E. A. Ogryzlo and Gerald B. Potter University of British Columbia
Vancouver, Canada
Contour Surfaces for Atomic and Molecular Orbitals
Models of atoms which show electrons moving in well defined orbits around the nucleus are not consistent with a vast body of experimental evidence or with current theories of atomic structure. The wave mechanical description of the electron in an atom arose from de Broglie's (1) wave theory of matter, and is contained in a wave equation proposed by Schrodinger (8) in 1926 (the symbols are defined in (5)):
The equation describes a standing wave, and if some physical picture of the electron is to he retained an interpretation of $ (wave function) in terms of some property of the electron is essential. Since in Maxwell's wave theory of light $? was successfully related to the intensity of a light beam, Schrodingcr (3)initially suggested that $2 for an electron was a measure of its density a t any point. A few months later, Max Born (5) gave the presently accepted interpretation of $. In view of the Heisenberg uncertainty principle (4), Born identified (or $*$, i.e., the wave function times its complex conjugate when $ contains imaginary parts) with the probability of finding the electron a t that point. I t is interesting to notice that the concept of a moving electron nowhere enters the description given by the Schrodinger equation, and in fact it is not possible to conceive of any classical motion that would result in most of the probability distributions that arise. Nevertheless. because the probability function ($*$) is not time dependent, models of the atom in various stationary states can be constructed. The ideal model of an atomic or molecular orbital would he a cloud-like structure showing the prohahility of finding an electron at all points in space relative to the nucleus. Although it is conceivable that a model could be made with a colored suspension in a clear plastic, the difficulty of preparing such a model precludes its use. Somewhat less effective, but more easily constructed models can be fashioned from styrofoam to represent contour surfaces. The surface visible must then represent a contour surface of constant $ (wave function) or of constant $*$ (probability fnnction). There are a number of ways in which the surface could be labeled, e.g., (a) by the value of $ relative to,.$ , (b) by the value of $*$ compared with $*$-,, (c) by the fraction of $*$ enclosed by the contour surface. The last method is probably the most effective for labeling the surface since it gives the probability of finding the electron within the surface, or, as a less correct alternative, the fraction of the electron(s) enclosed by the contour surface. For purposes of campntation it is more convenient to use method (b). We 256
/ Journal of Chemical Education
will now present a method by which contour surfaces can be calculated, and show how these can be used to construct some models that are useful for introductory courses in atomic and molecular structure. Atomic Orbital Contour Surfaces
Cohen (4) has shown that these contours for a p orbital can he found by expressing the !Fare function in Cartesian coordinates and separating the variables. This cannot he done in general; hence it is appropriate to use polar coordinates in which the variables are always separable. The function $*$ is given in Table 1 for a number of atomic orbitals. Strictly, these are correct only for "one electron" atoms, but are extended to the general case of a many electron atom by using an effective nuclear charge, Z, for a particular orbital (6). Though the effective nuclear charge varies with r we shall use an average value to make the calculations less formidahle. The method of obtaining contours will be developed in detail for the 3d orbitals of Ti(III), which has Z = 9.2. We consider first a 3 d , ~orbital, the contours of which arc identical with those of the 3dzE,,, 3d,, and 3d,, orbitals except for their direction in the coordinate system. The electron density. $*$, along the x axis, shown in Figure 1, is calculated from the appropriate function of Table 1 with 0 = 90' and 4 = 0". The contours of constant electron density are dierent in the xy and xz planes, and muqt he evaluated separately. I n the xy plane, we have, with 0 = 90':
,.
in which a is the Bohr radius (0.529 A), and a = Zrla. It is convenient to assume values of a and calculate values of 4 which give a chosen value of $*$. This has been done for $*$ equal to 0.316 and 0.10 of the maximum $*$, with the results given in Figurc ?. The same result can be found by a graphical solution
? A Figure 1. axis.
Electron density in o Tilllll 3drL,,9 orbital along the x
[or yF
employing the curve of Figure 1. The nodes ($*$ = 0) are found a t values of 4 that make cos224 equal to zero (+ = 45', etc.). The same procedure in the xz plane (+ = 0) develops solutions of
with the results shown in Figure 3. Thus the contour surfaces of the dz2->,, orbital are oval in cross section (see models in ~ il i )~. . The 3dza orbital is independent of 6 (see Table 1) hence the contour surfaces have cylindrical symmetry about the z axis. The electron density distribution
*'
at 0.10 ond 0.316 of maximum for a Figure 2. Contours of constont Tilill) 3d,lvn orbitalin the xy plane. There mro h o n o d d planer perpendiculortothe pope, and bisecting the oxir
I
i . ;
Figure 3. Contours of constant * 2 a t 0.10 and 0.316 of maximum for o Tillil) 3d,L,I orbital in tho x r plane. The r-oxis is the intersection of the two nod01 planer shown in Figure 2.
Probability Functions for Hydrogen-Like Atoms
Table 1. 71
along the z axis is identical with that of Figure 1, except that the magnitude of the density is larger by a factor The contours are given in Figure 4 for the same fractious of the maximum electron density as above, although for the "doughnut" in the xy plane, the electron density does not reach 0.316 of the maximum. Nodes are found a t 0 = cos-' (1/3)'12 = *55'. A similar treatment for the 3p, orbital of the chlorine atom (Z 6.1) gives the electron density (along the 2 axis) shown in Figure 5 and the contour lines in Figure 6. The contour surfaces can be generated by rotation of this curve around the z axis. Aside from a node in the xy plane there isoalso a spherical node of radius a = 6 or r = 0.52 A. Similar curves are shown in Figures 7 and 8 for the 2p, orbital of a carbon atom, previously treated by Cohen (4).
rn
+*+
Orbital 1
1
0
0
1s
-1 (z/a)=e-c
2
0
0
2s
1/(2s r ) ( z / a ) >( 2 -
as)ec0
An Alternative Orthogonal Set of p Orbitals
An Alternative Orthogonal Set of d Orbitals
3d.' 3d.3d". 3dzv 3d(,*-,a) where --
c = -zr
a
' / 4 3 em' E - 112 4 slnP 0 eoss E CM* 4 1p.38 n ( z / a ) ~ 4e - ( 2 ~ / 3 ) 4 sinP E ease 8 cos' 4 sin4 E sin' 2 4 sm' E cosa 2 4
ha and a = 4r2ntc' Volume 40, Number 5, M a y 1963
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257
Charge Enclosed by Contour Surfaces
With a norn~alizedwave function, this becomes a problem in integration of $*$ over thc appropriate limits: f
= frariion enclosed =
Jn+*+
a fuuctioi~of the limiting value of r. This is the area under the curve 28$*fi2 from 0 to r. At each of the angles 0°, 20'. 40°, 60°, and 70°, the limits of r are
sin BdB& dr
7%
COntoUr
This equation call be integrated directly when the orbital concesned has spherical symmetry with the limits: 0 < 0 < a,0 < 6 < 2 8 and the values of r defining the contour. 111 this integratioil each variable may be treated iudrpel~dently. Wheu the coutour surface has a lo~versymmetry, graphical integration can be nsed. Even this method becomes quite co~nplex in dealing with an orbital such as the dZz - p. The graphical integration of the carbon 2p, orbital will be described. Since there is cylindrical symmetry arouud the 2 axis the integration
can be included in the I)*$ function, hcncc we must cvaluate the double integral: f =
JJ 2s+*+rs sin BdB dr
oontour
In Figure 7 is shown the integral over
T
(but not 0 ) as
Figure 6. Contours of constant a1 0.031 6 and 0.10 ofmaximum for o Ci 3 m orbital. The xv done and o where of radius 0.52 A are nodal surfacet.
Figure 7.
o f the integrol2r
+'
Figure 4. Contours of constant at 0.10 and 0.316 of maximum for o Tillll) 3di* orbit01 in m y plane which includes the z oxir. The nodes are ronicrurface~mokingon angle = 55'withther axis.
vA Figure 5.
,." J,
Electron density in a C 3p,orbitol d o n g the= oxis and the value
Eledmn density in a CI 3pIorbitalalong t h e r axis.
258 / Journol of Chemical Educafion
( + ( r ) ) 2 r P d r along t h e r axis for this orbital
Figure 8. Contours of constant +% at 0.031 6. 0.10 and 0.316 of maximum for a C 2p,orbitol. The xy plane i s o nodal surface.
found from the electron density curve of Figure 7, and then the integral over r obtained a t that angle. For example, a t 8 = 10°, with a contour a t 0.10 of the limits of r are found by drawing a horizontal line a t 0.10 ($*$,., cos2 40') = 0.213 (dashed line in Fig. 7). The limits of r are thus found to be 0.065 and 0.975 A. Transferring thcse limits to the area curve, we find
Xow t,his function is to be integrated over the angle 8:
This is accoinplished by multiplying cach value of .1(8) found from Figure 7 by cosZ8and plotting the result against cos 0 . In Figure 9 the resulting curves for contours a t 0.316, 0.10, 0.0316, and 0.00 are shomn. The last is inrluded to provide a check on the method since with normalized wave functions, the final integration in t,his case must yield unity. The integration of these curves gives the fraction of the total electronic chargc within a specific contour surface (the areas in Fig. O are to be doubled since the graph only extends from 0 t.o ?r/2 rathrr than to-k). The results are 0.34, 0.66, 0.85, and 1.01, respectively, the last value indicating that the mrt,hod is satisfactory. -4 smooth curve
can be drawn thmugh thcsr points (Fig. lo), from which we find that to include 99% of the charge, a contour surface a t $*$ = 0.002$*J..,., mould be required. This contour intersects the z axis a t about r = 2 A. This number provides all interesting compariso~~ with the van der Waals radius of carhon of about 1.7 A. Atomic Orbital Models
Thrce dimensional modcls illustrating the contour.: described in the previous sections can ~ a s i l ybe fashioned from stysofoam. hiodels of some orbitals of chemical intemst are shomn in Figure 11. Such models suffer from the disadvantage of not displaying the presence of some nodes (in the 2s and 3s orhitals for example) and also uot showing the positions of highest clectron density. Figure 12 shows how it, is possiblc to ovclromc thrsc deficiencies by cutting wdges out of the inodela. Areas of low electron dcnsity can then be darkened.' The contours in Figure 12 represent thc complete set of orthogonal orbitals which can be idrntificd xith t,he quautum numbers n, 1 and ml for n = 1, 2, and 3. They are the orbitals obtained by t,hc method of solution described hy I'anling and Wilson in which tllr On these models h h c k spray paint was used on the outside nnrl a black marking pencil was used t o darken the interior.
I
Figure 11. gen atom.
I
8
.4
.6
cos
Figure 9.
constant
$ ' c o n t f o u r r of atomic orbifolr or t h e hydro-
I
I
10
Modelr o f
.2
0
e
Valuer of A (8) co~'8verwscor 8.
0
2
1 "I*.
d
b
VV'
.S
10
Figwe 12. Atomic orbital models w h c h h a v e b e e n r u f a w a y to show the presence o f hidden nodes ond positioniof maximum elcctron density.
"OX
Figure 10. Percentage o f electron enclosed b y a constant ijrz contour vertvrthe volues of ijrP/ijrP m m .
Volume 40, Number
5, May 1963
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259
variable (r, 9, and +) are assumed separable (6). Since other orbitals which can be obtained by taking linear combinations of these are useful for the description of polyatomic structures, they are given in Figures 13 and 14 as alternative orthogonal sets of 2p and 3d orhitals respectively. It is worth noting that values of the quantum number ml cannot be identified with most of these orbitals. This is still a very common textbook error. In Figure 15 somewhat schematic models of the i s , 2p, and 3d and 4f orbitals are arranged to show how they are related to one auother by the addition of successive planar nodes.
cylindrical charge symmetry around the molecular axis provide a better description. Even more schematic are the models of molecules shown in Figure 17. Unfortunately niodels that attempt to show contour surfaces enclosing large fractions of $*$ cannot show the individual orbitals. On the other hand, contours enclosing small fractions of $*$ often do not show the build-up of electron dew sity in the region between the nuclei. Yo actual calculations have been attempted for these models hence, even in one model, thc contour surfaces will represent different fractions of the total chargc. With these
Molecular Orbital Models
Molecular orbitals forn~ulatedby the linear combination of atomic orbitals (LCAO) approximation are shown in Figure 16. These models are principally useful as guides in demonstrating the symmetry of the orbitals and in providing a general idea of the distribution of electron density. The bonding and antibonding orbitals which are found in first and second period homonuclear diatomic molecules are shown. The two types of a orbitals are analogous to the atomic p, and p + ~orbitsls. In a non-linear polyatomic molecule for which the nuclei define a reference plane (e.g., ethylene), the d p , orbital is the logical one to use, while in linear molecules, the 72 p,, orbitals with Figure 16. Models of molecular orbitair proximotion.
a b t u i n e j by
Antibonding
the LCAO cp-
Bonding
rl r
o"1 r
Figure 13.
Figure 14.
An alternative orthogonal set o f 3d orbitolr.
3d,,
3d2 3dVl
3dru
3drLu2
Figure 1 1 Somewhat x h e m a t i c modelr of 1 r, 2 p , 3d ond 4 f orbitolr.
260
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Journal of Chemical Educofion
Figure 17. Somewhat rchematic molecular models, c u t o w a y to ,how inner o r b i t d s o n d nuclei. H2 N? CzHz
0s NO
Li? Ft HF
limitations in mind, the models have been cut away to show the u molecular orbitals around the molecular axis. All of the molecules in Figure 13 have cylindrical symmetry. The description of cylindrically symmetric n orbitals is possible in most cases by using either the real p, and p, atomic orbitals or the complex p , ~ orhitals in the LCAO description. 111the NO molecule, however, one electron occupies an antibonding orbital.
For cylindrical symmetry this must be placed in one of the p+, orbitals. The change in shape of the r contours of the molecules N2,NO, and O2 arises from the additional contribution from the x antibonding orbitals, but the bonding and antibonding orbitals cannot, because of overlap, be shown separately (see Fig. 12). Except for the acetylene molecule, these models have been formulated by the LCAO method without prior hybridization of the atomic orbitals. The point is controversial as indicated in recent calculations showing very little mixing of atomic orbitals even for HF, hence the concept of hybridization is used here only where required for structural reasons.
Acknowledgment
The author would like to express his thanks to Dr. Alan Bree for valuable discussions. Literature Cited (1) DEBROGLIE, L.,Ann. dePhys.,3,22(1925). E.,Ann. d . Phys., 7 9 , 360, 489 (1926); 81, ( 2 ) SCHB~~DINGER, 109 (1926). (3) BORN,M., 2.8.P h p . , 38,803 (1926). ( 4 ) COHEN,I., J. CHEM.EDUC.,38.20 (1962). L., A N D WILSON,E.,"Introduction to Quantum ( 5 ) PAULINQ, Mechanics," McGraw Hill, New York, 1935. ( 6 ) SLATER,J. C., "Quantum Theory of Matter," MeGrawHill, New York 1953. Chap. 6 .
Volume 40, Number 5, May 1963
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