Search

The interaction between CO2 and N2, both as essential components of the Earth’s atmosphere, plays a crucial role in investigating the greenhouse effect. In this work, we sampled 40,930 data points within the full-dimensional configuration space of CO2 and N2 and performed calculations at the level of explicitly correlated coupled cluster single, double, and perturbative triple level with the augmented correlation corrected valence triple-ζ basis set (CCSD(T)-F12a/AVTZ). To ensure computational accuracy while reducing computational costs, we employed the recently proposed Δ-machine learning (Δ-ML) method based on Permutation Invariant Polynomial-Neural Network (PIP-NN) for basis set superposition error (BSSE) correction. By leveraging the limited extrapolation capability of NN, efficient sampling was performed within the existing dataset, enabling the construction of the potential energy surface (PES) incorporating BSSE correction with only a small number of data points for BSSE calculations. A total of approximately 1100 data points were selected from the initial dataset to construct a BSSE correction PES. Utilizing this correction PES, BSSE predictions were carried out for all remaining data points, resulting in the successful development of a high-precision full-dimensional PES with BSSE correction for the CO2 + N2 system. The PIP-NN based Δ-ML method significantly reduced the required BSSE calculations by approximately 97.2%, resulting in a final PES with a fitting error of merely 0.026 kcal/mol.

A representation for learning potential energy surfaces (PESs) in terms of permutationally invariant polynomials (PIPs) using the Hartree-Fock expression for electronic energy is proposed. Our approach is based on the one-electron core Hamiltonian weighted by the configuration-dependent elements of the bond-order charge density matrix (CDM). While the previously reported model used an s -function Gaussian basis for the CDM, the present formulation is expanded with p -functions, which are crucial for describing chemical bonding. Detailed results are demonstrated on linear and cyclic C n clusters ( n = 3-10) trained on extensive B3LYP/aug-cc-pVTZ data. The described method facilitates PES learning by reducing the root mean squared error (RMSE) by a factor of 5 relative to the s -function formulation and by a factor of 20 relative to the conventional PIP approach. This is equivalent to using CDM and an sp basis with a PIP of order M to achieve the same RMSE as with the conventional method with a PIP of order M + 2. Implications for large-scale problems are discussed using the case of the PES of the C 20 fullerene in full permutational symmetry.

The singly hydrated hydroxide anion OH - (H 2 O) is of central importance to a detailed molecular understanding of water; therefore, there is strong motivation to develop a highly accurate potential to describe this anion. While this is a small molecule, it is necessary to have an extensive data set of energies and, if possible, forces to span several important stationary points. Here, we assess two machine-learned potentials, one using the symmetric gradient domain machine learning (sGDML) method and one based on permutationally invariant polynomials (PIPs). These are successors to a PIP potential energy surface (PES) reported in 2004. We describe the details of both fitting methods and then compare the two PESs with respect to precision, properties, and speed of evaluation. While the precision of the potentials is similar, the PIP PES is much faster to evaluate for energies and energies plus gradient than the sGDML one. Diffusion Monte Carlo calculations of the ground vibrational state, using both potentials, produce similar large anharmonic downshift of the zero-point energy compared to the harmonic approximation of the PIP and sGDML potentials. The computational time for these calculations using the sGDML PES is roughly 300 times greater than using the PIP one.

In this work, the permutation invariant polynomial neural network (PIP-NN) approach is employed to construct a quasi-diabatic Hamiltonian for system with non-Abelian symmetries. It provides a flexible and compact NN-based diabatic ansatz from the related approach of Williams, Eisfeld, and co-workers. The example of H 3 + is studied, which is an (E + A) × (e + a) Jahn–Teller and Pseudo-Jahn–Teller system. The PIP-NN diabatic ansatz is based on the symmetric polynomial expansion of Viel and Eisfeld, the coefficients of which are expressed with neural network functions that take permutation-invariant polynomials as input. This PIP-NN-based diabatic ansatz not only preserves the correct symmetry but also provides functional flexibility to accurately reproduce ab initio electronic structure data, thus resulting in excellent fits. The adiabatic energies, energy gradients, and derivative couplings are well reproduced. A good description of the local topology of the conical intersection seam is also achieved. Therefore, this diabatic ansatz completes the PIP-NN based representation of DPEM with correct symmetries and will enable us to diabatize even more complicated systems with complex symmetries. [ABSTRACT FROM AUTHOR]

Permutationally invariant polynomial (PIP) regression has been used to obtain machine-learned potential energy surfaces, including analytical gradients, for many molecules and chemical reactions. Recently, the approach has been extended to moderate size molecules with up to 15 atoms. The algorithm, including "purification of the basis," is computationally efficient for energies; however, we found that the recent extension to obtain analytical gradients, despite being a remarkable advance over previous methods, could be further improved. Here, we report developments to further compact a purified basis and, more significantly, to use the reverse differentiation approach to greatly speed up gradient evaluation. We demonstrate this for our recent four-body water interaction potential. Comparisons of training and testing precision on the MD17 database of energies and gradients (forces) for ethanol against numerous machine-learning methods, which were recently assessed by Dral and co-workers, are given. The PIP fits are as precise as those using these methods, but the PIP computation time for energy and force evaluation is shown to be 10–1000 times faster. Finally, a new PIP potential energy surface (PES) is reported for ethanol based on a more extensive dataset of energies and gradients than in the MD17 database. Diffusion Monte Carlo calculations that fail on MD17-based PESs are successful using the new PES. [ABSTRACT FROM AUTHOR]

The exact expressions for the dipole, quadrupole, and octupoles of a collection of N point charges involve summations of corresponding tensors over the N sites weighted by their charge magnitudes. When the point charges are atoms (in a molecule) the N‐site formula is an approximation, and one must integrate over the electron density to recover the exact multipoles. In the present work we revisit the N(N + 1)/2‐site point charge density model of Hall (Chem. Phys. Lett. 6, 501, 1973) for the purpose of fitting ab initio derived multipole moment hypersurfaces using permutationally invariant polynomials (PIP). We examine new approaches in PIP‐fitting procedures for the dipole, quadrupole, octupole moments, and polarizability tensor surfaces (DMS, QMS, OMS and PTS, respectively) for a non‐polar CCl4 and a polar CHCl3 and show that compared to the primitive N‐site model the N(N + 1)/2‐site model appreciably improves the relative RMSE of the DMS and does much more substantially so, by an order of magnitude, for the corresponding ones of QMS and OMS. Training datasets are obtained by sampling potential energies up to 18 000 cm−1 above the global minima, generated by molecular dynamics simulations at the DFT B3LYP/aug‐cc‐pVDZ level of theory. [ABSTRACT FROM AUTHOR]

A linearly parameterized functional form for a Cartesian representation of molecular dipole polarizability tensor surfaces (PTS) is described. The proposed expression for the PTS is a linearization of the recently reported power series ansatz of the original Applequist model, which by construction is non-linear in parameter space. This new approach possesses (i) a unique solution to the least-squares fitting problem; (ii) a low level of the computational complexity of the resulting linear regression procedure, comparable to those of the potential energy and dipole moment surfaces; and (iii) a competitive level of accuracy compared to the non-linear PTS model. Calculations of CH 4 PTS, with polarizabilities fitted to 9000 training set points with the energies up to 14,000 cm -1 show an impressive level of accuracy of the linear PTS model obtained with ~1600 parameters: ~1% versus 0.3% RMSE for the non-linear vs. linear model on a test set of 1000 configurations., (© 2022 Wiley Periodicals LLC.)

Δ-machine learning, or the hierarchical construction scheme, is a highly cost-effective method, as only a small number of high-level ab initio energies are required to improve a potential energy surface (PES) fit to a large number of low-level points. However, there is no efficient and systematic way to select as few points as possible from the low-level data set. We here propose a permutation-invariant-polynomial neural-network (PIP-NN)-based Δ-machine learning approach to construct full-dimensional accurate PESs of complicated reactions efficiently. Particularly, the high flexibility of the NN is exploited to efficiently sample points from the low-level data set. This approach is applied to the challenging case of a HO 2 self-reaction with a large configuration space. Only 14% of the DFT data set is used to successfully bring a newly fitted DFT PES to the UCCSD(T)-F12a/AVTZ quality. Then, the quasiclassical trajectory (QCT) calculations are performed to study its dynamics, particularly the mode specificity.

In this article, a time-invariant polynomial model of fixed-frequency pulsewidth modulation dc–dc converter is proposed. Through normalized coordinate transformation, the time-varying model of the converter is transferred into a time-invariant model, whose state matrix and input matrix are obtained by the addition and multiplication of matrices simply and expressed by the polynomial of duty cycle concisely. The proposed model has the following advantages. Compared with the state-space average (SSA) model, the ripple of the state vector and the switching period are considered, thereby improving the accuracy. Contrary to other time-invariant models, the state matrix and input matrix instead of the state vector are expanded into series. Therefore, neither the number of state vectors nor the order of the matrices increases, and the proposed model has a simple structure similar to the SSA model. In addition, as a time-invariant polynomial model, the calculation and analysis of the proposed model are much simpler than the time-variant exact model. In order to prove the above advantages of the proposed model, the general expression of the proposed model is derived and the proposed model is compared with other models. These analyses and comparisons are verified through the simulation and experimental results. [ABSTRACT FROM AUTHOR]

Abstract Random matrix products arise in many science and engineering problems. An efficient evaluation of its growth rate is of great interest to researchers in diverse fields. In the current paper, we reformulate this problem with a generating function approach, based on which two analytic methods are proposed to compute the growth rate. The new formalism is demonstrated in a series of examples including an Ising model subject to on-site random magnetic fields, which seems very efficient and easy to implement. Through an extensive comparison with numerical computation, we see that the analytic results are valid in a region of considerable size.The formulation could be conveniently applied to stochastic processes with more complex structures.

In this paper, we study the family of quadratic Riccati differential systems. Our goal is to obtain the complete topological classification of this family on the Poincaré disk compactification of the plane. The family was partially studied before but never from a truly global viewpoint. Our approach is global and we use geometry to achieve our goal. The geometric analysis we perform is via the presence of two invariant parallel straight lines in any generic Riccati system. We obtain a total of 119 topologically distinct phase portraits for this family. Furthermore, we give the complete bifurcation diagram in the 12-dimensional space of parameters of this family in terms of invariant polynomials, meaning that it is independent of the normal forms in which the systems may be presented. This bifurcation diagram provides an algorithm to decide for any given quadratic system in any form it may be presented, whether it is a Riccati system or not, and in case it is to provide its phase portrait. [ABSTRACT FROM AUTHOR]

A full-dimensional analytical potential energy surface (PES) is developed for the Cl + CH3CN reaction following our previous work on the benchmark ab initio characterization of the stationary points. The spin–orbit-corrected PES is constructed using the Robosurfer program and a fifth-order permutationally invariant polynomial method for fitting the high-accuracy energy points determined by a ManyHF-based coupled-cluster/triple-zeta-quality composite method. Quasi-classical trajectory simulations are performed at six collision energies between 10 and 60 kcal mol−1. Multiple low-probability product channels are found, including isomerization to isonitrile (CH3NC), but out of the eight possible channels, only the H-abstraction has significant reaction probability; thus, detailed dynamics studies are carried out only for this reaction. The cross sections and opacity functions show that the probability of the H-abstraction reaction increases with increasing collision energy (Ecoll). Scattering angle, initial attack angle, and product relative translational energy distributions indicate that the mechanism changes with the collision energy from indirect/rebound to direct stripping. The distribution of initial attack angles shows a clear preference for methyl group attack but with different angles at different Ecoll values. Post-reaction energy distributions show that the energy transfer is biased toward the products' relative translational energy instead of their internal energy. Rotational and vibrational energy have about the same amount of contribution to the internal energy in the case of both products (HCl and CH2CN), i.e., both of them are formed with high rotational excitations. HCl is produced mostly in the ground vibrational state, while a notable fraction of CH2CN is formed with vibrational excitation. [ABSTRACT FROM AUTHOR]

In this work, the dynamics of a prototypical heavy–light–heavy abstract reaction, Cl(2P) + HCl → HCl + Cl(2P), is investigated both by constructing a new potential energy surface (PES) and by rate coefficient calculations. Both the permutation invariant polynomial neural network method and the embedded atom neural network (EANN) method, based on ab initio MRCI-F12+Q/AVTZ level points, are used for obtaining globally accurate full-dimensional ground state PES, with the corresponding total root mean square error being only 0.043 and 0.056 kcal/mol, respectively. In addition, this is also the first application of the EANN in a gas-phase bimolecular reaction. The saddle point of this reaction system is confirmed to be nonlinear. In comparison with both the energetics and rate coefficients obtained on both PESs, we find that the EANN is reliable in dynamic calculations. A full-dimensional approximate quantum mechanical method, ring-polymer molecular dynamics with a Cayley propagator, is employed to obtain the thermal rate coefficients and kinetic isotopic effects of the title reaction Cl(2P) + XCl→ XCl + Cl(2P) (H, D, Mu) on both new PESs, and the kinetic isotope effect (KIE) is also obtained. The rate coefficients reproduce the experimental results at high temperatures perfectly but with moderate accuracy at lower temperatures, but the KIE is with high accuracy. The similar kinetic behavior is supported by quantum dynamics using wave packet calculations as well. [ABSTRACT FROM AUTHOR]

We develop a full-dimensional analytical potential energy surface (PES) for the F− + SiH3Cl reaction using Robosurfer for automatically sampling the configuration space, the robust [CCSD-F12b + BCCD(T) − BCCD]/aug-cc-pVTZ composite level of theory for computing the energy points, and the permutationally invariant polynomial method for fitting. Evolution of the fitting error and the percentage of the unphysical trajectories are monitored as a function of the iteration steps/number of energy points and polynomial order. Quasi-classical trajectory simulations on the new PES reveal rich dynamics resulting in high-probability SN2 (SiH3F + Cl−) and proton-transfer (SiH2Cl− + HF) products as well as several lower-probability channels, such as SiH2F− + HCl, SiH2FCl + H−, SiH2 + FHCl−, SiHFCl− + H2, SiHF + H2 + Cl−, and SiH2 + HF + Cl−. The Walden-inversion and front-side-attack-retention SN2 pathways are found to be competitive, producing nearly racemic products at high collision energies. The detailed atomic-level mechanisms of the various reaction pathways and channels as well as the accuracy of the analytical PES are analyzed along representative trajectories. [ABSTRACT FROM AUTHOR]

We have developed a full-dimensional analytical ab initio potential energy surface (PES) for the Cl− + CH3I reaction using the Robosurfer program system. The energy points have been computed using a robust composite method defined as CCSD-F12b + BCCD(T) – BCCD with the aug-cc-pVTZ(-PP) basis set and have been fitted by the permutationally invariant polynomial approach. Quasi-classical trajectory simulations on the new PES reveal that two product channels are open in the collision energy (Ecoll) range of 1–80 kcal/mol, i.e., SN2 leading to I− + CH3Cl and iodine abstraction (above ∼45 kcal/mol) resulting in ICl− + CH3. Scattering angle, initial attack angle, product translational energy, and product internal energy distributions show that the SN2 reaction is indirect at low Ecoll and becomes direct-rebound–back-side (CH3-side) attack-type, as Ecoll increases. Iodine abstraction mainly proceeds with direct stripping mechanism with side-on/back-side attack preference. Comparison with crossed-beam experiments and previous direct dynamics simulations shows quantitative or qualitative agreement and also highlights possible theoretical and/or experimental issues motivating further research. [ABSTRACT FROM AUTHOR]

Ion–molecular reactions play a significant role in molecular evolution within the interstellar medium. In this study, the entrance channel reaction, H3+ + C2H4 → H2 + C2H5+, was investigated using classical molecular dynamic (classical MD) and ring polymer molecular dynamic (RPMD) simulation techniques. We developed an analytical potential energy surface function with a permutationally invariant polynomial basis, specifically employing the monomial symmetrized approach. Our dynamic simulations reproduced the rate coefficient of 300 K for H3+ + C2H4 → H2 + C2H5+, aligning reasonably well with the values in the kinetic database commonly utilized in astrochemistry. The thermal rate coefficients obtained using both the classical MD and RPMD techniques exhibited an increase from 100 K to 300 K as the temperature rose. Additionally, we analyzed the excess energy distribution of the C2H5+ fragment with respect to temperature to investigate the indirect reaction pathway of C2H5+ → H2 + C2H3+. This result suggests that the indirect reaction pathway of C2H5+ → H2 + C2H3+ holds minor significance, although the distribution highly depends on the collisional temperature. [ABSTRACT FROM AUTHOR]

We consider the 4-dimensional N = 1 Lie superconformal algebra and search for completely "symmetric" (in the graded sense) 3-index invariant tensors. The solution we find is unique and we show that the corresponding invariant polynomial cubic in the generalized curvatures of superconformal gravity vanishes. Consequently, the associated Chern-Simons polynomial is a non-trivial anomaly cocycle. We explicitly compute this cocycle to all orders in the independent fields of superconformal gravity and establish that it is BRST equivalent to the so-called superconformal a-anomaly. We briefly discuss the possibility that the superconformal c-anomaly also admits a similar Chern-Simons formulation and the potential holographic, 5-dimensional, interpretation of our results. [ABSTRACT FROM AUTHOR]

We report a new permutationally invariant polynomial (PIP) fit to CCSD(T)/aug-cc-pVTZ energies for $ \hbox {CH}_5^{+} $ CH 5 + that provides fast analytical forces, unlike the original PIP fit of 2006. The new fit, which is slightly more precise than the previous one, is used in classical molecular dynamics, diffusion Monte Carlo and path integral molecular dynamics calculations of CH and HH radial distribution functions (RDFs) at temperatures of 10 and $ 50\,\hbox {K} $ 50 K . Corresponding RDFs are also reported for $ \hbox {CD}_{3}\hbox {H}_{2}^{+} $ CD 3 H 2 + . The well-known quantum localisation of the H atoms to the diatomic site from previous and present DMC calculations is maintained at $ 10\,\hbox {K} $ 10 K but shows additional delocalisation to the ' $ \hbox {CH}_{3}^{+} $ CH 3 + ' site at $ 50\,\hbox {K} $ 50 K . [ABSTRACT FROM AUTHOR]

Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) $ \mathcal {V} $ V , and let $ {\Bbbk [\mathcal {V}]^{G}} $ k [ V ] G be the corresponding algebra of invariant polynomial functions. A separating set $ S \subseteq {\Bbbk [\mathcal {V}]^{G}} $ S ⊆ k [ V ] G is a set of polynomials with the property that for all $ v,w \in \mathcal {V} $ v , w ∈ V , if there exists $ f \in {\Bbbk [\mathcal {V}]^{G}} $ f ∈ k [ V ] G separating $v$ and $w$, then there exists $ f \in S $ f ∈ S separating $v$ and $w$. In this article, we consider the action of $ G = \operatorname {GL}_2(\mathbb {C}) $ G = GL 2 ⁡ (C) on the $ \mathbb {C} $ C -vector space $ {\mathcal {M}}_2^n $ M 2 n of n-tuples of $ 2 \times 2 $ 2 × 2 matrices by simultaneous conjugation. Minimal generating sets $ S_n $ S n of $ \mathbb {C}[{\mathcal {M}}_2^n]^G $ C [ M 2 n ] G are well known and $ |S_n| = \frac 16(n^3+11n) $ | S n | = 1 6 (n 3 + 11 n). In recent work, Kaygorodov et al. [Kaygorodov I, Lopatin A, Popov Y. Separating invariants for $ 2 \times 2 $ 2 × 2 matrices. Linear Algebra Appl. 2018;559:114-124.] showed that for all $ n \geq ~1 $ n ≥ 1 , $ S_n $ S n is a minimal separating set by inclusion, i.e. that no proper subset of $ S_n $ S n is a separating set. This does not necessarily mean that $ S_n $ S n has minimum cardinality among all separating sets for $ \mathbb {C}[{\mathcal {M}}_2^n]^G $ C [ M 2 n ] G . Our main result shows that any separating set for $ \mathbb {C}[{\mathcal {M}}_2^n]^G $ C [ M 2 n ] G has cardinality $ \geq ~5n-5 $ ≥ 5 n − 5. In particular, there is no separating set of size $ \dim (\mathbb {C}[{\mathcal {M}}_2^n]^G) = 4n-3 $ dim ⁡ (C [ M 2 n ] G) = 4 n − 3 for $ n \geq ~3 $ n ≥ 3. Further, $ S_3 $ S 3 has indeed minimum cardinality as a separating set, but for $ n \geq ~4 $ n ≥ 4 there may exist a smaller separating set than $ S_n $ S n . We show that a smaller separating set does in fact exist for all $ n \geq ~5 $ n ≥ 5. We also prove similar results for the left–right action of $ \operatorname {SL}_2(\mathbb {C}) \times \operatorname {SL}_2(\mathbb {C}) $ SL 2 ⁡ (C) × SL 2 ⁡ (C) on $ {\mathcal {M}}_2^n $ M 2 n . [ABSTRACT FROM AUTHOR]

A simple formula is proven for the maximal amplitudes of N-phase hyperelliptic solutions of the modified nonlinear Schrödinger equation. The simple formula for the maximal amplitude depends only on the roots of an invariant polynomial. [ABSTRACT FROM AUTHOR]

Let G = \operatorname {exp}(\mathfrak {g}) be a connected and simply connected real nilpotent Lie group with Lie algebra \mathfrak g, H = \operatorname {exp}(\mathfrak {h}) an analytic subgroup of G with Lie algebra \mathfrak h, \chi a unitary character of H and \tau = \text {ind}_H^G \chi the monomial representation of G induced by \chi. Let D_{\tau }(G/H) be the algebra of the G-invariant differential operators on the fiber bundle over the base space G/H associated to the data (H,\chi). We prove the polynomial conjecture due to Corwin-Greenleaf stating that if \tau is of finite multiplicities, the algebra D_{\tau }(G/H) is isomorphic to the algebra {{\mathbb C}[{\Gamma }_{\tau }]}^H of the H-invariant polynomial functions on the affine subspace {\Gamma }_{\tau } = \{\ell \in {\mathfrak g}^*; {\ell }|_{\mathfrak h} = -\sqrt {-1}d\chi \} of {\mathfrak g}^*. In this case, we show that any non-zero element of D_{\tau }(G/H) admits a fundamental tempered solution. [ABSTRACT FROM AUTHOR]

We described a minimal separating set for the algebra of O 2 + (F q) -invariant polynomial functions of m -tuples of two-dimensional vectors over a finite field F q. [ABSTRACT FROM AUTHOR]

The recently proposed permutationally invariant polynomial-neural network (PIP-NN) based Δ-machine learning (Δ-ML) approach (PIP-NN Δ-ML) is a flexible, general, and highly cost-efficient method to develop a full dimensional accurate potential energy surface (PES). Only a small portion of points, which can be actively selected from the low-level (often DFT) dataset, with high-level energies are needed to bring a low-level PES to a high-level of quality. The hydrogen abstraction reaction between the methanol and hydroxyl radical, OH + CH3OH, has been studied using theories and experiments for a long time due to its great importance in combustion, atmospheric and interstellar chemistry. However, it is not trivial to develop the full dimensional accurate PES for it. In this work, the PIP-NN Δ-ML method is successfully applied to the title reaction. The DFT PES was fitted by using 140 192 points. Only 5% of the DFT dataset was needed to be calculated at the level of UCCSD(T)-F12a/AVTZ, aiming to improve the DFT PES to the target high-level, UCCSD(T)-F12a/AVTZ. More than 92% of the original unaffordable calculation costs were saved. The kinetics, including rate coefficients and branching ratios, were then studied by performing quasi-classical trajectory calculations on this newly fitted PES for the title reaction. [ABSTRACT FROM AUTHOR]

A new, elementary algorithm for constructing complete, minimal sets of generating invariants for finite or, more generally, discrete group actions, both linear and nonlinear, is proposed. The resulting fundamental invariants are piecewise analytic and endowed with a rewrite rule that enables one to immediately express any other invariant (polynomial, rational, smooth, analytic, etc.) as a function thereof. The construction is inspired by the method of equivariant moving frames for Lie group actions. [ABSTRACT FROM AUTHOR]

We report a new permutationally invariant polynomial (PIP) fit to CCSD(T)/aug-cc-pVTZ energies for $ \hbox {CH}_5^{+} $ CH 5 + that provides fast analytical forces, unlike the original PIP fit of 2006. The new fit, which is slightly more precise than the previous one, is used in classical molecular dynamics, diffusion Monte Carlo and path integral molecular dynamics calculations of CH and HH radial distribution functions (RDFs) at temperatures of 10 and $ 50\,\hbox {K} $ 50 K . Corresponding RDFs are also reported for $ \hbox {CD}_{3}\hbox {H}_{2}^{+} $ CD 3 H 2 + . The well-known quantum localisation of the H atoms to the diatomic site from previous and present DMC calculations is maintained at $ 10\,\hbox {K} $ 10 K but shows additional delocalisation to the ' $ \hbox {CH}_{3}^{+} $ CH 3 + ' site at $ 50\,\hbox {K} $ 50 K . [ABSTRACT FROM AUTHOR]