Critical binding and electron scattering by symmetric-top polar molecules W. R. Garrett Citation: The Journal of Chemical Physics 141, 164318 (2014); doi: 10.1063/1.4898730 View online: http://dx.doi.org/10.1063/1.4898730 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/16?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Low-energy photoelectron imaging spectroscopy of nitromethane anions: Electron affinity, vibrational features, anisotropies, and the dipole-bound state J. Chem. Phys. 130, 074307 (2009); 10.1063/1.3076892 Nonlinear harmonic components of the electric polarization of symmetric-top molecules J. Chem. Phys. 117, 1750 (2002); 10.1063/1.1488578 Structures and electron affinities of indole–(water) N clusters J. Chem. Phys. 112, 3726 (2000); 10.1063/1.480938 Electron binding to valence and multipole states of molecules: Nitrobenzene, para- and meta-dinitrobenzenes J. Chem. Phys. 111, 4569 (1999); 10.1063/1.479218 Dissociative electron attachment in cyclopentanone, γ-butyrolactone, ethylene carbonate, and ethylene carbonate- d 4 : Role of dipole-bound resonances J. Chem. Phys. 110, 11376 (1999); 10.1063/1.479078
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THE JOURNAL OF CHEMICAL PHYSICS 141, 164318 (2014)
Critical binding and electron scattering by symmetric-top polar molecules W. R. Garretta) Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA
(Received 15 August 2014; accepted 8 October 2014; published online 28 October 2014) Quantum treatments of electron interactions with polar symmetric-top rotor molecules show features not present in the treatment of the linear-polar-rotor model. For symmetric tops possessing non-zero angular momentum about the symmetry axis, a new critical dipole can be defined that guarantees an infinite set of dipole-bound states independent of the values of the components of the inertial tensor. Additionally, for this same class, the scattering cross section diverges for all nonzero values of dipole moments and inertial moments, similar to solutions for the fixed linear dipole. Additional predictions are presented for electron affinities and rotational resonances of these systems. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4898730] I. INTRODUCTION
The detailed features of the interaction of a charged particle with a long-range electric dipole field have been the subjects of theoretical and experimental interest over many years. The quantum mechanics of the problem involves features dictated by the long range nature of the potential function that describes the interaction. To be specific, consider the electron as the charged particle. Though the description of electron interactions through the potential created by fixed dipolar charges is non-central and thus more complicated than the long range Coulomb potential, the quantum features of the two problems have important things in common. Both potentials support an infinite number of electronic bound states, but for the dipolar field, characterized by dipole moment μ, the bound states exist only for moments greater than a critical value μc = 0.639ea0 (atomic units, with e the electronic charge and a0 the Bohr radius). For dipole moments smaller than this critical value no bound states exist.1 The number of bound states is independent of the dipole length (positivenegative charge separation) or any other short range forces, thus it is relatively easy to show that all molecular orbital or other quantum chemistry calculations of polar anion spectra in the Born-Oppenheimer approximation should yield an infinite number of bound exited states in the limit of infinite numerical accuracy.2 For charged particle scattering from a fixed dipole target the continuum states yield an infinite total scattering cross section3 for all nonzero values of dipole moments, similar to the behavior in the Coulomb scattering problem. Again, this divergence is produced by the long range dipolar interaction for all finite moments, μ, and independent of short range forces. The electron-binding-feature of a fixed electric dipole field does not guarantee that any strongly polar molecule with μ > μc can support a stable negative ion. At zero electron binding the rotational energy is greater than the binding energy, hence electron-rotational coupling plays a critical role in the quantum mechanical dynamics. Garrett showed that ina) E-mail: [email protected]
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clusion of rotational degrees of freedom in the quantum treatment of a polar negative ion caused significant changes in the critical binding and scattering features of the problem.4, 5 For a linear dipole rotor, critical moments still exist but the values are larger than that for fixed charges and critical moments are dependent on the moment of inertia, the short range components of the electron–molecule potential, and the rotational state of a linear rotor molecule. The number of excited dipole-bound states is now finite, not infinite, where for small to modest sized systems, the truncation leads to only one or at most a small number of electronic states.6, 7 The total electron scattering cross section that was infinite for all non-zero μ in the fixed-nucleus approximation,3 remains finite when rotational degrees of freedom are treated properly. The quantum mechanics of electric dipole-bound systems has an extensive history which will not be reviewed here. But new properties of such systems will be shown to belong to polar symmetric tops possessing non-zero internal rotational quantum numbers, including the existence of a new critical dipole moment. Indeed such systems can have properties similar to those of the fixed linear dipolar problem where infinite numbers of bound states and infinite scattering cross sections appear.
II. CRITICAL BINDING FOR DIPOLAR SYMMETRIC-TOP ANIONS
Surprisingly, the general problem of critical electron binding to a dipolar symmetric top (ST) has not been fully treated. That is, neither ground nor excited bound states have been fully examined. Near-threshold photo-detachment from a dipolar symmetric top has been studied by Engelking8 in an adiabatic approximation and extended by Herrick and Engelking,9 where anomalous threshold behavior was noted. Clary10 has examined shape resonances in dipolar ST molecules, in a similar approximation and with a pseudopotential similar to that of Garrett.11 Green12 has described the general formalism of molecular scattering by a symmetrictop scatterer, which would also apply to the symmetric top anion problem, and Chang and Fano13 provide some useful
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relationships for the electron-symmetric-top problem. However, theoretical solutions for electron binding to polar ST molecules or low energy photo-detachment profiles of such systems are available only through adiabatic treatments.10 The polar ST anion problem is of fundamental interest because of some novel features that arise in the underlying physics. An ab initio treatment of an electron very weakly bound to a symmetric top can be described in a manner similar to that presented by Chang and Fano.13 The model Hamiltonian for the problem of an electron with coordinate r in the field of a symmetric top molecule is written as H = Hrot − 1/22r + V (r, β),
(1)
where the rotational energy operator Hrot (αβγ ) and potential energy, V (r , β), contain Eulerian angles αβγ of the symmetric top. The model problem can be described by a wave function for a single electron moving in the field of the polar symmetric top rotor with principal moments of inertia Iaa , Ibb , and Icc. The parent molecule has total rotational angular momentum J, J2 = J (J + 1), and with projection Jz = mJ along lab coordinate z, where J is the sum of rotational angular momentum j perpendicular to the symmetry axis and internal angular momentum K about the symmetry axis of the rotor (see Fig. 1). The rotational wave functions for the rotor are ST eigenfunctions14 that are defined by J (J + 1) − K 2 K2 Hrot − JmKj (αβγ ) = 0. (2) + 2Iaa 2Icc The moment of inertia about the symmetry axis is Icc (the other two inertial moments are each equal to Iaa ). The dipole ˆ moment μ lies along the symmetry axis with unit vector R, ˆ so J · R = K. The single valence electron has angular momentum l where l2 = l(l + 1) and lz = ml , having r, ϑ, ϕ coordinates in the lab frame with origin at the center of mass. This is coupled to the angular momentum of the ST rotor
by the non-central interaction potential with asymptotic form ˆ 2 . The total angular momentum of → −μ(ˆr · R)/r V (r , R) = J + l with L 2 = L(L + 1) and Lz the anion system is L = M both conserved. For present considerations it can simply be noted that a system of equations for the ST anion, or for electron scattering, can be obtained by expanding the total wave function in suitably coupled angular and radial functions where L and M are diagonal. With this procedure, which will not be repeated here, the Shrödinger equation for the anion12 leads a set of coupled equations for radial functions FjLK l (r) for each L, K. These close-coupling (CC) sets have the form:
d2 l(l + 1) − + 2(E − ε ) FjLK l (r) jK dr 2 r2 LK VjLK =2 lj l (r)Fj l (r).
The potential function matrix element VjLK lj l (r) has been evaluated by Green12 and by Herrick and Engelking.9 The coupling between electron and dipolar top depends only on the orientation of the electron relative to the rotor, thus the matrix elements in the close coupling equations (6) are independent of the quantum number M. Now consider critical binding of an electron in the field of a dipolar symmetric top, including critical binding to rotationally excited states. On the basis of results for the linear rotor with finite inertial moment, rotationally excited states will exhibit binding with critical moments somewhat larger than that for the ground state. These moments converge to that for the fixed dipole in the limit of infinite inertial moments.4, 15 For the ST the dipole moment lies along the symmetry axis, ˆ and the angular momentum that is, along the unit vector K, and their sum J all precess about the laboratory vectors j, K, Z axis. With the addition of the electron of angular momen = J + l with projection quantum tum l the system total is L number M. For the polar ST top rotor the dipole moment has an orientation with respect to the axis of quantization when K > 0 (Fig. 1). Thus from a distance in lab coordinates the field of the polar object appears on temporal average as that due to a permanent dipole with an effective dipole moment whose value is the average of μcos (θ ) where θ is the angle between the symmetry axis and the laboratory Z axis. This average (the orientation angle) has a value cos (θ ) = KmJ /J(J + 1).14 Thus for a ST top with dipole moment μ, the time-averaged long range field appears as that due to fixed dipole with dipole moment μ¯ = μ
FIG. 1. Angular momentum components of a symmetric top rotor in laboratory coordinates. Total angular momentum J has projection K about the symmetry axis and j perpendicular to this axis. Precession occurs about the laboratory z axis with projection quantum numbers M, mk , and mj , respectively.
(3)
j l
KmJ . J (J + 1)
(4)
For K = 0 the average long range field vanishes as for the linear rotor, but for K > 0 a fraction of the moment is oriented on average giving the appearance of a “fixed” dipolar component in the long range potential. In the critical binding problem this feature allows one to carry out an averaging when analyzing dipole-bound excited states that are even more “remote” than the dipole-bound ground state. This averaging is not possible for linear or K = 0 ST rotors.
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FIG. 2. Vector model of the rotating symmetric top with angular momenta K about the symmetry axis and angular momentumj perpendicular to this axis coupled to give total angular momentum J.
The analytical characteristics of the CC equations can be rather tedious. However, the problems of interest here can be adequately described while avoiding some of the complexity of the set defined in Eq. (3). Indeed as a result of special characteristics of the electron-dipole critical binding problem, the features of interest here can be established without having to solve the full close-coupling equations. Specifically these features can be established by consideration of a vector model in place of the more detailed close coupling description. In this vein the symmetric top rotor is depicted in the vector diagram of Fig. 2. Rotations perpendicular to the symmetry axis with angular momentum j and projection quantum number mj are added to angular momentum K about the symmetry axis, with projection quantum number mk , to give total angular momentum J. Within the vector model, consider the behavior of vectors j and K in the context of a critically bound electron in the field of the dipole moment μ. In the limit of zero electron binding (E → 0), the dominant component of the radial wave function approaches infinity in extent and the average speed approaches zero. Thus for the case where K > 0 a dipole bound electron in an excited state (extremely remote) experiences the long range dipolar field produced by the timeaverage of the dipole moment vector as it precesses about J (Fig. 2). (When K = 0 this averaging is not possible since the time-average value of the long range potential goes to zero in this case, as with the linear rotor.) This is another example described by Garrett16 as a reverse Born-Oppenheimer approximation. That is, in this context the nuclear motion is fast as compared to the electron’s motion. Thus from the viewpoint of certain critical bindings the electron moves in the field of a time averaged dipolar field as produced by a “static” dipole moment aligned along J with a value determined by the projection of K on J. This average moment is given by Thus a μ¯ = μ cos(θ ) where θ is the angle between J and K. critically bound electron at extremely great distances experiences a “static” field produced by a projected moment (5) μ¯ = μK/ J (J + 1). This averaged value is smaller than the actual dipole moment, but it is “fixed” in space and “oriented” along the direction of J. Thus this averaged dipole moment behaves as a station-
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ary dipole in the context of the reverse Born-Oppenheimer problem of interest here. And it is a property that holds for finite moments of inertia of the symmetric top (Iaa and Icc of Eq. (2)). Indeed under conditions where the temporal rotational average is appropriate, the averaged value is independent of the values of these inertial moments. This stationary feature has important consequences for critical binding and electron scattering. Now consider the problem of critical binding of an electron to a polar symmetric-top molecule in the present context. As just described, in the E → 0 limit the effective long range potential function at sufficiently large distances can be described by a stationary dipole with a fixed moment given by Eq. (5). And importantly, the critical moment for a fixed dipole is independent of the nature of any short range forces that would be present in a real molecule.17 Thus for calculating the moment required to guarantee binding to the ST, the short range forces will differ for symmetric tops as compared to linear rotors, but these differences do not matter. Critical binding results from long range dipole coupling between components of the r-θ decomposed wave function. Indeed all of the short range forces can be set equal to zero or to infinity for r ≤ rc , where rc is an arbitrary cutoff for short range forces. The critical moment for a fixed dipole is independent of the value of rc .17 In this context the dominant components omitted in an interaction potential are attractive and of shorter range. Thus a critical moment resulting hereby is an upper limit on that required for binding. On the basis of the present argument one can establish a new critical dipole moment for the ST rotor. Since the long range potential which governs critical binding is the same as that of a fixed dipole with the reduced moment of Eq. (5), one can find the value of a critical ST moment by noting that it will be the same as that for a fixed dipole with the projected dipole moment μ¯ of Eq. (5). And since the critical moment for a fixed dipole is known, with no further effort we have the result that a critical moment that yields an infinite number of bound states will be realized when this “projected” moment μ¯ ≥ 0.639ea0 . So, from Eq. (5), the actual physical moment μc required to give this new critical value is μc > 0.639 J (J + 1)/Kea0 .
(6)
This new critical moment is larger than that for a linear static dipole but it applies to the dynamic system and is independent of the magnitudes of the Iaa , Ibb , Icc components of the moment of inertia. The relationship expressed in Eq. (6) defines new critical moments for symmetric top rotors that guarantee an infinite bound state spectrum. Since this μc is always greater than 0.639ea0 these new critical moments apply to excited states (J and K > 0). Thus a real anion satisfying condition (6) will exhibit an infinite number of excited bound states, again independent of inertial moments. Indeed it is obvious that the smaller the inertial moments the better the inverse BO approximation becomes. So the problem of critical binding to a ST dipole-rotor with finite inertial moments can show a finite bound state spectrum for K = 0, as with the linear rotor, but this same system in an excited K > 0 state can support an infinite bound
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state spectrum simply by increasing the dipole moment to reach the new critical value μc for a given system. This is a new and interesting result that represents a new class of quantum systems for which an infinite set of states are supported. The present result has been extracted while avoiding treatment of the full close coupling set expressed in Eq. (3). Note that Green has shown12 that the coupled set becomes independent of the spatial quantum number M. So the vector model utilized here yields results consistent with the full CC treatment. (Solutions to the appropriate CC equations for the same system with no time averaging and including short range details (e.g., dipole length) should yield the lower K = 0 critical moment that is associated with a finite spectrum of bound states with a value that depends on the magnitudes of the inertial moments.) We note that the effective stationary moment of Eq. (6) and the coupling matrix elements on the right side of Eq. (3) scale upward with K for a given J. This says that for a given total-system angular momentum, J, the critical moment for a real ST rotor will be smaller for K ⇒ J (i.e., for small j, nonzero K) than for the same J but with j ⇒ J (i.e., for small K, non-zero j). This is counterintuitive, since rotational energies of ST rotors increase with K at constant J for prolate and decrease with oblate rotors, but this higher critical moment is not influenced by these differences in internal energies. It is interesting to note that behavior consistent with the present result has been seen in an early experimental study, where Lykke et al.18 found that auto-ionization rates in CH2 CN− are more rapid for j = 40, K = 0 than for j = 30, K = 8, even though the (prolate) energy is larger for the one with the lower autoionization rate. Additionally it is worth noting that in nozzle jet expansions the external rotational cooling (i.e., reduction in magnitude of the quantum number j) is much more effective than internal rotational cooling (reducing quantum number K). Thus it is experimentally possible to drive√a symmetric top dipole or μ¯ → μK/ K(K + 1). For large K rotor toward J → K, this approaches a limit μ¯ → μ or μc → μc = 0.639ea0 . It should be noted also that this analysis is not applicable for a limiting case where the inertial moments go to infinity. In this instance the ST is fixed, requiring no temporal average, and the critical moment for all rotationally excited states becomes μc = 0.639ea0 . One way to see this is to let K → ∞ in Eq. (6) (J also goes to infinity) with infinite inertial moments, which gives μc → μc = 0.639ea0 . Finally, the case for the finite ST with K = 0 should behave as in the linear rotor, with critical moments and finite spectra dependent on inertial moments and short range details. This is true since it is only the long range part that establishes the critical binding feature, and this is the same for the linear and the K = 0 ST. III. SCATTERING CROSS SECTIONS FROM POLAR SYMMETRIC-TOP MOLECULES
Garrett first showed3 that an infinite cross section resulted for electron scattering by a fixed, linear, dipolar scatterer, including point-dipole and finite-dipole geometries. Earlier results for a stationary point dipole scatterer had shown a diver-
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gent cross section within the Born approximation19 and in an exact treatment,20 which was attributed to the point dipole singularity. However, the later result3 showed that the divergent scattering cross section resulted from the infinite sum of high angular momentum components of the scattering event, independent of short range forces. In an early study, Crawford21 demonstrated divergence of the cross sections for scattering from linear and symmetric-top point-dipolar systems in the limit of infinite moments of inertia (space-fixed dipoles). Additionally, the elastic cross section for scattering by a polar symmetric-top of finite moment of inertia was shown to diverge for angular momentum component K > 0. But the problems were treated only within the first Born approximation.19 We are now in a position to make a stronger statement about electron scattering from a polar symmetric-top where rotational degrees of freedom are included. In the analysis of the critical binding problem use was made of the fact that in the E = 0 limit with K > 0, the time-averaged long range potential for an oriented fixed dipole can be used at large distances to replace the exact problem. (Were the electron moves very slowly as compared to nuclear motions.) For low energy electron scattering an analogous step can be invoked. Consider the infinite set of close coupling equations for the symmetric top, Eq. (3), for K > 0. For the high angular momentum components the centripetal repulsion is strong. Consequently, only the long range part of the interaction potential is effective at very large values of the electronic an Moreover, for arbitrargular momentum quantum number l. ily large values of l the nuclei can undergo many vibrations and rotations while the long range potential for the moving electron undergoes negligible change in value. Thus for very high l values the target symmetric top can be described with vanishing error as a stationary dipole with an averaged dipole moment given by Eq. (5). With this recognition one can simply invoke the previous results for scattering from a stationary dipole.3 That is, the elastic scattering cross section diverges as a result of the sum over the high angular momentum components independent of short range features of the interaction potential or of the contribution of low angular momentum components. Indeed the potential function at short range can be replaced by zero or an infinitely repulsive core and the divergence persists. Thus without having to solve the exact problem we can easily see that the total scattering cross section for the polar symmetric top (with K > 0) diverges if treated in the CC analysis, though this result is not easily discerned from Eq. (3) when applied to electron scattering. Indeed direct solutions of the CC equations for K > 0 would produce a finite cross section as a result of summing a finite number of a slowly diverging series, similar to the case revealed in the history of the fixed linear dipole.4 Solutions for a ST system K = 0 would produce no such divergence. It is worth recognizing that the divergence for symmetric tops noted by Crawford21 actually holds beyond the first Born approximation which he utilized. Indeed a second way of obtaining the divergent result in a short proof is to note that the Born approximation to the high l components of a partial wave expansion become indistinguishable from the exact values in the limit. Moreover, since it is the partial waves in the limit that cause the divergence, we can say immediately that
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the result obtained by Crawford in the Born approximation is also valid in an exact treatment. IV. CONCLUSIONS
Historically, the quantum mechanical problems associated with bound and continuum states of electrons in electric dipolar fields have been extensively studied.22 Though its spatial average is zero, the long range 1/r2 nature of the potential function for an electron in an electric dipole field nevertheless produces an infinite scattering cross section and can produce an infinite number of bound states as in the Coulomb problem. But unlike the Coulomb problem, the bound states exist only when the dipole moment exceeds a critical value, below which no bound states are supported. In the context of electron interactions with real linear polar molecules, the simple stationary finite linear dipole corresponds to a system with infinite moment of inertia. When finite inertial moments are introduced, the dipole moments required for critical binding are modified upward, and the scattering cross sections become finite for all values of dipole moment. However, it has been demonstrated here that the situation is different for polar symmetric top systems. Under circumstances where the angular momentum projection about the ST symmetry axis is non-zero (K > 0), it has been established that (1) a new critical moment can be defined wherein an infinite excited state spectrum is realized. This value is independent of the magnitude of moments of inertia, even for very small components of the inertial tensor; and (2) the scattering cross section diverges for all magnitudes of the electric dipole moment. The problem becomes similar in character to the linear dipole with infinite inertial moment. With nozzle jet rotational cooling the new critical dipole moment required to produce an infinite bound state spectrum can be driven from above toward the static dipole value of 0.639ea0. Infinite bound states can be supported for symmetric top systems with finite moments of inertia, even for unrealistically small values of these moments. A final feature is of formal interest. For “realistic” polar ST rotors, a finite number of bound states will be produced with a lowest critical moment, μc, for the ground state which is somewhat larger than 0.639ea0 , where the actual value depends on the magnitude of the inertial tensor. For ST systems with K > 0, a second, higher, critical moment μc will produce an infinite bound state spectrum, which is independent of the inertial tensor. But the value of the K = 0 ground state critical moment μc increases rapidly as the inertial moments are decreased toward zero. Thus for ST with K > 0 one can produce (unrealistic) systems where μc > μc . That is, the finite spectrum can be made to disappear by decreasing the moment of
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inertia, while preserving the infinite spectrum associated with μc (K > 0) since this is independent of the inertial moment. This feature could be exhibited in full close coupling calculations for the polar ST rotor. The present results are of interest in the context of quantum mechanical features of model systems since the appearance of infinities in numbers of bound states and in scattering phenomena are often cited in textbook discussions. The simple polar symmetric top model is closer to a real system than other examples that exhibit such behavior (with the atomic hydrogen exception). It is interesting that some features, which are so readily established, have gone unrecognized for so long since the initial studies of the subject. Though a very large number of studies of specific polar anions have appeared in the context of quantum chemistry (Born-Oppenheimer (BO) approximation), almost no attention has been given to non-BO effects until the very recent appearance of a detailed study17 of a number of linear dipolar anions. In this work a new and powerful technique has been invoked in applications to critical binding, dipole bound excited states, and the revelation of a rich spectrum of rotational resonances for the HCN− anion.23 1 E.
Fermi and E. Teller, Phys. Rev. 72, 406 (1947); M. H. Mittleman and V. P. Myerscough, Phys. Lett. 23, 545 (1966); J. E. Turner and K. Fox, ibid. 23, 547 (1966); W. B. Brown and R. E. Roberts, J. Chem. Phys. 46, 2006 (1967); C. A. Coulson and M. Walmsley, Proc. Phys. Soc. 91, 31 (1967); J.-M. Levy-Leblond, Phys. Rev. 153, 1 (1967). 2 W. R. Garrett, Chem. Phys. Lett. 62, 325 (1979). 3 W. R. Garrett, Phys. Rev. A 4, 2229 (1971). 4 W. R. Garrett, Chem. Phys. Lett. 5, 393 (1970); Phys. Rev. A 3, 961 (1971). 5 W. R. Garrett, Mol. Phys. 24, 465 (1972). 6 W. R. Garrett, J. Chem. Phys. 73, 5721 (1980). 7 W. R. Garrett, J. Chem. Phys. 77, 3666 (1982). 8 P. C. Engleking, Phys. Rev. A 26, 740 (1982). 9 D. R. Herrick and P. C. Engleking, Phys. Rev. A 29, 2421 (1984). 10 D. C. Clary, J. Phys. Chem. 92, 3173 (1988). 11 W. R. Garrett, J. Chem. Phys. 69, 2621 (1978). 12 S. Green, J. Chem. Phys. 64, 3463 (1976). 13 E. S. Chang and U. Fano, Phys. Rev. A 6, 173 (1972). 14 R. N. Zare, Angular Momentum (John Wiley and Sons, New York, 1988), p. 123. 15 K. Fossez, N. Michel, W. Nazarewicz, and M. Ploszajczak, Phys. Rev A 87, 042575 (2013). 16 W. R. Garrett, J. Chem. Phys. 133, 224103 (2010). 17 O. Crawford and A. Dalgarno, Chem. Phys. Lett., 1, 23 (1967). 18 K. R. Lykke, D. M. Neumark, T. Anderson, V. J. Trapa, and W. C. Linberger, J. Chem. Phys. 87, 6842 (1987). 19 S. Altshuler, Phys. Rev. 107, 114 (1957). 20 M. H. Mittleman and E. von Holt, Phys. Rev. 140, A726 (1965). 21 O. H. Crawford, J. Chem. Phys. 47, 1100 (1967). 22 A review of extensive MO calculations for dipole-bound anions is provided by K. D. Jordan and F. Wang, Ann. Rev. Phys. Chem. 54, 367 (2003). 23 K. Fossez, “Reactions de capture radiative et spectroscopie d’anions multipolaires dans le cadre du Gamow Model,” Thesis, Universite de Caen Basse-Normandie (2014).
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THE JOURNAL OF CHEMICAL PHYSICS 141, 164318 (2014)
Critical binding and electron scattering by symmetric-top polar molecules W. R. Garretta) Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA
(Received 15 August 2014; accepted 8 October 2014; published online 28 October 2014) Quantum treatments of electron interactions with polar symmetric-top rotor molecules show features not present in the treatment of the linear-polar-rotor model. For symmetric tops possessing non-zero angular momentum about the symmetry axis, a new critical dipole can be defined that guarantees an infinite set of dipole-bound states independent of the values of the components of the inertial tensor. Additionally, for this same class, the scattering cross section diverges for all nonzero values of dipole moments and inertial moments, similar to solutions for the fixed linear dipole. Additional predictions are presented for electron affinities and rotational resonances of these systems. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4898730] I. INTRODUCTION
The detailed features of the interaction of a charged particle with a long-range electric dipole field have been the subjects of theoretical and experimental interest over many years. The quantum mechanics of the problem involves features dictated by the long range nature of the potential function that describes the interaction. To be specific, consider the electron as the charged particle. Though the description of electron interactions through the potential created by fixed dipolar charges is non-central and thus more complicated than the long range Coulomb potential, the quantum features of the two problems have important things in common. Both potentials support an infinite number of electronic bound states, but for the dipolar field, characterized by dipole moment μ, the bound states exist only for moments greater than a critical value μc = 0.639ea0 (atomic units, with e the electronic charge and a0 the Bohr radius). For dipole moments smaller than this critical value no bound states exist.1 The number of bound states is independent of the dipole length (positivenegative charge separation) or any other short range forces, thus it is relatively easy to show that all molecular orbital or other quantum chemistry calculations of polar anion spectra in the Born-Oppenheimer approximation should yield an infinite number of bound exited states in the limit of infinite numerical accuracy.2 For charged particle scattering from a fixed dipole target the continuum states yield an infinite total scattering cross section3 for all nonzero values of dipole moments, similar to the behavior in the Coulomb scattering problem. Again, this divergence is produced by the long range dipolar interaction for all finite moments, μ, and independent of short range forces. The electron-binding-feature of a fixed electric dipole field does not guarantee that any strongly polar molecule with μ > μc can support a stable negative ion. At zero electron binding the rotational energy is greater than the binding energy, hence electron-rotational coupling plays a critical role in the quantum mechanical dynamics. Garrett showed that ina) E-mail: [email protected]
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clusion of rotational degrees of freedom in the quantum treatment of a polar negative ion caused significant changes in the critical binding and scattering features of the problem.4, 5 For a linear dipole rotor, critical moments still exist but the values are larger than that for fixed charges and critical moments are dependent on the moment of inertia, the short range components of the electron–molecule potential, and the rotational state of a linear rotor molecule. The number of excited dipole-bound states is now finite, not infinite, where for small to modest sized systems, the truncation leads to only one or at most a small number of electronic states.6, 7 The total electron scattering cross section that was infinite for all non-zero μ in the fixed-nucleus approximation,3 remains finite when rotational degrees of freedom are treated properly. The quantum mechanics of electric dipole-bound systems has an extensive history which will not be reviewed here. But new properties of such systems will be shown to belong to polar symmetric tops possessing non-zero internal rotational quantum numbers, including the existence of a new critical dipole moment. Indeed such systems can have properties similar to those of the fixed linear dipolar problem where infinite numbers of bound states and infinite scattering cross sections appear.
II. CRITICAL BINDING FOR DIPOLAR SYMMETRIC-TOP ANIONS
Surprisingly, the general problem of critical electron binding to a dipolar symmetric top (ST) has not been fully treated. That is, neither ground nor excited bound states have been fully examined. Near-threshold photo-detachment from a dipolar symmetric top has been studied by Engelking8 in an adiabatic approximation and extended by Herrick and Engelking,9 where anomalous threshold behavior was noted. Clary10 has examined shape resonances in dipolar ST molecules, in a similar approximation and with a pseudopotential similar to that of Garrett.11 Green12 has described the general formalism of molecular scattering by a symmetrictop scatterer, which would also apply to the symmetric top anion problem, and Chang and Fano13 provide some useful
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J. Chem. Phys. 141, 164318 (2014)
relationships for the electron-symmetric-top problem. However, theoretical solutions for electron binding to polar ST molecules or low energy photo-detachment profiles of such systems are available only through adiabatic treatments.10 The polar ST anion problem is of fundamental interest because of some novel features that arise in the underlying physics. An ab initio treatment of an electron very weakly bound to a symmetric top can be described in a manner similar to that presented by Chang and Fano.13 The model Hamiltonian for the problem of an electron with coordinate r in the field of a symmetric top molecule is written as H = Hrot − 1/22r + V (r, β),
(1)
where the rotational energy operator Hrot (αβγ ) and potential energy, V (r , β), contain Eulerian angles αβγ of the symmetric top. The model problem can be described by a wave function for a single electron moving in the field of the polar symmetric top rotor with principal moments of inertia Iaa , Ibb , and Icc. The parent molecule has total rotational angular momentum J, J2 = J (J + 1), and with projection Jz = mJ along lab coordinate z, where J is the sum of rotational angular momentum j perpendicular to the symmetry axis and internal angular momentum K about the symmetry axis of the rotor (see Fig. 1). The rotational wave functions for the rotor are ST eigenfunctions14 that are defined by J (J + 1) − K 2 K2 Hrot − JmKj (αβγ ) = 0. (2) + 2Iaa 2Icc The moment of inertia about the symmetry axis is Icc (the other two inertial moments are each equal to Iaa ). The dipole ˆ moment μ lies along the symmetry axis with unit vector R, ˆ so J · R = K. The single valence electron has angular momentum l where l2 = l(l + 1) and lz = ml , having r, ϑ, ϕ coordinates in the lab frame with origin at the center of mass. This is coupled to the angular momentum of the ST rotor
by the non-central interaction potential with asymptotic form ˆ 2 . The total angular momentum of → −μ(ˆr · R)/r V (r , R) = J + l with L 2 = L(L + 1) and Lz the anion system is L = M both conserved. For present considerations it can simply be noted that a system of equations for the ST anion, or for electron scattering, can be obtained by expanding the total wave function in suitably coupled angular and radial functions where L and M are diagonal. With this procedure, which will not be repeated here, the Shrödinger equation for the anion12 leads a set of coupled equations for radial functions FjLK l (r) for each L, K. These close-coupling (CC) sets have the form:
d2 l(l + 1) − + 2(E − ε ) FjLK l (r) jK dr 2 r2 LK VjLK =2 lj l (r)Fj l (r).
The potential function matrix element VjLK lj l (r) has been evaluated by Green12 and by Herrick and Engelking.9 The coupling between electron and dipolar top depends only on the orientation of the electron relative to the rotor, thus the matrix elements in the close coupling equations (6) are independent of the quantum number M. Now consider critical binding of an electron in the field of a dipolar symmetric top, including critical binding to rotationally excited states. On the basis of results for the linear rotor with finite inertial moment, rotationally excited states will exhibit binding with critical moments somewhat larger than that for the ground state. These moments converge to that for the fixed dipole in the limit of infinite inertial moments.4, 15 For the ST the dipole moment lies along the symmetry axis, ˆ and the angular momentum that is, along the unit vector K, and their sum J all precess about the laboratory vectors j, K, Z axis. With the addition of the electron of angular momen = J + l with projection quantum tum l the system total is L number M. For the polar ST top rotor the dipole moment has an orientation with respect to the axis of quantization when K > 0 (Fig. 1). Thus from a distance in lab coordinates the field of the polar object appears on temporal average as that due to a permanent dipole with an effective dipole moment whose value is the average of μcos (θ ) where θ is the angle between the symmetry axis and the laboratory Z axis. This average (the orientation angle) has a value cos (θ ) = KmJ /J(J + 1).14 Thus for a ST top with dipole moment μ, the time-averaged long range field appears as that due to fixed dipole with dipole moment μ¯ = μ
FIG. 1. Angular momentum components of a symmetric top rotor in laboratory coordinates. Total angular momentum J has projection K about the symmetry axis and j perpendicular to this axis. Precession occurs about the laboratory z axis with projection quantum numbers M, mk , and mj , respectively.
(3)
j l
KmJ . J (J + 1)
(4)
For K = 0 the average long range field vanishes as for the linear rotor, but for K > 0 a fraction of the moment is oriented on average giving the appearance of a “fixed” dipolar component in the long range potential. In the critical binding problem this feature allows one to carry out an averaging when analyzing dipole-bound excited states that are even more “remote” than the dipole-bound ground state. This averaging is not possible for linear or K = 0 ST rotors.
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FIG. 2. Vector model of the rotating symmetric top with angular momenta K about the symmetry axis and angular momentumj perpendicular to this axis coupled to give total angular momentum J.
The analytical characteristics of the CC equations can be rather tedious. However, the problems of interest here can be adequately described while avoiding some of the complexity of the set defined in Eq. (3). Indeed as a result of special characteristics of the electron-dipole critical binding problem, the features of interest here can be established without having to solve the full close-coupling equations. Specifically these features can be established by consideration of a vector model in place of the more detailed close coupling description. In this vein the symmetric top rotor is depicted in the vector diagram of Fig. 2. Rotations perpendicular to the symmetry axis with angular momentum j and projection quantum number mj are added to angular momentum K about the symmetry axis, with projection quantum number mk , to give total angular momentum J. Within the vector model, consider the behavior of vectors j and K in the context of a critically bound electron in the field of the dipole moment μ. In the limit of zero electron binding (E → 0), the dominant component of the radial wave function approaches infinity in extent and the average speed approaches zero. Thus for the case where K > 0 a dipole bound electron in an excited state (extremely remote) experiences the long range dipolar field produced by the timeaverage of the dipole moment vector as it precesses about J (Fig. 2). (When K = 0 this averaging is not possible since the time-average value of the long range potential goes to zero in this case, as with the linear rotor.) This is another example described by Garrett16 as a reverse Born-Oppenheimer approximation. That is, in this context the nuclear motion is fast as compared to the electron’s motion. Thus from the viewpoint of certain critical bindings the electron moves in the field of a time averaged dipolar field as produced by a “static” dipole moment aligned along J with a value determined by the projection of K on J. This average moment is given by Thus a μ¯ = μ cos(θ ) where θ is the angle between J and K. critically bound electron at extremely great distances experiences a “static” field produced by a projected moment (5) μ¯ = μK/ J (J + 1). This averaged value is smaller than the actual dipole moment, but it is “fixed” in space and “oriented” along the direction of J. Thus this averaged dipole moment behaves as a station-
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ary dipole in the context of the reverse Born-Oppenheimer problem of interest here. And it is a property that holds for finite moments of inertia of the symmetric top (Iaa and Icc of Eq. (2)). Indeed under conditions where the temporal rotational average is appropriate, the averaged value is independent of the values of these inertial moments. This stationary feature has important consequences for critical binding and electron scattering. Now consider the problem of critical binding of an electron to a polar symmetric-top molecule in the present context. As just described, in the E → 0 limit the effective long range potential function at sufficiently large distances can be described by a stationary dipole with a fixed moment given by Eq. (5). And importantly, the critical moment for a fixed dipole is independent of the nature of any short range forces that would be present in a real molecule.17 Thus for calculating the moment required to guarantee binding to the ST, the short range forces will differ for symmetric tops as compared to linear rotors, but these differences do not matter. Critical binding results from long range dipole coupling between components of the r-θ decomposed wave function. Indeed all of the short range forces can be set equal to zero or to infinity for r ≤ rc , where rc is an arbitrary cutoff for short range forces. The critical moment for a fixed dipole is independent of the value of rc .17 In this context the dominant components omitted in an interaction potential are attractive and of shorter range. Thus a critical moment resulting hereby is an upper limit on that required for binding. On the basis of the present argument one can establish a new critical dipole moment for the ST rotor. Since the long range potential which governs critical binding is the same as that of a fixed dipole with the reduced moment of Eq. (5), one can find the value of a critical ST moment by noting that it will be the same as that for a fixed dipole with the projected dipole moment μ¯ of Eq. (5). And since the critical moment for a fixed dipole is known, with no further effort we have the result that a critical moment that yields an infinite number of bound states will be realized when this “projected” moment μ¯ ≥ 0.639ea0 . So, from Eq. (5), the actual physical moment μc required to give this new critical value is μc > 0.639 J (J + 1)/Kea0 .
(6)
This new critical moment is larger than that for a linear static dipole but it applies to the dynamic system and is independent of the magnitudes of the Iaa , Ibb , Icc components of the moment of inertia. The relationship expressed in Eq. (6) defines new critical moments for symmetric top rotors that guarantee an infinite bound state spectrum. Since this μc is always greater than 0.639ea0 these new critical moments apply to excited states (J and K > 0). Thus a real anion satisfying condition (6) will exhibit an infinite number of excited bound states, again independent of inertial moments. Indeed it is obvious that the smaller the inertial moments the better the inverse BO approximation becomes. So the problem of critical binding to a ST dipole-rotor with finite inertial moments can show a finite bound state spectrum for K = 0, as with the linear rotor, but this same system in an excited K > 0 state can support an infinite bound
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state spectrum simply by increasing the dipole moment to reach the new critical value μc for a given system. This is a new and interesting result that represents a new class of quantum systems for which an infinite set of states are supported. The present result has been extracted while avoiding treatment of the full close coupling set expressed in Eq. (3). Note that Green has shown12 that the coupled set becomes independent of the spatial quantum number M. So the vector model utilized here yields results consistent with the full CC treatment. (Solutions to the appropriate CC equations for the same system with no time averaging and including short range details (e.g., dipole length) should yield the lower K = 0 critical moment that is associated with a finite spectrum of bound states with a value that depends on the magnitudes of the inertial moments.) We note that the effective stationary moment of Eq. (6) and the coupling matrix elements on the right side of Eq. (3) scale upward with K for a given J. This says that for a given total-system angular momentum, J, the critical moment for a real ST rotor will be smaller for K ⇒ J (i.e., for small j, nonzero K) than for the same J but with j ⇒ J (i.e., for small K, non-zero j). This is counterintuitive, since rotational energies of ST rotors increase with K at constant J for prolate and decrease with oblate rotors, but this higher critical moment is not influenced by these differences in internal energies. It is interesting to note that behavior consistent with the present result has been seen in an early experimental study, where Lykke et al.18 found that auto-ionization rates in CH2 CN− are more rapid for j = 40, K = 0 than for j = 30, K = 8, even though the (prolate) energy is larger for the one with the lower autoionization rate. Additionally it is worth noting that in nozzle jet expansions the external rotational cooling (i.e., reduction in magnitude of the quantum number j) is much more effective than internal rotational cooling (reducing quantum number K). Thus it is experimentally possible to drive√a symmetric top dipole or μ¯ → μK/ K(K + 1). For large K rotor toward J → K, this approaches a limit μ¯ → μ or μc → μc = 0.639ea0 . It should be noted also that this analysis is not applicable for a limiting case where the inertial moments go to infinity. In this instance the ST is fixed, requiring no temporal average, and the critical moment for all rotationally excited states becomes μc = 0.639ea0 . One way to see this is to let K → ∞ in Eq. (6) (J also goes to infinity) with infinite inertial moments, which gives μc → μc = 0.639ea0 . Finally, the case for the finite ST with K = 0 should behave as in the linear rotor, with critical moments and finite spectra dependent on inertial moments and short range details. This is true since it is only the long range part that establishes the critical binding feature, and this is the same for the linear and the K = 0 ST. III. SCATTERING CROSS SECTIONS FROM POLAR SYMMETRIC-TOP MOLECULES
Garrett first showed3 that an infinite cross section resulted for electron scattering by a fixed, linear, dipolar scatterer, including point-dipole and finite-dipole geometries. Earlier results for a stationary point dipole scatterer had shown a diver-
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gent cross section within the Born approximation19 and in an exact treatment,20 which was attributed to the point dipole singularity. However, the later result3 showed that the divergent scattering cross section resulted from the infinite sum of high angular momentum components of the scattering event, independent of short range forces. In an early study, Crawford21 demonstrated divergence of the cross sections for scattering from linear and symmetric-top point-dipolar systems in the limit of infinite moments of inertia (space-fixed dipoles). Additionally, the elastic cross section for scattering by a polar symmetric-top of finite moment of inertia was shown to diverge for angular momentum component K > 0. But the problems were treated only within the first Born approximation.19 We are now in a position to make a stronger statement about electron scattering from a polar symmetric-top where rotational degrees of freedom are included. In the analysis of the critical binding problem use was made of the fact that in the E = 0 limit with K > 0, the time-averaged long range potential for an oriented fixed dipole can be used at large distances to replace the exact problem. (Were the electron moves very slowly as compared to nuclear motions.) For low energy electron scattering an analogous step can be invoked. Consider the infinite set of close coupling equations for the symmetric top, Eq. (3), for K > 0. For the high angular momentum components the centripetal repulsion is strong. Consequently, only the long range part of the interaction potential is effective at very large values of the electronic an Moreover, for arbitrargular momentum quantum number l. ily large values of l the nuclei can undergo many vibrations and rotations while the long range potential for the moving electron undergoes negligible change in value. Thus for very high l values the target symmetric top can be described with vanishing error as a stationary dipole with an averaged dipole moment given by Eq. (5). With this recognition one can simply invoke the previous results for scattering from a stationary dipole.3 That is, the elastic scattering cross section diverges as a result of the sum over the high angular momentum components independent of short range features of the interaction potential or of the contribution of low angular momentum components. Indeed the potential function at short range can be replaced by zero or an infinitely repulsive core and the divergence persists. Thus without having to solve the exact problem we can easily see that the total scattering cross section for the polar symmetric top (with K > 0) diverges if treated in the CC analysis, though this result is not easily discerned from Eq. (3) when applied to electron scattering. Indeed direct solutions of the CC equations for K > 0 would produce a finite cross section as a result of summing a finite number of a slowly diverging series, similar to the case revealed in the history of the fixed linear dipole.4 Solutions for a ST system K = 0 would produce no such divergence. It is worth recognizing that the divergence for symmetric tops noted by Crawford21 actually holds beyond the first Born approximation which he utilized. Indeed a second way of obtaining the divergent result in a short proof is to note that the Born approximation to the high l components of a partial wave expansion become indistinguishable from the exact values in the limit. Moreover, since it is the partial waves in the limit that cause the divergence, we can say immediately that
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the result obtained by Crawford in the Born approximation is also valid in an exact treatment. IV. CONCLUSIONS
Historically, the quantum mechanical problems associated with bound and continuum states of electrons in electric dipolar fields have been extensively studied.22 Though its spatial average is zero, the long range 1/r2 nature of the potential function for an electron in an electric dipole field nevertheless produces an infinite scattering cross section and can produce an infinite number of bound states as in the Coulomb problem. But unlike the Coulomb problem, the bound states exist only when the dipole moment exceeds a critical value, below which no bound states are supported. In the context of electron interactions with real linear polar molecules, the simple stationary finite linear dipole corresponds to a system with infinite moment of inertia. When finite inertial moments are introduced, the dipole moments required for critical binding are modified upward, and the scattering cross sections become finite for all values of dipole moment. However, it has been demonstrated here that the situation is different for polar symmetric top systems. Under circumstances where the angular momentum projection about the ST symmetry axis is non-zero (K > 0), it has been established that (1) a new critical moment can be defined wherein an infinite excited state spectrum is realized. This value is independent of the magnitude of moments of inertia, even for very small components of the inertial tensor; and (2) the scattering cross section diverges for all magnitudes of the electric dipole moment. The problem becomes similar in character to the linear dipole with infinite inertial moment. With nozzle jet rotational cooling the new critical dipole moment required to produce an infinite bound state spectrum can be driven from above toward the static dipole value of 0.639ea0. Infinite bound states can be supported for symmetric top systems with finite moments of inertia, even for unrealistically small values of these moments. A final feature is of formal interest. For “realistic” polar ST rotors, a finite number of bound states will be produced with a lowest critical moment, μc, for the ground state which is somewhat larger than 0.639ea0 , where the actual value depends on the magnitude of the inertial tensor. For ST systems with K > 0, a second, higher, critical moment μc will produce an infinite bound state spectrum, which is independent of the inertial tensor. But the value of the K = 0 ground state critical moment μc increases rapidly as the inertial moments are decreased toward zero. Thus for ST with K > 0 one can produce (unrealistic) systems where μc > μc . That is, the finite spectrum can be made to disappear by decreasing the moment of
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inertia, while preserving the infinite spectrum associated with μc (K > 0) since this is independent of the inertial moment. This feature could be exhibited in full close coupling calculations for the polar ST rotor. The present results are of interest in the context of quantum mechanical features of model systems since the appearance of infinities in numbers of bound states and in scattering phenomena are often cited in textbook discussions. The simple polar symmetric top model is closer to a real system than other examples that exhibit such behavior (with the atomic hydrogen exception). It is interesting that some features, which are so readily established, have gone unrecognized for so long since the initial studies of the subject. Though a very large number of studies of specific polar anions have appeared in the context of quantum chemistry (Born-Oppenheimer (BO) approximation), almost no attention has been given to non-BO effects until the very recent appearance of a detailed study17 of a number of linear dipolar anions. In this work a new and powerful technique has been invoked in applications to critical binding, dipole bound excited states, and the revelation of a rich spectrum of rotational resonances for the HCN− anion.23 1 E.
Fermi and E. Teller, Phys. Rev. 72, 406 (1947); M. H. Mittleman and V. P. Myerscough, Phys. Lett. 23, 545 (1966); J. E. Turner and K. Fox, ibid. 23, 547 (1966); W. B. Brown and R. E. Roberts, J. Chem. Phys. 46, 2006 (1967); C. A. Coulson and M. Walmsley, Proc. Phys. Soc. 91, 31 (1967); J.-M. Levy-Leblond, Phys. Rev. 153, 1 (1967). 2 W. R. Garrett, Chem. Phys. Lett. 62, 325 (1979). 3 W. R. Garrett, Phys. Rev. A 4, 2229 (1971). 4 W. R. Garrett, Chem. Phys. Lett. 5, 393 (1970); Phys. Rev. A 3, 961 (1971). 5 W. R. Garrett, Mol. Phys. 24, 465 (1972). 6 W. R. Garrett, J. Chem. Phys. 73, 5721 (1980). 7 W. R. Garrett, J. Chem. Phys. 77, 3666 (1982). 8 P. C. Engleking, Phys. Rev. A 26, 740 (1982). 9 D. R. Herrick and P. C. Engleking, Phys. Rev. A 29, 2421 (1984). 10 D. C. Clary, J. Phys. Chem. 92, 3173 (1988). 11 W. R. Garrett, J. Chem. Phys. 69, 2621 (1978). 12 S. Green, J. Chem. Phys. 64, 3463 (1976). 13 E. S. Chang and U. Fano, Phys. Rev. A 6, 173 (1972). 14 R. N. Zare, Angular Momentum (John Wiley and Sons, New York, 1988), p. 123. 15 K. Fossez, N. Michel, W. Nazarewicz, and M. Ploszajczak, Phys. Rev A 87, 042575 (2013). 16 W. R. Garrett, J. Chem. Phys. 133, 224103 (2010). 17 O. Crawford and A. Dalgarno, Chem. Phys. Lett., 1, 23 (1967). 18 K. R. Lykke, D. M. Neumark, T. Anderson, V. J. Trapa, and W. C. Linberger, J. Chem. Phys. 87, 6842 (1987). 19 S. Altshuler, Phys. Rev. 107, 114 (1957). 20 M. H. Mittleman and E. von Holt, Phys. Rev. 140, A726 (1965). 21 O. H. Crawford, J. Chem. Phys. 47, 1100 (1967). 22 A review of extensive MO calculations for dipole-bound anions is provided by K. D. Jordan and F. Wang, Ann. Rev. Phys. Chem. 54, 367 (2003). 23 K. Fossez, “Reactions de capture radiative et spectroscopie d’anions multipolaires dans le cadre du Gamow Model,” Thesis, Universite de Caen Basse-Normandie (2014).
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