12_variation_perturbation_matrixhq

2 0 0 1 2 3 4 5 r radial distance bohr trial

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Unformatted text preview: l Distance (Bohr) • Trial wavefunction (blue) gets ~85% of true energy • integrations are easier with Gaussians e r / a0 than simple exponentials e so Gaussian functions form the basis sets of many ab initio calculations 2 2 r / a0 How do we use Gaussian functions to make a better basis set? true Trial - one Gaussian Three Gaussians true typical basis sets of programs like Gaussian are constructed like this Cramer, Essentials of Computational Chemistry, p.158 0.5642 Three Gaussians trial 0.2765 . 5 10/7/2013 Variational Method: Example 2 – Harmonic Oscillator 1 Trial Wavefunction Lorentzian (true form is Gaussian) 1 Hamiltonian Need second derivative of , so try Mathcad symbolics. Type in function, put the cursor around the x, and go to “Symbolics”, “Variable”, “Differentiate”. You get . Again, put the cursor around the x, and go to “Symbolics”, “Variable”, “Differentiate”. You get 2 8 1 1 Find Expression for Trial Energy ∗ 1 1 ∗ 1 2 2 1 1 substitute unnormalized wavefunction and Hamiltonian 1 1 use 2 1 1 2 1 1 1 2 1 2 1 3 / 4 16 2 / 2 1 1 8 1 8 8 1 23 dx 3 8 1/ 2 This integral has a typo on p. 246 1 22 1 1 x / 4 2 x2 1 x 2 2 dx 1 1 x 2 2 2 3/...
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