Using the Perturbed Harmonic Oscillator To Introduce Rayleigh-Schrodinger Perturbation Theory Karl Sohlberg and David Shrelner University of Delaware, Newark. DE 19716 The perturbed harmonic oscillator is a problem often used to introduce Rayleigh-Schrodinger perturhation theory in introductory quantum chemistry texts.' In some respects it is an ideal example prohlem. There are no degenerate states or continuum states, which eliminates confusing complications. and the necessarv integrals can he solved analvticallv. It is perhaps this pleasing simplicity that has led many to neglect some important suhtleties in this classic prohlem. ?he problem usually posed in terms of moleEular vihrations. If the molecular interaction potential is expressed as a polynomial centered about the equilibrium separation, the quadratic term represents a harmonic oscillator potential, and the harmonic oscillator can he used as the zero-order unperturbed system. The higher order terms in the polynomial expansion can he taken as the perturhation to the Hamiltonian. The problem then is to use the RayleighSchrodineer .. .. .oerturhation treatment to improve the aoproximate energies given by the harmonic oscillator eigenstates. A notahlv comolete mathematical solution of this problem is given h ; ~ l u ~ g e .If~ the expansion is truncated, ieaving no terms higher than x4, the perturbation H' is: H' = bx3 + cx4
The bx3 term contributes nothing to the first-order enerm correction (E')and the expression for E' involves only c. TO second order both the bx3and cx' termscontribute. The first prohlem that arises is that the energy after the first-order correction is always a poorer approximation of the true energy than the harmonic oscillator approximation. When the corrections are carried out to second order, the approximation to the "true" enerw will onlv i m ~ r o v eif the second~-~~~ order correction is great;; in magnkudd than the first-order correction. This failure of the perturbation corrections series to converge smoothly is a consequence of the fact that the perturhation is expanded in terms of a set of functions which have definite parity. Since the goal of applying a perturbation treatment to anv ~ r o h l e mis to imorove the accuracy of the result over what ;an be achieved k i t h a simple model whose exact solutions are known, in any complete presentation of the perturhation approach it is important to show that the way in which the perturhation is descrihed is critical. ~~
~
~
Solutlon of the Problem A wide variety of molecular interaction potentials can he very adequately described in one dimension with the Morse ~otential.~
This will then he used as the "true" potential while admitting that though it is usually an excellent approximation, in some rare cases it may he unrealistic. Expanding this function in a Taylor series truncated to three terms yields:
Let:
The quantum mechanical Hamiltonian then becomes:
Where h = Planck's constant and p = oscillator reduced mass. This can he separated into a "zero-order" Hamiltonian HO and a perturbation H'.
Schrodinger's wave equation then becomes: The zero-order problem is that of the linear harmonic oscillator, and the solutions are well known.4 In the RaleighSchr6dinger perturhation treatment the energies can be expressed as a series:
E = EQ E' + E n + higher order corrections The first-order energy correction is:'
En= j V,H'V,& For simplicity only the ground state will he considered here, although the treatment extends directly to higher eigenstates. When the integration is performed (see Appendix) the first-order energy correction is: 3c E' = -
4n2
' Pauling. L; Wilson. E. 8. Introduction to Ouantum Mechanics;
Dover: New York, 1985. Flugge, S. Practical Ouanium Mechanics; Springer: New York. 1974. ~~~
~~
Morse. P.M. Phys. Rev. 1929.34, 57. * SchrMinger. E. Collected Papers on Wave Mechanics: Blackie and Sons: London. 1928.
wherea = ,kmlh.Note that the first-orderenergycorrection will always be positive in sign, 3ince D is positive for a Morse Volume 68
Number 3 March 1991
203
oscillator, 6and cr are real numbers raised to even powers, and the numerical coefficients are all positive. The "exact" energy will always be less than that given by the harmonic approximation however.3 The energy corrected to first order will then always be a poorer approximation of the "true" energy of the Morse oscillator eigenstate than the zero-order harmonic annroximation. Furthermore. this will be true even if all & terms in the Taylor expansion of the Morse potential are included in the perturbation H'. This is due to the fact that the odd-order terms have odd parity and will contribute nothing to the first-order energy correction. The even-order terms, which have even parity and do contribute, are all nositivelv signed and will give . positive contribution to the first-orderine;gy correctionsince they effectively steepen the wall of the potential well. The second-order energy correction is given as:' ~
h.
Appendix We wish to solve the integral
where: For the Morse oscillator, the second-order energy correction is:
-
+
Figure
1. Flrst-crder correction lo the energy (F = F ) as a parameters Dand 8.
Journal of Chemical Education
where
a.
where ~~-~~b.c. . .and h are eiven as before and o = Ry inspection one can see that this correction will always be negative in sien. T o second-order, Rayleigh-Schrodinger pert&bation thloorycan only give a better approximation to the exact enerev -.levels of the Morse oscillator than the zeroorder harmonic approximation, provided IE" > 15% Inorder toexhibit the conditions under which thisoccurs, one must compare the correction factors E' and Ex. When b = f(D,B) and c = g(D,@)are substituted into the E' and E" expressions, these corrections then become functions of D and& the Morse oscillator constants. The "correction surfaces" E'(D,fl) and E"(D,p) are shown in Figures 1and 2. The "total correction surface," through second-order, E' E" = F(D,@)is shown in Figure 3. Only in regions where the function is negative is it possible for the second-order perturbation treatment to give a more exact approximation to the "true" energy of the quantum Morse oscillator than the harmonic approximation. The precise shape of this surface will depend on the various constants (a,o,etc.) that will scale various aspects of the surface. They are, however, only scaling factors and will not change the general shape of this "total correction surface."
204
Conclusions When Ra~leigh-Schrdinger perturbation theory is used to approximate the energy levels of the Morse oscillator, the zero-order harmonic approximation is always more accurate than those values corrected to first-order if the perturbation is renresented bv the terms above second-order in a ~ o l v n o m i a i e ~ ~ a n s i o n . ~arises ~ h i s from the lack of contrib;ti& to the nerturbation correction bv the odd-ordered terms due to t h e i odd parity. Even when the corrections through secondorder are considered, the energy levels can only be improved by the erturbation corrections under the condition that IE' > When this inequality is true depends on the values of D and 0,the Morse oscillator constants. This example shows that care must be exercised in the application of perturbation theory in order to besure that the perturbation corrections indeed improve the accuracy of the calculation over that achieved with the zero-order approximation.
function of the
" "
F
0 020
o
010
, ,,, -O
1 00
-0 0 4 0
o o i o I.
0
0 0 0
Figure 2. Second-order energy correctton (F = F ) as parameters Dand 8.
Figwe 3. Total correction to the energy (F = F parameters Dand 8.
a
function of
UKI
+ F )as a function of Me
E = fir
and
and
H'
=
x-
The integral in the previous equation
I
c,x2
=
i=o
can be solved in closed form by making the following substitution. Let m = 1 n k - 2 ( i j). This integral can then be solved casewise on the parity of m. For m odd, I = 0 by symmetry. If m is even, then let
+ +
i=o )
( (k
- 2j)!j!
after suitable algebraic manipulation, this reduces to: H"k=
-(= 2-
=
n+k 1-0
2
2 1
IntlnAl Intlkll) i=o
j=o
-,,, +,,
e-8x,rt+,,+~
J--
Now making all the proper substitutions, this expression becomes
(
/-
I I
) I
+
W=-
dl
rn 2
then I can be solved in the following way
I=
e-8~'x2sdz = 2
lo-
e-h'x2"&
According to CRC's Standard Mathematical Tables (28th ed., p 292, eq 666),
Volume 68
Number 3
March 1991
205
