J . Phys. Chem. 1993, 97, 2464-2466
2464
Dimensional Scaling for Hz+ without the Born-Oppenheimer Approximation Carol A. Traynor and David Z. Goodson'*+ Department of Chemistry, Harvard University, Cambridge, Massachusetts 02138 Received: September 24, 1992; In Final Form: November 18, 1992
The standard dimensional continuation of the Schradinger equation, in which the kinetic energy is generalized to arbitrary D, while the potential energy is left unchanged, leads to dissociation of H2+ in the limit of large D if the nuclei are allowed to move freely. In general, the definition chosen for the Pdimensional Schrddinger equation is arbitrary as long as the correct equation results at D = 3. We propose a mild dimension dependence for the potential of the internuclear repulsion. This yields a stable and physically reasonable chemical bond in the limit D = and can therefore be used as the starting point for a dimensional perturbation theory without the Born-Oppenheimer approximation.
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Herschbach's pioneering work in applying dimensional scaling to atomic and molecular has amply demonstrated some unique advantages of this new technique. Formally, the method is a perturbation theory in terms of the parameter 1/D, where D is the dimensionality of space treated as a variable parameter. In the zeroth-order approximation, the limit D - =, the electrons and nuclei arrange themselves in a rigid configuration. (Thus, one derives a Lewis structure3 directly from the SchrWnger equation.) Within first-order perturbation theory, the particles vibrate harmonicallyabout their infinite-D positions, with effective masses proportional to D2. This model allows us to employ our intuitive notions of positions and trajectories in a manner that is fully consistent with quantum mechanics. Thevibrating Lewisstructureis a model for quantumchemistry that does not depend on the Hartree-Fock approximation. The large-dimension model gives a clear and direct description of electron dynamics. Applied to the He atom, the prototype for the study of electron correlation, it has provided explanation^^.^ for subtle correlation effects that are very difficult to understand from an independent-electron approach. Furthermore, even the lowest orders of the perturbation theory can yield greater numerical accuracy for total atomic energies than the HartreeFock approximation,Is6and the inclusion of higher-order terms leads to extremely accurate ab initio results.' Another standard simplification that is not needed for the large-D analysis is the Born-Oppenheimer approximation. Although the magnitude of the errors introduced by this approximation is usually smaller than that from the Hartree-Fock approximation, there are many important chemical systems,such as transition states and van der Waals molecules, in which the coupling between nuclear and electronic motion ought to be significant,*and the corrections are very difficult to calculate by using a basis of atomic orbitals. In the large-dimension limit the electrons and the nuclei all become infinitely massive, so there is no particular advantage to imposing a separability assumption. Within first order in 1 / D the system will, in principle undergo harmonic oscillations, which can be analyzed in terms of normal modes representing collective motions of all of the particles. However, there is an obstacle that has discouraged the use of dimensionalperturbation theory without the Born-Oppenheimer approximation: systems, such as H2+,that are only weakly bound at D = 3 tend to dissociate in the limit D m.9 This would appear to present a major limitation, since molecules with weak bonds are precisely the systems for which correctionsto the BornOppenheimer approximation are most important. It appears that the standard dimensional continuation of the Schradinger
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' Present address: Department of Chemistry, Brown University, Providence, RI 02912.
equation,'O in which the kinetic energy is generalized to D dimensions, while the potential energy is left unchanged, slightly overemphasizesthe effects of particle correlations. In this paper we propose an alternative dimensional-continuation scheme appropriate for weakly bound systems and thereby obtain a stable large-dimension limit for H2+without the Born-Oppenheimer approxmation. The inspiration behind our approach is the discovery by Frantz and Herschbachl' that a new dimensional scaling procedure of Lbpez-Cabrera et a1.'2 can lead to an adiabatic potential curve for Hz+with a stable minimum. Dimensional scaling is the key to obtaining useful results from dimensional continuation. The effects of varying D can be quite severe. It was Herschbach's realization' that the adverse effects can largely be scaled away, through an appropriate choice of dimension-dependent distance units, that made dimensional perturbation theory practicable. Recently, Upez-Cabrera et a1.'2 have devised an elegantprocedure that avoids distance scalings, instead introducing dimensiondependent scale factors directly into the hamiltonian. Upon transformation to an internal coordinate system, the Pdimensional hamiltonian for the ground state of H2+takes the form13
H=T+U+V where Vis the usual potential energy function, and the kinetic energy has been expressed as the sum of a centrifugal potential, U,and an operator Tthat includes partial derivativeswith respect to the internal coordinates. To leading order in 1/D, the centrifugal term is'3
with the coordinates p , I, and R as shown in Figure 1 and the proton mass m, a 1836 in atomic units. p and z describe the position of the electron, while R is the internuclear distance. T becomes insignificant in the limit of large D, leaving an effective potential U + V that determines the Lewis structure.' In the limit D the centrifugal potential Ubecomes infinite due to the factor of D2,and one can showI4that Vdiverges as ( D - 1)-1 in the limit D 1. To counteract this behavior, Lbpez-Cabrera et al. introduce dimension-dependentscale factors a K and avinto the effective potential,
--
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(3)
Within the Born-Oppenheimer approximation, m, in q 2 is set
0022-3654/93/2Q97-2464So4.QQ/Q Q 1993 American Chemical Society
Dimensional Scaling for H2+
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2465
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2
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Figure 1. Internal-coordinate system (p,z,R) for H2+. Figure 3. Large-dimension effective potential V,/Xp,z,R)with internuclear repulsion modified according to eq 8, shown as a function of z and R in the neighborhood of the minimum of Verf.
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E
Figure 2. Large-dimension effective potential, Vr/Xp,z,R)from eq 7, for H2+as a function of I and R , in atomic units. p is set at 1.3778, which
-0
corresponds to the minimum of the effective potential within the BornOppcnheimer approximation. I I
d
E: 0
0.5[
-0.0
to infinity and Vis given by
+
+
V(p,z) = -[p2 ( z - R / 2 ) ] - ' / 2- [ p 2 ( z + R / 2 ) ] - ' / 2 ( 5 ) with R treated as a parameter. Then the adiabatic internuclear potential is
0.3 0.4 0.5 0.6 0.7 0.8 0.9
t Figure 4. Variation with the nuclear-repulsion scale factor f of the coordinates p, I , and R corresponding to minima of the large-dimension effective potential.
nuclear potential of the form where pm(R)and z,(R) are the values of p and z that correspond to the minimum of Ve//for a given value of R. This internuclear potential has a stable and physically reasonable minimum, which would not have been the case if a straightforward distance scaling had been used.]' The scaling procedure is less successful if the Born-Oppenheimer approximation is not invoked. Then the internuclear potential is given by the full effective potential function, Ve,&p,z,R) aKU(p,z,R)+ ad-[p2 + ( z - R/2)]-'l2[p2 + ( z + R/2)]-'l2 1/Rj (7)
+
Here Ve//is a nonadiabatic potential function depending on three nonseparable degrees of freedom. Figure 2 shows that this function does not have a stable minimum for R in the vicinity of the minimum of the adiabatic effective potential. There is a minimum for R z 20 atomic units, but this is ten times the most probable value of R at D = 3.15 The source of this discrepancy is the fact that the potential of the internuclear repulsion, 1 / R , is multiplied by the scale factor aY in eq 6 but was not multiplied by this factor in the BornOppenheimer case. In eq 7 the coordinate R is a dynamical variable, so its scaling is justified in order to prevent divergence at D = 1 . Within the Born-Oppenheimer approximation, the term 1 / R is a parameter, the dimensiondependence of which can be arbitrarily defined. In effect, the analysis of Frantz and Herschbach" implicitly assumed a dimension-dependent inter-
The reduction of the repulsive force between the protons by a factor of */3 in the large-D limit is sufficient to ensure a chemical bond. This is true even without the Born-Oppenheimer approximation. Substituting eq 8 for 1/R in eq 7 does indeed lead to a stable minimum of Ve&,z,R) with respect to all three coordinates. Figure 3 shows Vegas a function of z and R in the neighborhood of the minimum. There does not appear to be any a priori justification for choosing the particular form for Vnvcl given by eq 8, and, in fact, one might object to it on the grounds that it becomes infinite at D = 1. We propose instead
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Equation 9 introduces an arbitrary scale factor {in the limit D 0 3 , reduces to 1/R at D = 3, and is well behaved at D = 1 . We will only consider here the zeroth-order approximation, ignoring terms proportional to 1/D. In Figure 4, we show the f dependence of the location of the minimum of V,/Xp,z,R).For sufficiently small { there exists a global minimum in Veri for which z = 0, corresponding to a symmetric configuration, with the electron equidistant from the protons. As f increases, this minumum becomes increasingly shallow, and then for { in the neighborhoodof 0.75 there occurs a symmetry-breakingtransition above which there is a global minima corresponding to config-
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2466 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 -0.5684 -0.5686 -0.5688
5 -0.5690
>
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Traynor and Goodson minimum of Vegis symmetric, in which case it is only necessary to consider classical motions of the parti~1es.l~In the doublewell case one must also include tunneling effects, but this does appear to be feasible.I* The choice of t would probably affect the dimensional singularity structure of the problem,7J7 which could have a strong effect on the convergence properties of the expansion. Considerations of this kind might provide criteria for choosing an optimal value for [. One can expect better results from a perturbation theory if the zeroth-order approximation is made to more accurately model the exact solution. We suggest in general that dimensional continuations of the Schradinger equation be defined so that the large-dimension limit resembles the desired D = 3 solution as closely as possible.
c '
-0.5696 -0.4 -0.3 -0.2 -0.1
= 0.8
0.0
0.1
0.2
0.3
0.4
z
Acknowledgment. We dedicate this paper to Dudley Herschbach on the occasion of his 60th birthday. Support for this work was provided by the Office of Naval Research, Grant N0001490-5-4025.
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References and Notes (1) Herschbach, D. R. J . Chem. Phys. 1986,84,838. (2) For an excellent review, see: Herschbach, D. R. In Dimensional
-0.6772l I I I -0.4 -0.3 -0.2 -0.1
I
0.0
I
0.1
I
0.2
I
0.3
I
0.4
z
Figure 5. Large-dimension effective potential Ve,jjp,z,R)with internuclear repulsion modified according to eq 9, shown as a function of z in the neighborhood of its minima, for two different values of the nuclearrepulsion scale factor {.
urations with the electron closer to one proton than to the other, as illustrated by Figure 5 . (A similar transition occurs in Hz+ within the Born-Oppenheimer approximation as R is increased" and in the two-electron atom16 as the nuclear charge is decreased below 1.23.) Further increase of {leads to rapid increase R, and Iz,~, representing dissociationof the molecule. At D = 3 the most probable value of R isis approximately 2, which at large D would result from {very close to the symmetry-breaking point. Since Vnue, is constrained to assume its correct form at D = 3, any choice of [ that gives a stable large-D minimum should in principle yield the same results for all expectation values once higher-order terms in the 1/D expansion are included. The computation of the higher-order terms will be simpler if the
Scaling in Chemical Physics; Herschbach, D.R., Avery, J., Goscinski, O., Eds.; Kluwer: Dordrecht, 1992; Chapter 1. (3) Lewis, G . N. J . Am. Chem. Soc. 1916, 38, 762. (4) Goodson,D. 2.; Herschbach, D. R. J. Chem. Phys. 1987,86,4997. Avery, J.; Goodson, D. 2.; Herschbach. D.R.Int. J . Quantum Chem. 1991, 39, 657. (5) Herschbach, D. R.;Loeser, J. G.; Watson, D. K. Z . Phys. D 1988, 10, 195. (6) Loeser, J. G.; Herschbach, D. R. J. Phys. Chem. 1985, 89, 3444. Loeser, J. G.J. Chem. Phys. 1987, 86, 5635. (7) Goodson.D. Z.; Herschbach, D. R. Phys. Rev. Letr. 1987,58.1628. Goodson,D. Z.; L6pe.z-Cabrera, M.; Herschbach, D. R.;Morgan, J. D., I11 J . Chem. Phys. 1992, 97, 8481. (8) Monkhorst, H. J. Phys. Rev. A 1987, 36, 1544. (9) van der Merwe, P.du T. Phys. Rev. A 1987,36, 3446. (10) Herrick, D. R.;Stillinger, F. H. Phys. Rev. A 1975, I ! , 42. (11) Frantz, D.D.; Herschbach, D. R . Chem. Phys. 1988, 126, 59. (12) L6pr.z-Cabrera, M.; Tan, A.; Loeser, J. G. J . Phys. Chem., this issue. (13) Avery, J.; Goodson, D. Z.; Herschbach, D. R. Theor. Chim. Acra 1991, 81, 1. (14) Doren, D. J.; Herschbach, D. R. Phys. Reu. A 1986, 34, 2654. (15) Bishop, D. M.; Cheung, L. P. Adv. Quantum Chem. 1980,12, 1. (16) Doren, D. J.; Herschbach, D. R. J. Phys. Chem. 1988, 92, 1816. (17) L6pez-Cabrera. M.; Goodson. D. Z.; Herschbach, D. R.;Morgan, J. D., 111 Phys. Rev. Letr. 1992. 68, 1992. (18) Kais, S.;Morgan, J. D., 111; Herschbach, D. R. J. Chem. Phys. 1991, 95, 9028.
