In the Laboratory
Solution of the Schrödinger Equation for a Diatomic Oscillator Using Linear Algebra An Undergraduate Computational Experiment Zbigniew L. Gasyna Department of Chemistry, The University of Chicago, Chicago, IL 60637;
[email protected] At the undergraduate or college level, physical chemistry introduces quantum mechanics as a precise theoretical description of chemical bonding, spectroscopy, and molecular interactions. Problems that successfully model simple but realistic molecular systems and can be solved computationally without great difficulty may be useful for setting up computational experiments. Undergraduate projects dealing with various quantum mechanical problems have frequently been described in this Journal (1–9). We have developed (10) a computational experiment in which the Wentzel–Kramers–Brillouin (WKB) approximation is used in the solution of the Schrödinger equation. In this article, we describe a computational experiment in which students examine the solution of the Schrödinger equation for a diatomic oscillator based on the linear algebra approach implemented in Mathematica.
where V{l, rot}(r) is the rotational energy. For a diatomic molecule the exact rotational quantum states are given by quantization of the rotational angular momentum, l, and V{l, rot}(r) is equal to
Scientific Background
We propose an approach based on the method of discrete variable representation (DVR) (11) for solving the Schrödinger equation numerically. This method provides an expedient numerical formula for setting up the Hamiltonian matrix, H. The Hamiltonian matrix is most easily set up in the so-called universal DVR basis according to Colbert and Miller (12). Their results of the grid point representation of kinetic energy in the case of the potential energy function spanning the interval of 0 < r < ∞, is given by
The binding of atoms into molecules is accompanied by a decrease of the electronic energy of the system as the nuclei assume a stable geometry of molecular structure. The evaluation of the electronic energy as a function of the interatomic distances yields a “potential energy surface” of the molecule. The absorption and emission of light by molecules changes their vibrational and rotational states and permits the analysis of molecular geometry and other properties of molecules. To determine or interpret energies of these states one must first determine the potential energy surface and then set up and solve the Schrödinger equation for the quantized vibration– rotation levels. A simple situation to examine is the potential energy surface, V(r), for a diatomic molecule as a function of the internuclear distance, r. For stable molecules, V(r) will have a minimum, V0, at the equilibrium bond distance, re, go to zero as r → ∞, and become very large as r → 0 owing to electron overlap and the Coulomb repulsion of the nuclei. We describe an approximation method for determining the energy levels of a diatomic molecule given an arbitrary analytic or numerical form for intermolecular potential, V(r). The vibration–rotation energy of a molecule is determined by solving for the eigenvalues, En, of the Schrödinger equation:
H : En:
(1)
The Hamiltonian, H, is the sum of the potential energy, V, and kinetic energy, K: H K V (2) The potential energy can be expressed as
V
V r V l , rot r
(3)
V l , rot r
l l 1 h 2 2Nr2
l 0, 1, 2, ...
(4)
where r is the internuclear separation and μ is the reduced mass [m1m2/(m1 + m2)] of the molecule. For homonuclear diatomic molecules the l value may be restricted to odd or even integers by symmetry constrains. The kinetic energy operator is given by h 2 v2 K 2 N vr 2
i ib
h 2 1
Ki i b
2 N %r 2
Q 2 1 3 2 i2
i ib
h 2 1
2 N %r
2
(5)
for i = i b
2
i
i b
(6)
2
2
i
i b
2
for i y i b
where the grid points are equally spaced
ri i %rr, i 1, 2 , ... , N
(7)
Potential energy will only contribute to the diagonal part of the Hamiltonian matrix:
Vii b E ii b V r i V l , rot ri
(8)
where δii′ is the Dirac delta function. Thus the matrix elements are computed as
Hi j Ki j Vi j Ei j
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 6 June 2008 • Journal of Chemical Education
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845
In the Laboratory
UT H U En, l
diag
(10)
The eigenvalues, En,l , are the vibration–rotation energies of the oscillator corresponding to the n and l quantum numbers, and the columns of the eigenvector matrix U provide the respective wavefunction coefficients. Application to Diatomic Molecules: Vibration–Rotation Spectrum of HCl We illustrate application of the method with the HCl molecule for which the potential energy has been obtained by electronic structure calculations. The potential energy curve for HCl has been computed using the Gaussian 98 program (13) at the QCISD(T) level with the 6-311++G(3df,3pd) basis set. Based on this potential, the energies for several of the lowest vibrational transitions, that is of the fundamental Δn = +1 and weakly allowed Δn = +2, +3, +4, and +5 absorption bands (overtones) that are observed in the IR spectrum of HCl (14), have been calculated and the results are shown in Table 1. Figure 1 shows the IR absorption spectrum of HCl in the gas phase recorded on a Mathson Polaris FTIR spectrometer with a 0.5 cm‒1 spectral resolution, which allows for the resolution of transitions due to H35Cl and H37Cl. The absorption spectrum in this spectral range is due to the allowed Δl = ± 1 rotational components of the fundamental transition (Δn = +1) forming the P branch (Δl = ‒1) and the R branch (Δl = +1) of lines. The calculated vibration–rotation spectrum for H35Cl is shown in the bottom part of Figure 1. The experimental spectrum indicates that the shorter spectral lines just to the left of each larger line are due to H37Cl, and the larger lines and the calculations are for H35Cl. The computational experiment has been successfully carried out for several years in various modifications in computational chemistry courses at this university. In one of the projects, vibration–rotation spectra of HF and HCl were computed for the fundamental vibrational transition 0 → 1, as well as for the 1 → 2 transition and compared to the published experimental data. In another experiment, students have examined the probability of electron tunneling between bound states in energy wells separated by one-dimensional potential barriers. Conclusions A simple numerical method based on a DVR technique is proposed in the solution of the eigenvalue problem for a diatomic oscillator described by an arbitrary potential energy function. The vibration–rotation spectrum of the HCl molecule calculated using this method demonstrates an excellent agreement with the experimental data. Although implemented in Mathematica, other algebraic programs including Matlab and Mathcad as well as programming languages such as Fortran or C, can be easily adapted for this computational experiment. Literature Cited
1. 2. 3. 4.
Knudson, S. K. J. Chem. Educ. 1991, 68, A39. Rioux, F. J. Chem. Educ. 1991, 68, A282. Veguilla-Berdecía, L. A. J. Chem. Educ. 1993, 70, 928. Hansen, J. C.; Kouri, D. J.; Hoffman, D. K. J. Chem. Educ. 1997, 74, 335.
846
Table 1. Vibrational Transition Frequencies for H35Cl Vibrational Calculated Calculateda Observedb Observeda,b Transition ν/cm–1 Δν/cm–1 ν/cm–1 Δν/cm–1 Δn +1 2887.4 2885.9 2782.6 2782.1 +2 5670.0 5668.0 2683.3 2678.9 +3 8353.3 8346.9 2588.2 2576.2 +4 10,941.5 10,923.1 2496.1 2473.4 +5 13,437.6 13,396.5 aIf the potential were harmonic all Δν would be identical. bData from
ref 14.
1.2 1.0 0.8
Absorbance
The eigenvalues of the Hamiltonian matrix can be obtained by diagonalization
0.6 0.4 0.2 0.0 0.2
2600
2700
2800
2900
3000
Wavenumber / cmź1
3100
Figure 1. Experimental IR spectrum of HCl gas (pressure ~130 mmHg) at 0.5 cm−1 resolution (top), and the vibration–rotation spectrum of H35Cl calculated for the Δn = +1, Δl = ±1 transitions (bottom). The calculated frequency of the Δn = +1, Δl = 0 (forbidden) transition is indicated by a thicker bar.
5. Tanner, J. J. J. Chem. Educ. 1990, 67, 917−921. 6. Varandas, A. J. C.; Martins, L. J. A. J. Chem. Educ. 1986, 63, 485−486. 7. Rioux, F. J. Chem. Educ. 1992, 69, A240. 8. Hansen, J. C. J. Chem. Educ.1996, 73, 924. 9. Gasyna, Z. L.; Rice S. A. J. Chem. Educ. 1999, 76, 1023−1029. 10. Gasyna, Z. L.; Light, J. C. J. Chem. Educ. 2002, 79, 133−134. 11. Lill, J. V.; Parker, G. A.; Light J. C. Chem. Phys. Letters 1982, 89, 483–489. 12. Colbert, D. T.; Miller, W. H. J. Chem. Phys. 1992, 96, 1982−1991. 13. Frisch M. J.; et al. Gaussian 98, Revision A.11, Gaussian, Inc. 14. Herzberg, G. Molecular Spectra and Molecular Structure I. Spectra of Diatomic Molecules, 2nd ed.; Van Nostrand: New York, 1950; pp 55–56.
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Journal of Chemical Education • Vol. 85 No. 6 June 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
