Describing Electron Distribution in the Hydrogen Molecule A New Approach Christopher J. Willis University of Western Ontario, London, ON, Canada, N6A 5B7
One of the most difficult tasks facing the instructor presenting an introduction to quantum mechanics is to give the student a graphical representation of the electron in the atom or molecule as a wave, and many articles in this Journal, and elsewhere, have addressed this problem in recent years ( I ) . After introducing the uncertainty principle, we tell the class that we must never ask "Exactly where is the electron?" Instead. we ask: "How is the neeative charge of the electron spread out in the atom or themolecule? Where is the charge density at a maximum? How does the distribution of charge vary between different orbitals?" Each of these auestions is best answered bv drawing a picture. I t is natural to start with a discussion of the concepts in relation to atomic orbitals, then leadinto molecular orbitals usinn LCAO theow. However. there is a simificant difference& our approaihes to the two situations~whichcanlead to auestions from alert students. with atoms, we are careful to distinguish between two a ~ ~ r o a c hto e sthe description of electron distribution. We ogen talk of electron density, the concentration of charge per unit volume a t a given point around the nucleus. Electron density at any point is defined as the square of the wave function,. (udZ, . a t that point. Alternatively, we may try to describe how the total negative charge of the -smearedout" electron is distributed throughout the space around the nucleus. The term distribution function or probability distribution function is often applied, with the value .. ~RI.~(I+I)~. Many students find it helpful to envisage a particle-like picture of the electron and speak in terms of "probability", but careful wording is needed to apply these concepts to these two functions. In regions of high electron density, we may say that there is a high probability of "finding" the electron in unit uolume ofsoace. But if we wish to indicate the probability of 'findin2 ;he electron anywhere a t agiven distance from the nucleus. we have to allow for the s ~ a t i a l factor. For spherical atok, the space available to a'ccommodate the electron increases on going outward from the nucleus, leading to the 4rrP term in the probability distribution function. The distribution function, not the electron density, is needed to know the whereabouts of the total electronic charge. This is especially useful when different orbitals are comoared. For examole. we sav that the 2s orbital is more d i k s e than the lsor:inbrobabfiity terms, thatthe electron is "on averwe" farther from the nucleus in the 2s than in the 1s. yet-for each of these orbitals there is a sharp maximum in electron density at the nucleus. Only by taking into account the spatial term in the distribution function can we make it absolutelv clear that the negative charge corresponding to the electron is found farthe; from the nucleus in the orbital of higher principal . quantum number. When we turn to the discussion ofmolecules, this distinction is not usually applied. We develop wave functions for MO's by linear combination of atomic orbitals. Then we calculate electron density by squaring the wave function.
Contour plots of electron density for H were published in 1953 (21, and Wahl extended coverage the homonuclear diatomic molecules of the fust-row elements in 1966 (3). These results are useful in answering "where is the greatest probability of finding the electron per unit uolume." However. because thev do not include a soatial term. they cannot tell us where the total chargeofthe electron(s; is. relative to the nuclei ol'the molecule. Nevertheless. this connection is frequently implied. For example, Coulson writes with reference to plots of yZfor.a in the species Hz+
30
\-,.
(A).
These e w e s show that, in a bonding orbital, the charge is concentrated rather more between the nuclei than would be of the comnanent atomic orbital exoeded bv, suoeroosition . . dcnaitwr;. In n honding nrhitnl, the lntcrnl spread is not wry p e a t , risr that the efTwtive thlrknevs o f t h e charge-cloud is less than the internuclear distance. The outer contours of constant density resemble in generalappearancea set of confocalellipses with the nuclei as foci
Similarly, Cartmell and Fowles show plots of electron density for the Hz molecule and write (5): The diagrams for the (yt,)' (bonding)function show clearly that the increased electron charge density associated with this
a
Figure 1. Plots of (a) the radial wave function, yr; (b) the electron density, ( q r f , (c)the radial probability distribution function,4n?(W,)2, for the 1s orbital of the hydrogen atom. Volume 68 Number 9 Seotember 1991
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The radial probability distribution function for the atom combines electron density per unit volume with the term 4w2, allowing for the increase in available volume within thin concentric "shells" a s the radius increases. The total function, 4nP(yr,)2, for the 1s orbital of hydrogen is plotted (Fig. lc). The maximum occurs a t r = 52.9 pm, that is, a t a distance from the nucleus of a., the radius of the first Bohr orbit. The value a t the nucleus is zero because the spherical volume element is zero when r = 0. For other s orbitals, there will be one or more zero values in the value of yr, and thus more than one maximum in the plots. Figure 2, showing yr,, (yrJ2, and 4nP(yrJ2 for the 2s orbital of hydrogen, illustrates these trends. The electron densitv. (wJ2 shows a sharp maximum a t the nucleus and a mucii smaller maximum a t r = 2a. (106 pm), whereas 4nr2(wJ2 contains two maxima. a t 40.4 and 277.0 Dm. The heip,h; of the outer maximum is considerably grcaicr than that ofthe inner. The bulkofthenceati\wchar~e .. associated with an electron in this orbital is inthe outer rebeon, about five times farther from the nucleus than that of an electron in the 1s orbital. However, i t is the plot of the probability distribution function, 4nP(yrJ2, that helps us appreciate this. The electrondensity plot (Fig. 2b)does not containthis information. I t is not easy to explain these important concepts to a class beginningthe study ofatomic structure, but once they are understood the student might reasonably expect that the same principles will be used in describing molecules. This article shows how this problem may be approached. F~g.re 2 Plots of (a) the rad al wave funnon, y,, (b) tne e ectron oensdy, (y,j2.(c)the radal probab ry alsrr mton f~nctlon, 4n?(y,j2, for the 2s orbtal of the nyorogen atom linear combination is concentrated in the region between the nuclei, whereas the charge cloud is pushed away from this region in the v. (antibanding)combination.
The Shape of the Hydrogen Molecule By definition, all radial functions for a n atom are dependent on one parameter only, radial distance from the nucleus. I n a diatomic molecule, spherical symmetry is lost, and we must find some other way of deriving a probability distribution function. For the hydrogen molecule, the 01% and MO's are formed by combination of the spherical 1s atomic orbitals of the two atoms, so they have circular
Both of these writers use the term "charge cloud", but the function to which they are referring shows only the intensity of the charge point by point, not the total amount of charge contained within the cloud. This picture will be considerablv altered bv allowing for the availablespace, a s well ns theelectron density. ~ o m a k e this modificat~on.we must include a term describing the space around thenuclei in terms of the molecular s-etw. In this article, this problem is discussed using the hydrogen molecule. A method ol'describingcharge d i s k h u tion in its MO's is sugg~sted.We b e r h with a brief review of the concepts used indescribing orbitals in the hydrogen atom. Atomic Orbitals F i ~ w r r1 shows the usual three plots of important functions for the i s orbital of hydrogen: radial component ofthe ware function, electron density, and radial pn~hahilitydistribution function.' The plots are shown with the nucleus (at the origin) a t the center. They extend in both positive and negative directions to show the student that the functions change in an identical way regardless of direction from the nucleus. For the 1s orbital (or any other s orbital), plots of the radial com~onentof the wave function.. w.,.. . or the electron density, (I+$, show a maximum a t r = 0; there is a maximum in electron density per unit volume a t the nucleus. Since the purpose of the figures is to show only changes in the values of the various quantities as a function of position, no units are given.
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Journal of Chemical Education
Figure 3. The wave functions of the MO's in the H, molecule: (a) bonding, cr,,; (b)antibonding, a*,,. Vertical lines show the positions of the two nuclei.
"sausage shaped" for 01. and egg-shaped "blobs", separated by a nodal plane, for o*~.. These diagrams are very helpful in a general discussion ofMO theory, but they leave some key questions unanswered.
I I UU
-
1. If spatial and electron density concepts are eombined, where is the bulkof the electronic charge in the Hz
molecule? 2. We explain the difference
in energy between the bondine and antibondine MO's hv s&ig that, in the k m e r an
increase in the charge is
Figure 4. Plots of electron density, (&, for the bonding MO of the H, molecule: (a) along the molecular axis (b) contour maps in the plane of the molecule. symmetry about the molecular axis. The usual representations of the wave functions for the bonding and antibonding orbitals, ol, and o*~., are shown in Figure 3, and the corresponding electron density plots in Figures 4 and 5. The "contour maps" of Figures 4b and 5b are more informative. They clearly show that the regions of maximum electron density, centered on the two nuclei, extend into the region between the nuclei in the a], orbital and away from this region in the orbital. Both orbitals have circular symmetry and a three-dimensional picture can be envisaged by rotation about the molecular axis. The resulting models can be used as a basis for the boundary surfaces often used to represent the Hz molecule. These are drawings of three-dimensional surfaces assumed to contain most of the total electronic charge. These will be
Figure 5. Plots of electron density, (y,& for the antibonding MO of the Hz molecule: (a)along the molecular axis; (b) contour maps in the plane of the molecule.
in the
Figure 6. A circular ring, distance r from the molecular axis, crosssection rx r.
region between the two nuclei. Do these diagrams demonstrate this? 3. When a s~herical boundarv surfaceis drawn for the hvdrogen . atom, we know that the wave ?unction, the electron density, and the nrnhahilitv function have the same value at everv ..-. -.= -.. ~ .distrihnt,inn " .--point bn that surface.For the Hzmolecule, is it possible to draw a surfaceover which a comparable probability distributionfunction has a constant value? ~~~~~
~~~
~
~~~
~
An alternative picture, which answers these questions, can be built up if we define an additional function to describe the MO's in the hydrogenmolecule. With the atom, the nucleus is the center of symmetry of the species, and all parameters are related to this point. For the H? molecule, the molecular axis is obviously the most important symmetry element present, so all measurements are taken around this line. Since both honding and antibunding a MO's have circular symmetry ahout the molecular axis, we mav call this a circular orobabilitv distrihution function. Th$ will tell us how muih of the total electronic charge is in the annular soace within a narrow circular r i w - extending around the molecular axis. Clearly, t h s will depend on two factors: the electron density at that distance from the axis, and the annular volume $thin the ring. The latter is to its radius of the circle. directlv "oro~ortional .. A more s~ecific~ i c t u r of e a circular distribution function is obtained by considering Figure 6. A cylinder of radius r and length r is drawn around the molecular axis. The volume contained in the "ring" between this cylinder and one of radius (r + &), where Ar is small, is approximately equal to its circumference multiplied by its cross-sectional The amount of electronic charge contained area. or Z~r(Ai-1~. within it is the product of volume and electron density, or 2i~r(Ap)~(@, where yf is the wave function of the MO at distance r from the axis. We can then define the circular distribution function (CDF)as:
(The approximation made in calculating the volume becomes exact as &+O.) Clearlv. the value of this function will be zero on the molecul& axis, where r = 0. Just as, with the atom, no part of the electron charge was found at the nucleus (a point), so in the molecule no part of the charge is found on a onedimensional line. In each case, the volume term is zero in the region where there is a maximum in the electron density Volume 68 Number 9 September 1991
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Figure 8.Location of any point with respect to the hydrogen nuclei HI and HZ. and
Figure 7.Contour map of the circular distribution function for the Is atomic orbital of an isolated hydrogen atom before approach by a second atom along the H-H axis. Before wnsideriug the appearance of this function for the hydrogen molecule, it is helpful to evaluate it for the separate atoms. As two atoms approach from infinite separation, the line between their nuclei defines the axis of the future molecule. About this axis, the CDF for the 1s atomic orbital of each atom is given by:
whereZis atomic number (= 1 for hydrogen), a l and azare in pm, and a, is the radius of the first Bohr orbit, 52.9 pm. Combination of these by addition or subtraction gives ~ I B and y~m,from which the respective CDF functions 2nr(y1~)~ and 2 n r ( y 1 ~may ) ~ then be calculated. Results are shown in Figures 9 and 10 for bonding and antihonding MO's respectively. In Figure 9a, the relative values of the CDF for the ol,bonding MO are shown a t distances from the axis r = 10,20, and 40 pm. As in the plot of (yd2 along the axis, two maxima are seen a t small r values. These merge into one peak, equidistant from each nucleus, as the off-axis distance increases. The contour
where r is the off-axis distance and a is the radial distance of each point from the nucleus. Acontour plot of this function is shown as Figure 7.2Along a line through the nucleus and perpendicular to the axis, r = a a t every point. The maximum value of the CDF occurs om).. that is.. onlv a t r = a I2 (26.5 . . half as far from the nucleus 2s the maximum in the radial distribution function for the Is orbital in the isolated. soher~calatom (Fie. - Ic,. . In the Hz molecule, the variatibnofthe CDF with position will. of course. deuend on the MO under consideration. To a fikt approxi'maiion, the wave functions for the t w o MO's formed by overlap of the atomic 1s orbitals are: Bonding, oh
VB =
M h + $21
where and are the wave functions for the 1s atomic p and N, orbitals of the hydrogen atoms HI and H normalization factors (both taken as 11 2 m t h ~ d~scus s are ion).^ The location of any point around the molecular axis may he expressed in terms of the distance from the axis, r , and from the two nuclei, al and az, as shown in Figure 8. The internuclear distance is taken as 74.2 pm, the bond length in the Hz molecule. The total atomic wave functions, including the constant angular component (I/&%) are then given by:
are in Relative values of the function in all contour diaarams arbitrary units. Taking the overlap integral, S,as zero. This introduces appreciable error because S = 0.585 for Hz+.More accurate values are N = lNz(yr+ S )=O.562 and N = 1 / @ ( ~ - S ) = 1.098. However, this does not affect the susequent argument.
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Journal of Chemical Education
Figure 9.Circuiar distribution function for the ol, bonding MO: (a) at increasing distance from the molecular axis: (b) contour maps in the plane of the molecule.
F gure 1 0 . C ~ar c ~d~strbullon fLnnlonfor the a',, ant bondlng MO: (a)at lncreastng dlslance from the molecu ar a m ; (01 contoLr maps in the plane of the molecule
plot, F i ~ r Yb, e shows the value of the CDF out tqwhere it has droooed to about 205 of its maximum value. Since this functioklhas circular symmetry about the molecular axis, the negative charge of the electron in this orbital is concentrated in a circular "donut" around the middle of the H-H axis. The maximum in the distribution of negative charge occurs about 37 pm off axis, or 50-55 pm from each hydrogen nucleus. In Figure 10a, the relative values of the CDF for the antibondine MO. a',.. are shownat off-axisdistances r = 5. 10, and 2oVpm.kvomaxima, divided by a node half-way between the nuclei, are seen in each case. At small r values. their separation is close to the internuclear distance, as in the plot of (yfI2,Figure 5a, but as the off-axis distance increases the maxima move slightly farther apart. The contour dot. Fimre lob. clearlv shows the unfavorable location of the majority of the elekronic charge, with maxima about 23 Dm off-axis and outside the refion between the two nuc1ei:~otationof the CDF function for o * ~about , the molecular axis will &e - two circular donuts4 As with atomic orbitals, these plots of molecular distribution fnnctions look quite different from those of electron density for the same orbital. I n particle terms, the electron density plot shows that the of fmding the electron in unit volume is at a maximum on the molecular axis, in the region of the two nuclei. Note that this is the case for Since this treatment nealects electron-electron reDulsions. numerical data areaccurateonhforthe one-electron molecular-ion; Hz+, butthegeneral picture isequallyapplicabletotheneutralH,molecule.
both the bondingand the antibondingM0. But ifwe simply ask "At what distance from the nuclei is there maximum probability of finding the electron?" the answer from the CDF plot is that, for the bonding MO, we should look about 35-39 pm off-axis midway between the nuclei. For the antibonding MO, the probability of finding the electron reaches a maximum about 23 pm off-axis and 45 pmaway from the central nodal plane between the two atoms. As was the case in distinguishing the 1s and 2s atomic orbitals, it is the plot of the distribution function that gives the most graphic differentiation between the o-bonding and o*-antibonding MO's. Comparing the CDF plots in Figures 9b and lob with Figure 7, we find that, while the charge cloud appears to have spread out from the nuclei in going from the atomic 1s to the bonding ols MO, it appears to have contracted antibonding slightly toward the a s in going to the MO. However. this effect d e ~ e n d on s the value used for the effective nuclear charge, assumed to be 1.00 in this discussion. Detailed calculations show that the effective nuclear charge should be greater than unity for the al. MO; Coulson gives 1.40 for Hz+a t 74.2 pm separation and 1.20 for Hz (6). This contracts the charge cloud for the electron(s) in the bonding orbital; for Hzt, the maximum in the CDF midway between the nuclei moves in from 37 pm to about 30 pm off-axis, very similar to the location found for the CDF in the isol~ted~hydrogen amm. Thin pictureofelectron distribution i~consistentwith the usual explanation of the difference in energy between the bonding and antibonding orbitals: The negative charge in the 01, is more favorably distributed with respect to the attractive influence of the positive nuclei. Although this treatment has been carried out for the Hz+ molecular ion (and, by extension, the neutral Hz molecule), the principles developed are clearly applicable to any a bond. The charge of the electrons in the bonding a orbital will be concentrated in a donut-shaped band around the bond axis, while that of electrons in the antibonding a* orbital is in two donuts, away from the region between the bonded nuclei. Conclusion
This a ~ ~ r o a to c hthe descri~tionof electron distribution in the hyf&ogen molecule has the advantage of following the same course as that usuallv taken in discussing ., atomic orbitals. It clearly distinguishes between the concepts of electron densitv and electron distnbutlon. It !zives a better picture o f t h e d k r e n c e between the honding &d antibondine molccularorhitals in the distribution ofneaative charrre a& the consequent difference in their energy. yt is therefoTe suggested that the discussion of a circular distribution fundion would be of help in representing the structure of the hydrogen molecule a t an elementary level. Acknowledgment I am grateful to Colin Baird for helpful discussions of this
material. Literature - Cited -
1. Allendoerfer, R. D J Ckem. Edm. 1990, 67, 37-39; Douglm, J. E. J Chem.Educ 1990.67.42-44:end references cited therein. 2. Bates,D.R.:Ledsham,K.; Stewart,A.LPhil. Tmns.Roy.Sm. 19SS,A246,215-224. 3. Wahl, A.C. Seienc. 1W3, 151, 961-967. Reproduced in Huheey, J. E. Inoganb ChemLslry. 3rd ed.: Harper and Row: New Vork, 1983; p 112.113. 4. Codson, C.A.Volonn, 2nd ed.: Oxford U.,1961, p 85. 5. Ca*mell,E.;Fowles.G.W.A.VolPncvondMolpeolpelorSffffff ,3rded.;B"ttemmths: London, 1966;p92.
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