— — < < Natural Resonance Theory: II. Natural Bond Order and Valency E. D. GLENDENING, U F. WEINHOLD Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706 Received 15 July 1997; accepted 1 November 1997 ABSTRACT: Resonance weights derived from the Natural Resonance Theory Ž . NRT , introduced in the preceding paper are used to calculate ‘‘natural bond order,’’ ‘‘natural atomic valency,’’ and other atomic and bond indices reflecting the resonance composition of the wave function. These indices are found to give Ž significantly better agreement with observed properties empirical valency, . bond lengths than do corresponding MO-based indices. A characteristic feature of the NRT treatment is the description of bond polarity by a ‘‘bond ionicity’’ Ž . index resonance-averaged NBO polarization ratio , which replaces the ‘‘covalent-ionic resonance’’ of Pauling-Wheland theory and explicity exhibits the complementary relationship of covalency and electrovalency that underlies empirical assignments of atomic valency. We present ab initio NRT applications Ž . to prototype saturated and unsaturated molecules methylamine, butadiene , Ž . Ž polar compounds fluoromethanes , and open-shell species: hydroxymethyl . radical to demonstrate the numerical stability, convergence, and chemical reasonableness of the NRT bond indices in comparison to other measures of valency and bond order in current usage. Q 1998 John Wiley & Sons, Inc. J Comput Chem 19: 610 ] 627, 1998 Keywords: natural resonance theory; resonance theory; valency; bond order *Present address: Department of Chemistry, Indiana State University, Terre Haute, IN 47809 Correspondence to: F. Weinhold See the Editor’s Note for the first paper in this series, p. 593, this issue. ( ) Journal of Computational Chemistry, Vol. 19, No. 6, 610 ]627 1998 Q 1998 John Wiley & Sons, Inc. CCC 0192-8651 / 98 / 060610-18
Transcript
— —< <
Natural Resonance Theory: II. NaturalBond Order and Valency
E. D. GLENDENING,U F. WEINHOLDTheoretical Chemistry Institute and Department of Chemistry, University of Wisconsin,Madison, Wisconsin 53706
Received 15 July 1997; accepted 1 November 1997
ABSTRACT: Resonance weights derived from the Natural Resonance TheoryŽ .NRT , introduced in the preceding paper are used to calculate ‘‘natural bondorder,’’ ‘‘natural atomic valency,’’ and other atomic and bond indices reflectingthe resonance composition of the wave function. These indices are found to give
Žsignificantly better agreement with observed properties empirical valency,.bond lengths than do corresponding MO-based indices. A characteristic feature
of the NRT treatment is the description of bond polarity by a ‘‘bond ionicity’’Ž .index resonance-averaged NBO polarization ratio , which replaces the
‘‘covalent-ionic resonance’’ of Pauling-Wheland theory and explicity exhibits thecomplementary relationship of covalency and electrovalency that underliesempirical assignments of atomic valency. We present ab initio NRT applications
Ž .to prototype saturated and unsaturated molecules methylamine, butadiene ,Ž . Žpolar compounds fluoromethanes , and open-shell species: hydroxymethyl
.radical to demonstrate the numerical stability, convergence, and chemicalreasonableness of the NRT bond indices in comparison to other measures ofvalency and bond order in current usage. Q 1998 John Wiley & Sons, Inc.J Comput Chem 19: 610]627, 1998
Keywords: natural resonance theory; resonance theory; valency; bond order
*Present address: Department of Chemistry, Indiana StateUniversity, Terre Haute, IN 47809
Correspondence to: F. WeinholdSee the Editor’s Note for the first paper in this series,
p. 593, this issue.
( )Journal of Computational Chemistry, Vol. 19, No. 6, 610]627 1998Q 1998 John Wiley & Sons, Inc. CCC 0192-8651 / 98 / 060610-18
NATURAL RESONANCE THEORY: II
Introduction
n the preceding paper, we introduced a ‘‘natu-I Ž .ral resonance theory’’ NRT formalism thatallows practical calculation of effective weights� 4w of resonance structures a for a delocalizeda
‘‘resonance hybrid.’’ Unlike the Pauling-Wheland1
Žtreatment based on a valence bond superposition.assumption for the wave function C , the NRT
approach is based on the one-particle reduced den-ˆ 2sity matrix G,
ˆ U XŽ . Ž .G s N C 1, 2 . . . N C 1 , 2 . . . N d2 . . . dNHŽ .1.1
and its approximation in resonance-averaged form
ˆ ˆ Ž .G , w G 1.2aÝ a aa
Ž .w G 0, w s 1 1.2bÝa aa
ˆwhere G is an idealized density matrix for reso-aˆnance structure a. Each G is constructed froma
quantities that can be obtained from natural bondŽ . 3, 4orbital NBO analysis using a wave function of
arbitrary form, and the resonance weights w area
chosen by a least-squares variational criterion toˆdescribe the actual G in optimal fashion. In the
framework of ab initio molecular orbital theory,Ž .the NRT expansion 1.2 was shown to reproduce
the true delocalization pattern with high accuracyŽtypically 80]90% or more of the delocalizationdensity, itself a small fraction of the total electron
.density for a variety of molecules and basis setlevels.
Ž .Central to the resonance theory RT descriptionof a delocalized molecule is the concept of bond
Žorder b ‘‘weighted average’’ number of bondsAB.between A and B , which can be correlated with
empirical bond length, bond energy, and otherproperties.5 In its most basic form. the formal bondorder between atoms A and B is defined as6
Ža. Ž .b s w b 1.3ÝAB a ABa
Ža. Ž .where b is the integer number of bonds in theABidealized Lewis-type structural formula for reso-nance structure a. In accordance with the interme-diate character of the resonance hybrid, bond
properties corresponding to fractional b are ex-ABpected to be intermediate between those for ideal-ized integer bond orders of completely localizedstructures.
A prototype example of resonance mixing isŽ .furnished by the formamide molecule NH CHO ,2
which is expected to be primarily described by theŽ .parent resonance structure I , with strong admix-
Ž .ture of the charge-separated structure II ,
O H Oy Hq6
6¨ Ž .C N C N 1.4
H H H HI II
and lesser contributions from other resonancestructures. From empirical considerations, Paul-ing7 estimated the relative contributions of struc-tures I and II to be about 60% and 40%, respec-tively, leading to formal bond orders
Ž .Ž . Ž .Ž .b s 0.60 2 q 0.40 1 s 1.60CO
Ž .Ž . Ž .Ž . Ž .b s 0.60 1 q 0.40 2 s 1.40 1.5CN
corresponding to significant C5N double bondwcharacter. For comparison, NRT analysis of an
ab initio MP2r6-31GU wave function for for-Ž .mamide optimized C geometry gives leadings
weights w s 59.29% and w s 27.87% and bondI IIorders b s 1.7277 and b s 1.3289, in crudeCO CN
xagreement with Pauling’s analysis. As is wellknown,8 Pauling’s recognition of this partial bCN
Ždouble-bond character leading to planar peptide.groups played an important role in the discovery
of the a-helix in proteins. Pauling’s monograph1Žb .
presents many other illustrations of how RT andbond order concepts can be used to rationalizemolecular properties in a simple, chemically intu-itive manner.
A principal aim of the present paper is to pre-sent ab initio NRT weightings and bond indices forprototypical cases of resonance mixing, in order totest their consistency with general RT concepts,determine their general numerical characteristicsŽstability and convergence with respect to changes
.in level of theory, NRT program parameters, etc. ,and establish the degree of correlation with empir-ical valency and geometrical parameters. As waspointed out in the preceding paper, Pauling-Whe-land RT encountered formal and practical difficul-ties which effectively led to its demise for quanti-tative ab initio purposes, and formulas such asŽ .1.3 were eventually replaced by MO-based mea-sures of bond order in which ‘‘resonance struc-
JOURNAL OF COMPUTATIONAL CHEMISTRY 611
GLENDENING AND WEINHOLD
tures’’ and ‘‘resonance weights’’ play no directrole. Full assessment of the NRT method thereforeinvolves comparisons with alternative MO-basedbond indices that are widely employed in currentab initio and semiempirical studies.
The plan of the paper is as follows: We describethe NRT formulation of bond order and atomicvalency in the next section, showing how it differsfrom Pauling-Wheland bond order in the treat-ment of ‘‘covalent-ionic resonance.’’ Next, alter-native MO-based measures of bond order andvalency are reviewed. Thereafter, selected NRTapplications are presented to ‘‘ordinary’’ saturatedŽ . Ž .methylamine , unsaturated butadiene , polar co-
Ž .valent fluoromethanes , and open-shell speciesŽ .hydroxymethyl radical to illustrate characteristicnumerical features of the method in typical cases
Žof weak or moderate delocalization. Cases of ex-treme delocalization or ‘‘anomalous’’ bonding are
.described in the following paper. The final sectionpresents a summary and conclusions.
Bond Order and Atomic Valency:NRT Definitions
Given the natural resonance weights wŽN., asa
determined from the NRT variational functionŽ .preceding paper , one obtains the ‘‘natural bondorder’’ bŽN. by direct analogy with the classicalAB
Ž .expression 1.1 ,
nRSŽN . ŽN . Ža. Ž .b s w b 2.1ÝAB a AB
a
Ža. Ž .where b is the integer 0, 1, 2 . . . number ofABA—B bonds in the structural formula for the athnatural Lewis structure. Because alternative MO-
Ž .based methods see next section generally do notinvolve explicit prediction of resonance weightsw , the comparison of NRT results to these meth-a
ods must be carried out directly in terms of bondorder values, as will be numerically illustratedbelow for several prototype molecules.
Closely related to formal bond order is the no-tion of atomic valency9 V of atom A. This can beAregarded as a measure of total bond-making capac-ity exhibited by atom A in the molecule, obtainedby summing the formal bond orders over all possi-ble atoms B. Accordingly, we define the ‘‘naturalatomic valency’’ V ŽN. asA
ŽN . X ŽN . Ž .V s b 2.2ÝA ABB
where the prime denotes omission of the termA s B. Qualitatively speaking, the valencies V ŽN.
Afor elements of groups IV]VII are expected tomatch the respective empirical values 4, 3, 2, 1Ž . Ž8 y group number in all but exceptional hyper-
.valent, hypovalent cases.Ž . Ž . wThe definitions 2.1 and 2.2 in conjunction
Ž .xwith the positivity property 1.2b intrinsicallyprevent the appearance of negative bond order oratomic valency in the NRT formalism.10
ŽN . ŽN . Ž .b G 0, V G 0, all A, B 2.3AB A
The limiting values bŽN. s 0 and V ŽN. s 0 are at-AB Atained in the limit of vanishing bonding interac-
Ž .tions e.g., two He atoms at infinite separation .Weak ‘‘nonbonded interactions’’ between atoms indifferent molecular units are reflected in small
Ž ŽN. .bond orders 0 - b g 1 between formally non-ABŽ .bonded atoms. Whereas the quantities 2.1 and
Ž .2.2 are expected to exhibit near-integer values inordinary compounds, these formulas are intrinsi-cally free to represent smoothly the transition be-tween distinct limiting bond patterns.
As was remarked in the preceding paper, ourformulation differs markedly from that of Pauling-
Ž .Wheland RT and related MO-based methods inits treatment of covalent-ionic resonance and bondionicity. However, given the total NRT bond orderŽ . ŽN.or ‘‘coordination number’’ b between twoABatoms, one can readily define the fractional ioniccharacter i as the resonance-weighted averageAB
w i Ža.Ý a ABa Ž .i s 2.4AB bAB
where iŽa. is the corresponding fractional ionicABcharacter in resonance structure CŽL., defined froma
wthe NBO polarization coefficients c and c cf. Eq.A BŽ . x2.9 in the preceding paper as
2 2c y cA BŽa. Ž .i s 2.5AB 2 2c q cA B
Ž .averaged, if necessary, over multiple bonds . Interms of the fractional ionic character, one canformally partition the total bond order bŽN. into itsAB
Ž .‘‘electrovalent’’ ionic and ‘‘covalent’’ contribu-tions, eŽN. and cŽN., respectively:AB AB
ŽN . ŽN . Ž .e s b i 2.6AB AB AB
ŽN . ŽN . Ž . Ž .c s b 1 y i 2.7AB AB AB
VOL. 19, NO. 6612
NATURAL RESONANCE THEORY: II
so that
ŽN . ŽN . ŽN . Ž .b s e q c . 2.8AB AB AB
Insofar as the NRT covalent bond order cŽN. dimin-ABishes as the ionic character of the bond increases,this quantity should be most directly comparableto the Pauling-Wheland bond order bŽPW.,AB
ŽPW. ŽN. Ž .b , c , 2.9AB AB
which is considered to vanish in the ionic limit.Ž .By analogy with 2.2 , one can also formally
sum the NRT electrovalent and covalent bond or-ŽN. ŽN. Žders e and c over all bonded atoms B whetherB AB
.in ionic, dative, or covalent sense to give the totalelectrovalency EŽN. and covalency C ŽN. of atom A.9A A
ŽN. X ŽN. Ž .E s e 2.10ÝA ABB
ŽN. X ŽN. Ž .C s c 2.11ÝA ABB
Ž . Ž . ŽN.As is shown by 2.2 and 2.8 , the full valency VAŽis just the sum of its electrovalent ionic, ‘‘un-
. Ž .shared’’ and covalent ‘‘shared’’ contributions,
ŽN. ŽN. ŽN. Ž .V s E q C . 2.12A A A
Ž .For cases of low ionic character i g 1 , the fullAB
valency V ŽN. and covalency C ŽN. will approxi-A Amately coincide, but, in more ionic cases, only thelatter quantity corresponds to ‘‘valency’’ V ŽPW. inA
Ž .the Pauling-Wheland or related MO-based sense,
ŽPW. ŽN. Ž .V , C 2.13A A
Ž . Ž .The distinction drawn in 2.10 ] 2.12 betweenthe NRT definitions of electrovalency, covalency,
Ž .and full valency and the associated bond ordershave important consequences for comparisons withempirical properties. It is generally expected thatreduced bond order should be associated with
Žbond weakening i.e., with increase of bond length,reduction of bond energy, red-shifting of vibra-
. 5tional frequencies, etc. . However, covalent bondŽN. Ž .order c , as defined in 2.7 , can exhibit the oppo-AB
site trend in variations that hold the total bŽN.AB
constant, because it is known that a bond of higherŽ ŽN..ionic character and, thus, of lower c is oftenAB
strengthened with respect to analogous apolar11 Žbonds involving the same atoms a trend re-
flected in the characteristically high enthalpies offormation and high melting and boiling points of
.ionic compounds . Thus, an attempt to correlatebond lengths with cŽN. alone may lead to inconsis-AB
Žtencies the bond length sometimes increasing andsometimes decreasing with cŽN. or with the closelyAB
ŽPW..related b , whereas the full NRT bond orderABbŽN. should exhibit a more consistent correlationABwith empirical bond properties as ionic charactervaries. Similarly, neither the electrovalency EŽN.
Anor covalency C ŽN. can be expected to match theAempirical valency V Žemp., except in extreme ionicAor covalent limits, whereas the full NRT valencyV ŽN. should generally be in good agreement withAV Žemp. in compounds of widely varying ionic char-A
Ž ŽN. ŽN..acter. Whereas the partial bond orders e , cAB ABŽ ŽN. ŽN..and valencies E , C are introduced for com-A A
Žparison with Pauling-Wheland and related MO-. ŽN. ŽN.based values, only the full b , V are expectedAB A
to exhibit satisfactory correlations with empiricalproperties over a wide range of ionicity.
Alternative Measures of Bond Order
The difficulties of evaluating Pauling-Whelandresonance weights and bond orders by practicalquantum-chemical means have led to suggestionsfor alternative ‘‘bond order’’-like quantities, partic-ularly in the MO framework. These bond indicesare routinely provided by popular electronic struc-ture programs and have largely supplanted theolder RT values for quantitative purposes.
In the framework of simple minimal-basis piMO theory, Coulson12 originally suggested that ameasure of bond order is given by
occ
Ž .P s 2 c c 3.1ÝAB iA iBi
Ž .where c is the coefficient of AO x on atom AiA Ain the ith occupied molecular orbital f i
Ž .f s c x . 3.2Ýi iA AA
Ž .In an orthonormal basis, the right side of 3.1 is anˆŽ . ² < < :off-diagonal element G s x G x of theAB A B
one-particle density matrix between AOs x andAx , equivalent in this case to the Fock-Dirac den-B
Ž .sity matrix P ‘‘bond-order matrix’’ ; in the moregeneral case, these matrices are related by
Ž .G s SPS 3.3
where S is the AO overlap matrix. More generally,Ž .the LCAO-MO expansion 3.2 includes multiple
� 4AOs x on each center, and the analogous partialrbond order P must be summed over AOs on eachrs
JOURNAL OF COMPUTATIONAL CHEMISTRY 613
GLENDENING AND WEINHOLD
atom to give the total A—B bond order bŽMO.,AB
on A on BŽMO. Ž .b s P . 3.4Ý ÝAB rs
r s
Ž .We refer to 3.4 as the ‘‘MO bond order’’ toŽ .distinguish it from the RT-based quantity 1.1 .
Ž .Although the MO bond order 3.4 achieves theŽ ŽMO.expected classical values e.g., b s 2 for ethy-AB
.lene in limiting cases, it differs significantly fromthe RT value in other instances. In benzene, forexample, elementary MO theory leads to bŽMO. sCC
2 121 for each CC bond, which can no longer be3
easily related to a weighted average of plausibleresonance structures. Moreover, the summed bondorders lead to ‘‘atomic valency’’ values
ŽMO. X ŽMO. Ž .V s b 3.5ÝA ABB
that deviate markedly from the expected integer1ŽMO.Žvalues e.g., V s 4 for each C atom of ben-C 3
. ŽMO.zene . A further difference is that the total bABŽ .or the individual orbital contributions P mayrs
Ž .assume negative values, whereas 1.1 is intrinsi-cally positive.
Wilberg13 originally introduced an alternativebond index
on A on BŽW. 2 Ž .b s P 3.6Ý ÝAB rs
r s
as a sum of squared density matrix elements. The‘‘Wiberg bond index’’ bŽW. is intrinsically positiveABand is often in better agreement with classical RTvalues. The Wiberg bond index has been revivedin many guises,14 sometimes with confusing orconflicting terminology.15 Wiberg originally pro-
Ž .posed the definition 3.6 in the framework ofsemiempirical MO theory, where the AO basis is
wŽ . ximplicity orthonormal S s d , but in ars rsnonorthogonal basis one might modify this for-mula by inserting various factors of overlap16 ororthogonalizing the AOs in various ways.17 A par-ticularly straightforward ab initio implementationof the Wiberg formula involves use of natural
Ž .atomic orbitals NAOs , an orthonormal set whoseproperties have been shown to resemble closelythose assumed in semiempirical MO treatments.18
Such a ‘‘Wiberg NAO bond index’’ is routinelycalculated in the general NBO program as a gen-eral guide to bond order, so we employ this spe-cific form of the Wiberg bond index in the numeri-cal comparisons in the following section. As anexample of related nonorthogonal AO-based bond
indices, we employ the ‘‘Mayer-Mulliken bondorder’’16 in which P is replaced by PS in theWiberg formula.
Reed and Schleyer19 introduced a novel mea-sure of bond order based on the form of natural
Ž .localized molecular orbitals NLMOs in the NAOŽRS. Žbasis. The Reed-Schleyer bond order b denotedAB
.‘‘NPArNLMO bond order by Reed and19 .Schleyer is the sum over occupied NLMOs
LMOsŽRS. Ž .b s b 3.7aÝAB iAB
i
of partial bond orders b defined asiAB
Ž . Ž . Ž .b s min n , n ? sgn S 3.7biAB iA iB iAB
where n is the number of electrons on atom A iniAŽNLMO i based on the squared LC-NAO coeffi-
.cients c in the NLMO ,ij
on A2 Ž .n s c 3.7cÝiA ij
j
Ž .and sgn S s "1 is a sign factor related to theiABŽ . ŽRS.overlap pattern. As 3.7b indicates, b acquiresAB
contributions only from shared occupancy on AŽ .and B the lesser of n and n and should thusiA iB
be most directly comparable to the covalent bondŽN. Ž .order c , eq. 2.7 ,AB
ŽRS. ŽN. Ž .b , c . 3.8AB AB
20 Ž .More recently, Cioslowski and Mixon CMintroduced a definition of ‘‘covalent bond order’’in the framework of Bader’s density partitioningmethod.21 The CM bond order bŽCM. is based on aABWiberg-like formula using integrated electron den-sities from atomic ‘‘basins’’ defined by Bader’szero-flux criterion. The novel feature of the CMbond order is its relation to a configuration spaceŽ .rather than Hilbert space or orbital description ofatoms. Some values of bŽCM. for C—F bonds ofABfluoromethanes were recently obtained by Wibergand Rablen,22 and will be compared to NRT andMO-based indices below.
Brown and coworkers23 introduced a related‘‘bond valence,’’ which is primarily an empirical
Žindex essentially, a way of reexpressing empiricalbond length to predict atomic valencies for ionic
.species and is not considered further here. Stillanother group of bond order formulas are basedon permutation-algebraic or graph-theoretic meth-ods.24 These formulas are primarily topological in
VOL. 19, NO. 6614
NATURAL RESONANCE THEORY: II
nature, are restricted to idealized planar networks,and therefore do not take account of geometryvariations, polarity differences, and other molecu-lar details. Because these methods are not easilyrelated to ab initio quantities, we do not considerthem further in this paper.
Illustrative Numerical Applications
SATURATED MOLECULES: METHYLAMINE
We first illustrate the NRT algorithm for a sim-Ž .ple saturated molecule, methylamine CH NH ,3 2
with the atom numbering shown below.
H 7H6-
¨ Ž .N C 4.1- 1 2
H3 H5H 4
This case of weak resonance delocalization willserve to demonstrate the ability of NRT analysis todescribe quantitatively subtle details of delocaliza-tion beyond the usual qualitative RT framework.
To compare weights and bond orders for differ-ent theoretical levels, we fix the molecule in ideal-
Ž . 25ized Pople-Gordon PG geometry, with equal˚ ˚Ž . Ž .bond lengths R 1.01 A and R 1.09 A , andNH CH
tetrahedral valence angles. Table I shows the lead-ing NRT weights for three distinct basis levels of
w Žuncorrelated MO theory HFr3-21G split valence. U Ždouble zeta , HFr6-31G double zeta plus polar-
. UU Žization and HFr6-311qq G triple zeta withextended diffuse and polarization sets on all
.xatoms and two levels of correlation treatmentw Ž .MP2 second-order Møller-Plesset theory , QCISDŽquadratic configuration interaction with single
. Uand double excitations , both at 6-31G basisx 26level . As Table I illustrates the leading delocal-
TABLE I.( )NRT Resonance Weights w for Methylamine at Several Levels of Ab Initio Uncorrelated MO and Correlatedi
( )corr Theory, All at Idealized Pople-Gordon Geometry.
( )NRT resonance weights w %a
a b c d ei Structure MO-1 MO-2 MO-3 corr-1 corr-2
HH-
¨1. N C 97.88 97.67 97.45 92.48 91.00-H
H H
HH-
+2. N C 0.80 0.83 0.89 1.95 2.21-H yH H
+HH-
¨ ( )3. N C 2 0.26 0.31 0.33 1.31 1.65H
y HH
yHH-
¨ ( )4. N C 2 0.26 0.31 0.31 1.29 1.60H
+ HH
yHH-
+ ( )5. N C 2 0.14 0.12 0.19 0.18 0.14-H
H Ha HF / 3-21G.b HF / 6-31G*.c HF / 6-311 ++ G**.d MP2 / 6-31G*.e QCISD / 6-31G*.
JOURNAL OF COMPUTATIONAL CHEMISTRY 615
GLENDENING AND WEINHOLD
ization correction arises from the ‘‘double bond]nobond’’ resonance structure
HH-
q Ž .N C 4.2-H
yH H
associated with the n ª s U NBO interaction ofN CH 5
the nitrogen lone pair with the antiperiplanar s UCH
antibond. It can be seen that the three MO resultsagree closely, showing the excellent stability andconvergence of the NRT method with respect tobasis expansion. The two correlated treatments give
Žappreciably higher resonance delocalization about1% higher for each of the three most important
.delocalization structures , and this tendency ofelectron correlation to enhance delocalizationseems to be a rather general effect. There is goodoverall agreement in the pattern of relative weight-
Žings in correlated and uncorrelated treatments e.g.,structure 2 is weighted about 0.6% above structure
.3 in both treatments , demonstrating that qualita-tive aspects of resonance delocalization are alreadypresent at the uncorrelated MO level, with electroncorrelation providing quantitative enhancement,rather than qualitatively new types, of resonancedelocalization.
Table II shows the corresponding results for theNRT bond orders and atomic valencies, comparingthese with the RHFr6-31GU values of the WibergŽ . Ž .WBI , Mayer-Mulliken MM , and Reed-SchleyerŽ . wRS bond indices. The MM values would showconsiderable variations in higher basis sets,16 but
Ž .the NAO-based WBI and RS values are stable, sothe RHFr6-31GU values may be taken as represen-
xtative of other basis levels. The agreement amongthe various NRT quantities is excellent, with a
Žperceptibly enhanced CN bond order ca. 1.08 vs..1.02 and slightly diminished CH and NH bond
Ž .orders 0.97 vs. 0.99 at correlated levels. The MO-Žbased indices which should only be compared to
.uncorrelated NRT indices show rather less or-derly patterns. The NRT atomic valences are seento be in consistent, excellent agreement with the
Žexpected empirical valency viz. 4.00 for C, 3.01 ".0.01 for N, 0.98 " 0.01 for H , whereas the devia-
tions of WBI, MM, and RS valencies from expectedŽempirical values are much greater often by an
.order of magnitude or more . Note that the NRTŽhydrogen valency V essentially coincides exceptH
in cases such as H-bonding or bridge-bonding: see.the following paper with its formal bond order to
the directly bonded atom, whereas the WBI, MM,and RS valencies have numerous small contribu-
TABLE II.( )Natural Bond Orders and Atomic Valencies of Methylamine Corresponding to the Entries in Table I , with
U( ) ( ( ) )Comparison WBI, MM, and RS Bond Indices RHF ///// 6-31G see 4.1 for Atom Numbering .
NRT bond order b Other bAB ABA—Ba b c d e b b bbond MO-1 MO-2 MO-3 corr-1 corr-2 WBI MM RS
tions from nonbonded atoms that appreciably alterthe ‘‘expected’’ value.
Also, from the fully optimized bond lengths˚ ˚Že.g., CN s 1.4533 A, NH s 1.0015 A, CH s4 5
˚ ˚ U1.0908 A, CH s 1.0838 A at the RHFr6-31G6.level , one can recognize that the NRT bond orders
are in improved correlation with structural fea-tures.27 For example, at all uncorrelated MO lev-els, the NRT bond orders are slightly larger forCH than for CH , consistent with the observed6 5slight increased in optimized CH bond length,5whereas the WBI and RS indices predict the oppo-site trend. Further examples will be cited below todemonstrate the improved order-bond length cor-
Žrelations which are not globally linear; cf. Fig 4 in.the following paper to be an important advantage
of the NRT quantities.However, Table II also shows the somewhat
disturbing result that the bond orders are smallerfor CH than for CH at the correlated levels,6 5despite the greater CH bond length at all levels5Ž Uconsistent with the leading n ª s delocal-H CH 5
.ization that gives rise to structure 2 in Table I .This points to a characteristic difficulty of the NRTbond order]bond length correlations. Formally, thes ª s U NBO delocalization leads to an equalAB CDdecrease in A—B and C—D bond orders and asimultaneous equal increase in B—C bond orderw Ž . xcf. mnemonic 4.3 in the preceding paper . How-ever, there is no physical reason why such delocal-
< < < <ization should produce identical D R , D R ,AB BC< <and D R bond length changes. Indeed, experi-CD
ence has shown that removal of a small amount ofelectron density from a filled donor NBO sABseems to have a significantly weaker effect on bondlength than addition of the equivalent amount ofelectron density to an unfilled acceptor NBO s U ,CD
and neither D R nor D R is likely to be equal inAB CDmagnitude to the change in the connecting bondR as the s ª s U interaction ‘‘turns on.’’BC AB CDThus, the delocalization corresponding to structure3 in Table I is likely to have significantly less effecton C—H bond length than that of structure 4Žwhereas 3 has greater effect than 4 on N—H bond
.length , even though the formal bond orderchanges are identical. As is shown in Table I, atuncorrelated levels the weights of structures 3 and4 are sufficiently small compared to the leadingdelocalization 2 that no anomalies are evidentŽ .w q w - w , whereas at correlated levels the3 4 2relative roles are reversed. Subtle anomalies of thistype are expected to affect the details and limit theaccuracy of bond order]bond length correlationsthat crudely assume equal bond length changes for
Žequal changes in formal bond order regardless of.physical origin . Nevertheless, as the numerical
examples below will show, these subtleties do notappear to detract seriously from the qualitativevalidity of conventional bond order]bond lengthcorrelations.
It is also important to understand the depen-dence of individual resonance weights and bondorders on NRT program parameters, particularlythe NRT energetic threshold NRTTHR that con-trols the magnitude of delocalization effects to be
Žinvestigated and thus the number of resonance.structures n considered . Table III presents theRS
results of such threshold variations at the RHFr6-31GU level, showing the change in resonanceweights w as NRTTHR is systematically raisedi
Ž .from its default value 1.0 kcalrmol by 0.5kcalrmol increments. Table III illustrates a re-
wmarkable property of the NRT expansion cf. eq.Ž . x4.15 in the preceding paper , namely, that the
TABLE III.( )Variations of n Number of Resonance Structures and NRT Weightings w with Changes in NRTTHR EnergeticRS i
U( )Threshold for Methylamine RHF ///// 6-31G Level .
( )NRT resonance weights %NRTTHR( )kcal / mol n w w w w wRS 1 2 3 4 5
percentage weights of secondary structures areoften practically independent of other resonance struc-
Žtures included in the expansion each w being dic-i
tated primarily by its associated stabilization en-.ergy D E . As was remarked in the precedingi
paper this behavior is expected when a resonancestructure involves an acceptor NBO that is notshared with other resonance structures of compa-
Žrable weighting i.e., if two delocalizations are not.‘‘cooperatively coupled’’ . A rough rule of thumb
Žis that the resonance weight w in percentagei.units of a secondary structure is approximately an
order of magnitude smaller than the associatedŽ .D E in kcalrmol . The entries in Table II serve toi
illustrate this rough proportionality. This examplesuggests that one can rather freely adjust the NRT-THR threshold to suppress less important delocal-
Ž .ization and speed execution without appreciablyaltering the relative weights of the structures re-tained.
Finally, we examine the ‘‘discontinuity’’ inweighting that may occur when a resonance struc-
Žture is arbitrarily treated as a reference rather. 28than secondary structure. Table IV shows NRT
Ž U .weightings for methylamine RHFr6-31G levelwhen additional structures are successively speci-fied as reference structures with the $NRTSTR
Žkeylist though structures 3 and 4 were ultimately.rejected as reference structures in this example . In
the present example, each secondary w is appar-i
ently boosted by about 0.6% when promoted toreference structure status. This ‘‘jump’’ empha-sizes that the NRT weightings have relative ratherthan absolute significance and that valid compar-isons of two systems should be based on a consis-tent choice of reference structures.
TABLE IV.Variations of NRT Weightings w fori
U( )Methylamine RHF ///// 6-31G Level with Respect to(Choice of Reference Structures Specified by
$NRTSTR Keylist; Only Bracketed Entries Were)Accepted as Final Reference Structures .
( )NRT resonance weights %Referencestructures w w w w w1 2 3 4 5
As a typical unsaturated molecule, we considerthe example of 1, 3-butadiene. For the trans confor-
˚Žmation, with idealized PG geometry CC s 1.34 A,˚ .CH s 1.08 A, trigonal valence angles , we apply
the same five ab initio methods considered above,leading to the NRT weightings shown in Table V.29
The principal delocalization correction is the dipo-Ž .lar structure entry 2 in Table V that can be
crudely associated with the conventional ‘‘long-bond’’30 covalent structure of Pauling-Whelandtheory. As expected, weights of the leading reso-nance correlations are significantly higher thanin the saturated methylamine case, reflecting thestronger diene conjugative delocalization. Moststriking in Table V is the evident effectiveness ofelectron correlation in enhancing resonance delo-
Žcalization e.g., increasing the weight of the lead-.ing resonance correction from about 2.8% to 6.9% .
However, as in the saturated case, electron correla-tion enhances a pattern that is already present inthe uncorrelated MO treatment, rather than intro-ducing qualitatively new resonance structures.
For this conjugated system, Table V verifies theexcellent NRT numerical characteristics that werepreviously noted in the saturated case, includingthe practical invariance to choice of basis set. TableVI displays the corresponding butadiene NRT bond
Ž .orders for idealized PG geometry at each level,with the atom numbering shown below.
H H5 7
Ž .C C H 4.31 2 9
H C C3 46
H H10 8
Very satisfactory numerical stability is evident inŽthese results e.g., bond orders for all three MO
levels typically agree within a few parts per thou-.sand as well as good overall agreement in bond
order patterns for correlated and uncorrelatedwave functions. Table VI also includes correspond-
Ž Uing WBI, MM, and RS bond indices RHFr6-31G.level for comparison as well as the fully opti-Ž U .mized RHFr6-31G level bond lengths, R andCC
R , in the final column. Comparison of the NRTCHbond orders with optimized bond lengths shows
Žthe good correlation even to small details of CH.bond length; see below . In this case the WBI, MM,
and RS values show qualitatively similar bondŽorder patterns though the WBI and MM values
incorrectly suggest very similar values of CH ,5
VOL. 19, NO. 6618
NATURAL RESONANCE THEORY: II
TABLE V.( )Leading NRT Resonance Weights w for trans-Butadiene at Several Levels of Ab Initio Uncorrelated MO andi
( ) ( )Correlated corr Theory Idealized PG Geometry .
( )NRT resonance weights %a b c d ei Structure MO-1 MO-2 MO-3 corr-1 corr-2
mized geometries are very similar, accentuatingthe subtle variations that are already exhibited in
.idealized PG geometry. The central b bondC C2 3Žorder varies most strongly, dropping from bC C2 3
.s 1.0392 to nearly pure single bond characterŽ .b s 1.0098 as the pi bonds are twisted out ofC C2 3
Žconjugation to f s 908, then rising again to bC C2 3
.s 1.0295 in the planar cis arrangement, in goodagreement with the expected chemical behavior.Figure 2a displays the correlation of calculated CC
Ž .bond order idealized geometry with ‘‘relaxed’’Ž UCC bond lengths fully optimized RHFr6-31G
.level for both types of CC bonds, showing that thebond lengths change in response to bond orderdifferences in the expected manner. However, as
ŽFigure 1 shows, the CH bond orders especially.b undergo still more subtle variations that canC H2 7
also be qualitatively correlated with actual bondlength changes. For example, Figure 1 suggeststhat at the twisted 90T conformation, C —H and1 5C —H should be nearly identical, and signifi-1 6cantly shorter than C —H , whereas, at the 0T cis2 7arrangement, the three bond lengths should bemore nearly comparable, with C —H intermedi-1 6ate between C —H and C —H . The fully opti-1 5 2 7mized bond lengths
T ˚ ˚90 : C —H s 1.0758 A, C —H s 1.0761 A,1 5 1 6
˚C —H s 1.0793 A2 7
T ˚ ˚0 : C —H s 1.0748 A, C —H s 1.0755 A,1 5 1 6
˚C —H s 1.0777 A2 7
agree with these predictions. Figure 2b displaysŽ .the plot of NRT bond order idealized vs. CH
VOL. 19, NO. 6620
NATURAL RESONANCE THEORY: II
( ) (FIGURE 2. a Correlation of NRT bond order idealized) (PG geometry with CC bond lengths optimized RHF /
U) ( )6-31G for central C —C bond left and terminal2 3( ) ( ) ( )C 5C bond right of twisted butadienes cf. Fig. 1 . b1 2
( ) ( ) ( )Similar to a , for C —H circles , C —H squares ,1 5 1 6( )and C —H triangles .2 7
Ž .bond length optimized for all points along thetorsional coordinate, showing that there is a mod-est but significant correlation between quantitativeNRT values and small geometry variations even atthis greatly expanded scale.
We can also use this example to illustrate fur-ther the type of ‘‘discontinuity’’ that occurs whenthe default NRT program encounters a change inthe number or type of reference structures. For thispurpose, we consider the model ring closure thatresults when the two terminal methylene groupsof cis-butadiene are twisted face-to-face to formcyclobutene, breaking p , p to formC — C C — C1 2 3 4
new s , p bonds. According to theC — C C — C1 4 2 3
well-known Woodward-Hoffman rules, simultane-ous dihedral twists f , f can be carried out in a12 34
Ž . Žconrotatory f s f or disrotatory f s12 34 12.yf sense, the former leading to ‘‘allowed’’ bond34
rearrangements. As a simple model of such a pro-
cess, we considered synchronous conrotatory ordisrotatory dihedral rotations of rigid trigonal
Žmethylene groups all other variables retaining. Ufixed Pople-Gordon values at the RHFr6-31G
level. As expected, the conrotatory pathway leadsto more pronounced bond rearrangements; for ex-ample, near f s 110T , the weight of cyclobutene-like resonance structures is about twice as large in
Žthe conrotatory as in the disrotatory form 20.7%.vs. 10.4% . The variations of NRT bond order
Ž .default program settings; Fig. 3, solid lines alongthe conrotatory pathway are shown in Figure 3.One can recognize the characteristic butadiene-like
Ž . Ž .pattern at left and cyclobutene pattern at right ,Ž T .separated by a ‘‘transition region’’ near 104
where a two-reference optimization is performed.Because the default ‘‘boundary’’ between sec-ondary reference is an arbitrary program thresholdŽ U .w G 0.35 w , passage through the transitionref maxregion necessarily results in discontinuities at theseboundaries. This example emphasizes the need to
Ž .select and impose with the $NRTSTR keylist acommon set of reference structures to traverse adesired transition region. To illustrate this option,Figure 3 also shows the alternative two-reference
Ž .bond orders dashed lines that result when buta-
FIGURE 3. Variations of NRT b bond orders bCC 12( ) ( ) ( )circles , b squares , and b triangles , for idealized23 14butadiene ª cyclobutene isomerization by conrotatory
( U )ring closure RHF / 6-31G level , showing differences( )between default NRT treatment solid lines and uniform
( )two-reference treatment dashed lines specified with the$NRTSTR keylist.
JOURNAL OF COMPUTATIONAL CHEMISTRY 621
GLENDENING AND WEINHOLD
diene and cyclobutene are both treated as refer-ence structures at all angles, avoiding the ‘‘discon-
Ž T . Žtinuity’’ near 112 of the default treatment. Theremaining discontinuity near 100T reflects anabrupt change in the character of the MO wave
.function, not an NRT threshold artifact. This ex-ample emphasizes again that valid NRT compar-isons of distinct regions of the potential energysurface should be based on a consistent choice ofcommon reference structures.
POLAR COVALENCY ANDELECTROVALENCY: FLUOROMETHANES
To test the NRT description of atomic valency inpolar and nonpolar compounds, we consideredthe family of fluoromethanes, CH F , n s 1]4.n 4ynThis allows us to investigate the NRT covalencyŽ . Ž .C , electrocovalency E , and total valencyX XŽ .V of X s C, F, H atoms in molecules of varyingXpolarity, for comparison with expected empiricalvalues.
Figures 4a]c illustrates the calculated NRT val-Ž . Ž . Ž .ues of C triangles , E squares , and V circlesX X XŽ . Ž .for carbon Fig 4a , hydrogen Fig. 4b , and fluo-
Ž .rine Fig. 4c atoms in the fluoromethane series,U Ždescribed at the RHFr6-31G level idealized PG
.geometry . As is shown in this figure, the NRTcovalency and electrovalency vary in a chemicallyreasonable way, with the electrovalency increasingas the polar character of attached bonds increases.Thus, the NRT covalency of carbon decreases from
Ž . Ž .3.1317 in CH to 1.9495 in CF , while the elec-4 4trovalency correspondingly increases. Despite thesevariations, the total NRT valency is seen to be
Žremarkably constant throughout the series V sC.3.99 " 0.01, V s 0.99 " 0.01, V s 1.00 " 0.02 , inH F
excellent agreement with the expected empiricalŽ .tetra- or uni- valency. Thus, Figure 4a]c illus-trates the complementary character of electrovalentand covalent contributions to total atomic valencyand indicates why bond indices that neglect oneor the other aspect may fail to give consistentagreement with experimentally observed valencyvalues.
We also calculated C—F bond orders at theUU Ž 31.higher RHFr6-311qq G level PG geometry
in order to make comparisons with available CMvalues for this series.20, 22 Results are shown inTable VII, where we also include comparisons withcorresponding WBI and MM values. The NRT re-sults suggest the expected C—F single bonds
Ž .throughout the series b s 1.01 " 0.01 , and theCF
ŽWBI and MM values 0.85 " 0.02, 0.95 " 0.09, re-.spectively are crudely in agreement. But the CM
Žvalues vary in the range 0.54]0.75 with a trend.opposite that of the WBI and MM indices as fluo-
rine ligancy is varied. The CM value crudely ap-Žproximates the NRT covalency for CF 0.543 vs.4
.0.531 , but these values become widely discrepantŽ .and trend in opposite directions as fluorines areremoved. A particularly discrepant feature of the
ŽCM bond indices not apparent from the results ofWiberg and Rablen22 but explicity indicated by
20 .Ciowlowki and Mixon for CHF is the surpris-3Žingly large F—F bond order 0.171, nearly one-third
.as large as the C—F bond order , which has noŽ . Ž .counterpart in the WBI 0.03 , MM 0.01 , or NRT
Ž .0.00 descriptions.Similar conclusions can be drawn from atomic
Ž .valency values not tabulated , where the NRTresults give fairly ‘‘ordinary’’ values for carbonŽ . Ž .3.992 " 0.003 , hydrogen 0.985 " 0.007 , and flu-
Ž .orine 1.01 " 0.01 throughout the series. The alter-native MO-based valencies are much less orderly;for example, at the RHFr6-311qq GUU level, thevalues of V vary in the range 3.49]3.73 for theCWiberg index and 3.83]4.18 for the MM indexŽ .with opposite trends as fluorine ligancy is varied .For the CM method, tabulated20 orders for CHF3allow one to infer fairly reasonable values for
Ž .fluorine and hydrogen V s 0.979, V s 1.062 ,F HŽbut an unreasonably low value for carbon V sC
. 322.583 , well outside the range of other indices.These results seem to confirm the general superior-ity of the NRT description for polar covalent com-pounds and raise questions concerning the prem-ises of the CM treatment.
OPEN SHELL SPECIES:HYDROXYMETHYL RADICAL
Finally, to demonstrate the application of NRTanalysis to an open-shell species, we consider the
Ž .hydroxymethyl radical CH OH as described at2U Ž . UUHFr6-31G uncorrelated and UMP2r6-31G
Ž .correlated levels of theory. Because this species isknown to have ‘‘floppy’’ geometry, we optimized
Žthe radical structure at each level C symmetry:s.see Table VIII before performing NRT analysis.
Consistent with the ‘‘different Lewis structuresfor different spins’’ open-shell NBO picture thatwas previously applied to such species,33 the NRTprocedure is applied separately to density matricesfor alpha and beta spin. This leads to the NRTweightings shown in Table IX, which exhibits the
VOL. 19, NO. 6622
NATURAL RESONANCE THEORY: II
( ) ( ) ( ) ( )FIGURE 4. a Variations of carbon atom electrovalency squares , covalency triangles , and total valency circles inU ( )the fluoromethanes, CF H , calculated at RHF / 6-31G level idealized Pople-Gordon geometry , showingn 4y n
( ) ( )‘‘complementary’’ electrovalent and covalent contributions to total atomic valency. b Similar to a , for hydrogen atom.( ) ( )c Similar to a , for fluorine atom.
Ž Ža. Ž b ..separate weightings w , w for each structurei i
i in the two spin sets. As a notational device toavoid explicit depiction of all nonbonded ‘‘lone’’electrons, we have included ‘‘formal charge’’ la-bels to indicate how many nonbonded electronsmust be added or subtracted to obtain the fullLewis structure specification. For example, theleading alpha structure in Table IX has one lone a
Ž .electron on carbon to give formal charge of y1Žand two on oxygen the usual electroneutral pat-
.tern , to give the overall y1 formal charge of the‘‘anion-like’’ alpha spin set, whereas, the corre-
sponding ‘‘cation-like’’ beta structure has no loneŽ .b electrons on C formal charge q1 but the usual
Žtwo on O. Of course, the ‘‘formal charges’’ havebut loose connection to the physical charge distri-bution as described, e.g., by natural population
.analysis. Note that, in the NRT framework, a‘‘resonance structure’’ is simply a pattern of one-
Ž . Ž .center nonbonded and two-center bonded spinorbitals occupied by electrons of the given spin,and there is no intrinsic reason why these struc-tures should be identical for a and b spins. In-deed, because the open-shell spatial distributions
JOURNAL OF COMPUTATIONAL CHEMISTRY 623
GLENDENING AND WEINHOLD
TABLE VII.UU (Calculated C—F and C—H Bond Orders for Fluoromethanes at RHF ///// 6-311++ G Level PG Geometry Except
) ( ) ( )for CM Values: See Reference 31 , Showing the Comparison between NRT, Wiberg WBI , Mayer-Mulliken MM ,( )and Cioslowski-Mixon CM Values.
CH F 0.9844 0.8686 0.1158 0.9580 0.99882 2CH F 0.9920 0.8555 0.1365 0.9661 0.99133
a RHF / 6-31++ GUU values.20
of a and b electrons are different, the optimalNRT representations of a and b distribution areexpected to be unequivalent, as Table IX shows tobe the case.
ŽIt is apparent from Table IX that b minority.spin electrons are significantly more delocalized
Ž .than a majority spins electrons. This is chemi-cally reasonable; since the b system has a vacancyon carbon that interacts strongly with nonbondedelectrons of the adjacent oxygen atom to give theformal ‘‘dative pi bond’’ of structure 2.b. Thestronger b resonance delocalization is also mani-fested in significantly higher CO bond order in the
Ž .b system viz. 0.55 vs. 0.53 at UMP2 level as wellŽas higher CO polarity 45% vs. 40% ionic charac-
.ter . Table X shows that the percentage ionicitytends to be systematically higher in correlatedwave functions and to be noticeably increased byresonance delocalization.
The atomic valency of carbon is seen to besignificantly altered from its normal closed-shellvalue, consistent with the NRT resonance struc-
tures. For example, in the a set, carbon resemblesŽa nitrogen atom with filled nonbonding orbital; cf.
¨ .NH OH whereas, in the b set, it resembles boron2Ž .with empty p-type orbital , both nominally ‘‘tri-
Ž .valent’’ V s 3.05 at the correlated level . How-C
ever, the corresponding valencies of hydrogenŽ . Ž .0.98 and oxygen 2.05 are affected little by theradical delocalization. The tabulated valencies thusseem to reflect well the chemical behavior at eachatom.
Conclusions
We have introduced formal definitions of ‘‘nat-ural bond order’’ and ‘‘natural atomic valency’’ in
Ž .the framework of natural resonance theory NRTand illustrated the ab initio application of theseformulas to several prototype cases involving weakor moderate resonance delocalization. These appli-cations serve to document the convergence, stabil-
TABLE VIII.˚( ) ( )Optimized Bond Lengths R A and Valence and Dihedral Angles u, f in Degrees for Hydroxymethyl Radical
U U( ) ( ) ( )C -Symmetry at Uncorrelated UHF ///// 6-31G and Correlated UMP2 ///// 6-31G Levels.S
a ( )Note that the slight pyramidalization of the methylene group is of opposite sense in the two levels.
VOL. 19, NO. 6624
NATURAL RESONANCE THEORY: II
TABLE IX.( )Leading NRT Resonance Weights a and b Spin for Hydroxymethyl Radical at Uncorrelated MO
( ) ( ) (UHF ///// 6-31G* and Correlated UMP ///// 6-31G* Levels of Theory Nonbonded Electrons Are Suppressed in thec)Lewis Structure Representations .
(a) ( b )( ) ( )w %, a spin w %, b spini iAlpha Betaa b a bi structure MO corr structure MO corr
H Hy +[ ] [ ] [ ] [ ]1. C O 97.20 95.39 C O 93.91 89.28- -
H HH H
H H+- -2. C O 1.03 1.52 C O 2.80 4.65
H Hy HH
H Hy + + +( ) ( ) ( ) ( )3. C O 0.77 2 1.55 2 C O 0.78 2 1.55 2
y yH HH H
H Hy y ( ) -4. C O 0.11 2 -0.01 C O 1.72 2.98
+ HH+H H
a UHF / 6-31G*.b UMP2 / 6-31G*.c Brackets identify reference structures; parenthesized numbers give degeneracy factors for equivalent structure formulas.
ity with respect to basis set extension, dependenceon program parameters, and general practicality ofthe NRT method for typical open- and closed-shellspecies at both uncorrelated and correlated levelsof theory. Our results also demonstrate the higherlevel of quantitative detail and the improved cor-relations of bond order and valency indices withempirical structural and chemical properties thatare achieved by the NRT expansion.
Although the underlying formulation of NRTdiffers substantially from that of Pauling-Whelandtheory, the new method retains a strong associa-
tion with the classical resonance concepts. Thehigh percentage of delocalization density recov-
Žered by the NRT expansion typically 80]90%,with relatively small numbers of resonance struc-
.tures seems to attest to the validity and accuracyof resonance-theoretic assumptions that underliemany concepts of physical organic chemistry.
Compared to previous methods for calculatingresonance weightings, bond order, and atomic va-lency, the NRT approach offers several important
.advantages. 1 The NRT statistical weighting dis-tribution is derived from a variational criterion,
TABLE X.( )Open-Shell a and b NRT Bond Orders b and Atomic Valencies V with Percentage Ionicity in ParenthesesAB A
U U( ) ( )for Hydroxymethyl Radical, Calculated at Uncorrelated MO UHF ///// 6-31G and Correlated UMP2 ///// 6-31G( )Levels cf. Table IX .
based on minimizing the ‘‘error’’ d of the de-Wscription. Thus, the NRT parameters are derivedpurely from theory, and the accuracy of the NRTdescription can be assessed internally from theoptimal d value, without appeal to external em-W
.pirical data. 2 Electrovalent and covalent contri-butions to bond order are treated in a unified
Žmanner specifically avoiding the Pauling-Whe-.land ‘‘covalent-ionic resonance’’ concept . Thus,
the NRT description is uniformly applicable to awide variety of organic and inorganic species, per-mitting consistent comparisons among diverse sys-
.tems. 3 NRT analysis can be readily integratedinto popular electronic structure packages for rou-tine application to wave functions produced by
Žstate-of-the-art ab initio methods including SCF,MCSCF, CI, and Møller-Plesset perturbative treat-
.ments . Thus, the NRT method may be used tobring resonance theory concepts back into themainstream of ab initio studies.
Although the present paper has focused on doc-umenting the performance of the NRT method fortypical saturated and unsaturated species, themethod is readily applicable to a much broadervariety of exceptional or ‘‘anomalous’’ bondingtypes. Examples of such applications are presentedin the following paper.
Acknowledgments
We thank Mr. Jay K. Badenhoop for discussionsand assistance on many of the numerical exam-ples. Computational resources provided throughNSF grant CHE-9007850 and a grant of equipmentfrom the IBM Corporation are gratefully acknowl-edged.
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Ž .10. In principle, one might have e.g., for an excited statehigher occupancy of an antibond than of the correspondingbond, which would lead to a net negative contribution toformal bond order. However, this situation is not providedfor in the present version of the NRT program.
Ž .11. See, e.g., Reference 1 b pp. 88]95, for discussion of therelationship of bond energies to electronegativity differ-ences.
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15. For critical discussion of the terminology employed by Jugand coworkers,14 see I. Mayer, Theor. Chim. Acta, 67, 315Ž .1985 ; note that the orbitals called ‘‘natural hybrid orbitals’’and ‘‘natural bond orbitals’’ by Jug and coworkers in 1983
VOL. 19, NO. 6626
NATURAL RESONANCE THEORY: II
are unrelated to the corresponding quantities defined previ-ously in references 3 and 4.
16. The quantities introduced by I. Mayer14 are closely associ-ated with Mulliken population analysis and its well-known
wpathologies in larger basis sets. For example, Baker J.Ž .xBaker, Theor. Chim. Acta, 68, 221 1985 cites an example in
which the calculated Mayer valency of carbon changes fromŽ .q3.39 to y ! 4.86 when diffuse functions are added.
Ž .17. Jug and coworkers see citations in reference 14 havevariously suggested Wiberg’s formula with no overlap,with Lowdin orthogonalization, and with a modified¨scheme of occupancy-weighted symmetric orthogonaliza-tion.
18. F. Weinhold and J. E. Carpenter, J. Mol. Struct. Theochem.,Ž .165, 189 1980 .
19. A. E. Reed and P. v. R. Schleyer, Inorg Chem., 27, 3969Ž . Ž .1988 ; J. Am. Chem. Soc., 112, 1434 1990 .
20. J. Cioslowski and S. T. Mixon, J. Am. Chem. Soc., 113, 4142Ž .1991 .
21. R. F. W. Bader and T. T. Nguyen-Dang, Adv. QuantumŽ .Chem., 14, 63 1981 .
22. K. B. Wiberg and P. R. Rablen, J. Am. Chem. Soc., 115, 614Ž .1943 .
23. I. D. Brown and R. D. Shannon, Acta Crystall., A29, 266Ž .1973 ; I. D. Brown and D. Altermatt, Acta Crystall., B41,
Ž .244 1985 . For applications, see I. D. Brown, J. Solid StateŽ .Chem., 82, 122 1989 ; M.-H. Whangbo and C. C. Torardi,
Ž .Science, 249, 1143 1990 .Ž .24. W. G. Penny, Proc. R. Soc. London, A158, 306 1937 ; W. C.
Ž .Herndon, J. Am. Chem. Soc., 95, 2404 1973 ; I. Gutman andŽ .N. Trinajstic, Top. Curr. Chem., 42, 49 1973 ; I. Gutman andŽ .H. Sachs, Z. Phys. Chem., 268, 257 1987 , and references
therein.25. J. A. Pople and M. Gordon, J. Am. Chem. Soc., 87, 4253
Ž .1967 .26. For a comprehensive description of computational methods
and basis set designations used herein, see W. J. Hehre, L.Radom, P. v. R. Schleyer, and J. A Pople, Ab Initio MolecularOrbital Theory, John Wiley, New York, 1986. All wave
function calculations were performed with the Gaussian 92program system: M. J. Frisch, G. W. Trucks, M. Head-Gordon, P. M. W. Gill, M. W. Wong, J. B. Foresman, B. G.Johnson, H. B. Schlegel, M. A. Robb, E. S. Repogle, R.Gomperts, J. L. Andres, K. Raghavachari, J. S. Binkley, C.Gonzalez, R. L. Martin, D. J. Fox, D. J. Defrees, J. Baker,J. J. P. Stewart, and J. A. Pople, Gaussian 92, Revision A,Gaussian, Inc., Pittsburgh, 1992.
27. Note that bond order]bond length correlations are notnecessarily expected to be linear. For example, Coulson12
originally suggested a linear relation between bond lengthand P , but the same cannot be expected for MM, WBI,ABand Jug indices, which are quadratic in P .AB
28. Of course, it is quite artificial to consider such a ‘‘transition’’in the present case; the default NRTWGT criterion for
Ž .acceptance as a reference structure w G 0.35)w en-a maxsures that only the most strongly delocalized systems willhave more than a single reference structure.
29. The default NRT expansions for these wave functions in-Ž .clude about 20 additional structures not shown with
smaller weightings G 0.01%.30. The superficially high weight of ‘‘long bond’’ structures in
earlier RT treatments can be attributed to the ‘‘overcorrela-tion’’ artifact of Heitler-London valence-bond wave func-tions; see R. McWeeny, Proc. R. Soc. London, A223, 63, 306Ž .1954 .
31. This differs slightly from the geometries employed by20 Ž U .Cioslowki and Mixon optimized HFr6-31G and Wiberg
22 Ž Uand Rablen partially optimized MP2r6-31G , with C-F.bonds constrained to equal the value for CH F . These3
geometry differences are likely to be unimportant on thescale of the comparisons considered here. The large differ-ence between CHF values of b reported in references 203 CFŽ UU . Ž0.572, RHFr6-31qq G level and 22 0.616 RHFr6-311
UU .qq G level is therefore puzzling.Ž .32. The CM V is rather close to the NRT covalency 2.427 inC
this case, but the corresponding valencies for fluorine differby almost a factor of two.
33. J. E. Carpeter and F. Weinhold, J. Mol. Struct. Theochem.,Ž .169, 41 1988 .
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