An algorithm for large scale density matrix renormalization group calculations

Abstract

We describe in detail our high-performance density matrix renormalization group ͑DMRG͒ algorithm for solving the electronic Schrödinger equation. We illustrate the linear scalability of our algorithm with calculations on up to 64 processors. The use of massively parallel machines in conjunction with our algorithm considerably extends the range of applicability of the DMRG in quantum chemistry.

The ab-initio density matrix renormalization group in practice

“…This phenomenon is sometimes also referred to as having incorrect, or "losing" quantum numbers (i.e., not having the appropriate irrep labels for the states). 20 One way to detect an incorrect convergence to a local minimum is to increase M. A calculation with incorrect quantum numbers at small M will exhibit a very sudden lowering of the energy at larger M when the correct quantum number sector is finally recovered.…”

“…54,56 The entanglement structure of the MPS has even generated a niche in interpretative quantum chemistry. 52,[57][58][59][60] With the advent of publicly available quantum chemistry DMRG codes, 10,[20][21][22] the DMRG is transitioning into a method available not only to expert developers but also to the informed user. As with all complex methods, there is some degree of experience required to use it effectively.…”

“…Since then, many groups have independently implemented and improved on the ab-initio DMRG algorithm. Some of these improvements include parallelization, 8,20 nonAbelian symmetry and spin-adaptation, 7,[21][22][23] orbital ordering 5,[24][25][26] and optimization, 9,27-29 more sophisticated initial guesses, 5,24,25,30,31 better noise algorithms, 5,32 extrapolation procedures, 5,33,34 response theories, 35,36 as well as the combination of the DMRG with various other quantum chemistry methods such as perturbation theory, 37 canonical transformations, 38 configuration interaction, 39 and relativistic Hamiltonians. 40 In the ecosystem of quantum chemistry, the DMRG occupies a unique spot.…”

Hilbert space renormalization for the many-electron problem

“…57 The more accurate calculations. For a fair comparison with the HS-MPS results, the Fock-space DMRG calculations were performed with the B code 17,18,48 using only S z symmetry, unless otherwise stated.…”

“…The most widely used example of this latter class is the density matrix renormalization group (DMRG) method, [11][12][13][14] which employs the simple one-dimensional TNS, viz., the matrix product state (MPS). 15 DMRG has been successfully applied in quantum chemistry to compute near-exact many-electron wavefunctions of several systems with a very large number of valence quantum degrees of freedom, 10,[16][17][18][19][20][21][22][23] such as the oxygen-evolving complex (Mn 4 CaO 5 ) 24 and the iron-sulfur clusters. 25 Successfully combining these different representations for static and dynamic correlation is a nontrivial problem, and is an important research challenge.…”