Solving multi-reference problems with a single-reference coupled- cluster method Yingbin Ge Department of Chemistry Central Washington University 65 th Northwest/22 nd Rocky Mountain Regional Meeting of the American Chemical Society, June 20-23,

Contents Why solve multi-reference problems with a single- reference method? Background of the left-eigenstate completely renormalized coupled-cluster method (CCL). 1-3 Using the CCL method on o Bi-radicals (Eg. H 3 C–H  CH 3 + H). o Tri-radicals (Eg. H 2 C–H  3 CH 2 + H). o Tetra-radicals (Eg. H 2 Si=SiH 2  3 SiH SiH 2 ). o Unimolecular decomposition of ethylene (C 2 H 4 ) P. Piecuch and M. Wloch, J. Chem. Phys. 123, (2005). 2. P. Piecuch, M. Wloch, and J. R. Gour, Chem. Phys. Lett. 418, 467 (2006). 3. M. Wloch, P. Piecuch, and J. R. Gour, J. Phys. Chem. A 111, (2007).

Why solve multi-reference problems with a single-reference method? To study the mechanism of chemical vapor deposition such as silicon carbide CVD. >100 gas-phase reactions at 1000 – 2000 K; many of which involve bi/tri/tetra-radicals. 1 Need an accurate, size-extensive, and inexpensive black-box method to study CVD. 1.Y. Ge, M. S. Gordon, F. Battaglia, and R. O. Fox, JPCA, 111, 1462 & 1475 (2007). 2.Y. Ge, M. S. Gordon, F. Battaglia, and R. O. Fox, JPCA, 114, 2384 (2010). 3

Background of CCL “Left eigenstate completely-renormalized coupled-cluster singles, doubles, and non-iterative triples”, a.k.a. CR-CCSD(T) L or CR-CC(2,3), 1-3 abbreviated as CCL. An accurate, size-extensive, and relatively inexpensive black-box method. RHF and ROHF based CCL are implemented in GAMESS ($contrl cctyp=cr-ccl $end). 1. P. Piecuch and M. Wloch, J. Chem. Phys. 123, (2005). 2. P. Piecuch, M. Wloch, and J. R. Gour, Chem. Phys. Lett. 418, 467 (2006). 3. M. Wloch, P. Piecuch, and J. R. Gour, J. Phys. Chem. A 111, (2007). 4

5 Breaking bonds of closed-shell species in silicon carbide CVD 1 A–B  A + B A = H, Cl, CH 3, SiH 3 B = H, Cl, CH 3, SiH 3 1. Y. Ge, M. S. Gordon, and P. Piecuch, J. Chem. Phys. 127, (2007).

6 H 3 C–SiH 3  CH 3 + SiH 3 CCL, CCSD(T), and FCI energies MINI basis set

7 H 3 C–SiH 3  CH 3 + SiH 3 Errors (in mE h ) relative to FCI/MINI

Quantitative Assessment of the quality of the CCL method?  E CCL (i) = E CCL (i) – E FCI (i) at the i-th structure. NPE: nonparallelity error. NPE CCL = max[  E CCL (1),  E CCL (2), …,  E CCL (N)] – min[  E CCL (1),  E CCL (2), …,  E CCL (N)] REE: reaction energy error. REE CCL = |  E CCL (3R e ) –  E CCL (R e )| 8

999 H 3 C–SiH 3  CH 3 + SiH 3 Errors (in mE h ) relative to FCI/MINI

10 H 3 C–SiH 3  CH 3 + SiH 3 Errors (in mE h ) relative to FCI/MINI NPE REE

11 Nonparallelity error (in mE h ) a. Excluding the H 3 C−Cl data

12 Reaction energy error (in mE h ) a. Excluding the H 3 C−Cl data

Breaking bonds of open-shell species in silicon carbide CVD 1 A–B  A + B A = 3 CH 2, 1 SiH 2 B = H, Cl, CH 3, SiH 3 1. Y. Ge, M.S. Gordon, P. Piecuch, M. Wloch, and J.R. Gour,J. Phys. Chem. 112, (2008). 13

Methods to be compared ROHF-based CCL. UHF-based CCSD(T). Multi-reference MP2 (MRMP2). Basis Sets:  MINI  6-31G  6-31G(d)  cc-pVDZ  cc-pVTZ 14

Benchmark methods Full configuration interaction (FCI) gives exact energy within a given basis set. Full second-order configuration interaction (FSOCI). Internally-contracted multi-reference configuration interaction (MRCI). Davidson quadruple correction for FSOCI(Q) and MRCI(Q). FCI >> FSOCI(Q) >> MRCI(Q) in cost. 15

H 2 C–H  3 CH 2 + H Errors (in mE h ) relative to FCI/6-31G(d) 16

Nonparallelity error (in mE h ) of FSOCI(Q) and MRCI(Q) MRCI(Q) will be used as benchmark when larger basis sets are used. 17

ROCCL vs. UCCSD(T) ROCCL vs. MRMP2 UCCSD(T): often used to treat near- equilibrium open-shell species with little multi- reference character. MRMP2: can be used to treat bond-breaking reactions, bi-radical and tri-radical systems with significant multi-reference character. Eg. H 2 C–H  3 CH 2 + H 18

H 2 C–H  3 CH 2 + H Errors (in mE h ) relative to FCI/6-31G(d) 19

H 2 C–H  3 CH 2 + H Errors (in mE h ) relative to MRCI(Q)/cc-pVTZ 20

H 2 C–H  3 CH 2 + H Nonparallelity error (in mE h ) 21

H 2 C–H  3 CH 2 + H Reaction energy error (in mE h ) 22

Average (over several basis sets) nonparallelity error (in mE h ) 23

Average (over several basis sets) reaction energy error (in mE h ) 24

Average NPE and REE (in mE h ) for bond-breaking reactions of open-shell species with tri-radical character 25

How about tetra-radicals? Neither is good; CCL is better. 26 MINI Basis Set

Quality of the CCL method For bi-radicals, NPE & REE: R-CCL << R-CCSD(T). For tri-radicals, NPE: RO-CCL ≈ MRMP2 < U-CCSD(T). REE: RO-CCL ≈ U-CCSD(T) < MRMP2. CCL is ideal for calculations on PES study such as the unimolecular decomposition pathways of C 2 H 4. 27

Why C 2 H 4  C 2 H 2 + H 2 ? Chemical vapor deposition of diamond, graphite, carbon nanotubes, and silicon carbide. Lack of accurate potential energy surface (PES) of the decomposition of C 2 H 4. C 2 H 4  C 2 H 2 + H 2 : is there a direct path? 28

Computational Methods Geometry optimization calculations at the CCL/cc-pVTZ level. Hessian calculations at CCL/cc-pVTZ level. Harmonic-oscillator/rigid-rotor approximation. Single point energies obtained at the CCL/cc- pV5Z level. 29

30 cc-pVDZcc-pVTZ Unsigned Mean Error Bond length0.017Å0.002Å Bond angle0.7 o 0.3 o Unsigned Max Error Bond length0.025Å0.006Å Bond angle1.5 o 0.4 o Quality of the CCL geometry 1 1. Experimental data of H 2, C 2 H 2, C 2 H 3, and C 2 H 4 molecular geometries are used.

31 Quality of the CCL Energy 1-2 CCL error compared to exp. 1-2 (kJ/mol) cc-pVTZcc-pVQZcc-pV5Z C 2 H 4  C 2 H 2 + H C 2 H 4  H 2 C=C: + H C 2 H 4  C 2 H 3 + H C 2 H 4  C 2 H 2 + 2H -212 Unsigned Mean 433 Unsigned Max Mebel, A. M.; Morokuma, K.; Lin, M. C. JCP 1995, 103, Chang, N. Y.; Shen, M. Y.; Yu, C. H. JCP, 1997, 106, 3237.

32 Quality of CCL vibrational frequency 1 1. Experimental data of H 2, C 2 H 2, C 2 H 3, C 2 H 4, and H 2 C=C: vib. frequencies are used.

C 2 H 4  H 2 C=C: + H 2  C 2 H 2 + H 2 33 C 2 H 4  H 3 C−CH:  C 2 H 2 + H 2

C 2 H 4  H 2 C=CH + H  C 2 H 2 + H 2 34 C 2 H 4  C 2 H 2 + H 2 (2 imaginary frequencies) C 2 H 4  H 2 C=CH + H  C 2 H 2 + H + H

CCL and CASSCF(12,12) imaginary vibrational frequencies for C 2 H 4  C 2 H 2 + H G**6-311G**cc-pVDZ aug-cc- pVDZ cc-pVTZ CCL 1824i 887i 1623i 919i 1643i 861i 1347i 977i 1438i 947i CASSCF 2526i 385i 2331i 581i 2378i 555i 2083i 729i 2074i 663i

CCL/cc-pVTZ imaginary vibrational frequencies for C 2 H 4  C 2 H 2 + H i947i

37 C 2 H 4  H 2 C=C: + H 2  C 2 H 2 + H 2

38 C 2 H 4  H 2 C=CH + H  C 2 H 2 + H + H

Conclusions (I) For bi-radicals, CCL is much more accurate than R- CCSD(T). For tri-radicals, CCL is slightly better than U-CCSD(T) and MRMP2. CCL/cc-pVTZ predicts accurate molecule geometry (0.002Å, 0.3 o ). Both CCL/cc-pVQZ and CCL/cc-pV5Z predicts reaction energy within 4 kJ/mol error. 39

Conclusions (II) Accurate CCL/cc-pV5Z C 2 H 4 PES and Gibbs energy surfaces at CVD temperatures are obtained. There’s no direct path from ethylene to acetylene. The dominant reaction path below 1800 K is C 2 H 4  H 2 C=C: + H 2  C 2 H 2 + H 2. The dominant reaction path above 1800 K is C 2 H 4  H 2 C=CH + H  C 2 H 2 + H + H. 40

Acknowledgements T. Cameron Shore (Central Washington University) Mark S. Gordon (Iowa State University) Piotr Piecuch & Jeff R. Gour (Michigan State University) Marta Wloch (Michigan Tech. University) This work is financially partly supported by the Central Washington University, and partly by the U. S. Department of Energy, Grant No. DE-FC07-05ID14661 and Grant No. DE-FG02-01ER

Questions and Comments? 42

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Experimentally determined reaction energies for [C 2 H 4  C 2 H 2 + H 2 ], [C 2 H 4  H 2 C=C + H 2 ], [C 2 H 4  C 2 H 3 + H], and [C 2 H 4  C 2 H 2 + H + H] are 1.72 eV, 3.63 ‒ 3.76eV (3.63 is used in the table), 4.75 eV, and 6.20 eV, respectively. Chang, N. Y.; Shen, M. Y.; Yu, C. H. Journal of Chemical Physics 1997, 106, Shimanouchi, T. Tables of Molecular Vibrational Frequencies, Consolidated Vol. I, Natl. Stand. Ref. Data Ser.; Natl. Bur. Stand. (US),

47 CCL cc-pVTZcc-pVQZcc-pV5Zcc-pVQZ* C 2 H 4  C 2 H 2 + H C 2 H 4  H 2 C=C + H C 2 H 4  C 2 H 3 + H C 2 H 4  C 2 H 2 + 2H Abs. Mean 4332 Abs. Max. 8544

48 CCL/cc-pVDZ & errorCCL/cc-pVTZ & errorExp. C2H2C2H2 R(C-C) R(C-H) C2H3C2H3 R(C-C) =CHR(C-H) =CH 2 cisR(C-H) =CH 2 transR(C-H) C=CHθ(CCH) C=CH 2 cisθ(CCH) C2H4C2H4 R(C-C) R(C-H) θ(CCH) H2H2 R(H-H) Abs. Mean Error R θ Abs. Max Error R θ

49 C2H4C2H4 Symm.Exp 22,24 CCLExp/theory 1AgAg AgAg AgAg AuAu B 1u B 1u B 2g B 2u B 2u B 3g B 3g B 3u H 2 C=CSymm.Exp 25 CCLExp/theory 1A1A A1A A1A B1B B2B B2B H2H2 Symm.Exp 19 CCLExp/theory 1ΣgΣg C2H2C2H2 Symm.Exp 22 CCLExp/theory 1ΣgΣg ΣgΣg ΣuΣu ΠgΠg ΠuΠu C2H3C2H3 Symm.Exp 23 CCLExp/theory 1A' A' A' A' A' A' A' A" A"

How much data? 8 species: H 2 A–B  H 2 A + B A = C or Si; B = H, Cl, CH 3, or SiH 3 5 basis sets: MINI or MIX, 6-31G, 6-31G(d), cc-pVDZ, cc-pVTZ. 6 methods: FCI, FSOCI(Q), MRCI(Q), CCL, UCCSD(T), MRMP2. 8*5*6=240 potential energy surfaces. 50

How to evaluate the method X?  E X (i) = E X (i) – E benchmark (i) at the i-th structure. NPE: nonparallelity error. NPE X = max[  E X (1),  E X (2) …,  E X (N)] – min[  E X (1),  E X (2) …,  E X (N)] REE: reaction energy error. REE X =  E X (3R e ) –  E X (R e ) STD: standard deviation of errors (1/3 of NPE). STD X = 51

How to get geometries and potential energy surfaces? Optimize structures with the breaking bond distance fixed at every 0.2 Å from R e to 3R e. Use FCI geometries, if possible. Or, use full-valence CASSCF geometries. CCL, UCCSD(T), and MRMP2 single-point energies are obtained to construct PES. 52

CCL/cc-pVQZ* E CCL/cc-pVQZ* = E CCL/cc-pVTZ + (E MP2/cc-pVQZ – E MP2/cc-pVTZ ) Assuming additivity of basis set effect and correlation correction. Additivity approximation is used in G1-4 theory. 53

CCL/cc-pVTZ* E CCL/cc-pVTZ* = E CCL/cc-pVDZ + (E MP2/cc-pVTZ – E MP2/cc-pVDZ ) Assuming additivity of basis set effect and correlation correction. Additivity approximation is used in G1-4 theory. Does G-n like approximation apply to bond- breaking reactions of open-shell species? 54

CCL nonparallelity errors (in mh) relative to MRCI(Q)/cc-pVTZ Smallest errors in each row are in bold font. 55

E(cc-pVTZ)-E(cc-pVDZ) MP2 overestimates basis set effect in the middle; it cancels out CCL error humps. CCL/TZ* is better than CCL/TZ* due to this fortuitous error cancellation. 56

Computational cost of CCL/cc-pVTZ* on a 2GHz machine SiCl 3 + CH 3 SiCl 3  SiCl 4 + CH 3 SiCl 2 cc-pVDZ:173 basis functions. cc-pVTZ: 344 basis functions. CCL/cc-pVTZ*: 1 day. CCL/cc-pVTZ: 36 days. 57

MRCI(Q) nonparallelity error (in mE h ) relative to FSOCI(Q) To evaluate CCL: Small basis sets: FCI as benchmark. Larger basis sets: MRCI(Q) as benchmark, lower cost than FSOCI(Q). 58

H 3 C–H  CH 3 + H Errors (in mE h ) relative to FCI/MINI 59

H 2 Si–H  1 SiH 2 + H Errors (in mE h ) relative to FCI/6-31G(d) 60

H 2 Si–Cl  1 SiH 2 + Cl Errors (in mE h ) relative to MRCI(Q)/cc-pVTZ 61

H 2 Si–Cl  1 SiH 2 + Cl Nonparallelity error (in mE h ) relative to MRCI(Q) 62

H 2 Si–Cl  1 SiH 2 + Cl Reaction energy error (in mE h ) relative to MRCI(Q) 63

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65 H 3 C–SiH 3  CH 3 + SiH 3 Errors (in mE h ) relative to FCI/MINI NPE REE

How about tetra-radicals? 66 MINI Basis Set