Energy Surfaces Of Polyatomic Molecules Research Articles

Neural network Gaussian processes as efficient models of potential energy surfaces for polyatomic molecules

Kernel models of potential energy surfaces (PESs) for polyatomic molecules are often restricted by a specific choice of the kernel function. This can be avoided by optimizing the complexity of the kernel function. For regression problems with very expensive data, the functional form of the model kernels can be optimized in the Gaussian process (GP) setting through compositional function search guided by the Bayesian information criterion. However, the compositional kernel search is computationally demanding and relies on greedy strategies, which may yield sub-optimal kernels. An alternative strategy of increasing complexity of GP kernels treats a GP as a Bayesian neural network (NN) with a variable number of hidden layers, which yields NNGP models. Here, we present a direct comparison of GP models with composite kernels and NNGP models for applications aiming at the construction of global PES for polyatomic molecules. We show that NNGP models of PES can be trained much more efficiently and yield better generalization accuracy without relying on any specific form of the kernel function. We illustrate that NNGP models trained by distributions of energy points at low energies produce accurate predictions of PES at high energies. We also illustrate that NNGP models can extrapolate in the input variable space by building the free energy surface of the Heisenberg model trained in the paramagnetic phase and validated in the ferromagnetic phase. By construction, composite kernels yield more accurate models than kernels with a fixed functional form. Therefore, by illustrating that NNGP models outperform GP models with composite kernels, our work suggests that NNGP models should be a preferred choice of kernel models for PES.

Quantum Gaussian process model of potential energy surface for a polyatomic molecule.

With gates of a quantum computer designed to encode multi-dimensional vectors, projections of quantum computer states onto specific qubit states can produce kernels of reproducing kernel Hilbert spaces. We show that quantum kernels obtained with a fixed ansatz implementable on current quantum computers can be used for accurate regression models of global potential energy surfaces (PESs) for polyatomic molecules. To obtain accurate regression models, we apply Bayesian optimization to maximize marginal likelihood by varying the parameters of the quantum gates. This yields Gaussian process models with quantum kernels. We illustrate the effect of qubit entanglement in the quantum kernels and explore the generalization performance of quantum Gaussian processes by extrapolating global six-dimensional PESs in the energy domain.

Permutationally Invariant, Reproducing Kernel-Based Potential Energy Surfaces for Polyatomic Molecules: From Formaldehyde to Acetone

Constructing accurate, high-dimensional molecular potential energy surfaces (PESs) for polyatomic molecules is challenging. Reproducing kernel Hilbert space (RKHS) interpolation is an efficient way to construct such PESs. However, RKHS interpolation is computationally most effective when the input energies are available on a regular grid. Thus, the number of reference energies required can become very large even for pentaatomic systems making such an approach computationally prohibitive when using high-level electronic structure calculations. Here, an efficient and robust scheme is presented to overcome these limitations and is applied to constructing high-dimensional PESs for systems with up to 10 atoms. Using energies as well as gradients reduces the number of input data required and thus keeps the number of coefficients at a manageable size. The correct implementation of permutational symmetry in the kernel products is tested and explicitly demonstrated for the highly symmetric CH4 molecule.

Deep Learning for Nonadiabatic Excited-State Dynamics.

In this work we show that deep learning (DL) can be used for exploring complex and highly nonlinear multistate potential energy surfaces of polyatomic molecules and related nonadiabatic dynamics. Our DL is based on deep neural networks (DNNs), which are used as accurate representations of the CASSCF ground- and excited-state potential energy surfaces (PESs) of CH2NH. After geometries near conical intersection are included in the training set, the DNN models accurately reproduce excited-state topological structures; photoisomerization paths; and, importantly, conical intersections. We have also demonstrated that the results from nonadiabatic dynamics run with the DNN models are very close to those from the dynamics run with the pure ab initio method. The present work should encourage further studies of using machine learning methods to explore excited-state potential energy surfaces and nonadiabatic dynamics of polyatomic molecules.

Efficient non-parametric fitting of potential energy surfaces for polyatomic molecules with Gaussian processes

We explore the efficiency of a statistical learning technique based on Gaussian process (GP) regression as an efficient non-parametric method for constructing multi-dimensional potential energy surfaces (PESs) for polyatomic molecules. Using an example of the molecule N4, we show that a realistic GP model of the six-dimensional PES can be constructed with only 240 potential energy points. We construct a series of the GP models and illustrate the accuracy of the resulting surfaces as a function of the number of ab initio points. We show that the GP model based on ∼1500 potential energy points achieves the same level of accuracy as the conventional regression fits based on 16 421 points. The GP model of the PES requires no fitting of ab initio data with analytical functions and can be readily extended to surfaces of higher dimensions.

An adaptive density-guided approach for the generation of potential energy surfaces of polyatomic molecules

We present an adaptive density-guided approach for the construction of Born–Oppenheimer potential energy surfaces (PES) in rectilinear normal coordinates for use in vibrational structure calculations. The procedure uses one-mode densities from vibrational structure calculations for a dynamic sampling of PESs. The implementation of the procedure is described and the accuracy and versatility of the method is tested for a selection of model potentials, water, difluoromethane and pyrimidine. The test calculations illustrate the advantage of local basis sets over harmonic oscillator basis sets in some important aspects of our procedure.

Optical diabolos: Configurations, nucleations, transformations, and reactions

Generically, elliptically polarized optical beams contain C points, isolated points of circular polarization embedded in a field of elliptical polarization. At a C point the major axis a and minor axis b of the polarization ellipse are degenerate. As one recedes from a C point this degeneracy is lifted in an unusual manner: surfaces a and b form paired cones that touch at their apex, the C point, to form a double cone known as a diabolo. Diabolos appear in a many different areas of science ranging from the mathematics of curved surfaces (umbilic points) to the optics of biaxial crystals (conical refraction) to the degeneracies of chaotic systems (quantum billiards) to the energy surfaces of polyatomic molecules (conical intersections). The cones of a diabolo are classified as either elliptic or hyperbolic. We show that in optical fields only hyperbolics can nucleate or annihilate, and that elliptics arise from hyperbolic transformations. We also show that hyperbolics divide into two subclasses which give rise to five different diabolo types. We formulate loop rules based on topology and geometry for the various diabolos. These rules constrain the landscapes of all systems that contain diabolos. We use the loop rules to derive the generic transformations and reactions of diabolos and to enumerate the nine basic diabolo landscapes. The theoretical predictions are confirmed using computer simulations of diabolos in elliptically polarized laser beams.

Potential Energy Surfaces for Vibrational and Rotational Wave Packet Motion in Polyatomic Molecular Systems

We develop analytical functional forms for the generation of full-dimensional representations of potential energy surfaces for polyatomic molecules of the type AX1..Xn, with one central atom A and peripheral atoms Xi, and for diatomic molecules adsorbed on surfaces. Examples are shown for NH3 and CO(ads)/Cu(100). Such forms are useful for realistic derivations of potential energy surfaces with applications on the calculation of ro-vibrational spectra and time-dependent molecular quantum dynamics. As an example, the effects of rotational motion and orientation on the vibrational wave packet dynamics of HF merged in an external electric field are discussed.

Relative likelihood of encountering conical intersections and avoided intersections on the potential energy surfaces of polyatomic molecules

We prove that conical intersections are much more likely than local minima of the electronic energy gap. Therefore, if one encounters a very small electronic energy gap along a path through configuration space, it is much more likely to be associated with the neighborhood of a conical intersection than with an avoided intersection.

On the representation of potential energy surfaces of polyatomic molecules in normal coordinates: II. Parameterisation of the force field

By substituting the standard mass-weighted normal coordinates with either Morse-like or Gauss-like coordinates, it is demonstrated that significant improvements can be made to the vibrational spectra of polyatomic molecules calculated variationally. Quartic force fields in the form of Taylor expansions are generated by density functional theory for water, formaldehyde and methane, and their vibrational spectra calculated by the perturbation normal coordinate code SPECTRO. These are then compared with three sets of spectra arising from the variational code MULTIMODE. Initial spectra are obtained using the identical Taylor expansion force fields. A subsequent set of spectra are then obtained for which the symmetric normal coordinates of the force fields are replaced by Morse-like coordinates and a final set of spectra are obtained for which the asymmetric normal coordinates of the force field are replaced by Gauss-like coordinates. The restriction is imposed that the complete set of derivatives to quartic are preserved under these coordinate transformations.

On the representation of potential energy surfaces of polyatomic molecules in normal coordinates

We present a method for reduced, piecewise fitting of ab initio potential energy data for use in rovibrational calculations of polyatomic molecules. The procedure involves fitting of the 1-, 2-, 3-, and 4-mode potential fragments by a least squares refinement of polynomial terms which are automatically generated by the geometries of the data points. The method is tested for H 2CS, for which we have an analytical six-dimensional potential energy surface, which is used to generate potential data in order to mimic the interface between the variational code MULTIMODE and electronic structure packages.

A general property of the quantum mechanical Hamiltonians for constrained systems

Whereas model constraints (e.g., frozen bonds, rigidified atomic groups, etc.) are often resorted to for the calculation of the potential energy surfaces of polyatomic molecules, the derivation of exact expressions for the corresponding kinetic energy operators is a rather new topic. Indeed, the dimensions of the configuration spaces being altered, the differential operators should be modified accordingly, but not the multiplicative operators. In addition, the physical condition ℱ = 0 (where ℱ is the total angular momentum vector) which, from a mathematical viewpoint, is a set of three constraints, also is often considered. The particular question raised is: in the derivation of the kinetic energy operator, is the order in which we apply the model constraints on the one hand and the dynamic constraints ℱ = 0 on the other hand, immaterial? The answer is yes.

Multiperturbation approach to potential energy surfaces for polyatomic molecules

In Z-dependent perturbation theory, the lowest-order wave functions for a polyatomic molecule are not only independent of the nuclear charges, but also of the total number of nuclear centers and electrons in the molecule. The complexity of the problem is then determined by the highest order retained in the calculation. Choosing the simplest possible unperturbed Hamiltonian, we describe an n-electron, m-center polyatomic molecule as n ‘‘hydrogenic’’ electrons on a single center perturbed by electron–electron and electron–nucleus Coulomb interactions. With this H0, the first-order wave function for any polyatomic molecule will be a sum of products of hydrogenic orbitals with either two-electron, one-center or one-electron, two-center first-order wave functions. These first-order wave functions are obtained from calculations on He-like and H+2-like systems. Similarly, the nth-order wave function decouples so that the most complex terms are just the nth-order wave functions of all the p-electron, q-center subsystems (p+q=n+2) contained in the molecule. We illustrate applications of this method with some results, complete through third order in the energy, for H+3-like molecules. These are compared with accurate variational results available in the literature. We conclude that, through this order, this perturbation approach is capable of yielding results comparable in accuracy to variational calculations of moderate complexity. The ease and efficiency with which such results can be obtained suggests that this method would be useful for generating detailed potential energy surfaces for polyatomic molecules.

Molecular potential energy surfaces by interpolation

A moving interpolation technique which provides an accurate representation of potential energy surfaces for polyatomic molecules is presented. The method uses the ab initio energy, energy gradient, and second derivatives calculated at dynamically important configurations. The interpolant of the energy and its derivatives converges to the exact value with increasing number of data. A procedure is given for finding the optimum configurations at which ab initio calculations are performed. The method is demonstrated by application to the six-dimensional surface of a diatomic plus diatomic reaction.

Dynamical treatment of unimolecular decomposition reactions. The RRKM formula

To explore the range of validity of the RRKM theory of unimolecular reactions, we present a completely dynamical derivation of the RRKM expression for the decomposition rate of isolated molecules. This derivation avoids the usual equilibrium statistical assumptions and expresses the conditions for validity of the RRKM theory in terms of fundamental, static, molecular properties. To carry out this derivation we apply a treatment of energy transfer and decomposition which combines the Wigner–Eisenbud R-matrix approach to scattering with a technique we previously developed for studying the internal dynamics of nonreactive molecules. We obtain a molecular dissociation rate which agrees with the predictions of microcanonical transition state theory by introducing conditions which ensure statistical equilibration of all states describing the molecular fragments in close proximity of each other. We verify that, under the conditions of our derivation, the distribution of product states is statistical, i.e., in agreement with the implications of simple phase space theory, when the RRKM unimolecular rate formula is valid. Our treatment relies on a number of assumptions which may be regarded as sufficient (although not necessary) conditions for RRKM behavior. These include assumptions concerning the relative magnitudes of various interaction matrix elements (and thus the relative rates of intramolecular relaxation, passage along the reaction coordinate, and decomposition), the properties of intramolecular potential energy surfaces, and the energy dependence of certain quantities. One assumption belonging to the last category may not be entirely valid in all relevant cases but can probably be relaxed considerably. Other assumptions seem plausible but ultimately remain somewhat uncertain due to our ignorance about detailed features of potential energy surfaces for polyatomic molecules.

A new generalized expansion for the potential energy curves of diatomic molecules

A new generalized expansion for the potential energy curves of diatomic molecules is proposed. It is given by where λ (p) = sgn(p) [1 − (Re/R)p], and contains both the Dunham and the Simons−Parr−Finlan (SPF) expansions as special cases corresponding to p = −1 and p = 1, respectively. In order to justify the new expansion, a perturbation theory is developed which yields the Born−Oppenheimer potential as a series identical in form to the new expansion. The perturbation is a purely kinetic−energy perturbation at Re. Prescriptions are given for obtaining both the expansion coefficients and the optimal value of p either from perturbation theory or from spectral data. In terms of spectral data p = −a1 −1. Applications of the new expansion to CO, HF, and 20 alkali halides indicate that it predicts dissociation energies in much closer agreement with experiment than the SPF expansion while maintaining the same quality of agreement with the RKR curve as the SPF expansion provides. A number of possible extensions are discussed including potential energy surfaces for polyatomic molecules, techniques for analytic continuation of the new expansion, and expansions for properties other than the potential energy.

Avoided crossings in photochemistry

Abstract

On the non-crossing rule for potential energy surfaces of polyatomic molecules

It is shown that the non-crossing for the intersection of potential energy curves of diatomic molecule applies also to potential energy surfaces of polyatomic molecules.

United Atom Treatment of the Behavior of Potential Energy Surfaces of Polyatomic Molecules at Small Internuclear Distances

It is shown that the potential energy function of any state of a polyatomic molecule can be expanded for small internuclear distances in the form V= ∑ ∑ α>βZαZβ/Rαβ+Wu+ ∑ α(E2α·Rα2+E3α·Rα3+···),where Wu is the energy of the particular state u of the united atom (UA) to which the molecular state in question connects, and Rα is the distance of the αth nucleus from the UA. Explicit expressions for the coefficients E2α, E3α are given in terms of the electronic charge density of the UA. The derived energy expansion is applied to the molecules CH4, NH3, H2O, and HF which all correspond to the united atom Ne.