12_variation_perturbation_matrixhq

N 1 23 a n 1 1 2 2 1 3 5 2n 1 2 n ax 0 x

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Unformatted text preview: 3 2 1 1 1 2 2 3 1 1 2 2 3 2 1 22 2 1 22 0 2 e ax dx 4a 8 Useful integrals n! n 1, 2,3,... a n 1 1/ 2 2 1 3 5 (2n 1) 2 n ax 0 x e dx 2n1 a n a 0 2 1/ 2 0 x n e ax dx x 2 n 1e ax 2 n! dx n 1 2a n 1, n 0,1, 2,3,... 3 10/7/2013 Find Optimal Value of Parameter 3 2 8 parameterized energy expression Find Best Value of Parameter 3 2 8 3 2 81 2 0 18 3 0.2829 Bohr-2 plug this back into energy expression 3 , 8 38 29 2 8 8 9 1 Compare to True Result 1 2 2 0.4244 0.5000 true result variational method result gets ~85% of true result Get Trial Wavefunction 0.2829 Bohr-2 where ∗ Normalize 4 4 42 1 3 5 (2n 1) 2n 1 a n a 1/ 2 x 2 n e ax dx 2 / 1 2 2 / 2 0 0.2765 Compare to true wavefunction . n 1, 2,3,... not 1, so wavefunction gets divided by the square root of this Simplest Gaussian basis function is not that good. Gaussian software uses many. 0.5642 4 10/7/2013 wavefunction (Bohr^-3/2) Compare Trial Wavefunction to True Wave Wavefunction 0.5642 true 0.4 ( r) . 0.2765 trial ( r) 0.2 0 0 1 2 3 4 5 r Radia...
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