correlation energy contributions to reaction heats

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY VOL. XII, 61-81 (1977)

Correlation Energy Contributions to Reaction Heats

PHILIP GEORGE* Department of Biology, University of Pennsylvania, Philadelphia, Pennsylvania 191 74, U.S.A.

MENDEL TRACHTMAN Department of Chemistry, Weizmann Institute of Science, Rehovot, Israel

AND

ALISTAIR M. BRETT AND CHARLES W. BOCK Department of Chemistry, Philadelphia College of Textiles and Science, Philadelphia, Pennsylvania

19144, U.S.A.

Abstracts

New, more accurate, Hartree-Fock limit energies (EHF) for ethane and ethylene are obtained from SCF total molecular energies using Ermler and Kern’s procedure. These results, together with EHF values for other small closed shell molecules, are employed to calculate correlation energy (E,) contributions to reaction heats. Cancellation to within 98% of the total E, involved, and often to more than 99%, is found for a wide variety of chemical reactions, which strongly suggests that there are systematic regularities in the contribution to E, from the different kinds of electron pairs in the valence shell. Assuming strictly localized pairs occupying orbitals having strongly directional character, E, for the valence shell is evaluated in terms of E, per lone pair, E, per X-H bond, and E, per X/X shared pair for Ne and for molecules containing first row atoms, where X is C, N, 0, and F.

Des energies de Hartree-Fock (EHF), nouvelles et plus prkcises, ont kt6 obtenues pour I’kthane et l’kthylkne par le prockdk de Ermler et Kern. Ces rksultats avec des valeurs de EHF pour d’autres moldcules a couches ferm6es ont Ct6 utilisks pour calculer les contributions de I’knergie de corrklation (E,) a des chaleurs de rkaction. On trouve pour un grand nombre de rkactions chimiques que les contributions a E, se compensent, souvent jusqu’k 99% de Ec, ce qui suggere qu’il y a des rkgularitks systkmatiques dam les contributions a E, des paires d’electrons diffkrentes dans la couche de valence. Admettant des paires strictement localiskes d’orbitales occupies avec un caractere directionnel prononck, on a calculk E, pour la couche de valence en termes de E, par paire non liante, par liaison X-H et par paire X/ Y partagke pour Ne et pour des moldcules contenant des atomes de la premibre ligne, oh X est C, N, 0 et F.

Neue genauere Hartree-Fock-Energien (EHF) sind mittels des Ermler-Kern’schen Verfahrens fur Athan und Athylen erhalten worden. Diese Ergebnisse, zusammen mit EHF-Werten fur andere kleine Molekule mit abgeschlossenen Schalen, werden zur Berechnung der Korrelationsenergiebeitrage zu Reaktionswarmen angewandt. Fur eine grosse Anzahl von chemischen Reaktionen zeigt es sich, dass die E,-Beitrage sich weitgehend, oft bis 99% von E,, kompensieren. Diese Tatsache deutet an, dass systematische Regelmassigkeiten in den Beitragen zu E, von den verschiedenen Arten von Elek- tronenpaaren in der Valenzschale vorkommen. Ausgehend von streng lokalisierten Paaren von besetzten Orbitalen mit ausgepragtem Richtungscharakter, haben wir E, fur die Valenzschale als E, pro nichtbindendes Paar, pro X-H-Bindung und pro X/X-geteiltes Paar fur Ne snd fur Molekule mit Atomen von der ersten Zeile berechnet, wo X C, N, 0 und F ist.

61 @ 1977 by John Wiley & Sons, Inc.

62 GEORGE ET AL.

1. Introduction

It has been recognized for some time that Hartree-Fock (HF) total molecular energies of reactants and products do not necessarily result in a calculated reaction heat identical to the experimental value [ l , 21. This would only be the case if the correlation energy” of reactants and products were the same, and thus cancelled out. Very substantial cancellation to within a few kcal/mol has been anticipated for closed shell reactions of simple molecules [5-71. As Hurley put it, “the correlation energy should remain almost invariant in any change that preserves the number of electron pairs and, as far as possible, their local spatial relationship to each other” [6]. But reactions differ in the extent to which structural elements are matched in reactants and products. The closer the matching, the greater would be the similarity in the local spatial relationship of electron pairs. Hence one would expect the smallest correlation energy contributions to the reaction heat for homodesmotic reactions [8-111, which are the most matched, somewhat greater contributions for isodesmic reactions [ 12-15], which are not matched so well, and finally the largest contributions for anisodesmic reactions [16], which are the least matched.

At the time of Hurley’s study [6], too few reliable HF energies? were available for a sufficiently diverse group of molecules to make possible a systematic study of the role of correlation energy in various types of chemical reaction, especially those of the simple hydrocarbons.

Ermler and Kern, however, have recently proposed a simple procedure for estimating HF limit energies that gives values accurate enough for this purpose [17]. Provided a basis set of multiple zeta plus polarization quality is employed, the SCF total energies obtained are in a constant ratio to the HF limit energies, i.e., EHF/ESCF = f . The numerical proportionality factor f which is clearly dependent on the specific basis used for the SCF calculations, may also vary from one series of molecules to another depending on the period of the constituent atoms. Further- more, the degree of constancy off is dependent on the quality of the basis set employed.

We use this procedure to obtain more reliable E H F values for several key molecules, and examine the convergence of SCF reaction heats to the limiting HF values for various types of reaction. The classification employed stems from that suggested by Snyder and Basch [2], namely, whether there is a separation of the heavy (first-row) atoms, no separation, or an increase in the extent of the heavy atom bonding.

As anticipated in the literature, for the majority of the reactions in which both reactants and products are closed shell species correlation energy differences are found to contribute little to the reaction heat, usually less than 10 kcal/mol. Moreover the cumulative uncertainty arising from the limiting HF total molecular energies is similar in magnitude in many cases.

* Following Lowdin [3] and Clementi [4] the correlation energy is defined as the difference

t Better than 0.003 a x . between the exact nonrelativistic energy and the HF total energy.

CORRELATION ENERGY CONTRIBUTIONS 63

Cancellation to this extent is very suggestive of some simple regularity in the contribution of the various electron pairs in the valence shells to the correlation energy of the molecule as a whole. In the discussion we explore this possibility, first by re-examining the regularity in the contribution of electrons in the outer shell in atomic systems [18], and then by partitioning the correlation energy (Ec) of molecular systems according to the equation

E,(V.S.) =Ec(L.P.)+Ec(X/H)+Ec(X/X) (1) using the new EHF values. From left to right, the terms in Eq. (1) represent that part of the correlation energy due to valence shell electrons, and the individual contributions attributable to lone pairs of electrons, to electron pairs in X-H bonds, and to electron pairs shared between first-row atoms.

2. Basic Data

Table I lists the references from which we have obtained SCF total energies, together with the basis sets used and the geometry employed. Not every basis set could be used for all the reactions studied because in some cases the necessary

TABLE I. Basis sets

Basis Set Geome t r y References

STO-3G

STO a

ETG

LEMAO-6G

4-31G

DZ

6- 3 1G**

[4 ,2 /21p

PMZ

Standard

Experiment a 1

Experimental

Experimental

Completely Optimized [30,31)

Experimental ~ 3 2 1 Standard

Experimental,

Experimental

a Slater exponents were used, the only exception being the choice of 1.2 for the exponent of hydrogen.

total molecular energy of a reactant or product molecule has not yet been evaluated. However, for as many of the reactions as possible, basis sets of essentially single zeta, double zeta, and double zeta plus polarization (PMZ) quality have been used to explore the degree of convergence of calculated reaction heats to the HF limiting values.

Table I1 lists the m-limit energies including estimated uncertainties used in this study, and the references from which they were taken. Employing the procedure of Ermler and Kern [17], together with the SCF data of Clementi and

64 GEORGE ET AL.

TABLE 11. Hartree-Fock total energies.

Molecule

Hartree-Fock L i m i t Energy

(a.u.) Referencesa

H2 -1.1336 +0.0001 [ 347

HF

H2°

NH3 CHL

co NO

H2°2 HCN

H 2 C 0

co2 N20

-100.071 t0.0003 r 67

-56.225 20.002 r 361 -76.068 to.001 c 351

--40,219 ?O. 001 c171b

-198.780 t 0 . 0 0 3 r6 1

-108.998 k0.002 371 -149.670 i0.003 c 67 -149.621 k0 .003 ! 61 -112.791 kO.001 [ 381 -129.286 f0.002 [ 61

-150.861 t0.005 [171c -92.916 to.001 1381

-113.932 t0.009 [ 61

161 -187.729 k0 .002

-183.758 to. 002 [: 381

-76.858 t0 .002

-78.071 kO.002

-79.269 t0.002

-230.82 t0 .02

61 d

T h i s paper T h i s papere

[33,391

a Additional references to original papers can be found in Hurley [6]. Other estimates include -40.225+0.005 (401 and -40.220*0.005 [6]. See footnotes b of Table I of Ref. [17]. Other estimates include -150.825 *0.005 [41] and

Other estimates include -78.080*0.008 161. Other estimates include -79.265 +0.005 [41] and -79.270rt0.008 [6].

-150.850*0.009 [6].

Popkie [42, 431, new, and evidently more reliable, estimates for the HF-limit energies of ethylene and ethane have been calculated as follows.

Using the high quality basis set of van Duijneveldt, Clementi and Popkie [42, 43 J found the SCF total energies of CH4, CH=CH, CHz=CH*, and CH3-CH3 to be -40.2136, -76.8483, -78.0616, and -79.2587 a.u., respectively. Selecting CH=CH as the reference molecule, because its HF-limit energy is known to be


-76.858 with an uncertainty of only 0.002 a.u. [6], a proportionality factor of 1.000126 f 0.000026 is obtained. This value is significantly smaller than the 1.00027*0.00003 found by Ermler and Kern [17] using Dunning’s PMZ basis set [44,45]. So as to give a conservative estimate of the uncertainty, we adopt Ermler and Kern’s uncertainty of 0.00003 for the factor, and thus obtain HF-limit energies of -40.219*0.001, -78.071 f0.002, and -79.269 f 0.002 a.u. for CH4, CH2=CH2, and CH3-CH3, respectively.

The value for CH4 is in excellent agreement with that obtained by Ermler and Kern using the different basis set (the values differ by only 0.0003 a.u.). This not only serves to confirm their value, which is a little less negative than some other estimates, e.g., Rothenberg and Schaefer’s value of -40.225 f 0.0005 [40] but, since a different molecule was used as reference (CH=CH instead of H 2 0 [17]), it points to an internal consistency among the estimates of the m-limit energy for H20, CH=CH, and CH4.

3. Results

Table I11 lists A€P values for reactions in which there is separation of the heavy (first-row) atoms as a consequence of reduction by molecular hydrogen, while Table IV lists reactions in which there is no net separation. These latter reactions fall into two categories: first, those in which there is preservation of the heavy atom bonding, in the sense that the number of heavy atoms joined together remains the same; and secondly those in which there is an increase in the heavy atom bonding in the sense that one or more additional bonds between heavy atoms are formed. Table V gives the values for corresponding reactions of benzene. As is to be expected, the SCF values converge to the HF value as the more extended basis sets are employed. The most substantial improvement is generally found when a basis set of double zeta quality is used in place of a basis set of single zeta quality.*

But the ordering above the s ~ o - 3 G or STO level is not always the same. This comes about because within a particular basis set, for instance DZ or even 6-3 lG , different values of f in the Ermler and Kern equation wduld be obtained depending on the molecules selected.?

Furthermore, the basis set giving the more negative total molecular energies does not always give the reaction heat closest to the HF value. For example, comparing the results using LEMAO-~G and 4-31G for the benzene reactions, the

* It is to be noted, however, that the dramatic improvement found here with the double zeta basis sets relates to reactions that are anisodesmic [16], with no matchingof the X / X bondingor the number of hydrogen atoms attached to each kind of heavy atom. This is not always the case, for example, with homodesmotic [S-1 l]or isodesmic reactions [ 12-15] in which structural elements are matched at least according to the type of X / X bonding.

t The 6-31G** basis set is of double zeta plus polarization quality €or the valence shell (split- valence basis), but the numerical proportionality factor in the Ermler and Kern equalion, i.e., about 1.00049, is considerably larger than that given by the Dunning or van Duijneveldt basis sets. Furthermore it varies somewhat from one molecule to another, which suggests that care must be exercised in choosing the basis set to use in the Ermler and Kern procedure.

66 GEORGE ET AL.

TABLE 111. LL€T for reactions with separation of heavy atoms

Experimental Theorec ic 1 Dif fe rence Values a Values t AH^;^^^-

Reac t ion (AH:)zpE Basis h q h ‘%h

I . F i s s i o n o f Molecules Containing Two Heavy Atoms A . S i n g l e Bond (i) F-F + H2 2HF -133.9 STO -5.8

STO-3G -37.8 DZ -131.7 6-31G** -136.9 HF -143.3 k2 .3

(ii)HO-OH + H2 2H20 -86.8

( i i i ) CH3-CH3 + H2 2CH4 -18.1

B. Double Bond ( i ) CH2=C% + 2H2 -3 2CHq -57.2

( i i ) H2C=0 i- 2H2 CH4 + H20 -59.0

( i i i ) O=O + 2H2 3 2H20 (3C) -125.1

(‘A) -147.7

C . T r i p l e Bond ( i ) HCECH + 3H2 2CH4 -105.6

( i i ) HCSN + 3H2 CH4 + NH3 -75.6

( i i i ) WN + 3 H 2 2NH3 -37.5

6-31G** -93.1 DZ -90.0 HF -88.7 f 4 . 5 PMZ -88.6 STO-3G -38.6

DZ -24.9 ETG -23.4 HF -22.2 f 2 . 6 6-31G** -21.7 STO -19.6

STO -86.5 DZ -66.5 6-31G** -64.7 ETG -64.2 HF -62.6 +2.6

D Z -70.1 ETG -69.0 STO-3G -63.6 6-31G** -59.2 HF -55.1 i 7 . 0

STO-3G -33.1 6-31G**-105.5 HF -124.7 f 3 . 3

HF -155.5 t 3 . 3

STO -141.5 DZ -120.9 6-31G** -117.9 ETG -116.5 HF -112.4 f2 .7

STO -90.9 DZ -90.9 6-31G** -80.0 HF -79 .8 k2.7

DZ -58.7 PMZ -38 .1 6-31G** -33.7 S TO -33.0 HF -32.1 f 4 . 0

-128.1 -96.1

-2 .1 +3.1 +9.4 t 2 . 3

~ 6 . 3 +3.2 C1.9 t 4 . 5 +1.8

-48.2

+1.5 +3.6 c4.1 t 2 . 6 +5.3 e6.8

+29.3 +9.3 +7.5 +7.0 +5.4 t 2 . 6

+11.1 +10.0

+4.6 +0.2 -3 .9 f 7 . 0

-92.0 -19.6

-0.4 f 3 . 3

f 7 . 8 2 3 . 3

c35 .9 +15.3 +12.3 e11 .0 +6.9 t 2 . 7

e15 .3 c15 .3

44.4 +4.2 +2.7

-121.2 +0.6 -3.7 -4.4 -5.4 f 4 . 0


TABLE III-continued ( i v ) CO t 3H2 CH4 + H20 -65.4 ETG -100.6

DZ -81.5 PMZ -59.1 HF -59.1 +2.1 6-31G** -58.4

HF -108.5 +3.3 5 (v) NO + H2 NH3 + H20 -102.9

11. F i s s i o n of Molecules Conta in ing Three Heavy Atoms A . P a r t i a l S e p a r a t i o n of Heavy Atoms ( i ) O=C=O + H2 CO + H20 +8.8 DZ -9.8

( i i ) O = C = O 2H2 H2C=0 + H20 +2.4 ETG -35.4

ETG -3 .8 HF t2 .3 f 2 . 6

DZ -21.1 STO-3G -9.4 HF -2.4 ?rl . i ’

( i i i ) N 2 0 + H2 -3 G N + H 0 -80.1 HF -109.4 f3.;! 2

B. Complete Separa t ion of Heavy Atoms ( i ) O=C=O -6 4H2 + CH4 L 2H20 -56.6 ETG -104.3

D Z -91 .3 STO-3G -73.0

+35.1 c 1 6 . 1

-5 .1 - 5 . 1 f 2 . 1 -1.0

1 5 . 6 +3.3

+18.6 +12.6

~ 6 . 5 f 2 . 6

+37.8 +23.5 +11.8

+4.7 f 7 . 7

+-28.7 23 .2

147.7 +34. 6 -.16.4

HF -51.5 +3.4 ~ 0 . 8 +3.4

( i i ) N20 J- 4H2 -3 2 N H 3 C H20 -118.2 t1F -141.5 .?4.5 c 2 3 . 4 +4.5

a Sources of experimental data: heats of formation and heat capacities, Refs. [46, 471: zero-point vibrational energies, Refs. [48-501: see also Ref. [16] for newly calculated values for some of the organic compounds. (AH&PE is the reaction heat at 0 K corrected for the zero-point vibrational energies of reactant(s) and product(s).

Calculated from the total molecular energies of reactant and product species-see Table I for the references to the various basis sets, and Table I1 for the Hartree-Fock limiting energies.

LEMAO-~G energies are consistently more negative, whereas the 4-3 1G energies give the closer agreement, indicating the importance of flexibility in the valence shell.

Turning now to the differences between the experimental and HF reaction heats, which are identified as

AE, = E, (products)- E, (reactants) several tentative generalizations can be made. The “errors” resulting from the remaining uncertainties in the HF energies have to be born in mind throughout.

( I ) For the reductive fission of both singly and doubly bonded molecules containing two heavy atoms, AE, is small and positive, indicating that the correlation energy of the reactant species is greater numerically, i.e., a larger negative number, than that of the products, see Table 111, IA and B. The small value for H202 may be due to an underestimation of E H F , see Table 11. The small negative value for O2 (3Z) may be attributed to its unpaired electron structure, since a positive value is obtained for the ‘A state. The quite large positive value for F2, which appears to be well outside the uncertainties in E H F for F2 and HF, bears out the previous findings of Snyder and Basch [2].

68 GEORGE ET AL.

TABLE 1V. rw“ for reactions with no net separation of heavy atoms.

Experimental T h e o r e t i c a l D i f f e rence Valuesa Valuesb (“:izpE-

‘%h React ion AH^)^^^ as is AH&

I . Prese rva t ion of Heavy-atom Bonding A . Sing le Bond Exchange (i)F-F -I- 2H20 3 2HF i HO-OH -47.1

( i i ) F - F 4. 2CH4 3 2HF -1 CH3-CH3 -115.8

(iii)HO-OH 4- 2CH4 + -68.7

2H 0 .4 CH -CH3 2 3

B. Double Bond Sing le Bo d 9 (i)O=O I- H2 HO-OH ( X) -38.3

(‘A) -60.9

(ii)H2C=CH2 A H2 CH3-CH3 -39.1

C . Double Bond Exchange (i)o=O A CH 4 -+ H2C-0 -c H20 (32) -66.1

1

3

( A ) -88.7

(ii)O=O 4 2CH 3 ( C) -67.3 4 H2C=C%+ 2H20

(‘A) -90.5

( i i i ) o = o 1 H C=CH -+ 2 ~ ~ ~ 0 (3x) -66.5 2 2

1 ( A ) -89.1

-1.8

D . T r i p l e Bond Double Bond (i)HCsCH -t H j H ~ C = C H ~ -48.4 2

STO - 3G DZ 6-3 1G** HF

STO-3G DZ 6-31G** HF

STO-3G DZ HF 6-31G**

STO-3G 6-31G** HF HF

S M 6-3 1G** DZ ETG HF



STO-3G 6-31G”* HF HF

ETG DZ 6 - 3 1C** HF STO-3G

STO DZ 6- 3 1G** ETG

+0.8 -41.7

-54.6 f 6 . 7

-18.2 -106.8 -115.2 -121.1 24.8

-43. a

-19.0 -65.1 -66.5 +6.9 -71.4

4 . 8 -12.4 -36.0 f 5 . 1 -66.7 25 .1

-66.8 -42.9 -41.6 -40.7 -40.4 f2.6

L29.8 -46.2 -69.6 2 8 . 8

-100.4 28.8

-L.59. 5 -40.8 -62 .1 f 6 . 3 -92.8 26 .3

to.1

-47.9 -5.4 -3.3 +7 .5 56.7

-97.5 -8.9 -0 .6 +5.3 t4 .8

-49.7 -3.6 -2.2 f 6 . 9 +2.7

-43.1 -26.0

- 2 . 3 f 5 . 1 +5.8 f 5 . 1

-;27.8 +3.8 + 2 . 5 +1.7 ~ 1 . 3 f 2 . 6

-95.9 -19.9 23.5 f 8 . 8

C11.7 f 8 . 8

-127.4 -27.1

-5.8 t 6 . 3 +2.3 f 6 . 3

-66.4 -51.6 -14.9 -77.2 f 1 4 . 4 - ~ 1 0 . 7 f 1 4 . 4

-107.9 214.4 1-18.8 514.4

-4.8 +3.0 -3.6 -1.8 -5 .4 -7.2 e7.5 f 8 . 2 -9.4 f 8 . 2

4 .9 . 7 -31.5

-55.1 t 6 . 7 -54.4 c6.0 - 5 3 . 2 c4.8 -52.4 4 . 0

HF -49.8 ?2.6 t 1 . 4 f 2 . 6

CORRELATION ENERGY CONTRIBUTIONS

TABLE IV-continued

(ii)CO + H2 + H C=O -6.4 2

E. Triple Bond Exchange (i)EN 4- 2CH4 --3 C68.1

HCiCH 1 2NH3

(ii)HCEN 1 CH4 j r30.0

3 HCECH + NH

(iii)EN + HCZCN + 2HCrN +8.1

F. Bond DisporportionationIRearrangement (i)HCHCH + CH3CH3 -3 -9.3

2H 2 C=CH2

(ii)O=C=O + CH + 2H2C=0 t61.4 6

11. Formation of New Heavy-atom Bonding A . One New Bond (i)H2C=CH2 -I 2Cll4 --$ 2CH3-CH3 -21.0

(ii)HC'CH + 2CH4 3

3 H2C=CHZ CH -CH3

(iii)CO + 2CH4 + t11.7

3 H2C=0 A CH3-CH

(iv)O=C=O + 4CH4 38.5 H C=O 1. 2CH -CH + H20 2 3 3

B. Two New Bonds (i)HCcCH + 4CH4 3CH3-CH3 -51.3

(ii)CO + 5CH4 3 -11.2 3CH -CH + H 0 3 3 2

ETG -31.6 DZ -11.4 HF -4.6 f6.3 6-31G** 10.9 S TO +13.8

STO r108.7 6- 3 1G** HF DZ

STO 6-31G** HF DZ

DZ S TO 6-31G HF

DZ, ETG 6-31G- HF S M

En: DZ HF STO-3G

DZ ETG HF 6-31G** STO S TO 6-3 1G** DZ ETG HF

STO 6 - 3 1G** HF DZ ETG

ETG DZ STO-3G HF

S M 6- 3 1G** ETG DZ HF ETG DZ 6-31G** HF

a-84.2 180.3 f6.3 ~ 6 2 . 2

r50.8 e31.9 +32.6 f3.8 +30.0

f2.3 +7.2 e8.4 C15.1 f3.8

-12.8 -11.6 -10.3 -9.4 f5.0

~11.8

133.6

e52.7 f13.2 +54.2

+49.0

-16.7 -17.3 -18.2 f5.0 -21.2

69

+25.2 44.9 -1.8 f6.3 -7.3 -20.2

-40.6 -16.1 -12.2 f6.3 45.9

-20.8 -1.9 -2.6 23.8 0.0

t5.9 +O. 9 -0.3 -6.9 f3.8

+3.5 +2.3 +1.0 eO.1 f5.0 -21.2

t27.8 t12.4 +8.7 f13.2 ~7.2

-4.3 -3.7 -2.8 f5.0 t0.2

-47.2 126.3 -35.4 +5.1 -31.5 +1.2 -29.5 -0.8 -29.0 -1.3 - 2 7 . 6 f5.0 -2.7 25.0

c33.4 -21.8 e22.6 -10.9 e17.6 -18.8 -5.9 +8.8 113.5 -1.9 -8.2 +19.9

e11.4 e27.1 ~28.7 +9.9 e29.7 +8.9 c42.0 f12.L -3.5 f12.1

-82.1 +31.4 -52.7 ~1.4 -46.3 -5.0 -46.2 -5.1 -45.8 f7.5 - 5 . 5 f7.5 -30.3 +19.2 -6.8 -4.4 e6.8 -18.0 ~ 6 . 9 f8.2 -18.1 f8.2

a See footnote a Table 111. See footnote b Table 111.

70 GEORGE ET AL.

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m


(2) For triply bonded molecules containing two heavy atoms, AE, for the reductive fission apparently depends to a greater extent on the nature of the heavy atoms concerned, ranging from a small positive value for acetylene, through a smaller positive value for hydrogen cyanide, to a small negative value for nitrogen, see Table 111, IC. This overall trend to a negative value, which is also outside the present uncertainties, cannot be attributed simply to the increasing electronegativity of the heavy atoms because the number of bonded H atoms also changes, so the effect is certainly composite.

(3 ) For the triatomic COz the uncertainties in E H F obscure the sign and magnitude of all but AE, for the partial fission yielding CO and H20, which is again small and positive like that for the molecules containing two heavy atoms, see Table 111, I I A and B. However, the AE, values for both the partial and complete reductive fission of N 2 0 are not only positive but much larger in magnitude than any of the values found for other reactions. Both values are far beyond E H F uncertainties and presumably originate in the peculiar structure of N20 , which in any valence bond formulation has to be represented as a polar

form, N-N-0- or N=N-0-. On the other hand CO also has as anomalous

structure, C-GO or C=O, yet AE, for its reductive fission is in no way unusual. Likewise for NO, even though it has an open shell structure with a single unpaired electron, the AE, value for its reductive fission is relatively normal.

( 4 ) For the simple reactions in Table 111, therefore, despite the separation of heavy atoms and concomitant formation of H-X bonds, there is, in general, invariance of the correlation energy to within a few kcal per mol-less than 10, usually between 0 and 5 . Considering that the correlation energies of these two and three heavy atom molecules range from 300 to 600 kcal/mol, the cancellation involved is to within at least 3%.

(5) The reactions in Table IV, I are, in principle, less drastic than those in Table I11 because no separation of heavy atoms occurs. Hence in many cases it is not surprising to find AE, having quite small positive or negative values, less than 4 kcal/mol, see reactions B(ii), C(i) and (ii), D(i) and (ii), and E(ii) and F(i). Larger values can occur however, for example, for the triple bond exchange reaction E(i),

+ +

+

NGN + 2CH4 + HC=CH+ 2NH,

But in such a reaction there is very extensive relocation of the electron pairs in the valence shells of the heavy atoms, so this finding is not altogether unexpected.

(6) The reactions in Table IV, I1 involve the formation of new heavy atom bonding and are thus the opposite of those in Table 111. Leaving aside the uncertainties due to the EHF values, there would seem to be systematically negative AEc values in keeping with this distinction.

(7) The AE, values for the benzene reactions in Table V all entail larger uncertainties, in part because E H F for benzene is estimated to within only 0.02 a.u., and in part because many more molecules are involved. For example, there are as many as 12 and 21 molecules beside the benzene in reactions B(i) and

72 GEORGE ET AL.

C(i). More appropriate comparisons are therefore to be made in terms of the C2 unit (CH=CH-). On this basis, the values in the last column can be seen, in the main, to follow the trends established above for the various types of reaction. The magnitude of the values gives little indication that substantial changes in correlation energy accompany the disruption of the aromatic structure. The observation that the sign of the values can apparently be either positive or negative argues even more strongly for this conclusion.

4. Partition of Valence Shell Correlation Energy in Terms of Contributions from Lone Pairs of Electrons, Electron Pairs in X-H Bonds, and Electron Pairs in

X-X Bonds

The remarkable extent to which correlation energies of reactant and product species cancel in almost all the reactions studied suggests very strongly that there are systematic regularities in the contributions of the various electron pairs in the valence shells. Following Clementi we assume the wave function for the 1s electrons to be unperturbed by rearrangements induced in the valence shells during bonding, and take the valence shell contribution as the appropriately corrected difference between Eexp and E H F less the ls2 contribution for each first-row atom in the molecule [51].

Further assuming the C-H bonds in CH4 are described by strictly localized electron pairs occupying orbitals having strongly directional character, Clementi evaluated the contribution per electron pair in the C-H bonds [5 11. Moskowitz took over this value to estimate upper and lower limits for the contribution of the shared pairs in the double and triple C-C bonds of ethylene and acetylene [52]. We now recalculate these values using the newer E H F data, obtain values for the C/C shared pairs in ethane and in benzene, and extend the calculation procedure to deal with molecules containing N, 0, and F atoms which contain lone pairs of electrons.

The new value of E H F for CH4 gives E, (V.S.) = 0.2499 a.u., hence, E, (C/H) per shared pair is 0.0625 a.u.* Using this value we obtain E,(C/C)per shared pair in the C/C single, double, and triple bonds of C2H6, C2H4, and C2H2, as 0.0905, 0.0883, and 0.0875 a.u., respectively. The slight trend towards a lower value as the bond multiplicity increases amounts to only 3% overall, and is almost within the uncertainties in E H F , so for the present purposes we adopt the average value 0.0888 a.u. (standard deviation, 0.0016 a.u.) for hydrocarbons in general. Ben- zene has nine C/C shared electron pairs, and using Ermler and Kern’s EHF of 230.82 a.u. [39] we find 0.0872 a.u. per shared pair, in close agreement with the value for the aliphatic hydrocarbons. Moreover, if the uncertainty of 0.02

* The full arithmetic value of 0.062475 a.u. has been used in subsequent calculations to avoid multiplication errors. This value differs from that derived by Clementi, 0.074 a.u., because he used a different calculation procedure, namely, assuming that the correlation energy for each electron pair in CH4 is equal to the molecular extra correlation energy in the HF molecule. Using the calculation procedure we have adopted, he found 0.0747 a.u. using Krauss’ SCF energy of -40.1668 a.u. as the HF-limiting energy [51,53].


underestimates the true EHF value, then the value per shared pair would rise giving even closer agreement. It would thus appear that within the accuracy of the present mlimiting energies, E, (V.S.) for hydrocarbons can be estimated with just these two parameters, E, (C/H) and E, (C/C).

The other simple hydrides, NH3, H20, and HF present a rather more complicated electron distribution since there are electron pairs in both X-H bonds and in the lone pair configuration. Clearly a distinction has to be made between them because it has been known for some time that there is a progressive increase in E, along the isoelectronic series CH4, NH3, H20, HF, and Ne [5l, 54,551. Revised values, based upon Eexp and the values for EHF listed in Table 111, prior to making corrections for relativistic energy, first-order Lamb shift, spin-spin, and spin-orbit energy [4], and for the ls2 electron pair, are 0.3057, 0.3611, 0.4096, 0.4586, and 0.5090a.u., respectively, see Table VI.

To partition the correlation energy into contributions from these two kinds of electron pair we make the simplifying assumption that the contribution per lone pair for any of the first-row atoms concerned is independent of the particular atom it belongs to, and equal to E,(V.S.) for Ne divided by 4. Hence we obtain 0.08605 a.u., see Table VI.

Some justification for this assumption is afforded by the data in Table VII for the atoms N, 0, and F, which contain 1,2, and 3 lone pairs and 3,2, and 1 unpaired electrons in the outer shell, respectively. Using the above value for E, per lone pair, E, per unpaired electron is found to be very nearly constant, 0.019* 0.001, see the last column. As a further test, to explore whether this simple additivity principle holds, the same kind of calculation has been carried out for the corresponding second-row atoms, P, S, and C1. In this case allowance has to be made, of course, for the filled 2s2 and 2p6 orbitals, in addition to the ls2 orbital, and E, per lone pair has to be calculated from the data for argon. The data in the lower part of Table VII shows that additivity is still approximately maintained. In fact, considering the numerous energy terms that go into each of these calculations, it is surprising to find such clear evidence for any quantitative regularity. It may be noted, too, that the value of E, per lone pair in A, 0.0845 a.u., is remarkably close to that for Ne, 0.0861 a.u., i.e., to within 2%: and the values per unpaired electron in P, S, and C1 are quite close to the average of 0.019* 0.001 for N, 0, and F. Since the valence shell of the second-row atoms is a good deal farther away from the nucleus than that of the first-row atoms, the similarity in the values of E, per lone pair, and per unpaired electron, from one row to another can be taken as a demonstration that correlation energy is somewhat insensitive to the exact details of the orbitals [7].

Returning to the analysis of the data for the simple hydrides, the contribution arising from the electron pairs in the X-H bonds has been calculated for HF, HzO, and NH3, see Table VI, column 7, giving per X-H bond 0.0808, 0.0746, and 0.0690 a.u., respectively. These values are to be compared with the 0.0625 a.u. per C-H, and 0.041 1 a.u. for H-H, calculated from the data for H2 [34]. There is a progressive decrease in E, per X-H bond as the electronegativity of X decreases, F+O+N+C, amounting to 29% of the value of C. In chemical

4

P

TA

BL

E VI.

Eval

uatio

n of

E,(V

.S.),

tha

t par

t of

the

corr

elat

ion

ener

gy of

mol

ecul

es d

ue to

val

ence

shel

l ele

ctro

ns: a

nd it

s sub

divi

sion

in

to c

ontri

butio

ns fr

om lo

ne p

airs

of e

lect

rons

, E,(L

.P.),

fro

m e

lect

ron

pairs

in X

-H

bond

s, E

, (X

/H),

and

from

ele

ctro

n pa

irs s

hare

d be

twee

n fir

st-r

ow a

tom

s, E

,(X

/X),

acc

ordi

ng to

the

equa

tion,

E,(

V.S

.) = E

,(L

.P.)

+E,(

X/H

)+E

, (X

/X).

Com

poun

d -E

a

-ZE

2c

-Ec (

V.

S . )

-E~(

L.P.

)~

-Ec (

X/H

)f

-Ec (X/X)'

L,r

,m,l

s ex

p

1.17

47

H2

Ne

129.

056

HF

100.

5296

76.4

776

56.5

861

40.5

247

H20

NH

3 CH

L

C2H6

79

.845

9

C2H4

78

.608

9

C2H

2 77

.357

1

232.

3138

'g

H6

199.

67 1

2

H20

2 15

1.64

76

F2

1.13

36

128.

5470

100.

071

76.0

68

56.2

25

40.2

19

79.2

69

78.0

71

76.8

58

230.

82

1981

780

150.

861

-

0.16

48

0.11

96

0.08

84

0.06

80

0.05

58

0.11

15

0.11

15

0.11

15

0.33

46

0.23

93

0.17

67

0.04

11

0.34

42

0.33

90

0.32

12

0.29

31

0.24

99

0.46

54

0.42

64

0.38

76

1.15

92

0.65

19

0.60

99

0.34

42

0.25

82

0.17

21

0.08

61

- - - -

0.51

63

0.34

42

0.04

11

0.08

08

0.14

91

0.20

70

0.24

99

0.37

49

0.24

99

0.12

50

0.37

49

0.15

91

0.09

05

0.17

65

0.26

26

0.78

43

0.13

56

0.10

66

109.

5886

93.4

664

113.

3767

114.

5616

188.

6964

184.

7656

150.

41 1

9

150.

3759

129.

9658

108.

998

92.9

16

112.

791

113.

932

187.

729

183.

7 50

149.

670

149.

621

129.

286

0.13

60

0.12

38

0.14

41

0.14

41

0.23

25

0.22

44

0.17

67

0.17

67

0.15

64

0.45

46

0.42

66

0.44

16

0.40

55

0.73

49

0.78

32

0.56

52

0.57

82

0.52

34

0.17

21

0.08

61

0.17

21

0.17

21

0.34

42

0.34

42

0.34

46

0.34

42

0.25

82

0.28

25

0.06

25

0.27

80

0.26

95

0.12

50

0.18

84

0.39

07

0.43

90

0.22

10

0.23

40

0.26

52

a So

urce

s of

expe

rimen

tal d

ata:

an

d ze

ro-p

oint

vib

ratio

nal e

nerg

ies,

see

foot

note

s Ta

bles

111

-V:

atom

ic d

ata,

see

Cad

e an

d H

u; [

56].

See

Tabl

e 11

. 2L

,,,m

,ls2 = E

(La

mb

corr

ectio

n) +

E (r

elat

ivis

tic e

nerg

y) +

E (m

ass c

orre

ctio

n)+E

(co

rrel

atio

n en

ergy

of

the Is

z el

ectro

ns).

Val

ues f

or th

e in

divi

dual

ener

gy te

rms o

n th

e ri

ght-

hmd

side

hav

e bee

n ta

ken

from

Cle

men

ti, g

ivin

g for

C, N

, 0, F

, and

Ne,

-0.0

5576

, -0

.i680

0,

-0.0

8836

, -0

.119

63,

and

-0.1

6484

a

x, r

espe

ctiv

ely

[4].

E, (

V.S.

)= E

exp-

EH

F-E

L..

.~~

~~

. E

, per

lone

pai

r ta

ken

to b

e ,E,(L.P.) f

or N

e, i.

e., -

0.08

605

a.u.

E

,(X

/H) c

alcu

late

d fr

om E

,(V

.S.)

for t

he h

ydrid

es w

ith E

, pe

r lo

ne p

air

from

foo

tnot

e e.

E

,(X

/X) c

alcu

late

d fr

om E

,(V

.S.)

, with

E,

per

lone

pai

r fro

m fo

otno

te e

, and

, whe

re n

eces

sary

, E(X

/H) fr

om fo

otno

te f.

76 GEORGE ET AL

TABLE VII. The partition of E,(O.S.), that part of the correlation energy of atoms due to outer shell electrons, into contributions from lone pairs of electrons, E,(L.P.), and from unpaired electrons in the

p-orbitals, E,(U.E.), according to theequation,E,(O.S.)= E,(L.P.)+E,(U.E.).

Atom -Ec ( O . S . ) a -E (L.P.) -Ec (U. E. ) -E per U.E. b

Ne 0.3442 0.3442

F 0.2764 0,2582 0.0182 0.018

0 0.2124 0.1721 0.0403 0.?20

N 0.1433 0.0861 0.0572 0.019

A 0.3381 0.3381

c 1 0.2723 0.2536 0.0187 0.019

S 0.2003 0.1691 0.0312 0.016

P 0.1273 0.0845 n. 0428 0.014

a E,(O.S.) =Eexp- EHF- E (Lamb correction)-E (relativistic energy) -E (mass correction) -E (correlation energy of the 1s’ electrons in the case of the first-row atoms, and the sum of the correlation energies of the Is’, 2s2, and 2 p 6 electrons for the second-row atoms). Values for the individual energy terms on the right-hand side have been taken from Clementi [4].

E, per lone pair taken to be :E, (L.P.) for Ne for the first row atoms, i.e., -0.08605 a.u., and i E , (L.P.) for A for the second row atoms, i.e., 0.08453 a.u.

reactions in which there is a charge in the X/H bonding, these differences will obviously lead to a AE, contribution to the reaction heat.

With the value of E, per lone pair, and values of E, per shared pair in X-H bonds, and assuming no overlap, it is a simple matter to calculate values for the shared pairs in the other kinds of X / X bonding. These results are gathered together in Table VIII. All the values for X / X which involve N, 0, and F atoms are significantly greater than the 0.0888 f 0.0016 for C/C bonding. As with the X/H bonding there is a progressive decrease in E, as the electronegativity of the atoms in the X / X bonding decreases. In this case, however, the effect is considerably more pronounced, amounting to 53% of the value for carbon.

TABLE VIII. E, (a.u.) per X-Xshared electron pair.

-0.1356 F2

H2°2

N2

-0.1066

02( A ) -0.1170

-0.0942

HC N -0.0927

1

CO -0.0898

H2C0 -0.0942

C02 -0.0977

N20 -0.1098


In the same vein, the following points of detail may be noted. The value for C/O from H2C0 lies between the values for C/C and O/O, and likewise the value for C/N lies between the values for C/C and N/N, and is itself a little le$s than the C/O value.

Finally the data for the unpaired electron molecules, 02(3E) and NO, call for comment. Taking the H202 value for the (single) shared pair in 02(3E) gives a contribution of 0.1144 a.u. for the brace of unpaired electrons-a value very much like that for the shared pairs in H202 or 02( lA) . If, for the shared pairs in NO, the average of the values for 02( ’A) and N2 is used, then the contribution from its single unpaired electron amounts to 0.0540-a value which is about half that for the N/O shared pair. It is to be noted, however, that this value is substantially greater than that for unpaired electrons in the free atoms, 0.019* 0.001 for N, 0, and F, and 0.016*0.003 for P, S, and C1, see above.

5. Analysis of the Contribution of Correlation Energy to Reaction Heats

The above treatment of the contribution of the various electrons in the valence shell to the molecular correlation energy enables us to specify those structural factors responsible for the invariance, or near invariance, of correlation energy in chemical reactions. It shows how in some cases, but not in others, the preservation of electron pairs per se will favor invariance: and it leads to a greater under- standing of the significance of preserving their “local spatial relationships” [6] . The following generalizations can be made.

(1) For those reactions, in which the number of C-H, N-H and O-H bonds and also the number of shared pairs in C/C, C/N, C/O bonding, etc. are the same in reactants and products, then to a first approximation, AE, = 0. All isodesmic [ 12-15] (and hence homodesmotic [S-111) reactions come into this category, e.g., reaction B(i) of Table V,

C6H,5 i- 6CH4 --* 3C2H4 2 C & j

and reaction IF(ii) of Table IV,

both of which are isodesmic. The values of E(C/O) per shared pair for COz and H2C0, however, would tend to suggest that simple cancellation no longer holds for such a deep-seated change in carbon/hetero-atom bonding, see Table VIII.

In addition, anisodesmic [16] reactions such as IF(i) in Table IV,

C ~ H Z + C ~ H ~ --* 2C2H4

and A(i) in Table V, C6H6 --* 3C2H2

which involve reactant and product species that contain the same heavy atom, come into this category, and, within the uncertainty attached to the EHF values, AE, is zero.

78 GEORGE ET AL.

(2) For hydrogenation reactions, which are characterized by a partial or complete separation of heavy atoms, X / X bonding is destroyed and replaced by X/H bonding, i.e.,

X-X+Hz -+ 2X-H

Thus there can be no simple generalization about AEc: its sign and magnitude will clearly depend on the relative magnitude of E, for reactants and products. For example, using the general values for the various electron pair contributions, AE, for the reduction of C2H6,

CH,-CH,+Hz + 2CH4

is +3.1 kcal/mol. On the other hand, for the complete reduction of nitrogen,

N=N+3HZ 4 2NH3

AE, = -5.1 kcal/mol. (3) For partial hydrogenation, in which there is no net separation of the heavy

atoms, there is a similar interplay between the E, values for the reactant and product species, and the same considerations apply.

(4) For reactions in which there is an exchange of X / X bonding, e.g.,

F-F+ 2CH4 -+ CH3-CH3 + 2HF

HZC=O + CH4 + HzC=CHz +HzO

NGN+2CH4 -+ HC=CH+2NH3

i.e., reactions IA(ii), IC(iv), and IE(i) in Table IV, there is a change in the location of the shared pairs with respect to the heavy atoms, and a concomitant change in X/H bonding. AE,, with regard to both sign and magnitude will depend on the interplay between E, (X/H) and E, ( X / X ) values.

(5) The formation of new heavy atom ( X / X ) bonding is brought about through the destruction of X/H bonds, see Table IV, IIA and B. In general AE, will be negative for such reactions since equal numbers of electron pairs change location from X/H to X / X , and E(X/H) per electron pair is less than E ( X / X ) per electron pair.

(6) Another type of reaction in which AE, would be expected to contribute to the reaction heat is that in which lone pairs of electrons become shared pairs, or vice-versa, since, in general, E, per X/H or E, per X / X would have values different from E, per lone pair. None of the reactions dealt with in the various tables are of this kind, but they are not uncommon, e.g., protonation

H++NH, + N&+

the formation of addition complexes,

NH,+BH, -+ H3N-BH3

and the formation of coordination complexes of metal ions. All these reactions have a feature in common, namely, one or the other of the reactants has an incomplete outer valence shell.


6. Estimation of Limiting HF Energies

For hydrocarbons in general, C,H,, Hurley [6] proposed a two parameter equation to estimate E H F :

EHF = (AH?),, - ZPE - 37.9039m - 0.56185n where (Ak&, and ZPE are the standard enthalpy of formation at 0°K and the zero-point vibration energy, respectively. Our treatment of the molecular correlation energy also provides two parameters, E,(C/H) and E,(C/C) per shared electron pair, from which E, (V.S.) can be determined, and, hence, by reversing the calculation procedure employed in Table VI, a value for E H F can be achieved.

The formal connection between the two methods is that in the hydrocarbon C,H, there are (4m - n ) / 2 shared pairs of electrons in the C/C bonding, and n shared pairs in the C/H bonds.

Table IX gives estimated E H F values for a selection of both aliphatic and aromatic liydrocarbons which have been the object of extensive ab initio SCF

TABLE IX. Estimated Hartree-Fock-limit energies.

E e x p - b ~ c a l c ~ E m ~ b

bEca lc Compound E a exp

me thane

e thane e thy l e ne ace ty l ene

propane propene cyc lopropane a l l e n e ProPYne cyc lopropene

n-bu t ane 1-butene t r a n s - 2 -but e ne 2-me t hylpr opene cyc lobut ane t r a n s - 1,3- but ad iene 1 -but yne 2-butyne

tT t -1 ,3 ,5 -hexa t r i ene

benzene naphthalene an thracene

-40.5247

-79.8459 -78.6089 -77.3571

-119.1695 -117.9380 -117.9262 -116.6886 -116.6900 - 116.6552

- 158.4954 -157.2626 -157.2670 -157.2687 - 157.2522 -156.0359 -156.0141 -156.0217

-233.4706

-232.3138 -386.0036 -539.6878

-0.3057

-0.5752 -0.5390 -0.5028

-0.8446 -0.8085 -0.8085 -0.7722 -0.7722 -0.7722

-1.1141 -1.0778 - 1.0778 -1.0778 -1.0778 -1.0417 - 1.0417 -1.0417

- 1.5445

- 1.5084 -2.4776 -3.4469

-40.219

-79.271 -78.070 -76.854

- 118.325 -117.130 -117.118 -115.916 -115.918 -115.883

-157.381 -156.185 -156.189 -156.191 -156.174 -156.994 - 154.972 - 154.980

-231.926

-230.806 -383.526 -536.241

a See footnote a Table VI. bSE,,,,=ZEL,,,m,,s2+E,(V.S).ForHE,.,.,.,,*andE, (V.S.)seefootnotescanddTableVII. E,

per C/H and C/C shared electron pairs taken to be 0.062475 and 0.0888 ax., respectively. Based on the difference between Ermler and Kern's value of 230.82 and the estimated values of

230.805 for benzene [39]. The uncertainty in the estimated values for the other hydrocarbons can be put at C1, 0.002; C2, 0.005; CJ, 0.007; C.,, 0.009; C,, 0.014; Clo, 0.023; and CI4, 0.033 a.u.

80 GEORGE ET AL.

calculations [20-33, 57-62]. The uncertainty in these estimated values is reck- oned to be proportional to t1.e number of C atoms in the molecule, see footnote c of Table IX. It may be noted that the value for benzene is well within the uncertainty of Ermler and Kern’s estimate [39], which would suggest that the values reported here for the first time for naphthalene and anthracene are reliable to much the same extent.

Like Hurley’s method, this alternative method can also be used for molecules containing other first-row atoms, making use of the values obtained for E, per lone pair, and E, (X/H) and E, ( X / X ) per shared pair.

Furthermore, with E, per unpaired electron, obtained from the atomic correlation energy data, EHF values can be estimated for free radical species. For example, for the OH radical we find EHF = -75.426, which may be compared with Cade and HUO’S estimated limiting value of -75.4218*0.0003, based on an extensive SCF calculation using a large set of Slater-type functions [56]. Consider- ing the diverse set of data employed in our estimation procedure, agreement to within 0.004 a.u. is quite satisfactory, and comparable accuracy may be anticipated in calculating values for NH2, 02H, hydrocarbon free radicals, etc.

Acknowledgment

The authors would like to thank Dr. A. Komornicki for supplying them with the 4-31G geometry optimized energy of benzene, and Drs. W. C. Ermler, I. Shavitt, A. Veillard, and W. J. Hehre for helpful references and comments.

Bibliography

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Received July 15, 1976 Revised October 13, 1976

correlation energy contributions to reaction heats

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