Diatomic Vibrations Revisited P. L. Goodfriend University of Maine, Orono, ME 04469
Although it is both traditional and productive to approach the prohlem of the vihrations of diatomic molecules starting from the model of the harmonic oscillator, useful insights into the motion of atoms in molecules, the nature of auantum mechanics, and the nature of the harmonic oscillator auproximation itself can he obtained hv usina other choices of basis functions and exploiting the variation theorem. Indeed, in this way some convenient, practically useful results can he obtained that augment the tools offered by the traditional approaches. The object of this paper is to present such an alternative approach, examine its relationship to other approaches to vibrational problems, and to point out some simple, useful applications. The material presented can also serve as a source of tractable, hut physically meaningful problems for elementary courses in quantum mechanics. Dlatomic Vibratlons and the Harmonic Oscillator Let us consider a diatomic molecule. After transforming the coordinates into those for the motion of the center of mass and a set of internal coordinates, the Born-Oppenheimer approximation yields a Schrodinger equation for the relative motion of nuclei, which can be written using Hartree atomic units as
where p is the reduced mass and V ( r )is a potential function for the motion of the nuclei consisting of the nuclear repulsion energy plus the electronic energy, written as a function of the internuclear distance. Equation 1 is a central field prohlem. Consequently, the wavefunctions can he written $ = xJr) Y,JR,d
(2)
tion to being intuitively appealing, the harmonic oscillator is a quantum mechanical prohlem with exact solutions in closed form. It is a very good zero-order approximation and lends itself readily for use in treating the anharmonic prohlem through perturbation theory. I t also allows polyatomic vihrations to he treated in terms of normal coordinates to zero order, because the quadratic forms of the kinetic and potential energies can he simultaneous diagonalized. Thus the harmonic oscillator as a zero-order approximation is the foundation for the analvsis of sDectra and vields the conventions for reporting spectroscopic constants as tabulated in reference works such as Huher and Herzherg (2). A Different Approach .. Another way of dealing with eq 3 is to choose suitable basis functions containing variational parameters and minimize the energy with respect to these parameters just as one does in dealing with electronic states. One major point of this paper is to call attention to the fact that one can choose very simple forms for such trial variation functions based upon the laree masses of nuclei. Let i s consider the lowest S (1 = 0)state for eq 1, that is, the lowest nonrotatine " vibrational state. It is clear from the semiclassical behavior of entities with atomic masses that thev are hinhlv .. . localized. In the case of the lowest S state. thii means rhar the wavefunction yieldsa \.cry thin spheri~nl shell of prdx~i)ilitvdensir\.. Amonr the basis fi~nrrionforms satisfying this coidition are
xo = M, e x p [ - b ( r
- a)']/r
(6)
and
where the angular parts are spherical harmonics. The equations for the radial functions can then he written as
The essence of the harmonic-oscillator, zero-order approach to the prohlem is the observation that, for hound molecules, there is a minimum in V ( r )at an "equilibrium" value of r , r,. Using Rayleigh's theory of small oscillations, V ( r )is expanded in a Taylor series about re:
The linear term in the expansion does not appear, because the first derivative vanishes a t the minimum. Using eq 4, eq 3 can he expanded (e.g., as done by Dunham (I)),in such a way that the leading terms can be viewed as a "rigid rotor" term and a term that can he converted into a one dimensional harmonic oscillator prohlem by writing
x, = X H O ~ (5) and approximating the interval for the radial displacement as [-m, + m ] rather than [-re, +=I. Using only these leading terms corresponds to truncating the expansion of V ( r )after the quadratic term. Many benefits result from taking this approach. In addi-
with very~. large values of q and D, a usuallv in the hundreds. and p greater than 10 and som&es in the hundreds. The form given by eq 6 is iust the conventional functional form for theground state ofthe harmonic oscillator. In this paper we will focus on the form given by eq 8, an S T 0 function, because of the very simple integrals, expressible as simple formulas, that result for expectation values and matrix elements for many different potentials and other operators; p and q are treated as variational parameters. Slngle-Function Ground State Calculations The expectation value of the kinetic energy in Hartrees for functions given by eq 8 is
Specific expressions for the expectation value of the potential energy depend upon the form of the potential function used. An advantaee of the choice of a basis function in the S T 0 form of eq 8 ;s the fact that for many different forms of the potential it is a trivialmatter to evaluate the inteerals for \ ' snalyt~cnllywith(rut expanding rlic pt,tcntinl in a puwer itriri. For example. t'or the .\lmse I'utentinl. which c m he written Volume 64
Number 9
September 1987
753
Larger Bask Sets
The success of the single-function calculations leads immediately to the possibility of dealing with excited states as well by extending the basis set to include a number of linearly independent functions of the S T 0 type. Let us choose a set of radial basis functions
we have
It has previously been pointed out (3)for van der Wads molecules obeying the LennardJones Potential, The values of p and q are taken as those from the singlefunction calculation with u = 0. We will consider a linear variation function containing three terms (u = 0,1,2).
that we can write
In obtaining eq 13, although it is not necessary, for convenience the Stirling approximation was used in dealina with the factorials. hi large values of q expected makes'this a good approximation. Once an expression for the total energy is obtained, the values of p and q for which the energy is a minimum are then found. Because of the simple expressions for the integrals, this can be carried out using easily written programs on either a main frame computer or PC. The entire process, from the derivation of the integral formulas to the writing and running of the programs. should be within the ca~ahilities of elementary &a&un mechanics students. If it is assumed that the vibrational function for the other rotational states for the V = 0 vibrational state are the same as that for the S state, then one may calculate the rotational constant, Bo, directly from the wavkfunction,
We obtain (in Hartree units)
The value of we for the ground state can also be estimated from the wavefuuction alone by using the approximate result (rigorous for the harmonic oscillator) that
Again the integrals for the Hamiltonian and overlap matrices are easily constructed, although one must take care to avoid overflows and underflows inthe actual computations. The eigenvaluea and eigenvect~~rs of the matrix corres~onding to the secular equation
IH,,- ES,I = o
are then obtained. In the results reported here, an IMSLS standard library routine for the generalized eigenvalue ~roblem was used. Carrying this out for the ground electronic state of the Hz molecule represented by a Morse potential gave the results given in Table 2. The calculated value of Do is 4.47798 eV, not significantly different from the single-function results. The energy spacing between the ground and first excited states in cm-I is 4154.9 cm-I c o m ~ a r e dwith the observed value including a single anharmonic term a t 4158.54 cm-1. The spacing between the first and second excited states is 4121.9 rrn-:. Thus the results must be considered very good. Carrying out hes same calculation for LiH gave resultsalso shown inl'able 2. Summarizing the results without giving all ot the details, the first excited state vields fair results: those for the second excited state are poor. For all cases examined with larger values of q, spurious, nonsensical results were obtained. For example, for N2 a state with energy much lower than the ground state was calculated. These spurious results are artifacts resulting
Table 1.
Table 1 contains results for calculations of the type described above for four molecules compared with the "observed" values obtained from Huber and Herzberg (12).I t will be noted that the results are excellent. No matter what form one uses for the potential, if one uses the expansion as shown in eq 4, truncates after the quadratic term, and uses the resultine ~ o t e n t i ain l the same kind of calculation, the results are very close to those obtained using the full potential. Thus, we have a vivid illustration of how good the harmonic oscillator approximation is near the bottom of the potential well. As will be pointed out below, i t is sometimes desirable t o estimate the wavefunction parameters without a detailed computation. This can be done by essentially reversing the above calculation of observable quantities leading to the following approximate relations: P = upre
(21)
Slnale-Function Reruns lor V = 0 States N2IiZg+)
H21'Zgt)
Xe2('Zgt)
LiH(X'Zt)
Do obs. (eV1 Ba calc. (cm-') Ba obs. (em-') w calc. (em-') w obs. (cm-')
288.8 596 9.7604 9.7594 1.98598 1.98958 2371 2358.57
25.3 35 4.47797 4.47813 59.8341 59.3219 4308 4401.2
68.2 714 0.0231 0.0230
30.9 93.0 2.42969 2.42871 7.42127 7.4065 1395.2 1405.65
Potentlal
Morse
Morse
Species P 9 % L
calc. (eV)
Table 2.
Mol.
V
19.2 21.12 L-J
Morse
Three-Term Linear Variation Resuns
EM
Cn
C'
CI
(17)
and q=pre-1
with all quantities in Hartree atomic units. 754
Journal of Chemical Education
(18)
LiH (X'Z+) 0 -0.08928013
1.805280 1.253819 -0.6520274 1 -0.0831465 -21.08179 55.60501 -34.73543 130.6956 2 -0.08177251 132.8557 , -282.8831
from numerical errors, and there is a valuable object lesson hcrt ior students who use cuniputer program* as I~lackIwxts. The. rriawn that the prucedure fails i l l spiteofthe intrinsic suitabilit\. oi the 1,;iiis functions can Or srrn vi\.idlv . bs. examining the secular equation, eq 21. There is a well-known theorem that if two rows or columns of a determinant are the same, the determinant is identically zero. For Nz,the differences between the matrix elements involving ~
~
mates p and q for each of the states involved using eq 17 and eq 18 and calculates the overlap integrals directly. For the overlap integral (OLIO1'), for example, we have
~~
xo = ~
exp(-286.8r) ~
r
(22) ~
~
~
using Stirling's Approximation.
and x , = N,r5" exp(-286.8r)
(23)
are small. and the secular eauation is almost identicallv zero. This situation puts a seveie strain on the numericai algorithms used in the calculation. Thus, for numerical reasons, the simple linear variation function used above is not a generally useful way to deal with excited vibrational states. One can avoid these difficulties while using the same functions by first constructing an orthonormal set using the Schmidt procedure or by choosing a new set of functions as described below. Recall that solutions to the harmonic oscillator problem are 1
Table 3. Comparlson of Estimated Franck-Condon Factors wlth Published Valuer V'
0 1 0 1
(24) e p - -re2) where the H,s' are Hermite polynomials. We replace eq 24 by the expression, + (-r
N
V"
Estimated Factor
Hz
B'Z. - XIZS+
(4, 5)'
0
0.0044901 0.0238306 0.0223773
0.0044 0.0154
0
Literature
0.0804878
0.0304 0.0749
Ha
C ' r . - X'Zqt
(4. 5 )
cot
A ~ T, x2z+
(6)
I 1
Values
H -r
where
0 1 0 1
0 0 1 1 CO+
"Cornet Tail" 0.0440511 0.147085
0.125800 0.110214 B '2'
- xZZ+
0.042745 0.11506 0.15201 0.19325 (6)
"First Nagalive" and a is the expectation value of r in the ground state ( u = 0 ) . In Hartree atomic units a = po, (27) The states represented by eq 25 are essentially equivalent (hut not rigorously equal) to harmonic oscillator states and may themselves be used as approximations to the various vibrational functions, or they can he used as basis functions in the linear variation approach. A possible advantage they have over the harmonic oscillator functions themselves is again the ease of analytical integration. For the ground state, eq 25 reduces to our "single fnnction", which we have seen agrees well with the harmonic oscillator results. I t is easy to show that the ratio of the kinetic energy in the V = 1state to that in the ground state is given by
0 1 0 1
0 0 1 1
CN
0 1
0.560385 0.341359 0.304470
0.41496 0.43277 0.31476
0.0104643
0.011762
B2Zt-XZZ
0 0
"Violet" 0.907251 0.0876462
N,
BPs,
and for a harmonic oscillator potential with the zero of energy a t the minimum,
For large values of q both ratios approach 3, the exact harmonic oscillator results. A Slrnple Useful Appllcatlon
Franck-Condon factors, the square of the overlap integrals between vibrational wavefunctions in different electronic states are important in calculating the intensity distribution in electronic band systems. T h e ease of using the functions in the form of eq 25 in calculating such overlap integrals suggests the following way of making estimates of Franck-Condon factors from a knowledge of re1,weL,reL',and we" for the two electronic states involved. One merely esti-
-
A3Z.
(11)
"First Positive" 0 1 0 1
0 0 1 1
0 1
L~H 0 0 1 1
0 1
0.403772 0.353518 0.380149 0.00349069
0.3385 0.4071
0.3253 0.0024
A'Z+ - x IZ+ 0.000489034 0.00426316 0.00228100
0.0005
0.0153100
0.0185
(4. 5 ) 0.0036
0.0033
*Reference numbers apply fa entire column at values in each section.
Volume 64
Number 9
September 1987
755
of estimates Obtained in contains a this way with values from the literature obtained in a number of different ways.
Literature Cited L. Dunhsm, J. L. Phw-Reu. 1932.42.721. 2. tluber. K. P.: Horzberg,G.Sperrio dud Molsruiur Structure. Cunatanla Mo!scu!er: Van Nostrand: I'rineetun. 1979.
758
Journal of Chemical Education
0 ‘,
Diolomic
I
!I. ( h d l r i r n d . P. 1.. Chpm. Phys l . d l 1984, 1117. 595. 4. ?ddwell..!. W.: Curdun, M . S.J . Mot. Sprrlruac. 1980.84503. 6. i.i,,.c s . c a . . . ~ . ~ h y hIY~S,S:J.SIO. , 6. Nichollr. It.W . Can. J . P l l p . 1962.40, 1772. 7. Slindlrr. K. J. J . Quanl. Speclmsc.Rodiot. Tmnsler. 1965.6.165. 8. Nirhc4!r. K. W . J . Rea..Nol. Rur.Stond.. Wosh. L964,68A. 75. 9. Flinn. Il.d.:Spindler.H. J.;Fife
