ChE 551 Lecture 23 Quantum Methods For Activation Barriers 1

Last Time Three Models Polayni’s model:Marcus equation: 2 (10.14) (10.33) BM

Works Most Of The Time 3

Fails With Quantum Effects: 4 Figure A hypothetical four-centered mechanism for H 2 /D 2 exchange. The dotted lines in the figure denotes mirror planes which are preserved during the reaction (see the text). This reaction is symmetry forbidden.

If Reaction Quantum Limited, Need Quantum Mechanics To Calculate Rates Quantum Methods very useful to engineers codes usable. Today an overview of methods. 5

Quantum Methods For Reaction Rates Solve schroedinger equation (11.39) Calculate Saddle point properties 6

Approximation To Solve Schroedinger Equation Hartree Fock (HF) Approximation Treat each electron as though it moves independently of all others (i.e. In the average field of all others) Configuration Interaction (CI) Consider how motion of each electron affects the motion of all of the other electrons 7

Hartree Fock Approximation 8 = Wavefunction for molecule = Antisymmenizer … One electron wavefunctions

Solution Of HF Equation For Stationary Atoms Kinetic Energy Electron-Electron of Electrons Repulsions Electron Core Exchange Energy Attraction Exchange energy: Extra energy term that eliminates electron-electron repulsion when electrons pair up in a bond. 9 ( E = ( ) + ( ) ( ) ) -

Algebra For Exchange Plug into Schroedinger Equation: 10 (11.55)

Correlation Energy Missing From Hartee Fock Physics: When one atom moves into an area others move out of the way. Leads to a lowering of electron-electron repulsion. Correlation Energy – Lowers the total energy. 11

Approximations Used To Solve Schroedinger Equation ApproximationExchangeCorrelation HFExact0 MP2/3ExactAnalytical Expansion DFTApproximation CIExactExpand in infinite series solve for coefficients CCExactExpand in a series solve for first n coefficients, use analytic expression for higher terms 12

Common Approximations 13 MethodDescription HFOne electron wavefunctions, no correlation energy as described in section CIOne of a number of methods where the configuration interaction is used to estimate the correlation energy as described in section In the literature the CI keyword is sometimes erroneously used to denote the CIS method GVBA minimal CI calculation where the sum in equation includes 2 configurations per bond. The configurations are to improve bond dissociation energies.

Common Approximations Continued 14 MCSCF, CASSC F A CI calculation where the sum in equation includes the all the single excitations of the "active orbitals" and ignores excitations of the "inactive" orbitals. In the limit of large number of configurations, MCSCF gives to the exact result. However, Often people only use a few configurations and still call the calcualtion MCSCF. CISA CI calculation where the sum in equation includes all the single excitations. This method tends to not be very accurate. CIDA CI calculation where the sum in equation includes all the double excitations. CISDA CI calculation where the sum in equation includes all the single and double excitations.

Common Approximations Continued 15 CISDTA CI calculation where the sum in equation includes all the single double and triple excitations. CCSD, QCISD An improved version of a CISD calculation, where the single and double excitations are included exactly, and an approximation is used to estimate the coefficients for the higher order excitations. CCSD(T) QCISD(T) An improved version of a CCSD calculation, where triple excitations are included. MP2, MP3, MP4 A calculation where Moller-Plesset perturbation theory is used to estimate the correlation energy as described in section The various numbers refer to the level of perturbation theory used in the calculation.

Common Approximations Continued 16 G2(MP2)A combined method where you substitute a MP2 calculation for the MP4 calculation in the G2 calculation G2, G3A combined method where you compute the energy as a weighted sum of CCSD(T) with different basis sets, an MP4 calculation with a large basis set, plus other corrections. CBSA different combined method, where you use a series of intermediate calculations to extrapolate the CCSD(T) results to infinite basis set size.

Density Functional Methods: Need Exchange & Correlation 17

Correlation Approximation For DFT 18

Mixed Methods 19

How Well Does Methods Do? 20

Practical Calculations 21

11.A Sample AbInitio Calculations 22 Calculate the equilibrium geometry of ethane at the MP2/6-31+G(d) level.

Solution 23 I decided to solve this problem using GAUSSIAN98. GAUSSIAN calculations are simple. You specify 1) the geometry of the molecule and 2) the computational method. The GAUSSIAN98 program does the rest.

Gaussian Input File #MP2/6-31+g(D) OPT TEST Ethane molecule 0 CHARGE 1~SPIN MULTIPLIERS C C 1 RCC H 1 RCH 2 ACH H 1 RCH 2 ACH 3 DHH H 1 RCH 2 ACH 4 DHH H 2 RCH 1 ACH 3 DCT H 2 RCH 1 ACH 6 DHH H 2 RCH 1 ACH 7 DHH RCC 1.87 RCH 1.09 ACH 111. DHH 120. DCT

Spin Multipliers Spin Multipliers = 2(net spin) + 1 Spin = + ½ One electron spin = ½ spin multipliers – 2 Two electrons spin = 0 or 1 spin multiplier = 1 or 3 25

The “Z-matrix” 1) Define atoms 2) Define atoms and bonds RCC = the carbon-carbon bond length RCH = the carbon-hydrogen bond length ACH = the CH bond angle DCH = the rotation angle between one hydrogen atom and the adjacent carbon atom DCT = the rotation of the methyl group on one side of the molecule relative to the carbon atom on the other side of the molecule. 26

Gaussian Input File #MP2/6-31+g(D) OPT TEST Ethane molecule 0 CHARGE 1~SPIN MULTIPLIERS C C 1 RCC H 1 RCH 2 ACH H 1 RCH 2 ACH 3 DHH H 1 RCH 2 ACH 4 DHH H 2 RCH 1 ACH 3 DCT H 2 RCH 1 ACH 6 DHH H 2 RCH 1 ACH 7 DHH RCC 1.87 RCH 1.09 ACH 111. DHH 120. DCT

Output from GAUSSIAN For The Input Shown (Abbreviated) Final structure in terms of initial Z-matrix: C C,1,RCC H,1,RCH,2,ACH H,1,RCH,2,ACH,3,DHH,0 H,1,RCH,2,ACH,4,DHH,0 H,2,RCH,1,ACH,3,DCT,0 28

Output Continued H,2,RCH,1,ACH,6,DHH,0 H,2,RCH,1,ACH,7,DHH,0 Variables: RCC= RCH= ACH= DHH=120. DCT=60. 29

Output Continued 1\1\ NATIONAL CENTER FOR SUPERCOMPUTING APPLICATIONS- BILLIE\FOpt\RMP2-FC\6- 31+G(d)\C2H6\RMASEL\30-Jun- 1999\0\\#MP2/6-31+G(D) OPT TEST\\Ethane molecule\\0,1\C,0.,0., \C,0.,0., \H, ,0., \H, , , \H, , , \H, , , \H, , 0., \H, , , \\Version=HP- 30

Output Continued PARisc-HPUX-G98RevA.6\HF= \MP2= \RMSD=3.599e-09\RMSF=1.313e- 04\Dipole=0.,0.,0.\PG=D03D LOVE IS BLIND, THAT'S WHY ALL THE WORLD LOVES A LOUVER. Job cpu time: 0 days 0 hours 1 minutes 34.6 seconds. File lengths (MBytes): RWF= 14 Int= 0 D2E= 0 Chk= 6 Scr= 1 Normal termination of Gaussian

Example 11.B Energy Calculations Use the results in table 11.A.2 to estimate the total energy and the correlation energy of ethane. 32

Solution: GAUSSIAN lists the energies in the output block at the end of the program. The HF= is the hartree fock energy in hartrees, MP2= is the MP2 energy in hartrees, where 1 hartree = kcal/mole. 33

Solution: From the output MP2= hartrees. Therefore the total energy is given by Total energy = hartrees  (kcal/mole/hartree)= kcal/mole. Again from the output HF= HF= hartrees  (kcal/mole/hartree)= kcal/mole 34

Correlation Energy The correlation energy is the difference between the HF and MP2 energy Correlation energy = HF-MP2 = ( )= kcal/mole. The correlation energy is only / = 0.3% of the total energy. However, the correlation energy is still -168 kcal/mole so it cannot be ignored. 35

Query What did you learn new in this lecture? 36