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Ground state properties of first row atoms: Variational Quantum Monte Carlo

This research paper explores the use of Variational Quantum Monte Carlo (VMC) method to optimize wave functions for calculating ground state properties of first-row atoms. It discusses the VMC approach, trial wave function optimization using methods such as steepest descent and Newton's method, and the calculation of local energy. The paper also discusses the use of Slater determinants and Jastrow functions in VMC, as well as the application of genetic algorithms for wave function optimization. Results for the optimization of energy for Be, Li, and He atoms using both steepest descent and Newton's method are presented.

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Ground state properties of first row atoms: Variational Quantum Monte Carlo

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  1. Ground state properties of first row atoms: Variational Quantum Monte Carlo Ebrahim FoulaadvandZanjan University, Zanjan, Iran

  2. Outline Variational Quantum Monte Carlo Quantum Monte Carlo Wave functions Wave Functionoptimization by Steepest Descent method Optimization Trial Wave Function by Newton’s method

  3. Variational Principle Ground state energy Local energy Variational Quantum Monte Carlo (VMC)

  4. Hamiltonian of Many-body particle systems Trial Wave function Slater determinants Jastrow function Variational Quantum Monte Carlo

  5. Slater determinants Variational Quantum Monte Carlo

  6. Slater Type Orbitals (STO) Gaussian Type Orbitals (GTO) Quantum Monte Carlo Wave function

  7. Two body Jastrow-Pade function *Smith and Moskovitz approch K. E. Schmidt and J. W. Moskowitz, J. Chem. Phys. 93, 4172 (1990) Quantum Monte Carlo (QMC) Wave function

  8. Cusp Conditions Slater Determinat Jastrow function Quantum Monte Carlo Wave function

  9. Variational Quantum Monte Carlo Calculating Local Energy

  10. Initial Trial wave function Initial Configuration Initial Parameters Cusp conditions Propose a move Evaluate Probability ratio Update electron position Yes Metropolis No Calculate Local Energy Output Variational Quantum Monte Carlo

  11. Direct Methods In this method we must calculate first and second energy derivatives respect to parameters 1 Newoton’s method Steepest Descent Conjugate Gradient Indirect methods 2 Variance minimization Genetic algorithm as a new method in QMC wave function optimization 1 Xi Lin, Hongkai Zhang and Andrew M. Rappe, J. Chem. Phys. 112, 2650 (2000). 2 C. J. Umrigar, K. G. Wilson and J. W. Wilkins, Phys. Rev. Lett. 60, 1719 (1988). Optimization Quantum Monte Carlo Wave functions

  12. Optimization by Steepest Descent Steepest Descent Method

  13. Steepest Descent Method

  14. Optimization Trial Wave Function by Steepest Descent method Energy of Be atom versus iteration

  15. Optimization Trial Wave Function by Steepest Descent method Li He

  16. Optimization QMC Wave function by Steepest Descent Method

  17. Optimization QMC Wave function by Steepest Descent Method Error in Quantum Monte Carlo Calculations Be

  18. Newton’s Method

  19. Newton’s Method Second energy derivative respect to variational parameters

  20. Newton’s Method

  21. We must propose an algorithm to calculate these values Newton’s Method

  22. 1. Singular Value Decomposition (SVD) 2. We determine the eigenvalues of the Hessian and add to the diagonal of the Hessian the negative of the most negative eigenvalue plus a constant Newton’s Method Singularity in Hessian Matrix

  23. Newton’s Method Details in Calculations by Newton’s method We have chosen initial parameters randomly. The first six iterations employ a very small Monte Carlo samples, NMC=300000, and a-diag =2 For each of the next six iterations we increase NMC and decrease a-diag by a multiplicative factor of 0.1 The remaining 11 iterations are performed with the values at the end of this process, namely, NMC= 2000000, and a-diag = 0.002

  24. Optimization QMC wave function by Newton’s Method Be Li He

  25. Energy Minimization Results Ground State Energy by Newton’s method Ground State Energy by Steepest Descent method S. J. Chakravorty, S. R. Gwaltney, E. R. Davidson, F. A. Parpia, and C. F. Fischer, Phys. Rev. A 47, 3649 (1993). K. E. Schmidt and J. W. Moskowitz, J. Chem. Phys. 93, 4172 (1990)

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