Exactly solvable 1D model explains the low-energy vibrational level structure of protonated methane
TThe simplest model to explain the vibrational level structure of protonated methane?
Jonathan I. Rawlinson ∗ School of Mathematics, University of Bristol, Bristol, UK
Csaba F´abri and Attila G. Cs´asz´ar
Laboratory of Molecular Structure and Dynamics,Institute of Chemistry, E¨otv¨os Lor´and University,P´azm´any P´eter s´et´any 1/A, H-1117 Budapest, Hungary andMTA-ELTE Complex Chemical Systems Research Group, P.O. Box 32, H-1518 Budapest 112, Hungary (Dated: February 15, 2021)A new one-dimensional model is proposed for the low-energy vibrational quantum dynamics ofCH +5 based on the motion of an effective particle confined to a 4-regular (quartic) 60-vertex metricgraph Γ with a single edge length parameter. Within this model, the quantum states of CH +5 areobtained in analytic form and are related to combinatorial properties of Γ . In particular, we showthat the bipartite structure of Γ gives a simple explanation for curious symmetries observed innumerically exact variational calculations on CH +5 . Protonated methane, CH +5 , also called methonium, isconsidered to be the prototype of pentacoordinated non-classical carbonium ions [1–3]. The curious carboniumcations yielded an extremely rich chemistry and a Nobel-prize to their discoverer, George Olah [4]. Nevertheless,these are not the only sources of fame for carbonium ionsand in particular for CH +5 . Over the last two decades [5],CH +5 has been posing a formidable challenge to high-resolution spectroscopists [5–15]. The most outstandingissue is that the spectrum of CH +5 is exceptionally com-plex even at extremely low temperatures [9, 13] due toits quasistructural nature [16].On the computational side, accurate rovibrational en-ergy levels and eigenstates have been made available forCH +5 , owing to huge numerical efforts, in recent years[7, 12, 14]. These studies have revealed close-lying clus-ters in the rovibrational energy levels, with fascinatingsymmetries. These features have defied explanation byconventional means, motivating the development of novelmodels for CH +5 . The most important models put for-ward so far are as follows: (a) particle-on-a-sphere (POS)[17–23], (b) five-dimensional (5D) rotor (superrotor) [24–26] and (c) quantum graph [15, 27]. So far, the quantum-graph model seems to result in the most satisfactory ex-planation of the low-energy quantum dynamics of CH +5 ,including both vibrations [15] and rotations [27].Quantum graphs have a long history, dating backto Linus Pauling’s description of electrons in organicmolecules in the 1930s [28]. They have only recentlybeen introduced to the study of nuclear dynamics, wherethey have proved useful in high-resolution spectroscopy[15, 27] and also α -cluster dynamics in nuclear physics[29, 30]. Quantum graphs [31] are metric graphs, thatis each of their edges possesses a length. In the con-text of rovibrational dynamics of molecules, each vertexof the graph represents a version [32] of an equilibriumstructure. The vertices are connected by edges whichrepresent collective internal motions converting different versions into each other. Once a quantum graph is set up,one constructs the one-dimensional Schr¨odinger equationfor a particle confined to the graph and solves it to deter-mine the energy levels and eigenstates (see the Supple-mental Material). In this way, complex multidimensionalquantum dynamics is mapped onto the effective motionof a one-dimensional particle confined to a much simplerspace.In the case of CH +5 , the equilibrium structure is com-posed of a H unit sitting on top of a CH +3 tripod, anarrangement with C s point-group symmetry. The fiveprotons can be rearranged in 5!=120 ways, generating120 symmetry-equivalent versions, which become the 120vertices of a quantum graph Γ [15], illustrated in Fig.1. There are two types of motion interconverting the 120 FIG. 1. Illustration of the 3-regular quantum graph Γ . Inthis model of the quantum dynamics of CH +5 there are twodistinct edge lengths, corresponding to internal rotations ofthe H unit by 60 o and the flip motion that exchanges a pairof protons between the H and CH +3 units. a r X i v : . [ phy s i c s . c h e m - ph ] F e b TABLE I. The block structure characterizing the first 60 vi-brational states of CH +5 , revealed in variational computations[12, 14]. The numbers in parentheses give the total number ofpositive and negative parity states within a block, reflectingthe degeneracy of the states.Block 1 Block 20 −
60 cm − (15,15) 110 −
200 cm − (15,15) A +1 ⊕ G +1 ⊕ H +1 ⊕ H +2 ⊕ G +1 ⊕ H +1 ⊕ I + ⊕ G − ⊕ H − ⊕ I − A − ⊕ G − ⊕ H − ⊕ H − equivalent versions: the internal rotations of the H unitby 60 o (both clockwise and counterclockwise), and theflip motion that exchanges a pair of protons between theH and CH +3 units (see Fig. 2 for the local structure of Γ ). Given our knowledge about the stationary pointsof the potential energy hypersurface of CH +5 [33], it isplausible that motions other than the internal rotationand flip motions can be disregarded as long as one is in-terested in the low-energy rovibrational quantum dynam-ics of CH +5 , and so one takes these motions to correspondto the edges of Γ . As one flip edge and two internalrotation edges are connected to each vertex of Γ , eachvertex has a degree of three ( Γ is a 3-regular graph).The 120 internal rotation and 60 flip edges are assignedeffective lengths L rot and L flip , respectively.The quantum graph Γ reproduces the low-energyrovibrational energy levels of CH +5 , as well as of CD +5 ,remarkably well when optimized values of L flip and L rot are used [15, 27] (see the Supplemental Material for thenumerical values of the edge length parameters and thevibrational energy levels of CH +5 ). For instance, the Γ model perfectly reproduces the block structure (statesoccuring in groups of 30) of the vibrational eigenstatesof CH +5 , first noted in a variational study of Wang andCarrington [12] and later confirmed in [14] (see Table I).Rovibrational eigenstates of CH +5 are labelled by irre-ducible representations (irreps) of the molecular symme-try (MS) group S ∗ = S × { E, E ∗ } , generated by S permutations of the five protons together with spatialinversion E ∗ ( E denotes the identity operation).Beyond the existence of blocks, we have noticed otherclear symmetry relations for the first 60 quantum states.A comparison of the group-theoretic relation (cid:0) A +1 ⊕ G +1 ⊕ H +1 ⊕ H +2 (cid:1) ⊗ A − (cid:39) A − ⊕ G − ⊕ H − ⊕ H − (1)with the data in Table I suggests a direct correspondencebetween the 15 positive-parity states in Block 1 [appear-ing on the left-hand side (LHS) of Eq. (1)] and the 15negative-parity states in Block 2 [right-hand side (RHS)of Eq. (1)]. Likewise, (cid:0) G − ⊕ H − ⊕ I − (cid:1) ⊗ A − (cid:39) G +1 ⊕ H +1 ⊕ I + , (2)suggesting a link between the 15 negative-parity states in Block 1 and the positive-parity states in Block 2. Theseremarkable relations have been lacking any simple expla-nation, even in terms of the Γ model. One purposeof this Letter is to introduce an even simpler quantumgraph model, derived from Γ , which explains thesesymmetries.Let us start with the Γ model. Following our origi-nal study [15], we neglect the potential energy along theedges of Γ , since the barriers to the internal rotationand flip motions are small (about 30 cm − and 300 cm − ,respectively). The edge lengths L flip and L rot were fixedin Ref. [15] by an optimization procedure. The flip edgeswere assigned a much smaller effective length than theinternal rotation edges, with the optimized values satis-fying L flip /L rot ≈ / L flip /L rot is so small that it is tempting to imagineshrinking the flip edges to zero length, effectively identi-fying the two vertices at the endpoints of each flip edgeto give a single vertex. In this way the number of ver-tices is halved and we get a new graph Γ with only theinternal rotation edges remaining. It seems reasonable toidentify each new vertex with the midpoint of the (nowcontracted) flip edge, which is a C v -symmetric transitionstate, as illustrated in Fig. 2. There are 60 symmetry-equivalent versions of this configuration, and we proposethat the low-energy quantum states can be understood interms of motion between these versions. Each vertex isconnected to precisely four other vertices, as shown alsoin Fig. 2, giving rise to the 4-regular (quartic) quantum
24 531111 111 22 2 22 33 33 334 4444 4 55 55 525 22 221 11 13 3 334 444 5 55 5 ts tstststs 12(-)12(+) 12(-)12(+) 13(-)13(+) 13(-)13(+) 11 22 33
FIG. 2. Local structure of the quantum graphs Γ (blue andred edges) and Γ (black edges). The red edges correspondto the flip motion and the labels indicate which proton isexchanged from a H unit to a CH +3 unit. The blue edgescorrespond to an internal rotation and the labels indicate theH unit which rotates relative to the CH +3 unit in a clockwise(+) or anticlockwise ( − ) fashion. The midpoint of each redflip edge is a C v -symmetric transition state (ts). In goingfrom Γ to Γ , the red edges shrink so that we are left withjust the transition states connected by black edges. FIG. 3. Illustration of the 4-regular quantum graph Γ . Inthis model of the quantum dynamics of CH +5 there is a singleedge length, connecting versions of C v -symmetric transitionstates, midpoints of the flip edge of Γ . graph Γ , which is illustrated in Fig. 3.An alternative way of rationalizing this contractionprocedure is as follows: at the energies we are interestedin, one can show that the Γ wave functions for theenergy eigenstates are approximately constant along theflip edges. In this limit, the boundary conditions of Γ become equivalent to those of Γ (see the SupplementalMaterial for an explanation).We now seek the quantum states corresponding tomotion on the Γ graph. The eigenenergies are foundby solving the time-independent Schr¨odinger equationfor a free particle moving along the edges, with the so-called Neumann boundary conditions [31] imposed on theeigenstates. These conditions are that the wave functionshould be continuous everywhere, with zero total momen-tum flux out of each vertex. As we have already pointedout, Γ is a 4-regular graph with all edges having a com-mon length l = L rot . Perhaps surprisingly, these prop-erties imply that the structure of the quantum energylevels can be determined entirely through considerationof combinatorial properties of the graph.More precisely, given a wave function ψ defined on thegraph Γ and obeying the time-independent Schr¨odingerequation along each edge, −
12 d ψ d x = Eψ, (3)consider the vector of its values at each vertex v =( ψ ( v ) , ψ ( v ) , . . . ). It is straightforward to prove, seethe Supplemental Material, that ψ is an eigenfunctionwith energy E satisfying the Neumann boundary condi-tions if and only if A v = 4 cos (cid:16) √ El (cid:17) v , (4) i.e. , if and only if λ = 4 cos (cid:16) √ El (cid:17) is an eigenvalueof the adjacency matrix A for the graph Γ , with v in the corresponding eigenspace. A is simply a matrixwhose elements indicate whether given pairs of verticesare connected by an edge or not, and is a familiar conceptin elementary graph theory [34]. Its entries are given by( A ) ij = (cid:40) v i , v j connected0 otherwise . (5)The factor of 4 on the RHS of Eq. (4) corresponds tothe fact that each vertex connects to precisely 4 othervertices.Equation (4) therefore relates the quantum spectrum (the eigenvalues of the Hamiltonian) to the so-called com-binatorial spectrum (the eigenvalues of the adjacency ma-trix). The combinatorial spectrum is a concept alreadyutilized in molecular spectroscopy [35], and only dependson the connectivity of the graph as encoded in A .To find the combinatorial spectrum of Γ , we lookfor roots of the characteristic polynomial χ A ( λ ) =det ( λ I − A ) associated with the adjacency matrix A .An explicit expression for A is easily derived by consid-ering paths of the form illustrated in Fig. 2. In the end,we obtain χ A ( λ ) = (cid:0) λ − λ + 16 (cid:1) (cid:0) λ − λ + 16 (cid:1) (cid:0) λ − (cid:1) (cid:0) λ − (cid:1) , (6)and the full combinatorial spectrum is given in Table II.Table II also shows the dimensions of the correspondingeigenspaces and the irreps corresponding to the symme-try action of the MS group S ∗ by graph automorphisms.We pause here to note the striking similarity betweenthe symmetry information of Tables I and II. First, note TABLE II. The combinatorial spectrum of the quantum graph Γ , where dim ( λ ) gives the degeneracy of a given eigenvectorcorresponding to the eigenvalue λ [see Eq. (6)]. λ dim ( λ ) S ∗ irrep4 1 A +1 √ G − (cid:0) √ (cid:1) H +112 (cid:0) − √ (cid:1) H − − √ G +1 H +2 ⊕ I − − H − ⊕ I + − √ G − (cid:0) − √ (cid:1) H +112 (cid:0) − − √ (cid:1) H − − − √ G +1 − A − that the combinatorial spectrum splits into positive λ andnegative λ , with each corresponding to a total eigenspacedimension of 30. Moreover, the eigenspaces associatedwith positive λ transform in precisely the same irreps asBlock 1 of Table I, while those associated with negative λ transform precisely like Block 2. Thus, purely combi-natorial properties of the quantum graph Γ have cap-tured the block structure of the lowest vibrational statesof CH +5 . Even more interestingly, we seem to have anexplanation for the curious relationship between Block 1states and Block 2 states: this corresponds to λ → − λ (see the Supplemental Material). The symmetry of thespectrum of Γ under λ → − λ is actually a simple con-sequence [34] of the fact that the quantum graph Γ is bipartite : the set of vertices V can be divided into twodisjoint and independent sets A and B such that everyedge connects a vertex in A to one in B . The set A is mapped to the set B by any odd permutation of theprotons.Equation (4) relates the combinatorial spectrum to thequantum spectrum, which is nicely illustrated in Fig. 4.In particular, we can see the consequences of the λ →− λ symmetry for the quantum energy levels: each statein Block 1 comes with a partner in Block 2, with theircorresponding values of √ El being related by reflectionin the line √ El = π/
2. In particular, the sum of thesquare roots of their energies is independent of which pairwe pick, implying that the dimensionless ratios (cid:112) E ( I − ) + (cid:112) E ( I + ) (cid:113) E (cid:0) H +1 (cid:1) + (cid:113) E (cid:0) H − (cid:1) , (cid:113) E (cid:0) H +2 (cid:1) + (cid:113) E (cid:0) H − (cid:1)(cid:113) E (cid:0) H +1 (cid:1) + (cid:113) E (cid:0) H − (cid:1) , . . . (7)are all equal to 1 in the Γ model. This compares veryfavourably with the variational seven-dimensional model FIG. 4. Illustration of the block structure and the symmetryproperties of the spectrum of the quantum graph Γ . Blackdots indicate energies of the quantum states. [7, 12, 14] results (cid:113) E (cid:0) A +1 (cid:1) + (cid:113) E (cid:0) A − (cid:1)(cid:113) E (cid:0) H +1 (cid:1) + (cid:113) E (cid:0) H − (cid:1) ≈ √ √ √
20 + √ ≈ . , (8) (cid:113) E (cid:0) G − (cid:1) + (cid:113) E (cid:0) G +1 (cid:1)(cid:113) E (cid:0) H +1 (cid:1) + (cid:113) E (cid:0) H − (cid:1) ≈ √
10 + √ √
20 + √ ≈ . , (9) (cid:113) E (cid:0) H − (cid:1) + (cid:113) E (cid:0) H +1 (cid:1)(cid:113) E (cid:0) H +1 (cid:1) + (cid:113) E (cid:0) H − (cid:1) ≈ √
41 + √ √
20 + √ ≈ . , (10) (cid:113) E (cid:0) G +1 (cid:1) + (cid:113) E (cid:0) G − (cid:1)(cid:113) E (cid:0) H +1 (cid:1) + (cid:113) E (cid:0) H − (cid:1) ≈ √
49 + √ √
20 + √ ≈ . , (11) (cid:112) E ( I − ) + (cid:112) E ( I + ) (cid:113) E (cid:0) H +1 (cid:1) + (cid:113) E (cid:0) H − (cid:1) ≈ √
58 + √ √
20 + √ ≈ . , (12)and (cid:113) E (cid:0) H +2 (cid:1) + (cid:113) E (cid:0) H − (cid:1)(cid:113) E (cid:0) H +1 (cid:1) + (cid:113) E (cid:0) H − (cid:1) ≈ √
59 + √ √
20 + √ ≈ . . (13)In this Letter we have drastically simplified the quan-tum graph model of the low-energy rovibrational quan-tum dynamics of CH +5 by reducing the original 120-vertexquantum graph to a 60-vertex graph, Γ . Γ was con-structed by shrinking the edges corresponding to the flipmotion that exchanges a pair of protons between the H and CH +3 units of the equilibrium structure of CH +5 . Thequantum states of Γ are obtained in analytic form, withthe structure of the energy levels depending only on com-binatorial properties. The bipartite structure of Γ givesa natural explanation for symmetries in the vibrationalenergy levels of CH +5 , in good agreement with the resultsof numerically exact variational nuclear dynamics com-putations. Note that neither the variational quantum-chemical computations [7, 12, 14] nor the quantum-graphmodels [15, 27] yield only the Pauli-allowed states of CH +5 (states with A ± , G ± , and H ± symmetry have non-zerospin-statistical weights), and thus our discussion focusedon all possible states; the non-existing states can be fil-tered out a posteriori .The work of JIR was supported by the EPSRCgrant CHAMPS EP/P021123/1. The work performedin Budapest received support from NKFIH (grant no.K119658) and from the ELTE Institutional ExcellenceProgram (TKP2020-IKA-05) financed by the HungarianMinistry of Human Capacities. ∗ Corresponding [email protected].[1] G. A. Olah and R. H. Schlosberg, J. Am. Chem. Soc. ,2726 (1968).[2] G. A. Olah, G. Klopman, and R. H. Schlosberg, J. Am.Chem. Soc. , 3261 (1969).[3] G. A. Olah, Carbocations and Electrophilic Reactions (VCH-Wiley Publishers, Weinheim, 1974).[4] G. A. Olah, “My search for carbocations and their rolein chemistry,” in
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