An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics

Abstract

Abstract. This paper gives an error analysis of the multi-configuration time-dependent Hartree (MCTDH) method for the approximation of multi-particle time-dependent Schrödinger equations. The MCTDH method approximates the multivariate wave function by a linear combination of products of univariate functions and replaces the high-dimensional linear Schrödinger equation by a coupled system of ordinary differential equations and low-dimensional nonlinear partial differential equations. The main result of this paper yields an L 2 error bound of the MCTDH approximation in terms of a best-approximation error bound in a stronger norm and of lower bounds of singular values of matrix unfoldings of the coefficient tensor. This result permits us to establish convergence of the MCTDH method to the exact wave function under appropriate conditions on the approximability of the wave function, and it points to reasons for possible failure in other cases.Mathematics Subject Classification. 81V55, 58J90, 35F25.

Error Analysis of the Dynamically Orthogonal Approximation of Time Dependent Random PDEs

“…The decomposition (3.10) and the corresponding system (3.14)-(3.17) have been proposed in [9,10] for a dynamically low rank approximation of a time dependent differential matrix/tensor equation. An analogous formulation in an infinite dimensional setting is derived in [11,12], related to the MCTDH approach in the quantum dynamics framework. We remark that for time dependent SPDEs, a dynamically biorthogonal (DyBO) method, which has a close relation with the DDO approximation, has been introduced in [26,27].…”

“…This allows us to "import" some of the theoretical results developed in [12,9] to our situation of a parabolic equation with random parameters. In particular, we reinterpret the DO equations given in [17,18,19] as a Galerkin projection onto the tangent space to the manifold of the rank S functions of the form (1.1).…”

“…In particular, we reinterpret the DO equations given in [17,18,19] as a Galerkin projection onto the tangent space to the manifold of the rank S functions of the form (1.1). Using curvature bounds for such a manifold, given in [12], we show that the DO approximation error for a linear parabolic equation with random input data can be bounded in terms of the best rank S approximation of the solution (Karhunen-Loève expansion), at each time instant. The bound is applicable on the largest time interval in which the best S-terms approximation is continuously differentiable over time.…”

Instabilities and inaccuracies of multi-configuration time-dependent Hartree-Fock

“…This makes the validity of applying MCTDHF to such a system uncertain. These difficulties are known in the mathematical literature [8,9] and recently mentioned in the quantum chemistry literature [10] but are not considered seriously in the physics community.…”

“…This makes the validity of applying MCTDHF to such a system uncertain. These difficulties are known in the mathematical literature [8,9] and recently mentioned in the quantum chemistry literature [10] but are not considered seriously in the physics community.In this contribution, we will demonstrate the problem on the basis of a physically relevant example. Finally, we will discuss several possible approaches to cure this problem.…”