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2782 J . Am . Chem. SOC. 982, 104, 2782-2789The Relationship between Energy Additivity and theEquivalent GroupJoseph R. Murdoch* and Dou glas E. M agnoliContribution fr om the Department of Chem istry, University of California, Los Angeles,California 90024. Received April 10, 1980

Abstract: The idea of the equivalent group which can be transferred from molecule to molecule as an in tact unit has playeda key role in understanding molecular structure and reactivity. In the present paper it is shown that introducing a substituentinto a molecule can produc e substa ntial shifts in electron density, both in the su bstitu ent and in the rest of the molecule. Basedon cu rrent concepts, one would not expect to find a high degree of energy additivity in such a situation. However, these changesin electron density can be mutu ally compensating, so that energy and bond length additivity are observed anyway. A surprisingnew result is that energy additivity does not require equivalent, transferable groups which maintain con stant electronic structu rein different molecules. This finding has important imp lications for any theory attempting to describe multiple substituenteffects since it may be necessary to explicitly recog nize interactions between groups and changes in electronic structure evenwhen energy additivity is observed. These phenom ena are examined from the viewpoint of a theory of nuclear substitution,which has been presented previously. The spa tial distribu tions of electron density, orbita l energy, and kine tic energy are usedto probe the relationship between group transferability and energy additivity.

Introduction. The Equivalent Group: A Ubiquitous ConceptEver since Dalton' proposed tha t molecules are composed of

atoms combined in definite proportions, the concept of theequivalent group has provided a focal point in rationalizingmultiple substituent effects on molecular structure a nd reactivity.The stru cture of the equivalent group is thought to be an intrinsicproperty of the group and independ ent of the other structu ral unitsin the molecule. Conseque ntly, the heat of formation of a moleculecomposed of equivalent groups can be expressed in terms of ad-ditive contribution s of each group. This approach has beensurprisingly successful for a wide variety of compounds* and haseven been applied to bicyclic materials containing strained ringsS3It has also been recognized that interactions can occur betweenmolecular components and that these interactions require addi-tional contributio ns to the heat of formation . It is interesting thatinteractions between groups can frequently be treated as additiveeffects. For example, gauche interactions between alkyl groupsare often additive in such a w ay that each gauche interactioncontr ibutes a f ixed amount to the heat of f~rmation.~Molecular mechanics a nd force-field approachesSalso preservethe basic idea of the equivalent group. Nonbonded interactionsbetween two atoms may be described in terms of potentialfunctions5 so that the internuclear distance and type of atoms arethe only variables affecting the interaction. Nonbon ded inter-actions may result in deviations from stand ard bond lengths an dbond angles, and this effect also requires add itional increm entsto the heat of formation. Again, these contributions depend onlyon the distortion of a specific bond length or bond a ngle and a reindependent of other interactions or geometric changes in themolecule. Th e high degree of successS at predicting hea ts offormation has resulted in a general acceptance of the idea thatfragments can maintain constant electronic and geometricstructure in different molecules. The equivalent group has gainedfurther support from theoretical treatments which reproduce abinitio energies and properties of large molecules by transferringmatrix elements or localized orbitals from simpler structures.6

~~ ~

(1 ) J . R. Partington, 'A Sho rt History of Chem istry," 3rd ed.,Harper andBrothers, New York, 19 60, p 173.(2) (a) S. W. Benson, "The Foundations of Chemical Kinetics",McGraw-Hill, New York, 1960, p 670. (b) D. R. Stull, E. F. Westrum, Jr.,and G . C. Sinke, "The Chemical Thermodyanmics of Organic Compounds",Wiley, New York, 1969.(3) (a) J. Gasteiger and 0.Dammer, Tetrahedron, 34, 2939 (1978). (b)D. Van Vechten and J. F. Liebman, Isr. J . Chem. 21, 105 (1981).(4) A . J. Kalb, A. L. H . Chung, and T. . Allen, J . A m . C h em . SOC. , 8,1938 (1966).(5) N. L. Allinger, Ado. Phys . Org. Chem. , 13, 1 (1976).(6) B. O'Leary, B. J. Duke, and J. E. Eilers, Adu. Quantum Chem. 9, 1(1975).

0002-7863/82/1504-2782$01.25 IO

The equivalent group concept is also fundamental to theHa mm ett equation' and othe r forms of linear free-energy rela-tionships.* Ham met t suggested tha t the effect of replacing asubstituent on the pK, of benzoic acids could be described in term sof a "substituent" pa rameter (a) and a "reactionn parameter, p .According to Ham mett, ' the acid dissociation constant of asubstituted benzoic acid is given by

where KO s the acid dissociation co nstant of benzo ic acid. u isregarded as an intrinsic property of the substituent and inde-pendent of the struc ture of the rest of the molecule. Likewise,p is looked upon a s an intrinsic property of the group undergoingreaction (e.g., - C 0 2 H -COT) and is indepen dent of substituent.Numerous refinementssa" of the Hammett equation have ap-peared since Hamme tt's original proposal. Forsythsf has presenteda simple extension which provides a rather remarka ble correlationof rates for arom atic substitution an d side-chain solvolysis reactionsof benzene, naphthalene, furan, thiophene, and other arom aticderivatives.Forsyth treats field and resonance interaction s of the sub stit-uents in terms of two parameters (D nd E'). T he D and E+parameters are regarded as intrinsic properties of the substituentand a re assumed to be independent of both the specific aromaticring to which the substituents are attached a nd the relative im-portance of the field and resonance effects. It is hard to avoidquestioning these assumptions, but one is faced with the fact thatForsyth's model provides an impressive correlation of solvolyticand electrophilic substitution reactions in a wide variety of aro-matic systems.The equivalent group models all have one common element:characteristic p arameters ar e assigned to certain structural unitswithin the molecule, and these param eters remain unc hanged evenwhen interactions occur between the struc tural units. The

(7 ) L. P. H ammett, "Physical Organic Chem istry", McGraw-Hill, NewYork, 1940.(8) (a) J. E. Leffler and E. Grunwald, "Rates and Equilibria of OrganicReactions", Wiley, New York, 1963. (b) S.Ehrenson, Prog. Phys. Org.Ch em. , 195 (1965). (c) C. D. Ritchie and W . F. Sager, ibid., 323 (1965).(d) C. D. Johnson, Chem. Reu., 75 ,75 5 (1975). (e) B. Giese, Angew. Chem.,Int . Ed. Engl . , 16, 125 (1977). (0 D. A. Forsyth, J . A m . C h em . Soc., 95 ,3594 (1973).(9) On e approach to this problem is to apply equivalent group assumptionsto the derioatiue of the energy change and to obtain, after integration, anonlinear relationship between two free energy quantities. This leads to asimple model for a rate-equ ilibrium relationship which quantitatively accoun tsfor many features of proton transfer and other reactions. However, such asemiempirical solution is not as convincing as one derived from a rigorous abinitio foundation. J. R. Murdoch, J . A m . Chem. Soc., 94 , 4410 (1972).$31982 American C hemical Societv

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Relationship between Energy Additivity and the Equivalent Group J . Am. Chem. Soc.. Vol. 104, N o. 10, I982 2783equivalent group approach can account for additivity and evensome degree of nonadditivity. However, the range of nonadditivityis necessarily limited due to the lack of a mechanism for intro-ducing systematic changes in the substituent param eters as in-teractions become progressively largere9Theoretical treatm ent of nonlinear substituent effects will re-quire understanding of how interactions between structural unitsin a m olecule alter the properties of the equivalent grou p and limitthe degree of transferability. Th e solution to this problem hasassumed a significance that extends well beyond a simple re-finem ent of the equivalent group approx imatio n. In recent years,there have been an increasing number of curved rate-equilibriumrelationships for a wide variety of reactions, including protontransfer.I0 Th e curvature is an example of a nonlinear substituenteffect and of a breakdown in the equivalent group concept.Through various semiempirical relationships, such as Marcusequation:* it is possible to relate this curvaturelo to v arious factorscontributing to the overall observed barrier for the reac tion. Thesefactors include bond making a nd bond breaking , diffusion, solventreorganization, orientation, and molecular distortions. On e im-plication of these results is tha t making an d breaking chemicalbonds is often less impo rtant than these other factors as a con-tribu tor to the overall barr ier. If valid, this result is of greatsignificance for many aspects of mechanistic chemistry an d holdsimportan t implications for the problem of designing catalysts foraccelerating solution phase reactions. A catalyst which works byaltering the way bonds a re formed and broken involves funda-mentally different principles than a catalyst which compensatesfor reorganization of solvent, orientation, or conformationalchanges. Unfortun ately, no direct experimental evidence isavailable for assessing the relative importance of bond formationand oth er factors to the overall barrier, and so present conclusions0must remain as tantalizing deductions derived from a theory9,based on assumptions of questionable applicability.The motivation for examining nonlinear substituent effects isnot to simply reduce the scatter in force-field calculations andin Hammett-type relationships, but to provide a reasonable the-oretical framework for evaluating the present conclusions regardingthe relatively minor importance of bond formation to reactionbarriers for certain classes of reactions. The first step in thedevelopment of a theory of nuclear substitution is presented inthe previous paper,12and the results have been used13 to evaluatethe importance of various contributions to reaction barriers.Initially it was thought that nonlinear substituent effects couldbe treated by extending equations based on additivity and theequivalent group formalism. However, it became appare nt thatthe equivalent group approximatio n breaks down before energyadditivity, so that energy additivity persists even after the electronicstructure of the various molecular units has been altered by mutua linteractions. Suc h behavior is totally unexpected on the basis ofprevious models of substituent effects. Thi s phenomenon is ap -parently general, and in the present paper, the fundamentalprinciples behind this unusual behavior are examined through thehemistructural relationship.The Hemistructural Relationship

In th e previous paper,I2 he effect of a relatively simple geometryconstraint on the molecular wave function and energy was ex-plored. This geometry constraint relates a hybrid structure,A-B-C, to two pa rent structures, A-B-A and C-B-C. Acoordin ate system is chosen so tha t the nuclear coordinates of th eA fragm ent are the same in A-B-C and A-B-A. A similarconstraint operates on the B and C fragments. This type ofgeometric arrangem ent was termedI2 the hemistructural rela-tionship and c an be regarded as a fo rm of geom etry additivity.

(IO) For leading references, see: J. R. Murdoch, J . A m . Chem.Soc., 102,(11) (a ) R. A. Marcus, J . Phys. Chem.,72, 891 (1968). (b) A. 0.Cohen(12) J. R. Murdoch, J . A m . Chem.Soc., 104, 588 (1982).(13) (a) D. E. Magnoli and J. R. Murdoch, J . A m . Chem.SOC. 03,7465(1981). (b ) J. R. Murdoch and D. E. Magnoli, J . A m . Chem.Soc., in press.

71 (1980).an d R. A. Marcus, ibid., 72, 4249 (1968).

An important consequence of the hemistructural relationship isthe exact transferability of all integrals necessary for defining theH ar tr ee -F oc k w av e f ~ n c t i o n . ~ ~ ~Integral transferability leads to the useful result that the nu-clear-electron attraction matrix elements for C-B-C, (V,/neiJ)CBc,can be expressed12 as an additive sum of th ree contribu tions:

where( 3 )

(6 Vn/nei)ABC = (Vnnei)ABC - (VnLj)ABA (4 )It has also shown that these additive changes in V,/neijcould12 leadto a similar additivity in the M O coefficients for C-B-C (#.)

$ u = C(aovm + avm)+m ( 5 )mwhere the corresponding unperturbe d M O for A-B-A is givenby

+Yo = z a u m +m ( 6 )man d +, represents a complete set of nuclear-independent basisf ~ n c t i o n s . ~he coefficient additivity allows a,,, to be expresseda s

a,,, = a r y m+ alym ( 7 )where a, is the change in coefficients or the perturbation A-B-A- -B-C, while a,,,s the change in coefficients for A-B-A -When a,,, (e q 7 ) s substituted into the expression for the kineticC-B-A.energy

it is seen that the linear terms contributing to T (Le., avjTijaO,i)are a function only of avjand respond independently to separatechanges in nuclear ch arge and position. Consequently, the linearcontribution to T is additive.The behavior of the kinetic energy is important since the virialtheorem requires that the total energy and the kinetic energyrespond in equal but opposite fashion to substituent changes.12While the virial theore m requires that th e total potential energy(V = V,, + V, + V,,,,) show the same degree of additivity as Tand E, this does not apply to the individual terms V,,, V,, andV,,.l2 Conse quently , it is possible to observ e a high degr ee ofadditivity in E , T , and V, even though V,,, V,, and V,,, areessentially nonadditive. One consequence of these relationshipsis that the to tal energy ( E = T + V,, + V, + V,,) will alwaysbe more additive than the orbital energy (E o= T + V,, + 2V,).I2According to standard forms of perturbation theory,Is energyadditivity is associated with first-order energy differences and nochange in M O coefficients. Changes in M O coefficients producesecond-order changes in energy which are nonadditive.* Con-sequently, the degree of energy additivity has been thought todepend on the degree to which changes in MO coefficients andsecond-order energy corrections are negligible. Gro up transfer-ability has generally been regarded as a necessary prerequisite

(14) The integral transferability leads to a simplified behavior of theHartree-Fock wave function and energy. The transferability was demon-strated using a complete basis set where the basis functions are independentof nuclear charge and position. The transferability is basis set dependent, butthe resulting behavior of the Hartr ee-Fo ck wave function holds for al l com-plete basis sets regardless of integral transferability. The im mediate valueof integral transferability is not as a computational advantage, but as a meansof exposing the relationship between the wave functions and energy of ABA,ABC , and CBC . Integral transferability is obviously an appealing idea forexecuting actual calculations, but it involves a number of tradeoffs which arepresently undergoing careful consideration.( 1 5 ) (a) F. L. Pilar, Elementary Q uantu m Chemistry, McGraw-Hill,New York, 1968. (b) M. J. S.Dewar, The Mo lecular Orbita l Theory ofOrganic Chemistry, McGraw-Hill, New York, 1969.

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2784 J. A m . Chem. Soc., Vol. 104, N o . 10, 1982T able I. Calcu la t ed" E ner gy Compon en t s f o rP r o ton- Bound Rar e Gas Dimer s

NeHNe* NeHAr + Ar HAr +E,, au - 1 5 3 . 4 6 7 6 2 0 - 4 0 2 . 2 8 2 8 2 8 - 6 5 1 . 1 0 1 7 2 5T 253 . 510 356 6 4 8 . 1 2 8 2 4 3 1 0 4 2 .7 4 6 4 5 9V,e - 6 7 5 . 5 5 5 3 7 2 - 1 6 3 8 .3 4 2 3 4 5 - 2 6 1 6 . 7 4 9 6 1 0V ex - 2 4 . 6 7 5 1 1 5 - 4 2 . 7 5 3 5 1 2 - 6 0 . 8 5 0 3 7 9V,, 1 5 8 . 96 3 8 1 4 3 3 6 . 7 19 1 4 8 5 2 2 . 30 1 0 9 2V n n 3 4 . 2 4 5 9 6 1 4 8 . 1 1 9 9 5 2 6 9 . 8 05 9 2 817 1 . 0 1 0 3 9 3 8 7 1 . 0 0 0 0 1 6 9 9 8 b 1 . 0 0 9 6 7 5 9 5r N e H , 2 . 044 036 66 2 . 044 036 66'HAr, 2 . 836 435 33 2 . 836 435 33

E - 2 5 3 . 5 1 0 3 5 7 - 6 4 8 . 1 2 8 5 1 3 - 1 0 4 2 . 7 4 6 5 1 0

bohrbohr

Murdoch and MagnoliTable 11. Addit ivi ty of Energy Com ponen ts . Mean Deviat ions"f o r P r o ton- Bound Rar e Gas Dimer s (Ar HNe' )

mea n deviat ion, mea n deviat ion,kcal kcal+1 . 157 A v C o b - 2 4 5 5 . 5 9 8- - 0 . 103 A Vn n - 45 1O 1 0a 0A TA v n e + 4 9 0 0 . 8 6 7 AE - 0 . 0 5 0

A v e x b + 5 . 7 9 4" Mean deviat ion equals the quan t i ty ca l cu la t ed f o r Ar HNetminus t he co r r es pond ing average quan t i ty f o r Ar HArt andNeHNet .V ee= V e x + Vco,where Vex is the exchang e energy and V,, isthe Cou lom b ener gy.T ab le 111. Popula t ion Analysis" fo r NeHNe' ,NeHAr*, and ArHAr+

Th e total electron /electron repuls ion (Vee ) is given by

NeHNe NeHAr Ar HAr1s 1 . 999 393 52 1 . 999 392 162 s 1 . 9 7 3 5 6 7 6 7 1 . 9 7 3 2 1 6 7 52p, 1 . 7 3 7 1 1 8 3 3 1 . 7 0 5 9 5 5 6 12p, 2.00 2.002p, 2.00 2.00

2P,2PY2PZ3P x3p ,3p 2

1 s 1 . 999 973 25 1 . 999 973 362 s 1 . 9 9 9 1 3 9 8 8 1 . 9 9 9 1 4 3 8 21 . 9 9 3 9 5 1 6 5. 9 9 4 5 1 1 3 02.00 2.002 . 00 2 . 001 . 731 135 242.00 2.002.00 2.00

3s 1 . 986 947 36 1 . 986 137 191 . 6 9 0 5 7 5 0 1

H 1s 0 . 57 9 8 4 0 9 6 0 . 6 0 9 7 1 8 4 9 0 . 6 6 0 4 3 7 9 3

" Standar d 3G expon en t s a r e s cal ed by $. Gauss ian lobe func-t ions replace Car tes ian p Gauss ians (J . L. M i t t e n , J. Chem. Phys.,44, 3 5 9 ( 1 9 6 6 ) ) . T h e o r b i ta l e x p o n e n t s fo r NeHAr + were t akenf r om the c o r r es pond ing s ca l ed values f o r NeHNet and Ar HAr +( excep t f o r t he p r o ton whos e exp onen t s a r e t he aver age o f t hos ef o r t he p r o to n in the two s ymmet r i ca l s t r uc tu r es) and then s ca l edb y ( 1 . 0 0 0 0 1 6 9 9 8 ) 2 .for energy additivity.16 Th e problem with basing a theory ofnuclear substitution on this foundation is that there a re manyexamples of additivity to within a few tenths of a kilocalorie, eventhough it is apparent from other p roperties that substantial changesin M O coefficients have taken place. This prompted a newtheoretical treat men ti2 which shows that w hile energy additivitycan be associated with coefficient additivity (eq 7 ) , there is noabsolute requirement that constant electron density be maintainedin a pa rticular region of spac e or over a specific molecular orb ital.This carries the im portant implication th at energy additivity isnot linked to the degree of structu ral transferability. This resultis unprecedented and in the following sections, the relationshipbetween additivity and transferability is examined by analyzingthe behavior of the various energy components (T , V,,, V,, V,,)and by comparing the spatial distribution of electron density,orbital energy, and kinetic energy.Energy Additivity and Transferab ility. Proton-Bo und RareGas Atoms

Initially, it was desirable to examine a simple system wheretransferability might have a good chance of adequately describingenergy additivity. Consequently, proton-bound rare gas dimerswere the first molecules to be stud ied, since a minimum of elec-tronic and geometric reorganization is expected" in the seriesXHX', XHY', and YHY' (X and Y are rare gas atoms).Calculations were carried out with GAUSSIAN 701* and PRO-METHEUS x,I9using Pople's standard 3G basis set as well as arescaled version which is described in Ta ble I. The energies andoptimized geometries (3G basis) of X, H-X', and XHY ' (X , Y= He, N e, Ar) are reported in the following paper.13b Since thecalculated energy of NeH Ar' was found to be within 0.1 kcal ofthe mean energy of Ne HN e' and ArHAr', this system was ex-amined further. I t is interesting to note th at the optimized ge-omet ry of NeH Ar' is close to the hemistructural geometry: theH-N e distance is shorter by 0.004 A and the H-Ar distance islonger by 0.011 A relative to the hemi-structural geometry.In general, the virial theorem will not be satisfied for calcu-lations employing a minimal basis set such as 3 G , unless theexponents and bond distances are rescaled. Procedures for scaling

(16) (a ) T. L. Allen and H. Shull, J. Chem. Phys., 35, 644 (1961). (b )M . Levy, W. J. Stevens, H. Shull, an d S. Hagstrom, ibid.,61, 1844 (1974).(c) R . F. W. Bader and P. M . Beddall, Chem. Phys. Lett., 8, 29 (1971). (d )R. F. Bader and G. R. Runtz, Mol. Phys., 30, 17 (1975).(17) (a ) V. Bondybey, P. K. Pearson, an d H. F. Schaefer 111, J. Chem.Phys., 57, 123 (1972). (b) P. S. Julienne, M. Krauss, and A. C. Wa hl, Chem.Phys. Lett., 11, 16 (1971).(18) W. J . Hehre, W . A. Lathan, R. Ditchfield, M. D. Newton, and J . A.Pople, QCPE, 1, 236 (1973).(19) PROMETHEUS x is an experimental SC F- M O program now underdevelopment at U CLA. It will be described in detail elsewhere.

" Scal ing factors are the same as those f r om T able 1. Scaling hasa negl igible ef fect on the popu lat ion analys is .approx imate Hartree-Fock wave function s have been reported,*"but it was found that the scaled wave function and the wavefunction obtained from another calculation with the new scalingfactors were invariably in poor agreement.The calculations presented in Tabl e I use a new scaling pro-cedure which will be described elsewhere. For Pople's stand ard3G basis set, the scaling factors and bond distances reported inTable I yield wave functions satisfying the virial theo rem ( E / T+ 1 = 0) to better than 10".The sca led energies show a high degree of additivity: the energyof NeH Ar' is 0.05 kcal lower than the mean energy of NeHN e'and ArHAr'. In striking contrast, V,, for NeH Ar' deviates fromthe mean by +4900.87 kcal, while (Vee+ Vnn) iverges in theopposite direction by -4900.81 kcal. The near cancelation ofrepulsive and attractive terms is required by the virial theoremI2since the total potential energy and total energy are related asE = V/2.A. Energy Additivity. A Classical Electrostatic Viewpoint. Thehigh degree of additivity for E , T, an d V (Table 11) suggests tha twe exam ine several simple models for rationalizing additivity. On eapproach is to test the possibility that A r and N e in the threestructures are behaving as essentially neutral, unpolarized ato ms(or high-order mu ltipoles), so tha t Ar /Ne , Ar /Ar , and Ne /N einteractions ar e negligible. If this were the case, the classicalelectrostatic energy, representing the total attractions and re-pulsions of the nuclei and the quantum mechanical electrondistribution, should show a degree of additivity comparab le to thequa ntu m mechanical total energy. Th e classical electrostaticenergy can be obtained from E by subtracting the exchangecontribution (Vex)rom Vce.zi Vex or NeH Ar' deviates from themean V,, for NeHN e' and ArHAr' by +5.8 kcal, so that theclassical electrostatic energy is nonadditive by -5.85 kcal. Con-sidering that th e total range of energy covered by these structuresis over 490000 kcal, this degree of additivity is quite impressive,

(20) E. Scarzafava and L. C. Allen, J . A m . Chem. SOC. , 3, 11 (1971).(21) S. Wolfe and A. Rauk, J. Chem. SOC. , 136 (1971).

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Relationship befw een Energy Additiv ity and the Equivalent Group .I.m . Ch em . SOC.,Vol. 104. No. 10 . 1982 2785

Figure 1. Projected electron density difference plot fa r ArHNe. Thefigure is constructed by separa tely placing NeHNe, ArHNe, andAr H Ar + in the xy plane and integrat ing he electron density along thez axis from the surface of the plane to -. Each molecule is divided inhalf through the hydrogen and the difference between the electron densitydistribution of the Ar fragment in the parent and the Ar fragment in thehybrid is plotted in the left half of the grid. The corresponding differ-ence for N e appear in the right half of the grid. The maximum equals0.h1892 electron/bohr.but i t is also clear th at n ear additivity of th e classical electrostaticenergy does not explain th e even higher additivity observed forE, T , an d V.z2E. Energy Additivity. The Orbital Energy. Th e orbital energyof NeHAr is additive to within 1.16 kcal. Th e deviation is about20 times the discrepancy observed for the total energy an d is inqualitative agreement with previous expectations. Additive totalenergies cannot be explained by invoking additive orbital energies.lzC. Energy Additivity and the Equivalent Group. ElectronDensity Distribution. It would also be interesting to examine theequivalent group model a s a rationalization for additivity. In Table111, the Muiliken population analyses for NeHNe, Ne HA rf an dArH Ar+ ar e lis ted. Th e major orbitals contributing to bondingare the Ne 2p,. Ar 3p, an d H 1s where the f direction lies alongthe internuclear axis. Th e Mulliken analysis shows a reducedpopulation (by 0.031) for the Ne 2p, orbital of NeH Ar + relative

to NeHN e*, while the population for AI 3px increases (by 0.041)on going to the hybrid structure. Th e H Is population for Ne HA r+(0.6097) is bracketed by the corresponding populations forN e H N e + (0.5798) a nd ArHAr+ (0.6604). The overall patternis not consistent with a constant wave function and transferablegroups, but with small increases in density in one region andcompensating decreases in another region,Thi s conclusion is reinforced by exam ining differences in thespatial distribution of electron density as seen n Figure 1. Figure1 is constructed by placing each molecule in the xy plane andintegrating the electron density along the z axis from the surfaceof the plane to - . I 3 Each molecule is divided in half throughthe hydrogen and the difference between the electron density~~ ~

(22) The clas ieal dcctro static energy was also calculated in a relf-con-sistcnt fashion by allowing th e iterative Ha rtre cFo ck procedure to continuewithout the contribution of the exchange integrals. This mcthcd convergedfar the systems examined here and invariably resulted in a higher degree ofnonadditivity. The iterative calculations were pcrformed with the standard3G basis and are no t sealed. The total energy (quantum mechanical) ofNe H Ar +deviates by only 0.1 kcal from the mean. while th e classical energyfor th e quantum mechanical charge distribution dcviates by -6.0 kcal fromth e man. Thencmulu are comparable to t h m obtained with the scaled basisset. The self-consistent classical energy deviates by -I1.9 kcal from theman.(23) Integration of electron densitiesalong l i n e or over volumes and arwshas been used pmioudy. Examples include: (a) P. Politzcr and R. R. Harris,L A m . Chem.Sm..92,6451 (1970). (b) R. F. W. Badcrand P. M. Beddall,ibid., 5. 305 (1973). (c) A. Streitwicscr,Jr., J. E. Williams, Jr.. S. ler -andratos, and J. M. McKclvey. ibid.,98,4778 (1976). Stre i twiser has usedth e term electron projection function to describe th e line integral of th eclcctron density along a path perpendicular to th e molecular plane. (d ) K.B.Wiberg, ibid.. 102. 1229 (1980).

Figure 2. Projected orb ital energy difference plot for A r H N e + . Theorbital energy is evaluated as a function of the spatial mrdi na tes ac-cording to eq IO. and projected diffcrenuJ are platted as in Figure I . (a)Core orbitals, m aximum equals 0 .499 au/bohri. (b ) Valence orbitals.maximum equals 0.00868 aulbohr. The integration mentioned in thetext refers to the combined sum of core and valence orbitals.distribution of the Ar fragment in the parent and the Ar fragmentin the hybrid is plotted in the left half of the grid. The corre-sponding differences for N e appe ar in the right half of the grid.Th e overall pattern is one of small but compensating changes inelectron density in different spatial regions. For example, the dipin Figure I near th e AI region corresponds to about 0.04 e com-pared to I 8 e assigned to this half of the grid for ArH Ar+ . Th epronounced rise near the AI half of the proton corresponds toroughly 0.025 compared to 0.33 e assigned to the Ar half of theproton in ArHAr. These assignments have been based on theMulliken analysis and not on partial integration over the surfaceof the grid. Nonetheless, they serve to emphasize that the overallchanges in electron distribution are relatively small (0.2% to 8%)and t ha t decreases in one region are offset by increases in anotherregion.D. Energy Additivity and the Equivalent Group. Sp ati al Dis-tribution of Orbital Energy. Th e consequences of these com-pensating increases and decreases of electron density on energyadditivity can be seen by plotting out differences in the spatialdistribution of energy. Th e energy corresponding o the exact wavefunction is given by

E = JqHq d r ( 9 )The integrand, $H$ dT, can be interpreted as the energy con-tribution from volume element d r and can be assigned a spatialMordin ate just as charge density can be assigned a spatial co-ordinate. If H is the Fock operator and J.i is one of the Fockeigenfunctions,

dE , $,H& dT = $(E,$, d r = (Ej ) ($ jJ . j r ) (10)Equation I O gives us the interesting result that the orbital energycontribution from volume element d r is equal to the fraction ofelectron density in the volume element times the orbital energy

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2186 J . A m . Chem. SOC.,Vol. 104, No. 10 , 1982 Murdoch and MagnoliT a b l e IV . T ota l E ner g ies of Pr o ton- Bound Anions ( au )

X = F= H , C - X = H,N- X = HO-XHCH,- -78.585 491' -94.345 572 -113.8 6669 3 -137.482 368-79.530 231b -95.528 372 -115.364749 -139.378 861XHNH, - -110.103 735 -129.623 211 -153.237 201-155.374485

71 -172.750 97089 -175.215603-196.359605- 199.234 027

- 31.36 2 6 36111 5 24 6 83X H O H - -149.139-151.201X H F -

' t a n d a r d 3G bas is set , geome tr ies given in T a b l e V. S t a n d a r d 4-31 basis set.corresponding to $ i . In other words, the orbital energy of eachM O and th e electron density are partitioned in space in the sameproportional manne r. Equation 10 is strictly valid only when $,is an exact eigenfunction of the operator H . However, even atthe 3G level, gross deviations in electron density distribution andin the relative orbital energies are not expected, and since we areinterested in qualitativ e comparisons, eq 1 0 has been used as asimple alternative to evaluating the integrand of eq 9 at each pointin space.Plots showing differences in the spatial distribution of orbitalenergy can be easily constructed by weighting projected electrondensity plots for each M O with the o rbital energy, summing overoccupied MO s and forming differences as with Figure 1. Anexample of such a plot is shown in Figure 2, which can be regardedas a projected orbital energy difference (POE D) graph in analogywith the projected electron density difference (PE DD ) plot2j inFigure 1.Figure 2 is a useful supplem ent to PED D plots since the en-ergetic consequences of a charge redistribution can be more easilyevaluated. The main point is that the compen sation observed inthe PE DD plot also comes through in the PO ED plot. The orbitalenergy is nonadditive by 1.16 kcal and from Figu re 2 it is apparenttha t this degree of additivity in the total orbital energy is du e tocancelation of substantially larger positive and negative deviationsin different spatial regions. Each half of the grid in Figure 2integrates to about 1401 kcal.E. Energy Additivity and the Equivalent Group. Spat ial Dis-tribution of Kinetic Energy. Projected kinetic energy difference(PKE D) plots can be constructed by integrating

d T i = $iT$i d r ( 1 1 )along the z axis and forming differences between correspondingfragments. Since even the exact is not an eigenfunction of thekinetic energy operator, the kinetic energy density is not pro-portional to the electron density in the same volume element. Theproperties of th e kinetic energy a re of interest since it was shownvia the virial theorem'* tha t total energy additivity occurs to theextent that the kinetic energy is additive. The virial theorem doesnot apply to the individual regions corresponding to the Ar andN e fragments,%but it is still of in terest to examine the con tributionof each spatial region to the total kinetic energy. Since com-pensating shifts in electron density and orbital energy have beenpreviously noted, comp ensati ng differences in the P KE D plotswould also be expected.2s In Figu re 3 this expectation is realizedand it can be shown that the additivity in total kinetic energy( 4 . 1 0 kcal of the mean) is not due to a nearly constant distributionof kinetic energy over the Ar an d N e regions. Larg e positivechanges (-80 kca1)26 n the Ar area ar e canceled by nearly equal

(24) Bader has outlined conditions which are sufficient for dividing amolecule into 'virial" fragmen ts so that each fragment satisfies the virialtheorem (ref 16d).(25) Bader has also shown that the total kinetic energy within "virial"fragments (ref 24) can be uniquely defined, even though the kinetic energyat a given point is not necessarily unique. Bader has also shown that thekinetic energy of a volume element is a function of the charge density in thatelement so that the compensating pattern seen in PEDD plots should berepeated in the PKE D plots. The significance of the particular distributionrepresented by eq 11 is that it corresponds to the contribution of each volumeelement to the kinetic energy expectation value, ( $ i T $ i ) .

Figure 3. Projected kinetic energy difference plot for ArHNe' . Thekinet ic energ y is evaluated as a funct ion of the spat ial coordinates ac-cording to eq 11, and projected dif ferences are plot ted as in Figure 1.M aximum equa l s 2.216 au /bohr 2 .negative changes in the N e are a to give excellent overall additivity(0.1 kcal).F. Additive Total Energy Does Not Require Equivalent Groups.The conclusions are clear. Th e additivity seen in the calculatedtotal energy for ArH Ne + is not due to transferable groups of nearlyconstant electronic structure or to additivity of the orbital energyor of the classical electrostatic energy. The electronic environmentsof the Ar, Ne, and H regions are noticeably different in thehemistructural molecule and in the parents. These changes inelectronic structure account for energy differences one to threeorders of magn itude larger than the overall deviation from ad-ditivity. Additivity is observed because the changes in electroni cstructure are mutually compensating.It is important to place the differences in Figures 1-3 in properperspective. For example, the kinetic energy difference over theAr frag ment in Figure 3 is about 8 0 kcal. Since the total kineticenergy over the Ar fragment in ArHNe ' is -327 00 0 kcal, theperturba tion could be regarded as sma1L2' From this viewpoin t,

(26) T he integrations over the area of the grid are performed numericallyusing Gaussian quadratu re or Simpson's rule. The PKED function integratedover the Ar half of the grid gives 0.032169 au. The total kinetic energydifference on the Ar half is 4(0.0321 69) au, since allowance must be ma defor the PK ED function below the grid (factor of 2) and for two electrons perorbital. Th e PKE D function integrated over the N e half of the grid gives4. 03 19 88 a u, corresponding to a kinetic energy difference of -4(0.031988)au. Th e total deviation from additivity is given by 4(0.032169 - 0.031988)au = 0.45 kcal which is in reasonable agreement with 0.1 kcal determined bydirect calculation.(27) It was mentioned earlier that the energy difference between NeHNe+and ArHAr' amoun ts to over 495 OOO kcal, and one could legitimately wonderabout the applicability of perturbation arguments to such a large energychange . How ever, by employing a 'balancing" trick this huge difference canbe reduced. It is instructive to compare the energies of the isonuclear andisoelectronic structures [NeHN e' + Ar + Ar], [ArHNe' + Ar + Ne], and[ArHAr' + N e + Ne]. The total energies of the three structures are'3b-1295.930139, -1295.895290, and -1295,860773 au , respectively, and theenergy range is now about 44 kcal instead of 495 000 kcal. Expressed as afraction of the total energy, the 44 kcal perturbation is quite small: 10.006%.The changes in Mulliken population (Table 111) also indicate a small per-turbation: the variations are largest for the valence px orbitals of Ar (0.041e) and Ne(0.031 e) and for the H Is orbital (0.081 e). Based on two electrons,these changes amou nt to 4% or less. As one might expect from the smallpopulation variations, the changes in density matrix elements are also small.The largest change in diagonal elements is for the H 1s orbital (12%) and thechanges for the remaining elements are less than 4 %. The pa ttern is similarfor the off-diagonal elements. Overall, these results suggest that treatingNeHNe', ArHNe', and ArHAr' in terms of perturbatio n theory" is rea-sonable. A more comple te evaluation of the quantitative applicability ofperturbatio n theory is desirable and is currently in progress.

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Relationship between Energy Additivity and the Equivalent Group J . A m . C h e m. SOC., ol. 104, No . 10, 1982 2787Table V. Optimizeda Bond Lengths for Proton-Bo und Anions Table VI. Energy Addi t ivi ty (kcal). Proton-Bound Anions

rXH3 AH,C- Xc = H,N- Xc = HO- x c = 1:-x c =

XHCH,- 1.3784 1.2018 1.0747 1.0253XHNH,- 1.4516 1.27 13 1.1208 1.04 26XHOH- 1.4907 1.3306 1.1774 1.0760XHF- 1.5313 1.3750 1.2153 1.114Deviation of X-Y distan ce frommean of X-X and Y-Y distances. rC H = 1.1090 A , iHCH =101.01" (methyl hydrogens, C symmetry). T N H = 1.0310 A ,LHNH = 106.7" (all HNH angles). rOH = 0.9660 A , LHOH =104.5" (bridging HOH angle). Hydrogens o n fragm ents are alwaysin trans, staggered geometries where appropria te.

(+0 .0037)b (+0.0096)b (+0.0642)b(+ 0.0027)b (+0.0323)

(-o.oool)ba Standard 3G basis set.

Ar has a very similar electronic structure in ArHAr' and inAr HN e+ , and th e transferable group concept accounts for nearly99.98% of the total energy. As long as we do not worry aboutthe remaining 0.02%, the transferable group is a reasonable de-scription. However, it is fair to say that th e additivity seen forthe total kinetic energy of ArHN e' (0.1 kcal, Table 11) has littleor nothing to do with the degree of fragment transferability. The0.1 kcal is due to a near cancelation of large (-80 kcal) positiveand negative deviations in separate spatial regions. In this sense,the variations in the spatial distribution of kinetic energy aresubstantial, and additivity persists in spite of the breakdown intransferability.G. Variations in Total Energy Follow the Variations in KineticEnergy. It was shown I2 tha t th e kinetic energy will respond toa perturbatio n in a simpler fashion than th e individual potentialenergy components (V,,, V,,, an d V,). The fundamental reasonfor this behavior is that the kinetic energy change can be describedsolely in terms of th e chan ge in wave function since the kineticenergy operator do es not changeI2 for the type of perturbationrepresented by the hemistructural relationship. The potentialenergy variations involve changes in both the op erator an d wavefunction. The extra terms associated with the operator ensurea more complicated relationship between V,,, V,, and V,, thanfo r T.I2 The importance of understanding the behavior of thekinetic energy lies in the fact tha t if the hem istructural moleculecorresponds to an equilibrium geometry or transition sta te, thetotal energy ( E = T + V,, + V, + V,,) will also equal -T. Thisrelationship also requires th at E = V/2 where V = V,, + V, +V,. Satisfaction of these constraints requires that the complex itiespresent in the individual terms V,,, V,, an d V,, cancel away inthe sum. Since these restrictions do not apply to the orbital energy(E o= T+ V,, + 2V,) an d sin ce V, an d V, are generally unequaland no nadditive, the orbital energy w ill be less additive than E ,T, or V. These features of th e nuclear substitution theoryI2 comethrough in very striking fashion (Table 11).Proton-Bound First Row Anions

In Table IV , energies are reported for XHY - where X and Yare anions (H3C-, H2N-, HO-, F ) . The calculations have beenperformed using Pople's 3G and 4 -31G basis set. It is well knowntha t Hartree-Fock theory gives poor descriptions of anions28an dthat realistic computations must include correlation energy.Nonetheless, certain trends within the calculated properties clearlyillustrate some of the features predicted for hemistructuralmolecules.I2 Consequently, we feel that these results are wortha brief discussion.The energies reported in Table IV correspond to geometrieswhich have been carefully optimized at the 3-G level with respectto the X-H and H-Y distances. Internal coordinates of the Xan d Y fragments have not been optimized an d correspond to ou r

(28) W. A. Lathan, L. A. Curtiss, W. J. Hehre, J. B. Lisle, and J. A. Pople,Prog. Phys . Org. Chem. , 11, 17 5 (1974) .

X= H,N- X = HO- x = FXHCH,- -0.6" -2.7" -6.2'-0.6b +0.6b +2 .lbXHNH,- -1.lQ -3.5"+0.2b +3.1bXHOH- -1.P+1.3b

a Energy deviation of XHY- from mean energy of XHX- an dSame a s aHX-; 3G basis set; geometries listed in Table V.above, except calculationsar e performed a t th e 4-31 level.estimates of reasonable values. Th e complete fragment geom etriesare given in Table V. On e interesting result is that if the X-H-Ystructure contains adjacent a toms in the first row of the periodictable, then the energies show a high deg ree of additivity. Examplesare H2N-H-CH,-, HO-H-NH2-, and F-H-OH- where the de-viations from additivity are 0 .6 , l . , l .O kcal, respectively. Asthe X and Y fragments diverge further in structu re, additivitybegins to break down as seen in the sequence H2N-H-CH 3- (0.6kcal), HO -H-CH 3- (2.7 kcal), F-H-CH3- (6.2 kcal) (Table VI).The 3G basis set predicts the basicity order F > HO- > H3C-> H,N-, which is incorrect29 and is probably due to the pro-gressively poorer description of the cen tral at om of the anion w ithincreasing nuclear charge. However, it is interesting that thedeviations from the hemistructural geometry apparently correlatewith the electronegativity of the end atoms. In X-H-Y- forconstant Y, the X-H distance shortens and the H-Y distancelengthens as the difference in electronegativity between X andY increases. It is especially noteworthy th at the H coordinatesdeviate more from the hemistructural position than the X or Ycoordinates. For example , the 0-H dis tance in HO-H-Flengthens by 0.0379 A compared to HO-H-OH -, while the H-Fdistance shortens by 0.0380 8, ompared to F-H -F. As a result,the O-F distance is within 0.0001 8,of the mean 0-0 nd F-Fdistances. This behavior is somewhat analogousI2 to Johnston'spr oposa l o f bond or der conse r ~a t ion~~n transition states ofatom-transfer reactions, and has been predicted from an analysisof the nuclear forces at the hemistructural geometry.I2 Thispattern holds fairly well for X , Y = 0, F; N , 0;and C, N andprogressively breaks down as X, Y diverge further in structure.It is interesting to note that the deviations are greater for the X-Hand Y-H distances tha n for the X-Y distance, as predicted.12These basic results have also been observed when the calcu-lations are repeated at the 4-3 1 level. Full geometry optimizationsfor all systems have not yet been done, but the degree of energyadditivity for the h emistructural geometries is fully comparable.Similar results have been noted for other anions (e.g. , C H 3 0- ,with both 3-G and 4-31 basis sets. Hy drid e bound cations, in-cluding alkyl, cyclopropenyl,and azacyclopropeny l, have also beenexamined with similar results. The 3-G and 4-3 1 basis sets havebeen used in the present study as economic expedients. Energyadditivity is not substantially different in th e two basis sets, andthe 3-G m inimal basis set undoubtedly underestimates the changesin electronic structure of the various fragments. As a result, theconclusion that ad ditivity is due to m utua l cancelation of positiveand negative deviations in different spatial regions is unlikely torequire significant revision when these calculations are repeatedusing more refined theo retical techniques. Howe ver, from previoustheoretical considerations,12 here ar e strong indications that thevirial theorem and the Hellmann-Feynman theorem play a keyrole in this beh avior.I2 Consequently,we are interested in obtainingwave functions which satisfy the virial and Hellmann-Feynmantheorems and which yield energies close to the H artree-Fock limit.It will also be impo rtant to include correlation effects. Th e initialstages of this work are now underway with P R O M E T H E U S x.

CH,CH,O- , HC CC- , NE C- , HCzC- C H2- , NEC- CHI- )

(29) The order calculated in t he 4-31 basis se t is reversed: H,C- > H 2N-(30) H . S. Johnston an d C. Parr , J . A m . Chem. SOC., 5 , 2544 (1963).> HO- > F.

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2788 J . Am . Chem. Sac.. Vol. 104. N o . 10 , 1982Table VII. Calculated Energy Comp onents for Proton-Bound Anions

Murdoch and Moynolr

E , , au T , au Vne. au V e x ,au v,,, au V,,, au E , a"b~ H F- -104.654026 196.362699 -539.789251 -21.SX7721 1409699XS 77677591 -196 '267699.- . . -,...~~ ~~~~~el:OH- -93.900366 172.752830 -485.994399 -19.941 768 129.612370 30.816317 -172,754651dHOHOH- -83.011 915 149.142551 -432.450 969 -18.306 672 118.454 923 34.017 615 -149.142 51A," kcal -42.291 0.129 78.884 2.151 -62.803 -19.632 -1.271A represcentst h e deviation of the calculated component of THOH- from the mean of thc corresponding compancnts for THT- andHOHOH-. Allvaluesareinkcal. b r ~ H = 2 . L 1 3 6 2 3 2 4 u g ; r i = 0 . 9 9 6 0 1 2 6 5 8 . r~H=2.11362324uB;rH,,=2.23586809u~;roH=1.834418 91 ug ;LHOH= 103.286", = 1.00010 8 850. d r H O = 2.2358 6 8 0 9 ~ ~ ; r ~ ~1.834418091 wg;LHOH = 103.286";q=0.995 142 069. T h e exponents for FHOH- have been taken from the corresponding orbitals in the symmetrical structures, exccpi for the

bridging proton whose exponents are the average of the corresponding exponen ts for the symmetrical structures. The scaling factor of1.000 108 850 o r FHOH- indicates that this "hemistructural" basis set was scalcd by a factor of (1,000 08 850)' to give the energies repor t-cd abovc

Figure 4. Projected electron density difference plot for F-H-OH-.Maximum equals 0.00517 electron/bohr'.As mentioned earlier, F - H a H - shows a high degree of energyadditivity (-1 kcal). Before closing, it would be worthwhile toexamine this behavior somewhat further. In Table VII, the en-ergies for F- H- F, F -H-OH - and HO-H-O H- a re l isted. Thegeometries of th e two symmetrical s tructures were carefully op-timized (3G basis) with respect to X-H an d H-Y distances. T hescaling procedure was then applied to produce the scaling factorsand new bond lengths reported in Ta ble VII. Th e unsymmetricals t ructure F - H a H - was sca led, mainta ining the geometry a t thehem i-struc tural positions defined by the final geom etries of thescaled F-H -F and HO-H-OH-. Since the geometry of F-H-OH- is not varied during t he scaling procedure, the virial theoremis not exactly satisfied, but the error is small (- 1 kcal). Theindividual potential energies, Vnn, ,,, an d V,, are substantiallynonadditive (Z(t80 kcal). while E, T, and Va re additive to within

I .4 kcal. Th e additivity of the orb ital energy is relatively poor(-42 kcal). Th e exchange energy is additive to within 2.1 kcal,so hat the classical electrostatic energy is somewhat less additivethan E, T , and V. Th e overall patter n is similar to that observedfor ArHNe+.T he PEDD, POED. and PKED plots for F-H-OH- ar e shownin Figures 4-5. As with A rHN e+. the degree of additivity dependson mu tual cancelation of lar ge positive and negati ve deviationsin separate spatial regions. For example, the integrated orbitalenergy difference (Figu re 5 ) over the F fragment yields about -132kcal, while the corresponding quantity over th e OH fragment gives-83 kcal. Th e combined sum is roughly -4 9 kcal in reasonableagreement with the exact 3G scaled value of -42 kcal. Thediscrepancy is due to the numerica l in tegra t ion procedure(Simpson's rule) used for integrating the grids. Over the sam eregions, th e integrated kinetic energy difference (Fig ure 6) is -50kcal (F ) and +62 kcal (O H). The in tegrated to tal (2 kcal) is inreasonable agreem ent with the direct value (0.1 kcal). Onc e again,it is seen that additivity depen ds on mutual cancelation ofdifferentchanges in electronic structu re rather th an similarity of fra gme nt

Figure 5. Projected orbital energy difference plot for F-H -OH-. (a )Core orbitals, maximum equals 0.302 au/bohr'. (b) Valence orbitals,maximum equals 0.027 u/bohr'. The integration mentioned i n the textrefers to the com bined sum of core and valence orbitals.

Figure 6.Maximum equals 0.735 u/bohr2.Projected kinetic energy difference plot for F-H-H-.

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J . Am . Chem.SOC. 982, 104, 2789-2796 2789electronic stru cture in different environments.The Relationship between Structural Transferability andEnergy Additivity

Transferability has o ften been used t o rationalize observationsof additivity.16 For exam ple, Rothenberg3 has shown th at o rbitalsof methane an d ethane can be localized an d tha t one-electronproperties of the localized C-H orbitals vary by less than 2-3%.Th e kinetic energy associated with a C-H orbital in methan e is0.8690 au while the corresponding quantity for ethane in thestaggered conformation is 0.886 a ~ . ~ hese values are within2% of each other, but the difference corresponds to about 10 kcal.However, additivity holds to a much higher degree for unbranchedsaturated hydrocarbons. For exam ple, th e to tal e n e r g i e ~ ~ * * ~ ~ ~fpropane (-1 18.09211 au), ethane (79.11582 a u), and m ethane(-40.13978 au ) are additive to within 0.08 kcal (4-31 G basis set).This additivity is not appreciably dependent on basis set and isseen for minimum basis sets as ell.^*^^^ This comparison suggeststha t th e degree of transferability seen with localized orbitals isnot sufficient to accoun t for the additivity in total energy.have reproducedorbital energics for methane, ethan e, and propane to w ithin 0.1%by transferring Fock m atrix elements derived from a basis set oflocalized orbitals. Thi s comparison emphasizes tha t transferabilityis a reasonable description of the total wave functions, but the

In a sim ilar vein, Degand, Leroy, and

(31) S. Rothenberg, J . A m . Chem. Soc., 93, 68 (1971).(32) (a) L. Radom, W. J . Hehre, and J. A. Pople, J . A m . Chem.Soc., 94 ,2371 ( 197 2). (b) The 3G energies of methane, ethane, and propane are-39.72686, -78.30618, and -1 16.88580 au (L . Radom and J. A. Pople, ibid.,92, 4786 ( 1970) ). The additivity is within 0.10 kcal.(33) (a) Ph. Degand, G. eroy, and D. Peeters, Theor. Chim.Acta, 30 ,243 (1 973 ). (b) The ab initio orbital energies for methane, ethane, andpropane are -28.3 284, -55.3 636, and -82.5678 au. The orbital energy ofethane is 53 .02 kcal above the mean orbital energy of methane and propane.In the same basis set (r ef 33a), the transferable matrix element procedure gives-28.3000 , -5 5.37980, and -82.521 0 au . The absolute values agree reasonablywell, but the orbital energy for transferableethane is only 19. 26 kcal abovethe mean.

degree of additivity differs from the corresponding ab initiocalculation by over 30 kcaL3jbAnother method based on transferable Fock matrix elementsis the SAMO technique34awhich has been applied to ethane,propane, and butane. Th e direct ab initio calculation gives additivetotal energies to within 0.10 kcal, while the S A M O energies areon ly ad ditiv e t o 3 0 k ~ a 1 . j ~ ~Conclusions

In the previous paper,* it was demonstrated analytically tha tenergy additivity does not require wave function transferabilityor constant electronic structure. In the current paper, exampleshave been presented which show a high degree of additivity intotal energy, but not because of constant electronic structure.Lar ge positive kinetic energy deviations (-80 kcal) in one spatialregion are offset by compa rable negativ e deviations in othe r regionsto give a total k inetic energy which is additive to 0.1 kcal. Thepresent work has shifted the focus from grou p transferability tomutual cancelation as the root of energy additivity.

Acknowledgment. This work was supported in part by a grantfrom the donors of the Petroleum Research Fund , administeredby the A merican Chemical Society, by the Pennwalt CorporationGra nt of Research Corporation, by a USPHS Biomedical Re-search Gran t (4-521355-24739), by N SF , and by continued as-sistance from the University Research Committee (UCLA).J.R .M . also acknowledges a Regents Junior Faculty Fellowship(1978-1979) a nd a UC LA Faculty Caree r Development Award(1979-1980). Th e auth ors thank Professors William McMillan,William Ge lbart, and Daniel Kivelson and D r. Peter Ogilby forextensive discussions. Th e autho rs would also like to thankProfessors Joel Liebman and Paul von Rague Schleyer for pre-prints and helpful comments.

(34) (a) J . E. Eilers and D. R. Whitman, J . Am . Chem.Sor., 95, 2067(19 73) . (b) The ab initio energies of ethane, propane, and butane are-78.8196, -1 17.6776, and -156.535 3 au. The correspondingSAM O energiesare -78.8049, -117.6275, and -156.5479 au .

Catecholate and Phenolate Iron Complexes as Models for theDioxygenasesRobert H. Heistand 11, Randall B. Lauffer , Erol Fikrig, and Lawrence Que, Jr.*Contribution fr om the Department of Chem istry, Baker La boratory , Cornel1 University, Ithaca,New York 14853. Received September 28 , 1981

Abstract: The syntheses and physical properties of a series of Fe(salen)X and Fe(sa1oph)X complexes where X is phenolateor catecholate ar e reported. Magnetic susceptibility measurements as well as electronic, infrared, and N M R spectra indicatethat the catecho late in F e(salen)catH behaves very much like a phenolate and is concluded to be monodentate in its coordinationto the iron. Th e abstraction of a proton from Fe(sa1en)catH results in an anionic complex, [Fe(salen)cat]-, with markedlydifferent properties; the catecholate in this complex is chelated. Both monodentate and chelated catecholate complexes arehigh-spin ferric, demonstrating that catechola te coordination to a bis(pheno lato)iron(III) complex does not result in the reductionof the ferric center. This is in agreement with observations made on dioxygenase-substrate complexes. I n addition, studieson a series of Fe(salen)X complexes where X is phenolate, thiophenolate, benzoate, and catecholate show that the dominan tsalen-to-Fe(II1) ch arge-trans fer inte ractio n is modu lated by the coordination of these ligands. Comparisons with correspondingdioxygenase complexes show that th e tyrosin ate-to-iron(II1) charge-transfer interactio ns are similarly affected , thus indicatingtha t the salen ligand provides a reasonable approximation of the iron environment in the dioxygenases.

Th e intera ction of molecular oxygen with metalloenzymes is acids, catalyzed by the nonh eme iron enzymes catechol 1,2-di-curren tly an are a of considerable activity. On e interesting oxygenase and protocatechuate 3,4-di 0xyg enase .~-~pectroscopicreaction is the dioxygena tion of catechols to yield &,cis-muconic stud ies on these enzym es show the active site iron to be a mo-(1) Hayaishi, O ., Ed. Molecular Mechanisms of Oxygen Activation; (2) Nozaki, M., in ref l, Chapter 4.(3 ) Que, L., Jr . Srruct . Bonding (Berlin) 1980, 40 , 39-72.cademic Press: New York, 1974.

0002-7863/82/1504-2789$01.25/00 1982 American Chemical Society