A Quantum Monte Carlo Study of Lanthanum

doi:10.4236/wjcmp.2013.34034

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World Journal of Condensed Matter Physics, 2013, 3, 203-206

Published Online November 2013 (http://www.scirp.org/journal/wjcmp)

http://dx.doi.org/10.4236/wjcmp.2013.34034

Open Access WJCMP

203

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.

Email: [email protected]

Received August 25th, 2013; revised October 6th, 2013; accepted October 21st, 2013

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

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 Hartree−1, and −0.0104 Har-

tree for the largest time step, τ = 0.01 Hartree−1. 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

204

ated with a trial wavefunction ψT is given by

 

RH R

RRdR







 (1)

which provide an upper bound to the exact ground state

energy E0.

The VMC method rewrites the last integral in the fol-

lowing form:

 



ˆT

RdR











(2)

where



ˆT



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



VMC

1NTi

iTi

ENR







 (3)

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:

 

RVE

















(4)

where τ is the imaginary time, τ = it and ET is the energy

offset.

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









 







,1,

DLT

fR fR

fRv REE fR

















(5)

where

 



 

R

interpreted as a prob-

ability density and

 



ˆT

ER R



 is the local en-

ergy.

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:







 

,,;,

RGRRfR

 



 

dR



(6)

where the Green’s function



,;GRR



 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





,;GRR







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

LL T

GRR

ER ERE









 

























 







(7)

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



 (8)

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A Quantum Monte Carlo Study of Lanthanum 205

where





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 H−1.

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

HF VMC

HF DMC

100%







 (9)

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.

EHF σHF EVMCσVMC EDMC σDMCη%

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).

IPDMC (eV) IPEXP (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

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A Quantum Monte Carlo Study of Lanthanum

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206

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 H−1, we found a

time step error of 0.0019 H and for the largest time step,

τ = 0.01 H−1, 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.

REFERENCES

[1] M. D. Brown, J. R. Trail, P. LoPez Rios and R. J. Needs,

“Energies of the First Row Atoms from Quantum Monte

Carlo,” The Journal of Chemical Physics, Vol. 126, No.

22, 2007, Article ID: 224110.

http://dx.doi.org/10.1063/1.2743972

[2] J. Toulouse and C. J. Umrigar, “Full Optimization of

Jastrow-Slater Wave Functions with Application to the

First-Flow Atoms and Homonuclear Diatomic Mole-

cules,” The Journal of Chemical Physics, Vol. 128, No.

17, 2008, Article ID: 174101.

http://dx.doi.org/10.1063/1.2908237

[3] K. E. Schmidt and J. W. Moskowitz, “Correlated Monte

Carlo Wave Functions for the Atoms He through Ne,”

The Journal of Chemical Physics, Vol. 93, No. 6, 1990, p.

4172. http://dx.doi.org/10.1063/1.458750

[4] S. D. Kenny, G. Rajagopal and R. J. Needs, “Relativistic

Corrections to Atomic Energies from Quantum Monte

Carlo Calculations,” Physical Review A, Vol. 51, No. 3,

1995, pp. 1898-1904.

http://dx.doi.org/10.1103/PhysRevA.51.1898

[5] E. Buendia, F. J. Galvez and A. Sarsa, “Correlated Wave

Functions for the Ground State of the Atoms Li through

Kr,” Chemical Physics Letters, Vol. 428, No. 4-6, 2006,

pp. 241-244.

http://dx.doi.org/10.1016/j.cplett.2006.07.027

[6] L. Wagner and L. Mitas, “A Quantum Monte Carlo Study

of Electron Correlation in Transition Metal Oxygen

Molecules,” Chemical Physics Letters, Vol. 370, No. 3-4,

2003, pp. 412-417.

http://dx.doi.org/10.1016/S0009-2614(03)00128-3

[7] X. P. Li, D. M. Ceperley and R. M. Martin, “Cohesive

Energy of Silicon by the Green’s-Function Monte Carlo

Method,” Physical Review B, Vol. 44, No. 19, 1991, pp.

10929-10932.

http://dx.doi.org/10.1103/PhysRevB.44.10929

[8] A. Ma, N. D. Drummond, M. D. Towler and R. J. Needs,

“All-Electron Quantum Monte Carlo Calculations for the

Noble Gas Atoms He to Xe,” Physical Review E, Vol. 71,

No. 6, 2005, Article ID: 066704.

http://dx.doi.org/10.1103/PhysRevE.71.066704

[9] J. B. Anderson, “Quantum Monte Carlo. Origins, Devel-

opment, Applications,” Oxford University Press, Oxford,

2007.

[10] M. P. Nightingale and C. J. Umrigar, “Quantum Monte

Carlo Methods in Physics and Chemistry,” Kluwer Aca-

demic Publishers, Berlin, 1999.

[11] L. M. Sobol, “A Primer for the Monte Carlo Method,”

CRC Press, Boca Raton, 1994.

[12] C. J. Cramer, “Essentials of Computational Chemistry:

Theories and Models,” John Wiley & Sons Ltd, England,

2004.

[13] W. M. C. Foulkes, L. Mitas, R. J. Needs and G. Ra-

jagopal, “Quantum Monte Carlo Simulations of Solids,”

Reviews of Modern Physics, Vol. 73, No. 1, 2001, pp. 33-

83. http://dx.doi.org/10.1103/RevModPhys.73.33

[14] J. B. Anderson, “Quantum Chemistry by Random Walk.

H 2P, H+3 D3h 1A′1, H2 3Σ+u, H4 1Σ+g, Be 1SJ,” The

Journal of Chemical Physics, Vol. 65, No. 10, 1976, p.

4121. http://dx.doi.org/10.1063/1.432868

[15] A. Alkauskas, P. Deàk, J. Neugebauer, A. Pasquarello

and C. Van de Walle, “Advanced Calculations for Defects

in Materials,” Wiley-VCH Verlag & Co. K GaA, Wein-

heim, 2011. http://dx.doi.org/10.1002/9783527638529

[16] L. K. Wagner, M. Bajdich and L. Mitas, “Qwalk: A

Quantum Monte Carlo Program for Electronic Structure,”

Journal of Computational Physics, Vol. 228, No. 9, 2009,

pp. 3390-3404.

http://dx.doi.org/10.1016/j.jcp.2009.01.017

[17] M. W. Schmidt, J. A. Boatz, K. K. Baldridge, S. T. Elbert,

M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K.

A. Nguyen, S. Su, T. L. windus, M. Dupuis and J. A.

Montgomery, “General Atomic and Molecular Electronic

Structure System,” Journal of Computational Chemistry,

Vol. 14, No. 11, 1993, 1347-1363.

http://dx.doi.org/10.1002/jcc.540141112

[18] https://bse.pnl.gov/bse/portal

[19] C. E. Moore, “Ionization Potential and Ionization Limits

Derived from the Analyses of Optical Spectra,” National

Bureau of Standards, Washington, 1970.