Thomas C. W. M a k and w a i - K Li ~~
The Chinese University of Hong Kong Shotin, N.T., Hang Kong
II
Relative Shes of Hydrogenic Orbitals and the Probability Criterion
Pictorial representations of the hydrogen-like wave functions (hydrogenic orbitals) have been discussed extensively in the chemical literature. Generally speaking, emphasis has always been laid on the portrayal of orhital shapes by means of polar graphs (I), angular distribution functions (2, 3). contour surfaces (4), three-dimensional stereo plots (51, and three-dimensional perspective diagrams (6, 7). Although some idea of the size of an atomic orhital can he gained from a scaled graphical representation (8,9), a systematic treatment of relative orhital sizes does serve a useful purpose. In this article, several commonly used criteria for the assessment of orhital size are discussed and an attempt is made to correlate them in terms of the ~robabilitvconcent. The wave'functionsof thehhydrogen-likeatom have the analytical form
Table I. Average Distances, Most Probable Distances, and Associated Probabilities of Hydrogenic Orbitals
where the radial part R,L and the angular parts 81, and a, are separately normalized to unity. For any given orbital the probability of finding the electron in the volume element d r = rZ sinOdrdRd6 in the vicinity of the point (r,R,$) is given by
The i values for orhitals 1s through 4f and their size ratios, calculated relative to the 1s value, are given respectively in the second and third columns of Tahle 1. The formula for i clearly shows that the size ratios for ns orbitals are identical to those for Bohr orbits, and that i values for orbitals in the same shell decrease with increasing 1 values.
Integrating over all possible values of 0 and 6 leads to the expression
Most Probable Distance (r,,,,J
Atomic Orbital
Size Ratio
P
P
For a given orbital this is readily obtained by maximization of the corresponding D,(r). For those orbitals with is defined as the distance of the "principal n - 1 > 1, r,. maximum," namely the outermost and most prominent of the maxima, from the nucleus. I t is well known that r,,, has the value 1 for the 1s orbital. However, r,. values for other orhitals are not readily available in the literature; these are listed in the fifth column of Tahle 1. It is seen that the size ratios for is, Zp, 3d, and 4f orhitals, which
which represents the probability of finding the electron between the distances r and r dr from the nucleus regardless of direction. Although the definition of the radial probability distribution function D,,(r) (10) leaves no room for misinterpretation, a large number of modern textbooks continue to present it as 47r+R2,1 which has the extraneous factor 4a, or as 4ar2+2.1, which is correct only for - ~ s-orbitals. ~I t will be seen from subsequent discusTable 2. Cumulative Probability Functions for Hydrogenic Orbitalssions that the function D d r ) plays an important role in the description of orhital size, and for simplicity the unit of length is henceforth taken as ao/Z, where a0 is the familiar Bohr radius and Z the nuclear charge.
+
~
~~~~~
~~
Bohr Orbits
Bohr's model of the hydrogen-like atom makes no distinction between orbitals of different 1 values in the same quantum shell. The radius of an orbit is given by the square of its principal quantum number n, and the sizes of the K, L, M, and N shells are in the ratio 1:4:9:16. Average Distance
li)
For hydrogenic orbitals, the average distance i can be evaluated exactly and is given by 90
/ Journal ot Chemical Education
M V
2
z
4
( +P+P+P + P + L - L ~ - . - P / z(I + h +P+P+L 645120+ +-) 10321920 P+L+
l-.-~/?
1+
8
48
a
8
.
48
Aa defined in Probability Criterion -tion
384
38iO
46080
3S;O
46080
4
364
of k t .
184320
Table 3. Relative Sizes of Hydrogenic Orbitals Based on Different Probabilitv Criteria Atomic P =.0.50 P = 0.90 P = 0.95 P = 0.99
orbital
o
Ratio
0
Ratio
0
Ratio
.
Ratin
satisfy the relation 1 = n - 1, are exactly the same as those for the Bohr orbits. Furthermore, for orbitals in the decreases as 1 increases, as in the case for same shell r,. r. Probability Criterion A plausible estimate of the spatial extension of an atomic orbital is the radius of a spherical boundary surface within which there is a high probability of finding the electron. This simple idea has been propounded in a number of popular textbooks ( I I ) , and a very lucid diagrammatic representation for the 1s orbital is available (12). In order to develop this criterion on a quantitative basis, i t is useful to define a cumulatiueprobability function
which gives the probability of finding the electron a t a distance less than or equal t o p from the nucleus. The function P,i(p) can be expressed in closed form and the results for 1s through 4f orbitals are given in Table 2. Once a particular value for P has been chosen, i t can be equated to each of the expressions in turn and the resulting transcendental equation solved numerically. The p values thus obtained for P = 0.50, 0.90, 0.95, and 0.99 are tabulated in Table 3. The significant quantities are the size ratios, with the 1s p value as standard, which provide
a rational scale of relative orbital size based on any adouted probability criterion. As the prescribed probability increases ( P values of 0.999, 0.9999, 0.99999, 0.999999 have been tried), the size ratios gradually decrease in magnitude and orbitals in the same shell tend to converge to a similar size. The data in Table 1 show that orbital size ratios based on i are smaller than those based on r,.,. The probability criterion (P = 0.95 or 0.99) yields the most compact scale of orbital size, according to which the sizes of the first four quantum shells are approximately in the ratio 1:3:6:10. The cumulative probability functions given in Table 2 can be used to give some measure of correlation between the various criteria for orbital size. For any specified distance p, the probability of finding the electron within this distance from the nucleus can be obtained by direct substitution into these expressions. In the fourth and last columns of Table 1, the probabilities corresponding to i and r,,, are listed, respectively. Scrutiny of the data in Tables 1 and 3 reveals that adoption of i as an indication of orbital size corresponds fairly closely to a 0.5 probability criterion. The use of r,., however, may be roughly matched to a lower probability value of 0.4, hut the agreement is not as satisfactory. Literature Cited 11) Pauling. L.. and Wilum. E. 8.."introduction to Quantum Mechanics." McGraluHill BmkCo., NruYork, 1935. pp. 148-149. 12) Fnedman. H. G.. Choppin, G.R., and Feuorbacher, D. G.. J. CHEM. EDUC.. 41.
. .
17) ~freit&er, A:, and owen;,P. H:. '"Orbital and Electmn ~ ~ ~ ~ i f ~ ' ~ i ~ ~ ~ Maemillan Co.. New York. 1913. 18) Weh1.A.C.. Sci. Amer.. 222.54(1970). 19) Gerh01d.G.A.. McMurchie. L.,andTye,T.,Arner.J. my$.. 40,988(1972J.
1971. p. 60. (12) ~ i ~ e n t G. d . c.. and spratley, R. n.. '"chemical ond ding c~sri~led through quan. fum Mechanics."Holden-Day, San Francisco. 1969, p. 27.
Volume 52, Number 2. February 1975 / 91
