Use the Variational method to find the expression for first excited state energy of the hydrogen atom.
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Step 1: The variational method involves choosing a trial wavefunction and minimizing the expectation value of the Hamiltonian with respect to the parameters in the trial wavefunction. Show more…
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Use the variational method to determine the energies of the hydrogen atom and the helium atom with a sample function of a normalized 1s Gaussian basis function. The only parameter that can be optimized is the exponent alpha. The integrals needed for energy evaluation using ̈̈ are provided with atomic units: - Kinetic energy integral for one electron: - Electron-core potential energy integral for one electron: - The repulsive energy integral between two electrons: The trial energy expression of the hydrogen atom: E(alpha). a) Use the variation method to optimize the exponent alpha and find the minimum energy of the hydrogen atom. Compare with the analytical solution for E in atomic units. What is the relative error?
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