23 - Chem 3502/5502 Physical Chemistry II (Quantum...

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Chem 3502/5502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2010 Laura Gagliardi Lecture 23, November 08, 2010 (Some material in this lecture has been adapted from Cramer, C. J. Essentials of Computational Chemistry , Wiley, Chichester: 2002; pp. 204-206, 460-463.) Solved Homework We need to evaluate < H > for the open-shell singlet. Thus 1s2 s H s = 1s 1 ( ) 2s 2 ( ) + 1s 2 ( ) 2s 1 ( ) [ ] α 1 ( ) β 2 ( ) − α 2 ( ) β 1 ( ) [ ] H ( ) 2s 2 ( ) + ( ) 2s 1 ( ) [ ] α 1 ( ) β 2 ( ) − α 2 ( ) β 1 ( ) [ ] = ( ) 2s 2 ( ) + ( ) 2s 1 ( ) [ ] H ( ) 2s 2 ( ) + ( ) 2s 1 ( ) [ ] = 1 2 ( ) 1 2 1 2 ( ) + ( ) 2 r 1 ( ) + ( ) 1 2 2 2 ( ) + ( ) 2 r 2 ( ) + 2s 1 ( ) 1 2 1 2 2s 1 ( ) + 2s 1 ( ) 2 r 1 2s 1 ( ) + 2s 2 ( ) 1 2 2 2 2s 2 ( ) + 2s 2 ( ) 2 r 2 2s 2 ( ) + ( ) 2s 2 ( ) 1 r 12 ( ) 2s 2 ( ) + ( ) 2s 1 ( ) 1 r 12 ( ) 2s 1 ( ) + ( ) 2s 2 ( ) 1 r 12 ( ) 2s 1 ( ) + ( ) 2s 1 ( ) 1 r 12 ( ) 2s 1 ( ) = ε 1s + ε 2s + J 1s2s + K 1s2s Step 1: insert the actual determinant with normalization implicit. Step 2: integrate over spin to arrive at a factor of one since the singlet spin function is normalized. Step 3: Expand the Hamiltonian into its one-electron and two-electron terms. Since each one- electron term allows integration over the other electron's coordinates (to give one or zero) these integrals are simplified. Note, however, that because of the nature of the spatial wave function there are 4 terms involving the 1/ r 12 operator, every one of which is positive. The change in sign for the exchange integrals K , resulting from the "+" in the
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23-2 spatial wave function in contrast to the " " in the spatial wave function of the M S = 0 triplet, is what accounts for the change in sign of K in the final energy expression, and why the open-shell singlet is above the triplet in energy. Perturbation Theory Often in eigenvalue equations, the nature of a particular operator makes it difficult to work with. However, it is sometimes worthwhile to create a more tractable operator by removing some particularly unpleasant portion of the original one. Using exact eigenfunctions and eigenvalues of the simplified operator, it is possible to estimate the eigenfunctions and eigenvalues of the more complete operator. Rayleigh-Schrödinger perturbation theory provides a prescription for accomplishing this. In the general case, we have some operator A that we can write as A = A (0) + λ V (23-1) where A (0) is an operator for which we can find eigenfunctions, V is a perturbing operator, and λ is a dimensionless parameter that, as it varies from 0 to 1, maps A (0) into A . If we expand our ground-state (indicated by a subscript 0) eigenfunctions and eigenvalues as Taylor series in λ , we have Ψ 0 = Ψ 0 (0) + λ ∂Ψ 0 (0) ∂λ λ = 0 + 1 2! λ 2 2 Ψ 0 (0) ∂λ 2 λ = 0 + 1 3! λ 3 3 Ψ 0 (0) ∂λ 3 λ = 0 + (23-2) and a 0 = a 0 (0) + λ a 0 (0) ∂λ λ = 0 + 1 λ 2 2 a 0 (0) ∂λ 2 λ = 0 + 1 λ 3 3 a 0 (0) ∂λ 3 λ = 0 + (23-3) where a 0 (0) is the eigenvalue for Ψ 0 (0) , which is the appropriate normalized ground-state eigenfunction for A (0) . For ease of notation, eqs. 23-2 and 23-3 are usually written as Ψ 0 = Ψ 0 (0) + λΨ 0 (1) + λ 2 Ψ 0 (2) + λ 3 Ψ 0 (3) + (23-4) and a 0 = a 0 (0) + λ a 0 + λ 2 a 0 (2) + λ 3 a 0 (3) + (23-5)
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23-3
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23 - Chem 3502/5502 Physical Chemistry II (Quantum...

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