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

Abstract

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.

Dual vibration configuration interaction (DVCI). An efficient factorization of molecular Hamiltonian for high performance infrared spectrum computation

“…The PES was constructed using the adaptive density-guided approach (ADGA) introduced by Sparta et al [69,73,74]. The force constants, equilibrium geometry and normal coordinates where extracted from Madsen et al [75]. In their work they describe the construction of oxazole PES at CCSD(T)/cc-pVTZ level for the one-mode part and MP2/cc-pVTZ for the two-mode part.…”

“…‚ If ą 0 and even then H " H vib pqq`H CC pqq (3). The equilibrium geometry in Bohr, atomic masses in electron rest mass, and normal coordinate eigenvectors should be indicated in the section COORDINATES like in the following example where the values has been extracted from Madsen & al [75]. In accordance with the Wilson method [22], the normal coordinates are built from a set of 3N A´6 eigenvectors pQ iα q iPt1,...,N A u, αPt1,2,3u of the Hessian matrix…”

“…Vibration configuration interaction (VCI) method [1,2,12] permits a better precision with a much higher computational expense coming from the size of the variational space that is tenfold with the number of atoms of the molecule. To avoid this bottleneck, contraction techniques have been proposed with MCTDH [13,14], followed by the Alternating Least Square (ALS) procedure [15] formulated for wave function representation with the Vibrational Coupled Cluster (VCC) theory [18,75]. In the category of variational methods using perturbation criteria, the A k decomposition [3] originally designed for electronic structure calculation and introduced in the VCI context with the Vibrational Multi Reference Configuration Interaction (VMRCI) [19], has been able to identify relevant sub-blocks of the Hamiltonian matrix thereby reducing the size of the system.…”

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

“…Earlier work from our group emphasized the potential of tensor decomposition with respect to representing the same VCC or VCI wave function with fewer parameters. 24,25 In order to lower the scaling and reduce the memory requirements of the VCC algorithms, we have in a recent publication 26 introduced a new non-linear-equation solver, which solves the VCC equations with all tensors decomposed to the canonical polyadic (CP) [27][28][29] tensor format. In this work, we present the general algorithm for calculating the VCC error vector required in the equation solver in CP format without constructing any tensors in full dimension at any stage of the calculation, and we benchmark the scaling behavior and memory requirements of the CP-VCC algorithm.…”

“…We stress that the goal of the CP-VCC algorithm is to deliver results identical to the ones obtained from conventional, full-tensor VCC calculations within the numerical accuracy of the latter. Error control is essential during the CP-VCC calculations in order to obtain correct results and also for solving the non-linear equations, 26 and we describe how accurate results are obtained with CP decomposition. The performance of the CP-VCC algorithm depends strongly on the choice of the CP-decomposition algorithm and the method for choosing the rank of the representation.…”