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12_variation_perturbation_matrixhq

# Variational theory three examples matrix approaches

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Unformatted text preview: top look now? •Variational Theory, three examples •Matrix Approaches •Perturbation Theory 1 10/7/2013 The energy of any trial wavefunction is always an upper bound to the exact ground state energy, where are the true solutions ∑ Construct trial wavefunction ∗ ∗ by postulate 4, trial function may not be normalized ∗ plug in trial function ∑ ∗ ∗ ∗ ∑ ∑∑ ∑∑ ∑ ∗ ∗ ∑ ∗ ∗ ∑∑ ∗ ∗ ∑∑ ∗ ∑∑ ∗ ∗ ∑∑ ∗ ∗ ∗ ∑ Subtract , the ground state energy, from each side ∗ ∗ , ∗ ∑∑ ∑ Since ∗ ∑∑ ∗ , ∑ ∑ ∗ ∗ every term on rhs is positive, 0 Vary the approximate wavefunction until the energy is minimized. ∗ ∗ 0 that minimize the energy by Find the solving the resulting set of equations. 2 10/7/2013 Variational Method: Example 1 - H Atom Ground State Gaussian, is a variational parameter, Ab initio calculations use Gaussian basis Trial Wavefunction Hamiltonian 1 ℓℓ 2 2 1 in atomic units, 1, 1, 4 1. Note that the values are one, but the units remain 1 1 Hamiltonian for Ground State 1 because ℓ 2 0 Find Expression for Trial Energy ∗ 2 ∗ 2 2 pull out angular parts, cancel differentiate, rearrange 1 2 differentiate by parts 2 3 2 3 3 22 2 3 1 22...
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