Transition State Theory Calculations Research Articles

Predicting decomposition temperatures and carbonization feasibility of heterocycles with their decomposition mechanism

Pyrolysis of heterocyclic compounds such as Meldrum’s acid derivatives was studied with two key mechanisms analysing their energy barrier diagrams by the hybrid density functional theory method M06-2X. The feasibility of these decomposition reactions was calculated specifically for Meldrum’s acid, monomethyl Meldrum’s acid, dimethyl Meldrum’s acid (DMMA), methylene Meldrum’s acid (MeMA) and (dimethylamino)methylene Meldrum’s acid (DMAMeMA), based on previous experimental analysis. Their initial decomposition products (ketene or substituted ketenes with CO2 and acetone) were found endothermic requiring activation energies (∼180–––230 kJ/mol) at the decomposition temperatures of 280, 299, 301, 503 and 414 K respectively. The transition state theory calculations show MeMA and DMAMeMA are thermally more stable than other derivatives with low values for the rate constants and high energy barriers to produce initial decomposition products, suggesting they can produce fine carbon materials during carbonization. DMMA is found the best option for obtaining heteroatoms-doped carbon material formation.

Nucleoside Cation Radicals: Generation, Radical-Induced Hydrogen Atom Migrations, and Ribose Ring Cleavage in the Gas Phase.

Nucleoside ions that were furnished on ribose with a 2'-O-acetyl radical group were generated in the gas phase by multistep collision-induced dissociation of precursor ions tagged with radical initiator groups, and their chemistry was investigated in the gas phase. 2'-O-Acetyladenosine cation radicals were found to undergo hydrogen transfer to the acetoxyl radical from the ribose ring positions that were elucidated using specific deuterium labeling of 1'-H, 2'-H, and 4'-H and in the N-H and O-H exchangeable positions, favoring 4'-H transfer. Ion structures and transition-state energies were calculated by a combination of Born-Oppenheimer molecular dynamics and density functional theory and used to obtain unimolecular rate constants for competitive hydrogen transfer and loss of the acetoxyl radical. Migrations to the acetoxyl radical of ribose hydrogens 1'-H, 2'-H, 3'-H, and 4'-H were all exothermic, but product formation was kinetically controlled. Both Rice-Ramsperger-Kassel-Marcus (RRKM) and transition-state theory (TST) calculations indicated preferential migration of 4'-H in a qualitative agreement with the deuterium labeling results. The hydrogen migrations displayed substantial isotope effects that along with quantum tunneling affected the relative rate constants and reaction branching ratios. UV-vis action spectroscopy indicated that the cation radicals from 2'-O-acetyladenosine consisted of a mixture of isomers. Radical-driven dissociations were also observed for protonated guanosine, cytosine, and thymidine conjugates. However, for those nucleoside ions and cation radicals, the dissociations were dominated by the loss of the nucleobase or formation of protonated nucleobase ions.

The influence of heterogeneous catalytic processes on the heat flux to the surface and the chemical composition of the shock layer at high-speed flow around blunt bodies

Numerical modeling of the high-speed flow of a multicomponent dissociated gas around a blunt body was carried out for the flow regime corresponding to the thermally stressed point of the gliding trajectory of entry into the Earth's atmosphere in the surface temperature range of 800–2000 K. As boundary conditions, a stage-by-stage model of heterogeneous catalysis of the interaction of dissociated air with the thermal protection material SiO2 was used, previously developed by the authors on the basis of transition state theory and quantum mechanical calculations. The dependence of the modes of heterogeneous chemical reactions on the surface temperature and the density of surface adsorption centers and their influence on the chemical composition of the shock layer and the convective heat flux to the streamlined surface are analyzed. The contributions of individual mechanisms of heterogeneous recombination to the total rate of formation of molecules on the surface are estimated. The possibility of using the density of adsorption sites as an effective parameter when setting boundary conditions on a real thermal protection surface in flow problems is shown.

Competitive Radical Migrations and Ribose Ring Cleavage in Adenosine and 2'-Deoxyadenosine Cation Radicals.

We report a combined experimental and computational study of adenosine cation radicals that were protonated at adenine and furnished with a radical handle in the form of an acetoxyl radical, •CH2COO, that was attached to ribose 5'-O. Radicals were generated by collision-induced dissociation (CID) and characterized by tandem mass spectrometry and UV-vis photodissociation action spectroscopy. The acetoxyl radical was used to probe the kinetics of intramolecular hydrogen transfer from the ribose ring positions that were specifically labeled with deuterium at C1', C2', C3', C4', C5', and in the exchangeable hydroxyl groups. Hydrogen transfer was found to chiefly involve 3'-H with minor contributions by 5'-H and 2'-H, while 4'-H was nonreactive. The hydrogen transfer rates were affected by deuterium isotope effects. Hydrogen transfer triggered ribose ring cleavage by consecutive dissociations of the C4'-O and C1'-C2' bonds, resulting in expulsion of a C6H9O4 radical and forming a 9-formyladenine ion. Rice-Ramsperger-Kassel-Marcus (RRKM) and transition-state theory (TST) calculations of unimolecular constants were carried out using the effective CCSD(T)/6-311++G(3d,2p) and M06-2X/aug-cc-pVTZ potential energy surfaces for major isomerizations and dissociations. The kinetic analysis showed that hydrogen transfer to the acetoxyl radical was the rate-determining step, whereas the following ring-opening reactions in ribose radicals were fast. Using DFT-computed energies, a comparison was made between the thermochemistry of radical reactions in adenosine and 2'-deoxyadenosine cation radicals. The 2'-deoxyribose ring showed lower TS energies for both the rate-determining 3'-H transfer and ring cleavage reactions.

On the General Mechanism for the Gas-phase Reaction of Methanimine with a Radical Species in the Interstellar Medium: Some Failures and an Important Success

Abstract The gas-phase reactions of methanimine (CH2NH) with small radicals, such as CN, CP, CCH, and OH, have been extensively studied theoretically in the literature, and the presence of a common, general reaction mechanism has been postulated. Since methanimine is considered the main precursor of complex imines in the interstellar medium (ISM), the present study extends the investigation of its reaction with other small radicals that have already been detected in the ISM. These are SiN, SH, NO, NS, HCO, HCS, and C3N. The corresponding products are easily formulated on the basis of the aforementioned general mechanism, and to understand whether they can be formed in the ISM, a preliminary thermochemical study has been carried out. The only exothermic addition reaction is that occurring between CH2NH and the C3N radical. This reaction has been further investigated in order to accurately characterize its reactive potential energy surface, which has then been employed in ab initio transition state theory calculations to derive global rate coefficients. The products of the CH2NH + C3N reaction are new potential interstellar species, namely, the Z and E isomers of HNCHCCCN and CH2NCCCN. For the first time, their structural characterization has been reported. In addition, this work investigates the possibility of H-abstraction processes for each radical species considered, and re-examines the CH2NH + CP reaction to derive the corresponding rate constants, that were still missing in the literature.

Oxidation of the 1‐naphthyl radical C10H7• with oxygen: Thermochemistry, kinetics, and possible reaction pathways

AbstractThe reaction of the 1‐naphthyl radical C10H7• (A2•) with molecular (3O2) and atomic oxygen, as part of the oxidation reactions of naphthalene, is examined using ab‐initio and DFT quantum chemistry calculations. The study focuses on pathways that produce the intermediate final products CO, phenyl and C2H2, which may constitute a repetitive reaction sequence for the successive diminution of six‐membered rings also in larger polycyclic aromatic hydrocarbons. The primary attack of 3O2 on the 1‐naphthyl radical leads to a peroxy radical C10H7OO• (A2OO•), which undergoes further propagation and/or chain branching reactions. The thermochemistry of intermediates and transition state structures is investigated as well as the identification of all plausible reaction pathways for the A2• + O2 / A2• + O systems. Structures and enthalpies of formation for the involved species are reported along with transition state barriers and reaction pathways. Standard enthalpies of formation are calculated using ab initio (CBS‐QB3) and DFT calculations (B3LYP, M06, APFD). The reaction of A2• with 3O2 opens six main consecutive reaction channels with new ones not currently considered in oxidation mechanisms. The reaction paths comprise important exothermic chain branching reactions and the formation of unsaturated oxygenated hydrocarbon intermediates. The primary attack of 3O2 at the A2• radical has a well depth of some 50 kcal mol−1 while the six consecutive channels exhibit energy barriers below the energy of the A2• radical. The kinetic parameters of each path are determined using chemical activation analysis based on the canonical transition state theory calculations. The investigated reactions could serve as part of a comprehensive mechanism for the oxidation of naphthalene. The principal result from this study is that the consecutive reactions of the A2• radical, viz. the channels conducting to a phenyl radical C6H5•, CO2, CO (which oxidized to CO2) and C2H2 are by orders of magnitude faster than the activation of naphthalene by oxygen (A2 + O2 → A2• + HO2).

Highly Efficient Autoxidation of Triethylamine.

Autoxidation has been acknowledged as a major oxidation pathway in a broad range of atmospherically important compounds including isoprene and monoterpenes. More recently, autoxidation has also been identified as central and even dominant in the atmospheric oxidation of the rather small nonhydrocarbons dimethyl sulfide (DMS) and trimethylamine (TMA). Here, we find even faster autoxidation in the aliphatic amine triethylamine (TEA). The atmospherically dominating autoxidation leads to highly oxygenated and functionalized compounds. Products with as many as three hydroperoxy (OOH) groups and an O:C ratio larger than 1 are formed. We present theoretical multiconformer transition-state theory (MC-TST) calculations of the unimolecular reactions in the autoxidation following the OH + TEA reaction and calculate peroxy radical H-shift rate coefficients >20 s-1 for the first two generations of H-shifts. The efficient autoxidation in TEA is verified by the observation of the proposed highly oxidized products and radicals in flow-tube experiments. We find that the initial OH hydrogen abstraction at the α-carbon is strongly favored, with the β-carbon abstraction yield being less than a few percent.

Polyrate 2023: A computer program for the calculation of chemical reaction rates for polyatomics. New version announcement

Polyrate 2023: A computer program for the calculation of chemical reaction rates for polyatomics. New version announcement

Theoretical and experimental study of the OH radical with 3‐bromopropene gas phase reaction rate coefficients temperature dependence

AbstractIn this work, the rate‐determining steps of the OH radical + 3‐bromopropene gas phase reaction were studied, which could explain for the possible negative activation energy observed in experiments. To obtain new kinetic parameters and data for critical revisions, a reinvestigation of the rate coefficient (k) and its temperature dependence was carried out using the PLP‐LIF technique, in the 254‐ to 371‐K range. Moreover, quantum‐mechanical and canonical variational transition state theory calculations were performed, taking into consideration four OH addition and two β‐hydrogen atom abstraction reaction channels. The proposed kinetic model fits to the observed experimental Arrhenius behavior, and three not negligible reaction pathways are described for the first time.

Three-Body Collisions Driving the Ion-Molecule Reaction C2- + H2 at Low Temperatures.

We report on the three-body reaction rate of C2- with H2 producing C2H- studied in a cryogenic 16-pole radio frequency ion trap. The reaction was measured in the temperature range from 10 to 28 K, where it was found to only take place via three-body collisions. The experimentally determined termolecular rate coefficient follows the form of with T0 = 20 K, where a = 8.2(3) × 10-30 cm6/s and b = -0.82(12) denotes the temperature dependence. We additionally performed accurate ab initio calculations of the forces between the interacting partners and carried out variational transition state theory calculations, including tunneling through the barrier along the minimum energy path. We show that, while a simple classical model can generally predict the temperature dependence, the variational transition state theoretical calculations, including accurate quantum interactions, can explain the dominance of three-body effects in the molecular reaction mechanism and can reproduce the experimentally determined reaction coefficients, linking them to a temperature-dependent coupling parameter for energy dissipation within the transition complex.

Structural, thermochemical and kinetic insights on the pyrolysis of diketene to produce ketene

Diketene (4-methylideneoxetan-2-one) is a precursor to the formation of either two molecules of ketene, or allene and CO2 using pyrolysis techniques. It is not known experimentally which of these pathways is followed, or indeed if both are, during the dissociation process. We use computational methods to show that the formation of ketene has a lower barrier than formation of allene and CO2 under standard conditions (by 12 kJ/mol). According to CCSD(T)/CBS, CBS-QB3 and M06-2X/cc-pVTZ calculations the formation of allene and CO2 is favoured thermodynamically under standard conditions of temperature and pressure; however, kinetically the formation of ketene is favoured from transition state theory calculations at standard and elevated temperatures.Graphical abstract

Thermal Decomposition Mechanism of Tetraethylsilane by Flash Pyrolysis Vacuum Ultraviolet Photoionization Mass Spectrometry and DFT Calculations: The Competition between β-Hydride Elimination and Bond Homolysis

Thermal decomposition of tetraethylsilane was investigated at temperatures up to 1330 K using flash pyrolysis vacuum ultraviolet photoionization mass spectrometry. Density functional theory and transition state theory calculations were performed to corroborate the experimental observations. Both experimental and theoretical evidence showed that the pyrolysis of tetraethylsilane was initiated by Si-C bond fission to the primary reaction products, triethylsilyl (SiEt3) and ethyl radicals. In the secondary reactions of the triethylsilyl radical, at lower temperatures, the β-hydride elimination pathway (producing HSiEt2) was found to be more favored than its competing reaction channel, Si-C bond fission (producing :SiEt2); as the temperature further increased, the Si-C bond fission reaction became significant. Other important secondary reaction products, such as EtHSi═CH2 (m/z = 72), H2SiEt (m/z = 59), and SiH3 (m/z = 31) were identified, and their formation mechanisms were also proposed.

Enhanced Sampling for Free Energy Profiles with Post-Transition-State Bifurcations

We present a strategy to explore the free energy landscapes of chemical reactions with post-transition-state bifurcations using an enhanced sampling method based on well-tempered metadynamics. Obviating the need for computationally expensive density functional theory-level ab initio molecular dynamics simulations, we obtain accurate energetics by utilizing a free energy perturbation scheme and deep learning estimator for the single-point energies of substrate configurations. Using a pair of easily interpretable collective variables, we present a quantitative free energy surface that is compatible with harmonic transition state theory calculations and in which the bifurcations are clearly visible. We demonstrate our approach with the example of the SpnF-catalyzed Diels-Alder reaction, a cycloaddition reaction in which post-transition-state bifurcation leads to the [4+2] as well as the [6+4] cycloadduct. We obtain the free energy landscapes for different stereochemical reaction pathways and characterize the mechanistic continuum between relevant reaction channels without explicitly searching for the pertinent transition state structures.

Kinetics of CN (v = 1) reactions with butadiene isomers at low temperature by cw-cavity ring-down in a pulsed Laval flow with theoretical modelling of rates and entrance channel branching.

We present an experimental and theoretical investigation of the reaction of vibrationally excited CN (v = 1) with isomers of butadiene at low temperature. The experiments were conducted using the newly built apparatus, UF-CRDS, which couples near-infrared cw-cavity ring-down spectroscopy with a pulsed Laval flow. The well-matched hydrodynamic time and long ring-down time decays allow measurement of the kinetics of the reactions within a single trace of a ring-down decay, termed Simultaneous Kinetics and Ring-down (SKaR). The pulsed experiments were carried out using a Laval nozzle designed for the 70 K uniform flow with nitrogen as the carrier gas. The measured bimolecular rates for the reactions of CN (v = 1) with 1,3-butadiene and 1,2-butadiene are (3.96 ± 0.28) × 10-10 and (3.06 ± 0.35) × 10-10 cm3 per molecule per s, respectively. The reaction rate measured for CN (v = 1) with the 1,3-butadiene isomer is in good agreement with the rate previously reported for the reaction with ground state CN (v = 0) under similar conditions. We report the rate of the reaction of CN (v = 1) with the 1,2-butadiene isomer here for the first time. The experimental results were interpreted with the aid of variable reaction-coordinate transition-state theory calculations to determine rates and branching of the addition channels based on a high-level multireference treatment of the potential energy surface. H-abstraction reaction rates were also theoretically determined. For the 1,2-butadiene system, theoretical estimates are then combined with literature values for the energy-dependent product yields from the initial adducts to predict overall temperature-dependent product branching. H loss giving 2-cyano-1,3-butadiene + H is the main product channel, exclusive of abstraction, at all energies, but methyl loss forming 1-cyano-prop-3-yne is 15% at low temperature growing to 35% at 500 K. Abstraction forming HCN and various radicals is important at 500 K and above. The astrochemical implications of these results are discussed.

Intrinsic kinetics mechanisms for the catalytic reduction of NO by Na-loaded char

An insight into the interaction between NO and Na-loaded char is essential to improve the catalytic ability of Na to NO reduction, which will be useful to lower NO emissions during thermal utilization of sodium-containing fuels. Here, the intrinsic kinetics mechanisms for the catalytic reduction of NO by Na-loaded char were discussed in details. Using density functional theory (DFT) calculations, possible reaction pathways were first obtained, followed by evaluation of the rate coefficients through transition state theory (TST) calculations. On this basis, the analyses of both sensitivity and rate of products (ROP) were performed to illustrate the intrinsic kinetic mechanism for the NO reduction by Na-loaded char in a certain combustion condition, with an emphasis on the effects of temperature and NO-to-CO stoichiometric ratio. Results indicated that the catalytic active center –ONa plays an important role in the catalytic reduction of NO by Na-loaded char. Specifically, in most cases, the interaction of NO with Na-loaded char largely depends on the elementary reaction of CNO-Na+NO+CO→21-IM3+CO2. As the stoichiometric ratio of NO to CO increases, the CO-Na+2NO→8-IM4+N2 becomes increasingly dominant. Moreover, higher temperature causes the CNO-Na+NO→20-P + N2O as the dominant reaction. Nonetheless, one thing that these reactions have in common is that they are all related to the catalytic active center –ONa. Therefore, the NO reduction Na-loaded char largely depends on the interaction of NO with the carbonaceous surface containing –ONa. Inspired by this, a conceptual approach was proposed to improve the catalytic performance of Na on NO reduction, and it has been shown to be theoretically feasible. To summarize, the combination of DFT, TST and kinetic calculations is useful to clarify the interaction between NO with Na-loaded char, and it gives a basis for the development of micro-kinetic model.

Reaction of OH with Aliphatic and Aromatic Isocyanates

Isocyanates are highly relevant industrial intermediates for polyurethane production. In this work, we used quantum chemistry and transition state theory (TST) to investigate the gas-phase reaction of isocyanates with the OH radical, which is likely one of the most significant chemical sinks for these compounds in the troposphere. para-Tolyl-isocyanate (p-tolyl-NCO) was chosen as a proxy substance for the large-volume aromatic diisocyanate species toluene diisocyanate and methylene diphenyl diisocyanate. Besides p-tolyl-NCO + OH, the model reactions CH3NCO + OH, H2C═CHNCO + OH, C6H5-NCO + OH, C6H5-CH3 + OH, and C6H6 + OH have been studied as well to analyze various substituent effects and to allow for comparison with literature. Quantum chemical computations at the CCSD(T)/cc-pV(T,Q → ∞)Z//M06-2X/def2-TZVP level were used as the basis for tunneling-corrected canonical TST calculations. For CH3NCO + OH, H abstraction by OH at the methyl group is the main reaction channel according to our calculations and predicted to be four orders of magnitude faster than OH addition at the NCO group. The calculated rate coefficient (8.8 × 10-14 cm3 molecule-1 s-1) at 298 K is in good agreement with experimental data from the literature. Likewise, for aromatic isocyanates, OH attack at the isocyanate group was found to be only a minor pathway compared to addition to the aromatic ring. In the OH + p-tolyl-NCO reaction, OH addition at the ortho-position relative to the NCO group has been identified as the main initial reaction channel (branching fraction: 53.2%), with smaller but significant branching fractions for the H abstraction at the methyl group (9.6%) as well as the other ring addition reactions (ipso: 2.3%, meta: 24.5%, para: 10.5%, all relative to the NCO group). By comparing OH addition to the aromatic ring in p-tolyl-NCO with the respective ring addition reactions of phenyl isocyanate and toluene, the site-selective reactivity trends observed for ring addition in the OH + p-tolyl-NCO could be rationalized by a dominating positive mesomeric effect of the NCO group and a positive electron-donating (inductive) effect of the CH3 group. Except for the OH ring adduct formed from OH addition in ipso-position to the NCO group, we estimate the first-generation radical intermediates in the OH + p-tolyl-NCO reaction to have sufficiently long lifetimes to react with O2 under atmospheric conditions and undergo typical oxidative reaction cascades like those known for benzene or toluene.

Computational studies on thermo-kinetics aspects of pyrolysis of isopropyl acetate and its methyl, bromide and hydroxyl derivatives

The gas-phase decomposition kinetics of isopropyl acetate (IPA) and its methyl, bromide and hydroxyl derivatives into the corresponding acid and propene were investigated using density functional theory (DFT) with the ωB97XD and M06–2x functionals, as well as the benchmark CBS-QB3 composite method. Transition state theory (TST) and RRKM theory calculations of rate constants under atmospheric pressure and in the fall-off regime were used to supplement the measured energy profiles. The results show that the formation of propene and bromoacetic acid is the most dominant pathway at the CBS-QB3 composite method, both kinetically and thermodynamically. There was a good agreement with experimental results. Pressures greater than 0.01 bar, corresponding to larger barrier heights are insufficient to ensure saturation of the measured rate coefficient when compared to the RRKM kinetic rates.Natural bond orbitals (NBO) charges, bond orders, bond indices, and synchronicity parameters all point to the considered pathways taking place via a homogenous, first-order concerted, as well as an asynchronous mechanism involving a non-planar cyclic six-membered transition state. The calculated data exhibit that the elongation of the Cα−O bond length and subsequent polarization of the Cα+δ…O−δ bond is the rate-determining step of the considered reactions in the cyclic transition state, which appears to be involved in this type of reaction.

Flash pyrolysis mechanism of trimethylchlorosilane

The thermal decomposition mechanism of trimethylchlorosilane at temperatures up to 1400 K was investigated using flash pyrolysis microreactor coupled with vacuum ultraviolet (118.2 nm) photoionization time-of-flight mass spectrometry. The main initiation reaction of the parent molecule was identified to be HCl molecular elimination producing Me2Si=CH2, and its onset temperature was estimated to be around 1210 K. The methyl loss channel was also observed at an onset temperature of ∼1280 K. Other possible initiation pathways such as chlorine atom loss and methane molecular elimination were considered to be insignificant. Quantum chemistry calculations at the UCCSD(T)/cc-pVTZ//UM05–2X/aug-cc-pVDZ level of theory were performed to study the energetics of the possible initiation pathways. The theoretical calculations revealed that the HCl elimination channel via a van der Waals intermediate was the most energetically favored pathway among all the initiation channels, in agreement with the experimental observations. Transition state theory (TST) and variational transition state theory (VTST) calculations were employed to calculate unimolecular dissociation rate constants of the initiation reactions at elevated temperatures. The results supported the experimental observations and reaction energetics calculations that the HCl molecular elimination reaction was the most prominent initiation reaction and the CH3 loss channel was a significant initiation reaction channel. Secondary reactions of the initial products were identified, and the possible mechanisms were proposed.

Temperature responses of two synergistic pathways for low-temperature catalytic oxidation of high-concentration NO over ACFs: Nano-confinement and nitrogen-containing groups

This study reveals for the first time that low-temperature catalytic oxidation of high-concentration NO over ACFs achieved by two synergistic paths: the quasi-homogeneous NO oxidation (path 1) and the nitrogen-containing functional groups catalysis (path 2). In path 1, the kinetic properties of reaction transition state are stabilized by ultra-micropores due to its non-specific van der Waals interaction, accelerating the transfer from gaseous reactants to the transition state 100 ∼ 1000 times. The micropore confinement effect of activation enthalpy is strongest at ambient temperature therefore the catalytic efficiency is the highest 6.89 × 10-4 mol·L-1·S-1. In path 2, extra π electrons generated by the substitution of N for C in the aromatic ring transfer to NOx and O2. Pyridine and quaternary nitrogen serve as catalytical active sites to promote the adsorption of NO and the activation of O2. Temperature increasing leads to a significant negative entropy barrier, the catalytic activity decreases accordingly, this is the main reason for the fluctuation in NO oxidation efficiency caused by temperature. The approximate proportion of the two oxidation pathways and their changes with reaction temperature are quantitatively analyzed by transition state theory calculations. Raising the reaction temperature from 25℃ to 200℃ increased the activity contribution of the active functional group catalytic pathway by about 25 %, which to some extent compensated for the negative entropy barrier effect of the quasi-homogeneous NO oxidation pathway.

Nonlinear Arrhenius behavior of self-diffusion in β−Ti and Mo

While anomalous diffusion coefficients with non-Arrhenius like temperature dependence are observed in a number of metals, a conclusive comprehensive framework of explanation has not been brought forward to date. Here, we use first-principles calculations based on density functional theory to calculate self-diffusion coefficients in the bcc metals Mo and $\beta$-Ti by coupling quasiharmonic transition state theory and large displacement phonon calculations and show that anharmonicity from thermal expansion is the major reason for the anomalous temperature dependence. We use a modified Debye approach to quantify the thermal expansion over the entire temperature range and introduce a method to relax the vacancy structure in a mechanically unstable crystal such as $\beta$-Ti. Thermal expansion is found to weakly affect the activation enthalpy but has a strong effect on the prefactor of the diffusion coefficient, reproducing the non-linear, non-Arrhenius "anomalous" self-diffusion in both bcc systems with good agreement between calculation and experiment. The proposed methodology is general and simple enough to be applicable to other mechanically unstable crystals.