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.

A two-step quadrature-based variational calculation of ro-vibrational levels and wavefunctions of CO2 using a bisector-x molecule-fixed frame.

In this paper, we propose a new two-step strategy for computing ro-vibrational energy levels and wavefunctions of a triatomic molecule and apply it to CO2. A two-step method [J. Tennyson and B. T. Sutcliffe, Mol. Phys., 1986, 58, 1067] uses a basis whose functions are products of K-dependent "vibrational" functions and symmetric top functions. K is the quantum number for the molecule-fixed z component of the angular momentum. For a linear molecule, a two-step method is efficient because the Hamiltonian used to compute the basis functions includes the largest coupling term. The most important distinguishing feature of the two-step method we propose is that it uses an associated Legendre basis and quadrature rather than a K-dependent discrete variable representation. This reduces the cost of the calculation and simplifies the method. We have computed ro-vibrational energy levels with J up to 100 for CO2, on an accurate available potential energy surface which is known as the AMES-2 PES and present a subset of those levels. We have converged most levels up to 20 000 cm-1 to 0.0001 cm-1.

Theory cracks old data: Rovibrational energy levels of orthoH2-CO derived from experiment.

Measurements of rovibrational spectra of clusters provide physical insight only if spectral lines can be assigned to pairs of quantum states, and further insight is obtained if one can deduce the quantitative energy-level pattern. Both steps can be so difficult that some measured spectra remain unassigned, one example is orthoH2-CO. To extend the scope of spectroscopic insights, we propose to use theoretical information in interpretation of spectra. We first performed high accuracy, full-dimensional calculations of the orthoH2-CO spectrum, at the highest practically achievable levels of electronic structure theory and quantum nuclear dynamics. Then, an iterative, theory-guided method developed here allowed us to fully interpret the spectrum of orthoH2-CO, extending the range of van der Waals clusters for which spectroscopy can provide physical insights.

Computing vibrational energy levels using a canonical polyadic tensor method with a fixed rank and a contraction tree.

In this paper, we use the previously introduced Canonical Polyadic (CP)-Multiple Shift Block Inverse Iteration (MSBII) eigensolver [S. D. Kallullathil and T. Carrington, J. Chem. Phys. 155, 234105 (2021)] in conjunction with a contraction tree to compute vibrational spectra. The CP-MSBII eigensolver uses the CP format. The memory cost scales linearly with the number of coordinates. A tensor in CP format represents a wavefunction constrained to be a sum of products (SOP). An SOP wavefunction can be made more accurate by increasing the number of terms, the rank. When the required rank is large, the runtime of a calculation in CP format is long, although the memory cost is small. To make the method more efficient, we break the full problem into pieces using a contraction tree. The required rank for each of the sub-problems is small. To demonstrate the effectiveness of the ideas, we computed vibrational energy levels of acetonitrile (12-D) and ethylene oxide (15-D).

Computing vibrational spectra using a new collocation method with a pruned basis and more points than basis functions: Avoiding quadrature.

We present a new collocation method for computing the vibrational spectrum of a polyatomic molecule. Some form of quadrature or collocation is necessary when the potential energy surface does not have a simple form that simplifies the calculation of the potential matrix elements required to do a variational calculation. With quadrature, better accuracy is obtained by using more points than basis functions. To achieve the same advantage with collocation, we introduce a collocation method with more points than basis functions. Critically important, the method can be used with a large basis because it is incorporated into an iterative eigensolver. Previous collocation methods with more points than functions were incompatible with iterative eigensolvers. We test the new ideas by computing energy levels of molecules with as many as six atoms. We use pruned bases but expect the new method to be advantageous whenever one uses a basis for which it is not possible to find an accurate quadrature with about as many points as there are basis functions. For our test molecules, accurate energy levels are obtained even using non-optimal, simple, equally spaced points.

Computing excited OH stretch states of water dimer in 12D using contracted intermolecular and intramolecular basis functions.

Due to the ubiquity and importance of water, water dimer has been intensively studied. Computing the (ro-)vibrational spectrum of water dimer is challenging. The potential has eight wells separated by low barriers, which makes harmonic approximations of limited utility. A variational approach is imperative, but difficult because there are 12 coupled vibrational coordinates. In this paper, we use a product contracted basis whose functions are products of intramolecular and intermolecular functions computed using an iterative eigensolver. An intermediate matrix F facilitates calculating matrix elements. Using F, it is possible to do calculations on a general potential without storing the potential on the full quadrature grid. We find that surprisingly many intermolecular functions are required. This is due to the importance of coupling between inter- and intra-molecular coordinates. The full G16 symmetry of water dimer is exploited. We calculate, for the first time, monomer excited stretch states and compare P(1) transition frequencies with their experimental counterparts. We also compare with experimental vibrational shifts and tunneling splittings. Surprisingly, we find that the largest tunneling splitting, which does not involve the interchange of the two monomers, is smaller in the asymmetric stretch excited state than in the ground state. Differences between levels we compute and those obtained with a [6+6]D adiabatic approximation [Leforestier et al. J. Chem. Phys. 137 014305 (2012)] are ∼0.6 cm-1 for states without monomer excitation, ∼4 cm-1 for monomer excited bend states, and as large as ∼10 cm-1 for monomer excited stretch states.

Using Collocation to Solve the Schrödinger Equation.

We review the collocation approach to the solution of the Schrödinger equation and its uses in applications. Interrelations between collocation and other methods are highlighted. We also stress advantages and disadvantages of the rectangular collocation formulation. Using collocation makes it possible to use any, e.g. optimized, coordinates and basis functions, including nonintegrable basis functions, and provides a straightforward way of dealing with singularities in the potential. In addition, we stress that using collocation facilitates tuning the shape of basis functions and the placement of points, both of which can be done with machine-learning methods. Applications to electronic and vibrational problems are reviewed focusing on calculations for molecules on surfaces for which there are few variational calculations. Collocation has advantages when potential energy surfaces are unavailable, in particular, for molecule-surface systems, and for systems for which standard direct product quadrature grids, often used with variational methods, are costly.

Computing vibrational energy levels by solving linear equations using a tensor method with an imposed rank.

Present day computers do not have enough memory to store the high-dimensional tensors required when using a direct product basis to compute vibrational energy levels of a polyatomic molecule with more than about five atoms. One way to deal with this problem is to represent tensors using a tensor format. In this paper, we use the canonical polyadic (CP) format. Energy levels are computed by building a basis from vectors obtained by solving linear equations. The method can be thought of as a CP realization of a block inverse iteration method with multiple shifts. The CP rank of the tensors is fixed, and the linear equations are solved with an method. There is no need for rank reduction and no need for orthogonalization, and tensors with a rank larger than the fixed rank used to solve the linear equations are never generated. The ideas are tested by computing vibrational energy levels of a 64-D bilinearly coupled model Hamiltonian and of acetonitrile (12-D).

Using nondirect product Wigner D basis functions and the symmetry-adapted Lanczos algorithm to compute the ro-vibrational spectrum of CH4-H2O.

By doing calculations on the methane-water van der Waals complex, we demonstrate that highly converged energy levels and wavefunctions can be obtained using Wigner D basis functions and the Symmetry-Adapted Lanczos (SAL) method. The Wigner D basis is a nondirect product basis and, therefore, efficient when the kinetic energy operator has accessible singularities. The SAL method makes it possible to exploit symmetry to label energy levels and reduce the cost of the calculation, without explicitly using symmetry-adapted basis functions. Line strengths are computed, and new bands are identified. In particular, we find unusually strong transitions between states associated with the isomers of the global minimum and the secondary minimum.

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.

Using collocation to study the vibrational dynamics of molecules.

In this paper, I review collocation methods for solving the time-independent and the time-dependent Schroedinger equation. Unlike traditional variational methods, collocation methods do not require integrals and quadrature. Either collocation or quadrature is necessary if the potential does not have a special form. If the basis is a direct product of univariate bases and the quadrature grid is also a direct product, there exist variational methods that do not require quadrature approximations for potential energy matrix elements. These methods, however, do require storing, in computer memory, vectors with as many components as there are quadrature points. For this reason direct-product variational methods are poor for problems with more than five atoms. There are well established ideas for reducing the size of the basis in a variational calculation. Three such ideas are: 1) prune the direct product basis; 2) use basis functions that are products of multivariate functions; 3) optimise the basis functions (e.g. Multiconfiguration time-dependent Hartree). Reducing the basis size, however, is not enough to the make variational methods tractable because, for all three of these ideas, quadrature rears its ugly head. Collocation is an attractive alternative to variational methods.

Neural Network Potential Energy Surfaces for Small Molecules and Reactions.

We review progress in neural network (NN)-based methods for the construction of interatomic potentials from discrete samples (such as ab initio energies) for applications in classical and quantum dynamics including reaction dynamics and computational spectroscopy. The main focus is on methods for building molecular potential energy surfaces (PES) in internal coordinates that explicitly include all many-body contributions, even though some of the methods we review limit the degree of coupling, due either to a desire to limit computational cost or to limited data. Explicit and direct treatment of all many-body contributions is only practical for sufficiently small molecules, which are therefore our primary focus. This includes small molecules on surfaces. We consider direct, single NN PES fitting as well as more complex methods that impose structure (such as a multibody representation) on the PES function, either through the architecture of one NN or by using multiple NNs. We show how NNs are effective in building representations with low-dimensional functions including dimensionality reduction. We consider NN-based approaches to build PESs in the sums-of-product form important for quantum dynamics, ways to treat symmetry, and issues related to sampling data distributions and the relation between PES errors and errors in observables. We highlight combinations of NNs with other ideas such as permutationally invariant polynomials or sums of environment-dependent atomic contributions, which have recently emerged as powerful tools for building highly accurate PESs for relatively large molecular and reactive systems.

Computational study of the rovibrational spectrum of CO2-N2.

The CO2-N2 complex is formed from two key components of Earth's atmosphere, and as such, has received some attention from both experimental and theoretical studies. On the theory side, a potential energy surface (PES) based on high level ab initio data was reported [Nasri et al., J. Chem. Phys., 2015, 142, 174301] and then used in more recently reported rovibrational calculations [Lara-Moreno et al., Phys. Chem. Chem. Phys., 2019, 21, 3550]. Accuracy of about 1 percent was achieved for calculated rotational transitions of the ground vibrational state of the complex, compared with previously reported microwave spectra. However, a very recent measurement of the geared bending mode frequency [Barclay et al., J. Chem. Phys., 2020, 153, 014303] recorded a value of 21.4 cm-1, which is wildly different from the corresponding calculated value of 45.9 cm-1. To provide some insight into this discrepancy, we have constructed a new more accurate PES, and used it to perform highly converged variational rovibrational calculations. Our new results yield a value of 21.1 cm-1 for that bending frequency, in close agreement with the experiment. We also obtain significantly improved predicted rotational transitions. Finally, we note that a very shallow well, previously reported as a distinct second isomer, is not found on our new PES, but rather a transition structure is seen in that location.

A variational calculation of vibrational levels of vinyl radical.

We report the vibrational energy levels of vinyl radical (VR) that are computed with a Lanczos eigensolver and a contracted basis. Many of the levels of the two previous VR variational calculations differ significantly and differ also from those reported in this paper. We identify the source of and correct symmetry errors on the potential energy surfaces used in the previous calculations. VR has two equivalent equilibrium structures. By plotting wavefunction cuts, we show that two tunneling paths play an important role. Using the computed wavefunctions, it is possible to assign many states and thereby to determine tunneling splittings that are compared with their experimental counterparts. Our computed red shift of the hot band at 2897.23 cm-1, observed by Dong et al. [J Chem. Phys. 128, 044305 (2008)], is 4.47 cm-1, which is close to the experimental value of 4.63 cm-1.

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.

Machine Learning Optimization of the Collocation Point Set for Solving the Kohn-Sham Equation.

The rectangular collocation approach makes it possible to solve the Schrödinger equation with basis functions that do not have amplitude in all regions in which wave functions have significant amplitude. Collocation points can be restricted to a small region of space. As no integrals are computed, there are no problems due to discontinuities in the potential, and there is no need to use integrable basis functions. In this paper, we show, for the Kohn-Sham equation, that machine learning can be used to drastically reduce the size of the collocation point set. This is demonstrated by solving the Kohn-Sham equations for CO and H2O. We solve the Kohn-Sham equation on a given effective potential which is a critical part of all DFT calculations, and monitor orbital energies and orbital shapes. We use a combination of Gaussian process regression and a genetic algorithm to reduce the collocation point set size by more than an order of magnitude (from about 51 000 points to 2000 points) while retaining mhartree accuracy.

Computational Study of the Rovibrational Spectra of CH2D+ and CHD2.

In this paper, we present rovibrational energy levels of CH2D+ and CHD2+. They are computed with a large basis and the Lanczos algorithm. CH2D+ and CHD2+ are believed to play an important role in interstellar space, but so far, there are no definitive observations. The predictions of this paper should facilitate detection. For CH2D+, two CH stretch bands have been studied at high resolution. Compared to our calculated energies, the root-mean-square error is 0.08 cm-1. For CHD2+, one CH stretch band has been studied at high resolution. Compared to our calculated energies, the root-mean-square error is 0.5 cm-1. Errors are larger, for both isotopologues, for bend states. We attribute these errors to the potential energy surface. Wave function and probability distribution plots are used to make assignments. The ν1 band of CHD2+ is significantly perturbed, and according to our calculations, the 3ν3 state is closest and might be the most important perturber.

Computational study of the ro-vibrational spectrum of CO-CO2.

An accurate ab initio ground-state intermolecular potential energy surface (PES) was determined for the CO-CO2 van der Waals dimer. The Lanczos algorithm was used to compute rovibrational energies on this PES. For both the C-in and O-in T-shaped isomers, the fundamental transition frequencies agree well with previous experimental results. We confirm that the in-plane states previously observed are geared states. In addition, we have computed and assigned many other vibrational states. The rotational constants we determine from J = 1 energy levels agree well with their experimental counterparts. Planar and out-of-plane cuts of some of the wavefunctions we compute are quite different, indicating strong coupling between the bend and torsional modes. Because the stable isomers are T-shaped, vibration along the out-of-plane coordinates is very floppy. In CO-CO2, when the molecule is out-of-plane, interconversion of the isomers is possible, but the barrier height is higher than the in-plane geared barrier height.

Using collocation and a hierarchical basis to solve the vibrational Schrödinger equation.

We show that it is possible to compute vibrational energy levels of polyatomic molecules with a collocation method and a basis of products of one-dimensional harmonic oscillator functions pruned so that it does not include functions for which the indices of many of the one-dimensional functions are nonzero. Functions with many nonzero indices are coupled only by terms that depend simultaneously on many coordinates, and they are typically small. The collocation equation is derived without invoking differences of interpolation operators, which simplifies implementation of the method. This, however, requires inverting a matrix whose elements are values of the pruned basis functions at the collocation points. The collocation points are the points on a Smolyak grid whose size is equal to the size of the pruned basis set. The Smolyak grid is built from symmetrized Leja points. Because both the basis and the grid are not tensor products, the inverse is not straightforward. It can be done by using so-called hierarchical 1-D basis functions. They are defined so that the matrix whose elements are the 1-D hierarchical basis functions evaluated at points is lower triangular. We test the method by applying it to compute 100 energy levels of CH2NH with an iterative eigensolver.

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.