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Simple Energy Corrections for Precise Atomization Energies of CHON Molecules. Pablo F. Salazar† and Jorge M. Seminario*,†,‡. Department of Chemi...
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J. Phys. Chem. A 2007, 111, 11160-11165

Simple Energy Corrections for Precise Atomization Energies of CHON Molecules Pablo F. Salazar† and Jorge M. Seminario*,†,‡ Department of Chemical Engineering and Department of Electrical and Computer Engineering, Texas A&M UniVersity, College Station, Texas 77843 ReceiVed: June 25, 2007; In Final Form: August 5, 2007

We test a few ways to improve atomization energies of CHON molecules and found that the best way to do it is simply by correcting the atom energies of the participating atoms. Extraordinary improvement on the average errors is obtained. For the HF/6-31G** level of theory an average error of 271.2 kcal/mol on 115 molecules is improved to 6.7 kcal/mol simply by correcting the atomic energy of the four chon atoms. The corrections to density functional methods allow us to reach a chemical accuracy of 2 kcal/mol.

I. Introduction Fitting parameters are commonly used to create new functionals; however, an alternative approach to simply correct the calculated energies, without attempting to redesign or modify the functionals is less common but very important to quickly calculate large systems. In the former case, the parameters actually modify the energy functionals from either theoretical or experimental feedback, whereas in the latter case, the parameters correct the energies found using the original procedures. As in previous work,1 we follow the latter, noninvasive, methodology for our energy corrections; this approach consists of choosing contributions to the total energy thought to be the cause of errors; then, coefficients or weights, which we call correctors, are assigned to them. These correctors are found by minimizing the deviations with respect to precise theoretical or experimental values. The nature of errors with one level of theory is usually different from the errors with other levels of theory, but they could be separately estimated. Also, errors due to the finite nature of the basis sets can be decomposed in contributions with respect to the angular moment of the basis functions. In most of the cases, the contributions are additive.2,3 Similar work has been already performed using correctors, for instance, Duan4 used this methodology to correct for the electron correlation energy missing in the HF method by adding weighted descriptors to the HF energy, the weighting values were found by fitting to the experimental values of heat of formation energy using a linear regression; instead, we choose to fit to the dissociation energy by using a Newton minimization scheme. Also, Cioslowski5 developed a bond density functional scheme, in which he introduced and calibrated additional parameters for the prediction of enthalpy of formation of large systems. Earlier work6-8 toward chemical accuracy has been crucial to build present sophisticated correctors. More sophisticated minimization methods have also been tried; Wang9 used a neural network-based scheme to find the weighting coefficients. Recently, we introduced a posteriori multiplicitybased corrections to ab initio energies to reproduce experimental atomization energies.1 This simple approach, compared to † ‡

Department of Chemical Engineering. Department of Electrical and Computer Engineering.

the alternative ones to improve density functionals and standard correlated methods, also required fewer computational resources than higher levels of theory did. In this work, we tested our approach with 115 molecules of the G3/99 set.10-12 The accuracy of the experimental energies of the test set was less than 1 kcal/mol, which is important for the development and validation of new methods. We limit our study to molecules composed of carbon, hydrogen, nitrogen, and oxygen, also known as CHON molecules. II. Methodology Several ab initio methods have been used in this work, aiming to evaluate the energy correctors through a broad span of levels of accuracy, from the inexpensive HF, going through the postHF (MP2), through the DFT (B3PW91), and to the highly accurate G3x method. The Hartree-Fock method is the first approximation and the least expensive of the ab initio methods; the repulsion between electrons is taken into account as an average effect but not the specific repulsion interaction between electrons, which is known as the correlation effect. However, this approximation is still useful for first-order predictions of many systems. The correlation between electrons is considered by the Møller-Plesset MP2 theory,13 which adds a second-order energy correction to the HF electronic energy. The B3PW91 uses the Becke-3 term hybrid exchange functional and the generalized-gradient approximation (GGA) Perdew-Wang 91 correlation functional.14-16 The G3 theory is a composite electronic structure method that is based on a sequence of ab initio molecular-orbital calculations. It approximates to a QCISD(T) high level of calculation at the MP2/6-31G* geometry and uses scaled HF/ 6-31G* for zero-point energies. The G3X theory17 is an extension of the G3 theory that uses B3LYP/6-31G(2df,p) geometries and zero-point energies scaled by 0.9854, taking about 10%-15% more time than G3 due to a B3LYP/6-31G(2df,p) frequency calculation. Besides the G3 theory, there are few other computational protocols for small molecules with high accuracy on the order of sub-kcal/mol: for instance, the “Wn theory” developed by Weizmann18,19 and the highly accurate extrapolated ab initio thermochemistry (HEAT).19 These two protocols show an unprecedented accuracy of 0.16 kcal/mol root-mean-square deviation between theory and experiment. This is less than the 0.6 kcal/mol deviation obtained with the G3x

10.1021/jp074940i CCC: $37.00 © 2007 American Chemical Society Published on Web 10/10/2007

Precise Atomization Energies of CHON Molecules

J. Phys. Chem. A, Vol. 111, No. 43, 2007 11161

TABLE 1: Total Electronic Energies for Different Level of Theory molecule

name

G3x (Ha)

H (2S) C (3P) N (4S) O (3P) H2 CH (2P) CH2 (3B1) CH2 (1A1) CH3 (2A2) CH4 NH (3Sg) NH2 (2B1) NH3 OH OH2 HCCH H2CCH2 H3CCH3 CN(2) HCN CO HCO (2A′) H2CO H3COH N2 H2NNH2 NO(2) O2 (3Sgg) HOOH CO2 CH2dCdCH2 CH3CCH HCOOCH3 C3H4 CH3CHdCH2 C3H6 C3H8 CH2CHCHCH2 C4H6 C4H6 C4H6 C4H6 C4H8 C4H8 C4H10 C5H8 CH3CN CH3ONO HCOOH NCCN (CH3)2NH CH2CO C2H4O CH3CHO HCOCOH CH2)CHCN CH3COOH (CH3)2CHOH (CH3)3N C4H4O C4H5N C5H5N CCH (2A′) C2H3 (2A′) CH3CO (2A′) H2COH (2A) C2H5 (2A′) (CH3)2CH (2A′) (CH3)3C (2A1) NO2 (2) C4H6 C5H8

hydrogen carbon nitrogen oxygen gas hydrogen gas methylidyne methylene methylene methyl radical methane imidogen amino radical ammonia hydroxyl radical water acetylene ethylene ethane cyano radical hydrogen cyanide carbon monoxide formyl radical formaldehyde methyl alcohol nitrogen gas hydrazine nitric oxide oxygen gas hydrogen peroxide carbon dioxide allene propyne methyl formate cyclopropene propylene cyclopropane propane butadiene 2-butyne methylene cyclopropane bicyclobutane cyclobutene cyclobutane isobutene trans butane spiropentane methyl cyanide methyl nitrite formic acid cyanogen dimethylamine ketene oxirane acetaldehyde glyoxal acrylonitrile acetic acid isopropyl alcohol trimethylamine furan pyrrole pyridine ethynyl radical vinyl acetyl radical hydroxymethyl radical ethyl radical isopropyl radical tert-butyl radical nitrogen dioxide methylallene isoprene

-0.50097 -37.82831 -54.56490 -75.03224 -1.17724 -38.46536 -39.13566 -39.12003 -39.82220 -40.50182 -55.20079 -55.85778 -56.54157 -75.70437 -76.40430 -77.30429 -78.55861 -79.79731 -92.68250 -93.39411 -113.27498 -113.80663 -114.45956 -115.68142 -109.49354 -111.82992 -129.84069 -150.25622 -151.49339 -188.51583 -116.61032 -116.61218 -228.97853 -116.57363 -117.86195 -117.84908 -119.09646 -155.93206 -155.91857 -155.90033 -155.88818 -155.91224 -157.15113 -157.16684 -158.39583 -195.19512 -132.70716 -244.91551 -189.69375 -185.58943 -135.11378 -152.54102 -153.72969 -153.77197 -227.73499 -170.76907 -229.00475 -194.29116 -174.40799 -229.93288 -210.08961 -248.18412 -76.58155 -77.87039 -153.11880 -115.01578 -79.12299 -118.42610 -157.73080 -204.99262 -155.91220 -195.23656

B3PW91/ 6-31G** (Ha) Training Set -0.50218 -37.82569 -54.56393 -75.03133 -1.17752 -38.45947 -39.13037 -39.11058 -39.82752 -40.50869 -55.20039 -55.85708 -56.53751 -75.69942 -76.39255 -77.29452 -78.56139 -79.80958 -92.66988 -93.38281 -113.26028 -113.80451 -114.45657 -115.68061 -109.47710 -111.82834 -129.83582 -150.26254 -151.48602 -188.50695 -116.61529 -116.61088 -228.97910 -116.58215 -117.87044 -117.86399 -119.11228 -155.93828 -155.92530 -155.91467 -155.90576 -155.92595 -157.17061 -157.17888 -158.41496 -195.21808 -132.70395 -244.91763 -189.68924 -185.57208 -135.12415 -152.54067 -153.73575 -153.77548 -227.72946 -170.76362 -229.00479 -194.29720 -174.42144 -229.93998 -210.09891 -248.19560 -76.57238 -77.87411 -153.12322 -115.01734 -79.13604 -118.44521 -157.75412 -204.99491 -155.92227 -195.24541

B3PW91/ cc-pVTZ (Ha)

HF/6-31G* (Ha)

HF/3-21G (Ha)

MP2(full)/ 6-31G* (Ha)

-0.50398 -37.83683 -54.58043 -75.06125 -1.17894 -38.47372 -39.15400 -39.12739 -39.84165 -40.52128 -55.22008 -55.88010 -56.56283 -75.73347 -76.43068 -77.32507 -78.58780 -79.83264 -92.70166 -93.41685 -113.30509 -113.84911 -114.50040 -115.72638 -109.51944 -111.87306 -129.88633 -150.32181 -151.55190 -188.58282 -116.65394 -116.65105 -229.06267 -116.61874 -117.90708 -117.89895 -119.14583 -155.98766 -155.97503 -155.96194 -155.95085 -155.97140 -157.21373 -157.22567 -158.45895 -195.27393 -132.74809 -245.00711 -189.76390 -185.63549 -135.16693 -152.59858 -153.78991 -153.82981 -227.81342 -170.82004 -229.08972 -194.36323 -174.47355 -230.01669 -210.16667 -248.27075 -76.60069 -77.90104 -153.17847 -115.06394 -79.16052 -118.47991 -157.79892 -205.07605 -155.97108 -195.30490

-0.49823 -37.68086 -54.38544 -74.78393 -1.12683 -38.26493 -38.92150 -38.87237 -39.55899 -40.19517 -54.95942 -55.55770 -56.18436 -75.38228 -76.01075 -76.81783 -78.03172 -79.22875 -92.20483 -92.87520 -112.73788 -113.24766 -113.86633 -115.03542 -108.94395 -111.16937 -129.24788 -149.61791 -150.76479 -187.63418 -115.86110 -115.86432 -227.78942 -115.82305 -117.07147 -117.05887 -118.26365 -154.91965 -154.90925 -154.88734 -154.87177 -154.89962 -156.09720 -156.11067 -157.29841 -193.91776 -131.92753 -243.66864 -188.76231 -184.59122 -134.23885 -151.72467 -152.86736 -152.91597 -226.59218 -169.76802 -227.81065 -193.11542 -173.26930 -228.62521 -208.80785 -246.69582 -76.15009 -77.39029 -152.29798 -114.40876 -78.59715 -117.63614 -156.67501 -204.03149 -154.89938 -193.95723

-0.49620 -37.48107 -54.10539 -74.39366 -1.12296 -38.05191 -38.70907 -38.65185 -39.34261 -39.97688 -54.65924 -55.24538 -55.87220 -74.97023 -75.58596 -76.39596 -77.60099 -78.79395 -91.68475 -92.35408 -112.09330 -112.60380 -113.22182 -114.39802 -108.30095 -110.55001 -128.50064 -148.76909 -149.94582 -186.56126 -115.21991 -115.22539 -226.51428 -115.16201 -116.42401 -116.40121 -117.61330 -154.05946 -154.05365 -154.01873 -153.98664 -154.03072 -155.23136 -155.24714 -156.43247 -192.82114 -131.19180 -242.28636 -187.70020 -183.56292 -133.49485 -150.87653 -152.00070 -152.05525 -225.30696 -168.82040 -226.53424 -192.04826 -172.31027 -227.35008 -207.64757 -245.31201 -75.73206 -76.96280 -151.43851 -113.77382 -78.16365 -116.98609 -155.80910 -202.84346 -154.04148 -192.87756

-0.49823 -37.73651 -54.45945 -74.88200 -1.14414 -38.34240 -39.00744 -38.97401 -39.67303 -40.33704 -55.06142 -55.69375 -56.35738 -75.52321 -76.19924 -77.07622 -78.29429 -79.50397 -92.44196 -93.16694 -113.02818 -113.54033 -114.17496 -115.35330 -109.26157 -111.50440 -129.56446 -149.95432 -151.13492 -188.11836 -116.24852 -116.25624 -228.43398 -116.21957 -117.46966 -117.46284 -118.67441 -155.44171 -155.43515 -155.41602 -155.40948 -155.42933 -156.63706 -156.64640 -157.84497 -194.58924 -132.35131 -244.33911 -189.25187 -185.17464 -134.67810 -152.16008 -153.31569 -153.35897 -227.20373 -170.31611 -228.43398 -193.70655 -173.84646 -229.33278 -209.50418 -247.51068 -76.35347 -77.61326 -152.72265 -114.70340 -78.84466 -118.01923 -157.19579 -204.56859 -155.42290 -194.61762

11162 J. Phys. Chem. A, Vol. 111, No. 43, 2007

Salazar and Seminario

TABLE 1: Continued molecule C5H10 C5H12 C5H12 C6H8 C6H8 C6H12 C6H14 C6H14 C6H5CH3 C7H16 C8H8 C8H18 C10H8 C10H8 CH3COOCH3 (CH3)3COH C6H5NH2 C6H5OH C4H6O C4H8O C5H8O C6H4O2 C4H4N2 NdC-CH2-CH2-CdN C4H4N2 CH3-C(dO)-CCH CH3-CHdCH-CHO CH3-C(dO)-OC(dO)-CH3 (CH3)2CH-CN CH3-CO-CH2-CH3 (CH3)2CH-CHO C4H8O2 C4H8NH CH3-CH2-CH(CH3)-NO2 CH3-CH2-O-CH2-CH3 CH3-CH(OCH3)2 (CH3)3C-NH2 cyc-CHdCH-N(CH3)CHdCHC5H10O CH3-CH2-CO-CH2-CH3 C5H10O2 cyc-C5H10NH (CH3)2CH-O-CH(CH3)2 C6H5 (2A1) (CH3)3C-O-CH3 C6H6 CH3NO2 C4H10 CH3NH2 CH3CONH2 CH3CH2NH2 CH3COCH3 C2H4NH C2H5OCH3 CH3CH2OH CH3OCH3 CH3CH2O (2A′′)

name

G3x (Ha)

B3PW91/ B3PW91/ 6-31G** (Ha) cc-pVTZ (Ha)

HF/6-31G* (Ha)

HF/3-21G (Ha)

MP2(full)/ 6-31G* (Ha)

-196.48266 -197.69520 -197.70229 -233.33068 -233.33039 -235.79261 -236.99460 -236.99550 -271.45783 -276.29402 -309.46732 -315.59346 -385.73775 -385.68030 -268.28912

-196.50359 -197.71759 -197.71910 -233.34346 -233.34313 -235.81539 -237.02021 -237.01801 -271.47487 -276.32283 -309.47438 -315.62546 -385.75680 -385.70195 -268.29377

-196.55724 -197.77198 -197.77325 -233.41187 -233.41223 -235.87841 -237.08502 -237.08262 -271.55409 -276.39806 -309.56776 -315.71111 -385.86847 -385.81399 -268.38764

-195.16358 -196.33310 -196.33382 -231.83190 -231.83358 -234.20801 -235.36779 -235.36430 -269.74017 -274.40249 -307.52422 -313.43718 -383.35505 -383.28262 -266.83683

-194.08847 -195.25156 -195.25674 -230.54323 -230.54358 -232.91691 -234.07067 -234.06936 -268.24021 -272.88977 -305.80706 -311.70888 -381.21581 -381.13780 -265.34743

-195.84010 -197.01553 -197.02326 -232.62282 -232.62087 -235.02161 -236.18612 -236.18660 -270.66287 -275.35670 -308.56609 -314.52729 -384.66461 -384.45974 -267.59323

crotonaldehyde acetic anhydride

-233.59722 -287.49496 -307.35194 -231.10928 -232.36337 -270.47191 -381.30584 -264.21714 -264.20900 -264.20975 -229.88931 -231.14461 -381.58694

-233.60388 -287.50748 -307.36112 -231.11375 -232.37635 -270.48618 -381.30682 -264.22580 -264.19766 -264.21907 -229.88518 -231.15181 -381.58938

-233.67969 -287.59724 -307.46161 -231.19359 -232.44996 -270.57010 -381.43360 -264.30741 -264.28339 -264.30056 -229.96494 -231.22875 -381.72352

-232.15347 -285.73082 -305.55806 -229.76072 -230.97645 -268.86641 -379.23557 -262.69349 -262.69024 -262.68301 -228.58911 -229.80397 -379.58350

-230.87376 -284.14352 -303.86010 -228.48803 -229.69916 -267.36810 -377.10068 -261.20619 -261.22345 -261.19750 -227.31173 -228.51499 -377.45878

-232.88475 -286.68343 -306.52345 -230.46190 -231.69229 -269.71011 -380.36128 -263.53629 -263.53383 -263.53042 -229.28772 -230.50614 -380.64634

2-methylpropanenitrile methyl ethyl ketone 2-methylpropanal 1,4-dioxane tetrahydropyrrole nitro-sec-butane diethyl ether dimethylacetal tert-butylamine n-methylpyrrole

-211.30738 -232.38239 -232.37222 -307.54636 -212.50635 -362.82890 -233.57537 -308.75986 -213.73552 -249.38316

-211.30972 -232.39412 -232.38116 -307.55383 -212.52152 -362.83904 -233.58947 -308.76727 -213.74557 -249.39621

-211.37448 -232.46862 -232.45612 -307.65637 -212.58470 -362.96146 -233.66413 -308.87086 -213.81004 -249.47347

-209.99800 -230.99805 -230.98580 -305.82531 -211.14533 -360.77464 -232.14489 -306.98426 -212.32183 -247.83878

-208.83265 -229.70935 -229.69708 -304.12996 -209.97508 -358.72770 -230.86173 -305.28599 -211.15083 -246.46132

-210.69465 -231.71266 -231.70196 -306.71829 -211.85005 -361.86967 -232.86743 -307.88511 -213.04083 -248.67238

tetrahydropyran diethyl ketone isopropyl acetate piperidine diisopropyl ether phenyl radical tert-butyl methyl ether benzene nitromethane

-271.67077 -271.68242 -346.89843 -251.81419 -312.18341 -231.46083 -272.87831 -232.15434 -244.91970

-271.68595 -271.69769 -346.90884 -251.83231 -312.20063 -231.47942 -272.89010 -232.16805 -244.92208

-271.76875 -271.78222 -347.02272 -251.90408 -312.29483 -231.54862 -272.97415 -232.23735 -245.01440

-270.01791 -270.03365 -344.91508 -250.18871 -310.21885 -230.06507 -271.17563 -230.70314 -243.66199

-268.52560 -268.53118 -342.99695 -248.80326 -308.50855 -228.78331 -269.68338 -229.41945 -242.25586

-270.87133 -270.88427 -345.94628 -251.03088 -311.21918 -230.77775 -272.04192 -231.48719 -244.34534

isobutane methylamine acetamide ethylamine acetone aziridine methoxyethane ethanol dimethyl ether ethoxy radical

Validation Set -158.39842 -158.41577 -95.82440 -95.82869 -209.14096 -209.14471 -135.12630 -135.13510 -193.08233 -193.09031 -133.87226 -133.88157 -194.27102 -194.28148 -154.98554 -154.98883 -154.96662 -154.97336 -154.30395 -154.31847

-158.45971 -95.86254 -209.21952 -135.17909 -193.15482 -133.92515 -194.34578 -155.04506 -155.02727 -154.37124

-157.29898 -95.20983 -207.97601 -134.24761 -191.96224 -133.03856 -193.10487 -154.07574 -154.06475 -153.46094

-156.43447 -94.68166 -206.81580 -133.50415 -190.88722 -132.27801 -192.03755 -153.22268 -153.21321 -152.61640

-157.84777 -95.51444 -208.58499 -134.68825 -192.54087 -133.47309 -193.69148 -154.52896 -154.51555 -153.86706

cyclopentane n-pentane neopentane 1,3-cyclohexadiene 1,4-cyclohexadiene cyclohexane n-hexane 3-methylpentane toluene n-heptane cyclooctatetraene n-octane naphthalene azulene acetic acid methyl ester tert-butyl alcohol aniline phenol divinyl ether tetrahydrofuran cyclopentanone benzoquinone pyrimidine butanedinitrile 1,4-dipyridine

theory. Certainly the set of molecules with such small tolerance is very limited. Comparing theoretical and experimental energies requires of a mixture of corrections to perform a fair comparison, as both are relative energies with respect to different references at different conditions. Theoretical energies are obtained at 0 K in vacuum having as reference the individual atoms composing the molecule; however, the experimental energies are taken at ambient conditions of pressure and temperature with respect to

reference states that can be simple gas molecules, liquids, or solids depending on each particular case. The atomization energy of a molecule is defined as

Do ) (

∑

Ei) - Emolecule - EZPE

(1)

i)atoms

where Ei and Emolecule are the individual energies of all atoms in the molecule and the total energy of the molecule, respec-

Precise Atomization Energies of CHON Molecules

J. Phys. Chem. A, Vol. 111, No. 43, 2007 11163

tively, and EZPE is the zero-point vibrational energy correction of the molecule. Thus Do is the energy stabilization due to the formation of the molecule from the participant atoms. Table A1 in the Supporting Information reports the atomization energy of the G3/99 set for different levels of theory using the above formula. On the other hand, the enthalpy of formation of a molecule is defined as

TABLE 2: Corrector Factors for the Training Set (115 Molecules)a (a) only Bi

B0

B1

B2

B3

G3x B3PW91/6-31G** B3PW91/cc-pvtz HF/6-31G* HF/3-21G MP2(full)/6-31G*

-0.5 2.4 1.8 20.0 29.4 11.6

-0.5 3.1 1.7 -4.1 2.8 4.7

-0.6 3.7 4.3 -32.1 -37.8 9.9

-0.8 0.4 2.2 -51.8 -53.6 13.2

∆fH0°molecule ) ∆fH298° - (H298° - H0°)molecule +

∑ (H298° - H0°)

(2)

atoms

where (H298° - H0°)atoms experimental values for H, C, O, and N are 1.01, 0.25, 1.04, and 1.04 kcal/mol, respectively.

(H298° - H0°)molecule ) H°corr - ZPE

(3)

where H°corr - ZPE(M) is calculated for each method. And then, the experimental atomization energy, as suggested by Curtiss,17 is calculated using the following formulas:

DoExperimental ) (

∑

∆fH0°) - ∆fH0°molecule

(4)

atoms

where ∆fH0°atoms experimental for H, C, O, and N are 51.63, 169.98, 58.99, and 112.53 kcal/mol, respectively.

(b) Bi and 

B0

B1

B2

B3



G3x B3PW91/6-31G** B3PW91/cc-pvtz HF/6-31G* HF/3-21G MP2(full)/6-31G*

-0.5 2.3 1.8 17.7 22.9 12.9

-0.7 2.6 2.6 -34.9 -43.6 -2.4

-0.9 3.0 5.7 -75.9 -93.3 -2.5

-1.2 -0.6 4.0 -104.8 -126.0 -2.9

0.998 0.995 1.009 0.613 0.440 0.909

(c) ∆Ei

∆E4

∆E3

∆E2

∆E1

G3x B3PW91/6-31G** B3PW91/cc-pvtz HF/6-31G* HF/3-21G MP2(full)/6-31G*

-0.1 0.5 -1.2 61.8 78.2 8.0

0.0 -1.3 -1.4 47.7 57.1 2.0

0.0 -0.7 0.7 11.7 10.4 9.4

0.1 -4.5 -1.9 -2.4 -10.6 -0.7

(d) ∆Ei, 

∆E4

∆E3

∆E2

∆E1



G3x B3PW91/6-31G** B3PW91/cc-pvtz HF/6-31G* HF/3-21G MP2(full)/6-31G*

-1.0 5.8 3.8 70.3 88.5 13.4

-1.3 5.3 5.5 59.8 80.1 8.7

-0.6 2.8 4.1 19.5 24.9 12.7

-0.3 -1.6 1.3 5.2 6.4 1.4

1.009 0.955 0.955 0.880 0.767 0.952

III. Effect of the Correctors The calculation of the experimental atomization energy following this procedure is reported in Table A1 (Supporting Information) for all molecules; then the average error for a level of theory is obtained from the mean absolute error average of the atomization energy with respect to experimental values. All calculations are performed using the program Gaussian-03,20 Table 1 reports the total electronic energies of the G3/99 set for different levels of theory. Table A2 (Supporting Information) reports the zero point energy corrections calculated using their corresponding level of theory for the calculation of their optimized geometries and energies. The results for the training and validation set, with and without the correctors described in the next section, are reported in Tables 3 and 4, respectively. Several correctors are used in this work, and as in our previous work, we introduce a multiplicative factor that compensate for the deficiencies to predict the total electronic energy. Then, the corrected total energy (Ecorrected) for each atom or molecule is given by

Ecorrected ) EE + EZPE corrected total energy using the multiplicative factor (5) This multiplicative factor allows us to make proportional changes to the energy of each molecule. However, to improve the atomization energies, we introduce corrective factors such that they minimize the average errors for a specific level of theory (method/basis-set). Complex bond additive correction procedures were developed since the 1980s to reduce the demand of computational resources or improve accuracy,21 instead we use even simpler additive factors ∆Bi, called boundary factors, that compensate for the deficiencies to predict the bond energies. We consider a different factor for each type

(e) 



G3x B3PW91/6-31G** B3PW91/cc-pvtz HF/6-31G* HF/3-21G MP2(full)/6-31G*

1.000 0.994 1.000 1.357 1.421 1.073

(f) atom factors

Ah

Ac

An

Ao

G3x B3PW91/6-31G** B3PW91/cc-pvtz HF/6-31G* HF/3-21G MP2(full)/6-31G*

-0.1 0.0 1.2 12.6 12.0 9.7

0.2 -2.0 -2.2 46.1 54.7 1.5

0.1 1.1 -1.1 62.0 82.3 7.7

-0.5 1.2 0.6 51.8 72.0 2.6

(g) atom factors and  G3x B3PW91/6-31G** B3PW91/cc-pvtz HF/6-31G* HF/3-21G MP2(full)/6-31G* a

Ah

Ac

An

Ao



-0.1 1.4 0.5 11.3 12.2 15.0

0.2 1.8 -3.9 43.2 55.3 13.8

0.1 3.3 -2.1 60.0 82.7 15.5

-0.5 3.4 -0.6 50.0 72.4 10.1

1.000 0.997 1.011 1.020 0.996 0.927

All quantities are in kcal/mol except for , which is unitless.

of bond; thus, the corrected total energy (Ecorrected) for each atom or molecule is given by

Ecorrected ) E + EZPE + ∆Ebound corrected total energy using the boundary factors (6) 3 where ∆Ebound ) ∑i)0 Bi*Ni; B0, B1, B2, and B3 are factors for no bond, single, double, and triple bonds; and Ni is the number of bonds of type i. This means that for the atoms C, H, O, and N, B0 equals 1, and for the molecules B0 equals zero. Because our test of molecules is dominated by single bonds, our factors B0 and B1 yield changes proportional to the number of atoms or the energy

11164 J. Phys. Chem. A, Vol. 111, No. 43, 2007

Salazar and Seminario

TABLE 3: Average Errors Using the Training Set for Several Levels of Theory level of theory

no correctors

Bi

Bi & 

∆Ei

∆Ei & 



atomic factors

atom factors &

G3x B3PW91/6-31G** B3PW91/cc-pVTZ HF/6-31G* HF/3-21G MP2(full)/6-31G*

0.7 6.4 3.1 271.2 313.5 66.3

0.7 2.1 1.9 20.1 30.2 5.6

0.7 2.1 1.9 14.8 17.3 4.9

0.7 2.6 2.6 6.8 12.4 5.6

0.7 2.1 2.0 4.7 5.7 5.2

0.7 3.0 3.1 37.3 57.9 13.0

0.6 2.4 2.1 6.8 7.1 4.4

0.6 2.4 2.0 6.7 7.1 3.8

TABLE 4: Average Errors Using the Validation Set for Several Levels of Theory level of theory

no correctors

Bi

Bi & 

∆Ei

∆Ei & 



atom factors

atom factors &

G3x B3PW91/6-31G** B3PW91/cc-pvtz HF/6-31G* HF/3-21G MP2(full)/6-31G*

0.6 3.9 1.9 241.0 276.1 72.5

0.6 1.8 1.4 13.7 18.2 4.7

0.6 1.8 1.3 8.1 9.6 3.3

0.6 1.2 1.0 4.5 6.6 4.9

0.6 1.4 1.3 3.1 4.7 3.7

0.6 2.2 1.8 23.4 37.2 14.6

0.6 1.3 1.3 5.2 6.1 3.7

0.6 1.5 1.3 5.2 6.1 2.8

of each molecule. Also, as in our previous work, we introduce additive factors ∆Emultiplicity, such as ∆E1, ∆E2, ∆E3, and ∆E4, for singlet, doublet, triplet, and quartet systems, respectively, to compensate for the errors in the multiplicity or total spin of the atoms and molecules. Considering these correctors, the corrected total energy (Ecorrected) for each atom or molecule is given by

Ecorrected ) E + EZPE + ∆Emultiplicity corrected total energy using the multiplicity factors (7) Due to the characteristics of our test set (most of the molecules are spin singlets), the factor ∆E1 represents an additive factor to all the molecules, independently of the magnitude of its energy. Also, notice ∆E4 represents an additive factor to the energy of nitrogen, ∆E2 is an additive factor to the energy of hydrogen, and ∆E3 is an average corrector among carbon and oxygen. These last three multiplicity factors yield also a multiplicative nature because their corrections increase as the number of atoms increases. We now introduce additive factors to the energy of the four elements of our set of molecules, Ac, Ah, Ao, An, further simplifying the corrections. Considering this, just the energy of each atom changes to

Eicorrected ) Ei + Ai corrected total atomic energy using the atomic factors (8) and the energy of the molecules remains unchanged. These atomic factors are expected to be equally as effective as the multiplicity factors, because these also share an additive and multiplicative nature and are even more specific to the correction of the energies of carbon and oxygen. Analyzing only the multiplicative factor, , from Table 2e, we can have an idea of the quality of each level of theory (method/basis). Values close to 1 would correspond to the more exact methods, and distant values from 1 would correspond to poor methods. We can rank our additive factors by comparing the multiplicative factor . Thus, the most effective factors are the atomic factors, followed by the multiplicity factors, and the boundary factors. The multiplicative factor also allows us to test the stability of our corrective factors; the smaller the change of each type of additive correctors when combined with the multiplicative corrector (), the better the stability. These changes are shown in Table 2 sections b, d, and g. This stability

is also proved in the validation set (Table 4), showing that the atomic factors are the most robust. There is a similarity between the multiplicity and atomic factors, due to the characteristics of our set of molecules. Most of the molecules in the training set have multiplicity 1; therefore, the factors ∆E2, ∆E3, and ∆E4 yield corrections to the energies of hydrogen, carbon, and hydrogen, respectively. The relationship between these factors is the same in both types of corrections. Moreover, although the difference among the atomic factors of carbon and oxygen is larger, the accuracy using atomic factors is better than using multiplicity factors. The source of the error for the MP2 (full)/6-31G* level of theory is mainly due to the prediction of the hydrogen and oxygen energies (Table 2 section f). Comparing the 6-31G** and cc-pVTZ basis sets, we notice that the hydrogen energy needs stronger corrections when an extended basis is used. The 6-31G** and cc-pvtz basis sets underestimate and overestimate, respectively, the nitrogen energy, and the error in the calculation for the carbon energy is the same with both. We can conclude that the accuracy of cc-pVTZ is due mostly to the calculation for the oxygen energy. Moreover, comparing the HF/6-31G* and HF/3-21G levels of theory, we also notice that the hydrogen energy needs stronger corrections when an extended basis set is used. The atomic corrections increase quasi-linearly with the energy, in agreement with the results in Table 2 section f. We can conclude observing the results in Table 3 that the best correction is the one when the atom energies are corrected by an additive energy, not only for simplicity and accuracy but also for stability. These implies only four parameters need to be include on the atomic energies; the molecular energies are untouched by these corrections. Tables 3 and 4 show that the average error without corrections is smaller in the validation set than in the training set, because our training set contains more large molecules than small ones. Then, because our methodology is size consistent, it is expected that the corrected absolute errors in all the methods be larger for a set of larger molecules, as shown in Tables 3 and 4. However, the relative corrections (per bond, atom, or percentage) are almost the same in both sets, which means, for instance, that the error per bond is maintained as the size increases. An important detail of the atomic factors is that their corrections are better for bigger molecules. For instance, the average error for the first 25 low-energy molecules (the small molecules of the training set) yields average errors of 6.5 kcal/ mol, and the average error for the 25 first high-energy molecules (the large molecules of the training set) is 2.5 kcal/mol. This is

Precise Atomization Energies of CHON Molecules because our methodology takes advantage of the fact that, as the energies of the atoms are already extremely well-corrected, the larger systems benefit because the larger basis set helps each individual atom to reshape their structure much better than when a smaller number of atoms (thus smaller basis set) is used. To validate the use of the reported energy correctors, we use them in 10 representative molecules from the G3 set shown in Table 1 that are not used in the training set; the results are shown in Table 4, which is practically self-explanatory, as it shows similar and better results than those obtained with the training set. The selection of just 10 molecules (compared with the 115 molecules for the training set) proves the specificity of these correctors. Certainly the G3x cannot be improved further because all corrections imposed on the methods are in some way considered in the correctors. Having the B3PW91 with errors of only 1.5 and 1.3 with the small 6-31G* and the cc-pVTZ basis sets, respectively, is really encouraging for the application of these methods to large systems. IV. Conclusions We report correctors for the total energies of a few levels of theory commonly used in quantum chemistry calculations. It is found that these correctors can make strong effects on the accuracy of practically all methods and that the strategic evaluation of additive and multiplicative correctors provides a good description of the quality of the correctors. The most practical and easiest of the corrections is simply to correct the atom energies by an additive corrector. For instance, for the popular HF/6-31G* method the error in the atomization energies is on average 271.2 kcal/mol, which is reduced to only 6.7 kcal/ mol when the energy of the CHON atoms are corrected by additive energies. This correction certainly makes the HF/631G* level of theory very useful. On the other hand, the B3PW91/cc-pVTZ yield corrected average errors of 2.0 kcal/ mol, which are already within chemical accuracy not even attainable by any of the uncorrected standard ab initio methods. These results justify the use of ab initio energy differences where the energy of the atoms is cancelled and we hope this work triggers its extension to molecules with other atoms in the periodic table. Acknowledgment. We acknowledge financial support from the Department of Energy/Basic Energy Sciences (DE-FG0205ER15729), as well as from the U.S. Defense Threat Reduction Agency (DTRA) through the U.S. Army Research Office (ARO), projects W911NF-06-1-0231 and W911NF-07-1-0199. Note Added after ASAP Publication. This Article was posted ASAP on October 10, 2007. The headings for the second columns in Tables 3 and 4 were changed from “no. of

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