Clifford Becker' College of St. Thomas St. Paul, Minnesota
f Orbitals
Geometry of the
EDITOR'SNOTE: The coincidence of receiving two articles on the same theme a t the same time justifies mention. Mr. Becker has shown consideration far editor and reader by modifying his original manuscript by drastic reduction to avoid obvious repetition. The subtle differences in the two presentations here provided (e.g., the choice of normalization factor and the choice of axis designations) should serve as a forus rather than a dispersion of reader interest. The comparison of figures should also add to the reader's interst and understanding.
Models of atomic orbitals obtained from the angular portions [Y(B,d)] of the hydrogenic wave functions [+(r,O,d)] have been very useful in understanding the geometry of chemical bonds. A geometric representation of an orbital, however, usually couveys a physical significance of probability distribution of an electron. The probability of finding an electron in a particular space is given as: f $Z(vO+)dV, or in this consideration, f f Y2(Od) sin Odddo, where the probability is associated with the square of the amplitude of the wave f u n ~ t i o n . ~ Approximate orbital representations can be achieved by graphing characteristic cross sections of the squared augular functions; the Styrofoam models pictured in Delivered in part a t an ACS Undergraduate Chemistry Symposiunl held a t Macalester College, St. Paul, Minnesota, April 27, 1963. The author gratefully acknowledges the guidance and encouragement of Professors James A. Ryan and James 3. Carney. 'Present address: Department of Chemistry, Ohio State University, Columbus, Ohio. DAY,M. CLYDE,JR., AND SELBIN,JOEL,"Theoretical Inorganic Chemistry," Reinhold Publishing Carp., New York, 1962, p. 32. EISENSTEIN, J . C., J . Chem.Phys., 25,142 (1056).
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this article are based on such graphs. The spherical coordinates used for these functions are defined by Figure Table 1.
Angulor Functions of the f Orbitols
Y,r.a - 39, Y.ra,, _ .*I Yam*- r,q YZ1,,2_ Y , w - .? ,q, Y Y,,,
= =
..,
I/,
= '/2 = I/,
= = =
'/$
I/?
I/,
(70)1/2sin38 COB 39 (iO)'~? sins0 sin 39 ( 7 ) 1 1 2(5 CO& - 3 cos 8) (42)'iP sin 8 (. cos2 i 8 - 1) cos 4 (42)'J2sin 8 ( 5 ensa 8 - 1) sin 4 (105)u8 sin2 8 cos 8 cos 2rn (108)111sin2 8 cos 8 sin 24
The angular distribution functions4 for the f orbitals normalized to 4 r (Table 1) describe four basic orbital
)f
.-
I
Y
Figure 1.
Editor's Note: The reader who is making comparisons with the accompanying H. G., CHOPPIN,G. article by FRIEDMAN, R., and FEUERBACAER, D. G.,T A ~ SJOURNAL, 41, 351(19G4) should note the difference in definition of coordinate systems. Rather than change one or the other set of designations, the correlation of the two representations remains as an "exercise for the reader." For example, Beeker's f , i j , ~ , * l is the Friedman, Choppin, and Feuerbarker f i ( j . ~ , > ) , also designated as f,,~.
shapes. The photographs in Figures 3-10 show Styrofoam models of the Y Zfunctions for the orbitals described in Table 1. (A) and (B) give complementary six-lobed lemniscates in the xy-plane, i.e., (cos 3$)%and ..
,".
.
,"
,,.,
(sin 34)%,with (A) projecting lobes along the x-axis hut none in the yz-plane (Fig. 3), while the xz-plane is nodal for (B) and two of (B)'s lobes are directed along the y-axis (Fig. 4). As a second cross section, taken perpendicular to the xy-plane, e.g., 4 = O0 or 4 = 00°, the nonsuperimposable (sin30)2is obtained, making the individual lobes rather flattened. These orbitals are concentrated in the region of the xy-plane and have no projection along the z-axis (maximum magnetic quantum number). (C), (Fig. 5 ) , exhibits the trace (5 cos3 0 - 3 cos 0)% in all planes passing through the z-axis since it is independent of 6. Figure 2 shows the graphs of these
-
.
.
-
,-
1
i
- \ :;
Figwe 9.
Y'model of fcxs~,?j;.
rtgure ic.
!
r - model of f,,:.
Figure 2. Graphs of the x r and yz cros rections of the angular function for f a m - r u * ) (left1 and Ihe rqvored angular function (right).
-
Figure 4.
Y'
model of f,,r:b?.g?).
Figure I 1 .
Y' model o f f ;
,,; ,.
-Figure 12.
YZmodei o f f . , ~ ~ q . .
Figure 14.
Y2modeloff,,ir,,r_a,2,.
4
Figure 13. "">
Figure 5.
Y' model of f* . . >L~,Z).
....,.u....,......
.......
Y'modeloff~,.! r.,.
,
Figure 6. Y' model of anologavr 3d orbital,dz!.
for Y and Y2. The xy-plane is nodal, and the function becomes zero for 0 = 3g014', 90°, 140°46', 21'Jo14', 270°, and 320°46', gives relative maxima for 0 = 64O, l l G O , 244", and 296', and absolute maxima for 0 = 0°, 180' (z-axis) for all 6. The orbital representation of as based on YZ,,,,L,,3), appears rather analogous to that of d,. (Fig. 6) since both are concentrated along the z-axis and are independent of $ (zero magnetic quantum number). 1 Sin 0 (5 cos20 - 1)1 is the principal section for (I)) in the xz-plane (4 = 0") (Fig. 7) and for (E) (Fig. 8) in the yz-plane ($ = 90°), while (D) has no projection in the yz-plane and (E) no projection in the xz-plane. The principal section exhibits zeros for 0 = Oo, 63.s0, 116.5', 180°, 243.5O, and 2'36.5', and relative maxima Volume
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for 8 = 31°, 90°, 14g0, 211°, 270°, and 32g0, for the for ( G ) . The maximum cross sections perpendicular appropriate constant @. A cross section taken perto the major section can only be estimated. The more pendicular to the smaller lobes a t 0 = 90°, i.e., in the widely separated adjacent lobes interestingly exhibit xy-plane, is ( ~ o s $ )but ~ , the oblique cross section of the the tetrahedral angle 109'28'. (F) and (G) are the larger lobes for 8 = 31' is an approximated ( ~ o s ~ $ ) ~first eight-lobed atomic orbitals encountered. since this section cannot be taken directly. These An alternate set of f orbitals, especially applicable rather flat orbitals are confined to the regions of the to cubic systems, is also sometimes l i ~ t e d . ~In this set, xz- and yz-planes respectively. fe, fs,, and ,f, are identical with f=v., fs+as,, and (F), (Fig. 9), exhibits the cross section (sin28 cos 8)Z f+y~,I respectively; ,f,,+~,,) or and f y , S V ~ S r l ) 01with unsymmetrical lobes in both the xz- and yz-planes, fr,have shapes equivalent to f,,,,.-,,zjbut are located while (G), (Fig. 101, is nodal for these planes, but gives along the x and y axes respectively; and f,+v2,s, i.e. that function for @ = 45" and 135'. Maxima are objr,, and I,,.-,.),or f, have shapes equivalent to f,,.-,2,,, served for 8 = 54"44', 125"16', 234'44', and 305O16' but project their lobes about the y and z axes and x and in their respective planes, and the function zeros occur on the axes, i.e., for 8 = 0°, 90°, 180°, and 270°, and z axes respectively. Both sets o f f orbitals are found also for 4 = 45' and 135O for (F) and 4 = 0' and 90° useful.
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