Potential Energy Matrix Elements Research Articles

A non-hierarchical multi-layer multi-configurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces.

The correlation discrete variable representation (CDVR) enables multi-layer multi-configurational time-dependent Hartree (MCTDH) quantum dynamics simulations on general potential energy surfaces. In a recent study [R. Ellerbrock and U. Manthe, J. Chem. Phys. 156, 134107 (2022)], an improved CDVR that can account for the symmetry properties of a tree-shaped wavefunction representation has been introduced. This non-hierarchical CDVR drastically reduces the number of grid points required in the time-dependent quadrature used to evaluate all potential energy matrix elements. While the first studies on the non-hierarchical CDVR approach have been restricted to single-layer calculations, here the complete theory required for the implementation of the non-hierarchical CDVR approach in the multi-layer MCTDH context will be presented. Detailed equations facilitating the efficient recursive computation of all matrix elements are derived, and a new notation adapted to the symmetry properties of the tree-shaped representation is introduced. Calculations studying the non-adiabatic quantum dynamics of photoexcited pyrazine in 24 dimensions illustrate the properties of the non-hierarchical multi-layer CDVR.

New diabatic potential energy surfaces of KH2 system constructed using neural network method

New diabatic potential energy surfaces (PESs) for the KH2 system, related to both ground and first excited states, were constructed using only adiabatic electronic energies. The elements of the diabatic potential energy matrix were determined using neural network (NN) method. Prior to calculation, the symmetry constraints of the coupling term were incorporated into the NN code. A comprehensive comparison was undertaken between the topographic characteristics of the newly constructed diabatic PESs and those previously developed using the molecular properties method. To validate the accuracy of the new diabatic PESs, the dynamics calculations of K(4p2P) + H2 (v 0 = 1, j 0 = 0) reaction were performed based on both new and old diabatic PESs with same parameters. The results indicate that the values based on the new diabatic PESs generally agree well with those from the old ones. In addition, compared with the previous diabatic PESs, the new diabatic PESs can accurately describe the geometric phase effect.

Representation of Diabatic Potential Energy Matrices for Multiconfiguration Time-Dependent Hartree Treatments of High-Dimensional Nonadiabatic Photodissociation Dynamics.

Conventional quantum mechanical characterization of photodissociation dynamics is restricted by steep scaling laws with respect to the dimensionality of the system. In this work, we examine the applicability of the multi-configurational time-dependent Hartree (MCTDH) method in treating nonadiabatic photodissociation dynamics in two prototypical systems, taking advantage of its favorable scaling laws. To conform to the sum-of-product form, elements of the ab initio diabatic potential energy matrix (DPEM) are re-expressed using the recently proposed Monte Carlo canonical polyadic decomposition method, with enforcement of proper symmetry. The MCTDH absorption spectra and product branching ratios are shown to compare well with those calculated using conventional grid-based methods, demonstrating its promise for treating high-dimensional nonadiabatic photodissociation problems.

Direct product-type grid representations for angular coordinates in extended space and their application in the MCTDH approach.

Multi-configurational time-dependent Hartree (MCTDH) calculations using time-dependent grid representations can be used to accurately simulate high-dimensional quantum dynamics on general ab initio potential energy surfaces. Employing the correlation discrete variable representation, sets of direct product type grids are employed in the calculation of the required potential energy matrix elements. This direct product structure can be a problem if the coordinate system includes polar and azimuthal angles that result in singularities in the kinetic energy operator. In the present work, a new direct product-type discrete variable representation (DVR) for arbitrary sets of polar and azimuthal angles is introduced. It employs an extended coordinate space where the range of the polar angles is taken to be [-π, π]. The resulting extended space DVR resolves problems caused by the singularities in the kinetic energy operator without generating a very large spectral width. MCTDH calculations studying the F·CH4 complex are used to investigate important properties of the new scheme. The scheme is found to allow for more efficient integration of the equations of motion compared to the previously employed cot-DVR approach [G. Schiffel and U. Manthe, Chem. Phys. 374, 118 (2010)] and decreases the required central processing unit times by about an order of magnitude.

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.

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.

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.

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.

Efficient and accurate evaluation of potential energy matrix elements for quantum dynamics using Gaussian process regression.

Solution of the time-dependent Schrödinger equation using a linear combination of basis functions, such as Gaussian wavepackets (GWPs), requires costly evaluation of integrals over the entire potential energy surface (PES) of the system. The standard approach, motivated by computational tractability for direct dynamics, is to approximate the PES with a second order Taylor expansion, for example centred at each GWP. In this article, we propose an alternative method for approximating PES matrix elements based on PES interpolation using Gaussian process regression (GPR). Our GPR scheme requires only single-point evaluations of the PES at a limited number of configurations in each time-step; the necessity of performing often-expensive evaluations of the Hessian matrix is completely avoided. In applications to 2-, 5-, and 10-dimensional benchmark models describing a tunnelling coordinate coupled non-linearly to a set of harmonic oscillators, we find that our GPR method results in PES matrix elements for which the average error is, in the best case, two orders-of-magnitude smaller and, in the worst case, directly comparable to that determined by any other Taylor expansion method, without requiring additional PES evaluations or Hessian matrices. Given the computational simplicity of GPR, as well as the opportunities for further refinement of the procedure highlighted herein, we argue that our GPR methodology should replace methods for evaluating PES matrix elements using Taylor expansions in quantum dynamics simulations.

Kinetic-energy matrix elements for atomic Hylleraas-CI wave functions.

Hylleraas-CI is a superposition-of-configurations method in which each configuration is constructed from a Slater-type orbital (STO) product to which is appended (linearly) at most one interelectron distance rij. Computations of the kinetic energy for atoms by this method have been difficult due to the lack of formulas expressing these matrix elements for general angular momentum in terms of overlap and potential-energy integrals. It is shown here that a strategic application of angular-momentum theory, including the use of vector spherical harmonics, enables the reduction of all atomic kinetic-energy integrals to overlap and potential-energy matrix elements. The new formulas are validated by showing that they yield correct results for a large number of integrals published by other investigators.

Multivalley envelope function equations and effective potentials for phosphorus impurity in silicon

We propose a system of real-space envelope function equations without fitting parameters for modeling the electronic spectrum and wave functions of a phosphorus donor atom embedded in silicon. The approach relies on the Burt-Foreman envelope function representation and leads to coupled effective-mass Schroedinger equations containing smooth effective potentials. These potentials result from the spatial filtering imposed on the exact potential energy matrix elements in the envelope function representation. The corresponding filter function is determined from the definition of the envelope function. The resulting effective potentials and the system of envelope functions jointly reproduce the valley-orbit coupling effect in the doped silicon. Including the valley-orbit coupling not only of the 1s, but also for 2s atomic orbitals, as well as static dielectric screening is found crucial to accurately reproduce experimental data. The measured binding energies are recovered with a maximum relative error of 1.53 %. The computed wave functions are in a good agreement with experimental measurements of the electron density provided by scanning tunneling microscopy.

Ab initio multiple cloning algorithm for quantum nonadiabatic molecular dynamics.

We present a new algorithm for ab initio quantum nonadiabatic molecular dynamics that combines the best features of ab initio Multiple Spawning (AIMS) and Multiconfigurational Ehrenfest (MCE) methods. In this new method, ab initio multiple cloning (AIMC), the individual trajectory basis functions (TBFs) follow Ehrenfest equations of motion (as in MCE). However, the basis set is expanded (as in AIMS) when these TBFs become sufficiently mixed, preventing prolonged evolution on an averaged potential energy surface. We refer to the expansion of the basis set as "cloning," in analogy to the "spawning" procedure in AIMS. This synthesis of AIMS and MCE allows us to leverage the benefits of mean-field evolution during periods of strong nonadiabatic coupling while simultaneously avoiding mean-field artifacts in Ehrenfest dynamics. We explore the use of time-displaced basis sets, "trains," as a means of expanding the basis set for little cost. We also introduce a new bra-ket averaged Taylor expansion (BAT) to approximate the necessary potential energy and nonadiabatic coupling matrix elements. The BAT approximation avoids the necessity of computing electronic structure information at intermediate points between TBFs, as is usually done in saddle-point approximations used in AIMS. The efficiency of AIMC is demonstrated on the nonradiative decay of the first excited state of ethylene. The AIMC method has been implemented within the AIMS-MOLPRO package, which was extended to include Ehrenfest basis functions.

Communication: separable potential energy surfaces from multiplicative artificial neural networks.

We present a potential energy surface fitting scheme based on multiplicative artificial neural networks. It has the sum of products form required for efficient computation of the dynamics of multidimensional quantum systems with the multi configuration time dependent Hartree method. Moreover, it results in analytic potential energy matrix elements when combined with quantum dynamics methods using Gaussian basis functions, eliminating the need for a local harmonic approximation. Scaling behavior with respect to the complexity of the potential as well as the requested accuracy is discussed.

On the ground state calculation of a many-body system using a self-consistent basis and quasi-Monte Carlo: An application to water hexamer

Given a quantum many-body system, the Self-Consistent Phonons (SCP) method provides an optimal harmonic approximation by minimizing the free energy. In particular, the SCP estimate for the vibrational ground state (zero temperature) appears to be surprisingly accurate. We explore the possibility of going beyond the SCP approximation by considering the system Hamiltonian evaluated in the harmonic eigenbasis of the SCP Hamiltonian. It appears that the SCP ground state is already uncoupled to all singly- and doubly-excited basis functions. So, in order to improve the SCP result at least triply-excited states must be included, which then reduces the error in the ground state estimate substantially. For a multidimensional system two numerical challenges arise, namely, evaluation of the potential energy matrix elements in the harmonic basis, and handling and diagonalizing the resulting Hamiltonian matrix, whose size grows rapidly with the dimensionality of the system. Using the example of water hexamer we demonstrate that such calculation is feasible, i.e., constructing and diagonalizing the Hamiltonian matrix in a triply-excited SCP basis, without any additional assumptions or approximations. Our results indicate particularly that the ground state energy differences between different isomers (e.g., cage and prism) of water hexamer are already quite accurate within the SCP approximation.

Photodissociation of methyl iodide embedded in a host-guest complex: A full dimensional (189D) quantum dynamics study of CH3I@resorc[4]arene

Accurate full dimensional quantum dynamics calculations studying the photodissociation of CH(3)I@resorc[4]arene on an ab initio based potential energy surface (PES) model are reported. The converged 189D quantum dynamics calculations are facilitated by the multilayer multi-configurational time-dependent Hartree (ML-MCTDH) approach combined with the correlation discrete variable representation (CDVR) for the evaluation of potential energy matrix elements. The potential employed combines an established ab initio PES describing the photodissociation of methyl iodide in the A band with a harmonic description of the resorc[4]arene host and a bilinear modeling of the host-guest interaction. All potential parameters required in the description of the vibrations of the host molecule and the host-guest interaction are derived from ab initio calculations on the host-guest complex. Absorption spectra at 0 K and 300 K are calculated and the electronic population dynamics during the bond breaking process occurring in the first 20-30 fs after the photoexcitation is investigated. Weak but significant effects resulting from the host-guest interaction on this time scale are found and interpreted. The present study demonstrates that accurate fully quantum mechanical dynamics calculations can be preformed for systems consisting of more than 50 atoms using the ML-MCTDH/CDVR approach. Utilizing an efficient statistical approach for the construction of the ensemble of initial wavepackets, these calculations are not restricted to zero temperature but can also study the dynamics at 300 K.

A fast method for solving both the time-dependent Schrödinger equation in angular coordinates and its associated “m-mixing” problem

An efficient split-operator technique for solving the time-dependent Schrödinger equation in an angular coordinate system is presented, where a fast spherical harmonics transform accelerates the conversions between angle and angular momentum representations. Unlike previous techniques, this method features facile inclusion of azimuthal asymmetries (solving the "m-mixing" problem), adaptive time stepping, and favorable scaling, while simultaneously avoiding the need for both kinetic and potential energy matrix elements. Several examples are presented.

Energy and energy gradient matrix elements with N-particle explicitly correlated complex Gaussian basis functions with L=1

In this work we consider explicitly correlated complex Gaussian basis functions for expanding the wave function of an N-particle system with the L=1 total orbital angular momentum. We derive analytical expressions for various matrix elements with these basis functions including the overlap, kinetic energy, and potential energy (Coulomb interaction) matrix elements, as well as matrix elements of other quantities. The derivatives of the overlap, kinetic, and potential energy integrals with respect to the Gaussian exponential parameters are also derived and used to calculate the energy gradient. All the derivations are performed using the formalism of the matrix differential calculus that facilitates a way of expressing the integrals in an elegant matrix form, which is convenient for the theoretical analysis and the computer implementation. The new method is tested in calculations of two systems: the lowest P state of the beryllium atom and the bound P state of the positronium molecule (with the negative parity). Both calculations yielded new, lowest-to-date, variational upper bounds, while the number of basis functions used was significantly smaller than in previous studies. It was possible to accomplish this due to the use of the analytic energy gradient in the minimization of the variational energy.