Solving multi-reference problems with a single-reference coupled-

cluster method

Yingbin Ge

Department of Chemistry

Central Washington University

65th Northwest/22nd Rocky Mountain Regional Meeting of the American Chemical Society, June 20-23, 2010

1

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 ono Bi-radicals (Eg. H3C–H CH3 + H).o Tri-radicals (Eg. H2C–H 3CH2 + H).o Tetra-radicals (Eg. H2Si=SiH2 3SiH2 + 3SiH2).o Unimolecular decomposition of ethylene (C2H4).

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).

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, 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

Breaking bonds of closed-shell species in silicon carbide CVD1

• A–B A + B

• A = H, Cl, CH3, SiH3

• B = H, Cl, CH3, SiH3

1. Y. Ge, M. S. Gordon, and P. Piecuch, J. Chem. Phys. 127, 174106 (2007).

6

H3C–SiH3 CH3 + SiH3 CCL, CCSD(T), and FCI energies

MINI basis set

7

H3C–SiH3 CH3 + SiH3 Errors (in mEh) relative to FCI/MINI

Quantitative Assessment of the quality of the CCL method?

ECCL(i) = ECCL(i) – EFCI(i) at the i-th structure.

• NPE: nonparallelity error.• NPECCL = max[ECCL(1), ECCL(2),…, ECCL(N)] –

min[ECCL(1), ECCL(2),…, ECCL(N)]

• REE: reaction energy error.• REECCL = |ECCL(3Re) – ECCL(Re)|

8

999

H3C–SiH3 CH3 + SiH3 Errors (in mEh) relative to FCI/MINI

101010

H3C–SiH3 CH3 + SiH3 Errors (in mEh) relative to FCI/MINI

NPEREE

11

Nonparallelity error (in mEh)

CCL CCSD(T) H−Cl 0.2 6.8 Cl−Cl 0.9 8.7 H3C−H 1.1 1.2 H3C−Cl 1.1 182.8 H3C−CH3 2.5 2.7 H3C−SiH3 1.3 4.6 H3Si−H 0.1 1.8 H3Si−Cl 1.6 14.3 H3Si−SiH3 0.4 3.7 Average 1.0 25.2/5.5a

a. Excluding the H3C−Cl data

1212

Reaction energy error (in mEh)

CCL CCSD(T) H−Cl 0.1 6.6 Cl−Cl 0.2 8.4 H3C−H 1.0 1.0 H3C−Cl 0.6 172.2 H3C−CH3 2.1 2.3 H3C−SiH3 1.2 4.2 H3Si−H 0.1 1.6 H3Si−Cl 1.4 14.0 H3Si−SiH3 0.4 3.2 Average 0.9 23.7/5.2a

a. Excluding the H3C−Cl data

Breaking bonds of open-shell species in silicon carbide CVD1

• A–B A + B

• A = 3CH2, 1SiH2

• B = H, Cl, CH3, SiH3

1. Y. Ge, M.S. Gordon, P. Piecuch, M. Wloch, and J.R. Gour,J. Phys. Chem. 112, 11873 (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

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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

H2C–H 3CH2 + H Errors (in mEh) relative to FCI/6-31G(d)

16

Nonparallelity error (in mEh) of FSOCI(Q) and MRCI(Q)

• MRCI(Q) will be used as benchmark when larger basis sets are used.

Species Basis FSOCI(Q) MRCI(Q)

H2C−H 6-31G 0.10 0.04 6-31G(d) 0.14 0.05

H2Si−H 6-31G 0.06 0.06 6-31G(d) 0.40 0.28

Average 0.18 0.11

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. H2C–H 3CH2 + H

18

H2C–H 3CH2 + H Errors (in mEh) relative to FCI/6-31G(d)

0

4

8

12

1 1.5 2 2.5 3 3.5 4

E -

E(F

CI)

(m

Eh)

R (angstrom)

H2C-H CCL

UCCSD(T)

MRMP2

19

H2C–H 3CH2 + H Errors (in mEh) relative to MRCI(Q)/cc-pVTZ

0

4

8

12

1 1.5 2 2.5 3 3.5 4

E -

E[M

RC

I(Q

)] (

mE

h)

R (angstrom)

H2C-H

CCL

UCCSD(T)

MRMP2-4mh

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H2C–H 3CH2 + H Nonparallelity error (in mEh)

Benchmark

CCL UCCSD(T) MRMP2 FCI/6-31G 1.7 2.1 1.9 FCI/6-31G(d) 3.3 3.4 3.4 MRCI(Q)/cc-pVDZ 3.6 3.7 4.8 MRCI(Q)/cc-pVTZ 4.2 4.2 5.6 Average 3.2 3.4 3.9

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H2C–H 3CH2 + H Reaction energy error (in mEh)

Benchmark

CCL UCCSD(T) MRMP2 FCI/6-31G 0.1 0.0 1.5 FCI/6-31G(d) 0.2 0.0 3.1 MRCI(Q)/cc-pVDZ 0.2 0.0 4.5 MRCI(Q)/cc-pVTZ 0.2 0.1 5.3 Average 0.2 0.0 3.6

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Average (over several basis sets)nonparallelity error (in mEh)

Species

CCL UCCSD(T) MRMP2 H2C−H 3.2 3.4 3.9 H2C−Cl 5.6 5.1 2.3 H2C−CH3 2.8 3.7 1.1 H2C−SiH3 2.7 3.1 1.6 H2Si−H 0.8 1.3 2.4 H2Si−Cl 1.8 3.2 3.5 H2Si−CH3 0.4 0.7 1.3 H2Si−SiH3 1.5 2.0 0.7 Average 2.3 2.8 2.1

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Average (over several basis sets) reaction energy error (in mEh)

Species

CCL UCCSD(T) MRMP2 H2C−H 0.2 0.0 3.6 H2C−Cl 0.5 0.7 1.3 H2C−CH3 0.2 0.0 0.8 H2C−SiH3 0.1 0.2 1.2 H2Si−H 0.2 0.3 0.5 H2Si−Cl 0.4 0.4 2.2 H2Si−CH3 0.1 0.0 0.7 H2Si−SiH3 0.1 0.1 0.1 Average 0.2 0.2 1.3

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Average NPE and REE (in mEh) for bond-breaking reactions of open-shell

species with tri-radical character

CCL UCCSD(T) MRMP2 NPE 2.3 2.8 2.1

REE 0.2 0.2 1.3

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How about tetra-radicals?Neither is good; CCL is better.

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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 C2H4.

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Why C2H4 C2H2 + H2?

• Chemical vapor deposition of diamond, graphite, carbon nanotubes, and silicon carbide.

• Lack of accurate potential energy surface (PES) of the decomposition of C2H4.

• C2H4 C2H2 + H2: 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.

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30

cc-pVDZ cc-pVTZ

Unsigned Mean Error

Bond length 0.017Å 0.002Å

Bond angle 0.7o 0.3o

Unsigned Max Error

Bond length 0.025Å 0.006Å

Bond angle 1.5o 0.4o

Quality of the CCL geometry1

1. Experimental data of H2, C2H2, C2H3, and C2H4 molecular geometries are used.

31

Quality of the CCL Energy1-2

CCL error compared to exp.1-2 (kJ/mol)

cc-pVTZ cc-pVQZ cc-pV5Z

C2H4 C2H2 + H23 3 3

C2H4 H2C=C: + H2-4 -2 -2

C2H4 C2H3 + H -8 -5 -4

C2H4 C2H2 + 2H -2 1 2

Unsigned Mean 4 3 3

Unsigned Max. 8 5 4

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

Quality of CCL vibrational frequency1

1. Experimental data of H2, C2H2, C2H3, C2H4, and H2C=C: vib. frequencies are used.

• C2H4 H2C=C: + H2 C2H2 + H2

33

• C2H4 H3C−CH: C2H2 + H2

• C2H4 H2C=CH + H C2H2 + H2

34

• C2H4 C2H2 + H2 (2 imaginary frequencies)

• C2H4 H2C=CH + H C2H2 + H + H

CCL and CASSCF(12,12) imaginary vibrational frequencies for C2H4 C2H2 + H2

35

6-31G** 6-311G** cc-pVDZ aug-cc-pVDZ

cc-pVTZ

CCL1824i

887i1623i 919i

1643i861i

1347i 977i

1438i 947i

CASSCF2526i

385i2331i

581i2378i

555i2083i 729i

2074i 663i

CCL/cc-pVTZ imaginary vibrational frequencies for C2H4 C2H2 + H2

36

1438i 947i

37

C2H4 H2C=C: + H2 C2H2 + H2

38

C2H4 H2C=CH + H C2H2 + 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, 0.3o).

• Both CCL/cc-pVQZ and CCL/cc-pV5Z predicts reaction energy within 4 kJ/mol error.

39

Conclusions (II)• Accurate CCL/cc-pV5Z C2H4 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

C2H4 H2C=C: + H2 C2H2 + H2.

• The dominant reaction path above 1800 K is

C2H4 H2C=CH + H C2H2 + 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-FG02-01ER15228.

41

Questions and Comments?

42

43

44

45

• Experimentally determined reaction energies for [C2H4 C2H2 + H2], [C2H4 H2C=C + H2], [C2H4 C2H3 + H], and [C2H4 C2H2 + 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

CCL

cc-pVTZ cc-pVQZ cc-pV5Z cc-pVQZ*

C2H4 C2H2 +

H2

3 3 3 2

C2H4 H2C=C +

H2

-4 -2 -2 -3

C2H4 C2H3 + H -8 -5 -4 -4

C2H4 C2H2 +

2H -2 1 2 1

Abs. Mean 4 3 3 2

Abs. Max. 8 5 4 4

48

CCL/cc-pVDZ & error CCL/cc-pVTZ & error Exp.

C2H2 R(C-C) 1.228 +0.025 1.209 +0.006 1.203

R(C-H) 1.079 +0.016 1.063 +0.000 1.063

C2H3 R(C-C) 1.334 +0.018 1.316 0.000 1.316

=CH R(C-H) 1.096 +0.016 1.079 -0.001 1.080

=CH2 cis R(C-H) 1.105 +0.020 1.090 +0.005 1.085

=CH2 trans R(C-H)1.100 +0.015 1.085 0.000 1.085

C=CH θ(CCH)135.8 -1.5 136.9 -0.4 137.3

C=CH2 cis θ(CCH)121.3 -0.2 121.3 -0.2 121.5

C2H4 R(C-C) 1.351 +0.014 1.336 -0.003 1.339

R(C-H) 1.098 +0.013 1.083 -0.003 1.086

θ(CCH)121.5 +0.3 121.5 +0.3 121.2

H2 R(H-H)0.7610 +0.0196 0.7427 +0.0013 0.7414

Abs. Mean Error

R0.017 0.002

θ 0.7 0.3

Abs. Max Error

R0.025 0.006

θ 1.5 0.4

49

C2H4 Symm. Exp22,24 CCL Exp/theory

1 Ag 3026 3160  0.9576

2 Ag 1623 1677  0.9678

3 Ag 1342 1372  0.9781

4 Au 1023 1050  0.9743

5 B1u 2990 3142 0.9515

6 B1u 1444 1481 0.9747

7 B2g 943 947 0.9958

8 B2u 3106 3249 0.9558

9 B2u 810 825 0.9822

10 B3g 3103 3222 0.9631

11 B3g 1236 1244 0.9936

12 B3u 949 970 0.9786

H2C=C Symm. Exp25 CCL Exp/theory

1 A1 3025 3134  0.9652

2 A1 1635 1668  0.9802

3 A1 1165 1229  0.9479

4 B1 835  751 1.1119 

5 B2 3050  3225 0.9457 

6 B2 320  345 0.9275 

H2 Symm. Exp19 CCL Exp/theory

1 Σg 4401 4409  0.9982

C2H2 Symm. Exp 22 CCL Exp/theory

1 Σg 3374 3515  0.9599

2 Σg 1974 2009  0.9826

3 Σu 3289 3414 0.9634

4 Πg 612 585 1.0462

5 Πu 730 750 0.9733

C2H3 Symm. Exp23 CCL Exp/theory

1 A' 3235 3249  0.99572 A' 3164 3182  0.99433 A' 3103 3078  1.00814 A' 1700 1620  1.04935 A' 1277 1398  0.91346 A' 1099 1072  1.02527 A' 758 727  1.04268 A" 955 919  1.03929 A" 895 794  1.1272

How much data?

• 8 species: H2A–B H2A + B

• A = C or Si; B = H, Cl, CH3, or SiH3

• 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? EX(i) = EX(i) – Ebenchmark(i) at the i-th structure.• NPE: nonparallelity error.• NPEX = max[EX(1), EX(2)…, EX(N)]–

min[EX(1), EX(2)…, EX(N)]• REE: reaction energy error.• REEX = EX(3Re) – EX(Re)• STD: standard deviation of errors (1/3 of NPE). • STDX =

1

N[E X

i1

N (i)]2 [E X

i1

N (i)] 2

51

How to get geometries and potential energy surfaces?

• Optimize structures with the breaking bond distance fixed at every 0.2 Å from Re to 3Re.

• 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*

• ECCL/cc-pVQZ* = ECCL/cc-pVTZ +

(EMP2/cc-pVQZ – EMP2/cc-pVTZ)

• Assuming additivity of basis set effect and correlation correction.

• Additivity approximation is used in G1-4 theory.

53

CCL/cc-pVTZ*

• ECCL/cc-pVTZ* = ECCL/cc-pVDZ +

(EMP2/cc-pVTZ – EMP2/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?

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CCL nonparallelity errors (in mh) relative to MRCI(Q)/cc-pVTZ

Species CCL Nonparallelity Error

cc-pVDZ cc-pVTZ* cc-pVTZ H2C–H 10.1 2.4 4.3 H2C–Cl 11.5 5.0 8.4 H2Si–H 04.4 1.3 1.1 H2Si–Cl 14.4 3.0 2.7 Average 10.1 2.9 4.1

Smallest errors in each row are in bold font.55

E(cc-pVTZ)-E(cc-pVDZ)H2C-H

-48

-46

-44

-42

-40

-38

-36

-34

1 1.5 2 2.5 3 3.5

R (angstrom)

Basis

Eff

ect

( mh

)

MRCI(Q)MP2CCL

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.

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Computational cost of CCL/cc-pVTZ* on a 2GHz machine

• SiCl3 + CH3SiCl3 SiCl4 + CH3SiCl2• cc-pVDZ:173 basis functions.• cc-pVTZ: 344 basis functions.• CCL/cc-pVTZ*: 1 day.

•CCL/cc-pVTZ: 36 days.

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MRCI(Q) nonparallelity error (in mEh) 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).

Species Basis NPE

H2C−H cc-pVDZ 0.2 cc-pVTZ 0.3

H2C−Cl 6-31G 0.6 6-31G(d) 1.6 cc-pVDZ 1.6

H2Si−H cc-pVDZ 0.4 cc-pVTZ 0.8

H2Si−Cl 6-31G 0.6 6-31G(d) 1.5 cc-pVDZ 1.5

Average 0.9

58

H3C–H CH3 + H Errors (in mEh) relative to FCI/MINI

59

H2Si–H 1SiH2 + H Errors (in mEh) relative to FCI/6-31G(d)

60

H2Si–Cl 1SiH2 + Cl Errors (in mEh) relative to MRCI(Q)/cc-pVTZ

-7

-3

1

5

9

2 2.5 3 3.5 4 4.5 5 5.5 6E -

E[M

RC

I(Q

)] (

mE

h)

R (angstrom)

H2Si-Cl (cc-pVTZ)CCL

UCCSD(T)

MRMP2-30mh

61

H2Si–Cl 1SiH2 + Cl Nonparallelity error (in mEh)

relative to MRCI(Q)

Basis

CCL UCCSD(T) MRMP2 6-31G 0.7 2.1 1.9 6-31G(d) 1.9 3.6 3.3 cc-pVDZ 1.8 3.6 4.2 cc-pVTZ 2.7 3.6 4.5 Average 1.8 3.2 3.5

62

H2Si–Cl 1SiH2 + Cl Reaction energy error (in mEh)

relative to MRCI(Q)

Basis

CCL UCCSD(T) MRMP2 6-31G 0.5 0.3 1.9 6-31G(d) 0.1 0.5 2.0 cc-pVDZ 0.1 0.5 2.8 cc-pVTZ 1.0 0.2 2.0 Average 0.4 0.4 2.2

63

64

65

H3C–SiH3 CH3 + SiH3 Errors (in mEh) relative to FCI/MINI

NPEREE

How about tetra-radicals?

66

MINI Basis Set