A new method to generate spin-orbit coupled potential energy surfaces: effective relativistic coupling by asymptotic representation

A new method has been developed to generate fully coupled potential energy surfaces including derivative and spin-orbit coupling. The method is based on an asymptotic (atomic) representation of the molecular fine structure states and a corresponding diabatization. The effective relativistic coupling is described by a constant spin-orbit coupling matrix and the geometry dependence of the coupling is accounted for by the diabatization. This approach is very efficient, particularly for certain systems containing a very heavy atom, and yields consistent results throughout nuclear configuration space. A first application to a diatomic system is presented as proof of principle and is compared to accurate ab initio calculations. However, the method is widely applicable to general polyatomic systems in full dimensionality, containing several relativistic atoms and treating higher order relativistic couplings as well.     

On the representation of coupled adiabatic potential energy surfaces using quasi-diabatic Hamiltonians: a distributed origins expansion approach

In two previous papers we have introduced a method to generate coupled quasi-diabatic Hamiltonians (H(d)) that are capable of representing adiabatic energies, energy gradients, and derivative couplings over a wide range of geometries including seams of conical intersection. In this work, two new synergistic features are introduced. Firstly, the functional form of H(d) is generalized. Rather than requiring there to be a low energy point of high symmetry to serve as the unique origin, functions centered on points distributed in nuclear coordinate space are used in the polynomials that comprise the matrix elements in H(d). The use of functions with distributed origins, allows reproduction of the ab initio data with lower order expansions, and offers the possibility of describing multichannel dissociation. The fitting algorithm is combined with a three-step procedure in which the domain of H(d) is extended from a core set of nuclear configurations to a region of nuclear coordinate space appropriate for nuclear dynamics, with a prescribed accuracy. This significant extension of the domain of definition compared to our original work, which is facilitated by the distributed origin approach, is achieved largely through the use of surface hopping trajectories. The 1,2(1)A states of NH(3), which provide an archetypical example of nonadiabatic dynamics, are used to demonstrate the utility of this approach. The representation describes 21 points on the 1(1)A-2(1)A seam of conical intersection and their local topography flawlessly and on the entire domain, the electronic structure data is represented to an accuracy of 77.00 (46.90) cm(-1), as measured by the root mean square (mean unsigned) error for energies lower than 50 000 cm(-1). This error is a factor of 10 lower than that of the most accurate representation of high quality ab initio data, on a comparable domain, previously reported for this system.     

Improved direct diabatization and coupled potential energy surfaces for the photodissociation of ammonia

Diabatic potential energy surfaces are a convenient starting point for dynamics calculations of photochemical processes, and they can be calculated by the fourfold way direct diabatization scheme. Here we present an improved definition of the reference orbital for applying the fourfold way direct diabatization scheme to ammonia. The improved reference orbital is a geometry-dependent hybrid orbital that allows one to define consistent dominant configuration lists at all geometries important for photodissociation. Using diabatic energies calculated with the new reference orbital and consistent dominant configuration lists, we have refitted the analytical representations of the ground and the first electronically excited singlet-state potential energy surfaces and the diabatic coupling surface. Improved functional forms were used to reproduce the experimental dissociation energies and excitation energies, which will be important for subsequent simulations of photochemical dynamics. We find that the lowest-energy conical intersection point is at 5.16 eV, with C _2v symmetry. Electronic Supplementary Material The online version of this article (doi:) contains supplementary material, which is available to authorized users. This article is part of the special issue dedicated to the memory of the late Professor Fernando Bernardi.     

A random-sampling high dimensional model representation neural network for building potential energy surfaces

We combine the high dimensional model representation (HDMR) idea of Rabitz and co-workers [J. Phys. Chem. 110, 2474 (2006)] with neural network (NN) fits to obtain an effective means of building multidimensional potentials. We verify that it is possible to determine an accurate many-dimensional potential by doing low dimensional fits. The final potential is a sum of terms each of which depends on a subset of the coordinates. This form facilitates quantum dynamics calculations. We use NNs to represent HDMR component functions that minimize error mode term by mode term. This NN procedure makes it possible to construct high-order component functions which in turn enable us to determine a good potential. It is shown that the number of available potential points determines the order of the HDMR which should be used.     

Improving the accuracy of interpolated potential energy surfaces by using an analytical zeroth-order potential function

We present a method for improving the accuracy and efficiency of interpolation methods, in which an analytical zeroth-order potential-energy surface is employed as a reference surface. To investigate and test the method, we apply it to hydrogen peroxide where there exists an accurate analytical surface which we take as the "exact" surface for obtaining the energies and derivatives for fitting and assessing the accuracy. Examples are given for four-dimensional and six-dimensional surfaces interpolated by using either the modified Shepard or second-degree interpolating moving least-squares approach, with comparisons for cases with and without using the zeroth-order potential.     

Transition structure modeling by intersecting potential energy surfaces

A previously proposed method of modeling force field transition structures as the lowest energy structure on the seam of two intersecting potential energy surfaces is improved with respect to the force field expression and the efficiency of the optimization algorithm. Comparison with ab initio and experimental results show that the force field method may have the potential of achieving almost quantitative results. It appears that the results from transition state modeling are currently limited by the accuracy of the underlying force field. © 1994 by John Wiley & Sons, Inc.     

Calculating vibrational spectra using modified Shepard interpolated potential energy surfaces

A potential energy interpolation approach based on modified Shepard interpolation and specifically designed for calculation of vibrational states is presented. The importance of the choice of coordinates for the rate of convergence is demonstrated. Studying the vibrational states of the water molecule as a test case, a coordinate system comprised of inverse bond distances and trigonometric functions of the bond angle is found to be particularly efficient. Different sampling schemes used to locate the reference points in the modified Shepard interpolation are investigated. A final scheme is recommended, which allows the construction of potential energy surfaces to sub-wave-number accuracy.     

Fully relativistic calculations on the potential energy surfaces of the lowest 23 states of molecular chlorine

The electronic structure and spectroscopic properties (R(e), omega(e), omega(e)x(e), beta(e), and T(e)) of the ground state and the 22 lowest excited states of chlorine molecule were studied within a four-component relativistic framework using the MOLFDIR program package. The potential energy curves of all possible 23 covalent states were calculated using relativistic complete open shell configuration interaction approach. In addition, four component multireference configuration interaction with single and double excitation calculations were performed in order to infer the effects due to dynamical correlation in vertical excitations. The calculated properties are in good agreement with the available experimental data.     

Generalization of classical mechanics for nuclear motions on nonadiabatically coupled potential energy surfaces in chemical reactions

Classical trajectory study of nuclear motion on the Born-Oppenheimer potential energy surfaces is now one of the standard methods of chemical dynamics. In particular, this approach is inevitable in the studies of large molecular systems. However, as soon as more than a single potential energy surface is involved due to nonadiabatic coupling, such a naive application of classical mechanics loses its theoretical foundation. This is a classic and fundamental issue in the foundation of chemistry. To cope with this problem, we propose a generalization of classical mechanics that provides a path even in cases where multiple potential energy surfaces are involved in a single event and the Born-Oppenheimer approximation breaks down. This generalization is made by diagonalization of the matrix representation of nuclear forces in nonadiabatic dynamics, which is derived from a mixed quantum-classical representation of the electron-nucleus entangled Hamiltonian [Takatsuka, K. J. Chem. Phys. 2006, 124, 064111]. A manifestation of quantum fluctuation on a classical subsystem that directly contacts with a quantum subsystem is discussed. We also show that the Hamiltonian thus represented gives a theoretical foundation to examine the validity of the so-called semiclassical Ehrenfest theory (or mean-field theory) for electron quantum wavepacket dynamics, and indeed, it is pointed out that the electronic Hamiltonian to be used in this theory should be slightly modified.     

Finding minimum energy reaction paths on ab initio potential energy surfaces using the fast marching method

The fast marching method (FMM) for determining minimum-cost paths has been extended to compute the minimum-energy reaction coordinates in chemical reactions. This was accomplished by building an interface between FMM and the Gaussian program. We demonstrate the new method using an S _N 2 reaction, the isomerization of HSCN to HNCS, and a gas-phase rearrangement reaction of relevance in mass spectrometry.     

Molecular potential energy surfaces by interpolation: strategies for faster convergence

A method for interpolating molecular potential energy surfaces introduced [Ischtwan and Collins, J. Chem. Phys. 100, 8080 (1994)] and developed as an iterative scheme has been improved by different criteria for the selection of the data points. Refinements in the selection procedure are based on the variance of the interpolation and the direct exploration of the interpolation error, and produce more accurate surfaces than the previously established scheme for the same number of data points.     

On the fitting of analytical potential energy surfaces by constrained optimization

Analytical expressions for the He—H₂ potential energy surface have been obtained by non-linear constrained optimization techniques.