Viewpoint Cite This: J. Phys. Chem. Lett. 2019, 10, 26−29
pubs.acs.org/JPCL
Connection between a Classical Separatrix and Quantum Inflection Point
J. Phys. Chem. Lett. 2019.10:26-29. Downloaded from pubs.acs.org by 37.9.41.162 on 01/03/19. For personal use only.
T
he value of classical trajectories and the Old Quantum Theory (OQT) arises from the potential to assist in understanding quantum behavior, as proven in many applications.1−3 However, few applications involve electronic structure, as expected given the small mass of an electron compared to that of a nucleus, but electrons are not entirely precluded from consideration.4−6 Another reason that electronic behaviors have received little attention is that the OQT is well-known for its failure to yield a bound ground state for the prototype for chemical bonding, the hydrogen molecular cation. Strand and Reinhardt,7 in their seminal paper, showed that the major failure of the OQT stems from its inability to treat tunneling through the potential barrier and the associated resonance between the two connected states. In OQT, the existence of a classical separatrix, strictly dividing molecular and separated atom types of electron motion, signals the frailty. The importance is that away from that separatrix OQT treatments of electrons should be less inaccurate. Here we use OQT to investigate the behavior of the electronic (not the total) energy of H+2 at small internuclear distances where all σ states except s states have a minimum. This minimum is associated with a rapid change in the quantum density along the internuclear axis. The location of a classical separatrix serves as a good predictor of the occurrence of both the inflection point and density change, even at small quantum numbers, thereby providing an interpretation for both. With clamped nuclei,8 this problem is separable in both quantum9 and classical7,10−13 mechanics. The classical treatment of H+2 rests on the existence of three independent constants of the motion, taken here to be the electronic energy Eel, the axial component of the electronic angular momentum M, and a (classical) separation constant G, which are analogous to the three separation constants that permit analytic solution of the Schrödinger equation. The constants appear hereafter in reduced form, ϵ = −
Figure 1. Classically allowed and forbidden (gray) regimes in ϵ, γ space. Each of the dashed lines is a separatrix, dividing the allowed region into the SA, MR, and UA regimes. The trajectory in the lower right inset belongs to the MR regime and the upper left to the UA regime. The fine lines are eigenvalue traces; see the text. Red pluses mark nuclear locations.
The modern version14−17 of the OQT allows only those trajectories for which the classical actions satisfy quantum rules: integer or half-integer multiples of Planck’s constant. The fine lines in Figure 1 are the loci of OQT eigenvalues for the states 1902σ (green), 1909σ (red), and 1916σ (blue). The existence of three constants of motion means that all trajectories are regular; irregular trajectories do not occur. The upper inset in Figure 1 shows a classical trajectory characteristic of all trajectories (except periodic ones) in the UA regime, and the lower inset shows a characteristic classical trajectory for the MR regime. In these insets, the nuclei are located at x = 0, z = ±1, marked by plus symbols. The regime names are dictated not only by the appearance of the trajectory but by the fact that, for any given state, the value of the internuclear separation R increases with the value of the constant of the motion γ For convenience, the separatrix at γ = 0 is called the outer separatrix; the other is the inner separatrix. The outer separatrix, which divides the SA and MR regimes, has been extensively treated18 and is the separatrix that marks the tunneling. Errors in the OQT approach are largest near this separatrix because classical action is discontinuous there. This in turn means that some OQT bond energies and bond distances display large errors as low-lying (metastable) bound states tend to occur near the outer separatrix. Classical trajectories near the outer separatrix generate a classical density that is large along the axis perpendicular to the bond axis, in agreement with the quantum idea that covalent bonding is associated with a buildup of electron density between the nuclei.
( R2 )Eel /Zλ and γ = ( R2 )G/Zλ, where R
denotes the internuclear distance and Zλ the average atomic number. Note that the total energy (sometimes called the Born−Oppenheimer energy) includes the nuclear repulsion, Et = Eel + ZAZB/R, in atomic units, where ZA(B) denotes the atomic number of nucleus A(B). Here we consider only σ (M = 0) states, for which the classical motion is planar. For convenience, we take this to be the x−z plane, with the origin at the geometric center of the diatomic ion and z-axis coincident with the internuclear axis. Applied to σ states, the OQT generates two separatracies (dashed lines), which divide allowed classical motion into three regions, termed the separated atom (SA), molecular (MR), and united atom (UA) regimes, as shown in Figure 1. In this figure, the gray areas color those regions of the (ϵ, γ) space in which classical motion is forbidden; each of the regions with allowed motion, in yellow, is labeled. © 2019 American Chemical Society
Received: October 19, 2018 Accepted: December 17, 2018 Published: January 3, 2019 26
DOI: 10.1021/acs.jpclett.8b03201 J. Phys. Chem. Lett. 2019, 10, 26−29
Viewpoint
The Journal of Physical Chemistry Letters
From the facts that the energies agree and the inflection point in the energy derivative in the OQT approach is directly connected with the position of the separatrix, the relationship between the quantum and OQT features is more than a casual empirical association. The states with the lowest quantum numbers to which these results might apply are 2pσ, 3pσ, and 3dσ. The accuracy of OQT electronic energies of these states can most charitably be described as poor. Nonetheless, the location at which the OQT electronic energy crosses the separatrix still describes well the position of the inflection point in the electronic energy, as shown by the results in Figure 3. Therefore, the agreement
The inner separatrix divides trajectories of the UA and MR regimes, Figure 1. Classical actions are continuous across this boundary, although the derivative of one action is not. At large quantum numbers, the agreement between quantum and classical electronic energies in these two regimes is excellent in the UA regime and in the MR regime until close to the outer separatrix. The lines in Figure 2 show derivatives of the
Figure 2. Solid lines are derivatives of the quantum elecronic energy as a function of the internuclear distance R for states 19S σ with S = 10−18 of the n = 19 manifold. The arrows mark the OQT UA/MR boundary for these states. Figure 3. As per the Figure 2 caption but for the 2p, 3p, and 3d σ states of H+2 and dEel/dR for the 2p state divided by 5 for convenience.
quantum electronic energy in the UA and MR regimes for a number of states from the n = 19 united atom manifold: 19S σ, S = 10,11, ..., 18. In these computations, the value of a derivative of the electronic energy is approximated simply by the difference, dEQel /dR ≈ ΔEQel ΔR, the superscripted Q indicating that these are quantum values. The outermost minimum belongs to the 1918σ state, with the minima for successive states occurring in order of decreasing values of the quantum number S at smaller values of R. The arrows in the figure mark the location of the inner OQT separatrix for the states specified by the displayed value of S , with the order agreeing with the order of the quantum minima. The quantum electronic energies show a minimum, below the energy of the united atom, before increasing toward the separated atom value; therefore, a characteristic of the derivative curves is an inflection point, at which the electronic energy shifts from concave to convex. The agreement in location between the classical separatrix and the quantum electronic energy inflection point for these and all states in the n = 19 manifold except the s state (S = 0) provides a classical rationalization of the quantum behavior changing from decreasing to increasing values. As mentioned, the observation does not apply to the state with S = 0, the s state. The electronic energy of an nsσ quantum state does not display a minimum but rather decreases smoothly from the separated atom to the united atom limit. In contrast, the OQT result for the s state does cross the separatrix and so does display a minimum before rising to the UA limit. For states with a large principle quantum number, such as n = 19, the separatrix crossing occurs at very small values of the internuclear separation, and the OQT results agree well with quantum energies at larger values of R. To this extent OQT results for s states are not in good agreement with quantum results.
between OQT and quantum results for this type of relationship is not limited to highly excited states, nor, by the way, do very high quantum numbers ensure complete agreement between OQT and quantum results because regardless of quantum numbers the discontinuity in the classical actions at the outer separation remains to guarantee a disagreement in energies19,20 in its vicinity. Classically, crossing the inner separatrix means moving from one regime to another, UA ↔ MR, with a corresponding change in the character of the classical trajectories associated with the two regimes, shown in Figure 1. In the UA regime (small values of the internuclear separation), the classical trajectories avoid the internuclear axis due to the existence of a classical turning point in the UA regime. As a consequence, the classical density along the axis vanishes. While one would not expect the quantum density to vanish, if the classical behavior has a manifestation in the quantum results, one would expect the quantum density along the axis to decrease significantly. Using the density at the midpoint of the axis as a proxy for the axial density, this is confirmed by the data in Figure 4a. Each line is a quantum density at the midpoint, Ψ2(x = 0, z = 0; R), from accurate quantum wave functions, for the states 19S σ, S = 10, 12, 14, 16, and 18. The line for the state with S = 10 rises at the smallest internuclear distance, and those for higher values of S appear successively at larger internuclear distances. (States with odd values of S have a node at z = 0, disqualifying them for tracking a density change at the mipoint.) In the figure, each dot indicates the position of the inner separatrix for the corresponding state of the manifold. The crosses mark the location of the outer separatrix, which indicates the classical shift from molecular to separated atom states; in quantum 27
DOI: 10.1021/acs.jpclett.8b03201 J. Phys. Chem. Lett. 2019, 10, 26−29
Viewpoint
The Journal of Physical Chemistry Letters
Figure 4. Quantum density at the midpoint of the internuclear axis as a function of the internuclear separation R, in atomic units. The dots mark the position of classical inner separatrix for each of these states, and the crosses mark the position of the outer separatrix. (a) States 19S σg, where S = 10, 12, 14, 16, and 18. (b) 3dσg state.
show that the OQT provides the appropriate framework, even beyond the long-known concept of the separation of molecular from separated atom behavior. In particular, the influence of the inner separatrix on the quantum wave function seems to have been overlooked. The classical interpretation assists not only for highly excited states, where it may be expected, but also for low-lying states, although the special character of s states excludes them (and therefore the ground state) for these particular connections. Interpretation of the decrease of the axial density decrease moving toward smaller internuclear distances is facilitated by the same classical considerations, and for states with large quantum numbers, a good quantitative estimate may be obtained as well. The exclusion of s states from the connection is also interesting. The OQT s states results do cross the inner separatrix and show a minimum in the electronic energy, approaching the UA energy limit from below; quantum s states do not show this minimum in the electronic energy. The distinction between the quantum and classical behaviors can be traced to the behavior of the quantum separation constant, the third eigenvalue of the quantum molecule: in contrast to the OQT eigenvalue, the quantum version never becomes negative9 for s states. The classical motion of the electron in a hydrogen molecular cation with clamped nuclei is entirely regular; the motion of two electrons in molecular hydrogen, even with clamped nuclei, is not. The applicability of these concepts to molecules with more electrons is the subject of future research.
mechanics, the density decreases as the molecular ion splits into separated atom states in a less abrupt fashion. Of the three lowest-lying non-s states, only the 3dσ state is gerade; the corresponding graph for that state, Figure 4b, again shows the agreement between the OQT and quantum results, even with much lower quantum numbers. We do know that a number of states in the n = 19 manifold have a minimum in total energy in the clamped nucleus approximation according to both the quantum and OQT approaches, as listed in Table 1. These occur in the classical Table 1. Bond Lengths (au) for the n = 19 Manifold of H+2 S
18
17
16
15
14
Q OQT
364 365
359 358
350 352
352 348
342 347
13
12
349
343 348
MR regime, at an internuclear separation at which the electronic energy falls well below the united atom limit.21,22 This places these metastable states in a slightly different category than that of the ground state23 as the electronic energy of the ground state never falls below that of the united atom limit. It is probably no surprise that the OQT boundaries describe the location of the quantum density changes well for highly excited states. Less well anticipated would be the accuracy for low-lying states. The situation at the outer separatrix is different due in large part to the discontinuous nature of a classical action there. To note the obvious, for gerade states, the quantum density at the axial midpoint has a maximum there at smaller values of the internuclear separation R, including at the equilibrium bond length, and a node (or very close to a node) when the internuclear separation becomes sufficiently large. This imposes two requirements: at some value of R (i.e., γ), (i) the density becomes as small as may be specified, and (ii) the density has at least one stationary point of inflection. There have been numerous papers on the classical framework for quantum behavior, including a number of examples in which the classical approach provides an interpretation of quantum and experimental results. Here we
Stephen K. Knudson*
■
Department of Chemistry, College of William and Mary, Williamsburg, Virginia 23187, United States
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Stephen K. Knudson: 0000-0002-1607-2637 Notes
The author declares no competing financial interest. 28
DOI: 10.1021/acs.jpclett.8b03201 J. Phys. Chem. Lett. 2019, 10, 26−29
Viewpoint
The Journal of Physical Chemistry Letters
■
REFERENCES
(1) Noid, D.; Koszykowski, M.; Marcus, R. Quasiperiodic and Stochastic Behavior in Molecules. Annu. Rev. Phys. Chem. 1981, 32, 267−309. (2) Child, M. Semiclassical Mechanics with Molecular Applications; Oxford Science Publications: Oxford, 1991. (3) Johnson, B. R.; Kinsey, J. L. Dynamical interpretation of the Hartley absorption oscillations in ozone. Phys. Rev. Lett. 1989, 62, 1607−1610. (4) Hopkinson, B.; Kwee, E.; Knudson, S. A connection between quantum critical points and classical separatracies of electronic states. J. Chem. Phys. 2002, 117, 5660−5669. (5) Du, M. L.; Delos, J. Effect of closed classical orbits on quantum spectra: Ionization of atoms in a magnetic field. I. Physical picture and calculations. Phys. Rev. A: At., Mol., Opt. Phys. 1988, 38, 1896−1912. (6) Du, M.; Delos, J. Effect of closed classical orbits on quantum spectra: Ionization of atoms in a magnetic field. II. Derivation of formulas. Phys. Rev. A: At., Mol., Opt. Phys. 1988, 38, 1913−1930. (7) Strand, M. P.; Reinhardt, W. P. Semiclassical quantization of the low lying electronic states of H2+. J. Chem. Phys. 1979, 70, 3812− 3827. (8) Sohlberg, K.; Tuzun, R. E.; Sumpter, B. G.; Noid, D. W. Full Three-Body Primitive Semiclassical Treatment of H+2 . Phys. Rev. A: At., Mol., Opt. Phys. 1998, 57, 906−912. (9) Bates, D. R.; Ledsham, K.; Stewart, A. Wave functions of the hydrogen molecular ion. Philos. Trans. R. Soc., A 1953, 246, 215. (10) Pearson, R. G. The Hydrogen Molecule Ion in the Old Quantum Theory. Theor. Chim. Acta 1983, 64, 127−136. (11) Athavan, N.; Froman, P.; Froman, N.; Lakshmanan, M. Quantal Two-Center Coulomb problem treated by means of the phase-integral method. I. General Theory. J. Math. Phys. 2001, 42, 5051−5076. (12) Athavan, N.; Lakshmanan, M.; Froman, N. Quantal Two-center Coulomb problem treated by means of the phase-integral method. II. quantization conditions in the symmetric case expressed in terms of complete elliptic integrals. numerical illustration. J. Math. Phys. 2001, 42, 5077−5095. (13) Athavan, N.; Lakshmanan, M.; Froman, N. Quantal two-center Coulomb problem treated by means of the phase-integral method. III. quantization conditions in the general case expressed in terms of complete elliptic integrals. numerical illustration. J. Math. Phys. 2001, 42, 5096−5115. (14) Einstein, A. Zum Quantensatz von Sommerfeld und Epstein. Verh. Dtsch. Phys. Ges. 1917, 19, 82−92. (15) Brillouin, L. La mécanique ondulatoire de Schrödinger; une méthode générale de résolution par approximations successives. C. R. Acad. Sci. 1926, 183, 24−26. (16) Keller, J. B. Corrected Bohr-Sommerfeld Quantum Conditions for Nonseparable Systems. Ann. Phys. (Amsterdam, Neth.) 1958, 4, 180−188. (17) Keller, J. B. Semiclassical Mechanics. SIAM Rev. 1985, 27, 485− 504. (18) Waalkens, H.; Dullin, H. R.; Richter, P. H. The problem of two fixed centers: bifurcations, actions, monodromy. Phys. D 2004, 196, 265−310. (19) Knudson, S. K.; Palmer, I. C. Semiclassical electronic eigenvalues for charge asymmetric one-electron diatomic molecules: general method and sigma states. Chem. Phys. 1997, 224, 1−18. (20) Waalkens, H.; Junge, A.; Dullin, H. R. Quantum Monodromy in Simple Molecules. J. Phys. A: Math. Gen. 2003, 36, L307−L314. (21) Knudson, S. K. Bound Primitive Semiclassical Diatomic Electronic States. Chem. Phys. 2002, 283, 389−399. (22) Knudson, S. K. The Old Quantum Theory for H+2 : Some Chemical Implications. J. Chem. Educ. 2006, 83, 464−472. (23) Bacskay, G. B.; Nordholm, S.; Ruedenberg, K. The Virial Theorem and Covalent Bonding. J. Phys. Chem. A 2018, 122, 7880− 7893.
29
DOI: 10.1021/acs.jpclett.8b03201 J. Phys. Chem. Lett. 2019, 10, 26−29
