Tensor Decomposition and Vibrational Coupled Cluster Theory

Abstract

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.

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

“…This means that the approximate ranks of amplitude tensors describing weak and spatially separated interactions are small when the tensors are fitted to an absolute threshold as also shown numerically in Ref. 24. In the CP-VCC algorithm, the computational effort is thus adapted dynamically to the strength of the physical interactions between the vibrational modes.…”

“…The VCC amplitudes and error vectors can be viewed as a set of individual tensors, i.e., t = {t m } and e = {e m }, where t m is the amplitude tensor of the MC m. [24][25][26] The VCC wave function is thus parametrized in terms of many small tensors instead of one large tensor of coefficients for each state as used in other approaches. 46 The order or dimensionality of the tensor is equal to the number of modes in the MC, and the cost of the VCC tensor contractions and direct products scales very steeply with respect to the order of the amplitude tensors.…”

“…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.…”

“…it has recently gained interest in the field of quantum chemistry. [38][39][40][41][42][43][44][45] It has been applied to VCC calculations 24,25 and later to other types of vibrational-structure calculations [46][47][48] as well as for fitting PESs. 49,50 In our previous work of Refs.…”

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

“…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.…”