Ronald C. Johnson and R. R. Renew Emory University
Atlanta, Georgia
I
I
I
S h ~ ofp Atoms
T h e "shape" of an atom is due to the positions of the electrons of the atom with respect to the nucleus. The distribution of these electrons about the nucleus of an atom depends on the shapes of the orbitals in which they reside. The shape of an atom can therefore be simply viewed as the sum of the shapes of the occupied orbitals of the atom. Atoms are frequently visualized as spherical balls; this description is accurate in many cases. The spherical shape of many isolated atoms1 arises from a fact which is seldom emphasized, namely that filled and half-filled subshells of electrons have spherical symmetry2 about the atomic nucleus. For example an isolated manganese atom is spherical. I t contains filled is, 2s, 2p, 39, 3p, and 4s subshells and a half-filled 3d subshell. That a filled or half-filled p subshell has spherical symmetry is demonstrated in the Appendix to this article. The spherical syinmetry of filled and half-filled subshells is a fundamental and unifying feature of the distribution of electrons about a nucleus. By realizing and using this symmetry one can better understand the shapes of single orbitals. Moreover one can visualize the shape of the electron probability distribution which arises from a subshell in which more than one orbital (but not all) are occupied (e.g., one electron in both a 2 p, and 2 p. orbital but none in a 2 p,). The sums of the electron probability distributions of two or more
The arrangement of electrons around a. nucleus and consequently the shape of orbitals is markedly affected by the presence of atoms, charged particles, and electric fields. Therefore in this article it k assumed that atoms and orbitals itre isolated from any perturbing influence, A subshell or orbital has spherical symmetry about a nucleus if all points a t a given finite distance from the nucleus are identical (eg., the probability of finding an electron a t all these points is the same).
orbitals are called orbital combinations in this article, and shapes of these combinations can he shown to have important consequences. A consequence of -the spherical symmetry of N e d subshells is that the sum of the electron probability distributions of the individual orbitals which comprise the subshell must he spherically symmetric. In the case of a p subshell it means that the three p orbitals in the subshell must have a geometry such that their sum is spherically symmetric. In practice 2 p orbitals have a dumbbell-lilce shape (1); the three orbitals differ only in their spatial orientation (each one pointing along a Cartesian axis, x, y, or z, and being called the p., p,, or v, .- orbital, resvectivelv). - . The vrobabilitv of finding an electron i'n a orbital is maxirkm along the z axismd diminishes away from this axis until it is zero in the xu plane. The p, orbital alone contributes electron probability along the z axis, the p, along the x axis, and the p, along the y axis. All three orbitals contribute to some extent in the regions between the axes; the sum of the three orbitals contributions a t any angle between the axes is the same as the contribution of one orbital along the axis on which it is the sole contributor. An atom having an outer p subshell containing only one electron would have a shape derived from the geometry of the orbital of that p electron superimposed on the spherical shape of the completely filled inner subshells. An atom with two electrons in an outer p subshell would have a shave comvosed of the s~hericalcore of filled inner subshells with the doughnut shaped symmetry of this partially filled p subshell superinlposed upon it. The doughnut shaped geolnetrY of the electron probability arising from electrons in two of three p orbitals can be visualized if one recalls that a half-filled subshell (ps~p,lp;) is spherically symmetric, When one subtracts a p, orbital from the spherical subshell, one cuts a hole through the subshell along the x
p,
Volume 42, Number 3, March 1965
/
145
axis; the shape of the remaining electron probability distribution resembles a doughnut. One can readily generate this orbital combination in three dimensions by rotating the contour diagram in Figure 3 of reference (1) about the x axis (also see Figure 6 of reference ( I ) . On the basis of this discussion one can describe the shapes of isolated atoms of the boron ([rare gas] ns2np1)and oxygen ([rare gas] nshp4) families as prolate spheroids, and those of the carbon ([rare gas] ns2np2)and halogen families as oblate spheroids. In order to describe the shapes of atoms of the transition elements one must know the shapes both of individual d orbitals and of combinations of d orbitals. The shapes of individual d orbitals are clearly presented in several recent articles (8, 3, 4). The atomic shape results from the geometry of the single occupied d orbital superimposed upon spherical filled inner subshells, or from the shape of a combination of occupied d orbitals superimposed on the spherical inner core. The shape of combinations of d orbitals depends both on the number of d electrons involved and on the type of d orbitals in which they reside. For example, if the 3d,, and 3d,2-,a orbitals each contain an electron, a doughnut-like electron probability distribution results with its maximum orientation in the zy plane. The circular symmetry of this combination in the xy plane can he appreciabed by visualizing the subtraction of the 3d,*, 3d,,, and 3d,, orbitals from the spherically symmetric complete d subshell. Since neither the 3 4 , nor the 3d,, orbitals provide electron probability in the xy plane, their removal has no effect. Since the 3d,3 orbit,al is circularly symmetric in the xy plane, its removal from the spherical d subshell leaves a circularly symmetric distribution in the xy plane. The removal of these three orbitals provides the hole in the doughnut. If a d electron is present in both the d+,. and d , ~ orbitals the resulting combination will have a shape different from that of the dz.-,* d,, combination; the maxinun electron prohability density will lie along the x, y, and z axes. The shapes of other comhioations of two d orbitals can be different from those discussed here, but it will be left to the reader to visualize their geometries. An important set of d orbitals includes the 3d,,, 3dz,, and 3d,, orbitals. The shape of the electron distribution produced by a half-filled or filled set of these orbitals is complex, hut certain features are quite important. The probability of finding an electron along
the x, y, and 2: axes is zero. A surprising feature is that the points in this combination a t which the probability of finding an electron is maximum do not coincide with the points a t which the probability of finding an electron is greatest for the individual 3d,,, 3d,,, and 3 4 , orbitals. These orbitals have maxima a t the centers of the edges of the octahedron described in Figure 1, whereas the combination of orbitals has its maxima a t the centers of the octahedral faces. This results from the fact that all three d orbitals contribute electron probability at the octahedral faces, whereas only individual orbitals contribute on the edges. Appendix: The Spherical Symmetry of Filled or Half-filled Sets of p Orbitals
I t is relatively easy to show that a filled or half-filled p subshell has spherical symmetry, if ons uses hydrogen p orbital wave functions. That the d and f subshells are also spherical can be demonstrated in an analogous fashion. It is generally assumed that the wave functions (in particular the angular part) for orbitals of all atoms resemble those of hydrogen, and therefore p, d, and f subshells of all atoms presumably have spherical symmetry. The shape of an individual hydrogen orbital is determined from the square of the appropriate hydrogen wave function (V). The shape of the electron distribution arising from electrons in two or more orbitals is determined from the sum of the squares of the wave functions of the occupied orbitals. The shape of an atom containing an electron in each of the three 2p orbitals is therefore derived from J'2p.? JIzp: +zn2. The wave functions used in the following calculation are presented in spherical coordinates. Therefore, in order to prove spherical symmetry it is necessary only to demonstrate that the sum of the squares of these wave functions is independent of the angular coordinates 8 and @, i.e., the probability of finding a p electron in the subshell depends only on r, the distance from the nucleus, and not on the direction from the nucleus. The wave functions used are taken from reference (5);
+
In above equations p
=
+
Z r / a and
where Z is the nuclear charge and a. is a constant. Literature Cited
Figure 1. An octahedron. O'r represent t h e renten of three of the twelve edges. A represents the cenfer of one of sir facer.
146
/
Journal of Chemical Education
(1) COHEN, I., J. CHEM.EDUC., 38, 20 (1961). E. A,, AND PORTER, G. B., J. CHEM.EDUC., 40,256 (2) OQRYZLO, (1963). (3) MANCH.W., AND FERNELIUS, W. C., J. CAEM. EDUC., 38,192 I> l-O- R- -l ),.. (4) PEARSON, R. G., Chem. Eng. A'eus, 37. No. 26, 72 (1959). ( 5 ) COTTON, F. A., AND WILKINSON, G., "Advanced Inorganic Chemistry," Interscience Publishers (a. division of John Wiley and Sans, Inc.), New York, 1962, p. 15.
