Linear response formalism for the Douglas–Kroll–Hess approach to the Dirac–Kohn–Sham problem: First- and second-order nuclear displacement derivatives of the energy

2007 ◽  
Vol 107 (15) ◽  
pp. 3236-3249 ◽  
Author(s):  
Alexei V. Matveev ◽  
Vladimir A. Nasluzov ◽  
Notker Rösch
Keyword(s):  
Linear Response ◽  
Second Order ◽  
Derivatives Of

scholarly journals Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

2021 ◽  
Author(s):  
V. Calisti ◽  
A. Lebée ◽  
A. A. Novotny ◽  
J. Sokolowski
Keyword(s):  
Circular Inclusion ◽  
Elasticity Tensor ◽  
Second Order ◽  
Topological Derivative ◽  
Microstructural Changes ◽  
Topological Derivatives ◽  
Spatial Dimensions ◽  
Elasticity Tensors ◽  
Derivatives Of ◽  
Optimization Of Microstructures

AbstractThe multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

scholarly journals On the derivatives of the principal invariants of a second-order tensor

1986 ◽  
Vol 16 (2) ◽  
pp. 221-224 ◽  
Author(s):  
Donald E. Carlson ◽  
Anne Hoger
Keyword(s):  
Second Order ◽  
Order Tensor ◽  
Derivatives Of

On second-order directional derivatives of value functions

Optimization ◽  
2013 ◽  
Vol 64 (2) ◽  
pp. 389-407 ◽  
Author(s):  
L. Minchenko ◽  
A. Tarakanov
Keyword(s):  
Second Order ◽  
Directional Derivatives ◽  
Value Functions ◽  
Derivatives Of

DFT STUDY ON SECOND-ORDER NONLINEAR OPTICAL PROPERTIES OF THE DERIVATIVES OF DISUBSTITUTED SEVEN-VERTEX COBALTACARBORANE METALLOCENYL

2010 ◽  
Vol 09 (01) ◽  
pp. 219-231 ◽  
Author(s):  
XIAODONG LIU ◽  
YONGQING QIU ◽  
SHILING SUN ◽  
CHUNGUANG LIU ◽  
ZHONGMIN SU
Keyword(s):  
Nonlinear Optical ◽  
Membered Ring ◽  
Second Order ◽  
Meta Position ◽  
Nlo Response ◽  
B3lyp Method ◽  
Acceptor Group ◽  
Calculation Results ◽  
Optimum Model ◽  
Derivatives Of

DFT B3LYP method was employed to calculate the second-order nonlinear optical (NLO) responses of the derivatives of disubstituted seven-vertex cobaltacarborane metallocenyl. The results show that cobaltacarborane metallocenyl plays a pushing/pulling role and a bridge role to transfer electron in these molecules. The five-membered ring of cyclopentadiene is more beneficial to increase second-order NLO response than the five-membered ring composed of two C atoms and three B atoms in cobaltacarborane. Moreover, the second-order NLO response is more powerful when one substituent containing electron donor group and one substituent containing electron acceptor group are located at meta position. Accordingly, among the nine models, model c2 is the optimum model with largest value of βtot. The calculation results also show that cobaltacarborane metallocenyl and ferrocene parts play the same roles to increase second-order NLO response. Thus, cobaltacarborane metallocenyl can be a promising second-order NLO material.

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