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Canonical Tensors Applied to Ab Initio Electronic Structure: Linear Scaling for Metallic Systems

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Technical Report,01 Nov 2016,31 Jul 2017

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Texas Southern University Houston United States

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An ab initio electronic structure algorithm capable of achieving linear scaling for metallic systems, and utilizing hundreds of thousands of Cardinal Sine basis elements on dual core processor with 16Gb memory, is presented. This is achieved by the implementation of a dimensionally separated Canonical Tensor Representation CTR of the Hamiltonian Matrix, and associated Density Matrix. Rank reduction of the Canonical Tensor Representation, required for matrix multiplies, was first developed by G. Beylkin et. al. SIAM Journal on Scientific Computing 26 2005 2133. This algorithm allows us to represent the full six-dimensional tensor matrix as three separated rank two-dimensional tensors, where we have extended Beylkins method via the use of the Frobenius norm for the two-dimensional tensors. A significant advantage is achieved with the Cardinal Sine basis via the use of the Gaussian-Sinc integral technology, developed by J. Jerke et. al. J Chem Phys 143 2015 064108, which allows us to construct the Hamiltonian in a natural canonical tensor-product of rank-2 tensors. Byseparating space into three one- dimensional forms N3N, linear scaling is attained with dense matrix representations. The matrix-matrix multiply operations are now linear because the scaling of the matrix- matrix multiply is a cube of a cubic root of the number of basis elements and the rank of the canonical tensors are insensitive to the system size. Solving for a density matrix, or canonical density, is solved with aid of the Spectral Projection algorithm SP2 technologies developed by Niklasson et. al. Physical Review B 66 2002 155115. Efficient storage of density matrices into dimensionally separated primitives and extremely fast builds of the Fock matrices allow for novel Self Consistent Field SCF acceleration. The energy represented in the primitive basis is minimized linearly with exponential cost functions associated withidompotency. A dozen SCFs converge metallic systems

Subject Categories:

  • Numerical Mathematics
  • Electrical and Electronic Equipment

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