MO theory made visible

edited by JAMES P. BiRK

MO Theory Made Visible

Carlo Mealli and Davide M. Proserpio lstituto oer lo Studio della Stereochimica ed Energetica dei Composti di Coordinazione, C.N.R., Via J. Nardi 39, 50132 Florence, Italy

Our understanding of chemical bonding and geometrical preferences, hence of the reactivity of molecules, has re- ceived a tremendous push from qualitative MO theory ( I ) . Through the application of simple concepts, chemical-phys- ical meaning can be assigned even to the most intricate quantum-mechanical numerical results. For example, much simplification is introduced by the usage of symmetry prop- ertiks, by the application of second-order perturbation theory to orbital interactions, and by the sectioning of a molecule into ideal fragments that seem to have universal fea- tures [isolobal analogy concept (2)l. In addition, our understanding of chemical reactivity becomes simpler if the evolution of specific MO'sk monitored along reaction path- wavs. The latter technique is known as the Walsh diagram construction and, subjected to certain rules, allows the iden- tification of the site of chemical reactivity.

At the present state of the art, the amount of theoretical background and of computational work in dealing with delocalized MO theory is not insignificant and requires in any case the intervention of the specialist. For these reasons the majority of lab chemists remain anchored to a localized vision of electronic structures. By using pencil and paper, stick bonds, and lone pairs, resonance structures are easily drawn. Electrophilic and nucleophilic centers of attack, identified for atoms carrying residual charges are used to describe chemical reactivity. By contrast, the task of drawing molecular orbitals by hand and of extracting from them the necessary chemical information appears much more hur- densome, particularly when dealing with transition metal compoundi. Nonetheless, in the new era of laboratory com- puters, routine usage of MO methods to describe molecules and their reactivitvis not out of reach. -

Here we the package of MO programs that has been collected in our Institute and that has been intemated ~~~

by a number of handy routines and programs in o d e r to make readilv visible the MO results. Our idea is tostart from an easy-to-build molecular model, to run MO calculations (of the extended Hiickel type) and to represent the results graphically in order to gain an insight of structural stability and/or chemical reactivity. The examination of numerical quantities is deliberately limited as much as possible.

Geometry Bullder, Extended HBckel, Fragment Molecular Orbltal Analysls Programs

The original programs (CDNT, ICON, FMO) were ob- tained from the group of Roald Hoffmann at Cornell Univer- sity. The computational algorithms have not been substan- tially changed (3). The actual new feature is a number of routines that allow an automatic assignment of molecular symmetry and the character of each calculated MO. Such a knowledge is mandatory in order to draw Walsh diagrams (vide infra).

The user interactively builds up the molecular model by providing the classical set of bond distances, bond angles, and torsion angles (4). The program recognizes the presence of the symmetry elements and how the atoms are related to each other. Then the actual MO calculation is performed. The program analyzes the MO coefficients in order to assign the character (flag) of each orbital with respect to a particu- lar symmetry element. The information is also transferred to a file for later usage. As an example, part of the output for the trigonal hipyramidal model NiHs3- is reproduced in Figure 1.

The FMO program (Fragment Molecular Orbital analysis (5)) is integrated within the main extended Htickel package. Upon a simple indication of the desired fragmentation, the

ATOM X Y Z Nil .00000 .00000 .00000

H 2 .00000 .OOOOO 1 65000

H 3 .OOOOO ,00000 -1.65000

H 4 1.65000 .OOOOO ,00000

H 5 -.82500 1.42891 .OOOOO H 6 -.82500 -1.42891 .DO000

THE MOLECULE HhS li C3 AXIS (Z AXIS1 R E U T I O N BETWEEN THE ATOMS: 0 0 0 5 6 4

THE MOLECULE H E THE MIRROR PLANE W

RELATION BETWEEN THE ATOMS: 0 3 2 0 0 0

THE MOLECULE HAS THE MIRROR PLANE YE

REMTION BETWEEN THE ATOMS: 0 0 0 0 6 5

THE MOLECULE HAS THE ROTIITION ~ W I S n RELATION BETWEEN THE &TOMS: 0 3 2 0 6 5

~ n e r g y l eve ls t e w , Electron occupation and symmetry Flags

E ( 1) i 4.04851 .O TSSS E ( 81 - -12.99000 2.0 EnSA E ( 2) = 3.83780 .O TASA E( 91 = -12.99000 2.0 D A S

E( 31 = 1.93968 .O ESAA E(i0) = -14.02871 2.0 TASA

E ( 41 = 1.93968 .O ESSS E(lll = -14.80690 2.0 ESSS

E ( 51 = -8.39686 .O TSSS E(12) = -14.80690 2.0 ESAA

E ( 61 = -11.66816 2.0 ES- E(131 = -14.97852 2.0 TSSS

E ( 71 = -11.66816 2.0 ESSS E(141 = -16.07110 2.0 TSSS

SUM OF ONE-ELECTRON ENERGIES = -248.01701 eY

Figure 1. Selected output from an enended HOckei caicuhtlon for the model NiHs3- at bigonal bipyramidal geometry (8 = 90'. 4 = 120'). For each MO. the flags (S a A) identlty the character wilh respect lo the mlrra planes and the twofold axis (CZ. symmetry (see text)). For lha threefold axis, lhe flags T and E, identity the belonging eilhar to a, or s symmetry species.

Volume 67 Number 5 May 1990 399

routine transforms the basis set from atomic to fragmental orbitals. Very informative interaction diagrams can he built up, since the percentages with which various fragment orbitals participate in a certain molecular orbital become available. Take for example the metal complex bis-cyclopenta- dienyl-M~-~~-CSz. The frontier fragment orbitals of both the CpzM fragment (le) and of the bent CSs molecule are well known (Id). Even so, their resulting interaction diagram (6) may be difficult to interpret. In fact, due to the low molecular svmmetrv (CJ three- or ~olv-center interactions . . .. . . are at work and there is a lot of orbital mixing. Appropriate three-dimensional revresentations of the MO's aid the analysis. In Figure 2, seleked MO's and FMO's are represented in order to illustrate the advantages of using our new program CACAO (vide infra).

CACAO (Computer Alded ComposRion of Atomlc Orbitals) One important feature of the program is that each MO

function 'Z' can be represented either in its natural distribu- tion over the whole molecule (delocalized mode as in Figure 2a. which represents the second HOMO of the com~lex - -~ . ~~~~

Cp,Mo-?'-CS,) or in a form that allows no overlap hetween theorbitalsofdifferent atoms (contracted modeas in Fiaure 2h, 2c). Although the latter representation is an artifact, it simplifies the perception of the nature of the MO. In effect the single bricks, used to assemble the MO building, are seen separately from each other. I t is noteworthy that the contracted mode is already universally adopted for educational purposes in hand-written MO sketches. Conveniently CA- CAO can output this type of drawing as the final visible result of a MO calculation.

The great success of the Jorgensen and Salem book The Organic Chemist's Book of Orbitals (7) is partially due to an exhaustive three-dimensional representation of the MO's for manv oreanic molecules. This has shown the meat advan- " - tage of using the pioneering computer p r o g r a ~ ~ ~ ~ l l l (8). Nonetheless. the usane of the Dromam is neither widespread nor routinely applied to MO studies, in contrast to the OR- T E P program for molecular structure representations. Moreover, in spite of the increasing number of theoretical papers dealing with transition metal elements, the available version of PSI177 does not include a generalized use of d orbitals.

The program, CACAO includes the following facilities:

(1) The program, summing up to-2500 FORTRAN77 statements, was originally written for a main-frame SEL-GOULD 32/27 computer. A PC-DOS 3.3 version isnow available. Theuser can choose the graphic system between CALCOMP, TEK- TRONM (PLOT 10Termiual ControlSyatem), and VDI (Vir- tual Device Interface; Graphics Development Toolkit, version 1.2) plotting routines.

(2) The program can be run interactively with minimum intervention from the user. .~~~~~~ ~~~ ~~~~ ~~~~

(3) All of theelements with atomic number up to89 can he proper. ly treated. An extension off orbitals ia pmgrammed but not available as yet.

(4) Each orbital is delimited by contour lines calculated in two sets orthogonal planes (fixed at a certain value of the function W.

(5) The program draws delimiting orbital envelopes in planes per- oendicular to the view direction (thick lines in Fia. 2).

(6) ~ n y view direction can be selected, and rotation; are allowed ahout the ages of the Cartesian workine svstem. Ootions are ~~ ~~~ ~~~~ ~ ~ ~ ~ .. - available to draw the molecular framework in different styles, including the possibility of selecting trnly certain atoms. Alter-

Figure 2. CACAO drawings for Uw sewrd HOMO of Cp,Mo-q2-CS2; a, me MO in question as seen down ihe y axis (&localizedmode); b, the same view of me MO drawn in lhs aultractedrnode: c, Me MO viewed down the zaxis (conmnedmode) (for sake of ciaritythe orbital contribution 01 Me Cp rlngs is omitted); d. CS2-4q conbibuting FMO ( j d , p 158): e, CS2-3h contributing FMO (ld, p 158); f. Cp2Mo--h contributing FMO (le, p 395).

400 Journal of Chemical Education

natively, the orbital contribution of atoms appearing within the molecular framework can be deleted.

(7) The usage of cut-off parameters and of molecular symmetry (automatically determined by the MO package) speeds up the calculation. In routine work, by selecting the drawing of the external envelopes only (which are the most rich in information anyway), the frontier MO set for a certain molecule (say five MO's) can be drawn for viewing in a few minutes on a personal computer IBM PSI2 Mod. 50.

(8) The precision with which contour lines are drawn depends on the number of grid points calculated, and such a magnitude is fixed by the user a t run time.

(9) Options are available to plot the MO's by using either the atomic orbital or the FMO basis set. The advantage of comhin- ing the two options is readily seen in Figure 2a-2f. The drawings 2d-2f show, in the contracted mode, the three FMO's that contribute most to form the second HOMO of CpzMo-vZ-CSz. Three different representations of the resulting MO are also

reoorted in Fieure 2. In nar- titular 2a is related to'the delocnlized mode with contour lines at values of Y = 0.04. Figure 2b (contracted mode) illustrates haw the metal orbital (d,,, slightly hybridized) interacts with the resultant of the two different CSz orbitals (2d + 2e). Finally, in Figure 2e (eon- trocted mode) the exclusion of the Cp n orbitals and the top viewpoint underline a double feature of the second HOMO, i.e., the Mo-C o bonding nature and thelarge weight of the uncoordinated sulfur atom (lone pair a t

Figm 3. Stereo drawing of the LUMO of Cp2t.kwZ-CS2. sexo)1 (9).

I SYMMETRY rXY XZ CX

1 2 3 4 5 6 7 8 9

REACTION COORDINATE (STEPS]

Figure 4. A graphical representation of Um Berry pseudaotation process for Um NTH." model. The rlgM side shows a wmputer drawn Walsh diagram relative to the five metal dorbltals. Each step isdefined by ambination of Band mangles (define3 in Figure 1 and sirnultaneouslyvarying in the ranges 90-100' and 120-100'. re apectively). The left side shows CACAO views of all of the Walsh MO's at the steps 1, 5, 9.

Volume 67 Number 5 May 1990 40 1

Stereo drawings, obtainable by slightly rotating (+3') the same MO, magnify the 3D effect. As an example, Figure 3 reports a stereo view of the LUMO of CpzMo-v2-CS2. Another useful feature of CACAO is that one specific MO can be plotted in succession along a deformation (or reaction) ~athway. The feature is very informative as the actual MO evolution in Walsh-type diagrams can be monitored step by step.

Walsh Dlagrams and Thelr Graphical Representatlon .

Building Walsh diagrams was previously a tedious, time- consuming task. After running a sequence of MO calculations for different geometrical conformers, the energies of the selected levels had to be drawn on graph paper, then correlated according to their symmetry. This required a careful control of the MO coefficients by jumping back and forth in the computer output. Now the whole procedure is automatically performed by the computer.

As an example of how a diamam of this t.me can be drawn, we report in ggure 4 a g r a p h h represent&on of the Berry Dseudorotation process for the model NiHs3-. The trigonal bipyramid - square pyramid interconversion is well known and reported in textbooks (le). The five metal d orbitals, split a t the trigonal hipyramidal geometry (D3h. left side) in e", e' la{ levels, convert to the levels b2, e, a], and bl of the square pyramid. Along the pathway the symmetry is as low as CZ".

Fieure 4 shows nicelv how each orbital is gradually tran- formed along the pa tGay defined by steps-in which the 0 and d aneles (seeFia. 1) varvsimultaneousl~. For instance, i t is quite ~duc&ionaito'verify how the orhital d,z(lall a t the D3h geometry) transforms into the orbital dz~.yz(b~ a t the Ca, geometry). Also notice how the BASA member of e" and 6SAA member of e' become degenerate at Ca, geometry.

Interestingly, the drawings confirm the &ends already discussed by Rossi and Hoffmann (10) for u-bond strengh in transition metal ent taco ordinate comolexes. As .~---- - ~ - - ~ ~ ~ ~ ~

an example, in passing from d6 to d8 square pyramidal species. there is an inversion of the lenethenine trend for the M- ~ i & ~ bond. In fact, up to the a l level ( 7 % ~ ) no axial anti- bonding character is transparent, hut when a l becomes pop- ulated the M-L.,i,.l bond is naturally weakened.

Conclusions We have made a presentation of an automated package of

programs to perform MO calculations and their graphical illustration. Although our main purpose is not that of con- vincing people of the power of the MO method, we hope that a user-friendly approach to the technique can be fruitful both a t the educational level (teaching the basis of qualitative MO theory tofirst-year students becomes easier if pictorial tools are available) and a t the research level ( l l) , in the routine work a t the labs. For example, by looking at the second HOMO of the Cp2Mo-q2-CS2 complex, represented in Figure 2, one realizes that this level is predominantly centered at the S.,, atom and that the atom in question predictably becomes a nucleophilic center for the molecule. Analogously the electrophilicity of the CS2 carbon atom can he predicted by lookingat theLUMO level (Fig. 3). Based on these simple observations, the pictorial MO presented here is a useful tool, even for the nonexpert, to predict trends for orbital-controlled chemical reactiuity.

Acknowledgment Thanks are expressed to A. Sabatini and A. Vacca for the

help provided in implementing the PC-IBM version of the programs.

Literature Cned 1. Among the many mmprohensive textbooks, we quote: s. Dewr, M. J. S.: Dougherty, R.

C. The PMO Theory of Orgonic Chamisfry; Plenum: New York. 1975. b. Fleming, I. Frontier Orbitnla and Orgonic Chsmicol Reactions; Wiley: Landon. 1976. c. Yates. K. Huckoi Moiaculcr Orbilal Theory: Academic: New York. 1978. d. Gimarc, B. M. Molecular Structure ond Bonding; Academic: New York, 1979. e. Albright. T. A,; Burdetf, J. K.; Whangbo, M:H Orbitol Inlarocfions in Chemistry; Wiley: New ""-7. ,one .".-, .""".

2. Hoffmsnn, R. Angew. Cham.Int. Ed.Eng1. 1982.21.111-724. 3. Hoffmann, R.: Lipscomb, W. N. J. Chsm Phys. 1962, 36, 2179-2189, 3469.3493;

Hoffmann, R. J. Chem. Phy. 1963.39.1397-1412. a. clark, T. A ~ o n d b o o k ofCampuroriom1 Chemistry; Wiley: New York. 1985. 5. Hoffmsnn, R.; Fuyimato. H.:Swenron, J. R.: Wan, C-C. J. Am. Chem Soc. 1973,95,

764+7650. 6. Li, J.; Hoffmsnn, R.: Mealli, C.: Silvestre, J. Orgonomdailica, 1989,8,1929. 7. Jorgenaen. W. L.: Salem, L. The Orgonir Chemist's Hook 01 Orbirds: Academic: New

York. 1974. 8. Jowensen, W. L. PSIl77. Quantum Chemistry Program Exchange, QCPE Program

" 0 Q*" .. 9. For detailed description of the bonding in M-nZ-CS? complexes see: a. Mealii, C.;

Hoffmann, R.: Stockis, A. Inorg. Chrm. 1984.23.5665. b. Bianchini, C.; Mealli, C.; Moli, A.: Sahat, M. In Stemochami8Lry of Organomsfoliic and lnorgmir Com- pounds: Bernal, I.. Ed.: Elsevier: Amsterdam, 1986: Vo l 1, Chapter 3.

10. Rassi,A. R.:Hoffmenn, RInorg. Cham. 1915,14,365-371. 11. Mealli, C.; Proserpio, D. M. Comments Inorg. Chsm. 1989, VIII, 31-53; Mealli, C.;

Plo.erpia, D. M.:Fachinetti,G.:Funaioli,T.:Fochi,G.;Zan~zi, P. F. Inorg. Chom. 1989.28. 1122: Bianchini, C.; Lasehi, F.: Mkri, D.; Meslli, C.: Meli, A.; Ottaviani, F. M.:Proserpio, D. M.: Sabat, M.; Zanello. F. Inorg. Chem.. 1989.28.2552.

Iterations II: Computing in the Journal of Chemical Education

Iterations II, a collection of 46 articles that appeared in the Computer Series between 1981 and 1986, has been carefully selected by the editors, Russell Batt and John W. Moore, to bring up to date the collection of computer applications that appeared in its predecessor volume Iterations. In addition to covering all aspects of instructional computing from introductory to graduate level, Iterations II provides an annotated bibliography of all computer-related articles that have appeared in the Journal from 1981 to 1986. 1987 paperback, 160 pp; US. $16.50; foreign $17.50 (postpaid). Send prepaid orders to Subscriptions and Book Order Department, Journal of Chemical Education, 20th and Northampton Streets, Easton, PA 18042.

402 Journal of Chemical Education

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MO theory made visible