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Monte Carlo Variational Method and the Ground-State of Helium - PDF

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					     Monte Carlo Variational Method and the Ground-State of Helium


                            S. B. Doma1) and F. El-Gamal2)
1)
   Faculty of Information Technology and Computer Sciences, Sinai University, El-
Arish, North Sinai, Egypt.
E-mail address: sbdoma@yahoo.com
2)
   Mathematics Department, Faculty of Science, Menofia University, Shebin El-Kom,
Egypt.

Abstract
The Variational Monte Carlo method is used to evaluate the energy of the ground state of the helium
atom. Trial wave functions depending on the variational parameters are constructed for this purpose.
Energies as well as standard deviations are plotted versus the variational parameters. The experimental
data are presented for comparison.

1. Introduction                                        on sampling the integrand on a regular
The term Monte Carlo refers to group                   grid for problems involving more than
of methods in which physical or                        a few dimensions. Moreover, the
mathematical problems are simulated                    statistical error in the estimate of the
by using random numbers. Quantum                       integral decreases and the square root
Monte Carlo (QMC) techniques                           of the number of points is sampled,
provide a practical method for solving                 irrespective of the dimensionality of
the many-body Schrödinger equation.                    the problem. The major advantage of
   It is commonly used in physics to                   this method is the possibility to freely
simulate complex systems that are of                   choose the analytical form of the trial
random nature in statistical physics [1].              wave function which may contain
There are many versions of the QMC                     highly sophisticated term, in such a
methods which are used to solve the                    way that electron correlation is
Schrödinger equation for the ground                    explicitly taken into account. This is an
state energy of a quantum particle.                    important valid feature for QMC
Among them are the diffusion Monte                     methods,       which     are   therefore
Carlo method [2], which is used to                     extremely useful to study physical
solve the time-dependent Schrödinger                   cases where the electron correlation
equation. Another method is the fixed-                 plays a crucial role.
phase Monte Carlo method [3] which
is used for wave equations that                        2.    Variational     Monte      Carlo
consider a magnetic field.                             Calculations
   The simplest of QMC methods is the                  The Variational Mont Carlo method [5]
variational Monte Carlo (VMC)                          is based on a combination of two ideas
technique which has become a                           namely the variational principal and
powerful tool in Quantum Chemistry                     Monte Carlo evaluation of integrals
calculations [4]. It is based on                       using importance sampling based on
evaluating a high-dimensional integral                 the Metropolis algorithm. It is used to
by sampling the integrand using a set                  compute quantum expectation values
of randomly generated points. It can                   of an operator. In particular, if the
be shown that the integral converges                   operator is the Hamiltonian, its
faster by using VMC technique than                     expectation value is the varitional
more conventional techniques based                     energy EVMQ,


                                                  1
          ∫ Ψ (R) HΨ dR
               ∗  ˆ                                                the distribution function itself. A given
               T               T
 EVMC   =                                              (2.1)       ensemble is chosen according to the
           ∫ Ψ (R)Ψ dR
                ∗
                T          T                                       Metropolis algorithm [6]. This method
where, ΨT is a trial wave function and                             uses an acceptance and rejection
                                                                   process of random numbers that have a
R is the 3N dimensional vector of
                                                                   frequency probability distribution like
electron coordinates. According to the
Varitional principal, the expectation                               Ψ 2 . The acceptance and rejection
value of the Hamiltonian is an upper                               method is due to the work of von
bound on the exact ground state energy                             Neuman [7] and is performed by
                                                                   obtaining a random number from the
 E0 , that is, E v ≥ E0 .
                                                                   probability distribution, P (R ) , then


Ψ ( ) depending on variational
   To evaluate the integral in Eq. (2.1)
                                                                   testing its value to determine if it will
we construct first a trial wave function
                                                                   be acceptable for use in approximation
                                                                   of the local energy. Random numbers
parameters α = (α 1 , α 2 ,.........., α N ) and                   may be generated using a variety of
then varies the parameters to obtain the                           methods [8,9]. Finally, it is important
minimum energy.                                                    to calculate the standard deviation of
   Variational           Monte            Carlo                    the energy,
calculations determine EVMC by                                                                     2
                                                                              EVMC − EVMC
                                                                               2
writing it as                                                      σ=
 EVMC = ∫ P ( R ) E L ( R ) dR ,             (2.2)                                M ( N − 1)

                                       2                           3. The Statement of the Problem
                      ΨT (R )                                      For nucleus with charge Z and infinite
where P(R ) =
                    ∫Ψ                                             mass the non-relativistic Hamiltonian
                                   2
                       T   (R ) dR
                                                                   in atomic units (a. u) reads [10]:
is positive everywhere and interpreted
as a probability distribution and
                                                                                    h2 2      h2 2
      HΨT ( R )
       ˆ                                                           H = T +V = −        ∇1 − −    ∇2 +V
 EL =             is the local energy                                              2m         2m
        ΨT ( R )                                                                                      (3.1)
function.     The     value            of       EL        is       where r1 and r2 denote the relative
evaluated using a series of points, Rij                            coordinates of the two electrons with
                                                                   respect to the nucleus. The potential V
proportional to P (R ) . At each of these
                                                                   is defined as
                               HΨT ( R )
                                ˆ
                                                                          Ze 2 Ze 2 e 2
points the "Local energy ",              ,                         V =−        −      +     ,
                                ΨT ( R )                                   r1     r2    r12
is evaluated. After a sufficient number
of evaluations the VMC estimate of                                  r12 = r1 − r2
 EVMC will be:                                                     In Hylleraas coordinates the above
                                       N    M                      Hamiltonian can be written as:
                     1 1
EVMC    = lim lim
           N →∞ M →∞ N M
                                   ∑∑ E
                                   j =1 i =1
                                                L   ( Rhj )                1 ∂2
                                                                    H =− ( 2 +
                                                                                  2 ∂      ∂2
                                                                                       + 2 +
                                                                                                  2 ∂
                                                                           2 ∂r1 r1 ∂r1 ∂r12 r12 ∂r12
                                         (2.3)
where M is the ensemble size of                                                    ∂2      1 ∂2   2 ∂   ∂2
                                                                   + 2r 1 ⋅r12
                                                                      ˆ ˆ                )− ( 2 +      + 2
random                                numbers                                    ∂r1 r12   2 ∂r2 r2 ∂r2 ∂r12
{R1 , R2 ,..................RM } and N is the                           2 ∂                   ∂2
number          of        ensembles.    These                      +            − 2r 2 ⋅r12
                                                                                   ˆ ˆ              ) + V (r1 , r2 , r12 )
                                                                       r12 ∂r12             ∂r2 r12
ensembles so generated must reflect



                                                               2
        ˆ ˆ           ˆ
where r1 , r2 and r12 denotes the unit             of a symmetric correlation factor, f ,
vectors of the corresponding distance.             which includes the dynamic correlation
In writing Eq. (3.1) we have taken into            among the electrons, times a model
account the Coulomb interactions                   wave function, ϕ , that provides the
between the particles, but have                    correct properties of the exact wave
neglected small corrections arising                function such as spin and the angular
from      spin-orbit       and    spin-spin        momentum of the atom, and is
interactions.                                      antisymmetric in the electronic
   The electronic eigenvalue is                    coordinates ψ = ϕ f .
determined from the Schrödinger                       With this type of wave function, and
equation:                                          by using different correlation factor,
 HΨ (r1 , r2 ) = EΨ (r1 , r2 )         (3.2)       the atoms He to Kr have been
where, ψ (r1 , r2 ) is the electronic wave         extensively studied [11-14]. The aim
function. Our goal, now, is to solve the           of this work is to extend this
six-dimensional partial differential               methodology to obtain ground state,
eigenvalues Eq. (3.2) for the lowest               and similarly excited states, of the
eigenvalue.                                        helium atom. This will be done within
                                                   the context of the accurate Born-
4. The Trial Wave Function                         Oppenheimer approximation, which is
The choice of trial wave function is               based on the notion that the heavy
critical in VMC calculations. How to               nucleus move slowly compared to the
choose it is however a highly non-                 much lighter electrons.
trivial task. All observables are
evaluated with respect to the                      5. The ground state of the helium
probability distribution                           atom
                        2                          For the ground state, the trial wave
            ΨT (R )
P(R ) =                                            function used in this work is given by
          ∫Ψ
                    2
             T   (R ) dR                            ψ (r1 , r2 ) = ϕ (r1 )ϕ (r2 ) f (r12 ) (5.1)
generated by the trial wave function.              where ϕ (ri ) is the single-particle wave
The trial wave function must                       function for particle i , and f (r12 )
approximate an exact eigen state in                account for more complicated two-
order that accurate results are to be              body correlations. For the helium atom,
obtained. Improved trial wave function             we have placed both electrons in the
also improve the importance sampling,              lowest hydrogenic orbit 1s to calculate
reducing the cost of obtaining a certain           the ground state. A simple choice for
statistical accuracy. A good trial wave            ϕ (ri ) is [15]:
function should exhibit much of the
same features as does the exact wave                ϕ (ri ) = exp(ri / a ) ,            (5.2)
function. One possible guideline in                with the variational parameter a to be
choosing the trial wave function is the            determined. The final factor in the trial
use of the constraints about the                   wave function, f , expresses the
behavior of the wave function when                 correlation between the two electrons
the distance between one electron and              due to their coulomb repulsion. That is,
the nucleus or two electron approaches             we expect f to be small when r12 is
zero. These constraints are called                 small and to approach a large constant
“cusp conditions” and are related to the           value as the electrons become well
derivative of the wave function.                   separated.       A        convenient and
Usually the correlated wave function,              reasonable choice is
ψ , used in VMC is built as the product


                                               3
Table-1 Energy of the ground state in                  that    minimum       in   energy     is
(a. u.) units obtained in frame of (VMC)               accompanied with a minimum in the
method, together with the standard                     standard deviation. The numerical
deviation and the experimental data for                results are in good agreement with the
comparison.                                            experimental results [17].
                                                          Table-1 indicates that the energy
               Calculated          Experimental        minima are in agreement with the
 EVMC          -2.8689             2.9037              experimental data.
 Standard 0.0024                   NA                     Calculations of the radial wave
 deviation                                             functions and the excited states of the
                                                       helium atom by using the same
                                                       technique of the variational Monte
                      r       
   f (r ) = exp                          (5.3)       Carlo method gave results in good
                 α (1 + β r )                        agreement with the corresponding
                                                       experimental findings [18].
where α , β are additional positive
variational parameters. The variational
parameter β controls the distance over
which the trial wave function heals to
its uncorrelated value as the two
electrons separate. Using the cusp
conditions [16] we can easy verify that
the variational parameters α , β satisfy
the     transcendental    equations      :
         2                2
       h               2h
 a=        2
              and α =        . Thus β is
     2me               me 2
the only variational parameter at our
disposal. With the trial wave function
specified by Eq. (6), explicit
expression can be worked out for the
local energy E L (R) in terms of the
values and derivatives of ϕ , f . The
Monte Carlo process described here
has been employed for the ground state
of the helium atom.
   Figure-1 shows the variation of the
ground state energy with respect to the
variational parameter β .
   In Fig-2 we present the variations of
the standard deviation with respect to
the variational parameter β .
   The variational parameters appear in
the trial wave function for the 21 S state
are taken as τ 1 = .865 , τ 2 = .522 and
C = 1.2 .
    A Variational Monte Carlo (VMC)
has been used to obtain numerical
ground state energy. Figures-1,2 show


                                                   4
                          -2.85


                         -2.855
 Energy (Ground State)




                          -2.86


                         -2.865


                          -2.87
                             -9.99E-1 0.05    0.1    0.15    0.2       0.25    0.3   0.35    0.4    0.45
                                             The Variational Parameter β




Fig-1 The Ground state energy of helium as function of the vaiational parameter β .


                         0.0042


                         0.0038
 Standard Deviation




                         0.0034


                          0.003


                         0.0026


                         0.0022
                                  0   0.05    0.1    0.15   0.2        0.25   0.3    0.35   0.4    0.45

                                       The Variational Parameter β



Fig-2 The standard deviation versus the variational parameter β .

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                                                                   5
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