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Chan, Garnet Kin‐Lic; Ayers, Paul W.; Croot, Ernest S.

doi:10.1023/a:1019999930923

Cited by 15 publications

(16 citation statements)

References 12 publications

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“…The precise analytic form of the DMRG energy convergence as a function of M has been a matter of debate in the literature. 26,29,36,53,54 We have previously found good agreement with the proposed form of Okunishi et al,…”

Section: Sweep and Error Convergencesupporting

confidence: 86%

Multireference correlation in long molecules with the quadratic scaling density matrix renormalization group

Hachmann¹,

Cardoen²,

Chan³

2006

The Journal of Chemical Physics

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200

288

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We have devised a local ab initio density matrix renormalization group algorithm to describe multireference correlations in large systems. For long molecules that are extended in one of their spatial dimensions, we can obtain an exact characterization of correlation, in the given basis, with a cost that scales only quadratically with the size of the system. The reduced scaling is achieved solely through integral screening and without the construction of correlation domains. We demonstrate the scaling, convergence, and robustness of the algorithm in polyenes and hydrogen chains. We converge to exact correlation energies ͑in the sense of full configuration interaction, with 1-10 E h precision͒ in all cases and correlate up to 100 electrons in 100 active orbitals. We further use our algorithm to obtain exact energies for the metal-insulator transition in hydrogen chains and compare and contrast our results with those from conventional quantum chemical methods.

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Section: Sweep and Error Convergencesupporting

confidence: 86%

Multireference correlation in long molecules with the quadratic scaling density matrix renormalization group

Hachmann¹,

Cardoen²,

Chan³

2006

The Journal of Chemical Physics

Self Cite

200

288

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“…We have recently proved, 20 that under fairly general conditions ͑see also Okunishi et al for a simpler discussion specific to the Ising model 21 ͒, the thermal density matrix eigenvalues decay asymptotically like…”

Section: Convergence Of the Energymentioning

confidence: 87%

“…͑28͒ to the one-particle level spacing, as described in Ref. 20. In interacting systems, the dependence of the correlation length on the effective range of the two-particle interaction is complicated.…”

Section: ͑29͒mentioning

confidence: 99%

Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization group

Chan¹,

Head‐Gordon²

2002

The Journal of Chemical Physics

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516

656

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We study the recently developed Density Matrix Renormalization Group ͑DMRG͒ algorithm in the context of quantum chemistry. In contrast to traditional approaches, this algorithm is believed to yield arbitrarily high accuracy in the energy with only polynomial computational effort. We describe in some detail how this is achieved. We begin by introducing the principles of the renormalization procedure, and how one formulates an algorithm for use in quantum chemistry. The renormalization group algorithm is then interpreted in terms of familiar quantum chemical concepts, and its numerical behavior, including its convergence and computational cost, are studied using both model and real systems. The asymptotic convergence of the algorithm is derived. Finally, we examine the performance of the DMRG on widely studied chemical problems, such as the water molecule, the twisting barrier of ethene, and the dissociation of nitrogen. In all cases, the results compare favorably with the best existing quantum chemical methods, and particularly so when the nondynamical correlation is strong. Some perspectives for future development are given.

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“…If we happen to know the eigenspectra of reduced density operators of jwi, we can easily assess the quality a DMRG approximation can have; it simply depends on how quickly the eigenvalues w a decrease. In fact, such analyses have been carried out for some exactly solved systems in one and two dimensions [84][85][86][87][88]. They reveal that in one dimension for gapped systems eigenvalues w a generically decay exponentially fast (roughly as e Àcln 2 a ), which explains the success of DMRG, but in two-dimensional stripe geometries of size L Â W; L ) W; c / 1=W, such that with increasing width W (increasing two-dimensionality) the eigenspectrum decay is so slow as to make DMRG inefficient.…”

Section: Dmrg and Entanglement: Why Dmrg Work And Why It Failsmentioning

confidence: 99%

The density-matrix renormalization group in the age of matrix product states

2011

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a b s t r a c tThe density-matrix renormalization group method (DMRG) has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. In the further development of the method, the realization that DMRG operates on a highly interesting class of quantum states, so-called matrix product states (MPS), has allowed a much deeper understanding of the inner structure of the DMRG method, its further potential and its limitations. In this paper, I want to give a detailed exposition of current DMRG thinking in the MPS language in order to make the advisable implementation of the family of DMRG algorithms in exclusively MPS terms transparent. I then move on to discuss some directions of potentially fruitful further algorithmic development: while DMRG is a very mature method by now, I still see potential for further improvements, as exemplified by a number of recently introduced algorithms.

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Untitled

Cited by 15 publications

References 12 publications

Multireference correlation in long molecules with the quadratic scaling density matrix renormalization group

Multireference correlation in long molecules with the quadratic scaling density matrix renormalization group

Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization group

The density-matrix renormalization group in the age of matrix product states

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