Computational Chemistry 5650Spring 2014Test30 points total possibleYou may use software (e.g., GaussView, Gaussian09, Avogadro, Excel) to help answer the problems.

Scores ranged from 10 to 23 out of 30; average was 17.

1. (5 points) Run a Gaussian09 CCSD cc-pVTZ geometry-optimization and vibration-frequency calculation on the hydrogen molecule (restricted, not unrestricted). Fill in the blanks and email your Gaussian log file to me (psiders@d.umn.edu).

Bond length = ____________________ AngstromEnergy = _____________________ HartreeVibration frequency = ______________ cm-1

Answers: R = 0.7426 Angstroms , E=-1.17234 Hartree, frequency = 4409 cm-1.

2. (5 points) Here are Cartesian coordinates for the gas-phase magnesium carbonatemolecule. atom x y z C 0.0 0.000000 -0.701361 O 0.0 1.106170 0.061369 O 0.0 -1.106170 0.061369 O 0.0 0.000000 -1.889571 Mg 0.0 0.000000 1.528569The same coordinates are in the xyz file that is on the class web site. Avogadro (notGaussview) can open that file. www.d.umn.edu/~psiders/courses/chem5650/2014/MgCO3.html

The coordinates were calculated using restricted Hartree-Fock theory with the 6-31G(d) basis set.The coordinates represent a stationary point on the potential energy surface. (The RMS gradient is 0.000004 atomic units, which approximately equals zero.)Do these coordinates represent a local minimum, a transition state, or a hilltop? Run whatever Gaussian09 calculation you need in order to answer the question. Email your log file to me (psiders@d.umn.edu). Write your convincing explanation in words on the test.

Answer: All vibration frequencies are positive, so the stationary point is a local minimum.(Frequencies = 179, 544, 585, 763, 912, 931, 1052, 1271, 2000 cm-1)(E = -462.08132 )

3. (5 points) A small molecular-mechanics force field for the carbonate group in MgCO3 is

V =12

k str∑i=1

3

(RC−Oi−Re)

2

where Re=1.290 Angstroms and kstr=5400 kJ/mol/(Angstrom)2

and RC−Oi stands for the distance between Carbon and the ith Oxygen atom.

(This force field contains only bond-stretching energy, no bending, torsion, or non-bonded terms.)Evaluate V, in kJ/mol, using this force field and the coordinates given in problem 2.

Answer:The three C-O distances are 1.344, 1.344 and 1.188 Angstroms.The squared distances are (1.344-1.290)2=0.002916 (twice) and (1.188-1.290)2=0.010404 A2.V = (0.5)(5400 kJ/mol)( 2 X 0.002916 + 0.010404)V = 2700 X ( 0.016236) = 43.8 kJ/mol

4. (5 points) An extended-Huckel calculation for the BeH2 molecule ( H-Be-H ) using this basis set { 2s on Be, 1s on the first H, 1s on the second H}

led to the following matrices:

This calculation treats the Be 2s electrons and the H 1s electrons; 4 electrons total. All basis functions

are normalized (i.e., ∫ϕi2 dv=1 ). Energies are in units of eV.

The ith molecular orbital (MO) has the form ψi= ci , BeϕBe+ c i , H1ϕH1+ ci , H2ϕH2 .Let ψ1 stand for the lowest-energy MO.

Let ψ2 stand for the second-lowest-energy MO.

a. (2 points) Write the formulas (i.e., showing explicitly the coefficients) for ψ1 and ψ2.

b. (1 point) Calculate the overlap between ψ1 and ψ2, using your MO formulas from part a.

c. (2 points) Sketch the MO diagram; add arrows to representelectrons in the orbitals.

H = (−9.32 −6.02 −6.02−6.02 −13.60 0.00−6.02 0.00 −13.60) E = (

−14.61 0.00 0.000.00 −13.60 0.000.00 0.00 −4.54) C = (

−0.35 0.00 0.86−0.57 0.71 −0.54−0.57 −0.71 −0.54)

S = (1.00 0.30 0.300.30 1.00 0.000.30 0.00 1.00) S−1 /2

= (1.08 −0.17 −0.17

−0.17 1.04 0.04−0.17 0.04 1.04) D = (

1.000 0.000 0.0000.000 1.318 0.0000.000 0.000 0.838)

S1,2=∫ψ1 ψ2 d v =∫(−0.35ϕBe−0.57ϕH1−0.57ϕH2 )(0.71ϕH1−0.71ϕH2)d v

S1,2 =−0.35×0.71∫ϕBeϕH1 dv +0.35×0.71×∫ϕBeϕH2dv

−0.57×0.71∫ϕH12 dv −0.57×(−0.71)∫ϕH2

2 dv

Note: The S matrix shows that ∫ϕH1∫ϕH2dv=0.

S1,2 =−0.35×0.71(SBe , H1−SBe , H2) − 0.57×0.71×(1−1) = 0

ψ1 =−0.35ϕBe−0.57ϕH1−0.57ϕH2ψ2 = 0.71ϕH1−0.71ϕH2

Questions 5-14 are multiple-choice questions. Circle one answer for each.

5. (1 point) The Fock operator F = Hcore + Σ (2 J - K )operates on what?

a) atomic basis functionb) molecular orbitalc) molecular orbital energyd) nuclear spatial position

Answer: b

6. (1 point) The term 2J-K in the Fock operator is from what type of term(s) in the Hamiltonian?a) electron kinetic energyb) electron-electron potential energyc) electron-nuclear potential energyd) spin-orbit coupling

Answer: b

7. (1 point) The Roothaan-Hall equations come from a) diagonalizing the Hamiltonian in a complete basis of atomic orbitals b) minimizing electron-electron repulsionc) substituting a linear combination of atomic orbitals into the Hartree-Fock equationsd) the VWN fit to the exchange energy of a uniform electron gas

Answer: c

8. (1 point) What is true of the Hartree-Fock orbital energies?a) They do not include dynamic correlation energy.b) They equal electron affinities.c) They neglect electron-electron repulsion.d) Two times the sum of occupied-orbital energies equals total electronic energy.

Answer: a

9. (1 point) Here is a Gaussian basis set for the Carbon atom.**** C 0 S 4 1.00 0.000000000000 0.4869669280D+03 0.1772582290D-01 0.7337109420D+02 0.1234778670D+00 0.1641345790D+02 0.4338754000D+00 0.4344983560D+01 0.5615041970D+00 SP 3 1.00 0.000000000000 0.8673525310D+01 -0.1213837487D+00 0.6354538411D-01 0.2096619260D+01 -0.2273384975D+00 0.2982677571D+00 0.6046513290D+00 0.1185173917D+01 0.7621032281D+00 SP 1 1.00 0.000000000000 0.1835578298D+00 0.1000000000D+01 0.1000000000D+01 ****Which one of the following names suits the basis set?

a) 3-21G*b) 4-31Gc) 6-31G(d,p)d) STO-6G

Answer: b

10. (1 point) Which one of the following basis sets is too small for a post-Hartree-Fock calculation? Choose one answer, the smallest.

a) 3-21G*b) 4-31G(d)c) 6-311++G(d,p)d) cc-pVQZ

Answer: a. Answer b is close enough to be acceptable.

11. (1 point) According to the first Kohn-Sham theorem, what is true of the ground-state electron density ρ0?

a) Any ground-state property is a functional of ρ0.b) Density ρ0 describes spatial distribution of electrons but not positions of nuclei.c) Ground-state electron density is a functional of the ground-state energy.d) Potential energy (but not kinetic energy) is a functional of ρ0.

Answer: a (page 449)

12. (1 point) The DFT exchange-correlation functional should supply what energies?a) all electron-electron repulsion energyb) all electron-electron repulsion energy plus a kinetic-energy deviationc) electron-electron repulsion energy of a reference noninteracting-electron gasd) kinetic-energy deviation plus electron-electron repulsion other than classical Coulomb energy

Answer: d (pages 452-453)

13. (1 point) The LDA (or LSDA or SVWN) functional isa) a hybrid functionalb) a meta functionalc) a Minnesota functionald) a pure density functional

Answer: d

14. (1 point) A GGA functional depends upon a gradient. The "gradient" referred to is the gradient of what?

a) the electron densityb) the electronic wave functionc) the nuclear kinetic energyd) the total electronic energy

Answer: a