arXiv subjects

Tucker Carrington

Publications and source records attributed to Tucker Carrington.

4 recordsLinked to original sources

arXiv2021

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 5 atoms. One way to deal with this problem is to represent tensors using a tensor format. In this paper, we use 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 Alternating Least Squares method. There is no need for rank reduction, no need for orthogonalization, and tensors with 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).

physics.comp-ph↗

arXiv2019

Using quadrature and an iterative eigensolver to compute fine-structure ro-vibrational levels of Van der Waals complexes: NH($^3Σ^-$)-He, O$_2$($^3Σ^-_g$)-Ar and O$_2$($^3Σ^-_g$)-He

We introduce a new method for computing spectra of molecules for which a spin-spin term in the Hamiltonian has an important effect. In previous calculations, matrix elements of the spin-spin term and of the potential were obtained by expanding the potential and using analytic equations in terms of $ 3-j $ symbols. Instead, we use quadrature. Quadrature is simple and makes it possible to do calculations with a general potential and without using the Wigner-Eckart theorem. In previous calculations, the Hamiltonian matrix was built and diagonalized. Instead, we use an iterative eigensolver. It makes it easy to work with a large basis. The ideas are tested by computing energy levels of NH($^3Σ^-$)-He, O$_2$($^3Σ^-_g$)-Ar and O$_2$($^3Σ^-_g$)-He.

physics.chem-ph↗

arXiv2016

Comparison of different eigensolvers for calculating vibrational spectra using low-rank, sum-of-product basis functions

Vibrational spectra and wavefunctions of polyatomic molecules can be calculated at low memory cost using low-rank sum-of-product (SOP) decompositions to represent basis functions generated using an iterative eigensolver. Using a SOP tensor format does not determine the iterative eigensolver. The choice of the interative eigensolver is limited by the need to restrict the rank of the SOP basis functions at every stage of the calculation. We have adapted, implemented and compared different reduced-rank algorithms based on standard iterative methods (block-Davidson algorithm, Chebyshev iteration) to calculate vibrational energy levels and wavefunctions of the 12-dimensional acetonitrile molecule. The effect of using low-rank SOP basis functions on the different methods is analyzed and the numerical results are compared with those obtained with the reduced rank block power method introduced in J. Chem. Phys. 140, 174111 (2014). Relative merits of the different algorithms are presented, showing that the advantage of using a more sophisticated method, although mitigated by the use of reduced-rank sum-of-product functions, is noticeable in terms of CPU time.

physics.chem-ph↗

arXiv2014

Calculating vibrational spectra with sum of product basis functions without storing full-dimensional vectors or matrices

We propose an iterative method for computing vibrational spectra that significantly reduces the memory cost of calculations. It uses a direct product primitive basis, but does not require storing vectors with as many components as there are product basis functions. Wavefunctions are represented in a basis each of whose functions is a sum of products (SOP) and the factorizable structure of the Hamiltonian is exploited. If the factors of the SOP basis functions are properly chosen, wavefunctions are linear combinations of a small number of SOP basis functions. The SOP basis functions are generated using a shifted block power method. The factors are refined with a rank reduction algorithm to cap the number of terms in a SOP basis function. The ideas are tested on a 20-D model Hamiltonian and a realistic CH$_3$CN (12 dimensional) potential. For the 20-D problem, to use a standard direct product iterative approach one would need to store vectors with about $10^{20}$ components and would hence require about $8 \times 10^{11}$ GB. With the approach of this paper only 1 GB of memory is necessary. Results for CH$_3$CN agree well with those of a previous calculation on the same potential.

physics.comp-ph↗