General Potential Energy Surface Research Articles

Using a pruned basis and a sparse collocation grid with more points than basis functions to do efficient and accurate MCTDH calculations with general potential energy surfaces.

We propose a new collocation multi-configuration time-dependent Hartree (MCTDH) method. It reduces point-set error by using more points than basis functions. Collocation makes it possible to use MCTDH with a general potential energy surface without computing any integrals. The collocation points are associated with a basis larger than the basis used to represent wavefunctions. Both bases are obtained from a direct product basis built from single-particle functions by imposing a pruning condition. The collocation points are those on a sparse grid. Heretofore, collocation MCTDH calculations with more points than basis functions have only been possible if both the collocation grid and the basis set are direct products. In this paper, we exploit a new pseudo-inverse to use both more points than basis functions and a pruned basis and grid. We demonstrate that, for a calculation of the lowest 50 vibrational states (energy levels and wavefunctions) of CH2NH, errors can be reduced by two orders of magnitude by increasing the number of points, without increasing the basis size. This is true also when unrefined time-independent points are used.

Eigenstate calculation in the state-averaged (multi-layer) multi-configurational time-dependent Hartree approach.

A new approach for the calculation of eigenstates with the state-averaged (multi-layer) multi-configurational time-dependent Hartree (MCTDH) approach is presented. The approach is inspired by the recent work of Larsson [J. Chem. Phys. 151, 204102 (2019)]. It employs local optimization of the basis sets at each node of the multi-layer MCTDH tree and successive downward and upward sweeps to obtain a globally converged result. At the top node, the Hamiltonian represented in the basis of the single-particle functions (SPFs) of the first layer is diagonalized. Here p wavefunctions corresponding to the p lowest eigenvalues are computed by a block Lanczos approach. At all other nodes, a non-linear operator consisting of the respective mean-field Hamiltonian matrix and a projector onto the space spanned by the respective SPFs is considered. Here, the eigenstate corresponding to the lowest eigenvalue is computed using a short iterative Lanczos scheme. Two different examples are studied to illustrate the new approach: the calculation of the vibrational states of methyl and acetonitrile. The calculations for methyl employ the single-layer MCTDH approach, a general potential energy surface, and the correlation discrete variable representation. A five-layer MCTDH representation and a sum of product-type Hamiltonian are used in the acetonitrile calculations. Very fast convergence and order of magnitude reductions in the numerical effort compared to the previously used block relaxation scheme are found. Furthermore, a detailed comparison with the results of Avila and Carrington [J. Chem. Phys. 134, 054126 (2011)] for acetonitrile highlights the potential problems of convergence tests for high-dimensional systems.

Using collocation and solutions for a sum-of-product potential to compute vibrational energy levels for general potentials

It is easier to compute a vibrational spectrum when the potential energy surface (PES) is a sum-of-products (SOP). Many popular computational methods work only if the PES is a SOP. However, the most accurate PESs are not SOPs. We propose using collocation and solutions of the Schrödinger equation with a SOP PES to compute solutions on a corresponding general PES. This makes it possible to account for coupling and anharmonicity omitted from the SOP PES. We find that correcting energy levels computed on a SOP PES with collocation reduces differences with exact energy levels by about two orders of magnitude.

A rectangular collocation multi-configuration time-dependent Hartree (MCTDH) approach with time-independent points for calculations on general potential energy surfaces.

We introduce a collocation-based multi-configuration time-dependent Hartree (MCTDH) method that uses more collocation points than basis functions. We call it the rectangular collocation MCTDH (RC-MCTDH) method. It does not require that the potential be a sum of products. RC-MCTDH has the important advantage that it makes it simple to use time-independent collocation points. When using time-independent points, it is necessary to evaluate the potential energy function only once and not repeatedly during an MCTDH calculation. It is inexpensive and straightforward to use RC-MCTDH with combined modes. Using more collocation points than basis functions enables one to reduce errors in energy levels without increasing the size of the single-particle function basis. On the contrary, whenever a discrete variable representation is used, the only way to reduce the quadrature error is to increase the basis size, which then also reduces the basis-set error. We demonstrate that with RC-MCTDH and time-independent points, it is possible to calculate accurate eigenenergies of CH3 and CH4.

A collocation-based multi-configuration time-dependent Hartree method using mode combination and improved relaxation

Although very useful, the original multi-configuration time-dependent Hartree (MCTDH) method has two weaknesses: (1) its cost scales exponentially with the number of atoms in the system; (2) the standard MCTDH implementation requires that the potential energy surface (PES) be in the sum-of-product (SOP) form in order to reduce the cost of computing integrals in the MCTDH basis. One way to deal with (1) is to lump coordinates into groups. This is mode combination (MC). One way to deal with (2) is to reformulate MCTDH using collocation so that there are no integrals. In this paper, we combine MC and collocation to formulate a MC collocation multi-configuration time-dependent Hartree (MC-C-MCTDH) method. In practice, its cost does not scale exponentially with the number of atoms, and it can be used with any general PES; the PES need not be an SOP and need not have a special form. No integrals and, hence, no quadratures are necessary. We demonstrate the accuracy and efficiency of the new method by computing vibrational energy eigenstates of methyl radical, methane, and acetonitrile. To do this, we use MC-C-MCTDH with a variant of improved relaxation, derived by evaluating a residual at points. Because the MC basis functions are multivariate, collocation points in multi-dimensional spaces are required. We use two types of collocation points: (1) discrete variable representation-like points obtained from (approximate) simultaneous diagonalization of matrices and (2) Leja points, which are known to be good interpolation points, determined from a generalized recipe suitable for any basis.

Quasiclassical method for calculating the density of states of ultracold collision complexes

We derive a quasiclassical expression for the density of states (DOS) of an arbitrary, ultracold, $N$-atom collision complex, for a general potential energy surface (PES). We establish the accuracy of our quasiclassical method by comparing to exact quantum results for the K$_2$-Rb and NaK-NaK systems, with isotropic model PESs. Next, we calculate the DOS for an accurate NaK-NaK PES to be 0.124~$\mu$K$^{-1}$, with an associated Rice-Ramsperger-Kassel-Marcus (RRKM) sticking time of 6.0~$\mu$s. We extrapolate the DOS and sticking times to all other polar bialkali-bialkali collision complexes by scaling with atomic masses, equilibrium bond lengths, dissociation energies, and dispersion coefficients. The sticking times calculated here are two to three orders of magnitude shorter than those reported by Mayle et al. [Phys. Rev. A 85, 062712 (2012)]. We estimate dispersion coefficients and collision rates between molecules and complexes. We find that the sticking-amplified three-body loss mechanism is not likely the cause of the losses observed in the experiments.

A pruned collocation-based multiconfiguration time-dependent Hartree approach using a Smolyak grid for solving the Schrödinger equation with a general potential energy surface.

Standard multiconfiguration time-dependent Hartree (MCTDH) calculations use a direct product basis and rely on the potential being a sum of products (SOPs). The size of the direct product MCTDH basis scales exponentially with the number of atoms. Accurate potentials may not be SOPs. We introduce an MCTDH approach that uses a pruned basis and a collocation grid. Pruning the basis significantly reduces its size. Collocation makes it possible to do calculations using a potential that is not a SOP. The collocation point set is a Smolyak grid. Strategies using pruned MCTDH bases already exist, but they work only if the potential is a SOP. Strategies for using MCTDH with collocation also exist, but they work only if the MCTDH basis is a direct product. In this paper, we combine a pruned basis with collocation. This makes it possible to mitigate the direct-product basis size problem and do calculations when the potential is not a SOP. Because collocation is used, there are no integrals and no need for quadrature. All required matrix-vector products can be evaluated sequentially. We use nested sets of collocation points and hierarchical basis functions. They permit efficient inversion of the (large) matrix whose elements are basis functions evaluated at points, which is necessary to transform values of functions at points to basis coefficients. The inversion technique could be used outside of chemical physics. We confirm the validity of this new pruned, collocation-based (PC-)MCTDH approach by calculating the first 50 vibrational eigenenergies of CH2NH.

Designing Reactions with Post-Transition-State Bifurcations: Asynchronous Nitrene Insertions into C–C σ Bonds

Summary Discoveries of chemical reactions with potential energy surfaces with post-transition-state bifurcations are increasing in frequency. Although such potential energy surface features are often discovered accidentally, we demonstrate here that they can be designed rationally. We describe a process for designing such reactions and apply it to the design of a nitrene addition/alkyl-shift reaction with a post-transition state bifurcation. We examine substituent effects on this reaction and propose a dynamics-based model for selectivity control.

Modern theoretical chemistry: the legacy of Prof. John N. Murrell

![Graphic][1] Professor John Murrell FRS (1932–2016) was a theoretical chemist whose research continues to have a major influence on molecular science. He studied for his PhD with Christopher Longuet-Higgins FRS first at King's College London and then at the University of Cambridge. This gave him a deep grounding in how theory can be used to interpret molecular spectra. Then, in the last 40 years of the twentieth century, first at the University of Sheffield and subsequently at the University of Sussex, he directed research working in the field of theoretical chemistry. This was at a time when computers were starting to become available so that realistic quantum mechanical calculations could be carried out to help explain experimental results. In particular, John Murrell was one of the first theoretical chemists to realize that general potential energy surfaces for molecules could be constructed that enabled calculations to be carried out on both molecular spectra and chemical reaction dynamics. This has proved to be a very powerful and general procedure for performing simulations in molecular science. He was an inspiring teacher and book writer and many of his students started their own research groups working in this field. A Royal Society Biographical Memoir contains full details of John Murrell's outstanding career [1]. This theme issue of the … [1]: /embed/inline-graphic-1.gif

A new collocation-based multi-configuration time-dependent Hartree (MCTDH) approach for solving the Schrödinger equation with a general potential energy surface.

We present a new collocation-based multi-configuration time-dependent Hartree (MCTDH) approach for solving the Schrödinger equation required to compute (ro-)vibrational spectra, photodissociation cross sections, reaction rate constants, etc., that can be used with general potential energy surfaces. Collocation obviates the need for quadrature and facilitates using complicated kinetic energy operators. When the basis is good, the accuracy of collocation solutions to the Schrödinger equation is not sensitive to the choice of the collocation points. We test the collocation MCTDH equations we derive by showing that they can be used to compute accurate vibrational energy levels of CH3. It is possible to choose (imaginary) time-independent collocation points with which collocation-based MCTDH energies are accurate. It is therefore not necessary to calculate potential values many times during the propagation.

The multigrid POTFIT (MGPF) method: Grid representations of potentials for quantum dynamics of large systems

In this article, a new method, multigrid POTFIT (MGPF), is presented. MGPF is a grid-based algorithm which transforms a general potential energy surface into product form, that is, a sum of products of one-dimensional functions. This form is necessary to profit from the computationally advantageous multiconfiguration time-dependent Hartree method for quantum dynamics. MGPF circumvents the dimensionality related issues present in POTFIT [A. Jäckle and H.-D. Meyer, J. Chem. Phys. 104, 7974 (1996)], allowing quantum dynamical studies of systems up to about 12 dimensions. MGPF requires the definition of a fine grid and a coarse grid, the latter being a subset of the former. The MGPF approximation relies on a series of underlying POTFIT calculations on grids which are smaller than the fine one and larger than or equal to the coarse one. This aspect makes MGPF a bit less accurate than POTFIT but orders of magnitude faster and orders of magnitude less memory demanding than POTFIT. Moreover, like POTFIT, MGPF is variational and provides an efficient error control.

A multilayer multiconfigurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces

The multiconfigurational time-dependent Hartree (MCTDH) approach facilitates multidimensional quantum dynamics calculations by representing the wavepacket in an optimal set of time-dependent basis functions, called single-particle functions. Choosing these single-particle functions to be themselves multidimensional wavefunctions which are represented using a MCTDH representation, a multilayer MCTDH scheme has been constructed and used for quantum dynamics calculations treating up to 1000 degrees of freedom rigorously [Wang and Thoss, J. Chem. Phys. 199, 1289 (2003)]. The present work gives a practical scheme which facilitates the application of the multilayer MCTDH approach, which previously has only been employed to study systems described by model-type Hamiltonians, to molecular systems described by more complicated Hamiltonians and general potential energy surfaces. A multilayer extension of the correlation discrete variable representation (CDVR) scheme employed in MCTDH calculations studying quantum dynamics on general potential energy surfaces is developed and tested in a simple numerical application. The resulting multilayer MCTDH/CDVR approach might offer a perspective to rigorously describe the quantum dynamics of larger polyatomic systems.

General Potential Energy Surfaces for Catalytic Processes

Here we argue that one may describe, in general terms, how catalysts modify reaction mechanisms by entering and leaving a conversion sequence, using potential energy surfaces that are conceptually correct. Thermochemical and kinetic constraints providing bounds that limit catalytic processes are formulated. 3-D diagrams are proposed. These clearly illustrate the basic principle that catalysts initially associate with the substrates and thus directly participate in modified conversion pathways, but are regenerated in the final step, thereby providing for a turnover number greater than unity.

Locating transition states by quadratic image gradient descent on potential energy surfaces

An analysis is given of the so-called ‘‘image function’’ approach to finding transition states. It is demonstrated that, in fact, such functions do not exist for general potential energy surfaces so that a plain minimum search is inappropriate. Nonconservative image gradient fields do exist, however, and their field lines, defined by Euler’s equation, can lead to transition states as exemplified by quantitative integrations of these equations for the Müller–Brown surface. As do gradient fields, image gradient fields contain streambeds and ridges, but their global structure is considerably more complex than that of gradient fields. In particular, they contain certain singular points where the image gradients change sign without passing through zero. They are the points where the two lowest eigenvalues of the Hessian are degenerate. Some of them can act as singular attractors for the image gradient descent and any algorithm must contain safeguards for avoiding them. (Such regions are equally troublesome for quasi-Newton-type transition-state searches.) Image gradient fields appear to have considerably larger catchment basins around transition states than do quasi-Newton-type or gradient-norm-type transition-state searches. A quantitative quadratic image-gradient-following algorithm is formulated and, through applications to the Müller–Brown surface, shown to be effective in finding transition states.