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A full-dimensional, permutationally invariant polynomial potential energy surface for glycine recently reported (R. Conte et al., J. Chem. Phys.2020, 153, 244301) is used with the code MULTIMODE to determine the IR absorption spectra for Conformers I and II using a new separable dipole moment function. The calculated spectra agree well with the experimental ones. The full-dimensional nature of the potential allows us also to examine dynamical results, such as tunneling rates. Remarkably, using a one-dimensional path based on the potential energy surface to estimate the tunneling rate from Conformer VI to Conformer I, good agreement is found with the recent experimental measurement. Finally a brief comparison of our potential energy surface with a recently reported sGDML one is made.

For every simple finite-dimensional complex Lie algebra, I give a simple construction of all (except for the Pfaffian) basic polynomials invariant under the Weyl group. The answer is given in terms of the two basic polynomials of smallest degree.

A potential energy surface (PES) for high-energy collisions between nitrogen molecules is useful for modeling chemical dynamics in shock waves and plasmas. In the present work, we fit the many-body...

A general strategy is presented for constructing and validating permutationally invariant polynomial (PIP) expansions for chemical systems of any stoichiometry. Demonstrations are made for three categories of gas-phase dynamics and kinetics: collisional energy-transfer trajectories for predicting pressure-dependent kinetics, three-body collisions for describing transient van der Waals adducts relevant to atmospheric chemistry, and nonthermal reactivity via quasiclassical trajectories. In total, 30 systems are considered with up to 15 atoms and 39 degrees of freedom. Permutational invariance is enforced in PIP expansions with as many as 13 million terms and 13 permutationally distinct atom types by taking advantage of petascale computational resources. The quality of the PIP expansions is demonstrated through the systematic convergence of in-sample and out-of-sample errors with respect to both the number of training data and the order of the expansion, and these errors are shown to predict errors in the dynamics for both reactive and nonreactive applications. The parallelized code distributed as part of this work enables the automation of PIP generation for complex systems with multiple channels and flexible user-defined symmetry constraints and for automatically removing unphysical unconnected terms from the basis set expansions, all of which are required for simulating complex reactive systems.

An efficient method is described for generating a fragmented, permutationally invariant polynomial basis to fit electronic energies and, if available, gradients for large molecules. The method presented rests on the fragmentation of a large molecule into any number of fragments while maintaining the permutational invariance and uniqueness of the polynomials. The new approach improves on a previous one reported by Qu and Bowman by avoiding repetition of polynomials in the fitting basis set and speeding up gradient evaluations while keeping the accuracy of the PES. The method is demonstrated for CH3–NH–CO–CH3 (N-methylacetamide) and NH2–CH2–COOH (glycine).

International audience; Assuming the variety of a polynomial set is invariant under a group action, we construct a set of invariants that define the same variety. Our construction can be seen as a generalization of the previously known construction for finite groups. The result though has to be understood outside an invariant variety which is independent of the polynomial set considered. We introduce the symmetrizations of a polynomial that are polynomials in a generating set of rational invariants. The generating set of rational invariants and the symmetrizations are constructed w.r.t. a section to the orbits of the group action.

The study of quadratic polynomial differential systems on the plane have been shown a tough challenge, there exist hundreds of papers about them which are dated for over a century and until now there exist several topics to be studied and concluded. For instance, the complete characterization of phase portraits of quadratic systems remains unknown and the complete topological classification of such systems has been a complex work. It is well known that the greatest difficult of working with quadratic systems is the quantity of parameters. A (generic) quadratic system is defined by 12 parameters, however by using affine transformations and time rescaling one can reduce this number by five, but yet this is a very large number, once the corresponding bifurcation diagram is a fivedimensional euclidean space. So, it is convenient to use some tools (as the Invariant Theory) in order to study families of quadratic systems with specific properties (for instance, according to the structural stability or possessing classes of invariant algebraic curves) with the purpose of reducing even more (when it is possible) this quantity of parameters. The main goal of this thesis is to contribute to the classification of the quadratic systems on the plane. More precisely, we present the complete study (modulo islands) of the bifurcation diagram of two families of quadratic systems possessing specific properties on their singularities, we do the complete topological classification (modulo limit cycles) of all the phase portraits of two sets of quadratic systems of codimension two and we perform the classification of quadratic differential systems with invariant ellipses according to their configurations of invariant ellipses and invariant lines. It is worth mentioning that these three works represent three different approaches to the study of quadratic systems and each one of them uses different techniques, which all together are useful towards the final goal of classifying phase portraits. O estudo dos sistemas diferenciais polinomiais quadráticos no plano tem se demonstrado desafiador, existem centenas de artigos datados de mais de um século sobre esse tema e ainda existem muitos tópicos para serem estudados e concluídos. Por exemplo, a caracterização completa dos retratos de fase de sistemas quadráticos permanece desconhecida e a classificação topológica completa de tais sistemas tem sido um trabalho complexo. É bem sabido que a principal dificuldade de se trabalhar com os sistemas quadráticos é a quantidade de parâmetros. Um sistema quadrático (genérico) é definido por 12 parâmetros, entretanto, usando transformações afins e reescala temporal podese reduzir este número para cinco, mas ainda são muitos parâmetros, uma vez que o correspondente diagrama de bifurcação é um espaço euclideano de dimensão cinco. Desta forma, fazse conveniente utilizar algumas ferramentas (a Teoria dos Invariantes, por exemplo) de modo a estudar famílias de sistemas quadráticos com propriedades específicas (por exemplo, de acordo com a estabilidade estrutural ou possuindo classes de curvas algébricas invariantes) para reduzir ainda mais (quando possível) essa quantidade de parâmetros. Nesta tese objetivamos contribuir com a classificação dos sistemas quadráticos no plano. Mais precisamente, apresentamos o estudo completo (módulo ilhas) do diagrama de bifurcação de duas famílias de sistemas quadráticos com propriedades específicas em suas singularidades. Fazemos a classificação topológica completa de todos os retratos de fases (módulo ciclos limites) de dois conjuntos de sistemas quadráticos de codimensão dois e fazemos a classificação de todos os sistemas quadráticos que possuem elipses invariantes de acordo com a chamada configuração de elipses invariantes e retas invariantes. Vale a pena ressaltar que esses trabalhos representam três abordagens distintas para o estudo dos sistemas quadráticos, e cada um deles utiliza técnicas diferentes, que em conjunto são úteis para o objetivo final de classificar retratos de fases.

There has been significant progress in recent years in the use of machine learning techniques to model high-dimensional reactive potential energy surfaces using large-scale data obtained from ab initio electronic structure calculations. In these methods, the strategy used to gather data becomes a key issue as the molecular size increases. In this work, we examine the applicability of the reaction path search algorithm implemented in the Global Reaction Route Mapping (GRRM) code as a data-gathering approach. The electronic energies and gradients sampled by using the GRRM calculation are directly used in potential energy surface fitting to a permutationally invariant polynomial function. This simple approach was applied to the HNS and HCNO reaction systems, and we found that the fitted potential energy surfaces reasonably reproduce the features of the electronic structure calculations used in the GRRM calculations. This suggests that the GRRM sampling scheme can be used to construct an initial potential energy surface.

Improper and hybrid-improper ferroelectrics (FEs) host spontaneous polarization ($P$) owing to the anharmonic coupling between polar and nonpolar modes. In general, the unstable zone-boundary nonpolar mode is a primary order parameter that couples to the stable zone-center polar mode to induce $P$. However, in certain FEs, such as the Aurivillius family, the polar mode is also unstable. In such cases, the strength of coupling between polar and nonpolar modes governs whether FE phase transition is a single or double step. Here we investigate a single-step avalanche-like transition in ${\mathrm{SrBi}}_{2}{\mathrm{Nb}}_{2}{\mathrm{O}}_{9}$, an Aurivillius family compound, using inelastic neutron scattering and Raman spectroscopy combined with group theory and first-principles simulations. We find that a strong trilinear coupling constant $\ensuremath{\lambda}$ between the two nonpolar and one polar instabilities enables their single-step condensation at the FE transition temperature ${T}_{FE}$ and overcomes the effect of the positive bi-quadratic coupling that works against it. The ratio of $\ensuremath{\lambda}$ to the leading-order term in the energy invariant polynomial is two to six times larger than in iso-structural ${\mathrm{SrBi}}_{2}{\mathrm{Ta}}_{2}{\mathrm{O}}_{9}$, which undergoes a double-step transition. We identify and track polar and nonpolar instabilities and find their temperature dependence consistent with the theoretical prediction based on the strong trilinear coupling mechanism. From the sign ($\ifmmode\pm\else\textpm\fi{}$) of bi-quadratic coupling constants in the energy invariant polynomial, we further rule out the triggering mechanism to drive the FE transition. Moreover, we observe little change in phonons across ${T}_{FE}$ despite a first-order transition. However, a significant phonon broadening occurs even at 300 K, which we rationalize based on a large anharmonic vibrational amplitude of O atoms in the ${\mathrm{NbO}}_{6}$ octahedra.

In the article Llibre and Vulpe (Rocky Mt J Math 38:1301–1373, 2006) the family of cubic polynomial differential systems possessing invariant straight lines of total multiplicity 9 was considered and 23 such classes of systems were detected. We recall that 9 invariant straight lines taking into account their multiplicities is the maximum number of straight lines that a cubic polynomial differential systems can have if this number is finite. Here we complete the classification given in Llibre and Vulpe (Rocky Mt J Math 38:1301–1373, 2006) by adding a new class of such cubic systems and for each one of these 24 such classes we perform the corresponding first integral as well as its phase portrait. Moreover we present necessary and sufficient affine invariant conditions for the realization of each one of the detected classes of cubic systems with maximum number of invariant straight lines when this number is finite.

Agraïments: The first author has been supported by AGAUR FI-DGR 2010. The third author has been supported by AGAUR PIV-DGR-2010, FCT grant PTDC/MAT/117106/2010 and through CAMG SD. We provide normal forms and the global phase portraits in the Poincaré disk for all the Hamiltonian non-degenerate centers of linear plus cubic homogeneous planar polynomial vector fields.

We provide a constructive method to obtain polynomial vector fields in ℝ3 having infinitely many limit cycles starting from polynomial vector fields in ℝ2 with a period annulus. We present two examples of polynomial vector fields in ℝ3 having infinitely many limit cycles, one of them of degree 2 and the other one of degree 12. The main tools of our method are the Melnikov integral and the Hamiltonian structure.

We describe software to incorporate electronic energies and gradients to develop high-dimensional potential energy surfaces, using a permutationally invariant polynomial basis. The energies and gradients are obtained using direct dynamics, using the efficient B3LYP/6-31+G(d) level of theory. The new software is described along with extensive testing and assessment of the benefits of using gradients as well as energies for CH4. Starting with a data set of 9000 configurations, we examine training and testing on data sets of energies only and energies plus gradients with data sets as small as 50. In addition to standard root-mean-square fitting errors of energies and gradients, normal-mode analyses and diffusion Monte Carlo calculations are performed to examine the fidelity of the fits using gradients. We show that a precisely fitted potential surface can be obtained using energies and gradients with only 100 or even just 50 widely scattered configurations. Finally, several fits are done using all the data from direct-dynamics trajectories with 1000 steps. These are more demanding fits compared to the one based on pruning data sets. The results of these fits are encouraging.

Zeta functions for linear codes were defined by Iwan Duursma in 1999. They were generalized to the case of some invariant polynomials by the preset author. One of the most important problems is whether extremal weight enumerators satisfy the Riemann hypothesis. In this article, we show there exist extremal polynomials of the weight enumerator type which are invariant under the MacWilliams transform and do not satisfy the Riemann hypothesis., Comment: 9 pages

We introduce and study a cyclically invariant polynomial which is an analog of the classical tridiagonal determinant usually called the continuant. We prove that this polynomial can be calculated as the Pfaffian of a skew-symmetric matrix. We consider the corresponding Diophantine equation and prove an analog of a famous result due to Conway and Coxeter. We also observe that Chebyshev polynomials of the first kind arise as Pfaffians., Comment: 8 pages

``$\Delta$-machine learning" refers to a machine learning approach to bring a property such as a potential energy surface (PES) based on low-level (LL) density functional theory (DFT) energies and gradients to close to a coupled cluster (CC) level of accuracy. Here we present such an approach that uses the permutationally invariant polynomial (PIP) method to fit high-dimensional PESs. The approach is represented by a simple equation, in obvious notation $V_{LL{\rightarrow}CC}=V_{LL}+\Delta{V_{CC-LL}}$, and demonstrated for \ce{CH4}, \ce{H3O+}, and $trans$ and $cis$-$N$-methyl acetamide (NMA), \ce{CH3CONHCH3}. For these molecules, the LL PES, $V_{LL}$, is a PIP fit to DFT/B3LYP/6-31+G(d) energies and gradients, and $\Delta{V_{CC-LL}}$ is a precise PIP fit obtained using a low-order PIP basis set and based on a relatively small number of CCSD(T) energies. For \ce{CH4} these are new calculations adopting an aug-cc-pVDZ basis, for \ce{H3O+} previous CCSD(T)-F12/aug-cc-pVQZ energies are used, while for NMA new CCSD(T)-F12/aug-cc-pVDZ calculations are performed. With as few as 200 CCSD(T) energies, the new PESs are in excellent agreement with benchmark CCSD(T) results for the small molecules, and for 12-atom NMA training is done with 4696 CCSD(T) energies.

Kinetic rates for thermochemical nonequilibrium models are generally computed from quasiclassical trajectory (QCT) calculations on accurate ab initio potential energy surfaces (PES). In this article, we use a feed-forward artificial neural network (ANN) to fit existing single-point energies for ${\mathrm{N}}_{2}+{\mathrm{N}}_{2}$ interactions [Bender et al., J. Chem. Phys. 143, 054304 (2015)] to construct a PES suitable for molecular simulation of high-temperature gas flows. We then perform detailed comparisons with a widely used ${\mathrm{N}}_{4}$ PES that was built using the permutation invariant polynomials (PIP) method. Specific physical considerations in the construction of the ANN for this application are detailed. Translation, rotation, and permutation invariance are precisely satisfied by mapping the interatomic distances onto a set of permutation invariant inputs, known as fundamental invariants (FI) that generate the permutation invariant polynomial ring. The diatomic energy is imposed by decomposing the total potential energy into a sum of a two-body and a many-body energy contribution. To obtain the correct dynamical behavior with the most basic, yet computationally efficient ANN, spurious long-distance interactions must be removed to avoid incorrect physical behavior at the dissociation threshold. We use a simple apodization function to smoothly taper off to zero any residual many-body interaction at large separations. Both accuracy and performance of the FI-ANN PES are assessed. QCT calculations are used to compute dissociation probabilities and vibrational energy distributions at various equilibrium temperatures. Excellent agreement with the results obtained from the PIP PES is found. For our test case, the ANN PES is also significantly more computationally efficient than the PIP PES at comparable root-mean-square error levels.

Let $R$ be an invariant polynomial ring of a reductive group acting on a vector space, and let $d$ be the minimum integer such that $R$ is generated by those polynomials in $R$ of degree no more than $d$ . To upper bound such $d$ is a long standing open problem since the very initial study of the invariant theory in the 19th century. Motivated by its significant role in characterizing multipartite entanglement, we study the invariant polynomial rings of local unitary groups — the direct product of unitary groups acting on the tensor product of Hilbert spaces, and local general linear groups — the direct product of general linear groups acting on the tensor product of Hilbert spaces. For these two group actions, we prove explicit upper bounds on the degrees needed to generate the corresponding invariant polynomial rings. On the other hand, systematic methods are provided to construct all homogeneous polynomials that are invariant under these two groups for any fixed degree. Thus, our results can be regarded as a complete characterization of the invariant polynomial rings. As an interesting application, we show that multipartite entanglement is additive in the sense that two multipartite states are local unitary equivalent if and only if $r$ -copies of them are local unitary equivalent for some $r$ .

Multivariate polynomial interpolation plays a crucial role both in scientific computation and engineering application. Exploring the structure of the D-invariant (closed under differentiation) polynomial subspaces has significant meaning for multivariate Hermite-type interpolation (especially ideal interpolation). We analyze the structure of a D-invariant polynomial subspace Pn in terms of Cartesian tensors, where Pn is a subspace with a maximal total degree equal to n,n≥1. For an arbitrary homogeneous polynomial p(k) of total degree k in Pn, p(k) can be rewritten as the inner products of a kth order symmetric Cartesian tensor and k column vectors of indeterminates. We show that p(k) can be determined by all polynomials of a total degree one in Pn. Namely, if we treat all linear polynomials on the basis of Pn as a column vector, then this vector can be written as a product of a coefficient matrix A(1) and a column vector of indeterminates; our main result shows that the kth order symmetric Cartesian tensor corresponds to p(k) is a product of some so-called relational matrices and A(1).

We consider harmonic polynomials of real variables $x,y,z$ that are eigenfunctions of the rotations about the axis $z$. They have the form $(x\pm yi)^{n}p(x,y,z)$, where $p$ is a rotation invariant polynomial. Let ${\mathfrak R}_{m}$ be the family of the polynomials $p$ of degree $m$ which are reducible over the rationals. We describe ${\mathfrak R}_{m}$ for $m\leq5$ and prove that ${\mathfrak R}_{6}$ and ${\mathfrak R}_{7}$ are finite.

We report full and fragmented potential energy surfaces (PESs) for N-methyl acetamide that contain the cis and trans isomers and the saddle points separating them. The full PES uses Permutationally Invariant Polynomials (PIPs) in reduced symmetry which describe the three-fold symmetry of each methyl rotor. A more efficient PES is an extension of the fragmented PIP approach we reported recently. In this approach, the set of Morse variables is partitioned and the fragmented PIP basis is the union of the PIP basis for each set of variables. This approach is general and can be used with neural network fits. The fits are done using roughly 250 000 electronic energies and gradients obtained from direct dynamics, using the B3LYP/cc-pVDZ level of theory. The full PIP basis in 66 Morse variables, with a maximum polynomial order of 3, contains 8040 linear coefficients. The fragmented PIP basis, also with a maximum polynomial order of 3, contains 6121 coefficients. The root-mean-square errors of both PESs are roughly 100 cm−1 for energies and 15 cm−1/bohr per atom for gradients, for energies up to roughly 45 000 cm−1, relative to the trans minimum. Energies and normal mode frequencies of the cis and trans isomers for the full and fragmented PESs agree well with direct calculations. The energies of the two saddle points separating these minima are precisely given by both PESs. Diffusion Monte Carlo calculations of the zero-point energies of the two isomers are also reported.

Over the past decade, about 50 potential energy surfaces (PESs) for polyatomics with 4–11 atoms and for clusters have been calculated using the permutationally invariant polynomial method. This is a general, mainly linear least-squares method for precise mathematical fitting of tens of thousands of electronic energies for reactive and nonreactive systems. A brief tutorial of the methodology is given, including several recent improvements. Recent applications to the formic acid dimer (the current record holder in size for a reactive system), the H2—H2O complex, and four protonated water clusters [H+(H2O)n=2,3,4,6] are given. The last application also illustrates extension to large clusters using the many-body representation. Expected final online publication date for the Annual Review of Physical Chemistry Volume 69 is April 20, 2018. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.