edited by RUSSELL
H. BATI
Kenyon College
Gambier. OH 43022
Direct Solution of Schroedinger Equation for Vibration via Spreadsheet Stephen K. Knudson College of Willlam and Mary Williamsburg, V A 23185 The Sehroedinger equation often seems mysterious to many students, a t least in part because most of the enrollees in physical chemistry have not taken differential equations. The advent of an awrauriate combi.. . nation of computer hardware and software orovides s nowerful tool with which to conduet a simulatim in the phy~icalchemistry laboratory. The hardware cumponent of thp pnckage is n microcomputer with a numeric coprocessor and with large amounts of memory, 640K or more; the software cornponent is a spreadsheet with a critically important feature, graphics capabilities. On this campus, the functions are implemented via IBM A T class microcom~utera and the Quattru spreadsheet, hut undoubtedly there are other rombmatims that also have the necessary capabilities. Here, we discuss the use of a spreadsheet to determine the eigenvalues for a model of vibrational motion. The one-dimensional Schroedinger equation for vibrational motion may be written as
.
solve a number of problems (5, 6, 11) of interest in higher level chemistry courses. The figures included in this report were all generated and printed by the program. The numerical solution to a differential equation is based an a fundamental definition:
where
Byconvention,y, denotesy(x) andy,+~,y(x h). From initial conditions y h ) =yo and y'(xo) = yo', the solution can he stepped along the entire relevant range of the indeoendent variable z. It'rhe~tudenmhavesome familiarity with spreadsherts, they can rntpr the equations them*eIvra: n little guidance about a good format for the entries ran save some rime. One possible setup is rhoan m Table 1. As theexerrire iulikrlvto he rht student's first exposure to numerkal methods despite three semesters of calculus, it is prudent to begin with a known problem. In the process, the specification of initial conditions and the effect of step size h on the accuracy of the results can be introduced. A suitable model differential equation is
+
Therefore, if thr solution is known to he) lx I at the point x , an appronimal~value of the functionashart but finitedi*tanreh away is given by ) Idy(xVd4h Y(X+ h) = ~ ( x +
(3)
The Runge-Kutta (7) method is improved implementation of the basic scheme, in which the estimation of the derivative is carried out in a multistep process. As a result, the solution (8) to the second-order differential equation y" = f l r , y) is given by a series of relations:
y"(x) = -kZylx) (Continued on page A401
where R is the internuclear separation, p is the reduced mass of the diatom in kilograrns/molecule, and V(R) is the potential energy function for the diatom. The solution to this equation is normally presented for the harmonic oscillator potential, V(R) = k(R - Re)%,but rarely is the solution given for more realistic potentisl functions. Here we consider the Morse potential, V(R) = D,[exp(-B(R - RJI - 112, where R, is the eouilibrium internuclear seoaration (bond len,eth~.l),18the bond eneqy,and,ilsrelarrd to thc force r o n m n t l'h~r,p t w n t ~ a l1s suff~cirntlyct,mplcx that the numerical np. proaeh seems reasonable, but an analytic solution is in fact available and is used for comparison. The numerical solution of differential equations has been discussed previously ( I ) , and there are software packages ( 2 4 ) that efficiently solve and plot solutions to differential equations. The approach presented here has the advantage that the learning curve is minimal and offers a familiar interface for a more detailed solution. A spreadsheet performs calculations in a row-eolumn format, in a manner familiar to most. I t may surprise many, as it did the author, that aspreadsheet can solve the differential Schroedinger equation for the ground state of a Morse oscillator in the blink of an eye. In fact, spreadsheets can Volume 68
Number 2
February 1991
A39
the computer bulletin board for which the solution can easily be shown to he a linear combination of sine and cosine and to which the students have had some exposure in the guise of the particle-in-abox problem. In our experience, skepticism about the numerical method is nearly always overcome when the graph of the sine (and the cosine) appears on the screen. The students can then reset the spreadsheet and solve the vibrational equation using a Morse potential. Before using the spreadsheet, we recast the Schroedinger equation into a form suited for the RungeKutta equations above. T o do this, start with eq 1,put all the terms except the second derivative on the right, and multiply by the constant (-hz12rr). Next, scale the variables to an appropriate set of units, since molecular-sized quantities are so small. Define a reduced distance r , r = RIR., a reduced energy *, E = EID., and a reduced potential u, " ( 7 ) = V(r)lD,; finally, define two parameters
Table 1. Spreadaheel Image
De:
Mar: r
h:
0.01
yij
Psi
0.70 0.924 0.71 0.842 0,720,766 0.73 0.695 0.74 0.629 0.75 0.567 0.76 0.510 0.77 0.457 0.78 0.408 0.79 0.363 0.80 0.321 0.81 0.283
0,0000 0.0000 0.0000 0,0000 0,0000 0,0000 0,0000 0.0001 0,0001 0.0002 0.0003 0.0004
Runge-Kutta Solution of Simple DlHerenlial Equation 8.58E-19 beta: 1.73E+10 Re: 1.29E-10 1.05E-34 Mu: i.63E- 27 0.97957 gamma: -4207.55 b: 2.2455 eps: d(Ps1)Idr kl r+h/2 yr+h/2) M 0.0002 0.0002 0.0003 0.0006 0,0009 0.0014 0.0022 0.0033 0.0048 0.0069 0.0096 0.0131
0.0000 0.0001 0.0002 0.0003 0.0004 0.0006 0.0009 0.0013 0.0018 0.0024 0.0030 0.0038
0.7050 0.7150 0.7250 0.7350 0.7450 0.7550 0.7650 0.7750 0.7850 0.7950 0.8050 0.8150
0.8827 0.8036 0.7299 0.6612 0.5974 0.5381 0.4831 0.4322 0.3851 0.3418 0.3019 0.2652
and obtain the final version of the Schroedinger equation:
with Figure 1. The Morse pdential v(r)of eq 9 (solid line)and the harmonic potential of me same fwce constant (+ symbol).
A graph of the Morse potential u(r) in these units is shown in Figure 1. Table 1contains the initial text portion of a smeadsheet set uo to im~lementthe calculations described ahore. For the low-lying states o t t h e \lorue potential with h = 0.01. only ahour 70 liner (values of r ) are needed. Because the wave function is exponentially increasing from the origin, starting from too small a value of r leads to unnecessary integration and numerieal difficulties. For the lowest several states, a value of r = 0.7 works well. The initialvalue for 'T' is taken to be zero, as the wave function is in fact very small at small r. The initial value for q' is arbitrary and determines the normalization of the wave function. Just as the initial value of r should be chosen to he not much smaller than the inner classical turning point, the integration should proceed not too much past the outer turning point. T o find the eigenvalue, guess a value for e, and look at a plot of 'T'(r)versus r. Since the guess is unlikely to be the correct value, the wave function will "hlow up" either positively or negatively for the larger values of r, as shown by casual inspection of the graph; the result for an energy value c close to but not quite a t the ground state eigenvalue is shown in Figure 2. Subsequent guesses will
~.
A40
~~
Journal of Chemical Education
Flclure 2. A old of the around stale wave function G p u t e d at an energyvery close to the eigenvalw. The spreadsheet automatically scales bom the ordinate and me abscissa. This plot Is taken directly horn Quanro as printed on a Hewlen-Packard Desklet printer and corresponds very closely to me Screen display except that the screen will be In color If the monitor Is a color monitor.
0.0000 00001 0.0002 0.0003 0.0005 0.0008 0.0011 0.0015 0.0020 0.0027 0.0034 0.0042
0.034336
M 0.0001 0.0002 0.0003 0.0004 0.0007 0.0010 0.0014 0.0019 0.0025 0.0033 0.0041 0.0050
eventually provide a value of c for which the wave function has a aineularitv of the onnobracketedbvtjlkse site sien: an eieenvalue " two wluea of the reduced energy r and can be found hy locatmg the value or t for which the wave function does not have a singularity over the range of r being considered. Additional eigenvalues can he found similarly.
A useful estimate of the reduced eigenvalue can he found by computing the force constant for the Morse potential and using i t in the harmonic oscillator approximation to estimate the energy of a state. The Morse potential used here is somewhat broader than the HO potential, and so the exact energy is somewhat less than the harmonic aooroximation. A eraoh of the wavefunction for the state with quantum number u = 5 is shown in Figure 3.
..
Figure 3. A plot of ihe v = 5 wave functlan at ihe eigenvalue e = 0.344674. Note f i w extended rangg of the Independent variable.
".
We select Morse parameters (9) to correspond to HCI, since we also do the infrared spectrum of HCIiDCI in the laboratory course; students can then compare their numerical results with their experimentally measured values. The first few energies (eigenvalues) and transition frequencies computed by the procedure described above are listed in Table 2; the analytic eigenvalues tli)) are shown for wmparison. Clearly, the Kunge Kutra mpthod reproduces the anal n i r rolution quite well. The value of 2886 c k l found foi the oarameters used here agrees well with the experimental result. In addition, iaotopic subsiicut~onis possible, hoth I ) for H a n d the two wrnmon chlorine isotopes, and the results agree well with experiment.
Table 2.
Elgenvalues and Transnlon Frequencies
Y
f
E(cm-'1
Ea(cm-')
Nn. n+ l)(cm-'1
0 1 2 3
0.034336 0.101175 0.165819 0.227866 0.2873 17 0.344574
1482.5 4368.4 7150.9 9829.9 12405.5 14877.6
1481.8 4367.7 7150.1 9829.1 12404.5 14878.5
2885.9 2782.5 2679.0 2575.6 2472.1
4 5 Analytic result.
A Cumulative Count Method for Determining the Half-life of Barium137 and Gallium-68 Radioactive Isotopes: A Spreadsheet Application
Elvln Hughes, Jr. Soutneastern. a.bn verslty rlarnmond.LA 70402
Standard procedures for determining the half-lifeofradioactive isotopes are well documented (12. 13). There methods usually require that the investigator measure the rates of radioactive decay and plot the logarithm o f t h e rateasn functionuftime. From (Continued on page A42) Volume 68
Number 2
February 1991
A41
