Numerical Solution of Small Molecules Using OneDimensional Functions
用一維函數求解某些小分子體系的數值解
Student thesis: Doctoral Thesis
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Award date  6 May 2020 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(8793e0514f014a89b56f8940e66b46e1).html 

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Abstract
In molecular calculations since the potential energy terms of the Hamiltonian are inseparable into their Cartesian parts, studies within the RayleighRitz variational framework are customarily performed with functions defined not in the Cartesian coordinate system. In association with it, if onedimensional (1D) Cartesian functions are used, the analytical integration will become challenging and purely numerical in nature, hindering their developments for such purposes. Despite the drawbacks described above, numerical studies of 1D Cartesian functions for molecular calculations are still necessary. In this work, we obtain numerical solution of small molecules using onedimensional hydrogen (1D H) functions which are in the family of 1D Cartesian functions. We implement our scheme, for example to obtain H2 ground state, as follows: we discretize and generate the allelectron sparse Hamiltonian matrix representation of small molecules using the standard order finitedifference method, then we construct the electronic trial variation function to describe the ground state using 1D H functions following the molecular orbital treatment, and finally we implement the iteration process which is considerably effective to enhance systematically previously obtained results. Comparing with the results of several standard approaches, we find that the scheme yields significantly more accurate energies and equilibrium bond length with the electron correlation effect included than the standard oneelectron methods in the considered region. The current calculations also satisfy the virial theorem to an accuracy of –V/T=2.0. The result of density is presented. The results of these calculations also demonstrate that the scheme is capable of treating general potentials and basis functions, flexible, and easily implemented without any partitioning of molecular systems into singlecenter terms and without any Fourier transform required, and potential to be developed to calculate larger size molecules.
Additionally, we report on the successful use of onedimensional (1D) Slater functions in the density functional framework as shown in the test calculations of neutral and charged small molecules where total energies at equilibrium separation are comparable with the calculations of the same method performed with the splitvalence basis functions and are consistent with the accurate results. The implemented local density functional calculated initially with the 1D Slater based wavefunction density can identify the role of the loosely and the tightly bound electrons in the bond region and confirm fairly the molecular electronic virial theorem. Furthermore, we find in our algorithm that the required number of selfconsistency loop to converge depends on the grid spacing size to a certain factor and that the computational time scaling behaves as O(N^(5/3)) where N is the number of electrons. As future remarks, the implementation of the current method within DFT framework is very suitable and promising for larger molecules size calculation while the exact treatment of the electronelectron interaction is more proper to be implemented within the variational framework.
Finally, we demonstrate that the residual vector as the basis set serves as a powerful method to accelerate either the SCF convergence for the ground state calculation within the density functional formalism or large matrices diagonalization. The number of SCF loops is significantly reduced with the speedup elapsed time and with the improved accuracy as newly more accurate approximated wavefunction is found using the residual as the basis method where the residual vector is assigned as the basis function rather than simply interpreted as the Newton/relaxation step. Power operation of the Hamiltonian on the residual provides an efficient method to construct more accurate approximated wavefunction in the iterative subspace and is key to obtain more electronic features. From the point of view of matrix diagonalization, the residual as the basis method maps large matrix problem into small matrix problem.
Additionally, we report on the successful use of onedimensional (1D) Slater functions in the density functional framework as shown in the test calculations of neutral and charged small molecules where total energies at equilibrium separation are comparable with the calculations of the same method performed with the splitvalence basis functions and are consistent with the accurate results. The implemented local density functional calculated initially with the 1D Slater based wavefunction density can identify the role of the loosely and the tightly bound electrons in the bond region and confirm fairly the molecular electronic virial theorem. Furthermore, we find in our algorithm that the required number of selfconsistency loop to converge depends on the grid spacing size to a certain factor and that the computational time scaling behaves as O(N^(5/3)) where N is the number of electrons. As future remarks, the implementation of the current method within DFT framework is very suitable and promising for larger molecules size calculation while the exact treatment of the electronelectron interaction is more proper to be implemented within the variational framework.
Finally, we demonstrate that the residual vector as the basis set serves as a powerful method to accelerate either the SCF convergence for the ground state calculation within the density functional formalism or large matrices diagonalization. The number of SCF loops is significantly reduced with the speedup elapsed time and with the improved accuracy as newly more accurate approximated wavefunction is found using the residual as the basis method where the residual vector is assigned as the basis function rather than simply interpreted as the Newton/relaxation step. Power operation of the Hamiltonian on the residual provides an efficient method to construct more accurate approximated wavefunction in the iterative subspace and is key to obtain more electronic features. From the point of view of matrix diagonalization, the residual as the basis method maps large matrix problem into small matrix problem.