CHEMISTRY EDUCATION: INVITED THEME ISSUE:Structural concepts RESEARCHAND PRACfiCE IN EUROPE Contributionfrom science 2001, Vol. 2, No.2, pp. 91-104
Frank WEINHOLD and Clark R. LANDIS University of Wisconsin, Theoretical Chemistry Institute and Department of Chemistry (USA)
NATURAL DO ND ORBIT ALS AND EXTENSIONS OF LOCALIZED BONDING CONCEPTS
Received5 Apri/2001
ABSTRACI': We provide a brief overviewof.'natura1"localized bonding concepts, as implementedin the currentnatW'al 0000 orbital program(NBO 5.0), aM describe~t extensionsof dJeseconcepts to transitionmetal bonding. [Chern. Educ. Res. Pract. Eur.: 2001, 2, 91-104]
KEY WORDS: notural orbitaU; NBO ]N'ograIn; Lewis stnlClure; transition metal bonding; ad hybridization
WHAT ARE "NATURAL" LOCA~IZED ORBITALS? The concept of "natural'" orbitals was first introducedby LOwdinI to describethe unique set of orthonormal I-electron functions OJ(r) that are intrinsic to the N-electron wavefunction'V(l, 2, ..., N). Mathematically,the OJ'Scan be consideredas eigenorbitalsof", (or, more precisely, of""s first-order reduceddensity operator),and they are therefore"best possible" (most rapidly convergent,in the mean-squaredsense) for describingthe electron density p( r) of 'V. Comparedto many other choicesof orbitals that might be imagined or invented [e.g., the standard atomic orbital (AO) basis functions of electronic structure packagessuch as Gaussian981, the naturalorbitals are singled out by 'V itself as "natural" for its own description. One might supposethat natural orbitals would also be "best possible" for teaching chemistry studentsabout 'V in qualitative orbital language.This would be so, exceptfor the fact that natural orbitals (like the canonicalmolecular orbitals of Hartree-Focktheory) are necessarilysymmetry adapted. Suppose, for example,that the quantummechanical system of interestcorresponds to two H atoms,one on earthand the other on the moon,
'1'2= 'I'(HC8rth.Hmooo)
On physical groundswe might expectthat the orbital descriptionof '1'2is nearly identical to that for the correspondinglocalized wavefunctions
'!'Ie = '!'~ ). '!'Im = '!'(Hmoon,
However, due to the fact that '!'2 must incorporatethe superpositionsymmetry3 between ~, H- (even if theseatoms have no interactionsof physical significance),the natural orbitals of '!'2 will be found to differ qualitatively from those of '!'Ie or '!'Im-This has the 92 WEINHOW &; LANDIS unfortunate consequence of making the orbitals look "more delocalized" (and less transferable) than is physically meaningful, obscuring many simplicities of chemical bonding. (Similar spurious mixings occur when the atoms are merely separatedby a few links of an alkane chain.) Thus, the natural orbitals as originally defined include bogus "delocalization effects" that have no physical significance, 4 limiting the usefulness of these orbitals for pedagogical purposes. To remove spurious effects associated with symmetry adaptation, one can formulateS a localized criterion for orbitals that have the analogous maximum-occupancy (natural) character in localized I-center and 2-center regions of the molecule. Because die maximum occupancy of an orbital is inherently limited to a pair of electrons6 by die Pauli exclusion principle, local 1- and 2-center orbitals with occupancies sufficiently close to 2.000 can serve equally well as "true" natural orbitals for describing'll. As anticipated by G. N. Lewis,' localized orbitals of near-double occupancy can be found in the 1- and 2-center regions suggested by the elementary Lewis structure diagram. Such natural bond orbitals (NBOs) provide the most accurate possible "natural Lewis structure" picture of 'If, because all orbital details (polarization coefficients, atomic hybrid compositions, etc.) are madlematically chosen to include the highest possible percentage of the electron density. This percentage (denoted %-pL) gives an intrinsic measure of the accuracy of the natural Lewis structure picture, and is often found to be >99010for common organic molecules, dramatic testimony to the profound accuracy of Lewis's concept. The NBOs are one of a sequenceof natural localized orbital sets that include natural atomic (NAO), hybrid (NHO), and (semi- )localized molecular orbital (NLMO) sets, intermediate between basis AOs and canonical molecular orbitals (MOs)
AOs -+- NAOs -+- NHOs -+- NBOs -+- NLMOs -+- MOs (2)
All these natural localized sets are complete and orthonormal, able to exactly describe any property of 'If. Compared to standard AOs, e.g., the NAOs give a much more condensed description of 'If, with only a small number (i.e., corresponding to the formal "minimal basis") having appreciable occupancy. Thus, a "minimal" description in terms of core and valence-shell NAOs is often found adequate for chemical purposes, providing a compact representation of 'If that is intimately related to standard valence concepts. The mutual orthogonality8 of natural localized orbitals may seem to be a conceptual liability, inasmuch as the concept of "orbital overlap" seems to be lost. However, each orthogonal NAO (or NHO, NBO, etc.) can be uniquely associated with a corresponding "pre-orthogonal" PNAO (or PNHO, PNBO, etc.) which remains orthogonal to PNAOs on the same atom but has nonvanishing overlap integrals with those on other atoms. In accordance with the Mulliken approximation,9 the corresponding Hamiltonian interaction elements are found to be closely proportional to these overlap integrals. That is, if F denotes the effective orbital Hamiltonian (Fock or Kohn-Sham operator), the interaction strength
{JAB=cAhA + c&he
O'AB~= CDhA - cAhs
6.E(2) ~ ~j~!['I~ . 2 1-+. f --- 2 I !1 c (5) 6..-~J ' .. .
whore F i. fu, ,ff~ti" o,hila! Hwltonion (Fock 0' Kohn-Shomopen'"" ond ,~
NL ~= CiiOj + LCjjO; (6) j that reflect the irreducible physical effect of OJ-+- "oj. delocalizations.Despite the compact, recognizableforms ofNLMOs and their closeconnection to chemicalStructure concepts, it is importantto recognizethat a Slaterdeterminant of doubly occupiedNLMOs is equivalentto the usual MO wavefunction.Hence, the simplicity of NBO-basedexpansions such as (6) is achievedwith no loss of accuracyin the descriptionof",.
OVERVIEW OF THE NATURAL BOND ORBITAL PROGRAM 94 WElNHOID4.LANDIS
Comparedto earlier "generations"of the program (NBO 3.0 and 4.0), NBO 5.0 contains extensivenew capabilities related to magnetic propertiesand transition metal bonding (as discussedbelow), as well as numerousextensions and improvementsof establishedNBO analysistools. Some of theseadvanced options are mentionedbelow and in Sec. 3, but we primarily focus on generic input and output featuresthat are common to all recent NBO versions. Input to the NBO program is in the form of keywords within the SNBO...$END key/ist,usually following the host ESSinput file, m.,
$NBO keywordl keyword2 $END
Thesekeywords (about 95 in the current version) can direct output for analysesof specializedproperties, e.g.,
NJC (naturalchemical coupling18) NCS (naturalchemical shielding19)
NBO-based wavefunction or energy decompositions, e.g.,
NRT(natural resonance theo~~ NEDA (natural energy decomposition analysiS21)
printing and display options,e.g.,
PLOT (graphicalorbital display)
operatormatrices in localizedbases, including
F (Fock), K (kinetic), V (l-e potential), Dr (dipole), DM (density),5 (overlap)
as well as other options.For example,the keylist
$NBO STERIC FNBO PLOT FILE=C2H4 $END
would perfOrDlnatural steric analysis,22print the matrix of the Fock operator F in the NBO basis,and print disk files for the ORBPLOT or NBOView orbital viewersunder the filename "C2H4" for this job. Output from the NBO programmay consistof (i) printed tablesin the .LOG file, (ii) disk files for input to other programs(e.g., graphicalutilities), and/or (iii) modifications of the host ESS checkpointfile that can affect perfonnanceof subsequentjobs. An exampleof the latter is storage of the NBOs from a starting single-determinantself-consistent-field (SCF) treatmentinto the checkpointfile for use in post-SCFcorrelation corrections (e.g., of complete active spaceCAS/NBO type23).Most important is the default .LOG file output, whosemajor sectionswe briefly summarizebelow:
Natural population analysis This sectiondisplays a list of all NAOs and their "type," population,and energy(if one-electronHF/DFT Hamiltonian is available).The orbital populationsare then summedto NATURAL BOND ORBITALS 95
give the table of atomic natural charges and the effective natural electron configuration on each atom.
Natural bond orbital analysis This section displays the NBOs in terms of their constituenthybrids, polarization coefficients, occupancies,and NAO composition. Each NBO is labeled as being of core (CR), bond (BD), valencelone pair (LP), or extra-valenceRydberg (RY) type, with affixed asterisk(*) for non-Lewis orbitals. Thus, the label "LP(I) N 2" identifies a valencelone pair on nitrogen 2 (~2), and "BD*(I) C I-H 4" identifies a valenceCI-H4 antibond (cr*CIH4). Parenthes~edlabel numbers such as BD(I), BD(2), BD(3) distinguish multiple bonds betweenthe same atoms, and similar LP or RY* labels distinguish multiple lone pair or Rydbergorbitals at eachcenter.
NHO directionality and "bond bending" analysis This section displays the angular deviations betweenthe bonding hybrids and the direct line-or-sightbetween nuclei, showingwhere significant bond bendingis present.
2nd-order perturbation theory analysis This section tabulates, for all possible donor-acceptor pairs, the values of the donor- acceptor stabilization energy as estimated by Eq. (5). The table also includes the corresponding Fock matrix elements in the numerator and denominator of this equation. Where appropriate, interactions are separated into intra- and inter-molecular types for different "molecular units," i.e., contiguously bonded atomic networks (usually equivalent to "molecules" or "ions"). This section of the output is normally the first to be examined by the experienced NBO user in searching for significant delocalization effects.
NBD summary This section summarizes the NBOs, occupancies, energies, and principal delocalizing interactions grouped by molecular units. The table also includes net charge and valence/Rydberg occupancy statistics for each unit.
Furtherdetails ofNBO output tablesand their interpretationare providedon the NBO website24and in the NBO 5.0 programmanual, which is mandatoryfor intelligent useof the program.
EXTENSION OF LOCALIZED BONDING CONCEPTS TO TRANSmON METALS
G.N. Lewis's octet rule and sharedelectron pair concepts7underlie the most broadly acceptedmodels of localized bonding in common main group elements.However, it is importantthat a quantitativewavefunction analysis should not only conformto our prejudices in these cases, but also suggest useful extensionsof localized concepts to less well understoodspecies. In the presentsection we sketchempirical andNBO-based computational evidence for a far-reaching extension of Lewis-like diagrams and bonding concepts to transitionmetal compounds. As a practical empirical criterion for what transition metal compoundsmight be considered"common," Table I exhibits the formulas and numberof unpairedelectrons (eu) for the Group 3-12 binary oxides, chlorides, and alkyl compoundsthat are available most cheaplyand in largest quantitiesfrom a standardchemical supply catalog?5 From this table a ~ ,T~1 TiCJ. (0) TiO2 (0) Ti(benzyl).. (0) ~'1.IJ:. ZrCJ. (0) ~ (0) ZrM~ (0) Hf Hn (0) H~ (0) HfM~ (0) s V VC4 (1) V20s (0) Nb NbCl, (0) V(CH2TMS)4 (1) ~OS (0) NbMes (0) Ta TaCl, (0) Ta20S (0) TaMes (0) 6 Cr CICl3 (3) c~ (0) Cr(cyclohexyl)4 (2) Mo MaCI, (I) M~ (0) W WCl, (1)' Mo(cyclohexyl~ (2) W~ (0) WM~ (0) Mn MnC12 (5). MnO (5)* Mn[C(TMShh (5)- Tc TcC4 (1-2). TC2o, (1) Re ReCls (-2). Re207 (0) ReMC6 (1)- 8 Fe FeCI] (5). Fe2~ (5)* Ru RuCI] (0) Fe(norbomyl). (0) RUO2 (0) Ru(mesityl). (0) Os OsCI] (1) Os~ (O)C Os(mesityl). (0) 9 Co CoC12 (4). CO]O4 (-4). Co(norbomyl). (I) Rh RhC13 (0) ~~ (0) Rh(mesitylh (0) Ir IrC4 (0) Ir2~ (0) Ir(mesityl») (0) 10 Ni NiCb (2)* NiO (4)* Pd PdCb (0) PdO (?) Pt PtCl2 (0) PtO (?)
11 Cu CuCl (0) Cu2O (0) [Cu(CH2TMS)]. (0) As AgCl (0) Ai20 (0) - '-" Au AuCI (O)d AU203 (1) :' 12 Zo ZoC12 (0) ZoO (0) rr ; :t: -- ': ' Cd CdC12 (0) CdO (0) hC'~'; - e' H 0 H i .According to criterion of lowest cost or availability in largest quantities (Ref. 25). b Inferred from magneticsusceptibility measurements. cOSO4(0) is comparable. d AuCl3 (0) is comparable. NATURAL BOND ORBITALS - -- 97 remarkable regularity becomes apparent If ~ denotes the group number, n the stoichiometricMLn formula ratio, and VLthe valencyof the ligand, the relationship
Ow - 61 + n YL +eu - 6 (7)
can be seen to be satisfied for an overwhelming majority of common MLn species. We now assume, following Lewis, that each "bond" linkage to a monovalent ligand is associated with a shared electron pair, so that the total number (eIJp)of valence bond-pair electrons is ~ = 2nVL. We also recognize that the maximum possible number (elp) of nonbonded lone-pair electrons is elp = 2(6M - 6) for any group 6M ~ 6. Thus, for mid to late d-block elements, Eq. (7) becomes
M(elp+ ~)+eu = 6 (6 s~:S 11) (8)
This is analogousto the correspondingLewis-type formula for normal-valentelements
.K (elp + e-.)+e. = 4 (14 ~ Gw s 18) (9)
which for closed-shell species (eu = 0) becomes
elp + e.., = 8 (14~Gw~ 18;e.=O) '(lO)
the famous "Rule of S" (octet rule). Thus, for d-block 'elements the corresponding relationshipis elp+~= 12 (6.s~ .sll; eu= 0) (J J)~
which may be tenDed the "Rule of 12" (dodectet rule) for stable closed-shell transition metal species. Just as the four valence orbitals (s + 3p) of the p-block underlie the usual Lewis octet rule (10), so may the six valence orbitals (s + 5d) of the d-bock be expected to underlie the corresponding dodectet rule (11) for transition metals. The distribution of six electron pairs around a dodectet-conforming transition metal atom M can be represented with a hexagonally-shaped "dot diagram"
analogousto the usualsquare dot-di~gram for an octet-conformingmain-group element X - "i -0- or :-x-: .. I Somesimple examples of dodectetconfonning Lewis-like formulasinclude I-IV H .. .. :, I .,..CH3 :, I H '" H'kH PI H2& W= CH H"'I'8 ;'" I 'CH3 H""~=CH.z H - H I D m IV 98 WEINHOLD & LANDIS illustrating the extendedcapacity for multiple metal-metalbonding with main-groupligands. One can also considera still moreunusual possibility suchas V
H-W. W-H v illustrating the extended capacity for multiple metal-metal bonding. Like its octet counterpart (13), the dodectet diagram (12) depicts the valence electron configuration and bond connectivity, but not (necessarily) the 3-dimensional geometry about the transition metal. In the spirit of the Pauling-Slate~6 theory of main-group hybridization, the actual molecular geometry of transition metal compounds is expected to correspond approximately to the separation angles a of idealized scJ1bonding hybrids (~= n -1) that form the sigma skeleton. Equivalent scJ1hybrids can be shown27to allow two possible separation angles, the acute angle (aaccute)or obtuse angle (aootuse)satisfying
aKOlk = COSO
QobIIIIC =cos-l-
Numerical valuesof theseidealized "natural" bond anglesare shownin Table 2 for various ~ [correspondingto percentaged-character = J1(~+ 1)]. Note that acutehybrid anglesdown to -550 are perfectly possiblefor transitionmetal sd-hybridization,whereas hybrid angles< 900 are forbidden in main-groupsp-hybridization.28
TABLE 2. Natural bond angles (aec- ~ . degrees) and percentage d-character of equivalent sa" hybrids; cf. Eqs. (14a b).
hybrid p a.-ac aobtuae %-d 90.00 66.67 st/ 109.47 75.00 sa" 114.09 80.00 sri 116.57 83.33 s~O 121.09 90.91
d ~ 125.26 100.00 NATURAL BOND ORBITALS 99
(b) Pt(CH3)2 (II.)
(c) H~OsCH~ (III) (djHW(CH,,)(CH)... (IV)
FIGURE 1: Equilibrium B3LYP/LANL2DZgeometries of I-IV. OptimizedL-M-L' bond angles: (a) 63.7(3), 67.1 (3), 113.8(3), 119.7(6); (b) 102.9 ; (c) 94.1 (2), 105.8 .. (d) 94.2, 95.2, 100.5.
Optimized B3L VP/LANL2DZ equilibrium structures for I-IV are shown in Figure 1, illustrating the propensity toward strongly bent geometries (e.g., st/ -like -650 , -1160 in I; scf -like -900 bond angles in m, IV) that are quite surprising from a YSEPR-like steric or electrostatic perspective. NBO analysis confinns the aptness of the localized Lewis-like description in each case. Table 3 summarizes details of the optimal hybrids, bond polarities, and overall accuracy of the NBO Lewis-like description (in terms of %-PL, the percentage of valence electron density representable by a strictly localized Lewis-like wavefunction), showing the high accuracy (97-99%-PL) of this simple picture. The localized M-C, M-H bond NBOs are all found to be surprisingly apolar (38-53% on M) and to have high "pair" occupancies (1.92-1.98e), comparable to the values found in pure hydrocarbons. 100 WEINHOW & LANDIS
TABLE 3. NBO descriptorsof I-IV, showing overall accuracy of NBO Lewis-like valenceelectron density (%-Pt.) and metal atom hybridization (~), bond polarity toward M (100 I CM11, diatomic symmetrytype (0, K, ~), and occupancy(e) of eachM-L bond NBO OMI.~ cwhw+ M.
M-LbondNBO ,i:"'"-~ .
species %-PL type hw 100I CW 12 OCC.(~)
Figure 2 illustratesthe optimizedgeometry of H-W. W-H(V) and contour diagrams of the quintuple-bondNBOs (one ofaww, two ofxww and two of&ww type), eachdisplayed in a plane to best display its distinguishing characteristics. The shapesof d.-d. metal-metalx- bonds (Fig. 2b,c) differ strikingly from those of main-groupcompounds (as expected),and both hybridized (~(h),Fig. 2d) and unbybridized (~(u),Fig. 2e) types of &-bondshave no counterpartsin main-groupchemistry. Nevertheless, the localizedmetal-metal NBOs exbl'bit a degreeof recognizabilityand transferabilitythat is in many respectssimilar to that of main- group Lewis structures.This commonalitysuggests the generalefficacy of a unified Lewis- like picture of bonding in p- and d-block elements.The NBO 5.0 algorithmsfor automated NRT resonancestructure searches have thereforebeen extendedto include such Lewis-like dodectetstructures, greatly improving applicability of resonanceconcepts to transition metal species. The resultspresented here for I-V are broadly representativeof the virtually unlimited possibilities for dodectet-confonningcompounds that can be readily envisioned from the mnemonic diagram (12). Particularly interesting are the compoundsfonned by tetravalent Group 8 elementsthat representanalogs of commonorganic species,such as "methane-like" VI "allene-like" VII , and "butadiene-like"vm , H H )c.s=O: H r \. I H,I."H H 0.=0& H20s=0s= QsH2 r "\ H"'~H.. H H VI VII VIII NATURAL BOND ORBITALS 101 --
(a) 0'ww = (sdlo.3)w+ (sdl°.3>w
' .', \ : \ -:::~~~( r- " i\ I.~; . ,.-~,..:,t "".'
~4,"r~2\..;/ l: !'.' j ~ ~-
xy D (b) If ww = (dxz>w + (du>w (c) rrn = (~>w + (~y>W
" ...: ., ..' $( " . , ~ . " , , : .. .""--"" ... I .'., '. . . '.' . ::{ " ..:; ~ :\" ,..",.}J ,:;!I{Q2
" ,.,.-:::::~..'.' '" -.-=":=:::':'-:;::=j~:'~." (-'-" '-"'.-- ,~_.- . ~ : I ::: ", . 'I. 1-.-.I-"-:~_. . .- ," . ;' " . .. ,,1.. . '-.'" " --,' '.-' :,. , ~_. - .. " , , ,: : -. ..\ , . ' .' . . ' ," : ~ ~o'.
D xy
(e) ww - (~>w + (dy.>w
1%
FIGURE 2: Metal-metal bond NBOs of quintiply-bonded H-W8 W-H (O/o-PL= 99.87%), showing hybrid composition (above) and chosen contour plane (lower left) for each bond. 102 WEINHOLD & LANDIS
It is remarkable,for example,that the localizedNBO Lewis-like descriptionVI for Os~ is more accurate(as judged by %-PL)than the standardlocalized Lewis structurefor methane. Perhapseven more remarkableis the observationthat Os~ has three local minima (Td, C2v, C4vsymmetry), corresponding to the three possiblegeometries in which all bond anglesare either -109° or -70°, as allowedby sd3hybridization. Compared to analogousmain group compounds,the Lewis-like transition metal compoundsare much more susceptibleto 3-center, 4-electron hypervalent interactions29 ("ionic resonance,"X-M :Y X: M-Y) which increasethe formal coordinationat the metal center.NBO 5.0 now includes a searchfor 3-centerhyperbonds (keyword: 3CHB) to flag these interesting features of molecular structure. Other systematic differences in d.-d. interaction strength and transition metal electronegativity also contribute to the richly distinctive chemistry of the d-block.30Nevertheless, we believe that considerableinsight (including identification of likely synthetictarget species)can be gainedby recognizingthe many similarities betweenlocalized Lewis-like bonding in p- and d-block elements,while also acknowledgingtheir characteristicdifferences. The many new features of NBO 5.0 shouldallow standardcomputational chemistry packages to play an ever more importantrole in productiveextensions of localizedbonding concepts.
CORRESPONDENCE: Frank WEINHOW. Department of Chemistry. University of Wisconsin. Madison. Wisconsin53706. USA; e-mail: [email protected]
Frank Weinhold and Clark R. Landis are both Professorsof Chemistry at the University of Wisconsin,Madison, where they are currently collaboratingon Ref. JOb.F. Weinhold, a memberof the Physical ChemistryDivision and Theoretical ChemistryInstitute, completedgraduate studies at Harvard with £.8. Wilson, Jr. and postdoctoral studies at Oxford with C.A. Coulson. His CUlTent researchfocuses on the nature of hydrogen bonding and the quantum cluster equilibrium (QCE) model of H-bonded fluids. C.R. Landis, a member of the Inorganic Chemistry Division, completed graduatestudies at the University of Chicago,with Jack Halpern. His CUlTentresearch focuses on the nature of bonding at transition metal complexes,the developmentof empirical force field methods that spanthe periodic table, mechanisticstudies of organotransitionmetal catalysts,and novel parallel synthetic strategies for making phosphine ligands and accelerating new catalysts for making phosphineligands and acceleratingnew catalystdiscovery.
NOTES AND REFERENCES
P.-O. LOwdin,Phys. Rev. 97, 1474-1489(1955). While the presentwork is forolulated in the molecular orbital framework of Hartrce-Fock (HF) or density functional theory (DFT), the entire treatmentcan be readily generalizedto any wavefunction,including the exact'V. 2. Gaussian98, M. J. Frisch, G. W. Trucks, H. B. Schlegel,G. E. Scuseria,M. A. Robb, J. R. Cheeseman,V. G. Zakrzewski. J. A. Montgomery, Jr., R. E. Stratmann,J. C. Burant, S. Dapprich, J. M. Millam, A. D. Daniels, K. N. Kudin, M. C. Strain, O. Farkas,J. Tomasi, V. Barone,M. Cossi,R. Cammi, B. Mennucci, C. PomeUi,C. Adamo, S. Clifford, J. Ochterski, G. A. Petersson,P. Y. Ayala, Q. Cui, K. Morokuma, D. K. Malick, A. D. Rabuck, K. Raghavachari,J. B. Foresman,J. Cios10wski,J. V. Ortiz, A. G. Baboul, B. B. Stefanov,G. Liu, A. Liashenko,P. Piskorz, I. Komaromi, R. Gomperts,R. L. Martin, D. J. Fox, T. Keith, M. A. AI-I .a.harn,C. Y. Pens,A. Nanayakkara,C. Gonzalez,M. Cba11acombe,P. M. W. Gill, B. Johnson,W. Chen, M. W. Wong, J. L. Andres, C. Gonzalez,M. Head-Gordon,E. S. Replogle,and J. A. Pople,Gaussian, Inc., PittsburghP A, 1998. 3. F. Weinhold, J: Chem.Ed. 76, 1141(1999). 4. More precisely, the transformation from symmetry-adaptedto non-symmetry-adapted forol correspondsto a unitary transformation of orbitals that merely multiplies 'V by an overall phasefactor, and hence bas no effect on the energy, density, or other measurable NATURAL BOND ORBITALS 103
properties. .s. J. P. Fosterand F. Weinhold,J. Am. Chem.Soc. 102,7211 (1980); A. E. Reed.L. A. Curtiss, and F. Weinhold, Chem.Rev. 88, 899 (1988); F. Weinhold,Natural bond orbital metbodY,in, P. v. R. Schleyer, N. L. Allinger, T. Clark, J. Gasteiger, and P. A. Kollman (eds.), "Encyclopedia of Computational Chemistry" (Wiley, Chichester,UK. 1998) Vol. 3, pp. 1792-1811. 6 For open-shellsystems, electrons of different spin generally occupy different spin-orbitals, and the occupancylimit of eachspin-orbital is accordinglyreduced to one. We consideronly closed-shellsystems in the presentwork. 7. G. N. Lewis, J. Am. Chem.Soc. 38,762 (1916); G. N. Lewis, "Valence and the structureof atomsand molecules"(The ChemicalCatalog Co., New York, 1923). 8 Mutually orthogonality of a set of orbitals is an elementaryprerequisite for associatingthe orbitals with a physical (Hemlitian) operator as eigenfunctions. Overlapping orbitals thereforecannot be consideredas eigenfunctioosof any ..~ Hamiltonian" of physical significance. 9. R. S. Mulliken, J. Chem.Phys. 46,497 (1944); J. Hinze and H. H. Jaffe, J. Am. Chem.Soc. 84,540 (1962); R. Hoffinann,J. Chem.Phys. 39, 1397(1963). 10 R. S. Mulliken, J. Chem.Phys. 3, 573 (1935). II. Jaguar 3.5, Schroedinger,Inc., Portland,OR, 1998. 12. High PerformanceComputational Chemistry Group, NWCHEM;A computationalchemistry program for parallel computers. Version 4.0.1 (2001), Pacific Northwest National Laboratory,Richland Washington99352. 13 Q-Chem 1.1, B. G. Johnson, P. M. W. Gill, M. Head-Gordon,C. A. White, J. Baker, D. R. Maurice, T. R. Adams, J. Kong, M. Challacombe,E. Schwegler, M. Oumi, C. Ochsenfeld,N. Ishikawa,J. Florian, R. D. Adamson,J. P. Dombroski, R. L. Graham,and A. Warshel,Q-Chem, Inc., Export. PA, 1997. 14 ParallelQuantum Solutions, 2013 GreenAcres RoadSuite A, Fayetteville,Arkansas 72762 (http://www.pqs-chem.com) IS GAMESS(-US).M. W. Schmidt,K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen,S. Koseki, N. Matsunaga,K. A. Nguyen, S. So, T. L. Windus, M. Dupuis, and J. A. Montgomery,Jr., J. CompoChem. 14, 1347(1993). The PC-GAMESSversion is available throughA. A. GranOVSky(http://classic.chem.msu.su/gran/gamess/index.htmI) 16. NBO 5.0, E. D. Glendening,J. K. Badenhoop,A. E. Reed,J. E. Calpenter,J. A. Bohmann,C. M. Morales, and F. Weinhold, Theoretical Chemistry Institute, University of Wisconsin (2001). .7. The procedureto attach or "activate" NBO 5.0 capability varies with the ESS package.For Jaguar,NWChem, Q-Chem,and PQS, NBO 5.0 co~ pre-installed.NBO 5.0 also comes pre-installed in PC-GAMESS (to be activated by a passwordkey) but must be manually installed in GAMESS-US (by a proceduredescribed in the manual). An earlier "NBO 3.1" versionalso comespre-instalJed in Gaussian98, but this can be easily replacedby the current NBO 5.0, using instructionsprovided in the manual.Current order information for NBO 5.0 program and manual: TCI/NBO Software ([email protected],608-262-5153) or the NBO 5.0 website(Ref. 24). 18 S. J. Wilkens, W. M. Westler,J. M. Markley, and F. Weinhold(submitted forpublicatioo). 19. J. A. Bohmann,T. C. Farrar,and F. Weinhold,J. Chem.Phys. 107, 1173(1997). 20 E. D. Glendening and F. Weinhold, J. Comp. Chem. 19, 593-609, 610-627 (1998); E. D. Glendening,J. K. Badenhoop,and F. Weinhold, J. CompoChen!. 19,628 (1998). 21. E. D. Glendeningand A. Streitwieser,J.Chem. Phys. 100,2900 (1994). 22. J. K. Badenhoopand F. Weinhold,J. Chen!.Phys. 107,5406-5421,5422-5432(1997); Int. J. QuantumChem. 72,269 (1999). 23 A. V. Nemukhin and F. Weinhold,J. Chem.Phys. 97, 1095(1992). 24. http://www.chem.wisc.edu/-nbo5 25 A1fa AesarResearch Chemicals, "Metals and MaterialsCatalog"' (1999-2(XK». 26 L. Pauling,.l Am. Chem.Soc. 53, 1367(1931); J. C. Slater,Phys. Rev. 37,481 (1931). 104 WEINHOW &. LANDIS
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