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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 . References [1] K. Binder and D. W. Heermann, Monte Carlo Simulations in Statistical Physics, Springer, Berlin, (1988). [2] Yu. Li , Jan Vrbik, Stuart M. Rothstein, Chem. Phys. Lett, 445 (2008) 345-349. [3] Botton P, Fixed-phase quantum Monte Carlo method applied to interacting electrons in a quantum dot, Phys. Rev. B 54 (1996) 4780. [4] B.L. Hammond, W.A. Lester Jr., P.J. Reynolds, Monte Carlo Methods in Ab- Initio Quantum Chemistry, World Scientific, Singapore, (1994). [5] S. Pottorf, A. Puzer, M. Y. Chou, Eur. J. Phys. 20 (1999) 205-212. 5 [6] N. Metropolis, A. W. Rosenbluth, N. M. Rosenbluth, J. Chem. Phys. 21 (1953) 1087-92. [7] D. E. Knuth, Semi Numerical Algorithms, volume II of The Art of Computer Programming. Addison Wesley, Reading Mass., (1997). [8] R. Saha, S. P. Bhattacharyya, Current Science, 86 (2004). [9] A. Papoulis. Probability, Random Variables, and Stochastic Processes (New York: McGraw-Hill), (1965). [10] A. Flores-Riveros, A. Rodriguez-Contreras, Phys. Lett. A, 372 (2008) 6175- 6182. [11] F.J. Gálvez, E. Buendia, A. Sarsa, J. Chem. Phys. 115 (2001) 1166. [12] F.J. Gálvez, E. Buendia, A. Sarsa, Chem. Phys. Lett. 387 (2003) 330. [13] N. Umezawa, S. Tsuneyuki, T. Ohno, K. Shiraishi, T. Chikyow, J. Chem. Phys. 122 (2005) 224101. [14] E. Buendia, F. J. Gálvez, A. Sarsa, Cheml Phys Lett. 428 (2006) 241-244. [15] S. E. Koonin, Computional Physics, San Juan. Wokingham, 1986. [16] T.A. Galek, N. C. Handy, A. J. Cohen, G. K. Chan, Chem. Phys. Lett. 404 (2005) 156–163. [17] B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules, Addison Wisely, New York (1983). [18] S. B. Doma and F. El-Gamal, The Open Applied Mathematics Journal (2009), to appear. 6

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