Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm

Nakata, Masao; Nakatsuji, Hiroshi; Ehara, Masahiro; Fukuda, Mituhiro; Nakata, Keiichi; Fujisawa, Katsuki

doi:10.1063/1.1360199

Cited by 262 publications

(253 citation statements)

References 18 publications

(13 reference statements)

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“…However the method was soon abandoned because of the computational cost. Interest in the subject was renewed at the beginning of this century, when Nakata [12] and Mazziotti [13] realized that the v2DM problem can be formulated as a semidefinite program (SDP) for which general-purpose primal-dual SDP solvers can be used [14], and they calculated the ground-state properties of small atoms and molecules. Primal-dual interior point methods are the "Rolls Royce" of SDP algorithms, having several appealing features, but they require a lot of storage and are computationally expensive.…”

Section: Introductionmentioning

confidence: 99%

A primal–dual semidefinite programming algorithm tailored to the variational determination of the two-body density matrix

Verstichel¹,

Aggelen²,

Neck³

et al. 2011

Computer Physics Communications

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The quantum many-body problem can be rephrased as a variational determination of the twobody reduced density matrix, subject to a set of N -representability constraints. The mathematical problem has the form of a semidefinite program. We adapt a standard primal-dual interior point algorithm in order to exploit the specific structure of the physical problem. In particular the matrix-vector product can be calculated very efficiently. We have applied the proposed algorithm to a pairing-type Hamiltonian and studied the computational aspects of the method. The standard N -representability conditions perform very well for this problem. * brecht.verstichel@ugent.be

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Section: Introductionmentioning

confidence: 99%

A primal–dual semidefinite programming algorithm tailored to the variational determination of the two-body density matrix

Verstichel¹,

Aggelen²,

Neck³

et al. 2011

Computer Physics Communications

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“…Semidefinite programs (SDPs) have gained a lot of attention in recent years with applications arising from a diverse range of fields including structural optimization, system and control, statistics, financial engineering and quantum chemistry [3,13,22]. The solution of an SDP has been found using primal-dual interior-point methods.…”

Section: Introductionmentioning

confidence: 99%

A parallel primal–dual interior-point method for semidefinite programs using positive definite matrix completion

2006

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Abstract. A parallel computational method SDPARA-C is presented for SDPs (semidefinite programs). It combines two methods SDPARA and SDPA-C proposed by the authors who developed a software package SDPA. SDPARA is a parallel implementation of SDPA and it features parallel computation of the elements of the Schur complement equation system and a parallel Cholesky factorization of its coefficient matrix. SDPARA can effectively solve SDPs with a large number of equality constraints, however, it does not solve SDPs with a large scale matrix variable with similar effectiveness. SDPA-C is a primal-dual interior-point method using the positive definite matrix completion technique by Fukuda et al, and it performs effectively with SDPs with a large scale matrix variable, but not with a large number of equality constraints. SDPARA-C benefits from the strong performance of each of the two methods. Furthermore, SDPARA-C is designed to attain a high scalability by considering most of the expensive computations involved in the primal-dual interior-point method. Numerical experiments with the three parallel software packages SDPARA-C, SDPARA and PDSDP by Benson show that SDPARA-C efficiently solve SDPs with a large scale matrix variable as well as a large number of equality constraints with a small amount of memory.

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“…This approach treats the two-particle reduced density matrix (2-RDM) rather than the N-particle wave function as the fundamental object of interest [5][6][7][8][9]. A Hamiltonian of N fermions with at most two-body interactions can be written in second-quantized form aŝ then we can write the energy as a linear functional of only the 2-RDM [2,10,11] E É ¼ hÉjĤjÉi…”

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confidence: 99%

“…week ending 19 NOVEMBER 2010 accuracy [6,8,9,13]. To improve such calculations even further, additional constraints must be added.…”

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confidence: 99%

“…To improve such calculations even further, additional constraints must be added. Again, one obvious approach is to construct three-and four-particle analogues of the two-particle positivity constraints [6,7,9,14,15]. However, implementing these higher-order positivity constraints is extremely costly; the operational complexity of the 3-positivity constraints scales at minimum as L 9 , where L is the number of one-electron basis functions used in the calculation [16,17].…”

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confidence: 99%

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Active-SpaceN-Representability Constraints for Variational Two-Particle Reduced Density Matrix Calculations

Shenvi

Izmaylov

2010

Phys. Rev. Lett.

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The ground-state energy of a system of fermions can be calculated by minimizing a linear functional of the two-particle reduced density matrix (2-RDM) if an accurate set of N-representability conditions is applied. In this Letter we introduce a class of linear N-representability conditions based on exact calculations on a reduced active space. Unlike wave-function-based approaches, the 2-RDM methodology allows us to combine information from calculations on different active spaces. By adding active-space constraints, we can iteratively improve our estimate for the ground-state energy. Applying our methodology to a 1D Hubbard model yields a significant improvement over traditional 2-positivity constraints with the same computational scaling.

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Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm

Cited by 262 publications

References 18 publications

A primal–dual semidefinite programming algorithm tailored to the variational determination of the two-body density matrix

A primal–dual semidefinite programming algorithm tailored to the variational determination of the two-body density matrix

A parallel primal–dual interior-point method for semidefinite programs using positive definite matrix completion

Active-SpaceN-Representability Constraints for Variational Two-Particle Reduced Density Matrix Calculations

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