Atomic Orbital Contours-A New Approach to an Old Problem
D. B. Scaife The City University
London, England
All functions in Tables 1and 2 are normalized to unity. For hydrogenic systems the variable, r , in expressions for R(r) is replaced hy Zr whereZ is the nuclear charge; each radial function would also he required to he multiplied by Z3" to satisfy the normalizing condition. In the case of multielectron systems approximate wavefunctions have been obtained ( 1 ) by replacing Z by an average value of the effective nuclear charge, Zefl (7).
First year degree students in chemistry often experience difficulty in appreciating the relationship between the particular mathematical form of a wavefunction, $, and the various types of diagram used to represent both the function and the corresponding prohability function, P(=ll.*$). I t has been the author's experience that manipulation of the functions by the student to. construct polar diagrams and, in particular, probability function contours can he of great value in developing a proper understanding of orbital representations. Unfortunately, the manual computing methods (1, 2) normally used for plotting probability contour diagrams, while instructive, can he extremely tedious. This paper describes a novel approach to the computation of prohability contours which has led to the development of a method which can enable a student, equipped with a pocket calculator, to construct a hydrogen p or d orbital as a tutorial exercise with the minimum of effort. The simplicity of the method allows ready translation into short Basic or Fortran programs. Hydrogen Wavefunctions The time-independent Schroedinger equation for the hydrogen atom, in atomic unit form (3),can he written V2$
+ Z(E - V)$ = 0
Probability Function Contours The physical significance of a wavefunction regarding the "spatial distribution"of the electron is that Pdr(=$*(r,B,$)+(r,B,4)d~)is the prohability of finding the electron in the volume element d i surrounding the point (r,B,$).' A prohability contour surface, i.e., one containing points of constant P, chosen such that the total probability of finding the electron within the contour surface is, say, 0.99 can be regarded as the boundary surface of the orbital; this provides a useful pictorial representation of the atomic orbital. Before constructing exact contours associated with a particular function, an initial examination of the corresponding R2(r) versus r plot and the polar diagram representing the angular function can enable the qualitative form of the contour diagram to be inferred. I t is also a useful preliminary exercise to determine the values of r and the angles fl and 4 where maxima in P occur (this is done by partially differentiating the P function with resvect to the variables and eqwating to zero1 and 118 cnl(:t~lnwt hr rxnct values of P at thrsr points. Thli informution p m v i d ~ui < heck i m the tinnl contotlr
(1)
When polar coordinates are used, solutions of this equation are of the form $,l,(r,fl,4); these are listed in Table 1together with the corresponding angular parts (Y;"(fl,$)) and radial parts (Rnl(r)), the symbols having the usual connotation. Table 2 shows the set of alternative angular and complete functions (all real) which is more commonly used since the functions are better suited to bonding problems. The reader should consult references (4-6) for details of the method of solution of the Schroedinger equation for the hydrogen atom.
'When the wavefunction is expressed in atomic unit form, the dimensions of the probability function, P, are (a.u. of length),3i.e., oo-a where oo is the Bohr radius.
Table 1. Hydrogen Wavefunctions n
m
I 0 o 2 o o 2 1 0 2 1 1 2 1 -1 3 0 0 3 1 0 3 1 1 3 1 - 1 3 2 . 0 3 2 1 3 2 -1 3 2 2 3 2 -2
Radial Function
Angular Function
Complete Function
RdrJ
Yf'"".$J
Jindr.fl.6J= RdrJYPlR.$J
E,
-112 -118
..
R l o = 2e-' = (II~V'~HZ
- r)e-'I2
RST= (112&1re~"~
., -1118 ,.
R30 = (2181&)(27 R3, = (4/81V&o(
- r)m-v3
.,
''
,
R, = ( 4 / 8 1 6 ) + e - " ~
., ., .,
442 1 Journal of ChemicalEducation
+
- 181 2+)e-""0~
yoo= 112&
$TOO
yon = 1 1 2 6
$SOO
Y,O = ( 4 1 2 6 ) c o d Y,' = (&12&) side* sinde-4 Y7-' = (&12&)
$210 #211
Alternative Symbol
=(l~&)e-~ = (1/4&)(2 -r ) e ~ ' ~
$(I4 $425)
= (114d%)re-~'~ COSR = (118&)reC~~sinRe*
PO) $(2p+d $(2p-d $43~) $(3po)
= (118&)reC~'~~ i n R e - ~ - 18r+2P)e-'" Yq0 = (fi181&)(6 r J r e ~ "cos0 ~ = (1/81&)(6 r ~ r e - " sinReid ~ YT' $(3p+,) = (1/81&/6 r~re-"3sinRe-'* $(3p-,) Y,-' Yz0 = ( 6 / 4 & ) ( 3 cas2R 1) = (1/81&)?e-"~(3 c0s28 1) $(3do) = (1181&)Pe-~'~sinR codei* $(3d+,) Ye' = ( 6 1 2 6 ) sinRcosEe* Y,-' = (fi/2&) sin0 cosfle-@ $sz-, = (1181&)+e-"~ sin0 cosRe-* $134-,) y22 = ( f i 1 4 6 ) ~inZ$B2* $322 = (1/162&)\/;;)~e-"~sin28e2" iC(3dtn) Y,-2 = ( 6 1 4 G ) ~ i n ~ R e - ~ ' * $32-2 = ( 1 / 1 6 2 d & % F n 3~ i n ~ R e - ~ * $(3d-2) $21-1
$SO,,
-
= (1/81&)(27
-
-
diagram and allows the selection of useful P and r ranges required for the exact calculations to be readily made. Contour plots are usually first construct.rd in onequadrant only ot swne sultahly selected plant, (in which, for example, either R or 4 is constant); use is then made of the symmetry properties of the function to complete the two-dimensional representation. Three-dimensional contour surfaces can in some cases (but not all) be -generated from the two-dimensional diagram. A manual method for determining probabilitv contours for the 2p, orbital in the R = 0°-90' &adrant ii a plane containing the z axis is to calculate P for a range of values of r over several values of R from 0' to 90° and plot curves of P versus r for each value of 8. The intersection of a line of constant P with these curves gives values of the coordinate r of points (r,B) on the chosen contour. Even with the aid of a pocket calculator this procedure ( 2 )is extremely tedious. The essentials of this method have been incorporated into a number of computer programs (8-10) for plotting contour diagrams. Here, the basic procedure is to calculate and store values of P for a large number of noints in the selected auadrant. A "sort" oronam is then usei to search the data foipoints at which the values of P have a given value (within chosen limits); the search is then repeatld for different values of P. Clearly, the smoothness of the contours so obtained will depend on the number of points located for each value of P which, in turn, will depend on the number of evaluations of P initially computed. Programs of this type are expensive in terms of computer time, narticularlv if smooth contours are required. A different apprnarh to runtour ronstrurtion hiri heen used I w ('then ( 1 I I who expressed the pn~linhilityfunction I I a~ p oibital in Cartesian coordinates and separated the variables; this method. however, is not generally applicable. - The method of contour co~structionused in the present paper expresses probability functions in Cartesian form but retains the parameter, r. Manipulation of the functions leads to expressions for the coordinates of points on any selected contour a3 functions of r and P. The following examples, using some of the functions of Tables 1 and 2, will illustrate the procedure. ~~~~~~~
~
~
hence =2
hence (2) P = (32~)-'rZe-' cos28 Putting z = rcosR (to convert to Cartesian form), substituting in eqn. (2), and rearranging we obtain
r 2 = 32nPe' = t
(4)
For a given value of P the function t can be evaluated for a range of values of r and the coordinates of points (x = (r2 t)'/2,z = t1l2)on the contour in the +x/+z quadrant obtained. If a value of r is such that it lies outside the limits of r defining a particular rmtour then no point hnving this value of r can be loratid on rhe contuur. In this rase the value of z calculated from equ. (3) will be found to be greater than r, which is clearly meaningless; calculations yielding z > r are therefore rejected. Since the P function is cylindrically symmetrical about the z axis and the xy plane is a reflection plane, a point (x,z) on thr contuur will generats tht: contour points (-x,z). ( x , - 2 1 , and (-x.-z) allowing the two-dimt:nsional diagram in the rz nlsnr to hc cotnnleted. Partial difft:rrnttation of ean. (2) shows = 0.0053&ao-~) occur that prohahilityLfunctionmaxima (P,, at noints on the +z and -z axes where r = 2. The :lp, wavefunrtiun difftws frum the 2p, l'unrtion in that rct:'l'ablrs I und 21. C'ollowinr it containsa radial n d c at r = (i the same procedure as above, contour points (x = (r2- t)"C z = t'/2) are again obtained but, in this case, the function t = (812a/2)P(6 - r)-2e2r/3. 3dz2 Contours
therefore P = (118126r)r'ec2'~3(3 cos28- 112 Substituting z = r cosR and rearranging
( 3 9 - r2)2= 8126rPeZrf3 =t
~
Zp, Contours
-t
=
+
giving the two positive values of z as z = [(rz t '/2)/3]'/2 and z2 = [(r2- t1/2)/3]1/2.The rejection condition for zl is z l > r;za cannot be located if t1l2 > r2. In the xz plane the x coordinates corresponding to z l and z2 in the +x/+z quadrant are, respectively, XI = (r2 - z12)1/2and x2 = (r2- z22)1/2.AS in the case of the 2p, function the 3d,2 probability function is cylindrically symmetrical about the z axis. Maxima occur at points distance r = 6 along the +z and -z axes (P,,, = 0 . 0 0 0 7 6 7 7 ~ ~ -and ~ ) a t all points on a circle (r = 6) in the xy plane (P,,, = 0.0001919~~-~). 3d, Contours
(3)
In the rz plane
Table 2. Angular
Function
fim(m = 0) or ( Ylm
* Y,-"')
Alternative Set of Angular and Complete Functions Complete Function
(m f 0)
R,,,Ynm (m = 0) or
* Y,rm) (m
%(Ylm
f
0)
Usual Symbola
Volume 55, Number 7, July 1978 1 443
can each nart of the inteeration he treated se~aratelv.For orbitals of lower symmetry, graphical integration can he used (1) . . hut even in the simole case of a ZD, . - orbital, where the 4 integration can he separ&d (since the function is symmetrical about the z axis) hours of tedious calculation are necessaw to obtain a single value of p ~at;least 3 values of p~ in the r&ge 0.8 to 1.0 are required to allow interpolation of the P value corresponding to p~ = 0.99. An approximation method has been devised which avoids the lengthy calculations of the graphical integration procedure. The approximation is based on the observation that as the value of P for a contour surface decreases and approaches zero, the houndary of the contour approximates to a sphere; a good approximation to eqn. (6) would then he expected to he p$ =
li" So2"R ~ ( ~ ) Y * ( R , @ ) YsinRdmdRdr (B,@)~~
(7)
where r is the maximum value of r for the contour. phis the total probability of finding the electron inside a sphere of m, p~ p & 1.0). Equation (7) radius r. (In the limit as r can he separated and becomes
- - -
P&-=
from which p &can he evaluated directly. The approximation method for obtaining a houndary surface contour for a 2px orbital has been compared with the graphical method of Ogryzlo and Porter (I). Equations (3) and (4) were first used to plot small sections of probahility contours in the vicinity of the z axis in the rangeP = 0.000005 to 0.0003 and the maximum value of r (the distance from the origin to the point of intersection of the contour with the z axis) for each contour noted. Values of pk were then evaluated from eqn. (8) using the maximum values of r as the upper limit of integration. For a 2p, orbital, eqn. (8) becomes
_l(lT _l(lZz Y*(B,@)Y(B,@) sinRdgd0 ff12(r)r2dr
which, since the angular function is normalized to unity, reduces to
p &was plotted against P (Fig. 2). Values of DT for a number of contours in the same region were detrrmi& hy the graphical method ( 1 , and these are also indudrd i n Fircure 2 :md nmpared aith tht-approxim;~tt values. A true total probahility of 0.990 corresponds to a contour for which P = 0.000008 no-3 (maximum r = 12.0 a d ; this contour corresponds to an approximate total probability of 0.993. Conversely, selection of a houndary surface using the approximation method (i.e., puttingp', = 0.99) gives a contour for which P = 0.000013 (maximum r = 11.5ao) which is agood approximation to the true 0.99 probahility contour. The approximation to the total probahility at values of p~ less than 0.98, however, is poor. The divergence of the two curves of Figure 2 increases rapidly as p~ decreases since the approximation method overestimates the total probahility as r decreases; the two curves will, however, reconverge in the 0 and p~ p & 0. Provided that a boundary limit as r surface is selected such that the total probability is 0.99 or greater, the approximation method is seen to he an extremely rapid and satisfactory alternative to the graphical procedure.
-
- -
Literature Cited 111 Ogryzlo, E. A.,snd Pnrter,C.B.,J. CHEM. EDUC.40.256 11963). (2) 0sterheld.R. K., J.CHEM. EDUC.44.286 (1967). I31 Kauzmann. W.,"Qusntum Chemistry;AcadomicPresi.NewYork, 1957,~. 214. (41 Hanna, M. W.,"Quantum Mechanics inChemistry,"2ndEd., W. A. Benjamin. New Yurk. 1969,Chap. 6. ( 5 ) Andermn. E. E.."Modem PhysicsandQusntum Mechanin: W . B. Saundera,Philadelphia,
P x lo4 Figure 2. Total probability within a contour as a function of P. Approximation values denoted ph graphically integrated values denoted pr.
1971.Chap.7.
(61 T d i . N., and Pompiila, F. R.,"AWmicTheory-An Introduction W Wave Mechanics? McCraw-Hill, New York, 1969, Chap. 5. I71 Slater. J. C., "Quantum Theoryuf Matter: McCraw-Hill. New York, 1953, Chap. 6. (8) Bsder,M.,J.CHEM. EDUC.,48.175 119711. 19) Craig, N. C.,Shererfr,D. o..Carl~n,T.S.,andAckermann, M. N.,J.CHEM.EDUC., 48,310 119711. (10) Holmgron. S. L.. and Evans. J. S.. J. CHEM. EDUC..51.189(1974). (11) Cohan, l.,J. CHEM. EDUC..38.20 119611.
Volome 55, Number 7,July 1978 / 445
