Bruce E. Bursten- Some comments on approximate LCAO molecular orbital theory in organometallic...

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Pure & A p p / . Chem., Vol. 63, No. 6, pp . 839444,1991Printed in Great Britain.@ 1991 IUPACADONIS 206922209100090T

So me comm ents on approximate LCAO molecularorbital theory in organometallic chemistry: Gettingmore by doing less?Bruce E. BurstenDepartment of Chemistry, The Ohio State University, Columbus, Ohio 43210,U.S.A.

Abstract - The use of approximate LCAO molecular orbital theory as anelectronic structural tool in organometallic chemistry is brieflyreviewed.disadvantages of these approximate methods as compared to rigorousHartree-Fock calculations.

The author offers his opinions concerning the advantages and

I N T R O D U C T I O NThe year 1951 was one of tremendous advances in both theoretical chemistry andorganotransition metal chemistry. In that year, Roothaan published his important treatiseon quantitative LCAO molecular orbital theory (ref. l), the starting point for all modernbasis set expansion methods for calculating the electronic structure of molecules. Duringthe same year, organotransition metal chemistry entered a new age with the synthesis offerrocene and the recognition of its unusual sandwich structure (ref. 2 ) . From theseconcurrent beginnings, electronic structure theory and organometallic chemistry haveexperienced phenomenal growth and maturation.theoretical organometallic chemistry, also began in the early 1950's with the first reportsof molecular orbital treatments of ferrocene (ref. 3 ) . Quantum chemists have continued tobe fascinated by the challenges posed by organometallic systems, and theoreticalorganometallic chemistry has grown to be a thriving, well-recognized chemical discipline(ref. 4 ) .In its infancy, the application of electronic structure methods to organometallic compoundswas largely an exercise in approximation: The generally large size of organometallicmolecules, both in terms of numbers of atoms and numbers of electrons, rendered an exactsolution of the Roothaan equations untenable (ref. 5). Thus, theoretical organometallicchemistry (not unlike theoretical organic chemistry) developed through the use ofapproximate molecular orbital methods. As computational capabilities improved, the need forusing approximate methods lessened, to the point that it is now possible to do ab initioHartree-Fock and even correlated treatments of reasonably large organometallic molecules(ref. 6). In spite of these computational advances, however, approximate molecular orbitaltheory is still the most popular electronic structural probe used to study organometalliccomplexes. Part of the reason for this popularity is computational expense; while it isindeed possible to do "good" calculations on organometallics, these calculations still taxthe capabilities of even the most advanced supercomputer. The primary raison d'6tre forapproximate methods in organometallic chemistry, however, is even more pragmatic:Approximate methods continue to provide theoretical and experimental organometallic chemistswith useful, readily interpretable, intuitively reasonable results that often lead toadvances in the field.

The interface of these two fields,

In this contribution, the field of approximate LCAO molecular orbital theory as applied totransition metal organometallics will be briefly reviewed.opinions concerning the advantages and disadvantages of these methods, and provide somecaveats for those investigators who wish to use these methods with no prior training in thefield.recent contribution by Fenske to this journal (ref. 7 ) . I will limit the discussion toapproximate and rigorous methods based on the Hartree-Fock method as a starting point orend; the newer breed of XQ and local-density-functional LDF) methods (ref. 8), which arephilosophically rather different, will not be discussed here.

The author will offer his

Many of the points made in this paper are supplementary to those in the excellent

* LCAO = Linear Combination of Atom ic O rb ita ls839


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840 B. E. BURSTEN

W H A T I S A P P R O X I M A T E L C A O M O L E C U L A R O R B I T A L T H E O R Y ?In closed-shell olecular orbital theory (ref. 9 ) , the molecular wavefunction is expressedas a single Slater determinant of doubly-occupied molecular orbitals, d i .the one-electron artree-Fock equation: Each d i satisfies

f d i - id1 (1)In eq. 1, f is the so-called Fock operator and e i is the molecular orbital energy.are given physical meaning through Koopmans theorem (ref. lo), which allows the c i to berelated to ionization energies of the molecule (reg. 11).standard eigenvalue equation, the construction of f requires a knowledge of the d i . As aresult, eq. 1 is actually a pseudoeigenvalue equation, the solution of which requires aniterative grocedure: A guess is made for the form of the Q i , which are then used toconstruct f, which is then used to solve :or new b i , and so forth.continued until the bi used to construct f are sufficiently similar to the ,ji obtained bysolving equation 1, at which point we have a self-consistent-field SCF) solution.Roothaan (ref. 1) treated the case in which the MOs d i are expanded as a linear combinationof basis functions, x,:

The e ,Although eq. 1 looks like a

The iterations are

d i - = crix, ( 2 )In eq. 2 , the clli re expansion coefficients and the summation is over all x,.expansion in eq. 2 is substituted into eq. 1, followed by linear variational minimization ofthe total energy, the Hartree-Fock-Roothaan atrix equation results:

When the

FC = SCe ( 3 )In eq. 3 , F (the Fock matrix) and S (the overlap matrix) are matrices whose elements consistof integrals calculated over the basis functions x,:

FbV- < X , I f I x u > ( 4 )s, v - < X, I x u > ( 5 )

The matrix C (the coefficient or eigenvector matrix) is composed of the expansioncoefficients cpi, nd c is a diagonal matrix of the orbital energies e i .3 is also an iterative procedure, this time on the expansion coefficients cOi.guess of cPi is used to construct the Fock matrix F.techniques of linear algebra, leading to the secular equation

The solution of e q .An initialEq. 3 is then solved using standard

IF - cSI - 0 ( 6 )The eigenvectors of eq. 6 are the new expansion coefficients, and the procedure is repeateduntil self-consistency is achieved.The most computer-time intensive aspect of rigorous LCAO-MO calculations is the calculationof the Fock matrix elements in eq. 4 . Almost all modern approximate LCAO-MO methods involveapproximations only to these matrix elements; all other aspects of the solution are handledas they are in an ab initio treatment.(STOs) for the basis functions, rather than the Gaussian-type orbitals (GTOs) used in abinitio calculations.although overlap integrals involving STOs can be readily calculated, the rigorouscalculation of Fock matrix elements using STOs is much more difficult than with GTOs.addition, nearly all approximate methods use a minimal basis of STOs, i.e. each atomicorbital is represented by a single STO (or by a fixed linear combination of two or moreSTOs).flexibility; ecause the radial shape of an atomic orbital is constrained at the beginningof the calculation, the molecular orbitals are forced to resemble strongly the atomicorbitals that were presumed.The first approximate LCAO-MO treatment of transition metal systems was that of Wolfsbergand Helmholz (ref. 12). They used empirical values for the diagonal Fock matrix elements,F,,, and approximated the off-diagonal elements as an overlap-scaled average of thecorresponding diagonal elements:

Most approximate methods use Slater-type orbitalsSTOs are better representations of hydrogenic atomic orbitals, but,

In

A minimal basis provides ease of interpretation at the expense of orbital

F, - kS,(F,, + Fvv)/2 ( 7 )The constant k in eq. 7 was varied between 1.67 and 2.00.evolved into the extended Huckel (EH) method of Hoffmann (ref. 13). The EH method uses aminimal basis of STOs, empirically chosen values for F,,, and eq. 7 (k - 1.75), or a slightmodification thereof (ref, 1 4 ) , to calculate the values of FFV. The nonempirical Fenske-Hall method (ref. 15) is also loosely derivative of the Wolfsberg-Helmholz method. In the

The Wolfsberg-Helmholz method


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Approximate MO theory in organometallic chemistry 841

Fenske-Hall ethod, the diagonal Fock matrix elements, F,,,, are calculated approximately butnonempirically.eq. 7 , but the essential notion that the value of F,,, is related to the values of F,,, and F,,is retained. Like the EH method, the Fenske-Hall method employs a minimal basis of STOs andonly the F matrix elements are approximated.popular approximate LCAO-MO methods used to study organometallic complexes.

The off-diagonal elements are calculated by a more complex formalism than

The EH and Fenske-Hall methods are the most

D I S A D V A N TA G E S A N D A D V A N T A G E S O F A P P R O X I M A T E L C A O - M OC A L C U L A T I O N SWhat precautions are necessary in using approximate LCAO-MO methods?advantages of these approximate techniques?that an uninitiated investigator should ask before using a particular method.approximate methods described here suffer from the following disadvantages:

What are the intrinsicThese are among the most important questionsAll the

A method is only as good as its approximations.discussed above were developed for certain types of chemical systems.transferability of a given method to a chemical system for which it has not previouslybeen used is always a matter that should be carefully and systematically investigated.The methods are limited to a single-configuration escription. This disadvantage is,of course, intrinsic to molecular orbital theory. Systems that are inadequatelydescribed by a single configuration, such as very weak bonding at long bond lengths,will not be properly described by either approximate or rigorous Hartree-Fock MOtheory.Minimal basis sets can be woefully inadequate. Because the approximate methodsdescribed here use a minimal basis set, the choice of basis set is the most imortantdecision an investieator has to make. It is this author's opinion that poor basis setchoice is the most common error made in the application of approximate MO theory.Unfortunately, there are few well-founded means to determine the quality of a minimalbasis set for molecular calculations, particularly for unoccupied atomic orbitals (suchas the 4s and 4p AOs of a first-row transition metal). Within the confines of aminimal basis, the best procedure is to choose STO basis functions for a series ofatoms in a consistent, systematic fashion (ref. 16).The use of approximations invalidates the variation principle. The variationprinciple, as applied to rigorous electronic structure methods, guarantees that anychange to the wavefunction that leads to a lower energy is an improvement to thewavefunction. This guarantee is lost as soon as approximations are employed. As aresult, the use of approximate methods to determine the total energy of a molecule(and, perforce, properties of the total energy such as molecular geometry or potentialenergy curves) is on far less secure footing than when ab initio methods ar used.Open shell molecules are generally treated incorrectly. Open-shell CAO-MO theory,also first treated rigorously by Roothaan (ref. 17), is far more complex than closed-shell theory. The approximate methods discussed here treat open-shell molecules in aclosed-shell fashion, a formally incorrect albeit ofttimes useful approach (ref. 18).

All of the approximate methodsThe

Given this broad spectrum of disadvantages for approximate methods, why do so manyinvestigators choose to continue to use them? The answer, of course, is that thedisadvantages of these methods are often outweighed by the advantages of using them.major advantages of approximate LCAO-MO methods are the following:(1)

The

Approximate methods are computationally inexpensive. In the early days ofcomputational chemistry, a major impetus for the development of approximate methods wasthe need for electronic structural tools that could be applied to relatively largemolecules (more than 10 atoms); the limited computational capacity of the time ruledout the possibility of performing higher level calculations. While the modern-dayadvances in computing have made this advantage less critical, approximate methods are,of course, still less computationally demanding than rigorous methods, to the pointthat many of the codes can be run on advanced personal computers.( 2 ) Minimal basis sets lead to easily interpreted nos and wavefunctions. As mentionedabove, a minimal basis set can be a poor one for molecular calculations. However, awell-chosen minimal basis set can lead to an adequate qualitative (and sometimesquantitative) description of the bonding in a molecule. The use of a proper minimalbasis leads to a more straightforward analysis of MOs than is possible with an extendedbasis; the minimal basis better fosters the description of a molecule in terms ofconstituent atomic orbitals. Hence, it is generally easier to extract chemically-useful information from a minimal basis set calculation than from a calculation using amore correct basis set.


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( 3 )

( 4 )

( 5 )

The

Population analysis is easy and Straightforward to interpret.methods use Mulliken population analysis (ref. 19) as a means of "translating" MOresults into chemically intuitive concepts-atomic charges, composition of theorbitals, bonding or antibonding character, and the like. Although Mulliken populationanalysis is an additional approximation, it is an extremely useful one, especially whencoupled with a minimal basis set.analysis must be used with care; the populations can be very unreliable, especially ifvery diffuse basis functions are used.Fragment analysis is widely used. Most approximate MO methods readily facilitateviewing the MOs of a molecule as resulting from the interaction of MOs of smallerfragments of the molecule (ref. 2 0 ) . This fragment approach is extremely useful inorganometallic chemistry, wherein, for example, the important bonding interactions in acomplex can be traced to the interactions of the metal atom with a series of ligands(ref. 21). While fragment analysis is, in principle, equally straightforward inrigorous Hartree-Fock theory, most Hartree-Fock practicioners do not perform this typeof analysis.The orbital energies and compositions yielded by approximate methods are often morechemically reasonable than those of ab initio Hartree-Fock methods.advantage of approximate LCAO-MO methods concerns their correctness with respect toexplaining observable properties of molecules. This author need not expound on theusefulness of approximate methods; the tremendous body of literature by Hoffmann andcoworkers (ref. 4 ) , by Fenske and coworkers (ref. 7 , ll), and by others attests to theutility of approximate methods in explaining chemical trends in organometallicchemistry. In fact, the approximations used, while developed for computationalexpediency, often yield orbitals that are superior to those of ab initio SCFcalculations (ref. 22). We can, rather facetiously, consider the approximate methodsto be "ab intuitiono" techniques: The methods are advantageous in that they oftenprovide a connection between an orbital description and chemical intuition.

Most approximate LCAO-MO

As with all approximations, Mulliken population

The major

implications of these advantages, particularly the last one, are quite profound. Aninvestigator can be faced with the dilemma of performing either a "quick and dirty" minimalbasis set approximate calculation, or an extended basis set ab initio calculation with theinclusion of some electron correlation. In many instances, the advantages of the formerapproach, in terms of computational expense and chemical interpretability far outweigh theadvantages of the latter with respect to computational rigor.W H A T S H O U L D A N O N T H E O R E T I C I A N B E LIE V E?

Given the manifold advantages and disadvantages of approximate methods vis-A-vis b initiomethods, the nontheoretician is often placed in the position of not knowing whichcalculations to believe or which results to trust. To paraphrase from a recent talk byHoffmann: "Theoreticians are purveyors of methods; all of us are trying to sell you a billof goods (ref. 23)." In this last section, I will offer some of my thoughts on how anontheoretician should approach the question of what to believe.In any electronic structural investigation, there are two important questions that must beasked: (i) What approximations are inherent to the theoretical approach used? (ii) Whatapproximations are being made to the chemical model under investigation? To clarify thesequestions, consider electronic structural studies of the reactions of biscyclopentadienylcomplexes of the early transition metals. Is it best to do a correlated ab initiocalculation on a model complex in which the Cp rings are modeled by C1 atoms (ref. 24 ) , todo an ab initio single-configuration alculation in which Cp rings are actually used(ref. 25), or to do a more extensive study using approximate MO theory (ref. 26)?Unfortunately, there is not a simple answer to this question, for it must be recognized thatall methods are useful within their ow n limitation L Thus, an approximate method is veryuseful for obtaining a qualitative view of the bonding and the frontier orbitals of amolecule, whereas a more rigorous method is usually essential for the detailed and precisecalculation of a reaction pathway. Investigators must consider what information they wishto extract from a calculation before choosing a method.There is a general inverse correlation between the rigor of the electronic structural methodused and the correctness of the chemical model studied.represented as a hyperbola, as shown in Fig. 1. Computational advances move the hyperbolafarther from the origin. Thus, for a given chemical model, increases in computing powerallow more rigorous theory to be used, Likewise, for a given level of theory, computationaladvances have allowed truer chemical models to be applied. It is important to note that, ingeneral, the ease with which an electronic structure calculation can be interpretedincreases as the chemical model becomes more correct and as the theoretical model becomesless rigorous. The hyperbola of Fig. 1 is closely related to the "hyperbola of quantumchemistry" advanced by Pople (ref. 27) and recently expanded upon by Karplus (ref. 2 8 ) .Because proponents of methodologies at all points along this hyperbola attempt to sell theirmethods as the most appropriate, Fig. 1 can be considered the "hyperbola of hyperbole" forelectronic structure calculations. Caveat emptorl

This relationship can be


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Approximate M O heory in organometallic chemistry a43

1 mCorrectnessof theoreticalmodel

Fig. 1. The "hyperbola of hyperbole," which diagramsthe inverse relation between the correctness of thetheoretical model and the correctness of thechemical model used.

F I NA L C O M M E N T SApproximate LCAO-MO calculations have been and will continue to be a valuable tool inorganometallic chemistry. They originated as the only means by which the bonding inorganometallic compounds could be studied in the 1950's; in the spirit of Coulson'sdelightful 1960 summation of the state of molecular structure calculations (ref. 29),organometallic electronic structure originated among the less-than-rigorous ut highlypragmatic "group 11" quantum chemists. Although a relative newcomer to the field, I havehad the good fortune to be involved in many studies of organometallic electronic structure,at several levels of theory. Of these various investigations, the ones using approximateLCAO-MO methods generally yield the information that is most easily transferred from thecomputer to the laboratory. I hope this contribution encourages further applications ofapproximate theory in organometallic chemistry, and that such investigations are undertakenthoughtfully and with the requisite care.

Acknowledgements The author wishes to thank Dr. Melanie Pepper and Mr. WilliamSchneider for stimulating discussions, and Dr. Joseph Norbeck for organizing aSymposium on Organometallic Theory at the XIVth International Conference onOrganometallic Chemistry.

1.2.

3.

4.5 .6.7.8.

R E FE R E N C E SC.C.J. Roothaan,- 23, 69-89 (1951).(a) T.J. Kealy and P.L. Pauson, Nature 168, 1039 (1951). (b) S.A. Miller, J.A. Tebboth,and J.F. Tremaine, J . Chem. SOC, 632 (1952). (c) G. Wilkinson, M. Rosenblum, M.C.Whiting, and R.B. Woodward, J. Am. Chem. SOC,& , 2125 (1952). (d) G. Wilkinson,Oreanomet. Chem, m, 273-278 (1975).(a) H.H. Jaff6, Chem. Phv , 21, 156-157 (1953). (b) J.D. Dunitz and L.E. Orgel,Pature 171, 121 (1953). (c ) W. Moffitt, J . Am . Chem. SOC, x , 386 (1954). (d) J.D.Dunitz and L.E. Orgel, J. Chem. Phvs, 2 , 54-958 (1955).(a) R. Hoffmann, Science 211, 995-1002 (1981). (b) R. Hoffmann, Aneew. Chem. Int. Ed,It should be pointed out here that molecular orbital theory is itself an approximationin that electron correlation is ignored.See, for example: A . Veillard, ed., Quantum Chemistrv: The Challenee of TransitionMetals and Coordination Chemistry, Reidel, Dordrecht (1986).

21, 711-724 (1982).

R.F. Fenske, Pure & ADD1. Chem, a, 153-1162 (1988).(a) S . Borman, Chem. Ene. News 68 , No. 15 (April 9, 1990), 22-30 (1990). (b) J.K.Labanowski and J.W. Andzelm, eds., pensitv Functional A m roaches to Chemistrv, Springer-Verlag, New York (1990).


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9 . For an excellent detailed development of molecular orbital theory, see: A. Szabo andN.S. Ostlund, Modern Quantum Chemistrv. 1st ed. revised, McGraw-Hill, New York ( 1 9 8 9 )10. T.C. Koopmans, Phvsica 1, 0 4 ( 1 9 3 3 ) .11. R.F. Fenske, Proe;. Inore. Chem. 2 1 , 1 7 9 ( 1 9 7 6 ) .1 2 . M. Wolfsberg and L. Helmholz, J. Chem. Phvs, 20, 8 3 7 - 8 4 3 ( 1 9 5 2 ) .1 3 . R. Hoffmann, J. Chem. Phvs. 2 , 3 9 7 - 1 4 1 2 ( 1 9 6 3 ) .1 4 . J.H. Ammeter, H.-B. Burgi, J.C. Thiebault, and R. Hoffmann, J. Am. Chem. SOC. 100, 3 6 8 6 -3 6 9 2 ( 1 9 7 8 ) .1 5 . M.B. Hall and R.F. Fenske, Inore. Chem. 11, 6 8 - 7 7 5 ( 1 9 7 2 ) .1 6 . (a) J.W. Richardson, W.C. Nieuwpoort, R.R. Powell, and W.F. Edgell, J. Chem. Phvs. 36,1 0 5 7 - 1 0 6 1 ( 1 9 6 2 ) . (b) B.E. Bursten, J.R. Jensen, and R.F. Fenske, J. Chem. Phvs. 68,3 3 2 0 - 3 3 2 1 ( 1 9 7 8 ) .1 7 . C.C.J. Roothaan, Rev. Mod. Phvs.z,7 9 - 1 8 5 ( 1 9 6 0 ) .1 8 . See, for example: J.L. Petersen and L.F. Dahl, J. Am . Chem. SOC, 97, 6 4 2 2 ( 1 9 7 5 ) .1 9 . (a) R.S. Mulliken, J. Chem. Phvs, 2 , 8 3 3 - 1 8 4 0 ( 1 9 5 5 ) . (b) R.S. Mulliken, J. Chem.

Phvs. 2 3 , 1 8 4 1 - 1 8 4 6 (1 9 5 5 ) . (c) R.S. Mulliken, J. Chem. Phvs. 2 , 3 3 8 - 2 3 4 2 ( 1 9 5 5 ) .(d) R.S. Mulliken, J. Chem. Phvs, 2 , 3 4 3 ( 1 9 5 5 ) .2 0 . For a formalism that also allows the judicious "freezing" of fragment orbitals, see:D.L. Lichtenberger and R.F. Fenske, J. Chem. Phvs. 6 4 , 4 2 4 7 - 4 2 6 4 ( 1 9 7 6 ) .2 1 . T.A. Albright, J.K. Burdett, and M.-H. Whangbo, Orbital Interactions in Chemistry,Wiley, New York ( 1 9 8 5 ) .2 2 . B.E. Bursten, J.R. Jensen, D.J. Gordon, P.M. Treichel, and R.F. Fenske, J. Am . Chem.2 3 . R. Hoffmann, "Making and Breaking Bonds in the Solid State," Frank C. Mathers Lecture,2 4 . See, for example: A.K. Rappe, J. Am. Chem. SOC. 109, 5 6 0 5 - 5 6 1 3 ( 1 9 8 7 ) .2 5 . See, for example: C.A. Jolly and D.S. Marynick, J. Am . Chem. SOC. 111, 9 6 8 - 7 9 7 42 6 . S e e , for example: K. Tatsumi, A. Nakamura, P. Hofmann, P. Stauffert, and R. Hoffmann,2 7 . J.A. Pople, J . Chem. Phvs, 43, S 2 2 9 - S 2 3 0 ( 1 9 6 5 ) .2 8 . M. Karplus, J. Phvs. Chem. 94, 5 4 3 5 - 5 4 3 6 ( 1 9 9 0 ) .2 9 . C.A. Coulson, Rev. Mod. Phvs. 2, 7 0 - 1 7 7 ( 1 9 6 0 ) .

--OC. 1 0 3 , 5 2 2 6 - 5 2 3 1 (1 9 8 1 ) .Indiana University, October 8 , 1 9 9 0 .( 1 9 8 9 ) .J . Am. Chem. SOC. 107, 4 4 4 0 - 4 4 5 1 ( 1 9 8 5 ) .

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