Research: Science and Education
Investigating the Adiabatic Approximation in Quantum Mechanics through the Analysis of Two Coupled Harmonic Oscillators Anne B. McCoy Department of Chemistry, The Ohio State University, Columbus, OH 43210;
[email protected] Students in advanced undergraduate or beginning graduate level physical chemistry and quantum mechanics courses are typically introduced to two approximate treatments that are often employed in quantum mechanical calculations. These are perturbation theory and variational treatments. Discussions of these methods are often included in a chapter of the textbook entitled “Approximate Methods” (1–3) and are usually described in both words and equations. With the increasing ease with which numerical electronic structure calculations can be performed at a variety of levels of theory, it is becoming increasingly important for students to understand two other approximate treatments. These are the selfconsistent field (SCF) (or Hartree–Fock [HF]) treatment of electronic problems and the adiabatic (or Born–Oppenheimer) separation of the treatments of the electronic and nuclear degrees of freedom. In a recent article in this Journal, Messina discussed the use of a simple model system consisting of two coupled harmonic oscillators to illustrate properties of the SCF approximation (4 ). The discussion that follows presents an analysis of the adiabatic approximation, using the coupled harmonic oscillator system to illustrate some of the key features of this approach. Background In chemistry, the adiabatic approximation is most commonly applied in the form of the Born–Oppenheimer approximation. This approximation provides a connection between the quantum mechanical treatment of the electrostatic interactions between the electrons and nuclei and the idea of a single potential function, the Born–Oppenheimer potential, that describes the forces between the atoms that make up the molecule of interest. Within this approximation, it is assumed that because there is a three-order-of-magnitude difference between the masses of the electrons and the nuclei, the electronic wave function can adjust essentially instantaneously to the much slower motions of the nuclei (5). Therefore, we can approximate the electronic wave functions and energies for the molecule of interest by calculating these quantities for a set of fixed molecular geometries and interpolating between these results. The resulting coordinate dependent energy levels, which are often called the Born–Oppenheimer potentials, can then be used in classical or quantum mechanical studies of the rotational and vibrational dynamics of the molecule of interest. Within the Born–Oppenheimer approximation the contributions to the electronic wave functions and energies that come from the kinetic energy in the nuclear coordinates are neglected. This term introduces a kinetic coupling between the electronic and nuclear parts of the problem and results from the parametric dependence of the electronic wave functions on the molecular geometry. While the effects of this term
can be large if two or more electronic states are close in energy or if the magnitude of the couplings between these states changes rapidly with the vibrational coordinates, for the most part, for the ground state of closed shell systems, the Born– Oppenheimer approximation provides a powerful way to obtain very accurate descriptions of molecular systems. A second, although less common, application of the adiabatic approximation in chemistry comes in the separation of high- and low-frequency vibrational modes in studies of vibrational spectroscopy and collision dynamics. For example, the adiabatic approximation can be used to separate intermolecular and intramolecular vibrations in molecular clusters or the scattering coordinate from the other degrees of freedom in scattering problems. In addition to simplifying calculations, the adiabatic separation of several low-frequency vibrational coordinates from the higher-frequency ones enables one to generate effective potentials that can be used to aid in the interpretation of the spectroscopy and dynamics of these systems (6, 7 ). While the relationship between the adiabatic approximations in thermodynamics and quantum mechanics has been the topic of several papers in this Journal (8), little attention has been paid to applications of this approximation to the quantum mechanical treatments of chemical systems. With the development of computational software packages such as Mathcad and Mathematica, adiabatic treatments of model systems can be performed in a straightforward manner on a personal computer. In the discussion that follows the implementation of an adiabatic treatment of a general Hamiltonian will be outlined. With this treatment, the result that the ground-state energy, calculated within the adiabatic approximation, will be smaller than the zero-point energy of the system is derived. This result is surprising to students. Most of the other approximations commonly used in quantum mechanics are based on the variational theorem and in these cases, the approximate ground-state energy is always larger than the true ground-state energy. The second part of the paper focuses on applications of the adiabatic approximation to two model systems, and the results are discussed in the context of the discussion of the theory. Theory To start, consider the general Hamiltonian: = Tr + TR + V(r,R)
(1)
where r and R can be thought of as the electronic and nuclear coordinates of the molecule of interest. The kinetic energy for the two subsets of the coordinates is given by Tr + TR . The potential, V(r,R), is given by the sum of the electrostatic interactions among the electrons and nuclei in the system.
JChemEd.chem.wisc.edu • Vol. 78 No. 3 March 2001 • Journal of Chemical Education
401
Research: Science and Education
Within the adiabatic approximation, the Schrödinger equation is solved in two steps. In the first, the nuclear coordinates, R, are fixed and the electronic Schrödinger equation, [Tr + V(r,R)] Φ (r ;R) = (R )Φ (r ;R)
Based on the above treatment, the normalized ground state wave function for the system of interest is (4)
and the ground state energy, E, is approximated by EA. In this treatment, the approximation comes from the fact that TR was not included in eq 2, as discussed above. One interesting feature of the adiabatic approximation is that, in the absence of other approximations, EA ≤ E. This contrasts with the result obtained by approximations that are based on the variational principle for which E var ≥ E. To demonstrate this result, we follow the proof of Epstein (10). To start, we note that ∫dR ∫dr {Ψ *(r,R) Ψ (r,R)} = E
m
(2)
is solved. Since the equation is solved for a fixed value of R, the resulting wave function will depend parametrically on R, while the energy eigenvalues, {}, will be functions of R. In the discussion that follows, will be used to represent the ground electronic state. This function is used as the potential surface in setting up the Schrödinger equation for the nuclear degrees of freedom, [TR + (R)] φ (R) = E Aφ (R) (3)
Ψ (r,R) ≈ φ (R) Φ (r ;R)
VR(R) Vr(r )
(5)
M V(R,r )
Figure 1. An illustration of the model systems used to illustrate the adiabatic approximation.
Here R and r represent displacement coordinates of the two harmonic oscillators, P and p are the momenta conjugate to R and r, respectively, and the associated masses are M and m. The final term in eq 10 represents the potential couplings between the two harmonic oscillators. In the discussion that follows, we will focus on the cases in which n = 1 and 2. When n = 1 the energies and corresponding wave functions can be calculated analytically by performing a transformation to the normal mode coordinates. In this coordinate representation, the Hamiltonian becomes separable into the sum of two one-dimensional harmonic oscillator Hamiltonians, (11) with the ground state energy
E = h ω+ + ω 2
(11)
where
Based on the linear variational principle, if, for a given value of R, we consider Ψ (r,R) to be an approximation to Φ (r ;R) in eq 2,
ω± =
∫dr{Ψ *(r,R )[T r + V (r,R)]Ψ (r,R )} ≥ (R)∫dr {Ψ *(r,R ) Ψ (r,R )} (6)
ωr2 + ωR2 2
±
ωr2 – ωR2 4
V 12
2
+
1/2
Mm
In the first step of the adiabatic approximation,
If we add TR inside the integral on both sides of eq 6 and integrate both sides over R , we get
rΦ r;R = R Φ r;R
E = ∫dR ∫dr {Ψ *(r,R)[Tr + TR + V (r,R)] Ψ (r,R)} ≥ ∫dR ∫dr {Ψ *(r,R)[TR + (R)] Ψ (r,R)} (7)
p 2 mωr2 2 MωR 2 + r + R + V 1rR Φ r;R = R Φ r;R 2m 2 2
If, instead, we consider that, for any value of r, Ψ (r,R) can be thought of as an approximation to φ(R) in eq 3, we find
2
p2 2m
+
∫dr {Ψ *(r,R)[TR + (R)] Ψ (r,R)} ≥ EA ∫dR{Ψ *(r,R) Ψ (r,R)}(8) Since the above relationship is true for any value of r, the inequality will hold if we integrate over all r. Combining this result with that in eq 7, we find
mωr2 2
2V R V 2R 2 r 2+ 1 r + 1 mωr2 m 2ωr4
This means that in the absence of other approximations, the groundstate energy calculated by the adiabatic approximation will always be smaller than the true ground-state energy.
Φ r;R =
To further investigate the implications of the results presented above, we will consider a system consisting of two coupled harmonic oscillators, illustrated in Figure 1. Within this model, the Hamiltonian is given by 2 2 p2 P 2 mωr 2 MωR 2 = + + r + R + V n r nR n (10) 2m 2M 2 2 402
2
–
V 12 2m 2ωr4
(13)
R 2 Φ r;R =
is solved, giving
(9)
Model
+
MωR2
R Φ r;R
R =
E ≥ EA
(12)
2 hω r 1 V1 2 2 + mω R – R 2 2 mω r2
(14)
αr V 1R αr π exp 2 r + mω 2 r
(15)
4
2
where α r = m ωr /h⁄ . Next, (R) is used to construct the Schrödinger equation in R, R φ R = EA φ R (16) 2 P + hωr + 1 M ω 2 – V 1 R 2 φ R = E φ R R A 2 2 2M mω r2 2
Journal of Chemical Education • Vol. 78 No. 3 March 2001 • JChemEd.chem.wisc.edu
Research: Science and Education
giving
1.0
φ R =
4
where αR = M ω ′R /h⁄ and
ω′R = ωR
hω′R hω r + 2 2
1e+1 0.8
αR R 2 αR exp π 2
1–
(18)
V 12 mMωr2 ωR2
(20)
2
is solved, giving hω r 2V2 2 1+ R 2 mω r2
M ωR 2 R 2 2
+
(21)
Next, (R) is used to construct the Schrödinger equation in R:
R φ R = EA φ R (22) 1/ 2
1e−1
1e−3
0.4
p 2 mωr2 2 MωR 2 + r + R +V2 r 2R 2 Φ r;R = R Φ r;R 2m 2 2 1/ 2
0.6
1e−2
rΦ r;R = R Φ r;R
2 P + hωr 1 + 2V 2 R 2 2 2M mω r2
1e+0
(19)
Comparing the expressions for the energy in eqs 11 and 17, we find that the adiabatic approximation will be most accurate when ωr >> ωR and |V1/(ωr – ωR )| 0. This means that the adiabatic approximation to the ground state energy is
1e−4 1e−5
0.2
0
1
2
3
log10(ωr) Figure 2. The percent error of the adiabatic energies is plotted on a logarithmic scale, as a function of log10 (ω r) and V1 with M, m, ω R, and h⁄ all set equal to one.
Table 1. Adiabatic and Exact Energies When V (R,r ) = V2 R2r 2 ωr
V2
Ea
( E – EA)b
( E – E ′A)c
1
0.1
1.023398
0.000409
0.001006
1
0.5
1.023398
0.004268
0.014399
10
0.1
5.502492
0.000000
0.000002
10
1.0
5.524248
0.000010
0.000156
10
5.0
5.609680
0.000188
0.002693
100
0.1
50.500250
0.000000
0.000000
100
1.0
50.502494
0.000000
0.000000
100
10.0
50.524388
0.000001
0.000017
100
50.0
50.612065
0.000002
0.000307
NOTE: In these calculations, ωR, m, M, and h⁄ are all set equal to 1. a The zero point energy of the system. b The adiabatic approximation to E, which is the solution to eq 22. c An approximation to E , given by eq 24. A
always smaller than the ground state energy, as we proved in the previous section. These general trends are the same as we found for the case when n = 1. When an approximation is employed, to simplify the evaluation of the adiabatic energy, EA is no longer guaranteed to be smaller than the ground state energy. To illustrate this, we obtain an approximate value of EA by replacing the square root in eq 22 with a quadratic expansion. Making this substitution, 2
2 P + hωr 1 + V 2 R 2 + M ω R R 2 φ R = E ′ φ R (23) A 2 2 2M mω r2
and
E ′A =
hω r hω R hV 2 + 1+ 2 2 2 mω r M ω R
1/ 2
(24)
The results of these calculation are reported in the fifth column of Table 1.
JChemEd.chem.wisc.edu • Vol. 78 No. 3 March 2001 • Journal of Chemical Education
403
Research: Science and Education
In the present example, the approximate energies are always larger than the ground state energy, which is exactly the opposite of the trend expected when the adiabatic approximation is employed without further approximations. This result illustrates a very important point. While we can often prove that a certain approximation will provide an upper or lower bound to the ground-state energy of a system of interest, this will only be true if no other approximations are made in the calculation. Conclusions In this paper, the properties of the adiabatic treatment of quantum mechanical systems have been investigated for two model coupled harmonic oscillator systems. Through the analysis of the results, we find that the adiabatic approximation is most accurate when the difference between the ground state energies for the two oscillators is large. This is exactly the situation that is realized for the electronic and nuclear degrees of freedom in a molecular system. The energy differences between electronic states are two or three orders of magnitude larger than the differences between the energies of the vibrational states. We also investigated an analytically solvable approximation to the adiabatic ground state of the quarticly coupled harmonic oscillator system. The results of these calculations demonstrate that the statement that the adiabatic ground-state energy of a system is always smaller than the ground state energy is only valid if no other approximations are introduced.
404
Acknowledgments I gratefully acknowledge the National Science Foundation (CHE-9732998) and Camille Dreyfus Teacher-Scholar Awards Program for partial support of this work. Literature Cited 1. Atkins, P. W.; Friedman, R. S. Molecular Quantum Mechanics, 3rd ed.; Oxford University Press: New York, 1997. 2. McQuarrie, D. A.; Simon, J. D. Physical Chemistry, a Molecular Approach; University Science Books: Sausalito, CA, 1997. 3. Laidler, K. J.; Meiser, J. H. Physical Chemistry, 3rd ed.; Houghton Mifflin: New York, 1999. 4. Messina, M. J. Chem. Educ. 1999, 76, 1439–1443. 5. Kauzmann, W. Quantum Chemistry; Academic: New York, 1957. 6. Takayanagi, T.; ter Horst, M. A.; Schatz, G. C. J. Chem. Phys. 1996, 105, 2309–2316. 7. Lee, H.-S.; McCoy, A. B.; Harding, L. B.; Carter, C. C.; Miller, T. A. J. Chem. Phys. 1999, 111, 10053–10060. 8. Boulil, B.; Deumié, M.; Henri-Rousseau, O. J. Chem. Educ. 1987, 64, 311–315. 9. Morales, D. A. J. Chem. Educ. 1990, 67, 211–213. 10. Epstein, S. T. J. Chem. Phys. 1965, 44, 836–837. 11. Wilson, E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations; Dover: New York, 1955.
Journal of Chemical Education • Vol. 78 No. 3 March 2001 • JChemEd.chem.wisc.edu
