G. Dana Brabson Frank J. Seiler Research Laboratory lAFSCl USAFAcademy, Colorado 80840
C U ~ C U ~ Uof~ ~Morse O ~ Wave Functions with Programmable Desktop cUICUIU~O~S
I t is very helpful for the student of electronic transitions in diatomic molecules to be able to visualize vibrational wave functions, probability distributions, and the overlap of wave functions. As the result of recent advances in programmable calculator technology, it is now possible to calculate and plot these mathematical functions for the Morse potential using a programmable desktop calculator. The results shed light on and give insight into the characteristics of real potential energy functions.
The argument of the polynomial is
The normalization constant, N, is given by Morse4
Calculatlonal Methods
Equations
Philip M. Morse proposed the potential energy function which bears his name in 1929 (1) E(r)
=
D8-1"1r-'s'
- 2Dk-OiT-nl
(1)
In this equation, E(r) is the potential energy as a function of internuclear distance r, D, is the dissociation energy referred to the bottom of the potential energy curve, re is the equilibrium internuclear distance for the vibrationless molecule and a is a constant defined as follows
where fi is the reduced mass of the molecule and w , is the vibrational constant.' In the same paper, Morse gave the radial vibrational wave functionZ
In this equation, n is the vibrational quantum number. The constants d and k are defined by eqn. (4) in which h is Planck's constant.
L is the generalized Laguerre polynomial and can be computed from the following series given by Morse3
Presented at the 43rd Annual Meeting, Colorado-Wyoming Academy of Science. 'It is of course necessary to be conscious of units. As written, eqns. (21 and (4) require that D, and w, he exwessed in ergs - and sec-', respectively. The factor of \I=, which was in error. has heen omitted in the correct formulationgiven here. Note that z and x (both used by Morse in the third equation on page 60 of Reference ( I ) ) are equivalent. 'The sign error in the denominator of the summation in eqn. (12) in Reference (I) has been corrected. Wpectroscopie notation, in which upper state parameters are primed and lower state parameters are double-primed, is used here and throughout the remainder of this paper.
At this point it is appropriate to discuss the phase factor, exp[ia(k - n - I)], which appears in eqn. (5). Because of the phase factor, the wave function normally will have both real and imaginary parts. However, when the probability distribution is calculated, the phase factor is multiplied by its complex conjugate and the resulting product has a value of +LO. As a consequence, the phase factor may he ignored during calculation of the probability distribution. A somewhat different situation arises when the overlap of two wave functions is calculated. In this case the nhase factor for one wave function is multiplied by the complex conjugate of the phase factor for the other wave function to yield a term of the form exp[in(k' - k" - n' + n")]." Thus the overlap has both real and imaginary parts. Calculation of the relative intensity of an electronic transition requires computation of the square of the overlap integral (see eqns. (15) and (16) below). During this calculation, the term exp[is(k' - k" - n' + n")] is multiplied by its complex conjugate. Since the resulting product has a value of +LO, it is apparent that the phase factor may be ignored for this calculaLion. In as much as the ~ h a s efactor is needed neither for calculation of the probability distribution nor for estimation of the relative intensity of a transition. the ~ h a s efactor will not he evaluated in any of the work that iollows. As a consequence, it is necessary to label the ordinate "R/ exp[i*(k - n - I)]" for a plot of a wave function, and to label the ordinate "R'R"/exp[ir(k' - k" - n' + n")]" for a plot of the overlap of two wave functions. At first, this may disturb the student. However, it is helpful to note that, if by chance the k's were integers (indeed, a physically realizable situation), the phase factors would reduce to Zk1.0. By combining eqns. (3). (5),( 6 ) ,and ( I ) , the radial wave function can be written in terms of a new normalization factor, M, as follows
where
(k-n-l)(k-n-2)n(n-1)~"-' 2!
- +
,,,
]
(8)
The normalization factor can he conveniently set up for calculation as follows Volume 50, Number 6.June 1973 / 397
by the Wang 702 Plotting Output Writer. Figures 1-4 are typical examples of the plots obtained.? Results
For typical values of k and n, r ( k - 2n - 1) frequently is larger than 1099, the upper limit for numbers which can be handled by the calculator. Under these circumstances, each term in eqn. (10) can be handled as shown in the following example
The Gamma functions are calculated using the following relationships (2) r(n r(l
+X) =
+
X)
+ x - 1 ) ( n + x - 2) ... (1 + x ) r ( l + x), + a,x + a,* + a3x3 + a , f + a $ + & ) .
(n
= 1
(12)
(13)
where O j x _ < l Ie(x)/ j 5 x lo-" a, a2 a, a, a,
= = =
= =
-0.5748646 0.9512363 -0.6998588 0.4245549 -0.1010678
The finite series in eqn. (8)can be conveniently set u p for calculation as follows zn+ - (k - n - On 2
p - (k -
n - 2)(n - 1)
...
22
Calculator Program
The Wang 720B Electronic Calculator was selected to perform the required calculations. This calculator has two features which enable it to successfully accomplish the necessary tasks: (1) it has a large dynamic range (10-98 to 10+99), and (2) it has 12 digit capacity. Both of these features are of vital importance. A comment on the evaluation of expression (14) is in order a t this point. When the indicated subtractions are performed, the results obtained are very small with respect to 1.0. The 12 digit capacity of the Wang 720B permits satisfactory evaluation of expression (14) for vibrational quantum numbers n 5 8. However, for larger values of the vibrational quantum number, double precision arithmetic is needed. The program written for the Wang 720B has 1239 program steps and requires the use of 64 storage registers." The program includes the double precision arithmetic subroutines which are required for evaluation of expression (14), and a plotting subroutine which drives the Output Writer. Standard numerical techniques are used to evaluate the overlap integral and the integral of the prohability distribution. As each point is calculated, the results are either printed in tabular form or plotted in graphical form 6 A copy of the program for the Wang 720B/Wang 702 may be obtained by writing to the author. 'A less sophisticated program designed for the Wang 700B/ Wang 702 has also been written. This latter program uses single precision arithmetic throughout and can be used, to calculate and plot a wave function or its probability distribution provided n 5 I g.The Wang 700B/Wang 702 program cannot he used to compute an overlap integral. This program uses 687 program steps and 34 storage registers; a copy may be obtained by writing to the author.
398 /Journal of Chemical Education
52 X-STATE Three typical wave functions are illustrated by Figure 1.The solid curve is the Morse potential for these wave functions, and represents the classical turning points for the three wave functions. Upon plotting the wave function for n = 0, the student immediately observes two important facts: (1) the wave function does not vanish a t the classical turning points, and (2) the wave function has a maximum a t approximately the equilibrium intemuclear distance. From these facts the student can, of course, infer that, for small quantum numbers, the molecule behaves in a nonclassical manner. Figure 2 shows a vibrationINTERNUCUR DISTANCE l A l al wave function and its cor- Figure 1. Typical wave functions responding prohability dis- for the X3Z9- State of diatomic tribution. With the help of suilur. Note that the ordinate has the prohability distribution a dual significance: For the podiagram, the student can tential energy curve, the ordinate make quantitative statements is the potential energy: for each about the probabilitv of find- Wave function, the ordinate is ing the &olecule -with an R/exp[iz(k - n - I ) ] . internuclear separation which lies outside the region defined by the classical turning points. The student can, for example, cut out and weigh various pieces of the probahility distribution plot. Alternatively, the student can set the program to calculate, the integral of the probability distribution between the left and right classical turning points; since the wave function is normalized, the value obtained is the probability of finding the molecule with an internuclear distance between these two values. Figure 3, which illustrates the probability distribution for n = 20, required two hours of calculator time. One of the most striking features of this figure is the lack of symmetry. This of course results from the fact that the slope of the repulsive side of the potential energy curve is quite steep at the n = 20 level, while the slope of the attractive side of the potential energy curve is more gradual. As a
N2
A
- SiATE
GAS PHASE n - 2
INTERNUCUR DISTANCE I A l Figure 2. Wave function and its corresponding probability distribution tar theA3Z,+ state of nitrogen.
-
-
N2 - A STATE GAS PHASE n 20
[SRI*.,R".,,
For absorption bands: 1.a.
dr]'
(16)
=X where X is the wave length of the transition. As a case in point, consider the (0,n") progression of the B32,X 3 2 , transition of Sz. The most intense fluorescent bands in this progression terminate in the n" = 8 and n" = 9 levels (4). As illustrated by Figure 46, the n" = 8 vibrational level of the X-state has a classical turning point which is about equal to the equilibrium internuclear distance of the B-state. The student may well ask what happens for vibrational quantum numbers less than n" = 8 or 9. The answer is suggested by Figure 4a. Because of the poor overlap of the two states, the (0,3) fluorescent band would he expected to have a relatively low intensity; this is precisely what is observed experimentally. What then happens when the vibrational quantum number of the X-state is larger than n" = 8 or 9. Figure 4c indicates the answer. Note that, although there is substantial overlap of the two states, the regions of negative overlap are nearly equal to the regions of positive overlap. As a consequence, the (0,131 band should, and does, have a relatively low intensity. To verify the program described in this paper, overlap integrals were calculated for several transitions in the First Positive band system of N2. With a few exceptions, the data published by Jarmain and Nicholls (5) were reproduced exactly.
-
INIERNUCEAR DISTANCE lA1 Figure
3. Probability distribution for the n = 20 vibrational level of the
A 3 2 , + state of nitrogen.
consequence, there is a much greater probability of finding the molecule near the right hand classical turning point than near the left hand turning point. The student can also use Figure 3 to illustrate the fact that classical behavior, which predicts that an oscillator is most likely to he found a t its classical turning points, is approached in the limit of very large quantum numbers. Figure 4 illustrates the overlap of typical wave functions. The student can cut the plot of the overlap into appropriate pieces and use an analytical balance to determine the overlap integral. While doing this, he will discover that the regions of negative overlap must he suhtracted from the regions of positive overlap. The calculator, of course, computes the correct result for comparison. Overlap integrals are directly related to the relative intensities of electronic transitions. The appropriate relationships are (3) For emission bands I..
a
1 X'
[SR'*.,R"~,, drI2
Acknowledgment
The author is indebted to E. M. Henry and J. H. Head for helpful discussions during preparation of this paper. Literature Cied 111 Mor3e.P. M.,Phya. R o o , 34.57. 1929. 12) Abramouitz. M.. and Stcgun. I. A.. '"Handbmk of Msfhemsfieal Functions." U. S. Government Ptinfing Office. Washington. D. C., 1966, pp. 256-7. 131 Herrberg. G., "Molecular Spectra and Molecular Structure. I.Speetrs of Diatamic Molecules." Van Nmtrand Reinhold Company. New York (2nd E d ) , 1950. PO. 2W-I. 141 h e n , B.,"Consfante. Sdeetionnoes Dorm- Spectroaeapiquer Cancernsnt lea Molecules Distomiguer." Hermannet Cie.. Paris. 19% l5! Jarmain. W . R..andNichol1a.R. W., Can. J. Phys., 32.201 (1954).
(15)
S2 GAS PHASE
52 GAS PHASE
S2 GAS PHASE
B-STATE nP=O
* B-STATE "'-0 I
X-STATE n"- 8
X-STAR n'
.
.
:
OVERlAP INTEGRAL = -0. W5
i
* OVERlAP INTEGRAL-a403
:
.13
;
i
1
\
Q 16
1.8
20
22
INERNUCLEAR DISTANCE UI
24
L6
L8
20
22
24
INTERNUCLEAR DISTANCE IAI
L6
18
20
22
24
INTERNUCLEAR DISTANCE IAI
Figure 4. Overlap of the n' = 0 level of the B 3 Z u - state of 52 With several vibrational levels of the X 3 2 , state. The vertical lines represent the classical turning paints of the wave functions. The student should not be disturbed by a negative value far the overlap integral since it is the square of the overlap integral, rather than the overlap integral itself, which has physical significapce.
Volume 50, Number 6 , June 1973 / 399
