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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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