The derivative of a matric function

Rinehart, R. F.

doi:10.1090/s0002-9939-1956-0077499-x

Cited by 7 publications

(5 citation statements)

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“…376 For all bulk phase AIMD simulations within the current thesis, the ADMM purification method was used, where the MO derivatives were calculated by means of the Cauchy representation. 377 More details abot the method can be found in the literature 376 .…”

Section: Auxiliary Density Matrix Methods (Admm)mentioning

confidence: 99%

OH radical in water from ab initio molecular dynamics simulation employing hybrid functionals

Apostolidou

2019

The Journal of Chemical Physics

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In this thesis, complicated solvation phenomena that are impossible to observe via laboratory experiments are comprehensively investigated by molecular dynamics (MD) simulations. The studied system types are the OH radical in aqueous systems and keratin in ionic liquids (ILs), which are both important key contributors to the environment. The OH radical effectively cleanses the atmosphere and is also used in wastewater cleaning processes. Regenerated keratin is a biodegradable material that has been recently used as an ingredient in bioplastics. Its regeneration minimizes environmental pollution and a regeneration process by means of a greener route with ILs underpins the urgency of this research area.Within the last 15 years, only representative ab initio MD (AIMD) simulation studies with the generalized gradient approximation (GGA) functionals can be found for the OH radical in aqueous environments. However, these aforementioned functionals include the self-interaction error (SIE) that leads to artificial delocalization of unpaired electrons. Thus, a comprehensive, representative and accurate Born-Oppenheimer (BO) MD simulation study of OH -wn (w=water, n=1-5) clusters and OH -w31 was performed by employing hybrid functionals and the accurate diffuse basis set DZVP-MOLOPT-SR-GTH. In particular, one important research area was to investigate the existence of the hemibonded structure. This hemibonded structure is a threeelectron two-center interaction between the oxygen atoms of the OH radical and water with a characteristic low O -O distance of around 238 pm and a delocalization of the unpaired electron over these two centers. With the aid of this highly accurate BOMD simulation, the absence of the long-debated hemibonded configuration is shown. This clearly conveys that the hemibonded structure is an artifact from the SIE. Furthermore, this thesis reveals that a less accurate basis set such as the 6-31G basis set and a too low simulation temperature can also lead to the occurrence of the hemibonded structure. Since these two features were also used in recent AIMD simulation studies, this thesis paves the way for a more accurate description of this system type and thus sets a clear milestone within the whole topic. Finally, also the orientation of the OH radical in the gas phase up to bulk liquid water was studied by means of these BOMD simulations. It is shown that the OH radical has a less propensity for the bulk phase and that the highest occurrence is given by the OH -w1 complex in which the OH acts as a hydrogen bond donor toward water and which will therefore be a focal point for future tropospheric reactions.Additionally, a BOMD simulation of pure liquid water, involving 32 water molecules and simulated at the B3LYP-D3 and PBE0-TC-LRC-D3 levels of theory, showed very precise structural features that were in very good agreement with highly accurate experimental data.The former BOMD simulations of the OH radical in aqueous systems by using the B3LYP-D3/DZVP-MOLOPT-SR-GTH level of theory were further used to ...

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Section: Auxiliary Density Matrix Methods (Admm)mentioning

confidence: 99%

OH radical in water from ab initio molecular dynamics simulation employing hybrid functionals

Apostolidou

2019

The Journal of Chemical Physics

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“…An interesting property of this algorithm is that it can be implemented in a linear scaling fashion. , In the current context, we prefer an extension of the McWeeny procedure based on a Cauchy integral representation P̃ = S − 1 true[ 1 2 π i ∮ Θ ( z − 0.5 ) S − 1 z − P̂ d z true] S − 1 where Θ( z ) denotes the Heaviside function. This scheme yields a pure density matrix for all input matrices, is noniterative, but is not easily incorporated in a linear scaling procedure.…”

Section: Theorymentioning

confidence: 99%

“…Most of the calculations that follow take advantage of the Cauchy integral theorem for matrix functions . For an arbitrary matrix F, it states f ( F ) = 1 2 π i ∮ f ( z ) 1 z I − F d z which, since d d x F − 1 = − F − 1 d F d x F − 1 transforms into an explicit formula for the derivative of a matrix function d f ( F ) d x = 1 2 π i ∮ f ( z ) 1 F − z I d F d x 1 F − z I d z …”

Section: Wave Function Fittingmentioning

confidence: 99%

“…Most of the calculations that follow take advantage of the Cauchy integral theorem for matrix functions. 33 For an arbitrary matrix F, it states which, since transforms into an explicit formula for the derivative of a matrix function Applying this formula, matrix function derivatives can be calculated through residues of its eigenvalues. Applying this to f(x) ) Θ(z), where Θ(z) denotes the Heaviside function, the expression for the purified density matrix becomes or, after some rearrangements The evaluation of the contour integral can easily be performed via diagonalization.…”

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confidence: 99%

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Auxiliary Density Matrix Methods for Hartree−Fock Exchange Calculations

Guidon

Hutter

VandeVondele

2010

J. Chem. Theory Comput.

499

513

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The calculation of Hartree-Fock exchange (HFX) is computationally demanding for large systems described with high quality basis sets. In this work, we show that excellent performance and good accuracy can nevertheless be obtained if an auxiliary density matrix is employed for the HFX calculation. Several schemes to derive an auxiliary density matrix from a high quality density matrix are discussed. Key to the accuracy of the auxiliary density matrix methods (ADMM) is the use of a correction based on standard generalized gradient approximations for HFX. ADMM integrates seamlessly in existing HFX codes, and in particular can be employed in linear scaling implementations. Demonstrating the performance of the method, the effect of HFX on the structure of liquid water is investigated in detail using Born-Oppenheimer molecular dynamics simulations (300 ps) of a system of 64 molecules.Representative for large systems are calculations on a solvated protein (Rubredoxin), for which ADMM outperforms the corresponding standard HFX implementation by approximately a factor 20.

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“…This analysis, for generic functions of A + E, was applied by Rinehart in 1956 [41] (see Higham [24,Prob. 3.4]), and more recently by Davies [10].…”

Section: Spectral Analysismentioning

confidence: 99%

GMRES Convergence for Perturbed Coefficient Matrices, with Application to Approximate Deflation Preconditioning

Sifuentes¹,

Embree²,

Morgan³

2013

SIAM J. Matrix Anal. & Appl.

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Abstract. How does GMRES convergence change when the coefficient matrix is perturbed? Using spectral perturbation theory and resolvent estimates, we develop simple, general bounds that quantify the lag in convergence such a perturbation can induce. This analysis is particularly relevant to preconditioned systems, where an ideal preconditioner is only approximately applied in practical computations. To illustrate the utility of this approach, we combine our analysis with Stewart's invariant subspace perturbation theory to develop rigorous bounds on the performance of approximate deflation preconditioning using Ritz vectors.

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The derivative of a matric function

Cited by 7 publications

References 0 publications

OH radical in water from ab initio molecular dynamics simulation employing hybrid functionals

OH radical in water from ab initio molecular dynamics simulation employing hybrid functionals

Auxiliary Density Matrix Methods for Hartree−Fock Exchange Calculations

GMRES Convergence for Perturbed Coefficient Matrices, with Application to Approximate Deflation Preconditioning

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