Understanding Molecular Orbitals; Sigma Orbitals

Transcription

1 Universit of Connecticut Chemistr Education Materials Department of Chemistr April 7 Understanding Molecular Orbitals; Sigma Orbitals Carl W. David Universit of Connecticut, Carl.David@uconn.edu Follow this and additional works at: Recommended Citation David, Carl W., "Understanding Molecular Orbitals; Sigma Orbitals" (7). Chemistr Education Materials..

2 Understanding Molecular Orbitals; Sigma-Orbitals C. W. David Department of Chemistr Universit of Connecticut Storrs, Connecticut 9-3 (Dated: April, 7) I. SYNOPSIS Molecular Orbitals differ significantl from atomic orbitals. One of their major sources of confusion, at least to diatomic molecular orbitals, is the nature of the coördinate sstem emploed. The drawings herein are intended to help understand these orbitals. II. INTRODUCTION The standard coördinate sstem for treating diatomic molecules places one nucleus (A) at (,,R/) and the other (B) at (,,-R/), as has been discussed repeatedl in these pages. If we attempt to build a molecular orbital using. Instead, r A = + + ( R/) (.) r B = + + ( + R/) (.) so, for a trial (sigma) LCAO-MO (an approimate wave function, not an eigenfunction of the H + sstem) ψ trial σ = ( c A e αr A + c B e αr B ) (.3) We alread know that for the homonuclear case the coëfficients are equal, so we make them one for our purposes here. We then have ψ σ = ( e αr A + e αr ) B (.) which we are interested in representing. Before that, however, let s introduce the anti bonding orbital ψ σ = ( e αr A e αr ) B (.5) III. ONE DIMENSIONAL PLOTS (ALONG THE -AXIS) A p(,,)< >p(λ,µ,φ) r A (,,R/) r B A p(,,r) (,,R/) We emplo our (now) standard tricks for capturing the essence of these functions, starting with one dimensional plots of ψ σ (,, ) versus. The σ orbital is shown in Figure, while the σ orbital is shown in Figure 3. It is clear that the bonding orbital has maima centered about the nucleii, while the anti bonding orbital has a node, the plane, where =. B φ (,, R/) φ (,, R/) IV. PSEUDO 3D PLOTS The σ orbital is shown in Figure, while the σ orbital is shown in Figure 5. FIG. : The Elliptical Coordinate Sstem for Diatomic Molecules. The µ coordinate is not depicted. from hdrogenic s orbitals, b placing one on nucleus A and the other on nucleus B then their form is no longer the simple Cartesian form we ve become accustomed to V. 3D CONTOUR PLOTS The σ orbital is shown in Figure, while the σ orbital is shown in Figure 7. Tpeset b REVTEX

3 sigma (sigma) MO(,,) versus and. psi().... psi... FIG. : The ψ σ(,, ) orbital, plotted against FIG. : The ψ σ(,, ) orbital, plotted against and sigma^* psi().5 (sigma) MO(,,) versus and.5.5 psi.5 FIG. 3: The ψ σ (,, ) orbital, plotted against VI. THE p σ ORBITALS FIG. 5: The ψ σ (,, ) orbital, plotted against and There is another orbital pair which are labeled σ, but built from p orbitals rather than s orbitals. Remember, the bond ais is contained in the -ais. That means that p orbitals are parallel to the bond ais (we epected them to be at right angles to the bond, but those are the p and p orbitals, which become π and π respectivel). To continue, we put one p orbital on nucleus A and the other on nucleus B. The wave function now becomes ψ p± = (( R ) e αr A ± ( + R ) e αr B ) (.) which reminds us that the orbitals are centered on two different locations in phsical space. When we plot this we obtain Figure as our first, linear plot. This is, of course, ψ p σ (,, versus being plotted. weird this is vide infra. Notice how Net we plot the bonding orbital, which is better behaved. Figure 9 shows the smmetr of the bonding orbital. We return now to the anti-bonding orbital, and show the contributors in colors so that the odd function being plotted can be understood (see Figure ). Finall, we plot the contour (3D) map of this orbital, which shows the weird, interdigitated, positive and negative lobes. We note in passing that the Maple code for these last few plots is not included in the tet.

4 VIII. MAPLE CODE sigma MO ( + = blue) Here is the Maple code which generated the figures to follow: psi() p sigma^* FIG. : The contour surface of ψ σ(,, ) orbital, plotted against, and... sigma^* MO ( +(blue) and -(red)) FIG. : The anti-bonding p σ orbital versus p sigma psi(). FIG. 7: The contour surfaces of ψ σ (,, ) orbital, plotted against, and. VII. CONCLUSION The rigid understanding of the functionalit of the contributing atomic orbitals to the resultant molecular orbital leads to a broad comprehension of what these orbitals look like. > #MO-hbrid-plot > restart; > with(plots); > fsa := ep(-ra); > fsb := ep(-rb);ra := sqrt(^+^+(-r/)^); > rb := sqrt(^+^+(+r/)^); > psi_plus := ep(-ra) + ep(-rb); > psi_minus := ep(-ra) - ep(-rb); FIG. 9: The bonding p σ orbital versus

5 p sigma^*.. psi() FIG. : The anti bonding p σ orbital versus broken up into its constituent parts to show where the odd shape comes from. p sigma FIG. : The anti bonding p σ orbital versus broken up into its constituent parts to show where the odd shape comes from. Warning, the name changecoords has been redefined [animate, animate3d, animatecurve, arrow, changecoords, compleplot, compleplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, clinderplot, densitplot, displa, displa3d, fieldplot, fieldplot3d, gradplot, gradplot3d, graphplot3d, implicitplot, implicitplot3d, inequal, interactive, listcontplot, listcontplot3d, listdensitplot, listplot, listplot3d, loglogplot, logplot, matriplot, odeplot, pareto, plotcompare, pointplot, pointplot3d, polarplot, polgonplot, polgonplot3d, polhedra supported, polhedraplot, replot, rootlocus, semilogplot, setoptions, setoptions3d, spacecurve, sparsematriplot, sphereplot, surfdata, tetplot, tetplot3d, tubeplot] ra := fsa := e ( ra) fsb := e ( rb) + + R + R

6 + rb := + + R + R psi plus := e ( + + R+R ) + e ( R+R ) psi minus := e ( + + R+R ) e ( R+R ) > #note, un-normalied orbitals in use! > lim := ; > plot(subs(r=,=,=,psi_plus),=-lim..lim,labels=[, psi() ],titl > e= sigma ); > plot(subs(r=,=,=,psi_minus),=-lim..lim,labels=[, psi() ],tit > le= sigma^* ); > plot3d(subs(r=,=,psi_plus),=-lim..lim,=-lim..lim,aes=boxed,label > s=[,, psi ],title= (sigma) MO(,,) versus and ); > plot3d(subs(r=,=,psi_minus),=-lim..lim,=-lim..lim,aes=boxed,labe > ls=[,, psi ],title= (sigma) MO(,,) versus and ); > implicitplot3d(subs(r=,psi_plus)=.,=-lim..lim,=-lim..lim,=-li > m..lim,aes=boxed,labels=[,, ],color=blue,title= sigma MO ( > + = blue) ); > f := > implicitplot3d(subs(r=,psi_minus)=-.,=-lim..lim,=-lim..lim,=-l > im..lim,aes=boxed,labels=[,, ],color=red): > f := > implicitplot3d(subs(r=,psi_minus)=.,=-lim..lim,=-lim..lim,=-li > m..lim,aes=boxed,labels=[,, ],color=blue): > displa(f,f,title= sigma^* MO ( +(blue) and -(red)) );