Solving the eigenvalue equations of correlated vibrational structure methods: Preconditioning and targeting strategies

This perspective addresses selected recent developments in the theoretical calculation of vibrational spectra, energies, wave functions and properties. The theoretical foundation and recently developed computational protocols for constructing hierarchies of vibrational Hamiltonian operators are reviewed. A many-mode second quantization (SQ) formulation is discussed prior to the discussion of anharmonic wave functions. Emphasis is put on vibrational self-consistent field (VSCF) based methods and in particular vibrational coupled cluster (VCC) theory. Other issues are also reviewed briefly, such as inclusion of thermal effects, response theoretical calculation of spectra, and the difficulty in treating dense spectra.

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“…To specifically converge interior eigenvalues is challenging, see ref. 114 and 119 and references therein. Ref.…”

Section: Solving Vibrational Eigenproblems and The Many State Problemmentioning

confidence: 99%

Selected new developments in vibrational structure theory: potential construction and vibrational wave function calculations

Christiansen

2012

Phys. Chem. Chem. Phys.

136

137

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“…36 Indeed, for the sake of response theory that is detailed below, we usually solve for only the VCI ground state. Equation ͑2͒ is usually solved for just a few eigenvalues and eigenstates by iterative procedures.…”

Section: A Vibrational Configuration Interactionmentioning

confidence: 99%

A Lanczos-chain driven approach for calculating damped vibrational configuration interaction response functions

Hansen

Seidler

Győrffy

et al. 2010

The Journal of Chemical Physics

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We present an approach based on the Lanczos method for calculating the vibrational configuration interaction response functions necessary for evaluating the pure vibrational contributions to the polarizabilities and first hyperpolarizabilities of molecules. The method iteratively builds a tridiagonal representation of the central response matrix, which is subsequently used for solving the response equations. From the same chain, the response functions can be evaluated approximately for any frequency as well as using any complex damping factor. Applications to formaldehyde, cyclopropene, and uracil illustrate the concepts presented.

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“…In Section 2, we recall the definition and use of the Subspace Projected Approximate matrices [17], leading to their SPAM eigenvalue method -cf. also [4,8,11,14,20]. In Section 3, the selection mechanism based on the subspace-projected approximate matrices is developed and converted into an algorithm.…”

Section: Outlinementioning

confidence: 99%

A Subspace-Projected Approximate Matrix Method for Systems of Linear Equations

Brandts

Silva

2013

E Asian J Appl Math

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Given two n × n matrices A and A0 and a sequence of subspaces with dim the k-th subspace-projected approximated matrix Ak is defined as Ak = A + Πk(A0 − A)Πk, where Πk is the orthogonal projection on . Consequently, Akν = Aν and ν*Ak = ν*A for all Thus is a sequence of matrices that gradually changes from A0 into An = A. In principle, the definition of may depend on properties of Ak, which can be exploited to try to force Ak+1 to be closer to A in some specific sense. By choosing A0 as a simple approximation of A, this turns the subspace-approximated matrices into interesting preconditioners for linear algebra problems involving A. In the context of eigenvalue problems, they appeared in this role in Shepard et al. (2001), resulting in their Subspace Projected Approximate Matrix method. In this article, we investigate their use in solving linear systems of equations Ax = b. In particular, we seek conditions under which the solutions xk of the approximate systems Akxk = b are computable at low computational cost, so the efficiency of the corresponding method is competitive with existing methods such as the Conjugate Gradient and the Minimal Residual methods. We also consider how well the sequence (xk)k≥0 approximates x, by performing some illustrative numerical tests.

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Solving the eigenvalue equations of correlated vibrational structure methods: Preconditioning and targeting strategies

Cited by 18 publications

References 73 publications

Selected new developments in vibrational structure theory: potential construction and vibrational wave function calculations

Selected new developments in vibrational structure theory: potential construction and vibrational wave function calculations

A Lanczos-chain driven approach for calculating damped vibrational configuration interaction response functions

A Subspace-Projected Approximate Matrix Method for Systems of Linear Equations

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