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Calculating Vibrational Spectra with Symmetrized SumofProduct Basis Functions Leclerc, Arnaud
Description
The memory cost of representing vibrational wavefunctions of polyatomic molecules with more than 6 atoms is huge if direct product basis sets are used. However the $n^D$ numbers which represent a wavefunction ($n$ being the number of 1D basis functions for each of the $D$ coordinates) can often be deduced from a much smaller set of coefficients by using approximate, lowrank tensor decompositions. One possible choice is the CP format [1], i.e. a sumofproducts of 1D functions whose memory cost scales as $\mathcal{O}(nD)$. In the reducedrank block power method (RRBPM) introduced in [2], the rank of the SOP basis functions is reduced by iteratively optimizing the 1D functions from which they are made. The optimized SOP basis functions span a subspace close to that of the desired eigenfunctions. Reduction is done with alternating least square algorithm of ref. [3]. Only 1D operations are required and there is no need to store multidimensional vectors in memory. With this method one can compute vibrational eigenstates of systems up to $D=30$ using a few GB of memory. The RRBPM can be improved in several ways, one of which is the use of molecular symmetry [4]. Symmetry properties can be exploited in the RRBPM by using symmetryconstrained sumofproducts without jeopardizing the memory advantage of the algorithm. Since the Hamiltonian is totally symmetric, it is possible to compute states of a particular symmetry by using start vectors each of which has the same symmetry. This makes symmetry assignments easier and improves the accuracy in comparison with symmetryfree calculations. The ideas are illustrated by applying them to the acetonitrile molecule $CH_3CN$ (a 12D problem). [1] T. G. Kolda and B. W. Bader, SIAM Review {\bf 51} 455 (2009). [2] A. Leclerc and T. Carrington, J. Chem. Phys. {\bf 140} 174111 (2014). [3] G. Beylkin and M. J. Mohlenkamp, SIAM J. Sci. Comput. {\bf 26} 2133 (2005). [4] A. Leclerc and T. Carrington, submitted to Chem. Phys. Lett.
Item Metadata
Title 
Calculating Vibrational Spectra with Symmetrized SumofProduct Basis Functions

Creator  
Publisher 
Banff International Research Station for Mathematical Innovation and Discovery

Date Issued 
20160128T15:14

Description 
The memory cost of representing vibrational wavefunctions of polyatomic molecules with more than 6 atoms is huge if direct product basis sets are used. However the $n^D$ numbers which represent a wavefunction ($n$ being the number of 1D basis functions for each of the $D$ coordinates) can often be deduced from a much smaller set of coefficients by using approximate, lowrank tensor decompositions. One possible choice is the CP format [1], i.e. a sumofproducts of 1D functions whose memory cost scales as $\mathcal{O}(nD)$. In the reducedrank block power method (RRBPM) introduced in [2], the rank of the SOP basis functions is reduced by iteratively optimizing the 1D functions from which they are made. The optimized SOP basis functions span a subspace close to that of the desired eigenfunctions.
Reduction is done with alternating least square algorithm of ref. [3]. Only 1D operations are required and there is no need to store multidimensional vectors in memory. With this method one can compute vibrational eigenstates of systems up to $D=30$ using a few GB of memory.
The RRBPM can be improved in several ways, one of which is the use of molecular symmetry [4].
Symmetry properties can be exploited in the RRBPM by using symmetryconstrained sumofproducts without jeopardizing the memory advantage of the algorithm. Since the Hamiltonian is totally symmetric, it is possible to compute states of a particular symmetry by using start vectors each of which has the same symmetry. This makes symmetry assignments easier and improves the accuracy in comparison with symmetryfree calculations. The ideas are illustrated by applying them to the acetonitrile molecule $CH_3CN$ (a 12D problem).
[1] T. G. Kolda and B. W. Bader, SIAM Review {\bf 51} 455 (2009).
[2] A. Leclerc and T. Carrington, J. Chem. Phys. {\bf 140} 174111 (2014).
[3] G. Beylkin and M. J. Mohlenkamp, SIAM J. Sci. Comput. {\bf 26} 2133 (2005).
[4] A. Leclerc and T. Carrington, submitted to Chem. Phys. Lett.

Extent 
31 minutes

Subject  
Type  
File Format 
video/mp4

Language 
eng

Notes 
Author affiliation: Université de Lorraine

Series  
Date Available 
20160729

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0307236

URI  
Affiliation  
Peer Review Status 
Unreviewed

Scholarly Level 
Faculty

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Aggregated Source Repository 
DSpace

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Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International