Communication: Multiple-property-based diabatization for open-shell van der Waals molecules

Numerical Equivalence of Diabatic and Adiabatic Representations in Diatomic Molecules

The (time-independent) Schrödinger equation for atomistic systems is solved by using the adiabatic potential energy curves (PECs) and the associated adiabatic approximation. In cases where interactions between electronic states become important, the associated nonadiabatic effects are taken into account via derivative couplings (DDRs), also known as nonadiabatic couplings (NACs). For diatomic molecules, the corresponding PECs in the adiabatic representation are characterized by avoided crossings. The alternative to the adiabatic approach is the diabatic representation obtained via a unitary transformation of the adiabatic states by minimizing the DDRs. For diatomics, the diabatic representation has zero DDR and nondiagonal diabatic couplings ensue. The two representations are fully equivalent and so should be the rovibronic energies and wave functions, which result from the solution of the corresponding Schrödinger equations. We demonstrate (for the first time) the numerical equivalence between the adiabatic and diabatic rovibronic calculations of diatomic molecules using the ab initio curves of yttrium oxide (YO) and carbon monohydride (CH) as examples of two-state systems, where YO is characterized by a strong NAC, while CH has a strong diabatic coupling. Rovibronic energies and wave functions are computed using a new diabatic module implemented in the variational rovibronic code Duo. We show that it is important to include both the diagonal Born-Oppenheimer correction and nondiagonal DDRs. We also show that the convergence of the vibronic energy calculations can strongly depend on the representation of nuclear motion used and that no one representation is best in all cases.

Topology of Conical Intersection Seams and the Geometric Phase

In this paper, we demonstrate two topological properties of crossing seams, that is, the sets of points in the N-dimensional space of nuclear coordinates where two electronic eigenstates are degenerate. We shall examine the typical case of states of the same spin with accidental degeneracies, whereby the crossing seam is of dimension N - 2. The first property we demonstrate is that a crossing seam has no boundary, therefore, it must either extend asymptotically to infinite values of one or more coordinates or wrap on itself. The second property is that two (or more) crossing seams can intersect each other but in such a way that neither of them ends at the intersection. When N = 3, the crossing seam is a line in a 3D space; this is so in triatomic molecules but also in reduced dimensionality treatments of larger polyatomics. The above-mentioned rules then mean that the crossing seam is a line of infinite length or a closed loop and can split into three branches but not in two. The example of the first two excited 1A' states of H2Cl+ illustrates these rules and shows their usefulness for computational search and characterization of crossing seams.

Fast and accurate excited states predictions: machine learning and diabatization

The efficiency of machine learning algorithms for electronically excited states is far behind ground-state applications. One of the underlying problems is the insufficient smoothness of the fitted potential energy surfaces and other properties in the vicinity of state crossings and conical intersections, which is a prerequisite for an efficient regression. Smooth surfaces can be obtained by switching to the diabatic basis. However, diabatization itself is still an outstanding problem. We overcome these limitations by solving both problems at once. We use a machine learning approach combining clustering and regression techniques to correct for the deficiencies of property-based diabatization which, in return, provides us with smooth surfaces that can be easily fitted. Our approach extends the applicability of property-based diabatization to multidimensional systems. We utilize the proposed diabatization scheme to achieve higher prediction accuracy for adiabatic states and we show its performance by reconstructing global potential energy surfaces of excited states of nitrosyl fluoride and formaldehyde. While the proposed methodology is independent of the specific property-based diabatization and regression algorithm, we show its performance for kernel ridge regression and a very simple diabatization based on transition multipoles. Compared to most other algorithms based on machine learning, our approach needs only a small amount of training data.

An ab initio study of the rovibronic spectrum of sulphur monoxide (SO): diabatic vs. adiabatic representation

We present an ab initio study of the rovibronic spectra of sulphur monoxide (32S16O) using internally contracted multireference configuration interaction (ic-MRCI) method and aug-cc-pV5Z basis sets. It covers 13 electronic states X3Σ-, a1Δ, b1Σ+, c1Σ-, A''3Σ+, A'3Δ, A3Π, B3Σ-, C3Π, d1Π, e1Π, C'3Π, and (3)1Π ranging up to 66 800 cm-1. The ab initio spectroscopic model includes 13 potential energy curves, 23 dipole and transition dipole moment curves, 23 spin-orbit curves, and 14 electronic angular momentum curves. A diabatic representation is built by removing the avoided crossings between the spatially degenerate pairs C3Π-C'3Π and e1Π-(3)1Π through a property-based diabatisation method. We also present non-adiabatic couplings and diabatic couplings for these avoided crossing systems. All phases for our coupling curves are defined, and consistent, providing the first fully reproducible spectroscopic model of SO covering the wavelength range longer than 147 nm. Finally, an ab initio rovibronic spectrum of SO is computed.

Correlated rotational excitations in NO–CO inelastic collisions

We present a joint experimental and theoretical study of rotationally inelastic collisions between NO ( X2Π1/2, ν = 0, j = 1/2, f) radicals and CO ( X1Σ+, ν = 0, j = 0) molecules at a collision energy of 220 cm−1. State-to-state scattering images for excitation of NO radicals into various final states were measured with high resolution by combining the Stark deceleration and velocity map imaging techniques. The high image resolution afforded the observation of correlated rotational excitations of NO–CO pairs, which revealed a number of striking scattering phenomena. The so-called “parity-pair” transitions in NO are found to have similar differential cross sections, independent of the concurrent excitation of CO, extending this well-known effect for collisions between NO and rare gas atoms into the realm of bimolecular collisions. Forward scattering is found for collisions that induce a large amount of rotational energy transfer (in either NO, CO, or both), which require low impact parameters to induce sufficient energy transfer. This observation is interpreted in terms of the recently discovered hard collision glory scattering mechanism, which predicts the forward bending of initially backward receding trajectories if the energy uptake in the collision is substantial in relation to the collision energy. The experimental results are in good agreement with the predictions from coupled-channels quantum scattering calculations based on an ab initio NO–CO potential energy surface.

Diabatic States of Molecules

Quantitative simulations of electronically nonadiabatic molecular processes require both accurate dynamics algorithms and accurate electronic structure information. Direct semiclassical nonadiabatic dynamics is expensive due to the high cost of electronic structure calculations, and hence it is limited to small systems, limited ensemble averaging, ultrafast processes, and/or electronic structure methods that are only semiquantitatively accurate. The cost of dynamics calculations can be made manageable if analytic fits are made to the electronic structure data, and such fits are most conveniently carried out in a diabatic representation because the surfaces are smooth and the couplings between states are smooth scalar functions. Diabatic representations, unlike the adiabatic ones produced by most electronic structure methods, are not unique, and finding suitable diabatic representations often involves time-consuming nonsystematic diabatization steps. The biggest drawback of using diabatic bases is that it can require large amounts of effort to perform a globally consistent diabatization, and one of our goals has been to develop methods to do this efficiently and automatically. In this Feature Article, we introduce the mathematical framework of diabatic representations, and we discuss diabatization methods, including adiabatic-to-diabatic transformations and recent progress toward the goal of automatization.

Effective non-adiabatic Hamiltonians for the quantum nuclear motion over coupled electronic states

The quantum mechanical motion of the atomic nuclei is considered over a single- or a multidimensional subspace of electronic states which is separated by a gap from the rest of the electronic spectrum over the relevant range of nuclear configurations. The electron-nucleus Hamiltonian is block-diagonalized up to O(εⁿ⁺¹) through a unitary transformation of the electronic subspace, and the corresponding nth-order effective Hamiltonian is derived for the quantum nuclear motion. Explicit but general formulas are given for the second- and the third-order corrections. As a special case, the second-order Hamiltonian corresponding to an isolated electronic state is recovered which contains the coordinate-dependent mass-correction terms in the nuclear kinetic energy operator. For a multidimensional, explicitly coupled electronic band, the second-order Hamiltonian contains the usual Born-Oppenheimer terms and nonadiabatic corrections, but generalized mass-correction terms appear as well. These, earlier neglected terms, perturbatively account for the outlying (discrete and continuous) electronic states not included in the explicitly coupled electronic subspace.

Correlated energy transfer in rotationally and spin-orbit inelastic collisions of NO(X2Π1/2, j = 1/2f) with O2(X3Σg-)

We present a combined experimental and theoretical study of state-to-state inelastic scattering of NO(X2Π1/2, j = 1/2f) with O2(X3Σg-) molecules at a collision energy of 480 cm-1, focusing in particular on the observation and interpretation of correlated excitations in both NO and O2. Various final states of the NO radical, in both spin-orbit manifolds, were measured with high resolution using a crossed molecular beam apparatus which employs a combination of Stark deceleration and velocity map imaging. Velocity map imaging directly measures both the angular distribution and the radial velocity distribution of the scattered NO molecules, which probes the kinetic energy uptake or release and hence correlated excitations of NO-O2 pairs. Simultaneous excitations of NO and O2 were resolved for all studied final states of NO. In all cases, the experimental results excellently agree with the results of simulations based on quantum scattering calculations. Trends are discussed by analyzing the scattering wave functions from the calculations.