The purpose of the present paper is to describe a method,
in quantitative sense, of discussing the isolobal analogy of fragments
(or molecules) proposed by Hoffmann. The »Frontier Hybrid
Orbitals (FHO)« are used to represent the directional, fronti er
orbitals of the central atom, which have the same meaning as the
»lobes« involved in the concept of isolobal analogy. The calculatio
n of Frontier Hybrid Orbitals (FHO\u27s) is based on the natural hybrid orbitals which are the eigenvectors of density matrix blocks. The overlap integrals, bonding capabillties, and valence orbital energy levels of the FHO\u27s of fragment s are also calculated. The relationship between FHO and the isolobal analogy is discussed in this paper

Cao Yang

Wang Youliang

English

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CROATICA CHEMICA ACTA CCACAA 62 (4) 669-676 (1989)CCA-1896YU ISSN 0011-1643UDC 541Conference PaperFrontier Hyhrid Orbitals. I. Principles*Cao Yang and Wang YouliangDepartment of Chemistry, Suzhou University, Suzhou, People's Republic of ChinaRecei ved September 18, 1989The purpose of the present paper is to describe a method,in quantitative sense, of discussing the isolobal analogy of frag-ments (or molecules) proposed by Hoffmann. The »Frontier Hy-brid Orbitals (FHO)« are used to represent the directional, fronti erorbitals of the central atom, which have the same meaning as the»lobes« involved in the concept of isolobal analogy. The calcu-latio n of Frontier Hybrid Orbitals (FHO's) is based on thenatural hybrid orbitals which are the eigenvectors of densitymatrix blocks. The overlap integrals, bonding capabillties, andvalence orbital energy levels of the FHO's of fragment s are alsocalculated. The relationship between FHO and the isolobal analogyis discussed in this paper.INTRODUCTIONThe concept of the isolobal analogy was proposed by Hoffmann,"?according to the Extended Huckel Molecular Orbital (EHMO) study resultsof molecular fragments, and the relationship between the hybrid orbitalsof the valency bond picture introduced into chemistry by Pauling" and themolecular geometry. Two fragments are said to be isolobal if the number,symmetry properties, approximate energy and the shape of the frontierorbitals and the number of their electrons are similar. From this point ofview, the isolobal analogy is determined by the properties of the frontierorbitals of fragments or molecules. The utility of the isolobal analogy isthat if two fragments are isolobal, then they should have a very similarelectronic description, structural characteristic and reactivity in qualitativesense.In the present paper we describe away of discussing the isolobalanalogy of molecular fragments depending on the atomic hybrid orbitalsin quantitative sense. The »Iobe« involved in the isolobal analogy is astrongly directional orbital forming a chemical bond, which is the delo-calized frontier orbital and indicates the characteristic of electronic stru-cture, bonding and reactivity of the fragment or molecule. The orbital, whichhas maximum bonding ability and is directional, can be substituted by thehybrid orbital of the central atom which is the localized result of molecularorbitals. We propose that the name »Frontier Hybrid Orbital (FHO)« be* The project supported by National Natural Science Foundation of China.670 C. YANG AND W. YOULIANGused for the orbital. In this work, the ca1culation method of FHO, the over-lap integrals, bonding capabilities, and valence orbital energy level ofFHO are described. And the relationship between FHO and isolobal analogyis primarily discussed.CALCULATION METHOD(1) Natural Hybrid OrbitalsFor the closed-shell system, the LCAO MO's are of the form:'Pi = ~C"i e, (1;"In numerical applications, orthogonalized AO's are consistently used, i. e.S~~~=~ 00Now by definition,P!lV = ~ ni C\i c,i(3)Pl'l' = ~ni C2l'ii(4)The matrix P with the elements PI'V or Pl'l' is called the molecular densitymatrix. Consider the n X n density matrix P as divided into atomic and inter-atomic blocks as,rP AA P AB .•• P ALPBA PBB··• PBLP=lPu P" ... Pu1j(5)here the submatrix PAA of order m X m is associated with the m atomicorbitals on A. The n n transformation matrix is T, the new matrix p' isthen given byP' = TPT (6)Now suppose we make an orthogonal transformation of the atomicbasis orbitals {<P} into a new set {h} such that the intraatomic off-diagonalelements of the new density matrix p' are zero. The set {h}, originallyproposed by McWeeny'', are called the natural hybrid orbitals (NHO). Thetransformation which converts the AO basis to the NHO basis is simplya special case of the general transformation T, leading to the diagonalizationof block P and thus to the NHO's of A.The transformation considered here is similar to that of the well-knownJacobi matrix diagonalization procedure.(7)Where, n/Aj and h/A> represent the eigenvalues and vectors, respectively.The eigenvectros are called the natural hybrid orbitals (NHO) of atom A.FRONTIEH HYBRID ORBITALS 671(2) Frontier Hybrid OrbitalsFor the open-shell system, the LCAO-MO's are of the form:'1'/' = ~ CI'i" <I>"IJ.'I'} = ~ C!'i~ <I>I'IJ.(8)(S)Now by definition,(10)iP!")~= ~ni C!'i~ Cvi~i(11)The total density matrix elements p!,!, are their sum,p!'v = p!'v" + P!'} (12)But the spin density matrix elements P!,vs are defined as the subtraction ofPIlV~ from PIlV",(13)Here, ps and PAAs are used to denote the total spin density matrix andthe block which only involves the atomic orbitals of atom A, respectively.In the diagonalization procedure as mentioned above,(14)where, ni(A) and hi(A) represent the eigenvalues and vectors, respectively,on which the natural hybrid orbitals occupied by one-electron can bedetermined. According to Foster and Weinhold.!" the eigenvalues of mole-cular density matrix indicate the characteristic of the bond and the elec-tronic structure of atoms in the molecule. For a fragment, which has oneor more single-electrons, its spin density matrix can be obtained by mole-cular orbital calculation, and its natural hybrid orbitals occupied by single-electro n can be determined by eq. (14). In general, when ni(A) = 1, therelative eigenvector hi(A) is the natural hybrid orbital occupied by single-electron, for which we propose that the name »Frontier Hybrid Orbital« beused. For a fragment, which has no single-electron, but has empty orbitalsor lone-pairs, its density matrix can be obtained by MO calculation, andits NHO's occupied by empty orbitals or lone-pairs can be determined byeq. (7). In general, when n/Al = O, the relative eigenvector h;<A) is thelone-pair hybrid orbital. For these empty and lone-pair hybrid orbitals,we also propose that the name »Frontier Hybrid Orbitals« be used. It mustbe noted that the above discussion and conclusions are based on the frag-ments MO calculations of atomic valence orbitals and valence electrons.Since the density matrix contains pertinent information about thesystem and can be defined in ab i.nitio as well as semiempirical MO methods,the quantitative features of the definition do not depend on a particular MOapproximation. Depending on the accuracy of the density matrix, whichin turn depends on the wavefunction used, quantitative changes may occurfor the value of Frontier hybrid orbitals.672 C. YANG AND W. Y01.JLlANG(3) Overlap Integrals, Bonding Capabilities, and 11alence Orbital EnergyLevels of Frontier Hybrid OrbitalsThe frontier hybrid orbitals (FHO's) of a fragment are a linear com-bination of the valence orbitals of the central atom, whose exponents arechosen as the orbital exponents of the central atom. To calculate the overlapintegral between a FRO and another atomic orbital or another FRO, theoverlap integral between the atomic orbitals of which the FRO is composedand other atomic orbitals is to be calculated first. Let hk denote the FRO,nk = };Cike, (15)iwhere <Pi is the atomic orbitals of which the FRO is composed, so theoverlap integral between hk and <Pj is,Skj= <hk I <{lj> = };CikSiji(16)Symbol i indicates that the sum includes all atomic orbitals of which hk is,composed. Then the overlap integral between hk and other FRO which isdenoted by hI is,Skl= (hk I hI) = };~ CikCjlSiji j(17)Here, symbol i has the same meaning as mentioned above, and symbol jindicates that the sum includes all atomic orbitals of which hi is composed.According to Mulliken.v ?" the bonding capability of a hybrid orbitalcan be determined by the overlap integrals between the hybrid orbital andthe orbitals of another ligand. Based on the discussion of the bondingcapabilities of fragments made, by Hoffmann," we suggest that the overlapintegrals of FRO's of fragments with the atomic orbital of Ratom be usedas the relative bonding capabilities of the FHO's of fragments, representedby F. In general, for an organic fragment ALn, the value of F is calculatedwhen the distance of A-H is 1.10 A. And for an inorganic fragment MLn,the value of F is calculated when the distance of M-H is 1.50 A.The Valence Orbital Energy Level (VOEL) of a FHO is chosen as theweighted average of the energy levels of atomic orbitals of which the FHOis composed. For example, the FHO of CH is 0.354(s), 0.935(p), then itsVOEL is,VOEL (CH3) = 0.354 VOEL(Cs)+ 0.935 VOEL (Cp) = -0.757The atomic valence orbital energy level values which are dealt with in thispaper are listed in Table I, on the basis of CNDO/2 approximation.TABLE IAtomic VOEL values of CNDOj2 (eV)Atom ns np Atom ns np (n-1)dH -0.38887 Cr -0.0585 -0.0757 -0.798B -0.5887 -0.2258 Mn -0.0537 -0.0709 -0.817C -0.8116 -0.4999 Fe -0.140 -0.035 -0.943N -1.773 -1.330 Co -0.494 -0.317 -1.716Al -0.451 -0.166 Ni -0.875 -0.695 -2.573Si -0.579 -0.362 Cu -1.189 -1.002 -3.083FHONTIEIl HYBHID ORBJTALS 673RESULTS AND DISCUSSIONIn this work, the CNDO/2-UHF method is applied to calculating aseries of organic and inorganic fragments, whose geometrical details chosenare the same as Hoffrnann's.š-"? and the standard CNDO/2 atom parametersare used. In terms of the calculation method mentioned above, the FHO'sof these fragments are calculated and illustrated in Tables II and III. TheTABLE IIFHO's of some AH3 fragments and their propertiesCoefficiens of atomicFragment Feature" orbitals in FRO F VOELs PCR3 S 0.354 0.935 0.432 -0.757CHs+ E 0.175 0.985 0.392 -0.635CH3- L 0.579 0.815 0.461 -0.878BH3 E 0.243 0.970 0.519 -0.362BH3* S 0.417 0.909 0.558 -0.451BHst- L 0.571 0.821 0.579 -0.522NHs L 0.612 0.791 0.361 -2.137NH3+ S 0.278 0.961 0.320 -1.771NHs+2 E 0.116 0.993 0.287 -1.526sin, S 0.403 0.915 0.601 -0.565SiHs+ E 0.218 0.976 0.555 -0.480SiHs- L 0.558 0.830 0.624 -0.624* Note: The emty, single-electron, and lone-pair hybrid orbitals are symbolizedby E, S, and L, respectively.TABLE IIIFHO's of fragments shown in Fujure 1 and their propertiesCoefficients of atomicFragment Symmetry orbitals in the FHO F VOELs p dCHs Cs. 0.354 0.935 0.432 -0.755CH2 C2v 0.508 0.861 0.454 -0.8431.000 0.342 -0.500CH linear 0.646 0.441 0.354 -0.7451.000 0.342 -0.5001.000 0.342 -0.500Mn(CO); C4v 0.198 0.446 0.573 0.435 -0.511Fe(CO)4 Csv 0.481 0.665 0.571 0.673 -0.6290.284 0.959 0.307 -0.914CO(CO)3 C2v 0.627 0.779 0.717 -0.5570.204 0.979 0.253 -1.7450.204 0.979 0.253 -1.745CpFe(COh 0.104 0.298 0.949 0.360 -0.920CpCo(CO) 0.354 0.391 0.850 0.497 -1.7570.108 0.413 0.904 0.408 -1.736CpNi 0.803 0.593 0.057 0.702 -1.2610.141 0.990 0.204 -2.6450.073 0.140 0.987 0.236 -2.701Co(CO)s C2v 0.273 0.412 0.865 0.475 --':1.750Ni(COl2 C2v 0.693 0.720 0.046 0.722 -1.2250.172 0.985 0.221 -2.654Cu(CO) linear 0.789 0.602 0.126 0.707 -1.9301.000 0.562 -1.0000.162 0.006 0.987 0.188 -3.241674 C. YANG AND W. YOlJLIANG8 eleclron 18 electron 16 e l e c t r o nL~H, I L ,L, IC-H L-Cr ~L Mn L-Fe-=-LW"', L•••.•.I -- \ L"H L OC // Coe OC i ma!-HO 1-Lo+ J -CO+ J -Lo+-, I / ~ ..." I - C;:J,C:J O- - Mn"::) --U ...F.c O-// /1 //\) "b f j n!-H o !_Lo+ !-CO+ 1-Lo+-_0 1 ~ I;;Ni3--Fe 3 Co,.....~ U- - o • () .. /\y • o·19 k' ocj-H o 1-L. + !-co" j -L·+-c3 '::c~~ -[;3I-U 04 () •..~0-I o (jd h I PFigure 1. Isolobal Relationshipsbonding capabilities (F) and valence orbital energy levels (VOEL) of theFHO's are also listed in Tables II and III.The AH3 fragments which are proposed to have the same geometriesare calculated, and their FHO's including empty, single-electron, and lone-pairhybrid orbitals are obtained and illustrated in Table II. They differ inquantity from each other. The results show that for an empty hybridorbital, its p-orbital character is the highest, the value of F is the lowest,and the value of VOEL is the highest, and that for a lone-pair hybrid orbital,its p-orbital character is the lowest, the value of F is the highest, and thevalue of VOEL is the lowest. In term s of Hoffmann's isolobal analogy, theAH fragments listed in Table II are isolobal to each other. From Table II,it can be seen that the FHO's of these fragments and their properties arequalitatively similar, but quantitatively different.The isolobal analogy of particular fragments is shown in Figure 1.Breaking one C-H bond from methane in a homolytic sense generates theFRONTIEH HYBRID ORBITALS 675f g~~oFigure 2. Contour diagrarns of FHO's of fragments in Figure 1. Contours shownare: -0.04, 0.02, 0.08, 0.14, 0.20, and 0.26.methyl radical, b. It has one FHO pointed in the direction of the miss inghydrogen atom with one electron in ito Removal of another hydrogen atomgenerates methylene, c. There are two FHO's pointed to the two missinghydrogen atoms. Removal of still another hydrogen atom creates a methinefragment, d. It has three FHO's with three electrons partitioned betweenthem. Now breaking one Cr-L bond from octahedral complex CrL6 createsa CrL,: fragment, the neutral isoelectronic fragment of which is MrrL; f. Ithas one FHO directed toward the missing ligand with one electron in itoRemoval of a second ligand from MrrL, and moving one element to theright in the periodic table forms the FeL4 fragment, g. And removal of athird ligand generates the COL3fragment, h.The FHO's of fragments shown in Figure 1 and their properties areillustrated in Table III. The contour diagrams of FHO's of fragments (b, c, t,g, j, k, n, o) are shown in Figure 2. From Table III and Figure 2, it can beseen that the FHO's of the fragments shown in Figure 1 are similar from676 C. YANG AND W. YOULIANGthe left to the right in the same line, but different from each other inquantity.In terms of the above discussion of the FRO and isolobal analogy, itis certain that the FRO calculated by the method described in this paperis useful and can be used to discuss the similarity and difference of fragments,::;0 that the electronic structure, bonding capability, and reactivity of amolecule can be analyzed by the FRO of its fragments. We shall continuethis discussion in our following work.REFERENCES1. R. H off man n, Angew. Chem. Int. Ed. 21 (1982) 711.2. M. Eli a n, M. M. L. C h e n, D. M. M i n g o s, and R. H off man n, Inorg.Chem. 15 (1976) 1148.3. T. A. Al b r i g h t, J. K. B u r d e t t, and M.-H. W han g b o, Orbital Inter-actions in Chemistry, J. Wiley: New York, 1985.4. J. K. B u r d e t t, Molecular Shapes, Wiley-Interscience, New York, 1980.5. M. Eli a n and R. H off man n, Inorg .. Chem. 14 (1975) 1058.6. K. Tat s u m i and R. H off man n, J. Amer. Chem. Soc. 103 (1981) 3328.7. D. M. H o ff man n and R. H o ff man n, Inorg. Chem. 20 '(1981) 3543.8. L. P a u 1i n g, The Nature of the Chemical Bond, 3rd, Cornell University Press,Ithaca, New York, 1960.9. R. Mc W e e n s, Rev. Mod. Phys. 32 (1960) 335.10. J. P. F o ste r and F. W e i n n h old, J. Amer. Chem. Soc. 102 (1980) 7211.11. R. S. M u Il ike n, J. Amer. Chem. Soc. 72 (1950) 4493.12. Z. S. He r man n and L. P a u 1i n g, Croat. Chem. Acta 57 (1984) 765.SAŽETAKGranične hibridne orbitale. I. Princip iCao Yang i Wang YouliangUveden je pojam graničnih hibridnih orbitala i ustanovljena je njihova rela-cija s koncept om ekvivalentnih orbitalnih režnjeva. Granične hibridne orbitaledobivene su pomoću »prirodnih hibrida«.1

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