Gaussian process regression adaptive density-guided approach: Toward calculations of potential energy surfaces for larger molecules.

We present a new program implementation of the Gaussian process regression adaptive density-guided approach [Schmitz et al., J. Chem. Phys. 153, 064105 (2020)] for automatic and cost-efficient potential energy surface construction in the MidasCpp program. A number of technical and methodological improvements made allowed us to extend this approach toward calculations of larger molecular systems than those previously accessible and maintain the very high accuracy of constructed potential energy surfaces. On the methodological side, improvements were made by using a Δ-learning approach, predicting the difference against a fully harmonic potential, and employing a computationally more efficient hyperparameter optimization procedure. We demonstrate the performance of this method on a test set of molecules of growing size and show that up to 80% of single point calculations could be avoided, introducing a root mean square deviation in fundamental excitations of about 3 cm-1. A much higher accuracy with errors below 1 cm-1 could be achieved with tighter convergence thresholds still reducing the number of single point computations by up to 68%. We further support our findings with a detailed analysis of wall times measured while employing different electronic structure methods. Our results demonstrate that GPR-ADGA is an effective tool, which could be applied for cost-efficient calculations of potential energy surfaces suitable for highly accurate vibrational spectra simulations.

Tensor-decomposed vibrational coupled-cluster theory: Enabling large-scale, highly accurate vibrational-structure calculations.

A new implementation of vibrational coupled-cluster (VCC) theory is presented, where all amplitude tensors are represented in the canonical polyadic (CP) format. The CP-VCC algorithm solves the non-linear VCC equations without ever constructing the amplitudes or error vectors in full dimension but still formally includes the full parameter space of the VCC[n] model in question resulting in the same vibrational energies as the conventional method. In a previous publication, we have described the non-linear-equation solver for CP-VCC calculations. In this work, we discuss the general algorithm for evaluating VCC error vectors in CP format including the rank-reduction methods used during the summation of the many terms in the VCC amplitude equations. Benchmark calculations for studying the computational scaling and memory usage of the CP-VCC algorithm are performed on a set of molecules including thiadiazole and an array of polycyclic aromatic hydrocarbons. The results show that the reduced scaling and memory requirements of the CP-VCC algorithm allows for performing high-order VCC calculations on systems with up to 66 vibrational modes (anthracene), which indeed are not possible using the conventional VCC method. This paves the way for obtaining highly accurate vibrational spectra and properties of larger molecules.

Efficient algorithms for solving the non-linear vibrational coupled-cluster equations using full and decomposed tensors.

Vibrational coupled-cluster (VCC) theory provides an accurate method for calculating vibrational spectra and properties of small to medium-sized molecules. Obtaining these properties requires the solution of the non-linear VCC equations which can in some cases be hard to converge depending on the molecule, the basis set, and the vibrational state in question. We present and compare a range of different algorithms for solving the VCC equations ranging from a full Newton-Raphson method to approximate quasi-Newton models using an array of different convergence-acceleration schemes. The convergence properties and computational cost of the algorithms are compared for the optimization of VCC states. This includes both simple ground-state problems and difficult excited states with strong non-linearities. Furthermore, the effects of using tensor-decomposed solution vectors and residuals are investigated and discussed. The results show that for standard ground-state calculations, the conjugate residual with optimal trial vectors algorithm has the shortest time-to-solution although the full Newton-Raphson method converges in fewer macro-iterations. Using decomposed tensors does not affect the observed convergence rates in our test calculations as long as the tensors are decomposed to sufficient accuracy.

FALCON: A method for flexible adaptation of local coordinates of nuclei.

We present a flexible scheme for calculating vibrational rectilinear coordinates with well-defined strict locality on a certain set of atoms. Introducing a method for Flexible Adaption of Local COordinates of Nuclei (FALCON) we show how vibrational subspaces can be "grown" in an adaptive manner. Subspace Hessian matrices are set up and used to calculate and analyze vibrational modes and frequencies. FALCON coordinates can more generally be used to construct vibrational coordinates for describing local and (semi-local) interacting modes with desired features. For instance, spatially local vibrations can be approximately described as internal motion within only a group of atoms and delocalized modes can be approximately expressed as relative motions of rigid groups of atoms. The FALCON method can support efficiency in the calculation and analysis of vibrational coordinates and energies in the context of harmonic and anharmonic calculations. The features of this method are demonstrated on a few small molecules, i.e., formylglycine, coumarin, and dimethylether as well as for the amide-I band and low-frequency modes of alanine oligomers and alpha conotoxin.

Calculating vibrational spectra without determining excited eigenstates: Solving the complex linear equations of damped response theory for vibrational configuration interaction and vibrational coupled cluster states.

It is demonstrated how vibrational IR and Raman spectra can be calculated from damped response functions using anharmonic vibrational wave function calculations, without determining the potentially very many eigenstates of the system. We present an implementation for vibrational configuration interaction and vibrational coupled cluster, and describe how the complex equations can be solved using iterative techniques employing only real trial vectors and real matrix-vector transformations. Using this algorithm, arbitrary frequency intervals can be scanned independent of the number of excited states. Sample calculations are presented for the IR-spectrum of water, Raman spectra of pyridine and a pyridine-silver complex, as well as for the infra-red spectrum of oxazole, and vibrational corrections to the polarizability of formaldehyde.

Tensor decomposition techniques in the solution of vibrational coupled cluster response theory eigenvalue equations.

We show how the eigenvalue equations of vibrational coupled cluster response theory can be solved using a subspace projection method with Davidson update, where basis vectors are stacked tensors decomposed into canonical (CP, Candecomp/Parafac) form. In each update step, new vectors are first orthogonalized to old vectors, followed by a tensor decomposition to a prescribed threshold TCP. The algorithm can provide excitation energies and eigenvectors of similar accuracy as a full vector approach and with only a very modest increase in the number of vectors required for convergence. The algorithm is illustrated with sample calculations for formaldehyde, 1,2,5-thiadiazole, and water. Analysis of the formaldehyde and thiadiazole calculations illustrate a number of interesting features of the algorithm. For example, the tensor decomposition threshold is optimally put to rather loose values, such as TCP = 10(-2). With such thresholds for the tensor decompositions, the original eigenvalue equations can still be solved accurately. It is thus possible to directly calculate vibrational wave functions in tensor decomposed format.

Tensor decomposition and vibrational coupled cluster theory.

The use of tensor decomposition in the calculation of anharmonic vibrational wave functions is discussed. The correlation amplitudes of vibrational coupled cluster (VCC) and vibrational configuration interaction (VCI) theories are considered as tensors and decomposed. A pilot code is implemented allowing a numerical study of the performance of the canonical decomposition/parallel factors (CP) for three and higher mode couplings in computations on water, formaldehyde, and 1,2,5-thiadiazole. The results show that there is a significant perspective in applying tensor decomposition in the context of anharmonic vibrational wave functions, with the CP tensor decomposition providing compression of data and a computational convenient representation. The calculations also illustrate how the multiplicative separability of the VCC ansatz with respect to noninteracting degrees of freedom goes well together with a tensor decomposition approach. Tensor decomposition opens for adjusting the computational effort spent on a particular mode-coupling according to the significance of that particular coupling, which is guaranteed to decrease to zero in the case of VCC in the limit of noninteracting subsystems.

A band Lanczos approach for calculation of vibrational coupled cluster response functions: simultaneous calculation of IR and Raman anharmonic spectra for the complex of pyridine and a silver cation.

We describe new methods for the calculation of IR and Raman spectra using vibrational response theory. Using damped linear response functions that incorporate a Lorentzian line-shape function from the outset, it is shown how the calculation of Raman spectra can be carried out through the calculation of a set of vibrational response functions in the same manner as described previously for IR spectra. The necessary set of response functions can be calculated for both vibrational coupled cluster (VCC) and vibrational configuration interaction (VCI) anharmonic vibrational wave-functions. For the efficient and simultaneous calculation of the full set of necessary response functions, a non-hermitian band Lanczos algorithm is implemented for VCC, and a hermitian band Lanczos algorithm is implemented for VCI. It is shown that the simultaneous calculation of several response functions is often advantageous. Sample calculations are presented for pyridine and the complex between pyridine and the silver cation.