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Article
Accurate Ab Initio Thermal Rate Constants for Reaction of O(P) with H and Isotopic Analogs 3
2
Thanh Lam Nguyen, and John F. Stanton J. Phys. Chem. A, Just Accepted Manuscript • Publication Date (Web): 10 Jun 2014 Downloaded from http://pubs.acs.org on June 20, 2014
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
The Journal of Physical Chemistry A is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
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The Journal of Physical Chemistry
Accurate Ab Initio Thermal Rate Constants for Reaction of O( 3 P) with H 2 and Isotopic Analogs
Thanh Lam Ngu yen and John F. Stanton* Dep a rt men t o f Ch em is t r y, Th e Un iv er s ity o f Tex a s a t A u st in , Texa s 7 8 7 1 2 - 0 1 6 5 , US A
*Corresponding author: [email protected]
Abstract: Semi-classical transition state theory, in combination with high accuracy quantum chemistry, was used to compute thermal rate constants from first principles for the O( 3 P) + H 2 reaction and its isotopic counterparts. In the temperature regime of 298-3500K (which spans eight orders of magnitude for rate constants), our theoretical results are in excellent agreement (within 5 to 15%) with all available experimental data from 298 to 2500K, but somewhat too low (from 15 to 35%) at higher temperatures. A number of possible reasons that might cause the degradation at high temperatures are
discussed.
Vibrational
state-selected
rate
constants
and
their
correlations with normal thermal rate constant are derived and given in supporting information.
Keywords: HEAT, SCTST, VPT2, VSSRC, Hydrogen-Fuel, 2DME
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Introduction The reaction of a ground-state ox ygen atom with a hydrogen molecule to produce a h ydrox yl radical and h ydrogen atom is an important chainpropagation step in H 2 combustion and that of h ydrogen-containing fuels. 1 - 5
O( 3 P) + H 2
→ OH + H
(1)
O( 3 P) + H 2 (ν=1)
→ OH(ν=1) + H
(2a)
O( 3 P) + H 2 (ν=1)
→ OH(ν=0) + H
(2b)
In term of atmospheric chemistry, the high classical barrier for the reaction of about 13.0 kcal/mol, 6 - 11 and the small concentration of groundstate ox ygen atoms in air renders it unlikel y to be an important reaction in the troposphere. However, numerous recent studies 1 2 - 1 4 have shown that reactions of triplet ox ygen atoms with vibrationally ex cited h ydrogen molecules (ν H H =1, see reactions (2a) & (2b)) might play some roles in the upper mesosphere. Given that molecules in this reaction s ystem are quite small, very accurate theoretical and experimental methods can be applied. Therefore, this reaction is an interesting benchmark s ystem for comparing theory
and
experiment. 1 5 - 3 0 , 3 1 - 5 1
extensivel y reviewed 2 , 3 , 5 , 7 , 9 , 1 0 , 1 9 , 2 2 thermal
reaction
rate
constants
The
title
reaction
has
been
both
and studied. 1 - 5 1 Experimentall y, 1 5 - 3 0 have
been
measured
by
numerous
techniques for a wide range of temperature from 298 K to 3500 K. 1 5 - 3 0
2
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The Journal of Physical Chemistry
Reaction
d ynamics
studied
by
crossed
molecular
beams
are
also
available. 3 8 , 4 7 , 4 8 On the theoretical side, 6 - 1 4 , 3 1 - 4 7 , 4 9 - 5 1 global potential energy surfaces with an accuracy of better than 0.5 kcal/mol have recentl y become available. 6 - 1 4 Calculations of thermal rate constants, 7 , 3 3 - 3 6 quasiclassical trajectory (QCT) and quantum d ynamics calculations 11 , 3 7 - 4 7 , 4 9 - 5 1 have been reported using these surfaces. In general, the theoretical results are consistent with what has been observed experimentall y. In this work, we calculate thermal rate constants for the title reactions using semi-classical transition state theory (SCTST), 5 2 - 5 5 which naturall y includes multi-dimensional quantum mechanical tunneling and full y coupled anharmonic vibrations. SCTST has recentl y been implemented and used in a number of studies. 5 4 - 5 7 As in the previous work, the input data for SCTST were obtained using the high accuracy extrapolated ab initio thermochemistry (HEAT) protocol. 5 8 - 6 0 Our recent work for various reaction s ystems has shown that a combination of SCTST and HEAT can provide thermal reaction rate constants with an accuracy comparable to that from experiments. 5 5 , 6 1 - 6 5 As will be shown below, this behavior is also true for the title reactions. In addition, thermal rate constants for reactions of triplet ox ygen atoms with vibrationally ex cited h ydrogen molecules (ν H H =1 & 2) are computed and compared to the experimental work of Light. 6 6
Theoretical Methodology
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Quantum Chemical Calculations Energies of various species in the reaction of ground-state ox ygen atom with molecular h ydrogen were calculated using HEAT-345(Q) and HEAT456(Q) protocols, 5 8 - 6 0 which have been described in detail in previous papers. 5 8 - 6 0 In this work, we have slightl y modified two terms in the original HEAT protocol as detailed below: i)
Zero-point vibrational energies (ZPE) were calculated using the CCSD(T)
method
(in
the
frozen-core
approximation)
in
combination with the “NASA Ames” atomic natural orbital (ANO) basis set of Taylor and Almlöf. 6 7 - 6 9 Harmonic force fields were calculated using the ANO2 basis set, 6 9 while the ANO1 basis set 6 9 was used to obtain anharmonic force fields. Second-order vibrational perturbation theory (VPT2) 7 0 was then used to compute anharmonic constants and the anharmonic ZPE. ii)
Experimental spin-orbit (SO) corrections were used for the ground-state ox ygen atom (SO O = 78 cm - 1 ) and h ydrox yl radical (SO O H = 69.6 cm - 1 ). 7 1 A spin-orbit correction for the transition structure (TS) is neglected, however, as our calculations show that the predicted kinetics are very insensitive to the magnitude of the SO correction at the TS (see Table S1 in supporting information). Hence, the barrier we compute corresponds to the average of two components of the transition state. It should be
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mentioned that the earlier work of Wagner et al 7 2 included a treatment of the Renner-Teller effect at this linear electronicall y degenerate TS and indicated that the magnitude of the effect is significant at room temperature. 7 2 However, our SCTST rates without the Renner-Teller effect at around room temperature agree well with experiments (see Figure 2A) as well as with full quantum d ynamics rates of Balakrishnan 4 3 who used two GLDP surfaces ( 3 A″ and
3
A′), which are coupling. However, the
Renner-Teller effect is negligible at high temperatures t ypical of combustion. Therefore, neglect of the Renner-Teller effect in our model does not appear to affect our SCTST rates significantly. To include an appropriate treatment of the Renner-Teller effect (where Born-Oppenheimer approximation breaks down) on this linear electronicall y degenerate TS in our SCTST model, one must include an additional term of vibronic coupling into Hamiltonian surfaces
and
( 3 A″
solve
and
3
this
A′)
in
Schrodinger order
to
equation obtain
for
two
geometries,
rovibrational parameters, anharmonic constants, and energ y barriers. Such work is possible in our group, which works on both kinetics and vibronic coupling, but is currentl y not available and in progress.
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The CFOUR quantum chemistry program 7 3 was used for CCSD(T) and CCSDT calculations, while CCSDT(Q) calculations used the MRCC code, 7 4 as interfaced with the CFOUR.
Chemical Kinetics Calculations Thermal reaction rate constants of the title reaction as displayed in Figure 1 can be expressed as:
1 × = × × 2 + 1 !, exp −!/) *! ℎ ×
(3) Where h is Planck’s constant, k B is the Boltzmann constant; and are the translational and electronic partition functions of the TS, is equal to 3×2=6, in which 3 represents the triplet respectivel y.
electronic
state
degeneracy
and
2
is
for
the
twofold
degeneracy
(neglecting SO effects) of Π s ymmetry of TS. and are the complete partition functions for molecular h ydrogen and triplet ox ygen atom, respectivel y. The Morse vibrational potential energy levels are used for molecular h ydrogen in order to compute its vibrational partition function. Effects of coupling between the vibration and rotations in H 2 molecule were also investigated and found to be small ( >? ?@ A A ?@ A
24
= 24B
(12)
C
The current SCTST approximation to the exact rate constant, k(T) e , can be obtained as discussed earlier while the classical mechanical rate constant, k(T) c can be calculated within the same framework with the following caveat. Note that “classical” here means just for the reaction coordinate. Specificall y, after the value of θ is calculated (see Eq. 6 of Ref 55), which corresponds to the state-resolved transition probabilit y according to the semiclassical relationship P(θ) = [1 + ex p(2θ)] - 1 , the reaction probabilit y is set to unit y for θ0 (above the barrier). This preserves the coupled anharmonic treatment provided b y SCTST, but also enforces proper classical behavior (no tunneling or reflection of flux that crosses above the barrier). Basing the summation ! on this ansatz,
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vibrational partition function of TS can then be computed using a sum of vibrational states, Eq. (13).
= 8D >
. E !*! 23 4
(13)
Quantum mechanical tunneling corrections for the title reaction and its isotopic analogs are calculated and displayed in Figure 6. Figure 6 shows that the tunneling corrections for O + H 2 are similar to that for O + HD, but of course much larger than for O + D 2 . This behavior, of course, can be expected because the H-atom is lighter; as a result, it tunnels much more efficientl y than does the D-atom. As can be seen in Figure 6, tunneling corrections decrease nearl y exponentiall y with temperature. Considering O + H 2 , the calculated tunneling enhancement is 516 at 200K, drops to ca. 14 at 300K, and becomes about 2.5 at 500K and onl y 1.2 at 1000K. So, tunneling is extremel y important at low temperatures, but negligible at t ypical combustion temperatures. At room temperature, th e tunneling correction of 14 for O + H 2 reaction means that more than 90% of the total reactive flux is achieved b y tunneling.
Sensitivity Analysis Calculated rate constants are exquisitel y sensitive to the input data used, which include rotational constants, harmonic vibrational frequencies, imaginary (barrier) frequency, anharmonic constants, and an energ y
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barrier. 6 1 , 6 2 , 6 5
Of these parameters,
the
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energy barrier
and
barrier
frequency are most important, 6 1 , 6 2 , 6 5 especiall y at the low temperatures in the atmosphere. In addition to these factors, we also studied effects of various basis sets on ro-vibrational parameters that will in turn affect the calculated rate constants. We found that the effects of basis sets on rate constants are generally fairl y small for this reaction s ystem (see Figure S1 in the supporting information). Because the HEAT protocol was used, the calculated energy barrier in this work is unlikel y to be wrong b y more than 0.3 kcal/mol. The effects of such an energy error are negligible at very high temperatures (T > 2500K), but may be significant at very low temperatures. At room temperature, a change of barrier height b y 0.2 kcal/mol can result in an alteration of rate constant of about 40%. However,
our
calculated
rate
constants
agree
well
with
available
experiments at around room temperature, and we therefore conclude that our calculated barrier is quite accurate. In addition, there are no experimental data below room temperature, because the reaction rate under these temperature conditions is too slow to be measurable.
Conclusions In this work, we have used high accuracy quantum chemistry to compute ro-vibrational parameters and energies of the title reaction s ystem, followed b y calculation of thermal rate constants using the SCTST/VPT2
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approach. Some salient results of this stud y can be summarized as follows: (i)
The reaction of O( 3 P) + H 2 = OH + H is predicted to be endothermic b y 1.45 ± 0.10 kcal/mol, in excellent agreement with an ATcT value of 1.54 ± 0.01 kcal/mol.
(ii)
Classical barrier heights are predicted to be 13.1 ± 0.2 kcal/mol (without DBOC and SO corrections) and 13.5 ± 0.2 kcal/mol (including both DBOC and SO corrections). The former agrees well (within 0.1 kcal/mol) with all barriers reported recentl y in the literature, where the smaller SO and DBOC effects have heretofore been ignored.
(iii)
For T ≤ 2500K, calculated thermal rate constants for the title reaction and its isotopic analogs are in quantitative agreement (within 5-15%) with all experimental data where they are available.
For
T
>
2500K,
the
calculated
k(T)
values
s ystematicall y underestimate the experiment b y about 15 to 35%. This underestimation may be due to the fact that non-statistical population
fractions
of
vibrationall y
excited
H 2 (ν H H =1&2)
molecules exist, and/or because of shortcomings of the VPT2 model. Other possible sources of error include rovibrational parameters,
and/or
neglecting
coupling
rotations in TS.
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of
vibrations
and
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(iv)
Page 20 of 39
Thermal reaction rate constants of ground state ox ygen atoms with vibrationall y excited h ydrogen atoms are derived and computed (see the supporting information). We are able to match the calculated k(T) values with experiment b y increasing the thermal
population
h ydrogen
atoms
fractions
(ν H H =1)
with
of
vibrationall y respect
to
the
first-excited Boltzmann
distribution. However, this is not in an y way to be interpreted as convincing evidence of non-statistical behavior; in our view, a breakdown of VPT2 at very high temperatures is the more likel y source of error.
Supporting information Optimized geometries, rovibrational parameters, anharmonic constants, thermal rate constants, and energies for various species in the reaction of O( 3 P) with H 2 and its isotopic counterparts are given. This material is available free of charge via the Internet at http://pubs.acs.org.
Acknowledgements JFS and T LN are supported b y the Robert A. Welch Foundation (Grant F1283) and the Department of Energy, Office of Basic Energy Sciences (Contract Number DE-FG02-07ER15884). We would like to thank John R. Barker, Universit y of Michigan for bringing this reaction to our attention and his advice. We would like to thank an anon ymous reviewer for
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pointing us to Refs. 30, 43 and 72 and other comments that led to improvement of this work.
Table 1: Calculated relative energies (kcal/mol) of various terms in HEAT-345(Q) protocol
δHF
δCCSD(T)
δCCSDT
δCCSDT(Q) a)
δREL
δZPE
δDBOC b)
δSO
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
15.3913
-13.0763
0.0308
-0.1239
0.1255
-0.9420
0.0168
0.0241
32.1324
-18.7682
-0.1756
-0.1064
0.0405
-2.2853
0.1704
0.2228
Species 3
O( P) + H2 OH(2Π) + H 2
TS (O—H—H, Π)
δHEAT 0.0000 1.4463 (1.3351) c) 11.2306 (11.1799) c)
a) Ca lc u lat ed a t C CS DT ( Q ) /cc -p VT Z le v el o f t heo r y b ) Ob ta i ned wi t h C C SD /a u g -c c -p VT Z l e ve l o f t heo r y c) Ob ta i ned wi t h HE AT -4 5 6 ( Q) p r o to co l
Table 2: A comparison of classical barrier heights and geometrical parameters of collinear TS in the O( 3 P) + H 2 = OH + H reaction calculated using various levels of theory Method
Au t h o r s
MRCI+Q/ [ 5 s 5 p 3 d 2 f1 g / 4 s 3 p 2 d 1 f] J3 surface
Walch
Year
a)
J o s e p h - Tr u h l a r Garrett b) Peterson-Dunning Rogers et al d)
c)
1987
Vo ( k c a l / mo l ) 12.4
O—H ( Å) 1.225
H—H ( Å) 0.893
1988
13.03
1.217
0.915
1997 2000
13.1 13.26 ± 0.3
1.215 1.244
0.894 0.865
2000
13.04 ± 0.3
1.217
0.903
2004 2004 2012
13.08 ±
Article
Accurate Ab Initio Thermal Rate Constants for Reaction of O(P) with H and Isotopic Analogs 3
2
Thanh Lam Nguyen, and John F. Stanton J. Phys. Chem. A, Just Accepted Manuscript • Publication Date (Web): 10 Jun 2014 Downloaded from http://pubs.acs.org on June 20, 2014
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
The Journal of Physical Chemistry A is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
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The Journal of Physical Chemistry
Accurate Ab Initio Thermal Rate Constants for Reaction of O( 3 P) with H 2 and Isotopic Analogs
Thanh Lam Ngu yen and John F. Stanton* Dep a rt men t o f Ch em is t r y, Th e Un iv er s ity o f Tex a s a t A u st in , Texa s 7 8 7 1 2 - 0 1 6 5 , US A
*Corresponding author: [email protected]
Abstract: Semi-classical transition state theory, in combination with high accuracy quantum chemistry, was used to compute thermal rate constants from first principles for the O( 3 P) + H 2 reaction and its isotopic counterparts. In the temperature regime of 298-3500K (which spans eight orders of magnitude for rate constants), our theoretical results are in excellent agreement (within 5 to 15%) with all available experimental data from 298 to 2500K, but somewhat too low (from 15 to 35%) at higher temperatures. A number of possible reasons that might cause the degradation at high temperatures are
discussed.
Vibrational
state-selected
rate
constants
and
their
correlations with normal thermal rate constant are derived and given in supporting information.
Keywords: HEAT, SCTST, VPT2, VSSRC, Hydrogen-Fuel, 2DME
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Introduction The reaction of a ground-state ox ygen atom with a hydrogen molecule to produce a h ydrox yl radical and h ydrogen atom is an important chainpropagation step in H 2 combustion and that of h ydrogen-containing fuels. 1 - 5
O( 3 P) + H 2
→ OH + H
(1)
O( 3 P) + H 2 (ν=1)
→ OH(ν=1) + H
(2a)
O( 3 P) + H 2 (ν=1)
→ OH(ν=0) + H
(2b)
In term of atmospheric chemistry, the high classical barrier for the reaction of about 13.0 kcal/mol, 6 - 11 and the small concentration of groundstate ox ygen atoms in air renders it unlikel y to be an important reaction in the troposphere. However, numerous recent studies 1 2 - 1 4 have shown that reactions of triplet ox ygen atoms with vibrationally ex cited h ydrogen molecules (ν H H =1, see reactions (2a) & (2b)) might play some roles in the upper mesosphere. Given that molecules in this reaction s ystem are quite small, very accurate theoretical and experimental methods can be applied. Therefore, this reaction is an interesting benchmark s ystem for comparing theory
and
experiment. 1 5 - 3 0 , 3 1 - 5 1
extensivel y reviewed 2 , 3 , 5 , 7 , 9 , 1 0 , 1 9 , 2 2 thermal
reaction
rate
constants
The
title
reaction
has
been
both
and studied. 1 - 5 1 Experimentall y, 1 5 - 3 0 have
been
measured
by
numerous
techniques for a wide range of temperature from 298 K to 3500 K. 1 5 - 3 0
2
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The Journal of Physical Chemistry
Reaction
d ynamics
studied
by
crossed
molecular
beams
are
also
available. 3 8 , 4 7 , 4 8 On the theoretical side, 6 - 1 4 , 3 1 - 4 7 , 4 9 - 5 1 global potential energy surfaces with an accuracy of better than 0.5 kcal/mol have recentl y become available. 6 - 1 4 Calculations of thermal rate constants, 7 , 3 3 - 3 6 quasiclassical trajectory (QCT) and quantum d ynamics calculations 11 , 3 7 - 4 7 , 4 9 - 5 1 have been reported using these surfaces. In general, the theoretical results are consistent with what has been observed experimentall y. In this work, we calculate thermal rate constants for the title reactions using semi-classical transition state theory (SCTST), 5 2 - 5 5 which naturall y includes multi-dimensional quantum mechanical tunneling and full y coupled anharmonic vibrations. SCTST has recentl y been implemented and used in a number of studies. 5 4 - 5 7 As in the previous work, the input data for SCTST were obtained using the high accuracy extrapolated ab initio thermochemistry (HEAT) protocol. 5 8 - 6 0 Our recent work for various reaction s ystems has shown that a combination of SCTST and HEAT can provide thermal reaction rate constants with an accuracy comparable to that from experiments. 5 5 , 6 1 - 6 5 As will be shown below, this behavior is also true for the title reactions. In addition, thermal rate constants for reactions of triplet ox ygen atoms with vibrationally ex cited h ydrogen molecules (ν H H =1 & 2) are computed and compared to the experimental work of Light. 6 6
Theoretical Methodology
3
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Quantum Chemical Calculations Energies of various species in the reaction of ground-state ox ygen atom with molecular h ydrogen were calculated using HEAT-345(Q) and HEAT456(Q) protocols, 5 8 - 6 0 which have been described in detail in previous papers. 5 8 - 6 0 In this work, we have slightl y modified two terms in the original HEAT protocol as detailed below: i)
Zero-point vibrational energies (ZPE) were calculated using the CCSD(T)
method
(in
the
frozen-core
approximation)
in
combination with the “NASA Ames” atomic natural orbital (ANO) basis set of Taylor and Almlöf. 6 7 - 6 9 Harmonic force fields were calculated using the ANO2 basis set, 6 9 while the ANO1 basis set 6 9 was used to obtain anharmonic force fields. Second-order vibrational perturbation theory (VPT2) 7 0 was then used to compute anharmonic constants and the anharmonic ZPE. ii)
Experimental spin-orbit (SO) corrections were used for the ground-state ox ygen atom (SO O = 78 cm - 1 ) and h ydrox yl radical (SO O H = 69.6 cm - 1 ). 7 1 A spin-orbit correction for the transition structure (TS) is neglected, however, as our calculations show that the predicted kinetics are very insensitive to the magnitude of the SO correction at the TS (see Table S1 in supporting information). Hence, the barrier we compute corresponds to the average of two components of the transition state. It should be
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mentioned that the earlier work of Wagner et al 7 2 included a treatment of the Renner-Teller effect at this linear electronicall y degenerate TS and indicated that the magnitude of the effect is significant at room temperature. 7 2 However, our SCTST rates without the Renner-Teller effect at around room temperature agree well with experiments (see Figure 2A) as well as with full quantum d ynamics rates of Balakrishnan 4 3 who used two GLDP surfaces ( 3 A″ and
3
A′), which are coupling. However, the
Renner-Teller effect is negligible at high temperatures t ypical of combustion. Therefore, neglect of the Renner-Teller effect in our model does not appear to affect our SCTST rates significantly. To include an appropriate treatment of the Renner-Teller effect (where Born-Oppenheimer approximation breaks down) on this linear electronicall y degenerate TS in our SCTST model, one must include an additional term of vibronic coupling into Hamiltonian surfaces
and
( 3 A″
solve
and
3
this
A′)
in
Schrodinger order
to
equation obtain
for
two
geometries,
rovibrational parameters, anharmonic constants, and energ y barriers. Such work is possible in our group, which works on both kinetics and vibronic coupling, but is currentl y not available and in progress.
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The CFOUR quantum chemistry program 7 3 was used for CCSD(T) and CCSDT calculations, while CCSDT(Q) calculations used the MRCC code, 7 4 as interfaced with the CFOUR.
Chemical Kinetics Calculations Thermal reaction rate constants of the title reaction as displayed in Figure 1 can be expressed as:
1 × = × × 2 + 1 !, exp −!/) *! ℎ ×
(3) Where h is Planck’s constant, k B is the Boltzmann constant; and are the translational and electronic partition functions of the TS, is equal to 3×2=6, in which 3 represents the triplet respectivel y.
electronic
state
degeneracy
and
2
is
for
the
twofold
degeneracy
(neglecting SO effects) of Π s ymmetry of TS. and are the complete partition functions for molecular h ydrogen and triplet ox ygen atom, respectivel y. The Morse vibrational potential energy levels are used for molecular h ydrogen in order to compute its vibrational partition function. Effects of coupling between the vibration and rotations in H 2 molecule were also investigated and found to be small ( >? ?@ A A ?@ A
24
= 24B
(12)
C
The current SCTST approximation to the exact rate constant, k(T) e , can be obtained as discussed earlier while the classical mechanical rate constant, k(T) c can be calculated within the same framework with the following caveat. Note that “classical” here means just for the reaction coordinate. Specificall y, after the value of θ is calculated (see Eq. 6 of Ref 55), which corresponds to the state-resolved transition probabilit y according to the semiclassical relationship P(θ) = [1 + ex p(2θ)] - 1 , the reaction probabilit y is set to unit y for θ0 (above the barrier). This preserves the coupled anharmonic treatment provided b y SCTST, but also enforces proper classical behavior (no tunneling or reflection of flux that crosses above the barrier). Basing the summation ! on this ansatz,
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The Journal of Physical Chemistry
vibrational partition function of TS can then be computed using a sum of vibrational states, Eq. (13).
= 8D >
. E !*! 23 4
(13)
Quantum mechanical tunneling corrections for the title reaction and its isotopic analogs are calculated and displayed in Figure 6. Figure 6 shows that the tunneling corrections for O + H 2 are similar to that for O + HD, but of course much larger than for O + D 2 . This behavior, of course, can be expected because the H-atom is lighter; as a result, it tunnels much more efficientl y than does the D-atom. As can be seen in Figure 6, tunneling corrections decrease nearl y exponentiall y with temperature. Considering O + H 2 , the calculated tunneling enhancement is 516 at 200K, drops to ca. 14 at 300K, and becomes about 2.5 at 500K and onl y 1.2 at 1000K. So, tunneling is extremel y important at low temperatures, but negligible at t ypical combustion temperatures. At room temperature, th e tunneling correction of 14 for O + H 2 reaction means that more than 90% of the total reactive flux is achieved b y tunneling.
Sensitivity Analysis Calculated rate constants are exquisitel y sensitive to the input data used, which include rotational constants, harmonic vibrational frequencies, imaginary (barrier) frequency, anharmonic constants, and an energ y
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barrier. 6 1 , 6 2 , 6 5
Of these parameters,
the
Page 18 of 39
energy barrier
and
barrier
frequency are most important, 6 1 , 6 2 , 6 5 especiall y at the low temperatures in the atmosphere. In addition to these factors, we also studied effects of various basis sets on ro-vibrational parameters that will in turn affect the calculated rate constants. We found that the effects of basis sets on rate constants are generally fairl y small for this reaction s ystem (see Figure S1 in the supporting information). Because the HEAT protocol was used, the calculated energy barrier in this work is unlikel y to be wrong b y more than 0.3 kcal/mol. The effects of such an energy error are negligible at very high temperatures (T > 2500K), but may be significant at very low temperatures. At room temperature, a change of barrier height b y 0.2 kcal/mol can result in an alteration of rate constant of about 40%. However,
our
calculated
rate
constants
agree
well
with
available
experiments at around room temperature, and we therefore conclude that our calculated barrier is quite accurate. In addition, there are no experimental data below room temperature, because the reaction rate under these temperature conditions is too slow to be measurable.
Conclusions In this work, we have used high accuracy quantum chemistry to compute ro-vibrational parameters and energies of the title reaction s ystem, followed b y calculation of thermal rate constants using the SCTST/VPT2
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The Journal of Physical Chemistry
approach. Some salient results of this stud y can be summarized as follows: (i)
The reaction of O( 3 P) + H 2 = OH + H is predicted to be endothermic b y 1.45 ± 0.10 kcal/mol, in excellent agreement with an ATcT value of 1.54 ± 0.01 kcal/mol.
(ii)
Classical barrier heights are predicted to be 13.1 ± 0.2 kcal/mol (without DBOC and SO corrections) and 13.5 ± 0.2 kcal/mol (including both DBOC and SO corrections). The former agrees well (within 0.1 kcal/mol) with all barriers reported recentl y in the literature, where the smaller SO and DBOC effects have heretofore been ignored.
(iii)
For T ≤ 2500K, calculated thermal rate constants for the title reaction and its isotopic analogs are in quantitative agreement (within 5-15%) with all experimental data where they are available.
For
T
>
2500K,
the
calculated
k(T)
values
s ystematicall y underestimate the experiment b y about 15 to 35%. This underestimation may be due to the fact that non-statistical population
fractions
of
vibrationall y
excited
H 2 (ν H H =1&2)
molecules exist, and/or because of shortcomings of the VPT2 model. Other possible sources of error include rovibrational parameters,
and/or
neglecting
coupling
rotations in TS.
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of
vibrations
and
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(iv)
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Thermal reaction rate constants of ground state ox ygen atoms with vibrationall y excited h ydrogen atoms are derived and computed (see the supporting information). We are able to match the calculated k(T) values with experiment b y increasing the thermal
population
h ydrogen
atoms
fractions
(ν H H =1)
with
of
vibrationall y respect
to
the
first-excited Boltzmann
distribution. However, this is not in an y way to be interpreted as convincing evidence of non-statistical behavior; in our view, a breakdown of VPT2 at very high temperatures is the more likel y source of error.
Supporting information Optimized geometries, rovibrational parameters, anharmonic constants, thermal rate constants, and energies for various species in the reaction of O( 3 P) with H 2 and its isotopic counterparts are given. This material is available free of charge via the Internet at http://pubs.acs.org.
Acknowledgements JFS and T LN are supported b y the Robert A. Welch Foundation (Grant F1283) and the Department of Energy, Office of Basic Energy Sciences (Contract Number DE-FG02-07ER15884). We would like to thank John R. Barker, Universit y of Michigan for bringing this reaction to our attention and his advice. We would like to thank an anon ymous reviewer for
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pointing us to Refs. 30, 43 and 72 and other comments that led to improvement of this work.
Table 1: Calculated relative energies (kcal/mol) of various terms in HEAT-345(Q) protocol
δHF
δCCSD(T)
δCCSDT
δCCSDT(Q) a)
δREL
δZPE
δDBOC b)
δSO
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
15.3913
-13.0763
0.0308
-0.1239
0.1255
-0.9420
0.0168
0.0241
32.1324
-18.7682
-0.1756
-0.1064
0.0405
-2.2853
0.1704
0.2228
Species 3
O( P) + H2 OH(2Π) + H 2
TS (O—H—H, Π)
δHEAT 0.0000 1.4463 (1.3351) c) 11.2306 (11.1799) c)
a) Ca lc u lat ed a t C CS DT ( Q ) /cc -p VT Z le v el o f t heo r y b ) Ob ta i ned wi t h C C SD /a u g -c c -p VT Z l e ve l o f t heo r y c) Ob ta i ned wi t h HE AT -4 5 6 ( Q) p r o to co l
Table 2: A comparison of classical barrier heights and geometrical parameters of collinear TS in the O( 3 P) + H 2 = OH + H reaction calculated using various levels of theory Method
Au t h o r s
MRCI+Q/ [ 5 s 5 p 3 d 2 f1 g / 4 s 3 p 2 d 1 f] J3 surface
Walch
Year
a)
J o s e p h - Tr u h l a r Garrett b) Peterson-Dunning Rogers et al d)
c)
1987
Vo ( k c a l / mo l ) 12.4
O—H ( Å) 1.225
H—H ( Å) 0.893
1988
13.03
1.217
0.915
1997 2000
13.1 13.26 ± 0.3
1.215 1.244
0.894 0.865
2000
13.04 ± 0.3
1.217
0.903
2004 2004 2012
13.08 ±
