Diabatic Potential Matrix Research Articles

Beyond Born-Oppenheimer Constructed Diabatic Potential Energy Surfaces for HeH2.

First-principles based beyond Born-Oppenheimer theory has been employed to construct multistate global Potential-Energy Surfaces (PESs) for the HeH2+ system by explicitly incorporating the Nonadiabatic Coupling Terms (NACTs). Adiabatic PESs and NACTs for the lowest four electronic states (12A', 22A', 32A' and 42A') are evaluated as functions of hyperangles for a grid of fixed values of the hyperradius in hyperspherical coordinates. Conical intersection between different states are validated by integrating the NACTs along appropriately chosen contours. Subsequently, adiabatic-to-diabatic (ADT) transformation angles are determined by solving the ADT equations to construct the diabatic potential matrix for the HeH2+ system which are smooth, single-valued, continuous, and symmetric and are suitable for performing accurate scattering calculations for the titled system.

Beyond Born-Oppenheimer constructed diabatic potential energy surfaces for F + H2 reaction.

First principles based beyond Born-Oppenheimer theory has been implemented on the F + H2 system for constructing multistate global diabatic Potential Energy Surfaces (PESs) through the incorporation of Nonadiabatic Coupling Terms (NACTs) explicitly. The spin-orbit (SO) coupling effect on the collision process of the F + H2 reaction has been included as a perturbation to the non-relativistic electronic Hamiltonian. Adiabatic PESs and NACTs for the lowest three electronic states (12A', 22A', and 12A″) are determined in hyperspherical coordinates as functions of hyperangles for a grid of fixed values of the hyperradius. Jahn-Teller (JT) type conical intersections between the two A' states translate along C2v and linear geometries in F + H2. In addition, A' and A″ states undergo Renner-Teller (RT) interaction at collinear configurations of this system. Both JT and RT couplings are validated by integrating NACTs along properly chosen contours. Subsequently, we have solved adiabatic-to-diabatic transformation (ADT) equations to evaluate the ADT angles for constructing the diabatic potential matrix of F + H2, including the SO coupling terms. The newly calculated diabatic PESs are found to be smooth, single-valued, continuous, and symmetric and can be invoked for performing accurate scattering calculations on the F + H2 system.

Diabatic description of charmoniumlike mesons

We apply the diabatic formalism, first introduced in molecular physics, to the description of heavy-quark mesons. In this formalism the dynamics is completely described by a diabatic potential matrix whose elements can be derived from unquenched lattice QCD studies of string breaking. For energies far below the lowest open flavor meson-meson threshold, the resulting diabatic approach reduces to the well-known Born-Oppenheimer approximation where heavy-quark meson masses correspond to energy levels in an effective quark-antiquark potential. For energies close below or above that threshold, where the Born-Oppenheimer approximation fails, this approach provides a set of coupled Schr\"{o}dinger equations incorporating meson-meson components nonperturbatively, i.e. beyond loop corrections. A spectral study of heavy mesons containing $c\overline{c}$ with masses below $4.1$ GeV is carried out within this framework. From it a unified description of conventional as well as unconventional resonances comes out.

Diabatic neural network potentials for accurate vibronic quantum dynamics-The test case of planar NO3.

A recently developed scheme to produce high-dimensional coupled diabatic potential energy surfaces (PESs) based on artificial neural networks (ANNs) [D. M. G. Williams and W. Eisfeld, J. Chem. Phys. 149, 204106 (2019)] is tested for its viability for quantum dynamics applications. The method, capable of reproducing high-quality ab initio data with excellent accuracy, utilizes simple coupling matrices to produce a basic low-order diabatic potential matrix as an underlying backbone for the model. This crude model is then refined by making its expansion coefficients geometry-dependent by the output neurons of the ANN. This structure, strongly guided by a straightforward physical picture behind nonadiabatic coupling, combines structural simplicity with high accuracy, reproducing ab initio data without introducing unphysical artifacts to the surface, even for systems with complicated electronic structure. The properties of diabatic potentials obtained by this method are tested thoroughly in the present study. Vibrational/vibronic eigenstates are computed on the X̃ and à states of NO3, a notoriously difficult Jahn-Teller system featuring strong nonadiabatic couplings and complex spectra. The method is investigated in terms of how consistently it produces dynamics results for PESs of similar (fitting) quality and how the results depend on the ANN size and ANN topography. A central aspect of this work is to understand the convergence properties of the new method in order to evaluate its predictive power. A previously developed, high-quality model utilizing a purely (high-order) polynomial ansatz is used as a reference to showcase improvements of the overall quality which can be obtained by the new method.

Vibronically and spin-orbit coupled diabatic potentials for X(P) + CH4 → HX + CH3 reactions: General theory and application for X(P) = F(2P).

An approach to systematically construct vibronically and spin-orbit coupled diabatic potential energy surfaces (PESs) for X(P) + CH4 → HX + CH3 reactions is proposed. Permutational symmetry and permutational invariants of the S4 group and its S3 and S2 × S2 subgroups are used to construct a diabatic model which properly describes the reaction starting from reactants to products. As a first example, the approach is applied to the construction of diabatic potentials for the F(2P) + CH4 → HF + CH3 reaction. The description of the entrance channel relies on a set of vibronically and spin-orbit coupled diabatic PESs previously developed by Westermann et al. [Angew. Chem., Int. Ed. 53, 1122 (2014)]. The same set of diabatic electronic states is also used in the transition state region and all four exit channels. There the lowest adiabatic PES derived from the diabatic model reproduces the CSBB-PES of Czakó et al. [J. Chem. Phys. 130, 084301 (2009)]. Interesting aspects of the newly developed diabatic potential matrix and the corresponding adiabatic PESs are discussed.

Neural network diabatization: A new ansatz for accurate high-dimensional coupled potential energy surfaces.

A new diabatization method based on artificial neural networks (ANNs) is presented, which is capable of reproducing high-quality ab initio data with excellent accuracy for use in quantum dynamics studies. The diabatic potential matrix is expanded in terms of a set of basic coupling matrices and the expansion coefficients are made geometry-dependent by the output neurons of the ANN. The ANN is trained with respect to ab initio data using a modified Marquardt-Levenberg back-propagation algorithm. Due to its setup, this approach combines the stability and straightforwardness of a standard low-order vibronic coupling model with the accuracy by the ANN, making it particularly advantageous for problems with a complicated electronic structure. This approach combines the stability and straightforwardness of a standard low-order vibronic coupling model with the accuracy by the ANN, making it particularly advantageous for problems with a complicated electronic structure. This novel ANN diabatization approach has been applied to the low-lying electronic states of NO3 as a prototypical and notoriously difficult Jahn-Teller system in which the accurate description of the very strong non-adiabatic coupling is of paramount importance. Thorough tests show that an ANN with a single hidden layer is sufficient to achieve excellent results and the use of a "deeper" layering shows no clear benefit. The newly developed diabatic ANN potential energy surface (PES) model accurately reproduces a set of more than 90 000 Multi-configuration Reference Singles and Doubles Configuration Interaction (MR-SDCI) energies for the five lowest PES sheets.

Non-adiabatic quantum reactive scattering in hyperspherical coordinates.

A new electronically non-adiabatic quantum reactive scattering methodology is presented based on a time-independent coupled channel formalism and the adiabatically adjusting principal axis hyperspherical coordinates of T Pack and Parker [J. Chem. Phys. 87, 3888 (1987)]. The methodology computes the full state-to-state scattering matrix for A + B2(v, j) ↔ AB(v', j') + B and A + AB(v, j) → A + AB(v', j') reactions that involve two coupled electronic states which exhibit a conical intersection. The methodology accurately treats all six degrees of freedom relative to the center-of-mass which includes non-zero total angular momentum J and identical particle exchange symmetry. The new methodology is applied to the ultracold hydrogen exchange reaction for which large geometric phase effects have been recently reported [B. K. Kendrick et al., Phys. Rev. Lett. 115, 153201 (2015)]. Rate coefficients for the H/D + HD(v = 4, j = 0) → H/D + HD(v', j') reactions are reported for collision energies between 1 μK and 100 K (total energy ≈1.9 eV). A new diabatic potential energy matrix is developed based on the Boothroyd, Keogh, Martin, and Peterson (BKMP2) and double many body expansion plus single-polynomial (DSP) adiabatic potential energy surfaces for the ground and first excited electronic states of H3, respectively. The rate coefficients computed using the new non-adiabatic methodology and diabatic potential matrix reproduce the recently reported rates that include the geometric phase and are computed using a single adiabatic ground electronic state potential energy surface (BKMP2). The dramatic enhancement and suppression of the ultracold rates due to the geometric phase are confirmed as well as its effects on several shape resonances near 1 K. The results reported here represent the first fully non-adiabatic quantum reactive scattering calculation for an ultracold reaction and validate the importance of the geometric phase on the Wigner threshold behavior.

Full-dimensional diabatic potential energy surfaces including dissociation: The 2E″ state of NO3

A scheme to produce accurate full-dimensional coupled diabatic potential energy surfaces including dissociative regions and suitable for dynamical calculations is proposed. The scheme is successfully applied to model the two-sheeted surface of the (2)E″ state of the NO3 radical. An accurate potential energy surface for the NO₃⁻ anion ground state is developed as well. Both surfaces are based on high-level ab initio calculations. The model consists of a diabatic potential matrix, which is expanded to higher order in terms of symmetry polynomials of symmetry coordinates. The choice of coordinates is key for the accuracy of the obtained potential energy surfaces and is discussed in detail. A second central aspect is the generation of reference data to fit the expansion coefficients of the model for which a stochastic approach is proposed. A third ingredient is a new and simple scheme to handle problematic regions of the potential energy surfaces, resulting from the massive undersampling by the reference data unavoidable for high-dimensional problems. The final analytical diabatic surfaces are used to compute the lowest vibrational levels of NO₃⁻ and the photo-electron detachment spectrum of NO₃⁻ leading to the neutral radical in the (2)E″ state by full dimensional multi-surface wave-packet propagation for NO3 performed using the Multi-Configuration Time Dependent Hartree method. The achieved agreement of the simulations with available experimental data demonstrates the power of the proposed scheme and the high quality of the obtained potential energy surfaces.

The Jahn-Teller effect in ${\rm CH}_{\rm 3} {\rm Cl}^{\rm + } ( {{\rm \tilde X}{}^{\rm 2}{\rm E}} )$ CH 3 Cl +(X̃E2): A combined high-resolution experimental measurement and ab initio theoretical study

The energy levels of CH(3)Cl(+)X̃(2)E showing strong spin-vibronic coupling effect (Jahn-Teller effect) have been measured up to 3500 cm(-1) above the ground vibrational state using one-photon zero-kinetic energy photoelectron and mass-analyzed threshold ionization spectroscopic method. Theoretical calculations have been also performed to calculate the spin-vibronic energy levels using a diabatic model and ab initio adiabatic potential energy surfaces (PESs). In the theoretical calculations the diabatic potential energy surfaces are expanded by the Taylor expansions up to the fourth-order including the multimode vibronic interactions. The calculated spin-orbit energy splitting (224.6 cm(-1)) for the ground vibrational state is in good agreement with the experimental data (219 ± 3 cm(-1)), which indicates that the Jahn-Teller and the spin-orbit coupling have been properly described in the theoretical model near the zero-point energy level. Based on the assignments predicted by the theoretical calculations, the experimentally measured energy levels were fitted to those from the diabatic model by optimizing the main spectroscopic parameters. The PESs from the ab initio calculations at the level of CASPT2/vq(t)z were thus compared with those calculated from the experimentally determined spectroscopic parameters. The theoretical diagonal elements in the diabatic potential matrix are in good agreement with those determined using the experimental data, however, the theoretical off-diagonal elements appreciably deviate from those determined using the experimental data for geometric points far away from the conical intersections. It is also concluded that the JT effect in CH(3)Cl(+) mainly arises from the linear coupling and the mode coupling between the CH(3) deform (υ(5)) and CH(3) rock (υ(6)) vibrations. The mode couplings between the symmetric C-Cl stretching vibration υ(3) with υ(5) and υ(6) are also important to understand the spin-vibronic structure of the molecule.

Toward the modeling of the NO2(2A″) manifold

AbstractGlobal diabatization angles are obtained to construct a 3 × 3 diabatic potential matrix for the 2A″ states of NO2. Over 12,500 energies have been calculated to map ab initio, the configuration space of (at least) the first 3 adiabatic sheets. With the level of theory used, such angles could be modeled such as to ensure a proper description of all involved crossing seams as well as the merging of the diabats into the adiabats at the asymptotic limits. This result provides an extension to the 3‐state case of the method proposed earlier (Mota and Varandas, J Phys Chem A 2008, 112, 3768), and illustrates the essential steps toward the modeling of the NO2(2A″) manifold via double many‐body expansion theory. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Nonadiabatic Dynamics on the Two Coupled Electronic PESs: The H+ + O2 System

Multistate adiabatic and diabatic PESs were computed for the H+ + O2 collision system in Jacobi coordinates, (R,r,γ) using the cc-pVTZ basis set and the ic-MRCI level of theory. In addition, all possible interaction potentials and nonadiabatic coupling matrix elements among those different electronic states were also computed. Comparisons with earlier computed interaction potentials were made wherever possible, and the differences between them is attributed to the multistate diabatization and the chosen level of theory and basis set. Focusing our attention on the ground-state (GS) and the first excited-state (ES) PES, quantum dynamics were performed using the 2 × 2 diabatic potential submatrix obtained from the multistate (four) diabatic potential matrix within the VCC-RIOSA scheme at two experimentally reported collision energies, E(cm) = 9.5 and 23 eV. The scattering quantities were computed for two experimentally observed collision processes, namely, the inelastic vibrational excitation (IVE), H+ + O2 (X3Σg(−),v = 0) → H+ + O2 (X3Σg(−),v′), and the vibrational charge transfer (VCT), H+ + O2 (X3Σg(−),v = 0) → H (2S) + O (X2Πg,v′′). Comparisons were made with experimental results and found an overall improvement relative to the earlier computed results, and the discrepancies, if any, could be brought down to minimum by further modification in employed ab initio PESs and the interaction potential.

DIABATIC ELECTRONIC MANIFOLD OF HN2(2A′) AND N + NH REACTION DYNAMICS ON ITS LOWEST ADIABAT

A previously reported approach [J. Chem. Phys.97:867, (1997)] to back transform the diagonal adiabats into a 2 × 2 diabatic potential matrix has been utilized to generate a global multi-sheeted form for the title system. Global adiabatic dynamics calculations carried out on the new form using the quasi-classical trajectory method yield results that lie essentially within the statistical error of similar calculations performed on the best surface reported thus far for the title reaction. This makes it suitable for future adiabatic and nonadiabatic calculations carried out either using classical or quantum methods.

Single surface beyond Born–Oppenheimer equation for a three-state model Hamiltonian of Na3 cluster

When a set of three states is coupled with each other but shows negligibly weak interaction with other states of the Hilbert space, these states form a sub-Hilbert space. In case of such subspace [J. Chem. Phys. 124, 074101 (2006)], (a) the adiabatic-diabatic transformation (ADT) condition, nablaA + tauA = 0 [Chem. Phys. Lett. 35, 112 (1975)], provides the explicit forms of the nonadiabatic coupling (NAC) elements in terms of electronic basis function angles, namely, the ADT angles, and (b) those NAC terms satisfy the so-called curl conditions [Chem. Phys. Lett. 35, 112 (1975)], which ensure the removal of the NAC elements [could be singular also at specific point(s) or along a seam in the configuration space] during the ADT to bring the diabatic representation of the nuclear Schrodinger equation with a smooth functional form of coupling elements among the electronic states. Since the diabatic to adiabatic representation of the Hamiltonian is related through the same unitary transformation (nablaA + tauA = 0), it could be quite interesting to explore the nature of the nonadiabatic coupling terms starting from a diabatic Hamiltonian and, thereafter, to formulate the extended Born-Oppenheimer (EBO) equation for those adiabatic states transformed from diabatic ones. We consider a three-state diabatic potential matrix constructed for the excited states of Na(3) cluster [J. Chem. Phys. 88, 6068 (1988)] at the pseudo-Jahn-Teller model situation, which can reproduce experimentally measured vibrationally resolved absorption lines [Surf. Sci. 156, 770 (1985)] with appropriate choice of coupling parameters, analytically calculate the nonadiabatic coupling elements along with their curls, and numerically evaluate the ADT angles to explore the nature of its nonadiabaticity. While formulating the single surface beyond the BO equation, our theoretical derivation demonstrates that the existence of zero curls of the NAC terms is a necessity. Indeed, when the energy gap between the third state (1(2) A(1)(')/2(2) A(1)(')) and the doubly degenerate states (2(2) E(')/3(2) E(')) of the model Hamiltonian for Na(3) cluster is considered to be either identically or approximately zero, the curl for each NAC element naturally approaches zero, leading to a theoretically valid EBO equation. We demonstrate the numerical validity of the EBO equation by calculating the nonadiabatic effects on the photoabsorption spectrum starting with the initial wave function located on the ground electronic state and compare with the corresponding diabatic spectrum when the three states are either degenerate at a point or approaching to form three-state degeneracy at the same point. Finally, we calculate the vibrational eigenspectrum of the ground adiabatic state by using (so to say) theoretically and numerically valid EBO equation to compare with those experimentally measured and BO/geometric phase calculated spectra (Tables I-III).

Nonadiabatic effects in the H+H2 exchange reaction: Accurate quantum dynamics calculations at a state-to-state level

Real wave packet propagations were carried out on both a single ground electronic state and two-coupled-electronic states of the title reaction to investigate the extent of nonadiabatic effects on the distinguishable-atom reaction cross sections. The latest diabatic potential matrix of Abrol and Kuppermann [J. Chem. Phys. 116, 1035 (2002)] was employed in the present nonadiabatic quantum state-to-state scattering calculations over a total energy range-from threshold (the zero point of the reagent H(2)) to 3.0 eV. Based on the assumption that the hydrogen atoms are distinguishable in the collisions where the inelastic and elastic ones are excluded, no significant nonadiabatic effects have been found in the calculations of the full state-to-state integral and differential cross sections up to a total energy of 3.0 eV for product vibrational levels v(')=0, 1, 2, 3. Our results therefore confirm the recent and the previous studies of the geometric phase effects in H+H(2) employing a different diabatic double many-body expansion potential matrix or a different BKMP2 potential energy surface.

Long-range multipolar potentials of the 18 spin-orbit states arising from the C(P3)+OH(X Π2) interaction

We present multipolar potentials at large intermolecular distances for the 18 doubly degenerate spin-orbit states arising from the interaction between the two open-shell systems, C((3)P) and OH(X (2)Pi). With OH fixed at its ground vibrational state-averaged distance r(0), the long-range potentials are two-dimensional potential energy surfaces (PESs) that depend on the intermolecular distance R and the angle gamma = CGH, where G represents the mass center of OH. The 18x18 diabatic potential matrix elements are built up from the perturbation theory up to second order and from a two-center expansion of the Coulombic interaction potential, resulting in a multipolar expansion of the potential expressed as a series of terms varying in R(-n). The expressions for the long-range coefficients of the expansion are explicitly given in terms of monomer properties such as permanent multipole moments, and static and dynamic polarizabilities. Accurate values for the monomer properties are used to properly determine the long-range interaction coefficients. The diagonalization of the full 18x18 potential matrix generates adiabatic long-range PESs in good agreement with their ab initio counterparts.

Dynamics and mechanism of the E→D, D′, β, γ, and δ nonadiabatic transitions induced in molecular iodine by collisions with CF4 and SF6 molecules

Nonadiabatic transitions among the first-tier ion-pair states of the iodine molecule in collisions with CF(4) and SF(6) partners are investigated by detecting the luminescence following the optical-optical double resonance excitation of the E0(g) (+)-state to the vibrational levels v(E)=8, 13, and 19. Total and partial rate constants, as well as vibrational product state distributions, are determined. It is found that electronic energy transfer in all channels is predominantly assisted by excitation of the dipole-allowed nu(3) and nu(4) modes of the partner. The measurements are accompanied by quantum scattering calculations that implement a close coupling treatment for the electronic and vibrational degrees of freedom and combine diatomics-in-molecule and long-range models for diabatic potential energy surfaces and coupling matrix elements. The analysis of experimental and theoretical data shows that the transitions without excitation of the partner are due to short-range couplings, whereas the vibrational excitation of the partner in the D0(u) (+) channel originates from the long-range coupling of two transition dipole moments: electronic of the iodine molecule and vibrational of the partner. Unexpectedly efficient excitations of the partner in the other ion-pair states, which are not coupled to the initial E0(g) (+)-state by the transition dipole, are interpreted within the postcollision mechanism. Qualitatively, this implies that during a single collision the long-range nonadiabatic transitions to D, nu(3) and D, nu(4) channels are followed by secondary short-range transitions without changing the state of the partner.

Electronic Diabatic Framework: Restrictions Due to Quantization of the Nonadiabatic Coupling Matrix

In this article, two issues related to the size of the electronic diabatic potential energy matrix are treated. (a) We frequently mention the fact that the dimension of a diabatic matrix obtained by a unitary transformation from the adiabatic framework is determined by the way the nonadiabatic coupling matrix τ breaks up into blocks. In this article, we prove for the first time that the size of the diabatic matrix as obtained in a direct way is determined in the same way. In other words, if the dimension of the above-mentioned decoupled block is N, then the dimension of any diabatic potential energy matrix with physical relevance has to be N as well, regardless of how it was derived. This number, N, is also equal to the number of coupled diabatic−Schrödinger equations to be solved. (b) The second issue is, consequently, related to the actual required number of coupled Schrödinger equations to be solved to obtain a well-converged solution. Starting with the earlier introduced number N, we show that this number can be reduced, and in fact, it is most likely equal to the number of energetically open adiabatic states (for a given energy). While doing that, we rigorously derived the relevant diabatic potential matrix for this reduced case. We also worked out in detail an example related to a three-state case and derived the relevant 2 × 2 diabatic potential matrix.

Collision-induced non-adiabatic transitions between the ion-pair states of molecular iodine: A challenge for experiment and theory

The ion-pair states of molecular iodine provide a unique system for studying the efficiency, selectivity, and mechanisms of collision-induced non-adiabatic transitions. Non-adiabatic transitions between the first-tier ion-pair states in collisions with molecular partners and rare gases are analyzed and discussed. The qualitative features of the rate constants and product state distributions under single collision conditions are summarized and interpreted in terms of appropriate theoretical approaches. Two mechanisms for the non-adiabatic transitions are clearly identified. The first, operative for collisions involving molecular partners possessing permanent or transition electrostatic moments, is highly selective. It connects the initially prepared level in the E 0+g electronic state with the near-resonant vibronic level of the D 0+u state with a minimum change of the total angular momentum. In an extreme quasi-resonant case when the gap between initial and final rovibronic level is less than 1 cm−1, this mechanism has a giant cross section, 40 times that of a gas kinetic collision. An electrostatic model, which includes the coupling of the giant E–D transition dipole moment with a moment of the colliding partner and the semiclassical Born approximation, provides a plausible interpretation of this mechanism. A second mechanism is shown to govern collisions with rare gas atoms. It results in population of several ion-pair states and broad distributions over rovibronic levels. This mechanism is successfully interpreted by quantum scattering calculations based on the diatomics-in-molecule diabatic potential energy surfaces and coupling matrix elements. The calculations provide good agreement with experimental measurements and reveal different mechanisms for the population of different electronic states. Unexplained features of the non-adiabatic dynamics and directions of future work are outlined.

Necessary Conditions for a Rigorous Minimal Diabatic Potential Matrix

In this article, one of the more important dilemmas in molecular physics is considered: given a matrix of the nonadiabatic coupling terms of any desired dimension, what is the minimal sub-Hilbert space for which diabatization is still valid. This problem was addressed by one of us before (Baer, M. Chem. Phys. Lett. 2000, 329, 450), but it was recently established that the suggested criteria therein lead to subspaces that are too large to be of any use. In this article, we discuss the conditions that have to be satisfied to reach the minimal subspace. We have found that these conditions are related to the spatial distribution of the various nonadiabatic coupling terms. Thus, if nonadiabatic coupling terms for the relevant states overlap only slightly in configuration space, the required size of the subspace for diabatization can be reduced significantly. As an example, we consider the C2H molecule.

Conical intersection revisited: extension to an elliptic form

In this Letter we relate to a general 2×2 diabatic potential matrix for which is derived an analytic expression for the various non-adiabatic coupling terms. Its main advantage is in leading to a simple algorithm to determine the number and the kind of conical intersections one encounters in a (planar) region defined by a closed contour. This approach was tested with respect to two models, one of them is the elliptical linear model. This model was then, further, applied to simulate ab initio results obtained for the C 2H molecule.