N-mode Representation Research Articles

Pitfalls in the n-mode representation of vibrational potentials.

Simulations of anharmonic vibrational motion rely on computationally expedient representations of the governing potential energy surface. The n-mode representation (n-MR)-effectively a many-body expansion in the space of molecular vibrations-is a general and efficient approach that is often used for this purpose in vibrational self-consistent field (VSCF) calculations and correlated analogues thereof. In the present analysis, a lack of convergence in many VSCF calculations is shown to originate from negative and unbound potentials at truncated orders of the n-MR expansion. For cases of strong anharmonic coupling between modes, the n-MR can both dip below the true global minimum of the potential surface and lead to effective single-mode potentials in VSCF that do not correspond to bound vibrational problems, even for bound total potentials. The present analysis serves mainly as a pathology report of this issue. Furthermore, this insight into the origin of VSCF non-convergence provides a simple, albeit ad hoc, route to correct the problem by "painting in" the full representation of groups of modes that exhibit these negative potentials at little additional computational cost. Somewhat surprisingly, this approach also reasonably approximates the results of the next-higher n-MR order and identifies groups of modes with particularly strong coupling. The method is shown to identify and correct problematic triples of modes-and restore SCF convergence-in two-mode representations of challenging test systems, including the water dimer and trimer, as well as protonated tropine.

Time-dependent density matrix renormalization group coupled with n-mode representation potentials for the excited state radiationless decay rate: Formalism and application to azulene

We propose a method for calculating the non-radiative decay rates for polyatomic molecules including anharmonic effects of the potential energy surface (PES) in the Franck-Condon region. The method combines the n-mode representation method to construct the ab initio PES and the nearly exact time-dependent density matrix renormalization group method (TD-DMRG) to simulate quantum dynamics. In addition, in the framework of TD-DMRG, we further develop an algorithm to calculate the final-state-resolved rate coefficient which is very useful to analyze the contribution from each vibrational mode to the transition process. We use this method to study the internal conversion (IC) process of azulene after taking into account the anharmonicity of the ground state PES. The results show that even for this semi-rigid molecule, the intramode anharmonicity enhances the IC rate significantly, and after considering the two-mode coupling effect, the rate increases even further. The reason is that the anharmonicity enables the C–H vibrations to receive electronic energy while C–H vibrations do not contribute on the harmonic PES as the Huang-Rhys factor is close to 0.

Approximate high mode coupling potentials using Gaussian process regression and adaptive density guided sampling.

We present a new efficient approach for potential energy surface construction. The algorithm employs the n-mode representation and combines an adaptive density guided approach with Gaussian process regression for constructing approximate higher-order mode potentials. In this scheme, the n-mode potential construction is conventionally done, whereas for higher orders the data collected in the preceding steps are used for training in Gaussian process regression to infer the energy for new single point computations and to construct the potential. We explore different delta-learning schemes which combine electronic structure methods on different levels of theory. Our benchmarks show that for approximate 2-mode potentials the errors can be adjusted to be in the order of 8 cm-1, while for approximate 3-mode and 4-mode potentials the errors fall below 1 cm-1. The observed errors are, therefore, smaller than contributions due to missing higher-order electron excitations or relativistic effects. Most importantly, the approximate potentials are always significantly better than those with neglected higher-order couplings.

Multilevel Approach for Direct VSCF/VCI MULTIMODE Calculations with Applications to Large "Zundel" Cations.

We test existing efficient schemes for the "direct-dynamics" approach in building a potential energy surface (PES) in the code MULTIMODE. These are (1) the n-mode representation (nMR) approach to the PES, (2) the exploitation of the normal mode's symmetry to reduce the computational effort in constructing the PES, (3) the use of sparse grids for fitting the n-mode potentials, and (4) different levels of ab initio theory for these potentials. These schemes are applied to a four-dimensional calculation for the proton-bound methanol dimer (CH3OH)2H+. In addition to the major reduction in complexity obtained by considering only four modes, the combination of these schemes leads to a significant reduction in the computational effort without any major loss of accuracy. VSCF/VCI test calculations are presented for (CH3OH)2H+.

Optimizing Vibrational Coordinates To Modulate Intermode Coupling.

The choice of coordinate system strongly affects the convergence properties of vibrational structure computations. Two methods for efficient generation of improved vibrational coordinates are presented and justified by analysis of a model anharmonic two-mode Hessian and numerical computations on polyatomic molecules. To produce optimal coordinates, metrics which quantify off-diagonal couplings over a grid of Hessian matrices are minimized through unitary rotations of the vibrational basis. The first proposed metric minimizes the total squared off-diagonal coupling, and the second minimizes the total squared change in off-diagonal coupling. In this procedure certain anharmonic modes tend to localize, for example X-H stretches. The proposed methods do not rely on prior fitting of the potential energy, vibrational structure computations, or localization metrics, so they are unique from previous vibrational coordinate generation algorithms and are generally applicable to polyatomic molecules. Fitting the potential to the approximate n-mode representation in the optimized bases for all-trans polyenes shows that off-diagonal anharmonic couplings are substantially reduced by the new choices of coordinate system. Convergence of vibrational energies is examined in detail for ethylene, and it is shown that coupling-optimized modes converge in vibrational configuration interaction computations to within 1 cm(-1) using only 3-mode couplings, where normal modes require 4-mode couplings for convergence. Comparison of the vibrational configuration interaction convergence with respect to excitation level for the two proposed metrics shows that minimization of the total off-diagonal coupling is most effective for low-cost vibrational structure computations.

Calculation of the vibrational excited states of malonaldehyde and their tunneling splittings with the multi-configuration time-dependent Hartree method.

We report energies and tunneling splittings of vibrational excited states of malonaldehyde which have been obtained using full dimensional quantum mechanical calculations. To this end we employed the multi configuration time-dependent Hartree method. The results have been obtained using a recently published potential energy surface [Y. Wang, B. J. Braams, J. M. Bowman, S. Carter, and D. P. Tew, J. Chem. Phys. 128, 224314 (2008)] which has been brought into a suitable form by a modified version of the n-mode representation which was used with two different arrangements of coordinates. The relevant terms of the expansion have been identified with a Metropolis algorithm and a diffusion Monte-Carlo technique, respectively.

Vibrational dynamics of the ${\rm H}_5^+$H5+ and its isotopologues from multiconfiguration time-dependent Hartree calculations

Full-dimensional multiconfiguration time-dependent Hartree (MCTDH) computations are reported for the vibrational states of the H(5)(+) and its H(4)D(+), H(3)D(2)(+), H(2)D(3)(+), HD(4)(+), D(5)(+) isotopologues employing two recent analytical potential energy surfaces of Xie et al. [J. Chem. Phys. 122, 224307 (2005)] and Aguado et al. [J. Chem. Phys. 133, 024306 (2010)]. The potential energy operators are constructed using the n-mode representation adapted to a four-combined mode cluster expansion, including up to seven-dimensional grids, chosen adequately to take advantage in representing the MCTDH wavefunction. An error analysis is performed to quantify the convergence of the potential expansion to reproduce the reference surfaces at the energies of interest. An extensive analysis of the vibrational ground state properties of these isotopes and comparison with the reference diffusion Monte Carlo results by Acioli et al. [J. Chem. Phys. 128, 104318 (2008)] are presented. It is found that these systems are highly delocalized, interconverting between equivalent minima through rotation and internal proton transfer motions even at their vibrational ground state. Isotopic substitution affects the zero-point energy and structure, showing preference in the arrangements of the H and D within the mixed clusters, and the most stable conformers of each isotopomer are the ones with the H in the central position. Vibrational excited states are also computed and by comparing the energies and structures predicted from the two surfaces, the effect of the potential topology on them is discussed.

Trajectory-guided configuration interaction simulations of multidimensional quantum dynamics

We propose an approach to modelling multidimensional quantum systems which uses direct-dynamics trajectories to guide wavefunction propagation. First, trajectory simulations are used to generate a sample of dynamically relevant configurations on the potential energy surface (PES). Second, the sampled configurations are used to construct an n-mode representation of the PES using a greedy algorithm. Finally, the time-dependent Schrödinger equation is solved using a configuration interaction expansion of the wavefunction, with individual basis functions derived directly from the 1-mode contributions to the n-mode PES. This approach is successfully demonstrated by application to a 20-dimensional benchmark problem describing tunnelling in the presence of coupled degrees of freedom.

Chebyshev high-dimensional model representation (Chebyshev-HDMR) potentials: application to reactive scattering of H2 from Pt(111) and Cu(111) surfaces

Gas phase and surface reactions involving polyatomic molecules are of central importance to chemical physics, and require accurately fit potential energy surfaces describing the interaction in their systems. Here, we propose a method for generating a High Dimensional Model Representation (HDMR) of a multidimensional potential energy surface (PES) and apply it to reactive molecule-surface scattering problems. In the HDMR treatment, all N degrees of freedom (DOF) of an N-dimensional PES are represented but only n < N are explicitly coupled. The HDMR is obtained from Chebyshev polynomial expansions for each degree of freedom, where expansion coefficients are efficiently evaluated using discrete cosine transform (DCT) algorithms and properties of Chebyshev polynomials. HDMR surfaces constructed for the reactive scattering of H2 from Pt(111) and Cu(111) are used in quantum dynamics simulations; the resultant state-resolved reaction and scattering probabilities are compared to those from simulations using full (6D) PESs and n-mode PESs from previous work. The results are encouraging, and suggest that this method may be applicable to "late barrier" reactive systems for which the previously-used n-mode representation fails.

Theoretical studies of the tunneling splitting of malonaldehyde using the multiconfiguration time-dependent Hartree approach

Full dimensional multi-configuration time-dependent Hartree calculations of the zero point energy and the tunneling splitting of malonaldehyde using a recently published potential energy surface [Y. Wang, B. J. Braams, J. M. Bowman, S. Carter, and D. P. Tew, J. Chem. Phys. 128, 224314 (2008)] are reported. The potential energy surface has been approximated by a modified version of the n-mode representation and careful convergence check has been performed to ensure accurate results. The obtained value for the splitting (23.4 cm(-1)) is in acceptable agreement with the experimental value of 21.583 cm(-1). The computed zero-point-energy is 14,670 cm(-1) which is lower than previous results of Wang et al., but likely to be about 4 cm(-1) too low because of shortcomings of the n-mode representation of the potential. The energies reported in this abstract contain a correction to account for neglected vibrational angular momentum terms.

Full dimensional (15-dimensional) quantum-dynamical simulation of the protonated water-dimer III: Mixed Jacobi-valence parametrization and benchmark results for the zero point energy, vibrationally excited states, and infrared spectrum

Quantum dynamical calculations are reported for the zero point energy, several low-lying vibrational states, and the infrared spectrum of the H(5)O(2)(+) cation. The calculations are performed by the multiconfiguration time-dependent Hartree (MCTDH) method. A new vector parametrization based on a mixed Jacobi-valence description of the system is presented. With this parametrization the potential energy surface coupling is reduced with respect to a full Jacobi description, providing a better convergence of the n-mode representation of the potential. However, new coupling terms appear in the kinetic energy operator. These terms are derived and discussed. A mode-combination scheme based on six combined coordinates is used, and the representation of the 15-dimensional potential in terms of a six-combined mode cluster expansion including up to some 7-dimensional grids is discussed. A statistical analysis of the accuracy of the n-mode representation of the potential at all orders is performed. Benchmark, fully converged results are reported for the zero point energy, which lie within the statistical uncertainty of the reference diffusion Monte Carlo result for this system. Some low-lying vibrationally excited eigenstates are computed by block improved relaxation, illustrating the applicability of the approach to large systems. Benchmark calculations of the linear infrared spectrum are provided, and convergence with increasing size of the time-dependent basis and as a function of the order of the n-mode representation is studied. The calculations presented here make use of recent developments in the parallel version of the MCTDH code, which are briefly discussed. We also show that the infrared spectrum can be computed, to a very good approximation, within D(2d) symmetry, instead of the G(16) symmetry used before, in which the complete rotation of one water molecule with respect to the other is allowed, thus simplifying the dynamical problem.

Variational quantum approaches for computing vibrational energies of polyatomic molecules

In this article, we review state-of-the-art methods for computing vibrational energies of polyatomic molecules using quantum mechanical, variationally-based approaches. We illustrate the power of those methods by presenting applications to molecules with more than four atoms. This demonstrates the great progress that has been made in this field in the last decade in dealing with the exponential scaling with the number of vibrational degrees of freedom. In this review we present three methods that effectively obviate this bottleneck. The first important idea is the n-mode representation of the Hamiltonian and notably the potential. The potential (and other functions) is represented as a sum of terms that depend on a subset of the coordinates. This makes it possible to compute matrix elements, form a Hamiltonian matrix, and compute its eigenvalues and eigenfunctions. Another approach takes advantage of this multimode representation and represents the terms as a sum of products. It then exploits the powerful multiconfiguration Hartree time-dependent method to solve the time-dependent Schrödinger equation and extract the eigenvalue spectrum. The third approach we present uses contracted basis functions in conjunction with a Lanczos eigensolver. Matrix vector products are done without transforming to a direct-product grid. The usefulness of these methods is demonstrated for several example molecules, e.g. methane, methanol and the Zundel cation.

Automatic generation of potential energy and property surfaces of polyatomic molecules in normal coordinates

A procedure for the automatic construction of Born-Oppenheimer (BO) potential energy and molecular property surfaces in rectilinear normal coordinates is presented and its suitability and accuracy when combined with vibrational structure calculations are assessed. The procedure relies on a hierarchical n-mode representation of the BO potential energy or molecular property surface, where the n-mode term of the sequence of potentials/molecular properties includes only the couplings between n or less vibrational degrees of freedom. Each n-mode cut of the energy/molecular property surface is first evaluated in a grid of points with ab initio electronic structure methods. The ab initio data are then spline interpolated and a subsequent polynomial fitting provides an analytical semiglobal representation for use in vibrational structure programs. The implementation of the procedure is outlined and the accuracy of the method is tested on water and difluoromethane. Strategies for improving the proposed algorithm are also discussed.

Automatic generation of force fields and property surfaces for use in variational vibrational calculations of anharmonic vibrational energies and zero-point vibrational averaged properties

An automatic and general procedure for the calculation of geometrical derivatives of the energy and general property surfaces for molecular systems is developed and implemented. General expressions for an n-mode representation are derived, where the n-mode representation includes only the couplings between n or less degrees of freedom. The general expressions are specialized to derivative force fields and property surfaces, and a scheme for calculation of the numerical derivatives is implemented. The implementation is interfaced to electronic structure programs and may be used for both ground and excited electronic states. The implementation is done in the context of a vibrational structure program and can be used in combination with vibrational self-consistent field (VSCF), vibrational configuration interaction (VCI), vibrational Moller-Plesset, and vibrational coupled cluster calculations of anharmonic wave functions and calculation of vibrational averaged properties at the VSCF and VCI levels. Sample calculations are presented for fundamental vibrational energies and vibrationally averaged dipole moments and frequency dependent polarizabilities and hyperpolarizabilities of water and formaldehyde.

"Morphing" of ab initio-based interaction potentials to spectroscopic accuracy: Application to Cl-(H2O)

Abstract We present anharmonic vibrational calculations for the Cl-(H2O) cluster and their convergence with the n-mode representation of the interaction potential. Extension of this representation up to 4-mode couplings produces results that appear to be converged to within 10 cm-1 or less relative to the exact 6-mode representation for this system. This methodology, in conjunction with the "morphing" technique, which is based on the scaling of the internal coordinates, provides an effective means of fitting intermolecular potentials to measured vibrational spectra. Application of this approach to the chloride-water interaction produces a revision of a previously developed empirical interaction potential that reproduces the measured fundamental and first overtone frequencies to within an average absolute deviation of 1.75 cm-1 in the 4-mode coupling representation.

MULTIMODE: A code to calculate rovibrational energies of polyatomic molecules

This review focuses on the calculation of rovibrational energies of polyatomic molecules using the code MULTIMODE. This code, which uses normal coordinates and a hierarchical n-mode representation of the potential, aims to be applicable to a wide class of molecules and molecular complexes. The theoretical and computational methods used in this code are described, followed by a review of selected applications. These applications illustrate various features of the code and also point out some limitations of the current version of the code. The review concludes with some ideas about possible future directions in this area of research.

Extensions and tests of "multimode": a code to obtain accurate vibration/rotation energies of many-mode molecules

We describe extensions and tests of the code “multimode” which does vibrational self-consistent field method (VSCF) and two types of state-mixing (denoted VSCF-CI and V-CI) for rovibrational energies of many-mode systems. The extensions include an exact treatment of rotation, flexible approaches to perform the CI calculations, and the inclusion of a Davidson diagonalization routine to find low-lying eigenvalues of large matrices. The code is tested against previous exact variational calculations for non-rotating H2CN, and J=0 and J=1 rovibrational states of H2CS. The code represents the full potential by a hierarchical n-mode representation, where n is the number of normal coordinates that are coupled together. Tests are presented for the convergence and accuracy of this representation for n equal to 3 and 4, where 4 is the current maximum value. These tests are done at the VSCF and V-CI level, with very encouraging results.