John Steven Ewa Xuefei Tao JingjingAlecu Mielke Papajak Xu Yu Zheng

Annual ReportCombustion Energy

Frontier Research CenterPrinceton, NJSept. 23, 2010

Truhlar groupUniversity of Minnesota

Practical methods for including torsional anharmonicity in thermochemical calculations on complex molecules

• We proposed a new approximation called the multi-structural method with torsional anharmonicity (MS-T) to calculate conformational-vibrational-rotational partition functions.

• It is applicable for molecules with multiple torsions coupled with overall rotation, each other, and/or with other low-frequency modes

• Assigning each torsion to a specific normal mode is not required.

• It has low computational cost, requiring no torsion barriers or scans.

Effect of conformational entropy and torsional anharmonicity on the butanol partition function

8

12

16

20

24

28

32

0 500 1000 1500 2000 2500 3000

QM

S/Q

HO

T (K)

MS-HOMS-T

Ratio of the conformational-rovibrational partition function of butanol calculated by the multi-structural method to that calculated by the single-structure HO approximation at the global minimum.

27 structures

1-pentyl radical

5

7

9

11

13

15

0 1000 2000 3000

QM

S/Q

HO

T (K)

MS-HO

MS-T

Ratio of the conformational-rovibrational partition function of 1-pentyl radical calculated by the multi-structural method to that calculated by the single-structure HO approximation at the global minimum.

Equations: Multi-structural torsion method

Qcon-rovibMS-T = Qrot, j exp −βU j( )

j=1

J∑ Qj

HOZ j f j,ττ=1

t∏

f j,τ =ω j,τ 2πβI j,τ

M j,τexp(−βI j,τω j,τ

2 / M j,τ2 )I0(βI j,τω j,τ

2 / M j,τ2 )

Z j =gQrot, jQj

HO + (1− g)Qjimp

Qrot, jQjHO =

Qjimp = Qrot, jQj

HOZ jintZ j

coup

Z jint =

ω j,m−1

m=1

F−t∏ ω j,τ

−1

τ=1

t∏

ω j,m−1

m=1

F∏

Z jcoup =

det S j

det I jrot I j,τ

τ=1

t∏

1/2

g = tanhq j,τ

FR

Pj,τq j,τCHO

τ=1

t∏

1/ t

For other slides in this presentation, Zj is not included yet.

number of structures number of torsions

The 1, 4-Hydrogen Shift Isomerization of the 1-Pentyl Radical

Multistructural anharmonicty is dominated by torsions in the potential energy surface of the reaction.Reactant: 1-pentyl 4 torsions, 15 distinct structuresProduct: 2-pentyl 4 torsions, 7 distinct structures Transition state: 1 torsion, but 2 distinct structures (due to the ring inter-conversion)

1-pentyl 2-pentyl TS-1 TS-2

A practical approach to compute the rate constant

tunneling based on a single reaction path (ground-state

transmission coefficient)

k = κG(T )˜ k Th

mins

Q1GT,HO(T ,s)

Qcon−rovib‡,MS−T (T )

Q1‡,HO(T )

Qcon−rovibR,MS−T exp −V (s)/ ˜ k T( )

separable multi-structure anharmonicity factor

Multi-configuration Shepard Interpolation:

PES of 1-pentyl and 2-pentyl by MM3 force field 9 Shepard-points (energy, gradient, Hessian) by M06 functional

Reaction path for variational effects and tunneling

T (K) FR F‡ FP

300 6.8 2.0 11.0

600 10.0 2.2 14.1

1000 11.9 2.2 14.0

1500 12.9 2.0 11.9

2000 11.7 1.9 9.6

Multi-structure anharmonicity factors FMS-T

for reactant, TS, and product

-2 -1 0 1 2

85

90

95

100

105

110

s (bohr)

V aG

(kca

l/mol

)

0 1 2 3 4 5

-4

0

4

8

Calulated rate constant based on MS-VTST

Yamauchi et al. 1999 Miyoshi et al. 2002

1000/T (K-1)

logk

(s-1)

The 1, 4-Hydrogen Shift Isomerization of the 1-Pentyl Radical(cont.)

For this slide, Zj is not included yet.

The Reactions: CH3OH + HO2 → CH2OH + HOOH (R1a)

→ CH3O + HOOH (R1b)

CH3OH + CH3 → CH2OH + CH4 (R2a) → CH3O + CH4 (R2b)

John Alecu’s poster

Rel

ativ

e En

ergy

(kca

l mol

-1)

9.89

18.29

0.00

23.65

18.99

CH3OH + HO2

CCSDT(2)Q/CBS(aT+aQ) + CV + Rand experiment.

Rel

ativ

e En

ergy

(kca

l mol

-1)

-8.76

13.69

0.00

13.85

0.34

CH3OH + CH3

CCSDT(2)Q/CBS(aT+aQ) + CV + Rand experiment.

Accuracy vs. cost

• all calculations based on geometries optimized with M06-2X/MG3S• single-processor CPU time relative to equivalent HF/cc-pVDZ calculation

Electronic Model ChemistryMean Unsigned Error (MUE)

(kcal mol-1)∆E V‡ Average Cost

N9

CCSDT(2)Q/CBS(aT+aQ) + CV + R 0.3 0 (by def.) 78,000N8

CCSDT/CBS(T+Q) + CV + R 0.1 0.2 0.1 16,000N7

CCSD(T)-F12b/aug-cc-pVTZ 0.2 0.3 0.2 4,900CCSD(T)/CBS(D+T) 0.2 0.5 0.4 900

CBS-QB3 0.2 1.2 0.9 48MCG3-MPW 0.3 0.6 0.5 39

N6

BMC-CCSD 1.0 0.4 0.6 19N4

M08-HX/maug-cc-pVTZ 0.5 0.8 0.7 43

M08-HX/ma-TZVP 1.0 0.8 0.8 25

scaling

Fitting rate constantsConventional model

k = A T300 K

n

exp −E /RT( )

k = A T300 K

n

exp −E T +T0( )

R T 2 +T02( )

This new form may be a much better way to fit curved Arrhenius plots than the almost universally used first form. This could have significant effects on combustion mechanisms and the representation of rate data in such mechanisms

New modelproposed in J. Zheng and D. G. Truhlar, PCCP 12, 7782 (2010).