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.

[1]  Boris N. Khoromskij,et al.  Tensor Numerical Methods in Quantum Chemistry , 2018 .

[2]  Tucker Carrington,et al.  An intertwined method for making low-rank, sum-of-product basis functions that makes it possible to compute vibrational spectra of molecules with more than 10 atoms. , 2017, The Journal of chemical physics.

[3]  N. Madsen,et al.  Atomic-batched tensor decomposed two-electron repulsion integrals. , 2017, The Journal of chemical physics.

[4]  N. Madsen,et al.  Efficient algorithms for solving the non-linear vibrational coupled-cluster equations using full and decomposed tensors. , 2017, The Journal of chemical physics.

[5]  H. Najm,et al.  Low-rank canonical-tensor decomposition of potential energy surfaces: application to grid-based diagrammatic vibrational Green's function theory , 2017 .

[6]  Reinhold Schneider,et al.  Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations , 2016, Foundations of Computational Mathematics.

[7]  G. Rauhut,et al.  Tensor decomposition in potential energy surface representations. , 2016, The Journal of chemical physics.

[8]  Mike Espig,et al.  Tensor representation techniques for full configuration interaction: A Fock space approach using the canonical product format. , 2016, The Journal of chemical physics.

[9]  O. Christiansen,et al.  Linear-scaling generation of potential energy surfaces using a double incremental expansion , 2016, 1605.01674.

[10]  O. Christiansen,et al.  FALCON: A method for flexible adaptation of local coordinates of nuclei. , 2016, The Journal of chemical physics.

[11]  Gregory Beylkin,et al.  Randomized Alternating Least Squares for Canonical Tensor Decompositions: Application to A PDE With Random Data , 2015, SIAM J. Sci. Comput..

[12]  Tucker Carrington,et al.  Using Nested Contractions and a Hierarchical Tensor Format To Compute Vibrational Spectra of Molecules with Seven Atoms. , 2015, The journal of physical chemistry. A.

[13]  Emil Lund Klinting,et al.  Hybrid Optimized and Localized Vibrational Coordinates. , 2015, The journal of physical chemistry. A.

[14]  S. Hirata,et al.  Higher-order diagrammatic vibrational coupled-cluster theory. , 2015, The Journal of chemical physics.

[15]  O. Christiansen,et al.  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. , 2015, The Journal of chemical physics.

[16]  O. Christiansen,et al.  Automatic determination of important mode-mode correlations in many-mode vibrational wave functions. , 2015, The Journal of chemical physics.

[17]  P. Comon,et al.  A regularized nonnegative canonical polyadic decomposition algorithm with preprocessing for 3D fluorescence spectroscopy , 2015 .

[18]  S. Pal,et al.  Study of molecular vibration by coupled cluster method: Bosonic approach , 2015 .

[19]  O. Christiansen,et al.  Tensor decomposition techniques in the solution of vibrational coupled cluster response theory eigenvalue equations. , 2015, The Journal of chemical physics.

[20]  C. Pouchan,et al.  Toward linear scaling: Locality of potential energy surface coupling in valence coordinates , 2014 .

[21]  T. Carrington,et al.  Calculating vibrational spectra with sum of product basis functions without storing full-dimensional vectors or matrices. , 2014, The Journal of chemical physics.

[22]  A. Auer,et al.  Tensor decomposition in post-Hartree-Fock methods. II. CCD implementation. , 2013, The Journal of chemical physics.

[23]  O. Christiansen,et al.  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. , 2013, Physical chemistry chemical physics : PCCP.

[24]  O. Christiansen,et al.  Tensor decomposition and vibrational coupled cluster theory. , 2013, The journal of physical chemistry. A.

[25]  Robert M Parrish,et al.  Tensor hypercontraction. II. Least-squares renormalization. , 2012, The Journal of chemical physics.

[26]  Reinhold Schneider,et al.  Variational calculus with sums of elementary tensors of fixed rank , 2012, Numerische Mathematik.

[27]  Wolfgang Hackbusch,et al.  A regularized Newton method for the efficient approximation of tensors represented in the canonical tensor format , 2012, Numerische Mathematik.

[28]  Edward F. Valeev,et al.  Computing many-body wave functions with guaranteed precision: the first-order Møller-Plesset wave function for the ground state of helium atom. , 2012, The Journal of chemical physics.

[29]  Robert M Parrish,et al.  Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order Møller-Plesset perturbation theory. , 2012, The Journal of chemical physics.

[30]  O. Christiansen,et al.  Approximate inclusion of four-mode couplings in vibrational coupled-cluster theory. , 2012, The Journal of chemical physics.

[31]  Ove Christiansen,et al.  Vibrational absorption spectra from vibrational coupled cluster damped linear response functions calculated using an asymmetric Lanczos algorithm. , 2012, The Journal of chemical physics.

[32]  Edward F. Valeev,et al.  Low-order tensor approximations for electronic wave functions: Hartree-Fock method with guaranteed precision. , 2011, The Journal of chemical physics.

[33]  Daniel M. Dunlavy,et al.  A scalable optimization approach for fitting canonical tensor decompositions , 2011 .

[34]  Wolfgang Hackbusch,et al.  Tensor Decomposition in post-Hartree Fock Methods. , 2010 .

[35]  Ove Christiansen,et al.  New Formulation and Implementation of Vibrational Self-Consistent Field Theory. , 2010, Journal of chemical theory and computation.

[36]  Ove Christiansen,et al.  Automatic derivation and evaluation of vibrational coupled cluster theory equations. , 2009, The Journal of chemical physics.

[37]  W. Hackbusch,et al.  Black Box Low Tensor-Rank Approximation Using Fiber-Crosses , 2009 .

[38]  Guntram Rauhut,et al.  Toward large scale vibrational configuration interaction calculations. , 2009, The Journal of chemical physics.

[39]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[40]  J. Olsen,et al.  An efficient algorithm for solving nonlinear equations with a minimal number of trial vectors: applications to atomic-orbital based coupled-cluster theory. , 2008, The Journal of chemical physics.

[41]  Ove Christiansen,et al.  Towards fast computations of correlated vibrational wave functions: vibrational coupled cluster response excitation energies at the two-mode coupling level. , 2008, The Journal of chemical physics.

[42]  Guntram Rauhut,et al.  Configuration selection as a route towards efficient vibrational configuration interaction calculations. , 2007, The Journal of chemical physics.

[43]  Martin J. Mohlenkamp,et al.  Algorithms for Numerical Analysis in High Dimensions , 2005, SIAM J. Sci. Comput..

[44]  William W. Hager,et al.  A New Conjugate Gradient Method with Guaranteed Descent and an Efficient Line Search , 2005, SIAM J. Optim..

[45]  Ove Christiansen,et al.  A second quantization formulation of multimode dynamics. , 2004, The Journal of chemical physics.

[46]  Ove Christiansen,et al.  Vibrational coupled cluster theory. , 2004, The Journal of chemical physics.

[47]  Ove Christiansen,et al.  Møller–Plesset perturbation theory for vibrational wave functions , 2003 .

[48]  R. Bro,et al.  Practical aspects of PARAFAC modeling of fluorescence excitation‐emission data , 2003 .

[49]  Martin J. Mohlenkamp,et al.  Numerical operator calculus in higher dimensions , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[50]  Trygve Helgaker,et al.  Molecular Electronic-Structure Theory: Helgaker/Molecular Electronic-Structure Theory , 2000 .

[51]  M. Beck,et al.  The multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propa , 1999 .

[52]  Joel M. Bowman,et al.  Vibrational self-consistent field method for many-mode systems: A new approach and application to the vibrations of CO adsorbed on Cu(100) , 1997 .

[53]  M. Ratner,et al.  MOLLER-PLESSET PERTURBATION THEORY APPLIED TO VIBRATIONAL PROBLEMS , 1996 .

[54]  R. Benny Gerber,et al.  Vibrational wave functions and spectroscopy of (H2O)n, n=2,3,4,5: Vibrational self‐consistent field with correlation corrections , 1996 .

[55]  David J. Thuente,et al.  Line search algorithms with guaranteed sufficient decrease , 1994, TOMS.

[56]  Johan Håstad,et al.  Tensor Rank is NP-Complete , 1989, ICALP.

[57]  Joel M. Bowman,et al.  The self-consistent-field approach to polyatomic vibrations , 1986 .

[58]  Joel M. Bowman,et al.  Self‐consistent field energies and wavefunctions for coupled oscillators , 1978 .

[59]  J. Chang,et al.  Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition , 1970 .

[60]  J. Watson.  Simplification of the molecular vibration-rotation hamiltonian , 2002 .

[61]  F. L. Hitchcock.  Multiple Invariants and Generalized Rank of a P‐Way Matrix or Tensor , 1928 .

[62]  F. L. Hitchcock.  The Expression of a Tensor or a Polyadic as a Sum of Products , 1927 .

[63]  W. Heisenberg,et al.  Zur Quantentheorie der Molekeln , 1924 .