Diabatic Potential Energy Matrix Research Articles

Accurate description of the nonadiabatic proton-coupled electron-transfer process in a diabatic representation: A model study

Using a two-dimensional model system for the proton-coupled electron-transfer process, the eigenvalues and nonadiabatic dynamics have been notably investigated by means of diabatic representation in the Born-Oppenheimer picture. Different from a previously reported diabatic model, which was approximately determined by the eigenstates of the differently defined electronic Hamiltonians, the rigorous diabatization was achieved by a fitting method based on the adiabatic energies and derivative couplings. As expected, the rigorous diabatic models were found to be able to accurately reproduce the eigenvalues or nonadiabatic dynamics in the adiabatic representation for three different models. Furthermore, we proposed an approximate diabatic scheme, which was built from fitting the adiabatic energies solely but with a fixed form for the off-diagonal terms in the diabatic potential energy matrix. This approximate diabatic model without the aid of the derivative coupling is shown to be as accurate as the rigorous one, which provides a simple and efficient way to accurately describe the complex proton-coupled electron-transfer processes due to accurately computing derivative couplings that are quite challenging for realistic systems.

EOM-CCSD-based neural network diabatic potential energy matrix for 1πσ*-mediated photodissociation of thiophenol

A new diabatic potential energy matrix (PEM) of the coupled 1ππ* and 1πσ* states for the 1πσ*-mediated photodissociation of thiophenol was constructed using a neural network (NN) approach. The diabatization of the PEM was specifically achieved by our recent method [Chin. J. Chem. Phys. 34, 825 (2021)], which was based on adiabatic energies without the associated costly derivative couplings. The equation of motion coupled cluster with single and double excitations (EOM-CCSD) method was employed to compute adiabatic energies of two excited states in this work due to its high accuracy, simplicity, and efficiency. The PEM includes three dimensionalities, namely the S−H stretch, C−S−H bend, and C−C−S−H torsional coordinates. The root mean square errors of the NN fitting for the S1 and S2 states are 0.89 and 1.33 meV, respectively, suggesting the high accuracy of the NN method as expected. The calculated lifetimes of the S1 vibronic 00 and 31 states are found to be in reasonably good agreement with available theoretical and experimental results, which validates the new EOM-CCSD-based PEM fitted by the NN approach. The combination of the diabatization scheme solely based on the adiabatic energies and the use of EOM-CCSD method makes the construction of reliable diabatic PEM quite simple and efficient.

Unified Description of the Jahn-Teller Effect in Molecules with Only Cs Symmetry: Cyclohexoxy in Its Full 48-Dimensional Internal Coordinates.

The two lowest potential energy surfaces of cyclohexoxy which are coupled by conical intersections and the spin-orbit interaction are determined in the full 48-dimensional internal coordinate space using a feedforward neural network to fit a diabatic potential energy matrix. The electronic structure data are obtained at the multireference configuration interaction with single- and double-excitation level. Underlying parallels between these coupled surfaces and those of the alkoxy radicals methoxy and isopropoxy are established. Earlier work by Dillon and Yarkony is extended. While the parallels would have been challenging to appreciate using the concept of the Jahn-Teller active modes, they are readily seen in terms of two internal modes centered at the conical intersection: g the energy difference gradient vector and h the interstate coupling gradient vector. In other words, g and h vectors provide a unified description of the Jahn-Teller effect in molecules exhibiting C3v and quasi-C3v symmetries. A spectral simulation in the full 48-vibrational-internal coordinate space is reported. This spectrum is obtained using recently developed algorithms designed to increase the size of the systems that can be treated with a time-independent vibronic coupling approach.

Vibrational energy levels of the S 0 and S 1 states of formaldehyde using an accurate ab initio based global diabatic potential energy matrix

Low-lying vibrational energy levels of both the ground (S 0) and first excited singlet (S 1) states of formaldehyde have been determined using an exact kinetic energy operator and adiabatic potential energy surfaces derived from a recently developed full-dimensional global diabatic potential energy matrix (DPEM) based on a neural network representation of high level ab initio data. The agreement with available experimental band origins provides strong evidence in support of the accuracy of the DPEM, which can be used in the future for understanding the internal conversion of H2CO(S 1) and the subsequent roaming dynamics on its S 0 state.

Neural Network Representation of Three-State Quasidiabatic Hamiltonians Based on the Transformation Properties from a Valence Bond Model: Three Singlet States of H3+

A neural network (NN) approach was recently developed to construct accurate quasidiabatic Hamiltonians for two-state systems with conical intersections. Here, we derive the transformation properties of elements of 3 × 3 quasidiabatic Hamiltonians based on a valence bond (VB) model and extend the NN-based method to accurately diabatize the three lowest electronic singlet states of H3+, which exhibit the avoided crossing between the ground and first excited states and the conical intersection between the first and second excited states for equilateral triangle configurations (D3h). The current NN framework uses fundamental invariants (FIs) as the input vector and appropriate symmetry-adapted functions called covariant basis to account for the special symmetry of complete nuclear permutational inversion (CNPI). The resulting diabatic potential energy matrix (DPEM) can reproduce the ab initio adiabatic energies, energy gradients, and derivative couplings between adjacent states as well as the particular symmetry. The accuracy of DPEM is further validated by full-dimensional quantum dynamics calculations. The flexibility of the FI-NN approach based on the VB model shows great potential to resolve diabatization problems for many extended and multistate systems.

Three-dimensional diabatic potential energy surfaces of thiophenol with neural networks

Three-dimensional (3D) diabatic potential energy surfaces (PESs) of thiophenol involving the S0, and coupled 1ππ* and 1πσ* states were constructed by a neural network approach. Specifically, the diabatization of the PESs for the 1ππ* and 1πσ* states was achieved by the fitting approach with neural networks, which was merely based on adiabatic energies but with the correct symmetry constraint on the off-diagonal term in the diabatic potential energy matrix. The root mean square errors (RMSEs) of the neural network fitting for all three states were found to be quite small (<4 meV), which suggests the high accuracy of the neural network method. The computed low-lying energy levels of the S0 state and lifetime of the 0° state of S1 on the neural network PESs are found to be in good agreement with those from the earlier diabatic PESs, which validates the accuracy and reliability of the PESs fitted by the neural network approach.

A fundamental invariant-neural network representation of quasi-diabatic Hamiltonians for the two lowest states of H3.

The fundamental invariant neural network (FI-NN) approach is developed to represent coupled potential energy surfaces in quasidiabatic representations with two-dimensional irreducible representations of the complete nuclear permutation and inversion (CNPI) group. The particular symmetry properties of the diabatic potential energy matrix of H3 for the 1A' and 2A' electronic states were resolved arising from the E symmetry in the D3h point group. This FI-NN framework with symmetry adaption is used to construct a new quasidiabatic representation of H3, which reproduces accurately the ab initio energies and derivative information with perfect symmetry behaviors and extremely small fitting errors. The quantum dynamics results on the new FI-NN diabatic PESs give rise to accurate oscillation patterns in the product state-resolved differential cross sections. These results strongly support the accuracy and efficiency of the FI-NN approach to construct reliable diabatic representations with complicated symmetry problems.

Impact of Diabolical Singular Points on Nonadiabatic Dynamics and a Remedy: Photodissociation of Ammonia in the First Band.

Several recent publications have pointed out a potentially severe drawback in some widely used diabatization methods based on the electronic properties of molecules. In a diabatic representation defined by a property-based method, artificial singularities may arise due to the defining equation of the adiabatic-to-diabatic (AtD) transformation. Such diabolical singular points (DSPs) may seriously affect nuclear dynamics if they lie in the relevant configuration space. Their impact is demonstrated here using the A-band photodissociation of ammonia as an example. To this end, quantum dynamics calculations are performed based on a diabatic potential energy matrix (DPEM) constructed using the generalized Mulliken-Hush method, which is based on dipoles. These property-based results are compared with the results obtained with a DPEM determined using derivative coupling explicitly. A DSP seam is found near the Franck-Condon region, which results in a complete failure to reproduce the absorption spectrum. A modification of the generalized Mulliken-Hush method is proposed to remove the DSPs while preserving the conical intersection, which leads to an accurate reproduction of the absorption spectrum and the NH2(Ã)/NH2(X̃) product branching ratio.

Semiglobal diabatic potential energy matrix for the N-H photodissociation of methylamine.

We constructed an analytic diabatic potential energy matrix (DPEM) that describes the N-H photodissociation of methylamine; the electronic state space includes the ground and first excited singlet states. The input for the fit was calculated by extended multi-state complete active space second-order perturbation theory. The data were diabatized using the dipole-quadrupole diabatization method in which we incorporated a coordinate-dependent weighting scheme for the contribution of the quadrupole moments. To make the resulting potential energy surfaces semiglobal, we extended the anchor points reactive potential method, a multiscale approach that assigns the internal coordinates to categories with different levels of computational treatment. Key aspects of the adiabatic potential energy surfaces obtained by diagonalizing the DPEM agree with the available experimental and theoretical data at energies relevant for photochemical studies.

ADT: A Generalized Algorithm and Program for Beyond Born-Oppenheimer Equations of "N" Dimensional Sub-Hilbert Space.

The major bottleneck of first principle based beyond Born-Oppenheimer (BBO) treatment originates from large number and complicated expressions of adiabatic to diabatic transformation (ADT) equations for higher dimensional sub-Hilbert spaces. In order to overcome such shortcoming, we develop a generalized algorithm, "ADT" to generate the nonadiabatic equations through symbolic manipulation and to construct highly accurate diabatic surfaces for molecular processes involving excited electronic states. It is noteworthy to mention that the nonadiabatic coupling terms (NACTs) often become singular (removable) at degenerate point(s) or along a seam in the nuclear configuration space (CS) and thereby, a unitary transformation is required to convert the kinetically coupled (adiabatic) Hamiltonian to a potentially (diabatic) one to avoid such singularity(ies). The "ADT" program can be efficiently used to (a) formulate analytic functional forms of differential equations for ADT angles and diabatic potential energy matrix and (b) solve the set of coupled differential equations numerically to evaluate ADT angles, residue due to singularity(ies), ADT matrices, and finally, diabatic potential energy surfaces (PESs). For the numerical case, user can directly provide ab initio data (adiabatic PESs and NACTs) as input files to this software or can generate those input files through in-built python codes interfacing MOLPRO followed by ADT calculation. In order to establish the workability of our program package, we selectively choose six realistic molecular species, namely, NO2 radical, H3+, F + H2, NO3 radical, C6H6+ radical cation, and 1,3,5-C6H3F3+ radical cation, where two, three, five and six electronic states exhibit profound nonadiabatic interactions and are employed to compute diabatic PESs by using ab initio calculated adiabatic PESs and NACTs. The "ADT" package released under the GNU General Public License v3.0 (GPLv3) is available at https://github.com/AdhikariLAB/ADT-Program and also as the Supporting Information of this article.

On the nonadiabatic collisional quenching of OH(A) by H2: a four coupled quasi-diabatic state description.

A four-state diabatic potential energy matrix (DPEM), Hd, for the description of the nonadiabatic quenching of OH(A2Σ+) by collisions with H2 is reported. The DPEM is constructed as a fit to adiabatic energies, energy gradients, and derivative couplings obtained exclusively from multireference configuration interaction wave functions. A four-adiabatic-electronic-state representation is used in order to describe all energetically accessible regions of the nuclear coordinate space. Partial permutation-inversion symmetry is incorporated into the representation. The fit is based on electronic structure data at 42 882 points, described by over 1.6 million least squares equations with a root mean square (mean unsigned) error of 178(83) cm-1. Comparison of ab initio and Hd determined minima, saddle points, and energy minimized points on C2v, Cs, C∞v, and C1 (noncoplanar) portions of two conical intersection seams are used to establish the accuracy of the Hd.

Full-dimensional three-state potential energy surfaces and state couplings for photodissociation of thiophenol.

An analytic full-dimensional diabatic potential energy matrix (DPEM) for the lowest three singlet states of thiophenol (C6H5SH) at geometries accessible during photodissociation is constructed using the anchor points reactive potential (APRP) scheme. The data set used for modeling is obtained from electronic structure calculations including dynamic correlation via excitations into the virtual space of a three-state multiconfiguration self-consistent field calculation. The resulting DPEM is a function of all the internal coordinates of thiophenol. The DPEM as a function of the S-H bond stretch and C-C-S-H torsion and the diabatic couplings along two in-plane bend modes and nine out-of-plane distortion modes are computed using extended multiconfigurational quasidegenerate perturbation theory followed by the fourfold way determination of diabatic molecular orbitals and model space diabatization by configurational uniformity, and this dependence of the DPEM is represented by general functional forms. Potentials along 31 tertiary internal degrees of freedom are modeled with system-dependent, primary-coordinate-dependent nonreactive molecular mechanics-type force fields that are parameterized by Cartesian Hessians calculated by generalized Kohn-Sham density functional theory. Adiabatic potential energy surfaces (PESs) and nonadiabatic couplings are obtained by a transformation of the DPEM. The topography of the APRP PESs is characterized by vertical excitation energies, equilibrium geometries, vibrational frequencies, and conical intersections, and we find good agreement with available reference data. This analytic DPEM is suitable for full-dimensional electronically nonadiabatic molecular dynamics calculations of the photodissociation of thiophenol with analytic gradients in either the adiabatic or diabatic representation.

Direct diabatization and analytic representation of coupled potential energy surfaces and couplings for the reactive quenching of the excited 2Σ+ state of OH by molecular hydrogen.

We have employed extended multiconfiguration quasidegenerate perturbation theory, fourfold-way diabatic molecular orbitals, and configurational uniformity to develop a global three-state diabatic representation of the potential energy surfaces and their couplings for the electronically nonadiabatic reaction OH* + H2 → H2O + H, where * denotes electronic excitation to the A 2Σ+ state. To achieve sign consistency of the computed diabatic couplings, we developed a graphics processing unit-accelerated algorithm called the cluster-growing algorithm. Having obtained consistent signs of the diabatic couplings, we fit the diabatic matrix elements (which consist of the diabatic potentials and the diabatic couplings) to analytic representations. Adiabatic potential energy surfaces are generated by diagonalizing the 3 × 3 diabatic potential energy matrix. The comparisons between the fitted and computed diabatic matrix elements and between the originally computed adiabatic potential energy surfaces and those generated from the fits indicate that the current fit is accurate enough for dynamical studies, and it may be used for quantal or semiclassical dynamics calculations.

Vibronically and spin-orbit coupled diabatic potentials for X(2P) + CH4 → HX + CH3 reactions: Neural network potentials for X = Cl.

Vibronically and spin-orbit (SO) coupled diabatic potentials for the Cl(2P) + CH4 → HCl + CH3 reaction are constructed based on a recently developed approach [T. Lenzen and U. Manthe, J. Chem. Phys. 150, 064102 (2019)]. Diabatic potentials and couplings describing the entrance channel of the reaction are obtained based on ab initio data using a diabatization by an ansatz scheme. A detailed investigation of the electronic structure in the entrance channel using multireference configuration interaction (MRCI), coupled cluster [CCSD/CCSD(T)], and SO-MRCI calculations is presented. Neural networks using permutationally invariant polynomials as inputs are employed to represent the elements of the diabatic potential energy matrix. The same set of diabatic states is also used in the transition state region and all four exit channels. Here, the lowest adiabatic potential energy surface (PES) derived from the diabatic model is chosen to reproduce an adiabatic PES recently developed by Li and Guo. The accuracy of the resulting PES is evaluated, and the properties of the newly developed coupled diabatic potentials are analyzed in detail.

A Quasi-Diabatic Representation of the 1,21A States of Methylamine.

We report a full 15-dimensional two-state quasi-diabatic potential energy matrix (PEM) for the 1,21A states of methylamine (CH3NH2) suitable for the description of its two distinct photodissociation channels, CH3NH2(11A) + hν → CH3NH2(21A) → CH3 + NH2 or CH3NH + H. The PEM is fit to ab initio electronic structure data (energies, energy gradients, and derivative couplings) obtained from multireference configuration interaction single and double excitation wave functions at 7732 geometries, using a diabatic representation based on symmetry-adapted polynomials. The root-mean-square error is 78.34 cm-1 for the energies and 2.98% for the gradients. The computed T0(21A) = 42 461 cm-1 (41 669 cm-1), D0(CH3NH+H) = 33 639 cm-1 (34 250 cm-1), and D0(CH3+NH2) = 28 543 cm-1 (29 300 cm-1) are in good agreement with the experimental values given in the parentheses. The absorption spectrum of CH3NH2 is computed using this diabatic PEM and a normal mode based kinetic energy operator. Agreement with experimental data is quite satisfactory. Our accurate representation of the PEM over a wide range of nuclear configuration space offers the possibility for quality nonadiabatic dynamics simulations involving the CH3 + NH2 and CH3NH + H dissociation channels. Surface hopping trajectories are used to illustrate the types of nuclear motion supported by the PEM.

Nonadiabatic Dynamics in Photodissociation of Hydroxymethyl in the 32A(3p x) Rydberg State: A Nine-Dimensional Quantum Study.

The nonadiabatic predissociation dynamics of the hydroxymethyl radical (CH2OH) in its 32A(3p x) state is investigated using a nine-dimensional quantum mechanical model based on an ab initio three coupled diabatic state potential energy matrix. The calculated absorption spectrum, which is dominated by predissociative resonances, is in excellent agreement with experiment. The predissociation is facilitated by two conical intersection seams formed between the 32A(3p x) and 22A(3s) states near the Franck-Condon region. The h and g vectors of energy minimized points on these seams are analyzed using the normal modes of the 32A equilibrium structure. The low-lying predissociative resonances have been assigned and their lifetimes are less than 100 fs and moderately mode specific. The absorption spectrum is dominated by a CO vibrational progression, due apparently to the promotion of an electron from the ground state antibonding πCO* orbital to the carbon Rydberg orbital, which effectively increases the C-O bond order.

Two-state diabatic potential energy surfaces of ClH2 based on nonadiabatic couplings with neural networks.

A general neural network (NN)-fitting procedure based on nonadiabatic couplings is proposed to generate coupled two-state diabatic potential energy surfaces (PESs) with conical intersections. The elements of the diabatic potential energy matrix (DPEM) can be obtained directly from a combination of the NN outputs in principle. Instead, to achieve higher accuracy, the adiabatic-to-diabatic transformation (ADT) angle (mixing angle) for each geometry is first solved from the NN outputs, followed by individual NN fittings of the three terms of the DPEM, which are calculated from the ab initio adiabatic energies and solved mixing angles. The procedure is applied to construct a new set of two-state diabatic potential energy surfaces of ClH2. The ab initio data including adiabatic energies and derivative couplings are well reproduced. Furthermore, the current diabatization procedure can describe well the vicinity of conical intersections in high potential energy regions, which are located in the T-shaped (C2v) structure of Cl-H2. The diabatic quantum dynamical results on diabatic PESs show large differences as compared with the adiabatic results in high collision energy regions, suggesting the significance of nonadiabatic processes in conical intersection regions at high energies.

Beyond Born-Oppenheimer treatment on spectroscopic and scattering processes

We review the formulation of beyond Born-Oppenheimer (BBO) theory based on first principle for the construction of diabatic potential energy surfaces (PESs) both for few important spectroscopic systems, viz., Na3 cluster, NO2 radical as well as scattering process like D+ + H2. The essential theoretical development leading to the BBO equations are thoroughly discussed. It has been found that the above molecular systems posses numerous nonadiabatic interactions that range from Jahn-Teller, Renner-Teller types of conical intersections along with strong pseudo Jahn-Teller couplings between various electronic states. We have calculated the adiabatic PESs and nonadiabatic coupling terms for those systems and subsequently performed adiabatic-to-diabatic transformation to construct smooth, symmetric and continuous diabatic potential energy matrix. Nuclear dynamics has been performed on the diabatic PESs of Na3 and NO2 to simulate the photoelectron spectra that match quite well with the experimentally measured ones. Moreover, we have carried out reactive scattering dynamics on the adiabatic and diabatic surfaces of system to reproduce experimental cross sections for reactive charge and non-charge transfer processes.

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.

Neural network based coupled diabatic potential energy surfaces for reactive scattering.

An approach for the construction of vibronically coupled potential energy surfaces describing reactive collisions is proposed. The scheme utilizes neural networks to obtain the elements of the diabatic potential energy matrix. The training of the neural network employs a diabatization by the Ansatz approach and is solely based on adiabatic electronic energies. Furthermore, no system-specific symmetry consideration is required. As the first example, the H2+Cl→H+HCl reaction, which shows a conical intersection in the entrance channel, is studied. The capability of the approach to accurately reproduce the adiabatic reference energies is investigated. The accuracy of the fit is found to crucially depend on the number of data points as well as the size of the neural network. 5000 data points and a neural network with two hidden layers and 40 neurons in each layer result in a fit with a root mean square error below 1 meV for the relevant geometries. The coupled diabatic potential energies are found to vary smoothly with the coordinates, but the conical intersection is erroneously represented as a very weakly avoided crossing. This shortcoming can be avoided if symmetry constraints for the coupling potential are incorporated into the neural network design.