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F3a51a841b453b267287709386 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 <hA I F I hB> of bonding NHOs hA, hB can be approximated in terms of overlapping PNHOs h A, h B as <hAI F 1hB>:k<hAIha> "(3) where k is a proportionality constantof order unity. Thus, PNHO overlap diagramsremain highly effective for teaching studentsthe "principle of maximum overlap," but without encouragingthe frequentmisconception that geometricalorbital overlaps(rather than matrix elementsof the systemHamiltonian) are somehowthe origin of chemicalbonding. NATURAL BOND ORBn' ALS 93 {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 ,~<a;IFla;> 'r~<a;IFlaj> ",fu,re'pe,ti"om'tal~""~ofdono'ond ='P'"' NBO, C""id=tioo of ,"~" ontihoad, (4h) fu'refure I~" '" fMreoching 'XIoMion of "'m~"" Low'. ""'"" ,on"",, '" 'DOomp'" I~ding d,I~lization oon~tiOM in .imp" NBO P-"'ti" '.'=10, ..,h ~ Eo ('\ 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(*)
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