Ch120a: Nature of the Chemical Bond - Fall 2016 Problem Set 1 TA: Shane Flynn

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1 h120a: Nature of the hemical ond - Fall 2016 Problem Set 1 T: Shane Flynn (sflynn@caltech.edu) Problem 1: Dirac Notation onversion In Quantum Mechanics various integrals occur so often that Paul Dirac developed a notation system for writing them in a quick/compact manner. I: Dirac Notation onvert the Following Expressions to Dirac Notation h(t) = s(t) b = f (t)g(t)dt f(x) = n W n (x) W n(x )V (x )dx D ˆ φ(x) dx ψ (x ) E d dx e(x) = e (x )l(x )dx F dxf m(x) ˆF f n (x) II: More Dirac Take the following (Dirac) functions and express them in standard integral notation. Where: 1 = φ 1 (r 1 ) 2, 1 = φ 1 (r 2 )φ 2 (r 1 ) (Read this as e 1 at position 2 and e 2 at position 1). You could also define this in the other order which would just change the signs (ie 2, 1 = φ 2 (r 1 )φ 1 (r 2 ). 1 Ĥ 2 = φ 1(r)Ĥφ 2(r) I have inserted, between the functions to (hopefully) clarify the notation. Think of the notation as 2 wavefunctions associated with 2 electrons. The H operator acts on both electrons, a lower case h corresponds to a single electron. 1, 2 2, 1 1

2 1, 2 + 2, 1 1, 2 Ĥ 2, 1 D Now onsider a system where we have 3 electrons (and associated wavefunctions). Please provide all of the permutations for this system (all of the different combinations of the electrons with their functions also know as the antisymmaterizer). So our system has permutations of the form : φ 1 (r 1 )φ 2 (r 2 )φ 3 (r 3 ). III: Dirac Some More (1, 2, 3) Now Expand the Following Functions (your final answer should be in Dirac notation) Here h(1) is an operator acting on electron 1 only, (assume 1 and 2 are orthonormal), H(1,2) acts on both electrons and r 12 acts on both electrons. 1, 2 h(1) h(2) 2, 1 1, 2 + 2, 1 1 r 12 2, 1 e warned this has a-lot of algebra, but it should be simple (tedius) math!!! 1, 2 2, 1 h(1) + h(2) + H(1, 2) 1, 2 2, 1 IV: Normalization Now give the normalization constant for the following expressions assuming: 1 2 = S 1 1 = 2 2 = 1 1, 2 + 2, 1 1, 2 2,

3 V: Integrals onsider again (III.): 1, 2 2, 1 h(1) + h(2) + H(1, 2) 1, 2 2, 1. Take h(1) = , and h(2) = , and H(1, 2) = 1 r 12. Write out the integrals for the Kinetic Energy, oulomb, and the Exchange terms. Problem 2: Optimal Sized Wavefunction onsider the 3-Dimensional trial wavefuntion (un-normalized) for the Hydrogen atom. { 1 r ψ = r 0 r < r 0 0 otherwise Hint: We evaluate φ φ as ψ ψ = r0 0 ( 1 r r 0 ) 2 4πr 2 dr = 2π 15 r3 0 alculate The Kinetic Energy (expectation value) for this trial wavefunction. [ Hint: For the laplacian go to spherical coordinates and evaluate directly: 2 = ( 1 d )] r 2 dr r2 d dr alculate the Potential Energy (expectation value) for this trial wavefunction. Find the value of r 0 that minimizes the total energy. Is the resulting energy higher or lower than the energy of a H atom? What would you expect and why? Problem 3: One Electron Multiple Nuclei onsider the linear molecule H 2+ 3 as depicted below. What is the full Hamiltonian for this system (before the orn-oppenheimer approximation)? What is the electronic Hamiltonian for this system? Why is nuclear motion normally neglected on the timescale of electronic motions? 3

4 onsider the first 3 energy levels for this system from the perspective of the nodal theorem. onsider a basis of only 1s orbitals,, (centered on H, H, and H respectively). What are the (unnormalized) electronic wavefunctions for infinite R and R? (I am asking conceptually what happens to the linear combinations at distance (not for a specific linear combination). What are the LO wavefunctions appropriate for finite R = R? Hint: There are 3, think in terms of nodes and sketch each one. Order theses states by energy and call them ψ 0 (ground state), ψ 1 (first excited state), and ψ 2 (second excited state). The actual functions for this system are: ψ 0 = + + ψ 1 = ψ 2 = α + To make sure these functions are orthonormal solve for the coefficient α. You can write the final coefficients in terms of S =, S =, and S =. Hint ssume = 1, overlaps take the form, and S = S. Problem 4: elow are sketches of 2-Dimensional nodal patterns (think of a planar molecule). onstruct a table to compare the ordering of the energies for each case. My suggestion would be to make a table with each row and column corresponding to a case. Then fill in each cell with either,, =, or ID (insufficient description). In this way clearly the diagonal will be all =. note, ID is due to the fact that the nodal theorem can not be applied to systems that do not have a common type of node (even if they have a different total number of nodes). s was done above, use the nodal theorem to order the 3-Dimensional eigenfunctions having the following forms r = x2 + y 2 + z 2. onsider α to be a scaling paramater that can be reoptimized for each state. For each state sketch the nodal patterns and indicate the total number of nodal surfaces in each wavefunction. xye αr (1) (y 2 1)x 3 e αr (2) (x + y)e iαr (3) 4

5 (x 2 y 2 )e αr (4) (r 1)xyze αr (5) 5