We propose a new algorithm for calculation of vibrational spectra of molecules using tensor train decomposition. Under the assumption that eigenfunctions lie on a low-parametric manifold of low-rank tensors we suggest using well-known iterative methods that utilize matrix inversion (locally optimal block preconditioned conjugate gradient method, inverse iteration) and solve corresponding linear systems inexactly along this manifold. As an application, we accurately compute vibrational spectra (84 states) of acetonitrile molecule CH3CN on a laptop in one hour using only 100 MB of memory to represent all computed eigenfunctions.
@cite_4: In this paper we propose and test a method for computing numerically exact vibrational energy levels of a molecule with six atoms. We use a pruned product basis, a non-product quadrature, the Lanczos algorithm, and the exact normal-coordinate kinetic energy operator (KEO) with the πtμπ term. The Lanczos algorithm is applied to a Hamiltonian with a KEO for which μ is evaluated at equilibrium. Eigenvalues and eigenvectors obtained from this calculation are used as a basis to obtain the final energy levels. The quadrature scheme is designed, so that integrals for the most important terms in the potential will be exact. The procedure is tested on C2H4. All 12 coordinates are treated explicitly. We need only ∼1.52 × 108 quadrature points. A product Gauss grid with which one could calculate the same energy levels has at least 5.67 × 1013 points.
@cite_19: Theoretical calculation of rotation-vibration energy levels of polyatomic molecules is a topic with a long history, characterized by a close, symbiotic relationship with molecular spectroscopy on one side and quantum chem­ istry on the other. What brings them together is the notion of the potential energy surface, which plays a central role in our understanding of the molecular structure and dynamics. In the case of polyatomic molecules, the experimental spectra cannot be inverted directly to yield potential surfaces [see Ref. (1) for some recent efforts within the semiclassical SCF approach], but they do provide a stringent test for the theoretically obtained potential surfaces and observables derived from them. These surfaces, usually from ab initio calculations, seldom meet the standards of spectroscopic accuracy, especially if more extended, high-energy regions are of interest. The only practical way available to test and improve them is by comparing the calculated and the experimental spectra, and minimizing the difference between the two. The subject of the theoretical treatment of coupled molecular vibrations has undergone a real renaissance in the past decade. Significant conceptual advances have been made, particularly concerning highly vibrationally and
@cite_5: In this paper we propose a scheme for choosing basis functions for quantum dynamics calculations. Direct product bases are frequently used. The number of direct product functions required to converge a spectrum, compute a rate constant, etc., is so large that direct product calculations are impossible for molecules or reacting systems with more than four atoms. It is common to extract a smaller working basis from a huge direct product basis by removing some of the product functions. We advocate a build and prune strategy of this type. The one-dimensional (1D) functions from which we build the direct product basis are chosen to satisfy two conditions: (1) they nearly diagonalize the full Hamiltonian matrix; (2) they minimize off-diagonal matrix elements that couple basis functions with diagonal elements close to those of the energy levels we wish to compute. By imposing these conditions we increase the number of product functions that can be removed from the multidimensional basis without degrading the acc...
@cite_15: We present a new methodology to perform fast correlation-corrected vibrational self-consistent field (CC-VSCF) calculations using ab initio potential energy points calculated on the fly. Our method is based on the replacement of all-electron basis sets with a pseudo-potential basis for heavy atoms, and on an efficient reduction of the number of pair-coupling elements used in the CC-VSCF procedure. The method is applied to several test systems: H2O, NH3, and CH4, where it proves to be efficient, providing a speedup factor of 2 compared to a standard CC-VSCF calculation. We also apply our technique to the simulation of the vibrational spectrum of ethane and show that very accurate results can be obtained with a substantial speedup for this system.
@cite_10: We describe a new method to truncate and recouple basis functions in general variational calculations based on a direct‐product representation of multidimensional wave functions. The method is presented for molecular vibrations; however, the procedure is quite general and can be used in any basis set expansion method. The direct‐product Hamiltonian matrix H is decomposed into a block diagonal matrix H0 plus a remainder H1 . A new subset of basis functions is obtained by diagonalizing H0 . This subset of basis functions is shown to be eigenfunctions of a Hamiltonian in a reduced dimensionality space, ‘‘dressed’’ by the remaining degrees of freedom. These dressed eigenfunctions are then augmented by the component of the original direct‐product basis in which H0 is diagonal. The new basis is recoupled using an energy selection criterion, yielding a substantial reduction in the size of the final full Hamiltonian matrix. The method also suggests a generalization of the vibrational self‐consistent field method,...
The simplest basis set for representing unknown eigenfunctions is the direct product (DP) of one-dimensional basis functions. If a fast matrix-by-vector operation is given, then Krylov iterative methods are available and the only problem is the exponential storage requirements @math . Alternatively one can prune a direct product basis @cite_4 @cite_15 @cite_5 or use a basis that is a product of functions with more than one variable @cite_10 @cite_19 .