Bond Dissociation Energies of C10 and C18 Methyl Esters from Local

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Bond Dissociation Energies of C10 and C18 Methyl Esters from Local Multireference Averaged-Coupled Pair Functional Theory Victor B. Oyeyemi,† Johannes M. Dieterich,‡ David B. Krisiloff,§ Ting Tan,§ and Emily A. Carter*,‡,∥,⊥ †

Departments of Chemical and Biological Engineering, ‡Mechanical and Aerospace Engineering, §Chemistry, ∥Program in Applied and Computational Mathematics, and ⊥Andlinger Center for Energy and the Environment, Princeton University, Princeton, New Jersey, 08544, United States ABSTRACT: We previously developed a fast, local, reduced scaling Cholesky-decomposed multireference averaged-coupled pair functional (CD-LMRACPF2) method, which takes advantage of the locality of dynamic correlation and numerical approximations such as Cholesky decomposition and integral screening. Motivated by the desire to study large biodiesel methyl ester molecules, here we validate CD-LMRACPF2 for the computation of bond dissociation energies (BDEs) in a suite of oxygenated molecules, and show that the low-cost method is very accurate compared to the conventional variant. We then demonstrate the power of CD-LMRACPF2 for fast and accurate computation of energies of molecules containing up to 13 second-row atoms within a polarized triple-ζ (cc-pVTZ) basis set. We use biodiesel methyl esters as a chemically interesting model system and furnish BDEs of C10 and C18 methyl esters, with the latter performed within a cc-pVDZ basis set. We describe trends in the BDEs and explain how structural (isomeric) differences affect BDEs, as well as discuss implications of BDE trends for biodiesel physical and chemical properties. electrons.5,6 Thus, beyond a reasonable cutoff distance, some electron−electron interactions are effectively negligible. Local approximation approaches have been applied to MBPT,7−13 CC,14−23 and CI.24−29 Some of these methods have been used on systems with hundreds of atoms. For example, Riplinger, Neese, and co-workers tested their local CCSD(T) (single and double excitation CC with perturbative triples) method on crambin, a protein with 644 atoms.30,31 Maurer et al. applied a linear scaling MBPT method to molecules with over 1000 atoms.12 These local methods on average achieve chemical accuracy (i.e., ∼1 kcal/mol) in energy differences. The local CCSD(T) of Riplinger et al.,31 for instance, gave mean and maximum absolute deviations of 0.53 and 2.15 kcal/mol, compared to canonical CCSD(T), for a set of reaction energies. The current study is motivated by interests in elucidating biodiesel combustion chemistry. Renewable biodiesel is a cleanburning fuel used in small cars, buses, and trucks.32 Use in airplanes has also been successfully demonstrated.33 Biodiesel use supports the desire for environmentally friendly and sustainable alternatives to petroleum-derived fuels. Like any other fuel, properties of biodiesel combustion are determined by the chemical structure of the constituent molecules. Biodiesel is made from the transesterification reaction of plant-, animal-, or algal-derived fatty-acids with methanol (and sometimes ethanol).32 This produces large, long-chain methyl

1. INTRODUCTION Quantum chemistry has proven to be an indispensable simulation tool to determine physical and chemical properties in multiple fields, from atmospheric and combustion sciences to biochemistry. The accuracy and predictive capability of quantum chemistry is impressive and its application continues to grow. However, the most accurate methods of quantum chemistry are extremely computationally demanding for the increasingly large systems for which studies are often desired. Popular approaches such as density functional theory (DFT) are well-established for studying molecules and extended solids. More than any other quantum mechanics method, DFT approximations (DFAs) have pushed the application size limits the farthest. DFAs typically scale as O(N3) or less with system size, and can be used to treat hundreds of atoms.1 Unlike DFAs, ab initio correlated wave function (CW) methods, such as coupled cluster (CC) theory,2,3 many-body perturbation theory (MBPT),2 and configuration interaction (CI),4 have much smaller application size limits of a few dozen atoms because they usually scale as O(N6−7). However, these CW methods are generally more accurate than DFA. Thus, for applications such as thermochemical kinetics, where accuracy is paramount, CW methods are preferable if the cost of such methods can be reduced. One strategy that has been used to extend the size limits of CW methods is the development of local correlation methods. Local approximations in CW methods take advantage of the fact that the correlation between two electrons is asymptotically proportional to r−6, where r is the distance separating the two © XXXX American Chemical Society

Received: December 29, 2014 Revised: March 13, 2015

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DOI: 10.1021/jp512974k J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

Figure 1. Thermochemistry of large C16−C18 biodiesel methyl esters motivates the current study.

Figure 2. MRACPF2 BDEs in methyl linolenate (methyl octadeca-9,12,15-trienoate) obtained from calculations on methyl butanoate, butane, 1pentene, 3-hexene, and 1,4-pentadiene surrogates. Reprinted from ref 37. Copyright 2014 American Chemical Society.

(PESs), an area where CCSD(T) and MBPT fall short. In several previous studies, we employed a multireference averaged coupled-pair functional34,35 (MRACPF2)-based scheme to study thermochemistry and kinetics of molecules, such as hydrocarbons, aldehydes, carboxylic acids, and small oxygenated fuel molecules.36−39 We also analyzed approximate bond dissociation energies (BDEs) of C18 methyl ester biodiesel molecules. The C18 methyl ester BDEs were estimated indirectly from those of small molecule surrogates, using C1− C4 methyl esters for the ester end of the chain and small

esters (CH3OC(O)R) with 16 to 18 carbon atoms in the hydrocarbon chain (see Figure 1). While the exact composition of biodiesel differs by feedstock, most of the esters have 19 carbons atoms and the majority of the esters have between one and three CC bonds. Though the aforementioned local single reference CCSD(T) and MBPT methods boast good accuracy and impressive ability for application to very large molecules, we are interested in advancing multireference theories that are capable of consistently describing entire potential energy surfaces B


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CO and CC bonds. For those bonds, a CAS(4e,4o) active space consisting of the π and π* orbitals of CO or CC, and the σ and σ* orbitals of the dissociating bond is needed. Bis-allylic bonds, in which the breaking bond is β to two different double bonds, require CAS(6e,6o). Lastly, a CAS(8e,6o)36 active space is needed for acyloxyl-radical-producing bonds. For the CD-(L)MRACPF2 calculations of any BDE, we use a superset of references from important CASSCF configurations (those with CASSCF coefficients greater than 0.05) at the equilibrium geometries of the molecule and the supermolecule. Using a looser reference selection threshold (e.g., 0.1) will leave out important configurations and will therefore likely lead to erroneous BDEs. The reference energy is defined as the energy of a CI performed using just the reference configurations. It is only the same as the CASSCF energy when all the CASSCF configurations are used as references, which is the case for CAS(2e,2o)-based calculations. For larger active space calculations, only a subset of the full CASSCF configurations are retained after the 0.05 threshold is applied. We extrapolate the reference energies (Eref) and the correlation (Ecorr) energies to the complete basis set (CBS) limit, using cc-pVDZ (X = 2) and cc-pVTZ (X = 3) basis sets within Truhlar’s scheme:47−49

saturated and unsaturated hydrocarbons for the long carbon chain segment.37 Figure 2 shows one such set of BDE estimations. Using surrogates effectively allowed us to bypass the system size limit of conventional MRACPF2. Our group developed a local, reduced scaling MRACPF2 algorithm that has enabled the direct study of molecules with up to 50 second-row atoms.27,28,40,41 This opens up the opportunity to extend our previous studies of small biodiesel ester surrogates to more relevant larger esters, such as methyl decanoate (denoted as C10 for an ester with 10 carbon atoms in the alkyl chain) and four unsaturated C10 variants, as well as a C18 methyl ester. We first tested and verified the effectiveness of the local, Cholesky-decomposed MRACPF2 (CDLMRACPF2) theory for the computation of BDEs. This required optimizing the local parameters which control the cost and accuracy of the local approximation. Below, we report validation of the local approximation against conventional (i.e., nonlocal) MRACPF2 theory for relevant molecules and then demonstrate that the local approximation can be used to compute BDEs in molecules up to C18 esters. The rest of the paper is structured as follows. Section 2 describes our MRACPF2-based method for BDE computation, including approaches for geometry optimization and vibrational frequency calculations. Section 3 gives a brief description of the local correlation approximation and associated notes on computational cost improvements over conventional MRACPF2 theory. Sections 4 and 5 present results validating local MRACPF2 BDEs. Section 6 reports predicted BDEs in C10 and C18 methyl esters. Finally, section 7 concludes with a summary of key findings.

(1)

D0 = De + ΔZPE

(2)

D298 = D0 + ΔH 0 → 298

(3)

(4)

cor EXcor = E∞ + Acor X −β

(5)

CASSCF calculations were performed within MOLCAS 7.8.50 TigerCI24−28,40,41 was used for CD-(L)MRACPF2 calculations. Geometry optimization and vibrational frequency calculations were performed with ORCA.51

2. MULTIREFERENCE SCHEME AND COMPUTATIONAL DETAILS Our scheme for BDE calculations has been outlined previously.36−38 We define De (electronic energy change upon dissociation), D0 (enthalpy of dissociation at 0 K), and D298 (enthalpy of dissociation at 298 K) for a generic bond A− B as De = ΔEMRSDCI/MRACPF(2)

ref EXref = E∞ + Aref X −α

3. NOTES ON LOCAL MRACPF2 CALCULATIONS Localized orbitals provide the molecular orbital basis for local MRACPF2 theory, because spatially localized electrons are necessary to identify the long-range dynamic correlation that can be subsequently neglected. We localize the CASSCF orbitals using the Pipek−Mezey method52 for inactive orbitals and the Local Orthonormal Virtual Orbital method53 for virtual orbitals (though a newer localization approach may be fruitful as well54,55). Such localization is permissible because, like MRSDCI, MRACPF2 is invariant under unitary transformations of inactive or virtual orbitals. Nonlocal MRACPF2 energies obtained using natural CASSCF orbitals should be the same as nonlocal MRACPF2 energies obtained using localized orbitals. Care must be taken to preserve energy invariance in the orbital localization process. If all CASSCF orbitals, including inactive, active, and virtual orbitals, are localized together, the MRACPF2 energy ceases to be invariant. The correct approach is to localize each orbital class separately, as illustrated in Figure 3. Note that we freeze the core orbitals in the CD-(L)MRACPF2 calculations (i.e., no excitations are allowed out of these orbitals); these orbitals are therefore not localized with the other inactive orbitals because doing so would break invariance. To truncate nonlocal dynamic electron correlation, we identify the localized orbitals with spherical domains, as illustrated in Figure 4a. Here we give a general overview of our localization approach; see ref 24 and ref 56 for details. The center (C) of each sphere is defined as the centroid of the electron density of the orbital (a “pseudo” electron density for

ΔZPE and ΔH0→298 are the zero point energy and thermal correction differences between dissociated fragments (A• + B•) and the undissociated molecule (AB), respectively. Geometries, vibrational frequencies, ΔZPE, and ΔH0→298 are computed via DFT-B3LYP42 within the cc-pVTZ43 basis set. The frequencies are scaled by 0.985 to account for anharmonic effects and basis set incompleteness.44 Electronic energies are calculated with local and/or nonlocal multireference averaged coupled-pair functional (CD-LMRACPF2 and/or CDMRACPF2) methods.27,28 The complete active space self-consistent field (CASSCF)45 method uses the CD integrals and the resulting CD-CASSCF46 is used to generate the reference multiconfigurational wave function for CD-(L)MRACPF2 calculations. CASSCF active space sizes vary depending on the nature of the breaking bond.36 The CAS(2e,2o) active space consisting of two electrons in two orbitals (σ and σ*) of the dissociating bond is sufficient for most single bonds. π-Electron resonance effects arise when breaking bonds beta (β) to double bonds, including C


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more atoms. In those cases, a minimum sphere radius value is applied. To optimize the parameters we compared CDLMRACPF2/cc-pVDZ and CD-MRACPF2/cc-pVDZ BDEs of bonds in decane and methyl butanoate. The parameters were varied with BOBYQA57 (a constrained, local optimizer) to minimize the difference between the local and nonlocal BDEs. The optimized values are presented in Table 1. Along with the local approximations, significant cost savings is made possible with the use of Cauchy−Schwarz atomic orbital integral screening,26 Cholesky-decomposed integrals,26 parallelization,40 and integral-direct CI implementations.41 See refs 28, 40, and 41 for further discussions on the algorithms and code performance optimization. Figure 3. CASSCF orbital localization in subspaces preserves MRACPF2 energy invariance.

4. VALIDATION AND COST SAVINGS OF LOCAL MRACPF2 BDES The accuracy of nonlocal MRACPF2 BDEs relative to experimental and Active Thermochemical Table (ATcT)derived58,59 BDEs36 was previously demonstrated. In all, BDEs computed for bonds in hydrocarbons, alcohols, aldehydes, carboxylic acids, and methyl esters were accurate to within 1 kcal/mol on average.36−38 Here, we show that CDLMRACPF2 produces accurate BDEs compared to nonlocal CD-MRACPF2 BDEs for key hydrocarbon and oxygenated molecules. Our validation set of all bonds in pentane, butanol, butanal, and methyl butanoate is representative of the bonds/ molecules we have previously studied. The bonds in the molecules are diverse, but most require only a CAS(2e,2o) active space. Others need CAS(4e,4o) and CAS(8e,6o). Figure 5 shows deviations between BDEs computed using CDMRACPF2 and CD-LMRACPF2 for bonds in pentane, butanol, butanal, and methyl butanoate. Each graph shows BDE errors for the indicated molecule, with bonds numbered from left to right in the inset molecular structures. Errors (nonlocal CD-MRACPF2 BDE−CD-LRMRACF2 BDE) are shown for CD-(L)MRACPF2 BDEs computed within cc-

unoccupied orbitals). The radius of each sphere is adjustable and determines the extent of the local correlation. In our local truncation approximation, electrons in overlapping spherical domains correlate, while those in nonoverlapping spherical domains do not correlate. Additionally, a hemispherical-capped cylindrical domain is defined for orbitals in active spaces to allow preservation of excitations along a bond dissociation curve (Figure 4b). Local truncations fall in two categories: weak pairs (WP) and truncation of virtuals (TOV, Figure 4c).6 In WP, simultaneous excitations from two occupied orbitals are allowed only if those two orbitals’ domains overlap.56 In TOV, an excitation from an occupied orbital into a virtual orbital is allowed only if the two orbitals’ domains overlap.24 For maximum flexibility, different parameters control spherical domain sizes for the WP and TOV approximations, and it is possible to tune the radii separately for occupied and virtual orbitals. Special consideration is given to orbitals that are localized primarily on a single atom, such as core and lone pair orbitals, because these orbitals form very small spheres compared to the orbitals involved in bonding between two or

Figure 4. Local orbital domain construction and local truncation approximations. (a) shows steps from orbital localization to identification of localized orbitals with spherical domains of center C and radius R. (b) depicts a hemispherically capped cylinder domain for active orbitals that allow for a smooth, continuous bond breaking. (c) illustrates WP and TOV approximations. D


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The Journal of Physical Chemistry A Table 1. Optimized Local MRACPF2 Parameters (See Text for Details) parametersa

comments

values

occupation threshold, occ virtual occupation threshold, vocc WP default radius (bohr), wpdr TOV virtual default radius (bohr), tvdr TOV occupied default radius (bohr), todr WP radius multiplier, wprm TOV virtual radius multiplier, tvrm TOV occupied radius multiplier, torm TOV cylinder radius (bohr), tvcr WP cylinder radius (bohr), wpcr Cholesky decomposition threshold Cauchy−Schwartz threshold

sphere is extended until this charge occupation (normalized to 1.0) is reached. same as above, but for virtual orbitals. for highly localized occupied orbitals, a default sphere is used. only for weak pair (WP) approximation. default sphere radius for highly localized virtual orbitals. Only for TOV approximation. default sphere radius for highly localized occupied orbitals. Only for TOV approximation. each sphere determined by occ or vocc is scaled. Only for WP approximation. each virtual sphere for TOV approximation is scaled. each occupied sphere for TOV approximation is scaled. for virtual orbitals in active space. for occupied orbitals in active space. numerical accuracy of the Cholesky decomposition AO Integrals below this about are screened out.

0.80 0.80 2.65 2.65 2.65 1.70 2.00 1.95 2.00 2.00 1 × 10−5 1 × 10−5

a

Cylinder radii of 2 bohrs were used for CAS(2e,2o) calculations, while radii of 4 bohrs were used for CAS(4e,4o), CAS(6e,6o), and CAS(8e,8o) calculations to more adequately account for increased orbital delocalization and resonance in bond breaking involving allylic, bis-allylic, and acyloxyl bonds, respectively.

Figure 5. Deviations of CD-LMRACPF2 (local) D298s from conventional CD-MRAPCF2 (nonlocal) D298s in pentane, butanol, butanal, and methyl butanoate. Deviations in BDEs within cc-pVDZ and cc-pVTZ basis sets, as well as CBS-extrapolated BDEs, are shown.

because they are the most accurate when compared to experimental or ATcT reference BDEs. The accuracy of CDLMRACPF2 BDEs is very good for hydrocarbon bonds (e.g., pentane). Maximum BDE deviation (at the cc-pV∞Z basis set limit) from conventional MRACPF2 over all bonds is 0.38 kcal/mol and the MAD is only 0.17 kcal/mol. Average errors for oxygenated molecules are slightly higher than those of pentane. Maximum and average deviations for butanol are 0.50 and 0.18 kcal/mol, respectively. Average errors for butanal and methyl butanoate are 0.32 and 0.22 kcal/mol, respectively. Overall, deviations between local and nonlocal BDEs are fairly small. This lends credence to substituting nonlocal CD-

pVDZ, cc-pVTZ, and cc-pV∞Z basis sets, with the latter yielding CBS-extrapolated BDEs. For each molecule, the maximum absolute error (MAX) and mean absolute deviation (MAD) are shown under the graph. The most apparent observation from the graphs is that the errors increase as one moves from cc-pVDZ to cc-pVTZ and cc-pV∞Z bases, though there are a few exceptions within each molecule. The slightly higher cc-pVTZ error is probably because the local parameters we used were optimized for the cc-pVDZ basis set. Extrapolated BDEs tend to then have larger errors because they are propagated from cc-pVDZ and ccpVTZ. While we show cc-pVDZ and cc-pVTZ BDEs and statistics in Figure 5, we discuss only extrapolated BDEs below E


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molecule were kept fixed at the equilibrium structure while each bond was broken, i.e., no relaxation of the rest of the molecule was allowed at these points). The points fall in the ranges 1.1− 2.0 Å in increments of 0.1 Å, 2.4−3.5 Å in increments of 0.1 Å, and 4.0−9.5 Å in increments of 0.5 Å. All of the calculations were run on nodes with two Intel Xeon E5-2670 @ 2.60 GHz processors and 64 GB RAM. The CD-LMRACPF2 code was compiled with ifort/icpc version 13.0 and linked against Intel’s MKL version 11.0. All 16 cores were used. The nonlocal CDMRACPF2 wave function size is the same for all bonds and at all bond lengths, and has 159 428 015 configuration state functions (CSFs; see Figure 6a). Nonlocal CD-MRACPF2 computational times vary between points along the PES (Figure 6b): the minimum and maximum computation times are 19 600 and 27 200 s, respectively; the mean and standard deviation are 24 500 and 2600 s, respectively. Other than some oscillations, the computational times for nonlocal CDMRACPF2 for the five bonds are about the same at any particular stretch length (Figure 6b; the largest percentage difference in nonlocal computational time from the lowest time for each bond length is less than 6%). Turning to the local calculations, the size of the CD-LMRACPF2 wave function for all bonds decreases fairly smoothly as the bond elongates such that the maximum wave function size is at 1.1 Å and the minimum is in the 5−9.5 Å range (Figure 6a) where the bond is dissociated. The wave function size decrease occurs even though a stretchable cylinder is used as the local domain of the breaking bond. Nonbonding interactions that do not involve the active orbitals are subject to the local approximation and are truncated from the wave function as the fragments separate. We further analyze the computational cost of CDLMRACPF2 on decane (Figure 7a) in Figure 7b,c using two metrics: (i) speed-up, the ratio of CD-MRACPF2 wall time to CD-LMRACPF2 wall time, and (ii) wave function size, the CD-LMRACPF2 wave function size (in CSFs) as a percentage of CD-MRACPF2 wave function size. Considering the CDLMRACPF2 wave function size variation from bond to bond (Figure 7b), we observe that at bond lengths shorter than 1.53 Å, the equilibrium bond lengths, the wave function sizes are larger for interior bonds. The wave function sizes for interior bonds are larger than for the terminal bond by up to 1.5

MRACPF2 with CD-LMRACPF2 for calculating BDEs of large molecules. With accuracy established, we now turn attention to cost savings of the CD-LMRACPF2 method. Figure 6a,b, and

Figure 6. Absolute timing data for CD-LMRACPF2 vs nonlocal CDMRACPF2 for breaking different C−C bonds in decane. Inset decane structure shows the bonds numbered from 1 to 5. “1 n” denotes nonlocal CD-MRACPF2 for bond 1, and “1 l” denotes CDLMRACPF2 for bond 1, etc.

Figure 7a−c compare computational costs of CD-LMRACPF2 and CD-MRACPF2 for decane. We performed both CDMRACPF2 and CD-LMRACPF2 calculations at the cc-pVDZ basis sets for 35 points along the bond breaking PES of the five unique C−C bonds in decane (the internal coordinates of the

Figure 7. Cost, as characterized by relative wave function size and computational speedup, of CD-LMRACPF2 vs conventional nonlocal CDMRACPF2 for different C−C bonds in decane. (a) Bonds in decane are labeled from 1 to 5. Mean, minimum, and maximum (b) wave function size (% of CSFs) and (c) wall-time speed-ups of the CD-LMRACPF2 wave function relative to the CD-MRACPF2 wave function. The ratios are taken at 35 points from 1.1 to 9.5 Å along the bond-breaking PES of each bond. F


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The Journal of Physical Chemistry A percentage points and minimum speed-ups lower by up to 2.5, respectively. On the other hand, at bond lengths greater than 1.53 Å, wave function sizes are smaller and speed-ups (Figure 7c) are larger for interior bonds. Mean wave function sizes decrease by up to three percentage points and speed-ups increase by nine. The trend for lengths greater than the equilibrium bond length is intuitive because interior bonds dissociate into similarly sized fragments and thus take better advantage of the local approximations. For compressed interior bonds, fewer electron pair correlations are being neglected, which is why the wave function size increases and the speed-ups decrease. In all, the mean speed-up is between 10 and 15 for decane. Because of the reduced algorithmic scaling with system size of our method, the speed-ups are expected to be even larger for larger basis sets and larger molecules. For example, nonlocal CD-MRACPF2 with cc-pVTZ bases on decane is currently beyond our computational capacity. However, we show below that CD-LMRACPF2 calculations with cc-pVTZ bases on molecules larger than decane are well within our reach.

Figure 9. CD-LMRACPF2 D298s (top values) and conventional nonlocal CD-MRAPCF2 D298s (bottom values) in methyl butanoate and 1,4-pentadiene.

6. BDES OF C10 ESTERS Because of the large sizes of biodiesel methyl esters, both experimental and computational research studies of biodiesel have initially focused on understanding the combustion chemistry of small surrogates, including methyl formate, methyl acetate, methyl propanoate, and methyl butanoate.60−66 Of these, methyl butanoate combustion best represents observable biodiesel fuel properties, and therefore is the most common surrogate. However, several biodiesel combustion features, such as early CO2 production and low temperature chemistry, are only well-reproduced by larger methyl ester surrogates.67−70 Large surrogates molecules up to and beyond methyl decanoate are currently being studied.64,69,71−79 Methyl decanoate, with 10 carbon atoms in the alkyl chain, has been shown to reproduce more biodiesel fuel combustion properties than methyl butanoate.67−70 Here, we focus on C10 methyl esters because they are not only the latest biodiesel surrogates but they are also sufficiently large to demonstrate the accuracy and improved feasibility limit of the CD-LMRACPF2 method. Typical biodiesel methyl esters have structural characteristics shown in Table 2. Most of the esters have between one and three CC bonds in the hydrocarbon chain. Percentages of various esters in the fuel vary by feedstock. Those in soybean-derived biodiesel common in the United States and rapeseed-derived biodiesel common in Europe are shown in Table 2. The CC bond closest to the ester group is always at the eighth and ninth carbon atoms of the hydrocarbon chain. Subsequent CC bonds are two carbon atoms away, such that there is a bis-allylic C−H bond between two double bonds. The unsaturated C10 surrogates in Figure 10 are selected in an attempt to replicate the relative positioning of CC bonds as much as possible. Here, the first CC bond starts at the third carbon position of the hydrocarbon chain. Thus, in addition to methyl decanoate, we study methyl-4-decenoate and methyl4,7-decadienoate. Although biodiesel esters naturally occur with C−H bonds at CC in cis configurations, we include transconfigurations, as well. This allows us to assess the effect of isomeric differences on ester BDEs. Figure 11 shows CD-LMRACPF2 BDEs of the C10 methyl esters, along with estimates for several bis-allylic CAS(6e,6o) BDEs.81 The BDEs are largely consistent with those obtained from our earlier surrogate estimation approach. Some differences and similarities in BDE trends within and between molecules are recognized and will be discussed next. The methyl ester C−H BDE of methyl decanoate (100.1 kcal/mol)

5. ACCURACY OF CD-LMRACPF2 BDES FOR MOLECULES USED AS BIODIESEL SURROGATES We previously used BDEs of small ester and hydrocarbon surrogates to estimate BDEs of C18 esters found in biodiesel. Studying small surrogates also enables isolation of the effects of the ester group on biodiesel combustion. In actuality, the use of small surrogates in earlier research was unavoidable due to the practical difficulty associated with computational studies of large molecules. Figure 8 shows local (top) and nonlocal

Figure 8. CD-LMRACPF2 D298s (top values) and conventional nonlocal CD-MRAPCF2 D298s (bottom values) in butane, 1-pentene, and 3-hexene.

(bottom) CD-MRACPF2 BDEs in the surrogates butane, 1pentene, and 3-hexene, while Figure 9 shows BDEs for methyl butanoate and 1,4-pentadiene. These five surrogates are sufficient to estimate BDEs in all biodiesel methyl esters, as we demonstrated in ref 37. The local BDEs (top) and nonlocal BDEs (bottom) are nearly identical in all cases, with most local BDEs within 0.2 kcal/mol of the corresponding nonlocal BDEs. Apart from methyl butanoate, the molecules here are different from those used in our optimization and validation set in Figure 5. Thus, we further confirm the impressive accuracy of CDLMRACPF2. G


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The Journal of Physical Chemistry A Table 2. Biodiesel Methyl Esters and Their Proportions in Soybean and Rapeseed Biodiesel80 % in biodiesel common name methyl methyl methyl methyl methyl

palmitate stearate oleate linoleate linolenate

IUPAC name methyl methyl methyl methyl methyl

hexadecanoate octadecanoate (E)-octadec-9-enoate (9E,12E)-octadeca-9,12-dienoate (9E,12E,15E)-octadeca-9,12,15-trienoate

no. of carbons:no. of CC

soybean

rapeseed

C16:0 C18:0 C18:1 C18:2 C18:3

10−12 3−5 18−26 49−57 6−9

2−6 4−6 52−65 18−25 10−11

Figure 10. C10 ester surrogates of biodiesel.

is slightly lower than those in the unsaturated esters (100.6− 100.8 kcal/mol). C−O bonds α to CO have BDEs of 99.5− 100.0 kcal/mol, while the C−O bond producing acyloxyl radicals have lower BDEs of ∼87.5 kcal/mol. The terminal alkyl C−H bond is ∼102.1 kcal/mol, except for the lower value of 101.3 kcal/mol in trans-methyl-4-decenoate. As expected, the interior C−H and C−C BDEs of methyl decanoate are nearly constant (C−C, 87.5−88.4 kcal/mol; C−H, 99.6−100.1 kcal/ mol). BDEs of interior bonds of the unsaturated esters are less regular. Vinylic C−H bonds α to CC have high BDEs of 108.8−109.9 kcal/mol because the resulting radical is sp2 hybridized and the geometry does not relax much.37 Allylic C−H bonds β to CC have dramatically lower BDEs of 84.0− 87.7 kcal/mol when only one CC is involved, and ∼74 kcal/ mol (in cis-methyl-4,7-decadienoate) or ∼79 kcal/mol (in transmethyl-4,7-decadienoate) for bis-allylic C−H bonds due to resonance stabilization of the resulting radicals. Oxyallylic C−H bonds β to the carbonyl groups also produce resonancestabilized radicals, which weaken the incipient bonds, leading to BDEs of 91.7−92.5 kcal/mol−about 8 kcal/mol lower than energies of other secondary C−H bonds. The C−C bond β to CO in methyl decanoate has a BDE of 82.0 kcal/mol, while those in the unsaturated esters have BDEs of ∼66 kcal/mol for the cis configuration and ∼70 kcal/mol for the trans configuration, respectively, again attributable to resonance stabilization in the resulting radicals. The lower BDEs for the unsaturated esters are due to the additional resonance created by the adjoining CC bonds. Terminal C−C bond BDEs are 88.4−88.7 kcal/mol when there are no adjacent CC bonds (in methyl decanoate and methyl-4-decenoate) and lower values of 70.3 and 75.0 kcal/mol for allylic C−C bonds in cismethyl-4,7-decadienoate and trans-methyl-4,7-decadienoate, respectively. The discussion above shows that biodiesel BDEs are especially dependent on the location of the breaking bond relative to double bonds in the molecule. The BDE differences

Figure 11. CD-LMRACPF2/cc-pV∞Z BDEs in C10 esters. Italicized CAS(6e,6o) (bis-allylic) BDEs are estimates and are obtained as CAS(2e,2o) BDEs − 2 kcal/mol.81

due to unsaturation in the alkyl chain in particular affect autoignition behavior, thermal stability, and soot production, as discussed by Westbrook.82 Here we illustrate how BDEs in biodiesel esters and radical products affect some of these H


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Figure 12. Selected BDEs in methyl octadecanoate. Integral-direct CD-LMRACPF2/cc-pVDZ BDEs for methyl octadecanoate are shown in blue; CD-LMRACPF2/cc-pVDZ BDEs for surrogate molecules methyl butanoate and pentane are shown in black for the ester segment and hydrocarbon chain, respectively. Italicized numbers are estimated as “surrogate values − 1 kcal/mol,” in line with trends established for the computed data.

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ACKNOWLEDGMENTS This work was supported as part of the Combustion Energy Frontier Research Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award No. DE-SC0001198. The work reported was performed on the TIGRESS high performance computer center at Princeton University, which is jointly supported by the Princeton Institute for Computational Science and Engineering and the Princeton University Office of Information Technology. We thank Nari L. Baughman for critically reading this manuscript.

combustion properties. As discussed above, unsaturation weakens C−H bonds at allylic and bis-allylic sites. At relatively low temperatures (e.g., < 500 °C), allylic or bis-allylic C−H and C−C bonds easily break compared to nonallylic C−H and C− C bonds due to their lower BDEs. At combustion temperatures (e.g., 500−1500 °C), R + O2 ↔ R-O2 addition becomes disfavored for R produced by abstraction (R−H + X → R + XH) from allylic or bis-allylic sites because the R−O2 bond is also weakest at allylic and bis-allylic sites, with BDEs of about 27 and 16 kcal/mol, respectively, compared with 38 kcal/mol for a regular secondary C−H bond site.82 The weak R−O2 bonds of allylic and bis-allylic radicals reduce low temperature chemistry, thereby limiting autoignition in fuels with more unsaturated methyl esters. The accumulation of R radicals at low, precombustion temperatures also favors association of R radicals. These large R−R′ adducts then can create a viscous liquid that clogs engine components. We end by reporting a few BDEs in methyl octadecanoate to demonstrate the power of the CD-LMRACPF2 method (Figure 12). On the one hand, these few BDEs indicate that we can accurately compute large ester BDEs within ∼1 kcal/ mol compared to nonlocal BDEs of smaller surrogates. On the other hand, we have also validated the accuracy of the surrogate approach used earlier.37

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7. CONCLUSIONS We optimized new local truncation parameters for our CDLMRACPF2 theory and showed that BDEs computed with CD-LMRACPF2 are nearly the same as those computed with conventional CD-MRACPF2, while roughly a factor of 12 faster to compute for methyl decanoate. The reduced cost makes it possible to compute BDEs in C10 methyl ester surrogates of biodiesel. We analyzed BDE trends in methyl decanoate and unsaturated C10 variants. Directly computed BDEs of the C10 esters are consistent with those obtained using the surrogate similarity approach we implemented earlier37 to estimate BDEs of C18 esters. Similarly, directly computed BDEs of methyl octadecanoate also suggest that the surrogate similarity approach offers an efficient means in the future to estimate BDEs in large, complex molecules.

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Bond Dissociation Energies of C10 and C18 Methyl Esters from Local

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