1. ©2013, Jordan, Schmidt & Kable Lecture 15 Lecture 15 Density Functional Theory & Potential Energy Surfaces Chemistry = f(x 1,y 1,z 1,…x N,y N,z N )

2. ©2013, Jordan, Schmidt & Kable Lecture 15 Copyright Notice Some images used in these lectures are taken, with permission, from “Physical Chemistry”, T. Engel and P. Reid, (Pearson, Sydney, 2006); denoted “ER” throughout the lectures and other sources as indicated, in accordance with the Australian copyright regulations.

3. ©2013, Jordan, Schmidt & Kable Lecture 15 15.1 Familiar with existence of density function theory. 15.2 Can follow derivation of simple functional. 15.3 Understand that the “true” functional is not known, but that B3-LYP is popular. 15.4 Can paraphrase the adiabatic Born-Oppenheimer approximation. 15.5 Can interpret potential energy curves and surfaces, drawn in “3d” or as contour plots. Learning outcomes

4. ©2013, Jordan, Schmidt & Kable Lecture 15 Configuration Interaction for H 2 We do a complete active space self-consistent field (CAS-SCF) theory calculation, to allow both configurations while simultaneously optimizing the orbitals, with the same basis as before. 11 1  11 At 0.9Å ITER TOTAL ENERGY DEL(E) LAG.ASYMM. SQCDF MICIT DAMP 1 -1.125049420 -1.125049420 0.160266 1.377E-02 1 0.0000 2 -1.141745458 -0.016696038 0.004000 1.652E-04 1 0.0000 3 -1.141768656 -0.000023198 0.000164 7.577E-07 1 0.0000 4 -1.141768724 -0.000000069 0.000020 1.318E-08 1 0.0000 5 -1.141768726 -0.000000001 0.000003 2.313E-10 1 0.0000 -------------------- LAGRANGIAN CONVERGED -------------------- FINAL MCSCF ENERGY IS -1.1417687256 AFTER 5 ITERATIONS -MCCI- BASED ON OPTIMIZED ORBITALS ---------------------------------- CI EIGENVECTORS WILL BE LABELED IN GROUP=D2H PRINTING ALL NON-ZERO CI COEFFICIENTS STATE 1 ENERGY= -1.1417687256 S= 0.00 SZ= 0.00 SPACE SYM=AG ALPH|BETA| COEFFICIENT ----|----|------------ 10 | 10 | 0.9900962 01 | 01 | -0.1403909 HF energy was -1.11832. CASSCF energy is -1.14177 E h Variational Principle!!

5. ©2013, Jordan, Schmidt & Kable Lecture 15 Configuration Interaction for H 2 CI wavefunction decribes dissociation (chemistry!) because it accounts for CORRELATION. Wavefunction at equilibrium is well described by a single configuration (determinant). ALPH|BETA| COEFFICIENT ----|----|------------ 10 | 10 | 0.8803840 01 | 01 | -0.4742616 ALPH|BETA| COEFFICIENT ----|----|------------ 10 | 10 | 0.9947259 01 | 01 | -0.1025694

6. ©2013, Jordan, Schmidt & Kable Lecture 15 The Quantum Chemistry Landscape Level of correlation (# of configurations) Quality of basis set Sensible compromise The answer 1 st year chemistry Hartree-Fock limit pointless

7. ©2013, Jordan, Schmidt & Kable Lecture 15 Density Functional Theory There is a unique ground state electron density for a given configuration of nuclei. The reverse has been shown to be true: “The ground state properties are uniquely determined by the electron density,  (x,y,z)” – Hohenberg-Kohn Theorem (1938). Problem is, we don’t know the energy functional…

8. ©2013, Jordan, Schmidt & Kable Lecture 15 Density Functional Theory We do know the “how many electrons do we have?” functional: To see how one might find an energy functional, consider an electron in a 1-dimensional box:

9. ©2013, Jordan, Schmidt & Kable Lecture 15 Density Functional Theory Put N up spin, non-interacting electrons in the box:

10. ©2013, Jordan, Schmidt & Kable Lecture 15 Density Functional Theory Put N up spin, non-interacting electrons in the box: So, for non-interacting electrons we can know the kinetic energy functional.

11. ©2013, Jordan, Schmidt & Kable Lecture 15 Density Functional Theory What about a uniform electron gas in 3 dimensions? What functional of the density will give us the right answer?

12. ©2013, Jordan, Schmidt & Kable Lecture 15 Density Functional Theory What about a uniform electron gas in 3 dimensions? So, for non-interacting electrons we can know the kinetic energy functional in three dimensions.

13. ©2013, Jordan, Schmidt & Kable Lecture 15 Density Functional Theory The Thomas-Fermi uniform electron gas Application of this functional with additional integrals for potential energy comprises the Local Density Approximation (LDA). ITER EX TOTAL ENERGY E CHANGE DENSITY CHANGE DIIS ERROR INTEGRALS SKIPPED 1 0 -0.413722710 -0.413722710 0.151527477 0.000000000 55 0 2 1 -0.456028236 -0.042305525 0.005090780 0.000000000 55 0 3 2 -0.456371938 -0.000343703 0.006527127 0.000000000 55 0 4 3 -0.456671223 -0.000299285 0.005025954 0.000000000 55 0 5 4 -0.456796724 -0.000125501 0.003470249 0.000000000 55 0 6 5 -0.456854524 -0.000057800 0.003558977 0.000000000 55 0 7 0 -0.456900580 -0.000046056 0.002676611 0.000000000 55 0 8 1 -0.456911841 -0.000011260 0.002581048 0.000000000 55 0 9 2 -0.456917172 -0.000005331 0.001364825 0.000000000 55 0 10 3 -0.456917991 -0.000000819 0.001334560 0.000000000 55 0 11 4 -0.456918307 -0.000000316 0.000077150 0.000000000 55 0 12 5 -0.456918308 -0.000000001 0.000003032 0.000000000 55 0 13 6 -0.456918308 0.000000000 0.000000550 0.000000000 55 0 H atom energy with 6-311++G basis is not so great with LDA.

14. ©2013, Jordan, Schmidt & Kable Lecture 15 Density Functional Theory If LDA does not work, perhaps correct for gradients. H is not that uniform! Here is the Gill 96 functional. NONZERO BLOCKS ITER EX TOTAL ENERGY E CHANGE DENSITY CHANGE DIIS ERROR INTEGRALS SKIPPED 1 0 -0.467519527 -0.467519527 0.209366420 0.000000000 55 0 2 1 -0.497883555 -0.030364029 0.012770841 0.000000000 55 0 3 2 -0.498140065 -0.000256509 0.019213722 0.000000000 55 0 4 3 -0.498427939 -0.000287874 0.007190400 0.000000000 55 0 5 0 -0.498515857 -0.000087918 0.000514301 0.000000000 55 0 6 1 -0.498508611 0.000007246 0.001330888 0.000000000 55 0 7 2 -0.498525247 -0.000016636 0.003183988 0.000000000 55 0 8 3 -0.498562464 -0.000037217 0.006040991 0.000000000 55 0 9 4 -0.498627742 -0.000065278 0.006805444 0.000000000 55 0 10 5 -0.498695072 -0.000067330 0.010208524 0.000000000 55 0 11 6 -0.498796099 -0.000101027 0.007576321 0.000000000 55 0 12 7 -0.498856527 -0.000060428 0.006507510 0.000000000 55 0 13 8 -0.498894418 -0.000037891 0.005362436 0.000000000 55 0 14 0 -0.498915737 -0.000021318 0.010644004 0.000000000 55 0 15 1 -0.498930292 -0.000014555 0.001737926 0.000000000 55 0 16 2 -0.498931125 -0.000000834 0.000363614 0.000000000 55 0 17 3 -0.498931146 -0.000000021 0.000048746 0.000000000 55 0 18 4 -0.498931146 -0.000000001 0.000009367 0.000000000 55 0 19 5 -0.498931146 0.000000000 0.000001416 0.000000000 55 0 Gill works pretty well, but not perfect.

15. ©2013, Jordan, Schmidt & Kable Lecture 15 Density Functional Theory But, Gill96 takes no account of correlation, E(2.0Å)=-0.6370032368 But don’t worry, Gill knows what he’s doing. HF [2,2] CI

16. ©2013, Jordan, Schmidt & Kable Lecture 15 Density Functional Theory A very nice functional which take everything into account is B3-LYP Not perfect, but better dissociation than HF. Not rigorous, not variational. HF [2,2] CI B3-LYP

17. ©2013, Jordan, Schmidt & Kable Lecture 15 The adiabatic Born-Oppenheimer approximation Curves like this form a landscape on which molecules vibrate and react. Implicitly, we assume that the electrons instantaneously adapt to the positions of the nuclei. This is the adiabatic Born-Oppenheimer approximation. This approximation is the foundation of chemical physics, and only breaks under certain circumstances, for instance where two electronic states are close in energy. Internal conversion is an example of Born-Oppenheimer breakdown.

18. ©2013, Jordan, Schmidt & Kable Lecture 15 Potential energy surfaces – electronic states Here are pictured the various low- lying electronic states of C 2. Each has its own potential energy curve and is described well by one configuration at equilibrium. However, as the molecule dissociates, there are many instances where CI is important. The emission from d to a is responsible for the blue coloration of lean flames and was observed in comets since the 19 th century. Scott will teach you all about that stuff.

19. ©2013, Jordan, Schmidt & Kable Lecture 15 Potential energy surfaces – reactions Here is pictured the classic potential energy surface (PES) for a linear harpoon reaction with a barrier: A + BC → AB + C. The reaction barrier appears as a mountain pass, and is called a saddle point – it has one direction with negative curvature, like a saddle. If the PES is known with sufficient accuracy, then dynamical calculations can be made which predict the outcome of chemical reactions. Meredith like this. Scott likes Meredith doing this.

20. ©2013, Jordan, Schmidt & Kable Lecture 15 Contour Plots of acetylene H-C ☰ C-H Excited state acetylene Ground state acetylene (draw yourself)

21. ©2013, Jordan, Schmidt & Kable Lecture 15 Very strange world of Jahn-Teller In the E” excited state of phenalenyl radical, there are e’ vibrational modes which lift the electronic degeneracy. The potential energy surface splits into two at a conical intersection. An adiabatic walk around the conical intersection accompanies a change in sign of the electronic wavefunction… BO approximation is completely broken here. (Tim likes doing this…)