Research: Science and Education edited by
Advanced Chemistry Classroom and Laboratory
Joseph J. BelBruno Dartmouth College Hanover, NH 03755
Variational Principle for a Particle in a Box Juan I. Casaubon Facultad de Ciencias Exactas y Naturales, Universidad de Belgrano y CONICET, Villanueva 1324, 1426 Buenos Aires, Argentina Graham Doggett* Chemistry Department, University of York, Heslington, York YO10 5DD, UK;
[email protected] The particle in a one-dimensional infinite square well potential provides one of the simplest quantum mechanical model systems, in which the potential is constant in a box of length L. The position of the particle, x, may be defined with respect either to the left-hand edge of the box, 0 ≤ x ≤ L, or to the center of the box, L/2 ≤ x ≤ L/2. This “particle-in-abox” (PIB) model is widely used as an example in teaching chemistry undergraduates the derivation and properties of the solutions of the Schrödinger equation (1, 2). The PIB system also provides a very good paradigm for motivating the uses of the variational principle, symmetry constraints, and the concept of a basis set. The next section contains a brief summary of the equations associated with the variational principle, when a linear combination of functions is used to model the exact wave function. Subsequent sections describe how suitable approximate wave functions with zero, one, or two variational parameters may be formulated with and without imposing the requirements of symmetry. A new form of one-parameter wave function is also described. Results are obtained for several approximations for the first three energy states, using polynomial basis functions up to degree four, and links are made with the work of Bransden and Joachain (3). The Variational Method
Introduction of a Basis Set If the model wave function χ var is expressed as a linear combination of r suitably chosen basis functions, φ k, r–1
χvar = Σ a k φk k=0
then the ak are the variational parameters, and the (known) expansion functions, φ k , may be neither normalized to unity nor orthogonal. For any choice of r suitable basis functions then, as already noted, r upper bounds, ε k, are expected. To carry out the variational approach, the substitution of eq 3 into eq 1 and the use of r conditions analogous to eq 2, but with each of the ak values in turn replacing ξ , leads to the secular determinant (eq 4) for the energies: |H – ε S| = 0
εvar =
χ var |χ var
(1)
The optimum forms for χ var are obtained by minimizing ε var with respect to the n parameters contained in χ var, a procedure which yields n values for ε var, with the upper-bound property for each one that ε k ≥ Ek (k = 1, 2, …, n). Different choices of χ var may be made; if there are no variation parameters, then the energy is obtained directly from eq 1. In the case of a trial function depending upon an adjustable parameter, ξ, say, the optimum value of ξ is obtained by solving
∂εvar =0 ∂ξ
(2)
after using eq 1 to evaluate the energy. The question of constructing suitable approximate wave functions is best accomplished through the introduction of a basis set, as described in the next section.
(4)
and the secular equations (eq 5) for the coefficients r–1
Σ t=0
H k t – εS k t a t = 0;
k = 0, 1, 2, …, r – 1
(5)
The energy and overlap integrals Hkt and Skt , respectively, have the properties Hkt = Htk = φk||φt and Skt = Stk = φk|φt. For the PIB problem, the exact energies, in atomic units, are
In the Rayleigh–Ritz variational method (1, 3, 4), the trial function χ var is used to evaluate the expectation value, ε var, of the Hamiltonian, :
χ var χ var
(3)
En =
n2 π 2 2mL
2
;
n = 1, 2, 3, …
(6)
where m is the mass of the particle (for an electron, m = 1), and the associated eigenfunctions are
Ψn = for 0 ≤ x ≤ L.
2 sin nπ x L L
(7)
Choice of Basis Functions, φ k In general, the basis functions, φk, may be chosen to be eigenfunctions of part of the Hamiltonian operator, or of a model operator (as in the SCF method), or, in Dewar’s words (5), we can “choose the basis functions in any way we please, guided by intuition, prayer, or any other suitable source of inspiration’’. In this work, different sets of basis functions are used, in which the individual members are built out of (nonorthogonal) monomials of the form x k (k = 0, 1, 2, …). The most general polynomial form of degree r, for the approximate PIB wave function, is given by χvar = a0 + a1x + a2x2 + … + ar x r
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which has r + 1 terms, irrespective of the choice of origin for the position coordinate. However, in order that χvar be an acceptable model wave function for the PIB problem, it must satisfy the same boundary conditions as the exact solution: that is, χvar(0) = 0 and χvar(L) = 0, if the coordinate origin is taken at the left-hand box edge. These boundary conditions require that a0 = 0 and a1 = a2L – a3L2 – a 4L3 – … – ar Lr –1, respectively. Thus, χvar takes the form
The Symmetry of the Ground State Wave Function Under inversion, the point x is transformed into the point L – x and vice versa, leaving χvar invariant. The wave function (eq 12) is therefore even (g) under inversion. This property is more apparent if the coordinate origin is transferred to the center of the box, using the substitution x → x + L/2:
χvar = x(x – L){c1 + c2(x + L) + c3(x2 + xL + L2) + … + cr–1Gr–1(x,L)} (9)
where α = L/2. In this coordinate system, x → x under inversion, and the invariance of eq 13 is obvious. If the PIB problem is solved using the box center as origin, then it is appropriate to constrain χvar to be expanded over either even or odd monomials, x k, and also to satisfy the boundary conditions χvar(±α) = 0. In this situation, the expansion functions φk are given by (x 2 – α2)Gk(x 2, α2) (even parity), and xGk(x 2, α2) (odd parity).
in which ck = ak+1, and k
Gk x, L = Σ x k–s L
s–1
s=1
The Gk polynomials have the property G1(x,L) = 1; Gk+1(x,L) = xGk(x,L) + Lk;
k = 1, 2, …
and two degrees of freedom have been lost from the unconstrained expansion of eq 8. The original basis of r + 1 monomials therefore reduces to a new basis of r – 1 functions of the form φk = x(x – L)Gk(x,L);
k = 1, 2, …, (r – 1)
(10)
and the expansion (eq 9) becomes χvar = c1 φ1 + c2 φ 2 + … + cr–1 φr–1
(11)
An expansion of the form of eq 11 acts as a paradigm for the solution of the secular problem for three-dimensional systems where, in the case of atoms or molecules, for example, the expansion could be made using Slater-type atomic orbitals or Cartesian Gaussian functions (usually centered on the nuclei) (6 ) as the basis functions, φk . The only real difference between the PIB model system and an atom or molecule (assuming the clamped nucleus approximation) lies in the complexity and number of the energy integrals Hkt : boundary conditions and spatial symmetry requirements are common to all problems. The normalization requirement and symmetry constraints, if imposed, will lead to a further loss of degrees of freedom. A Model Wave Function with no Variational Parameters It follows from the above discussion that the simplest acceptable approximate wave function can possess only one basis function and is therefore represented by the quadratic polynomial χvar = Nφ1 = Nx(x – L) (12) where N is determined by the normalization requirement χvar| χvar = 1. This choice of wave function, in which χvar < 0 for 0 < x < L, stems from the definition of φk in eq 10; however, the sign of the φk may be changed if desired—without loss of generality—to ensure that χvar > 0. The model wave function in the form of eq 12, described by Levine (1) and Besalú and Marti (2), possesses no variational parameters, and yields an energy of ε1 = 5 2 mL with an error of 1.3% relative to the exact energy E1.
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χvar → N(x + L/2)(L/2 – x) = M(x 2 – α2)
(13)
One-Parameter Model Wave Functions If one degree of freedom is required, this is achieved by working with two basis functions of the form of eq 10, as is seen in eq 9: χvar = x(x – L){c1 + c2(x + L)}
(14)
But, since the c2L term in the braces is independent of x, the wave function (eq 14) may be written in the equally acceptable form χvar = Mx(x – L){1 + cx}
where c = c2 /(c1 + Lc2) and M is a constant. The optimization of c leads to a secular determinant (eq 4) of order two, with roots ε1 = 5 2 , ε2 = 21 2 5mL 5mL There is thus no improvement to the ground state, but an upper bound, ε2 (with a 6% error), is now obtained for the first excited state (odd parity), with wave function given by χvar = Nx(L –2x)(x – L) = M{LG1 – 2⁄3G2}
or χvar = Nx(x2 – α2) when transforming the origin to the center of the box. It is clear, therefore, that the extra degree of freedom has permitted the second energy state to be accessed without any corresponding improvement in the ground state— a direct consequence of the effects of symmetry, since the ground state can be improved only through the inclusion of the monomial x4 (i.e., three basis functions are required). However, it is possible to improve the ground state energy by working with a different quadratic form for the wave function, as is seen in the next section.
A New Ground State One-Parameter Wave Function Consider the following well-behaved function, depending upon the one parameter ξ
χvar =
M L – 2ξ x
0≤x ≤ξ
M ξ2 + x L – x
ξ≤x ≤L –ξ
M L – 2ξ L – x
L –ξ≤x ≤L
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Research: Science and Education
The energy of χ var and the derivative of the energy with respect to the variational parameter ξ are as follows:
5 εvar = m
3
2
16ξ – 12ξ L + L
3
32ξ5 + 40ξ4L – 10ξ2L 3 + L 5 (16)
∂εvar 1 = ∂ξ 2m
2
40ξ 16ξ + 4ξL – L
2
8ξ3 + 2ξ2L + 4ξL 2 + L 3
2
The derivative is zero for ξ = 0, 0.15451L, 0.40451L. The value ξ = 0 corresponds to the previous approximate wave function (eq 12); ξ = 0.15451L yields a ground state energy of 4.9442/mL2 , with an error reduction from 1.3 to 0.19%. The negative solution has no physical significance.
The Bransden and Joachain Approximation for χvar Bransden and Joachain (3) used the following fourthdegree polynomial to model the ground state of the PIB problem (origin of coordinates at the box center): χvar = N(α2 – x2)(1 + cx2)
(17)
where c is the variational parameter and α = L/2. As seen above, this even-parity wave function is built from the two basis functions (α 2 – x 2)G1(x 2, α 2) and (α 2 – x 2)G2(x 2, α 2), and provides upper bounds to E1 and E3. On using eq 17 in eqs 4 and 5, two values for c are obtained: one corresponding to the ground state (n = 1), with no nodes in (0, L) and energy (0.0015% error)
Thus, for r basis functions it is possible to determine upper bounds to r /2 even states and (r – 2)/2 odd states (r even) and (r – 1)/2 even and (r – 1)/2 odd states (r odd).
Use of a Ground State Wave Function to Model Excited States If the model wave function for the ground state (n = 1) is written as ψ1(x,L), which is equivalent to χvar(x,L), the wave function for the state with quantum number n, χvar may be built up from n suitably scaled and oriented contiguous segments derived from ψ1 —each of which is defined in a box segment of length L/n. Thus, for the second excited state (n = 3), for example, the box is divided into three segments of length L/3, and the model excited state wave function takes the following forms in the three segments: χvar
ψ1 x, L/3 x,L = ψ1 x – L/3, L/3 ψ1 x – 2L/3, L/3
In the general case, χvar is constructed from the n segment wave functions
1
ψ1 x –
i–1
ε3 = 51.065 mL 2 If a trial function of odd parity is taken by multiplying eq 17 by x , an upper bound for the n = 2 state may be found, as suggested in Problem 8.17 of Bransden and Joachain (3). The same values of ε1, ε2, and ε3 are obtained using χvar expanded over the first three basis functions, x(x – L)Gk(x,L), for k = 1, 2, and 3: χvar = x(x – L){c1G1 + c2G2 + c3G3} =
x(x – L){c1 + c2(x + L) + c3(x 2 + xL+ L2)} = Mx(x – L){1 + cx + c′x(L + x)} where c = c2/(c1 + c2L + c3L2) and c′ = c3/(c1 + c2L + c3L2). Use of Symmetry If the x(x – L)Gk(x,L) basis functions are used to describe χvar, symmetry-adapted wave functions can be constructed before using the variational principle by applying the projection operators (1+ î), (1 – î), where î is the inversion operator, to obtain even or odd wave functions, respectively. However, by avoiding the use of symmetry-adapted expansions, selected even and odd states can be accessed in a single calculation.
i –1 L , L/n , n
i = 1, 2, 3, …, n
It is now easy to show that the two integrals in the numerator and denominator of the expression for the expectation value of the Hamiltonian (eq 1) reduce to n3 and n times, respectively, the contributions obtained from ψ1(x,L). Hence eq 1 reduces to
εn =
ε1 = 4.934875 2 mL and the other to the second excited state (n = 3), with two nodes in (0, L) and energy (14% error)
0 ≤ x ≤ L/3 L/3 ≤ x ≤ 2L/3 2L/3 ≤ x ≤ L
n 2K mL 2
,
n = 1, 2, 3, …
where K = mL 2 ε1. Thus, in the calculations reported here, values of K = 5, 4.9442, 4.934875 were obtained, with the ground state wave functions of eqs 12, 15, and 17, respectively. However, as the approximate ground state wave function is improved in quality, K → π 2/2 (see eq 6); that is K → 4.934802. Conclusion The solution of the particle-in-a-box problem is shown to provide a useful quantum chemical paradigm for demonstrating the use of the variational principle. In this investigation, the approximate wave functions for ground and excited states are expanded in terms of a basis of monomials, x r. The secular determinant and associated equations may be solved easily, using Maple or Mathematica, for example, to compare the calculated energies and wave functions for basis set expansions of increasing size. The analysis given here may be readily adapted to form a class exercise for computer-literate undergraduates, and is a useful model system for illustrating the procedural basis for understanding the solution of more advanced electronic structure calculations. The wave functions produced can also be characterized by their even/odd symmetry characteristics under inversion, and the notion of a symmetry projection operator can be introduced. Finally, a different model for the PIB system is introduced, which permits the construction of excited state wave functions from
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suitably scaled and signed segmented functions built from a model ground state wave function (a piecewise wave function). Literature Cited 1. Levine, I. N. Quantum Chemistry; Prentice-Hall: Englewood Cliffs, NJ, 1991. 2. Besalú, E.; Martí, J. J. Chem. Educ. 1998, 75, 105.
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3. Bransden, B. H.; Joachain, C. J. Introduction to Quantum Mechanics; Longman: Harlow, UK, 1989. 4. Winter, R. G. Quantum Physics; Faculty Publishing: Davis, CA, 1986. 5. Dewar, M. J. S. The Molecular Orbital Theory of Organic Chemistry; McGraw-Hill: New York, 1969. 6. Davidson, E. R.; Feller, D. Chem. Rev. 1986, 86, 681.
Journal of Chemical Education • Vol. 77 No. 9 September 2000 • JChemEd.chem.wisc.edu
