Vibrational angular momentum level densities of linear molecules
aa r X i v : . [ phy s i c s . a t m - c l u s ] D ec Vibrational angular momentum level densities of linearmolecules
Klavs Hansen Center for Joint Quantum Studies and Department of Physics, School of Science,Tianjin University, 92 Weijin Road, Tianjin 300072, ChinaandQuantum Solid-State Physics, Department of Physics and Astronomy,KU Leuven, 3001 Leuven, Belgium
Piero Ferrari
Quantum Solid-State Physics, Department of Physics and Astronomy,KU Leuven, 3001 Leuven, Belgium18/12/2020, 01:49
Abstract
While linear molecules in their vibrational ground state cannot carry an-gular momentum around their symmetry axis, the presence of vibrationalexcitations can induce deformations away from linearity and therefore alsoallow angular momentum along the molecular axis. In this work, a recurrencerelation is established for the calculation of the vibrational level densities (den-sities of states) of linear molecules, specified with respect to both energy andangular momentum. The relation is applied to the carbon clusters of sizes n = 4 , , Corresponding author; email: [email protected] ntroduction It is a commonly accepted truth that linear molecules can only rotate around twoaxes. As stated this is true. The reason is the simple fact that the principal momentof inertia around the molecular symmetry axis is zero, or at least as close to zeroas the presence of off-axis electrons allows. Disregarding the rotational motion ofelectrons, which have quantum energies that are usually far beyond molecular rota-tional energies, no molecular rotation can occur around an axis on which all nucleiin a molecule are located. The statement is particularly transparent in quantumtheory with its quantized angular momentum [1].This simple result does not, however, imply that a linear molecule can onlycarry angular momentum in the two directions perpendicular to the symmetry axis.Excitation of the doubly degenerate perpendicular vibrational modes will inducedeformations of the molecule away from linearity and generate non-zero momentsof inertia, hence allowing for a non-zero angular momentum around the symmetryaxis.This is a special case of the general possibility of vibrations carrying angularmomentum which was already established for the eigenmodes of an elastic spherewith the work in ref. [2]. Some of these predicted vibrational modes were onlyconfirmed much later [3]. Also the presence of angular momentum in the elementaryexcitations of helium droplets play a role for their thermal properties, as analyzedin detail in [4].The spectroscopic implications of the phenomenon in molecular context has beenthe subject of a number of studies. Studies of non-linear molecules were reportedin [5–13], and linear molecules in [7, 14–16], including experimental studies of bothspectroscopic nature and fragmentation processes where the vibrational angular mo-mentum plays a role [14, 17]. Most of this previous work on the subject has beendedicated to the energies and degeneracies of the single modes for spectroscopic pur-poses. The question has also been treated for bulk matter [18], for which phononinteractions with spin is of interest [19]. 2n contrast, scant attention has been devoted to the effect of the vibrationalangular momentum of molecules in connection with reactivity (see [20], though).However, the energy and angular momentum resolved level densities are essential inorder to implement the relevant conservation laws in thermally activated reactionsof both bi- and unimolecular nature.For a linear molecule composed of n atoms, there are n − n displacements of n atoms along the molecular axis, of which one mode is a transla-tional motion) and 2( n −
2) bending modes (2 n modes, of which two with no nodesin the CM system are translational, and two with one node are rotations aroundthe two axes perpendicular to the molecular axis), in agreement with the stan-dard counting of non-vibrational modes (the term bending mode or perpendicularmode/motion will be used to designate all these 2( n −
2) modes, although a numberof these modes are really best described as transverse waves). The term molecularaxis will refer to the ground state axis around which the molecules oscillate in the2( n −
2) bending modes.The enumeration places the degrees of freedom of a linear molecule in four classes.One comprises the three translational degrees of freedom, which decouple rigorouslyfrom all other degrees of freedom by the translational invariance of the equations ofmotion. A second contains the two rotational motions around the axes perpendicularto the molecular axis. The third class contains the vibrational motion along themolecular axis (longitudinal motion, n − n −
2) bendingmodes of interest here, carrying both vibrational energy and angular momentum.The level density, which is the focus of this article, counts the number of statesfor a given energy and is, therefore, the microcanonical partition function. Forsmall, isolated systems it assumes a particularly important role because the differ-ence between canonical and microcanonical quantities become important for smallsystems. This holds for issues concerning the energy content but perhaps even morefor questions concerning angular momentum, as this quantity does not appear inthe description of the macrostate of a canonical system.3he total angular momentum resolved vibrational level density, ρ tot ( E, L z ), isobtained as a convolution of the function for the longitudinal modes, ρ l ( E ), and theones for the bending motion, ρ p ( E, L z ), as ρ tot ( E, L z ) = R ρ l ( E − ǫ ) ρ p ( E, L z )d ǫ . Animplementable procedure for this will be given below, and its use demonstrated witha calculation of three small carbon clusters. Angular momentum along the z-axis, L z The angular momentum eigenstates for a bending mode in the harmonic approx-imation, which will be considered sufficient here, are those of a two-dimensionalharmonic oscillator due to the double degeneracy of the bending modes. It is aninteresting fact that the large quantum number limits of such systems are not theclassical limits, and that semiclassical quantization does not give the correct answersin that limit [21]. As it is, the exact spectrum can be found in the harmonic approx-imation without taking this limit. To do so, each degenerate pair of bending modescan be considered separately. Orienting the coordinate system with the z -axis alongthe molecular axis, the operator for the z -projection of the angular momentum iswritten as L z = xp y − yp x (1)= i ~ (cid:0) a † x + a x (cid:1) (cid:0) a † y − a y (cid:1) − i ~ (cid:0) a † y + a y (cid:1) (cid:0) a † x − a x (cid:1) = i ~ (cid:0) a † y a x − a † x a y (cid:1) , where a † x , a x are the raising and lowering operators for the oscillator vibrating alongthe x -direction, and similarly for the operators with subscript y . L z commutes withthe Hamiltonian:[ L z , H ] = i ~ ω [ (cid:0) a † y a x − a † x a y (cid:1) , a † y a y + a † x a x + 1] = 0 . (2)The Hilbert space of the transverse modes is spanned by the vibrational states | n, m i , where the first integer gives the energy of the motion in the x -direction and4he second in the y -direction. It is clear from the expression for L z that | n, m i statesare not eigenstates of L z . An explicit calculation shows this: L z | n, m i = i ~ (cid:0) a † y a x − a † x a y (cid:1) | n, m i (3)= i ~ (cid:16) √ m + 1 √ n | n − , m + 1 i − √ n + 1 √ m | n + 1 , m − i (cid:17) . As the operator raises one vibrational quantum number and lowers the other, andthe two oscillators are degenerate, it is equally clear that the angular momentumeigenstates are composed of eigenstates with the same energy, n + m = constant,of which there are n + m + 1. A direct diagonalization of the three lowest levelswith energies 1, 2 and 3 times ~ ω (zero point energy included) gives the angularmomenta 0, ± ~ , and 0 , ± ~ . The general case can be calculated by construction ofthe operators [22, 23] a d ≡ √ a x − i a y ) , (4) a g ≡ √ a x + i a y ) , and the associated number operators n d ≡ a † d a d , (5) n g ≡ a † g a g . The Hamiltonian and the angular momentum operators can be written in terms ofthese as H = ~ ω ( n d + n g + 1) , (6) L z = ~ ( n d − n g ) . For a given energy, the angular momentum states therefore differ by 2 ~ , as thecalculated example also suggested. This spacing, together with the number ofstates, defines the angular momentum spectrum uniquely as having the eigenval-ues N ~ , ( N − ~ , ...., − N ~ for states with the total energy E = ~ ω ( N + 1).5 evel densities For molecules or clusters for which the vibrational spectrum is known, the vibrationalcontribution to the level density, ρ , can be calculated with the Beyer-Swinehartalgorithm [24]. The algorithm is a convolution of level densities of the individualuncoupled degrees of freedom in the form of normal modes. Due to the discretenature of vibrational excitations, this convolution takes the form of a sum over levels.The summation is repeated recursively for each new mode included. Energy is theonly physical argument in the procedure. The intermediate steps in the recurrenceare labeled by the number of modes, M , that have been included. This adds theinteger M as an argument to the function [24]: ρ = ρ ( M, E ) . (7)The procedure can be condensed by contracting terms on the right hand side andthe resulting recurrence written as [25] ρ ( M, E ) = ρ ( M, E − ~ ω M ) + ρ ( M − , E ) . (8)Adding angular momentum expands this array with that quantum number. In total,the procedure then makes use of the energy, E , the angular momentum L z , and theinteger M labeling the vibrational modes included in the recurrence: ρ = ρ ( M, E, L z ) . (9)As usual, angular momentum is most conveniently given in units of ~ and is thereforerepresented by integers. This will be used for the argument of ρ . For this choice,the dimension of ρ is energy to the power -1, where the unit of energy is determinedby the value chosen for the input vibrational frequencies. For other purposes thanindexing ρ , the dimension of L z will be that of Planck’s constant.The rotating bending modes are included by extending summations over modesto include all possible angular momenta. As the modes come pairwise, the modeindex M changes by two for each iteration. Adding one more quantum of energy6ill change the angular momentum by either +1 or -1: ρ ( M, E, L z ) = ρ ( M − , E, L z ) (10)+ X i =0 ρ ( M − , E − ~ ω M , L z + 2 i − X i = − ρ ( M − , E − ~ ω M , L z + 2 i )+ X i = − ρ ( M − , E − ~ ω M , L z + 2 i − X i = − ρ ( M − , E − ~ ω M , L z + 2 i ) ... The first term on the right hand side of this equation is the contribution from thepartitionings where there is no excitation in either of the two added modes. Thesecond term gives the two contributions from a single excitation in the two modes,distributed as a negative or a positive angular momentum quantum. The followingterms are generated after the same principle, with n + 1 different contributions tothe angular momentum for an added energy of n .Also this recurrence relation can be condensed. The sum contains a subset ofterms that adds up to ρ ( M, E − ~ ω M , L z − ρ ( M, E − ~ ω M , L z + 1). Collecting these two terms leaves out the first term on the right handside, ρ ( M − , E, L z ), and it double-counts terms that on inspection are found toadd up to ρ ( M, E − ~ ω M , L z ). By adding the missing and subtracting the double-counted terms, it is therefore possible to write the recurrence in the much morecompact form ρ ( M, E, L z ) = ρ ( M − , E, L z ) + ρ ( M, E − ~ ω M , L z −
1) (11)+ ρ ( M, E − ~ ω M , L z + 1) − ρ ( M, E − ~ ω M , L z ) . A numerical calculation with this expression comprises three loops nested in theorder of appearance of the arguments in ρ . Computationally, two matrices with7ndices E, L z are needed. If only the modes carrying angular momenta are required,the recurrence is started with the initial values ρ (0 , j, k ) = δ j, δ k, , where the δ ’s areKronecker’s δ . If the non-angular momentum-carrying modes should be includedinto the level density, Eq.8 is used with the angular momentum index set to zero.After convoluting the M non-angular momentum modes the result, ρ l ( M, E, ρ (0 , ,
0) = 1etc., i.e. changing the initial conditions to ρ (0 , E, j ) = ρ l ( M, E, j ) δ j, . Application to C , , , , , as representativeexamples of linear carbon clusters. For the neutral species, linear structures are thelowest energy carbon cluster conformers up to n = 7 [26], and all three clusters areexpected to have linear ground states, without any Jahn-Teller deformations.The ground state and the vibrational frequencies of the clusters C n ( n = 4 , , , together with the energy of the mode(in cm − ), calculated for the electronic ground state. At the top of the figure thevibrational ground-state structure is shown, with aligned and almost equidistantcarbon atoms.The calculated vibrational frequencies are listed in Table 1, together with theexperimental values, given in [30]. While the lowest-energy modes are well repro-duced by the calculations, some deviations are seen for the higher frequency modes.Overall, however, the calculated values are in fair agreement with the experimental8
03 cm -1
206 cm -1
369 cm -1
486 cm -1 Figure 1: The relative amplitudes of the normal coordinates for the fourdoubly degenerate vibrational modes of C . The representations of themodes are chosen so all motion of the atoms is in one plane. The topgeometry corresponds to the vibrational ground-state structure. values. Many of the modes have a dipole moment of zero, preventing detectionby infrared spectroscopy. For this reason, the analysis here will use the calculatedfrequencies.Fig.(2) shows the results of a calculation of the angular momentum specified leveldensities of C , , with Eq.11 for a series of total excitation energies. The highestenergies in the calculations were chosen to be on the order of magnitude of theexcitation energies of clusters that decay on typical mass spectroscopic time scalesof tens to hundreds of microseconds. A zoomed view of the values for C around L z = 0 is shown in Fig.(3). 9able 1: Calculated vibrational quantum energies of C , C and C in cm − . The doubly degenerate bending modes are indicated inbold. The experimental values [30] are shown below the theoreticalvalues calculated here.n ~ ω (cm − )4
169 346
937 1589 21204 Exp. 160, 339, 1549, 20326
103 206 369 486
668 1225 1729 2028 21796 Exp. 90, 246, 637, 1197, 1694, 1960, 20617
77 168 264 527
Microcanonical temperature, L z -resolved Although the values shown in Figs.(2,3) are microcanonical, it is nevertheless possi-ble and occasionally also convenient to define a temperature for such systems. Thistemperature is particularly useful for the interpretation of measured kinetic energyrelease distributions (KERDs), because it allows these distributions to be writtenwith what is effectively a Boltzmann factor multiplied by phase space factors etc.[31]. The definition is (with k B set to unity) T − ≡ ∂ρ ( E, L z ) ∂E (cid:12)(cid:12)(cid:12)(cid:12) L z . (12)Fig.(4) shows the microcanonical temperature for different angular momentumsectors of the level density. The slopes are clearly all very similar. They are alsovery close to the value of n − n harmonic oscillators (the -1is the difference between canonical and microcanonical heat capacities, see [32] fordetails). The two other cluster sizes calculated give similar results. Although heatcapacities are thus fairly insensitive to the precise value of the angular momentum,the temperature has a clear angular momentum dependence. The higher the angu-10ar momentum, the lower the temperature. In conjunction with the constant heatcapacity, this suggests that the angular momentum comes with a price. Effectively,it ties up energy and reduces the available phase space for the other vibrations. Forthe C clusters, the lowering of the microcanonical temperature corresponds to ashift in energy of about 250 cm − for each time two units of ~ are added, in the highenergy limit. This is reasonably close to the average frequency of 291 cm − of thebending modes to assign this shift to the excitation required to reach the specifiedangular momentum.As one application of the results derived, the effective Boltzmann factor that en-ters the kinetic energy release distributions will be calculated. A number of theoriesare available from the literature for the calculation of the rest of these distributions.A discussion of these theories will lead us too far astray and we will simply considerthe Boltzmann factor here.We will limit ourselves to consider the expression of a product arising froma reactant with zero initial vibrational angular momentum. The function to beapproximated is then ρ ( E − ε, L z ), where E is the product energy before deductionof ε , the kinetic energy released, and L z is the resulting vibrational mode angularmomentum. The logarithm of ρ is expanded to first order in ε to give ρ ( E − ε, L z ) ≈ ρ ( E, L z ) exp (cid:18) − εT ( E, L z ) (cid:19) . (13)The temperatures are given by the calculated values in Fig.(4) for C . The temper-atures decrease approximately with 11 cm − each time L z increases by one unit of ~ . The approximation sign in Eq.13 refers to the fact that the rate of decrease with L z for low values (below 20 ~ ) is less. As the values taken by L z cover a wider range(see top line of Fig.(3), we will ignore this and represent the temperature for thecase of C as: T ( E, L z ) = T ( E, L z = 0) −
11 cm − ~ L z . (14)The coefficient of 11 cm − is on the order of the price in energy per unit of angularmomentum, which we saw above to be about 250/2 cm − , divided by the number11f vibrational degrees of freedom, 3 n − − , in good agreement withthe fitted value of 11 cm − . For the C case we can therefore write Eq.13 as ρ ( E − ε, L z ) ≈ ρ ( E, L z ) exp − εT ( E, − cm − ~ L z ! . (15)The result in Eq.15 shows that the effective temperature as measured withKERDs for a given total excitation energy decreases with increasing angular mo-mentum. The L z dependence at high excitation energies can be fitted with thefunction ρ ( E, L z ) ≈ ρ ( E,
0) exp (cid:0) − a | L z | . (cid:1) , (16)around the peak at L z = 0. The value of the coefficient is a = 6 × − for thehighest energy curve shown in Fig.(3). This can then be used as is or expandedfurther with L z → L z + δL z .The factor ρ ( E, L z ) is part of the normalization of the KERD’s, and the entiredistribution will then be determined by the two exponentials and the prefactor whichwill not be discussed in detail here. Disregarding this prefactor, the decay will bebiased toward a reduction of the absolute value of the angular momentum.The L z distributions will vary with the experimental conditions. To neverthelessgive some information on the effect of the vibrational angular momentum, the effectwill be illustrated with a schematic calculation. The KER distributions will berepresented by the derived Boltzmann factor and the preexponential phase spacefactor combined with the kinematic speed factor [31]. These two factors combine toa factor of kinetic energy to the power one. For the product C with the final stateenergy E , L z and fragment translational kinetic energy ε this becomes: P ( E, L z , ǫ )d εproptoρ ( E, L z ) ε exp − εT ( E, − cm − ~ L z ! d ε, (17)where L z is the final state vibrational angular momentum. This is then the expres-sion for zero initial L z . The curves for L z = 0 , ~ , E = 20000 cm − are shown inFig.(5). 12 anonical values The canonical equilibrium values are likewise interesting. With the level structurefound above, the canonical partition function is easily calculated and the angularmomentum specified populations, P ( L z = n ~ ) ≡ P ( n ) found for one degenerate pairof modes with frequency ω to be P (0) = (cid:0) − e − β ~ ω (cid:1) − Z , (18) P ( ± n, n >
0) = e − nβ ~ ω P (0) , where the total partition function is Z = (cid:0) − e − β ~ ω (cid:1) − . (19)The zero of the energy is set equal to the zero point energy of the oscillators here.The mean value of L z given by these distributions is obviously zero. The meansquare is calculated to1 ~ h L z i = P (0) ∞ X n = −∞ n e −| n | β ~ ω (20)= 2 P (0) (cid:18) ~ ω (cid:19) ∂ ∂β (cid:0) − e − β ~ ω (cid:1) − = 2e − β ~ ω − β ~ ω (1 − e − β ~ ω ) (1 − e − β ~ ω ) . The square root of this function, h L z i / / ~ , shown in Fig.(6), is an almost straightline as a function of T / ~ ω , with a small offset close to zero. Unity is reachedat T = 0 . ~ ω . This means that the mode will contribute with angular momentaquanta even for temperatures below the value corresponding to the vibrational quan-tum energies. For the calculated lowest frequency of C the value T = 0 . ~ ω cor-responds to a temperature of 112 K, and for the highest frequency to 531 K. Bothof these are fairly modest temperatures for carbon chemistry.13 ummary Eq.11 is the main results of this work. The numerical prescription derived shouldbe applicable to any linear molecule with known vibrational frequencies. If thetotal vibrational level density specified with respect to both energy and angularmomentum is needed, this is calculated by a simple convolution of the level densitiesfor the modes that carry angular momentum with those that do not.
PF acknowledges a Postdoctoral grant from the Research Foundation – Flanders(FWO).
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Springer Series on Atomic, Optical, and Plasma Physics (Springer, Dordrecht,2018).[32] J. U. Andersen, E. Bonderup, and K. Hansen, J. Chem. Phys. , 6518 (2001),URL https://doi.org/10.1063/1.1357794 .16 igure 2: The angular momenta specified level density for different totalexcitation energies for the clusters indicated in the frames, all on iden-tical scales. The curves are from top to bottom for E = 44000 cm − down to 2000 cm − in steps of 2000 cm − in all three cases. L z is givenin units of ~ . igure 3: A zoomed view of Fig.(2) for C . igure 4: The microcanonical temperature for the angular momentumspecified states. The values are calculated numerically with a finitederivative over 1000 cm − , centered at the abscissa values, with Eq.12.The curves show the values for L z = 0 ~ to 100 ~ , in steps of 20 ~ (top tobottom). igure 5: The schematic kinetic energy release distributions calculatedfor C with final state energy E and the two L z / ~ values of 0 (full line)and 40 (dotted line). The energy is 20000 cm − . The microcanonicaltemperature is calculated to 2238 cm − for this energy. The values of ρ ( E,
0) and ρ ( E,
40) differ by a factor of ca. 3 for this energy. Thetemperatures differ by 20 %. The curves are normalized to identicalareas. igure 6: The root-mean-square of L z in units of ~ as a function of thereduced temperature for a single pair of degenerate oscillators.as a function of thereduced temperature for a single pair of degenerate oscillators.