1. 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, 2010 1

2. 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 2 + 3 SiH 2 ). o Unimolecular decomposition of ethylene (C 2 H 4 ). 2 1. P. Piecuch and M. Wloch, J. Chem. Phys. 123, 224105 (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, 11359 (2007).

3. 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

4. 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, 224105 (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, 11359 (2007). 4

5. 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, 174106 (2007).

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

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

8. 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

9. 999 H 3 C–SiH 3  CH 3 + SiH 3 Errors (in mE h ) relative to FCI/MINI

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

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

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

13. 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, 11873 (2008). 13

14. 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

15. 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

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

17. 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

18. 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

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

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

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

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

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

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

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

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

27. 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

28. 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

29. 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. 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. 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 2 333 C 2 H 4  H 2 C=C: + H 2 -4-2 C 2 H 4  C 2 H 3 + H -8-5-4 C 2 H 4  C 2 H 2 + 2H -212 Unsigned Mean 433 Unsigned Max. 854 1.Mebel, A. M.; Morokuma, K.; Lin, M. C. JCP 1995, 103, 7414. 2.Chang, N. Y.; Shen, M. Y.; Yu, C. H. JCP, 1997, 106, 3237.

32. 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.

33. 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

34. 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

35. CCL and CASSCF(12,12) imaginary vibrational frequencies for C 2 H 4  C 2 H 2 + H 2 35 6-31G**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

36. CCL/cc-pVTZ imaginary vibrational frequencies for C 2 H 4  C 2 H 2 + H 2 36 1438i947i

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

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

39. 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

40. 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

41. 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-FG02-01ER15228. 41

42. Questions and Comments? 42

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46. 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, 3237. Shimanouchi, T. Tables of Molecular Vibrational Frequencies, Consolidated Vol. I, Natl. Stand. Ref. Data Ser.; Natl. Bur. Stand. (US), 1972. 46

47. 47 CCL cc-pVTZcc-pVQZcc-pV5Zcc-pVQZ* C 2 H 4  C 2 H 2 + H 2 3332 C 2 H 4  H 2 C=C + H 2 -4-2 -3 C 2 H 4  C 2 H 3 + H -8-5-4 C 2 H 4  C 2 H 2 + 2H -2121 Abs. Mean 4332 Abs. Max. 8544

48. 48 CCL/cc-pVDZ & errorCCL/cc-pVTZ & errorExp. C2H2C2H2 R(C-C) 1.228+0.0251.209+0.0061.203 R(C-H) 1.079+0.0161.063+0.0001.063 C2H3C2H3 R(C-C) 1.334+0.0181.316 0.0001.316 =CHR(C-H) 1.096+0.0161.079-0.0011.080 =CH 2 cisR(C-H) 1.105+0.0201.090+0.0051.085 =CH 2 transR(C-H) 1.100+0.0151.0850.0001.085 C=CHθ(CCH) 135.8-1.5136.9-0.4137.3 C=CH 2 cisθ(CCH) 121.3-0.2121.3-0.2121.5 C2H4C2H4 R(C-C) 1.351+0.0141.336-0.0031.339 R(C-H) 1.098+0.0131.083-0.0031.086 θ(CCH) 121.5+0.3121.5+0.3121.2 H2H2 R(H-H) 0.7610+0.01960.7427+0.00130.7414 Abs. Mean Error R 0.0170.002 θ 0.70.3 Abs. Max Error R 0.0250.006 θ 1.50.4

49. 49 C2H4C2H4 Symm.Exp 22,24 CCLExp/theory 1AgAg 30263160 0.9576 2AgAg 16231677 0.9678 3AgAg 13421372 0.9781 4AuAu 10231050 0.9743 5B 1u 299031420.9515 6B 1u 144414810.9747 7B 2g 9439470.9958 8B 2u 310632490.9558 9B 2u 8108250.9822 10B 3g 310332220.9631 11B 3g 123612440.9936 12B 3u 9499700.9786 H 2 C=CSymm.Exp 25 CCLExp/theory 1A1A1 30253134 0.9652 2A1A1 16351668 0.9802 3A1A1 11651229 0.9479 4B1B1 835 7511.1119 5B2B2 3050 32250.9457 6B2B2 320 3450.9275 H2H2 Symm.Exp 19 CCLExp/theory 1ΣgΣg 44014409 0.9982 C2H2C2H2 Symm.Exp 22 CCLExp/theory 1ΣgΣg 33743515 0.9599 2ΣgΣg 19742009 0.9826 3ΣuΣu 328934140.9634 4ΠgΠg 6125851.0462 5ΠuΠu 7307500.9733 C2H3C2H3 Symm.Exp 23 CCLExp/theory 1A'32353249 0.9957 2A'31643182 0.9943 3A'31033078 1.0081 4A'17001620 1.0493 5A'12771398 0.9134 6A'10991072 1.0252 7A'758727 1.0426 8A"955919 1.0392 9A"895794 1.1272

50. 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

51. 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

52. 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

53. 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

54. 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

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

56. 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

57. 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

58. 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

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

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

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

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

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

64. 64

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

66. How about tetra-radicals? 66 MINI Basis Set