Vibrational mode and collision energy effects on reaction of H2CO+ with C2D4

We report the effects of collision energy (Ecol) and five different H2CO+ vibrational modes on the reaction of H2CO+ with C2D4 over the center-of-mass E(col) range from 0.1 to 2.1 eV. Properties of various complexes and transition states were also examined computationally. Seven product channels are observed. Charge transfer (CT) has the largest cross section over the entire energy range, substantially exceeding the hard sphere cross section at high energies. Competing with CT are six channels involving transfer of one or more hydrogen atoms or protons and one involving formation of propanal, followed by hydrogen elimination. Despite the existence of multiple deep wells on the potential surface, all reactions go by direct mechanisms, except at the lowest collision energies, where short-lived complexes appear to be important. Statistical complex decay appears adequate to account for the product branching at low collision energies, however, even at the lowest energies, the vibrational effects are counter to statistical expectations. The pattern of Ecol and vibrational mode effects provide insight into factors that control reaction and interchannel competition.     

Reaction of formaldehyde cation with molecular hydrogen: effects of collision energy and H2CO+ vibrations

The effects on the title reaction of collision energy (E(col)) and five H(2)CO(+) vibrational modes have been studied over a center-of-mass E(col) range from 0.1 to 2.3 eV. Electronic structure and Rice-Ramsperger-Kassel-Marcus calculations were used to examine properties of various complexes and transition states that might be important. Only the hydrogen abstraction (HA) product channel is observed, and despite being exoergic, HA has an appearance energy of approximately 0.4 eV, consistent with a transition state found in the electronic structure calculations. A precursor complex-mediated mechanism might possibly be involved at very low E(col), but the dominant mechanism is direct over the entire E(col) range. The magnitude of the HA cross section is strongly, and mode specifically affected by H(2)CO(+) vibrational excitation, however, vibrational energy has no effect on the appearance energy.     

State-selective preparation of NO2+ and the effects of NO2+ vibrational mode on charge transfer with NO

Two color resonance-enhanced multiphoton ionization (REMPI) scheme of NO(2) through the E (2)Sigma(u)(+) (3psigma) Rydberg state was used to prepare NO(2)(+) in its ground and (100), (010), (02(0)0), (02(2)0), and (001) vibrational states. Photoelectron spectroscopy was used to verify >96% state selection purity, in good agreement with results of Bell et al. for a similar REMPI scheme. The effects of NO(2)(+) vibrational excitation on charge transfer with NO have been studied over the center-of-mass collision energy (E(col)) range from 0.07 to 2.15 eV. Charge transfer is strongly suppressed by collision energy at E(col) < approximately 0.25 eV but is independent of E(col) at higher energies. Mode-specific vibrational effects are observed for both the integral and differential cross-sections. The NO(2)(+) bending vibration strongly enhances charge transfer, with enhancement proportional to the bending quantum number, and is not dependent on the bending angular momentum. The enhancement results from increased charge transfer probability in large impact parameter collisions that lead to small deflection angles. The symmetric stretch also enhances reaction at low collision energies, albeit less efficiently than the bend. The asymmetric stretch has virtually no effect, despite being the highest-energy mode. A model is proposed to account for both the collision energy and the vibrational state dependence.     

Elastic and charge transfer processes in H+ + CO collisions

Proton and hydrogen atom time-of-flight spectra in collision energy range of E(trans) = 9.5-30 eV show that the endoergic charge transfer process in the H+ + CO system is almost an order of magnitude less probable than the elastic scattering [G. Niedner-Schatteburg and J. P. Toennies, Adv. Chem. Phys. LXXXII, 553 (1992)]. Ab initio computations at the multireference configuration interaction level have been performed to obtain the ground- and several low-lying excited electronic state potential energy curves in three different molecular orientations namely, H+ approaching the O-end and the C-end (collinear), and H+ approaching the CO molecule in perpendicular configuration with fixed CO internuclear distance. Nonadiabatic coupling terms between the ground electronic state (H+ + CO) and the three low-lying excited electronic states (H + CO+) have been computed and the corresponding diabatic potentials have been obtained. A time-dependent wavepacket dynamics study is modeled first involving only the ground and the first excited states and then involving the ground and the three lowest excited states at the collision energy of 9.5 eV. The overall charge transfer probability have been found to be approximately 20%-30% which is in qualitative agreement with the experimental findings.     

Experimental and theoretical studies of charge transfer and deuterium ion transfer between D2O+ and C2H4

The charge transfer and deuterium ion transfer reactions between D(2)O(+) and C(2)H(4) have been studied using the crossed beam technique at relative collision energies below one electron volt and by density functional theory (DFT) calculations. Both direct and rearrangement charge transfer processes are observed, forming C(2)H(4) (+) and C(2)H(3)D(+), respectively. Independent of collision energy, deuterium ion transfer accounts for approximately 20% of the reactive collisions. Between 22 and 36 % of charge transfer collisions occur with rearrangement. In both charge transfer processes, comparison of the internal energy distributions of products with the photoelectron spectrum of C(2)H(4) shows that Franck-Condon factors determine energy disposal in these channels. DFT calculations provide evidence for transient intermediates that undergo H/D migration with rearrangement, but with minimal modification of the product energy distributions determined by long range electron transfer. The cross section for charge transfer with rearrangement is approximately 10(3) larger than predicted from the Rice-Ramsperger-Kassel-Marcus isomerization rate in transient complexes, suggesting a nonstatistical mechanism for H/D exchange. DFT calculations suggest that reactive trajectories for deuterium ion transfer follow a pathway in which a deuterium atom from D(2)O(+) approaches the pi-cloud of ethylene along the perpendicular bisector of the C-C bond. The product kinetic energy distributions exhibit structure consistent with vibrational motion of the D-atom in the bridged C(2)H(4)D(+) product perpendicular to the C-C bond. The reaction quantitatively transforms the reaction exothermicity into internal excitation of the products, consistent with mixed energy release in which the deuterium ion is transferred in a configuration in which both the breaking and the forming bonds are extended.     

Vibrational effects on the reaction of NO(2)(+) with C(2)H(2): effects of bending and bending angular momentum

NO(2)(+) in six different vibrational states was reacted with C(2)H(2) over the center-of-mass energy range from 0.03 to 3.3 eV. The reaction, forming NO(+)+C(2)H(2)O and NO+C(2)H(2)O(+), shows a bimodal dependence on collision energy (E(col)). At low E(col), the reaction is quite inefficient (<2%) despite this being a barrierless, exoergic reaction, and is strongly inhibited by E(col). For E(col)> approximately 0.5 eV, a second mechanism turns on, with an efficiency reaching approximately 27% for E(col)>3 eV. The two reaction channels have nearly identical dependence on E(col) and NO(2)(+) vibrational state, and identical recoil dynamics, leading to the conclusion that they represent a single reaction path throughout most of the collision. All modes of NO(2)(+) vibrational excitation enhance both channels at all E(col), however, the effects of bend (010) and bend overtone (02(0)0) excitation are particularly strong (factor of 4). In contrast, the asymmetric stretch (001), which intuition suggests should be coupled to the reaction coordinate, leads to only a factor of approximately 2 enhancement, as does the symmetric stretch (100). Perhaps the most surprising effect is that of the bending angular momentum, which strongly suppress reaction, even though both the energy and angular momentum involved are tiny compared to the collision energy and angular momentum. The results are interpreted in light of ab initio and Rice-Ramsperger-Kassel-Marcus calculations.     

H+ versus D+ transfer from HOD+ to CO2: bond-selective chemistry and the anomalous effect of bending excitation

Reactions of HOD(+) with CO(2) have been studied for HOD(+) in its ground state, and with one quantum of excitation in each of its vibrational modes: (001)--predominantly OH stretch, 0.396 eV; (010)--bend, 0.153 eV; and (100)--predominantly OD stretch, 0.293 eV. Integral cross sections and product recoil velocities were recorded for collision energies from threshold to 3 eV. The cross sections for both H(+) and D(+) transfer rise with increasing collision energy from threshold to ～1 eV, then become weakly dependent of the collision energy. All three vibrational modes enhance the total reactivity, but quite mode specifically. The H(+) transfer reaction is enhanced by OH stretch excitation, whereas OD stretch excitation has little effect. Conversely, the D(+) transfer reaction is enhanced by OD stretch excitation, while the OH stretch has little effect. Excitation of the bend strongly enhances both channels. The effects of the stretch excitations are consistent with previous studies of neutral HOD mode-selective chemistry, and can be at least qualitatively understood in terms of a late barrier to product formation. The fact that bend excitation produces the largest overall enhancement is surprising, because this is the lowest energy excitation, and is not obviously connected with the reaction coordinates for either H(+) or D(+) transfer. A rationalization in terms of the effects of water distortion on the potential surface is proposed.     

Vibrational mode and collision energy effects on reaction of H2CO+ with CO2

The effects of collision energy (Ecol) and five different modes of H2CO+ vibration on the title reaction have been studied over the center-of-mass Ecol range from 0.1 to 3.2 eV, including measurements of product ion recoil velocity distributions. Electronic structure and Rice-Ramsperger-Kassel-Marcus calculations were used to examine properties of various complexes and transition states that might be important along the reaction coordinate. Two product channels are observed, corresponding to Hydrogen Transfer (HT) and Proton Transfer (PT). Both channels are endothermic with similar onset energies of approximately 0.9 eV; however, HT dominates over the entire Ecol range and accounts for 70-85% of the total reaction cross section. Both HT and PT occur by direct mechanisms over the entire Ecol range, and have similar dependence on reactant vibrational and collision energy. Despite these similarities, and the fact that the two channels are nearly isoenergetic and differ only in which product moiety carries the charge, their dynamics appear quite different. PT occurs primarily in large impact parameter stripping collisions, where most of the available energy is partitioned to product recoil. HT, in contrast, results in internally hot products with little recoil energy and a more forward-backward symmetric product velocity distribution. Vibration is found to affect the reaction differently in different collision energy regimes. The appearance thresholds are found to depend only on total energy, i.e., all modes of vibration are equivalent to Ecol. With increasing Ecol, vibrational energy becomes increasingly effective, relative to Ecol, at driving reaction. For HT, this transition occurs just above threshold, while for PT it begins at roughly twice the threshold energy.     

H+ versus D+) transfer from HOD+ to N2: mode- and bond-selective effects

Reactions of HOD(+) with N(2) have been studied for HOD(+) in its ground state and with one quantum of excitation in each of its vibrational modes: (001)--predominately OH stretch, 0.396 eV, (010)--bend, 0.153 eV, and (100)--predominately OD stretch, 0.293 eV. Integral cross sections and product recoil velocities were recorded for collision energies from threshold to 4 eV. The cross sections for both H(+) and D(+) transfer rise slowly from threshold with increasing collision energy; however, all three vibrational modes enhance reaction much more strongly than equivalent amounts of collision energy and the enhancements remain large even at high collision energy, where the vibration contributes less than 10% of the total energy. Excitation of the OH stretch enhances H(+) transfer by a factor of ～5, but the effect on D(+) transfer is only slightly larger than that from an equivalent increase in collision energy, and smaller than the effect from the much lower energy bend excitation. Similarly, OD stretch excitation strongly enhances D(+) transfer, but has essentially no effect beyond that of the additional energy on H(+) transfer. The effects of the two stretch vibrations are consistent with the expectation that stretching the bond that is broken in the reaction puts momentum in the correct coordinate to drive the system into the exit channel. From this perspective it is quite surprising that bend excitation also results in large (factor of 2) enhancements of both H(+) and D(+) transfer channels, such that its effect on the total cross section at collision energies below ～2 eV is comparable to those from the two stretch modes, even though the bend excitation energy is much smaller. For collision energies above ～2 eV, the vibrational effects become approximately proportional to the vibrational energy, though still much larger than the effects of equivalent addition of collision energy. Measurements of the product recoil velocity distributions show that reaction is direct at all collision energies, with roughly half the products in a sharp peak corresponding to stripping dynamics and half with a broad and approximately isotropic recoil velocity distribution. Despite the large effects of vibrational excitation on reactivity, the effects on recoil dynamics are small, indicating that vibrational excitation does not cause qualitative changes in the reaction mechanism or in the distribution of reactive impact parameters.     

Experimental studies of the dynamics of the bond-forming reactions of CF2(2+) with H2O using position-sensitive coincidence spectroscopy

The dynamics of the product channels forming OCF(+)+H(+)+HF and HCF(2) (+)+H(+)+O following the collisions of CF(2) (2+) with H(2)O have been investigated with a new position-sensitive coincidence experiment at a center-of-mass collision energy of 5.6 eV. The results show the formation of OCF(+) occurs via the formation of a doubly charged collision complex [H(2)O-CF(2)](2+) which subsequently undergoes a charge separating dissociation to form H(+) and HOCF(2) (+). The HOCF(2) (+) monocation subsequently fragments to form HF+OCF(+). The lifetimes of the collision complex and the HOCF(2) (+) ion are at least of the order of their rotational period. The kinetic energy release in this reaction indicates that it involves the ground state of CF(2) (2+) and forms the ground electronic states of OCF(+) and HF. The mechanism for forming HCF(2) (+) involves the direct and rapid abstraction of a hydride ion from H(2)O by CF(2) (2+). The resulting OH(+) ion subsequently fragments to H(+)+O, on a time scale at least comparable with its rotational period.     

Velocity map imaging of ion-molecule reactive scattering: the Ar(+)+ N(2) charge transfer reaction

We present a velocity map imaging spectrometer for the study of crossed-beam reactive collisions between ions and neutrals at (sub-)electronvolt collision energies. The charge transfer reaction of Ar(+) with N(2) is studied at 0.6, 0.8 and 2.5 eV relative collision energies. Energy and angular distributions are measured for the reaction product N. The differential cross section, as analyzed with a Monte Carlo reconstruction algorithm, shows significant large angle scattering for lower collision energies in qualitative agreement with previous experiments. Significant vibrational excitation of N(+)(2) is also observed. This theoretically still unexplained feature indicates the presence of a low energy scattering resonance.     

Vibrational inelastic and charge transfer processes in H(+)+H2 system: an ab initio study

State-resolved differential cross sections, total and integral cross sections, average vibrational energy transfer, and the relative probabilities are computed for the H(+)+H2 system using the newly obtained ab initio potential energy surfaces at the full CI/cc-pVQZ level of accuracy which allow for both the direct vibrational inelastic and the charge transfer processes. The quantum dynamics is treated within the vibrational close-coupling infinite-order-sudden approximation approach using the two ab initio quasidiabatic potential energy surfaces. The computed collision attributes for both the processes are compared with the available state-to-state scattering experiments at E(c.m.)=20 eV. The results are in overall good agreement with most of the observed scattering features such as rainbow positions, integral cross sections, and relative vibrational energy transfers. A comparison with the earlier theoretical study carried out on the semiempirical surfaces (diatomics in molecules) is also made to illustrate the reliability of the potential energy surfaces used in the present work.     

Nonadiabatic time-dependent wave packet study of the D+ + H2 reaction system

A theoretical investigation on the nonadiabatic processes of the D(+) + H(2) reaction system has been carried out by means of exact three-dimensional nonadiabatic time-dependent wave packet calculations with an extended split operator scheme (XSOS). The diabatic potential energy surface newly constructed by Kamisaka et al. (J. Chem. Phys. 2002, 116, 654) was employed in the calculations. This study provided quantum cross sections for three competing channels of the reactive charge transfer, the nonreactive charge transfer, and the reactive noncharge transfer, which contrasted markedly to many previous quantum theoretical reports on the (DH(2))(+) system restricted to the total angular momentum J = 0. These quantum theoretical cross sections derived from the ground rovibrational state of H(2) show wiggling structures and an increasing trend for both the reactive charge transfer and the nonreactive charge transfer but a decreasing trend for the reactive noncharge transfer throughout the investigated collision energy range 1.7-2.5 eV. The results also show that the channel of the reactive noncharge transfer with the largest cross section is the dominant one. A further investigation of the v-dependent behavior of the probabilities for the three channels revealed an interesting dominant trend for the reactive charge transfer and the nonreactive charge transfer at vibrational excitation v = 4 of H(2). In addition, the comparison between the centrifugal sudden (CS) and exact calculations showed the importance of the Coriolis coupling for the reactive system. The computed quantum cross sections are also compared with the experimental measurement results.     

Quantitative (upsilon, N, Ka) product state distributions near the triplet threshold for the reaction H2CO --> H + HCO measured by Rydberg tagging and laser-induced fluorescence

In this paper, we report quantitative product state distributions for the photolysis of H2CO --> H + HCO in the triplet threshold region, specifically for several rotational states in the 2(2)4(3) and 2(3)4(1) H2CO vibrational states that lie in this region. We have combined the strengths of two complementary techniques, laser-induced fluorescence for fine resolution and H atom Rydberg tagging for the overall distribution, to quantify the upsilon, N, and Ka distributions of the HCO photofragment formed via the singlet and triplet dissociation mechanisms. Both techniques are in quantitative agreement where they overlap and provide calibration or benchmarks that permit extension of the results beyond that possible by each technique on its own. In general agreement with previous studies, broad N and Ka distributions are attributed to reaction on the S0 surface, while narrower distributions are associated with reaction on T1. The broad N and Ka distributions are modeled well by phase space theory. The narrower N and Ka distributions are in good agreement with previous quasi-classical trajectory calculations on the T1 surface. The two techniques are combined to provide quantitative vibrational populations for each initial H2CO vibrational state. For dissociation via the 2(3)4(1) state, the average product vibrational energy (15% of E(avail)) was found to be about half of the rotational energy (30% of E(avail)), independent of the initial H2CO rotational state, irrespective of the singlet or triplet mechanism. For dissociation via the 2(2)4(3) state, the rotational excitation remained about 30% of E(avail), but the vibrational excitation was reduced.     

Quantum dynamics of inelastic excitations and charge transfer processes in the H+ +NO collisions

State-resolved differential cross section, integral cross section, average vibrational energy transfer, and the relative transition probability are computed for the H(+)+NO system using our newly obtained ab initio potential energy surfaces (PES) at the multireference configuration interaction level of accuracy employing the correlation consistent polarized valence triple zeta basis set. The quantum dynamics is treated within the vibrational close-coupling rotational infinite-order sudden approximation using the coupled ground state and first excited state ab initio quasidiabatic PES. The computed collision attributes for the inelastic vibrational excitation are compared with the state-to-state scattering data available at E(c.m.)=9.5 eV and E(c.m.)=29.03 eV and are found to be in overall good agreement with those of the experiments. The results for the vibrational charge transfer processes at these collision energies are also presented.     

Reaction of HOD+ with NO2: effects of OD and OH stretching, bending, and collision energy on reactions on the singlet and triplet potential surfaces

Integral cross sections and product recoil velocity distributions were measured for the reaction of HOD(+) with NO(2), in which the HOD(+) reactant was prepared in its ground state and with mode-selective excitation in the 001 (OH stretch), 100 (OD stretch), and 010 (bend) modes. In addition, we measured the 300 K thermal kinetics in a selected ion flow tube reactor and report product branching ratios different from previous measurements. Reaction is found to occur on both the singlet and triplet surfaces with near-unit efficiency. At 300 K, the product branching indicates that triplet → singlet transitions occur in about 60% of triplet-coupled collisions, which we attribute to long interaction times mediated by complexes on the triplet surface. Because the collision times are much shorter in the beam experiments, the product distributions show no signs of such transitions. The dominant product on the singlet surface is charge transfer. Reactions on the triplet surface lead to NO(+), NO(2)H(+), and NO(2)D(+). There is also charge transfer, producing NO(2)(+) (a(3)B(2)); however, this triplet NO(2)(+) mostly predissociates. The NO(2)H(+)/NO(2)D(+) cross sections peak at low collision energies and are insignificant above ~1 eV due to OH/OD loss from the nascent product ions. The effects of HOD(+) vibration are mode-specific. Vibration inhibits charge transfer, with the largest effect from the bend. The NO(2)H(+)/NO(2)D(+) channels are also vibrationally inhibited, and the mode dependence reveals how energy in different reactant modes couples to the internal energy of the product ions.     

Reaction of C2H2(+) (n · ν2, m · ν5) with NO2: reaction on the singlet and triplet surfaces

Integral cross sections and product recoil velocity distributions were measured for reaction of C(2)H(2)(+) with NO(2), in which the C(2)H(2)(+) reactant was prepared in its ground state, and with mode-selective excitation in the cis-bend (2ν(5)) and CC stretch (n · ν(2), n = 1, 2). Because both reactants have one unpaired electron, collisions can occur with either singlet or triplet coupling of these unpaired electrons, and the contributions are separated based on distinct recoil dynamics. For singlet coupling, reaction efficiency is near unity, with significant branching to charge transfer (NO(2)(+)), O(-) transfer (NO(+)), and O transfer (C(2)H(2)O(+)) products. For triplet coupling, reaction efficiency varies between 13% and 19%, depending on collision energy. The only significant triplet channel is NO(+) + triplet ketene, generated predominantly by O(-) transfer, with a possible contribution from dissociative charge transfer at high collision energies. NO(2)(+) formation (charge transfer) can only occur on the singlet surface, and appears to be mediated by a weakly bound complex at low energies. O transfer (C(2)H(2)O(+)) also appears to be dominated by reaction on the singlet surface, but is quite inefficient, suggesting a bottleneck limiting coupling to this product from the singlet reaction coordinate. The dominant channel is O(-) transfer, producing NO(+), with roughly equal contributions from reaction on singlet and triplet surfaces. The effects of C(2)H(2)(+) vibration are modest, but mode specific. For all three product channels (i.e., charge, O(-), and O transfer), excitation of the CC stretch fundamental (ν(2)) has little effect, 2 · ν(2) excitation results in ～50% reduction in reactivity, and excitation of the cis-bend overtone (2 · ν(5)) results in ～50% enhancement. The fact that all channels have similar mode dependence suggests that the rate-limiting step, where vibrational excitation has its effect, is early on the reaction coordinate, and branching to the individual product channels occurs later.     

Reaction dynamics of OH+(3Sigma-)+C2H2 studied with crossed beams and density functional theory calculations

The reactions between OH+(3Sigma-) and C2H2 have been studied using crossed ion and molecular beams and density functional theory calculations. Both charge transfer and proton transfer channels are observed. Products formed by carbon-carbon bond cleavage analogous to those formed in the isoelectronic O(3P)+C2H2 reaction, e.g., 3CH2 + HCO+, are not observed. The center of mass flux distributions of both product ions at three different energies are highly asymmetric, with maxima close to the velocity and direction of the precursor acetylene beam, characteristic of direct reactions. The internal energy distributions of the charge transfer products are independent of collision energy and are peaked at the reaction exothermicity, inconsistent with either the existence of favorable Franck-Condon factors or energy resonance. In proton transfer, almost the entire reaction exothermicity is transformed into product internal excitation, consistent with mixed energy release in which the proton is transferred with both the breaking and forming bonds extended. Most of the incremental translational energy in the two higher-energy experiments appears in product translational energy, providing an example of induced repulsive energy release.     

H原子与CO2分子碰撞的化学反应与能量转移

Collisions between hot H atoms and CO2 molecules were studied experimentally by time-resolved Fourier transform infrared emission spectroscopy. H atoms with three translational energies, 174.7, 241.0 and 306.2 kJ/mol respectively, were generated by UV laser photolysis to initiate a chemical reaction of H+CO2!OH+CO. Vibrationally excited CO (v≤2) was observed in the spectrum, where CO was the product of the reaction. The highly efficient T-V energy transfer from the hot H atoms to the CO2 was verified too. The highest vibrational level of v=4 in CO2 (v≤3) was found. Rate ratio of the chemical reaction to the energy transfer was estimated as 10.     

Microcanonical statistical study of ortho-para conversion in the reaction H3+ + H2-->(H5+)*-->H3+ + H2 at very low energies

The ortho-para conversion of H(3) (+) and H(2) in the reaction H(3) (+)+H(2)-->(H(5) (+))(*)-->H(3) (+)+H(2) in interstellar space is possible by scrambling the five protons via (H(5) (+))(*) complex formation. The product distribution of the ortho-para conversion reaction can be given by ratios of cumulative reaction probabilities (CRP) calculated by microcanonical statistical theory with conservation of energy, motional angular momentum, nuclear spin, and parity. A statistical method to calculate the state-to-state reaction probabilities for given initial nuclear spin species, rotational states, and collision energies is developed using a simple semiclassical approximation of tunneling and above-barrier reflection. A new calculation method of branching ratios for given total nuclear spins and scrambling mechanisms is also developed. The anisotropic long-range electrostatic interaction potential of H(2) in the Coulomb field of H(3) (+) is taken into account using the first-order perturbation theory in forming the complex. The CRPs and the product distribution of the ortho-para conversion reaction at very low energies with reactants in their ground vibronic and lowest rotational states for given initial nuclear spin species are presented as a function of collision energy assuming complete proton scrambling or incomplete proton scrambling. The authors show that the product distribution at very low energies (or very low temperatures) differs substantially from the high energy (or high temperature) limit branching ratios.