper
open the way to the possibility for performing high ac-
curacy QMC calculations for the lanthanides.
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