In this proof-of-principle study, we apply tensor decomposition techniques to the Full Configuration Interaction (FCI) wavefunction in order to approximate the wavefunction parameters efficiently and to reduce the overall computational effort. For this purpose, the wavefunction ansatz is formulated in an occupation number vector representation that ensures antisymmetry. If the canonical product format tensor decomposition is then applied, the Hamiltonian and the wavefunction can be cast into a multilinear product form. As a consequence, the number of wavefunction parameters does not scale to the power of the number of particles (or orbitals) but depends on the rank of the approximation and linearly on the number of particles. The degree of approximation can be controlled by a single threshold for the rank reduction procedure required in the algorithm. We demonstrate that using this approximation, the FCI Hamiltonian matrix can be stored with N5 scaling. The error of the approximation that is introduced is below Millihartree for a threshold of ϵ = 10−4 and no convergence problems are observed solving the FCI equations iteratively in the new format. While promising conceptually, all effort of the algorithm is shifted to the required rank reduction procedure after the contraction of the Hamiltonian with the coefficient tensor. At the current state, this crucial step is the bottleneck of our approach and even for an optimistic estimate, the algorithm scales beyond N10 and future work has to be directed towards reduction-free algorithms.
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Especially in high dimensions it is a good advise to balance the representation system of every tensor which is stored in real computer implementations. Otherwise, one cannot avoid a number overflow in some entries of the representation system. For example consider the rank-one tensor with (vμ)l = 1 for all l ∈ {1, …, t}. If one introduces the following constrains for the representation system: ‖vμ‖ = 1 for all μ ∈ {2, …, d}, we have ‖v1‖ = tn. If n and t are large enough, we will produce a number overflow on every computer system. With (55) we have a balanced and solid representation system. Furthermore, with (55) we avoid unnecessary scaling influences in our objective function and consequently their second derivative with respect to the representation system.
Calculation done on one node with an Intel® Xenon® CPU X5650 @ 2,67 GHz.
The geometries are H2 rH–H = 0.7425 Å, LiH rLi–H = 1.5950 Å, H2OrO–H = 0.9572 Å, θHOH = 104.52°, H4 square geometry rH–H = 1.0584 Å, BeH2 rBe–H = 1.2905 Å θHBeH = 180.00°, HF rF–H = 0.9327 Å. Intermolecular distance is always 3.0 Å.
The geometries are LiH rLi–H = 1.5983 Å, BeH rBe–H = 1.3601 Å, BH rB–H = 1.2372 Å, CH rC–H = 1.1301 Å, NH rN–H = 1.0447 Å, OH rO–H = 0.9821 Å, FH rF–H = 0.9285 Å, H2 rH–H = 0.7425 Å, H2O rO–H = 0.9572 Å, θHOH = 104.52°, H4 square geometry rH–H = 1.0584 Å, BeH2 rBe–H = 1.2905 Å θHBeH = 180.00°. Intermolecular distance is always 3.0 Å.
