Potential Energy Matrix Research Articles

How the antisymmetrization affects a cluster–cluster interaction: Two-cluster systems

We study effects of the antisymmetrization on the potential energy of two-cluster systems. The object of the investigation is the lightest nuclei of p-shell with a dominant alpha-cluster channel. For this aim we construct matrix elements of two-cluster potential energy between cluster oscillator functions with and without full antisymmetrization. Eigenvalues and eigenfunctions of the potential energy matrix are studied in detail. Eigenfunctions of the potential energy operator are presented in oscillator, coordinate and momentum spaces. We demonstrate that the Pauli principle affects more strongly the eigenfunctions than the eigenvalues of the matrix and leads to the formation of resonance and trapped states.

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.

Two-layer Gaussian-based MCTDH study of the S1 ← S vibronic absorption spectrum of formaldehyde using multiplicative neural network potentials

The absorption spectrum of the vibronically allowed S1(1A2) ← S0(1A1) transition of formaldehyde is computed by combining multiplicative neural network (NN) potential surface fits, based on multireference electronic structure data, with the two-layer Gaussian-based multiconfiguration time-dependent Hartree (2L-GMCTDH) method. The NN potential surface fit avoids the local harmonic approximation for the evaluation of the potential energy matrix elements. Importantly, the NN surface can be constructed so as to be physically well-behaved outside the domain spanned by the ab initio data points. A comparison with experimental results shows spectroscopic accuracy of the converged surface and 2L-GMCTDH quantum dynamics.

Properties of a potential energy matrix in oscillator basis

Matrix elements of potential energy are examined in detail. We consider a model problem — a particle in a central potential. The most popular forms of central potential are taken up, namely, square-well potential, Gaussian, Yukawa and exponential potentials. We study eigenvalues and eigenfunctions of the potential energy matrix constructed with oscillator functions. It is demonstrated that eigenvalues coincide with the potential energy in coordinate space at some specific discrete points. We establish approximate values for these points. It is also shown that the eigenfunctions of the potential energy matrix are the expansion coefficients of the spherical Bessel functions in a harmonic oscillator basis. We also demonstrate a close relation between the separable approximation and L2 basis (J-matrix) method for the quantum theory of scattering.

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.

Representation of coupled adiabatic potential energy surfaces using neural network based quasi-diabatic Hamiltonians: 1,2 2A' states of LiFH.

An analytic quasi-diabatic representation of ab initio electronic structure data is key to the accurate quantum mechanical description of non-adiabatic chemical processes. In this work, a general neural network (NN) fitting procedure is proposed to generate coupled quasi-diabatic Hamiltonians (Hd) that are capable of representing adiabatic energies, energy gradients, and derivative couplings over a wide range of geometries. The quasi-diabatic representation for LiFH is used as a testing example. The fitting data including adiabatic energies, energy gradients and interstate couplings are obtained from a previously fitted analytical quasi-diabatic potential energy matrix, and are well reproduced by the NN fitting. Most importantly, the NN fitting also yields quantum dynamic results that reproduce those on the original LiFH diabatic Hamiltonian, demonstrating the ability of NN to generate highly accurate quasi-diabatic Hamiltonians.

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.

A Reduced Method of Coupled Vibrational Channels: Analysis of Regular Perturbations in the $${{c}^{3}}{\mathbf{\Sigma }}_{{\mathbf{\Omega }}}^{ + }$$ -State of a KRb Molecule

To reduce the dimension of a vibrational problem to be solved by the coupled channel method in the case of interacting electronic states of diatomic molecules, we propose to use the contact van Vleck transformations, which make it possible to take into account the nonadiabatic intramolecular interactions with distant states by modifying the initial potential energy matrix. The efficiency of the reduced coupled vibrational channel method (RCVCM) has been demonstrated by an example of analyzing the regular spin–orbital, electron–rotational, and spin–rotational perturbations discovered in the fine structure of vibrational–rotational levels of the $${{c}^{3}}{\Sigma\text{}}_{{\Omega\text{}}}^{ + }$$ -state of a KRb molecule by methods of high-precision laser-emission spectroscopy. The adiabatic potentials and nonadiabatic electron matrix elements, as functions of the internuclear distance necessary for application of the RCVCM, were obtained in the framework of the nonempirical high-level calculation. It has been demonstrated that the RCVCM has broad extrapolation possibilities and makes it possible to describe the position of the regularly perturbed energy levels of Ω-components of the triplet $${{c}^{3}}{\Sigma\text{}}_{{\Omega\text{}}}^{ + }$$ ‑state at the spectroscopic accuracy level.

Photodissociation dynamics in the first absorption band of pyrrole. II. Photofragment distributions for the 1A2(πσ*)←X̃1A1(ππ) transition.

The analysis of the total kinetic energy release (TKER) of the photofragments pyrrolyl + H-atom formed in the photodissociation of pyrrole in the low-lying state 1A2(πσ*) is presented. The TKER distributions contain complementary and often more precise information on the fragmentation process than the broad diffuse absorption spectra. The distributions are calculated quantum mechanically for the diabatic state 1A2(πσ*) either isolated or coupled to the ground electronic state at an exit channel conical intersection. The calculations use the novel ab initio quasi-diabatic potential energy matrix constructed in the work of Picconi and Grebenshchikov [J. Chem. Phys. 148, 104103 (2018)]. The approximate overlap integral-based adiabatic mapping approach is introduced with which the quantum mechanical TKER distributions can be efficiently and accurately reproduced. Finally, the calculated TKERs are compared with the experimental results. The main features of the measured vibrationally resolved distributions are reproduced, and the spectral peaks are assigned and interpreted in detail.

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.

Volkov basis for simulation of interaction of strong laser pulses and solids

An efficient and accurate basis comprised of Volkov states is implemented and tested for time-dependent simulations of interactions between strong laser pulses and crystalline solids. The Volkov states are eigenstates of the free electron Hamiltonian in an electromagnetic field and analytically represent the rapidly oscillating time-dependence of the orbitals, allowing significantly faster time propagation than conventional approaches. The Volkov approach can be readily implemented in plane-wave codes by multiplying the potential energy matrix elements with a simple time-dependent phase factor.

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.

Non-adiabatic effects in F + CHD3 reactive scattering.

The effect of non-adiabatic transitions on the F(2P) + CHD3(ν1) → DF + CHD2 and F(2P) + CHD3(ν1) → HF + CD3 reactions is investigated. The dynamics of the nuclei was simulated using trajectory surface hopping and a vibronically and spin-orbit coupled diabatic potential energy matrix. To facilitate the calculations, the fewest switching algorithm of Tully was adapted to the use of a complex diabatic potential energy matrix. For reactions of CHD3 with ground state fluorine atoms, F(2P3/2), the ratio between the previously computed adiabatic cross sections and the non-adiabatic ones was found to range from 1.4 to 2.1. The actual ratio depends on the translational energy and the initial vibrational state of CHD3. The total reactivity of CHD3(ν1 = 1) was found to be always larger than that of CHD3(ν1=0) mainly because of the increase in the cross sections for the HF + CD3 channel. Thus, the inclusion of non-adiabatic transitions in the theoretical treatment cannot resolve the existing disagreement between theory and experiment. Cross sections for the reaction of CHD3 with spin-orbit excited fluorine atoms, F(2P1/2), were found to be significantly smaller than the ones for reaction with F(2P3/2).

Theoretical Cross Sections of the Inelastic Fine Structure Transition M(2P1/2) + Ng ↔ M(2P3/2) + Ng for M = K, Rb, and Cs and Ng = He, Ne, and Ar.

Scattering matrix elements of the inelastic fine structure transition M(2P1/2) + Ng ↔ M(2P3/2) + Ng are computed using the channel packet method (CPM) for alkali-metal atoms M = K, Rb, and Cs, as they collide with noble-gas atoms Ng = He, Ne, and Ar. The calculations are performed within the block Born-Oppenheimer approximation where excited state VA2Π1/2(R), VA2Π3/2(R), and VB2Σ1/2(R) adiabatic potential energy surfaces are used together with a Hund's case (c) basis to construct a 6 × 6 diabatic representation of the electronic Hamiltonian. Matrix elements of the angular kinetic energy of the nuclei incorporate Coriolis coupling and, together with the diabatic representation of the electronic Hamiltonian, yield a 6 × 6 effective potential energy matrix. This matrix is diagonal in the asymptotic limit of large internuclear separation with eigenvalues that correlate to the 2Pj alkali atomic energy levels. Scattering matrix elements are computed using the CPM by preparing reactant and product wave packets on the effective potential energy surfaces that correspond to the excited 2Pj alkali states of interest. The reactant wave packet is then propagated forward in time using the split operator method together with a unitary transformation between the adiabatic and diabatic representations. The Fourier transformation of the correlation function between the evolving reactant wave packet and stationary product wave packet yields state-to-state scattering matrix elements as a function of energy for a particular choice of total angular momentum J. Calculations are performed for energies that range from 0.0 to 0.01 hartree and values of J that start with a minimum of J = 0.5 for all M + Ng pairs up to a maximum that ranges from J = 450.5 for KAr to J = 100.5 for CsAr. A sum over J together with an average over energy is used to compute thermally averaged cross sections for a temperature range of T = 0-400 K.