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12_variation_perturbation_matrixhq

# 8 27 in atomic units 8 8 1072013 variational method

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Unformatted text preview: Value of Parameter 2 27 8 27 16 0 effective nuclear charge; formal charge is 2 plug this back into energy expression , 27 8 27 16 27 27 8 16 27 16 Most Accurate Value (ignoring repulsion 2.8477 2.9037 4.000 ) Effect of Electron-Electron Interaction H-like 1s with Z=2 •whenever you add electrons to energy diagrams, the energy levels change •we generally don’t redraw these diagrams because it is understood that the energies change 9 10/7/2013 Wavefunction as a Column Matrix Given a basis set of functions, , that are orthonormal ( ∗ , ), wavefunctions ( can be expressed as coefficients multiplying the basis functions. as analytical function or as a column matrix ∑ ∗ Ket as vector | ∗ ⋮ ⋮ Recipe for getting the coefficients 10 10/7/2013 Hamiltonian Operator as a Square Matrix Given an orthonormal basis set, matrix elements are 1,2, … , the Hamiltonian for ∗ , Recipe for getting the matrix elements The Hamiltonian matrix will be ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Hamiltonian Operator as a Square Matrix Particle-in-Box Example: and evaluate matrix elements ∗ , ∗ 2 1 0 2 0 0 2 0 2 ⋮ 0 ⋯ 0 2 0 2 3 2 0 0 0 Eigenvalues, 0 ∗ 2 , When basis set is eigenfunction of , then H is diagonal (with the eigenvalues on the diagonal). 0 ⋱ ⋯ ⋮ 2 A diagonal H is a solved problem. Solving a problem is equivalent to diagonalizing H. , on diagonal 11 10/7/2013 Ramp Potential Hamiltonian Matrix whe...
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