ARTICLE pubs.acs.org/jchemeduc
Lorentz Trial Function for the Hydrogen Atom: A Simple, Elegant Exercise Thomas Sommerfeld* Department of Chemistry and Physics, Southeastern Louisiana University, Hammond, Louisiana 70402, United States
bS Supporting Information ABSTRACT: The quantum semester of a typical two-semester physical chemistry course is divided into two parts. The initial focus is on quantum mechanics and simple model systems for which the Schr€odinger equation can be solved in closed form, but it then shifts in the second half to atoms and molecules, for which no closed solutions exist. The underlying principle that bridges this chasm is the variational method, which is easily the most important guiding principle for the construction of approximate wavefunctions including molecular orbitals. Students frequently encounter difficulties crossing this bridge, even when they did well in the first half. One contributing factor is the dilemma of the variational method that variational problems tend to be either too trivial or mathematically too involved to clearly show the concepts at work. Here, a Lorentz trial function for the hydrogen atom is discussed, a very elegant, yet straightforward example that somewhat alleviates this dilemma. Together with the standard example of a Gauss trial function and the exact solution, the Lorentz function provides an opportunity for an in-depth study of the variational principle before applying it to heavier atoms and molecules. KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Textbooks/Reference Books, Quantum Chemistry, Theoretical Chemistry
T
he variational principle is a cornerstone of quantum mechanics, and the most important guiding principle for the construction of approximate wavefunctions such as molecular orbitals and valence-bond wavefunctions.14 It states that the trial energy, ET, defined as the energy expectation value computed with a suitable, normalized trial function, ΦT, is greater than or equal to the exact ground-state energy, E0, of a quantum system Z ^ ΦT g E0 : dτ ΦT H ð1Þ ET ¼ as
quantum chemistry as such, the variational principle and the variational method have been discussed in various contexts in this Journal (see refs 57 for recent contributions), notably as an area where the effort of learning to use a computational package can be justified in undergraduate teaching. In an introductory physical chemistry course, the variational principle acts as a bridge that crosses the chasm between systems for which the Schr€odinger equation can be solved in closed form (e.g., particle in a box, harmonic oscillator, H atom) and systems for which only approximate solutions exist (all other atoms, molecules, chemical bonding). Many students have difficulties to cross this bridge, and feel “lost” in the second half of the course. Possible reasons include the change in gear from finding, or at least discussing, solutions of differential equations to constructing a seemingly bewildering variety of trial functions, the uncertainty associated with not having the true but only an approximate answer, and the additional complications due to the many-body nature of the ensuing topics. However, one factor certainly contributing is the lack of good examples. Here an example is considered to be “good”, if both, the integrations required for steps 1 and 2 and the minimization of step 3, are nontrivial, yet manageable for typical students. The dilemma most instructors are only too familiar with is that variational problems tend to be either too trivial or too involved so that the underlying concepts are more obscured than clarified. Even the two most popular examples used in many undergraduate texts,13 the parabolic trial function, x (L 2 x), for the particle-in-the-box, and the Gauss trial function, eαr , for the hydrogen atom, cannot be considered to be good. The former is a
Here the integral is over “all space” (as), that is, over all degrees ^ is the Hamiltonian of freedom with their appropriate ranges, H of the quantum system and “suitable” means that the trial function fulfills the boundary conditions of the problem. The power of this simple expression unfolds as a criterion for judging different trial functions: A trial function is “better” if its trial energy is lower. It is then a small step to the variational optimization of trial functions, often referred to as the variational method. It consists of three steps: 1. Construct a suitable trial function that depends not only on the dynamical variables, but also on one or more parameters, a1, a2, ..., and normalize it. 2. Evaluate the integral in eq 1 yielding the trial energy as a function of the parameters: ET (a1, a2, ...). 3. Find the optimal parameters by minimizing ET with respect to the parameters. This is then considered to be the best approximation for the true wavefunction within the constraints of the trial function ansatz. Owing to their major role in the quantum semester of a typical two-semester undergraduate physical chemistry course and in Copyright r 2011 American Chemical Society and Division of Chemical Education, Inc.
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Figure 2. Integrals required for computing the trial energy associated with a Lorentz trial function.
Figure 1. Lorentz function. The full width at half the maximal value is indicated.
good demonstration for the variational principle, but cannot serve as an example for the variational method, because it lacks a parameter to be optimized, whereas the latter is too involved, because the integrals needed in steps 1 and 2 are highly nontrivial (see below). There are many examples, in particular in graduate-level textbooks, that undergraduates can successfully tackle when they are supplied with hints, helpful relations, or partial results, but what is lacking are nontrivial examples that students can grasp as a whole and feel in control over. In this article, the Lorentz trial function for the hydrogen atom is discussed, which is still not quite a “good” example, but more so than a Gauss trial function. This example somewhat alleviates the dilemma of the variational method and adds to the meager choice of doable problems for inclass activities, homework, and tests.
’ DISCUSSION In this section, the hydrogen atom as a variational problem is considered. The ground-state wavefunction, the 1s orbital, is of course known, which enables students to compare their optimized trial functions with the exact answer and to appreciate the limitations of the variational method associated with the flexibility of the trial function. Focusing on the radial part of the wavefunction, the relevant Hamiltonian is 2 2 ^ þ V^ ¼ p d r 2 d e ^ ¼ T H 2mr 2 dr dr 4π ε0r 1 d d 1 ð2Þ ¼ 2 r2 , 2r dr dr r ^ and V^ are the kinetic and potential energy operators, where T and the explicit expression of the Hamiltonian is given in SI and in atomic units. There is only one dynamical variable, the electronproton distance, r, and dτ = r2 dr, so that eq 1 becomes Z ∞ ^ Tðr Þ r 2 dr ΦT ðr Þ HΦ ð3Þ ET ¼ 0
The trial function used here is the Lorentz function, f ðxÞ ¼
1 a2 þ x2
ð4Þ
displayed in Figure 1. The Lorentz function is, similar to the Gauss function, a singly peaked function that is well-known from modeling line shapes of, for example, NMR spectra. The parameter a is a measure of its width (cf. Figure 1) and can be used for variational energy minimizations. At least one graduate text4 lists
the problem of using a Lorentz trial function for the harmonic oscillator, which leads to similar integrals. For the hydrogen atom, the range of r is from 0 to ∞, and thus, only the right half of a Lorentz function is used Φ T ðr Þ ¼
1 with r g 0: a2 þ r 2
ð5Þ
Computing the integrals required for steps 1 and 2 is not straightforward, but much easier than evaluating those of a Gauss trial function. Similar to the Gauss function case, evaluating the integrals becomes straightforward if students are supplied with a few standard integrals listed in Figure 2. In contrast to the Gauss case, however, one can alternatively demonstrate the trick of replacing the integration over r with an integration over an angle θ with cos 2 θ ¼
a2
x2 a2 and sin2 θ ¼ 2 , 2 þx a þ x2
ð6Þ
and let students do (some) of the integrations themselves. This tends to work well with students who already took their second physics course, calculus-based electromagnetism, where similar integrals occur. Doing a similar derivation for a Gauss function is somewhat more involved as one needs to consider a Gauss functions in two-dimensions, and to evaluate either a path or a shell integral in polar coordinates. A third alternative to do the integrations is to let students use a symbolic mathematics package (see Supporting Information). Regardless of how students perform the integrations, the results are exceptionally simple (see Table 1) and so is the expression for the trial energy, E T ðaÞ ¼
p2 1 e2 2 1 2 ¼ 2 , 2 4a πa m 4a 4π ε0 πa
ð7Þ
which is given in SI and in atomic units. Consequently, step 3, finding the optimal parameter, aopt, that minimizes the trial energy, is also a straightforward exercise, aopt ¼
π 4 p2 a0 and ET aopt ¼ 2 4 π ma0
ð8Þ
where a0 is the Bohr radius. Owing to the simplicity and elegance of the intermediate and final results, solving the variational problem of a Lorentz trial function for the hydrogen atom can be a deeply satisfactory experience. Because it is far from trivial, yet not too challenging, in particular, if the integrals in Figure 2 are provided, it is well suited for homework or guided inquiry activities. When broken down into four steps—(a) normalize the trial function, (b) show ^ is (1/4)a2, (c) show that the that the expectation value of T ^ expectation value of V is 2/πa, and (d) minimize the trial energy with respect to a—the problem even works well in a test. 1522
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Table 1. Comparison of Intermediate and Final Results of Variational Calculations for the Radial Part of the Hydrogen Wavefunction
Unnormalized trial function
~ T(r; a) Φ
Normalization integral
R∞ 2 ~ *T Φ ~T r dr Φ 0
Normalized trial function Kinetic energy
ΦT (r; a) R∞
^ ΦT r2dr Φ*T T
0
Potential energy
R∞
r2dr Φ*TV^ ΦT
0
a
Optimized parameter
aopt
Minimal energy
ET(aopt)
Lorentz Test Function
Gauss Test Function
Conversion Factor to SI Unitsa
1
er /a
—
pffiffiffiffiffiffi 3 2πa 16
—
2
a2 þ r 2 π 4a rffiffiffi a 2 π a2 þ r 2
4 ð2π Þ1=4 a3=2
1 4a2
er
2
=a2
3 2a2 rffiffiffiffiffi 2 2 a π
2 πa
3 pffiffiffiffiffiffiffi 2π 4
π 4
2
4 π2
4 3π
— p2 m e2 4π ε0 a0 p2 ma0
All intermediate results are listed in atomic units and the conversion factor to SI, or other units is given. Note that e2 p2 ¼ 4π ε 0 ma0
Figure 3. Comparison of the exact 1s orbital with the optimized trial functions (see Table 1 for the functional forms and the values of parameters).
Figure 4. Comparison of the exact radial density, r2Φ21s, of the 1s orbital with the densities of the optimized trial functions (see Table 1 for functional forms and values of parameters).
The hydrogen atom is particularly useful as an introductory example for the variational method because the computed trial energies and the optimized trial function can be compared with the exact result. Clearly both, a simple Gauss and a simple Lorentz function, provide at best a crude approximation of the exact solution, 2er/a0 (Figure 3), with the Lorentz function seemingly tracking the exact wavefunction somewhat better than the Gauss function. On the other hand, the trial energy of the Lorentz function, 11.0 eV, is higher than that of the Gauss function, 11.5 eV. Both are well above the exact value of 13.6 eV, but in contrast to Figure 3, the energies nevertheless suggest that the Gauss function should be considered the better of the two trial functions. This conflict is resolved when the radial densities, r2Φ2T, associated with the two trial functions are plotted (Figure 4). A comparison of the two plots is a strong reminder that visual impressions of radial wavefunctions can be deceptive, a point well worth driving home.
’ CONCLUSION In conclusion, the Lorentz trial function for the hydrogen atom is an overlooked beauty that deeply enriches the introduction of the variational method and prepares students with a more solid foundation for the second half of the semester. From a mathematical point of view, the problem is among the easiest nontrivial examples, helping students to clearly see the underlying concepts at work. Moreover, the elegance of the intermediate and final results resembles the closed solutions students are at that point familiar with, making it an excellent homework (or test) problem after the Gauss trial function has been studied in class. Regarding previously covered material, the problem revisits the hydrogen orbitals and reinforces many fundamental concepts, including normalization, expectation values, finding minima of functions, and the difference between radial wavefunctions and radial densities. Regarding future material, it introduces 1523
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the Lorentz function, which represents a natural line shape, and is therefore widely used in the numerical analysis of spectra. Moreover, visual comparison of optimized trial functions with the exact wavefunction (cf. Figures 3 and 4) invite a discussion of how trial functions can be improved leading naturally to linear combinations of primitive functions and basis sets in general.
’ ASSOCIATED CONTENT
bS
Supporting Information The whole variational problem of a Lorentz trial function for the hydrogen atom, including evaluation of the integrals required for steps 1 and 2, minimization of the trial energy in step 3, and visualization of the optimization procedure and the optimized trial function, can be done with the help of a symbolic mathematics package. As an example, a Sage worksheet is included. Sage is an open-source package that can be downloaded without charge from www.sagemath.org. An alternative to running a local copy of Sage is to use one of the Sage servers open to the public (e.g., at www.sagenb.org/). Also included is a pdf-printout of the worksheet that should allow readers familiar with other tools to migrate the worksheet. This material is available via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT I wish to acknowledge helpful discussions with my colleague William Parkinson, who suggested publishing this material. ’ REFERENCES (1) Atkins, P. W.; de Paula, J.; Physical Chemistry, 8th ed.; W. H. Freeman: New York, 2006. (2) Engel, T. Quantum Chemistry and Spectroscopy, 2nd ed.; Prentice Hall: Upper Saddle River, NJ, 2009. (3) Levine, I. R. Physical Chemistry, 6th ed.; McGraw-Hill: New York, 2008. (4) Levine, I. R. Quantum Chemistry, 4th ed.; Prentice Hall: Englewood Cliffs, NJ, 1991. (5) Yindra, L. J. Chem. Educ. 2001, 78, 1557. (6) Dunn, S. K. J. Chem. Educ. 2002, 79, 1378. (7) Metz, R. J. Chem. Educ. 2004, 81, 157.
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