12_variation_perturbation_matrixhq

2 e v 2 ev e v 1 05 e v 2 0 0 2 4 6 8 10 x 14 1072013

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Unformatted text preview: tors of H-ES (eigenvector coefficients multiply the corresponding basis functions to give the wavefunctions) in Mathcad E genvals ( H S) c genvecs ( H S) 15 10/7/2013 Variational Method: Example 4 – 2x2 Secular Equation Trial Wavefunction where the basis set has two functions, and which are not orthogonal, and are real Find Expression for Trial Energy ∗ multiply out ∗ matrix elements , , , , , , , ˆ H i , j f i Hf j d , , Si , j f i f j d take derivative with respect to c1 and c2 and set equal to zero Variational Method: Example 4 – 2x2 Secular Equation rearranging , 2 , , , 2 , , , derivative with respect to 2 , , , , 2 , 2 , 2 , , , 2 , , , 2 , , , 2 , , , 2 , , derivative with respect to 2 , , , , 2 , 2 , 2 , 2 , , , , 2 , rearranging the two resulting equations, collecting on , , , , , , , , only has solution if det Secular Equation and 0 matrix form , , , , , , , , 0 16 10/7/2013 H1,1 ES1,1 H1, 2 ES1, 2 0 determinant H ES H 2, 2 ES2, 2 1, 2 1, 2 H ES1,1 H 2, 2 ES 2, 2 H1, 2 ES1, 2 0 2 1,1 multiply out H1,1H 2, 2 EH1,1S 2, 2 ES1,1 H 2, 2 E 2 S1,1S 2, 2 H12, 2 2 EH1, 2 S1, 2 ES12, 2 0 collect terms E 2 S1,1S 2, 2 S12, 2 E 2H1, 2 S1, 2 H1,1S2, 2 H 2, 2 S1,1 H1,1H 2, 2 H12, 2 0 E2 2H E 1, 2 S1, 2 H1,1S 2 , 2 H 2 , 2 S1,1 S 2 1,1 S 2 , 2 S1, 2 H S 2 1,1 H 2 , 2 H1, 2 2 1,1 S 2 , 2 S1, 2 quadratic equation 0 ax 2 bx c 0 x E 2 H1, 2 S1, 2 H1,1S 2, 2 H 2, 2 S1,1 1 2 S1,1S 2, 2 S12, 2 2 b b 2 4ac 2a 2 H1, 2 S1,...
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