Research: Science and Education
Correlation in Simple Systems Carl W. David Department of Chemistry, University of Connecticut, Storrs, Connecticut 06269-3060;
[email protected] Introduction The question of correlation and its place in chemistry looms large when one starts to consider the quantum mechanics of atoms more complicated than the H atom and molecules more complicated than H2+. The question of correlation is rarely if ever actually addressed (1), other than by assigning an energy to it that has been obtained by complicated calculations. It seems appropriate to attempt discussing correlation in a simple example, so that students can at least understand why the subject is fraught with such peril. There exists a tradition of “tweaking” the potential energy functions (2) in the Hamiltonian to obtain “tractability” in the case of helium’s 2-electron wave function, where the explicit Hylleraas-type wave function (3) is known to fail. Thus, a wave function of the (1s2) form exp[Z(r1 + r2)], which is a priori wrong, when adjusted to include explicit r12 dependence, such as exp[Z(r1 + r2) + αr12] or exp[Z(r1 + r2)](1 + αr12 + …), remains just as wrong, no matter what the value of α (Z is the atomic number). Every correction of the form exp[β(r1 + r2) + αr12](γ + δr12 + …) fails to be an eigenfunction also, although the computed energy for such trial wave functions can be brought extraordinarily close to the almost exact (known) value (2.903 724 377 034 119 593 8(50) au [4 ]). Turn the problem around and ask, what potential energy would one need to have a pre-chosen correlated wave function be an eigenfunction of the appropriate Hamiltonian containing that potential energy function? This would explicitly illustrate how, when r12 is embedded in the eigenfunction, the disruption caused is catastrophic. Even excluding correlation, dealing with Coulomb’s law induces sufficient mathematical difficulties that we choose a different tack here.
interest in real atomic and molecular problems. We could equally well have chosen
Ψ = sin
Ψ = A sin
πx 1 πx 2 sin a a
(1)
provided that the domain of the problem was 0 ≤ x1, x2 ≤ a (A is a normalization constant). There are a host of interesting correlated wave functions we could invent (based on eq 1) to see how the potential energy function would have to change to make the correlated wave function an eigenfunction. Consider the following (where we have dropped the normalization constant for clarity):
Ψ = sin
πx 1 πx 2 sin exp α x 1 – x 2 a a 2
2
(2)
where α is an real arbitrary constant. This form emphasizes the dependence on “r12”, which is the “coordinate” of major
682
x1 – x2
2
which would have emphasized the r12 character in a slightly different way, but would have induced difficulties when x1 > x2 and x2 > x1. Any function, even
Ψ = sin
πx 1 πx 2 sin a a
x1 – x2
2
could be a suitable candidate. The idea is only to introduce an explicit functional dependence on r12. Our example emphasizes tractability. The kinetic energy part of the Hamiltonian is
1 ∂ 2 + ∂ 2 = 1 ∇2 2 2 ∂x 12 ∂x 22
(in “au”) so we have, employing elementary calculus:
πx πx ∂ sin 1 sin 2 exp α x 1 – x 2 a a ∂Ψ = 2 ∂x 1 ∂x 1 π cos πx 1 sin πx 2 exp α x – x 2 a a a 2 1 2
2
= + α x1 – x2 Ψ
so ∂2 Ψ = ∂x 12
The Illustration Consider the two-particle one-dimensional system where, in the absence of interaction (and correlation), the ground state wave function might be
πx 1 πx 2 sin exp α a a
πx πx ∂ π cos 1 sin 2 exp α x 1 – x 2 a a a 2 ∂x 1 π a
2
sin
+ α x1 – x2 Ψ
2
πx 1 πx sin 2 exp α x 1 – x 2 a a 2
π cos πx 1 sin πx 2 exp α x – x 2α x 1 – x 2 α a a 2 1 2
2
2
=
+ αΨ +
+ α2 x 1 – x 2 2 Ψ
or
∂2 Ψ = π a ∂x 12
2
+ α + α2 x 1 – x 2
2
Ψ+
πx 2 πx 1 πx 1 sin a π sin exp α x 1 – x 2 2α x 1 – x 2 cos a a πx 1 a 2 sin a
2
or ∂2 Ψ = ∂x 12
π 2 + α + α2 x 1 – x 2 a
2
+ 2α x 1 – x 2
Journal of Chemical Education • Vol. 78 No. 5 May 2001 • JChemEd.chem.wisc.edu
π a
πx 1 a Ψ πx 1 sin a
cos
Research: Science and Education
so, 1 ∇2 Ψ = 2
π 2 – α – α2 x – x 1 2 a
2
πx πx + α x 1 – x 2 π cot 1 – cot 2 a a a
Ψ
constants have been included in the “trial” wave function), one obtains (using the full six-dimensional Hamiltonian)
r ⋅r r ⋅r α – Z α – Z 1 – β α2 + β 2 + + – + αβ 1 12 – 2 12 r1 r2 r 12 r r r 2r 12 2 1 12
which means that a potential energy function of the form
V=
πx πx U = +α2 x 1 – x 2 2 – α x 1 – x 2 π cot 1 – cot 2 + U 0 (3) a a a
as the “potential energy function”, showing clearly the sources of difficulties this kind of trial wave function induces.
where U0 = 0 (the zero is to remind us that we are neglecting explicit interactions between particles, such as those that arise from Coulomb’s law) would allow this wave function “ansatz” to be an eigenfunction of the Hamiltonian. Strange as the apparent potential energy function appears, it is a relatively simple function that depends on the correlated motion of the two particles; that is, it contains “r12” explicitly. It is therefore especially disappointing to notice that although we have achieved our purpose, that is,
Ψ = sin
πx 1 πx sin 2 exp α x 1 – x 2 a a 2
2
is an eigenfunction of the Hamiltonian πx πx = KE + U= 1 ∇2 + α2 x 1 – x 2 2 – α x 1 – x 2 π cot 1 – cot 2 a a a 2
it is the only easily obtainable eigenfunction. If we redo the computation using
Ψ = sin
n 1πx 1 n πx sin 2 2 exp α x 1 – x 2 a a 2
2
hoping for an “easy win” (even if α depended on n1 and n2), we find, to our chagrin, that the apparent potential energy function (different from eq 3) contains n1 and n2, which is, of course, impossible. The search for other eigenfunctions, which surely exist, for the potential energy function given in eq 3 is a Herculean task if in fact it can be done at all. If one attempts the same kind of treatment for helium’s electrons, starting with a trial wave function such as exp[ α(r1 + r 2) + βr12] (where two [nonlinear] variational
Discussion When one considers that the simplest of correlated problems is incredibly difficult, and that the difficulty is apparent using nothing other than elementary differentiation, one comes to an understanding of how rigorous a requirement being an eigenfunction is. This seems enlightening when students are able to do quantum mechanical computations on desktop computers using commercially available software. Clearly, the constant warning that such programs do not report “the truth” is lost on many students, who need to be reminded that even the simplest of problems are beyond our abilities. We learn from this exercise that quantum chemistry, a priori, must be difficult. Chemistry is required to deal with polyelectronic systems, and the solutions of the Schrödinger equation for such cases, which must depend on ri,j, (where i and j label the i th and j th electrons, and the wave function depends on all i–j pairs) is much more difficult than one imagines, approximation schemes not withstanding. That this is well known to students and professionals alike need not deter us from noting what one of the sources of these difficulties is, regardless of current computational practices. Literature Cited 1. Summerfield, J. H.; Beltrame, G. S.; Loeser, J. G. J. Chem. Educ. 1999, 76, 1430–1438. 2. Kestner, N. R.; Sinanoglu, O. J. Chem. Phys. 1962, 128, 2687– 2692. 3. Hylleraas, E. Z. Phys. 1928, 48, 469. 4. Goldman, S. P. Int. Phys. Rev., A 1998, 57, R677–R680.
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