Analytic Energy Derivatives for the Direct Iterative Approach to the Generalized Bloch Equation

2001 ◽  
Vol 66 (8) ◽  
pp. 1164-1190 ◽  
Author(s):  
Holger Meissner ◽  
Josef Paldus
Keyword(s):  
Bloch Equation ◽  
Second Order ◽  
Iterative Approach ◽  
Vector Method ◽  
Energy Derivatives ◽  
Analytic Energy Derivatives

A general formalism for the analytic energy derivatives in the context of the recently developed state-selective version of the direct iterative approach to the generalized Bloch equation is presented. An explicit formalism is developed for both the gradients and the Hessian by exploiting the so-called Z-vector method. A procedure for the development of the corresponding algorithm for higher than the second-order properties is also briefly outlined.

Adiabatic second-order energy derivatives in quantum mechanics

1958 ◽  
Vol 54 (2) ◽  
pp. 251-257 ◽  
Author(s):  
W. Byers Brown ◽  
H. C. Longuet Higgins
Keyword(s):  
Quantum Mechanics ◽  
General Equation ◽  
Canonical Ensemble ◽  
Second Order ◽  
Order Derivative ◽  
Interacting Particles ◽  
Energy Derivatives ◽  
Order Energy ◽  
Classical Analogue ◽  
Image Position

ABSTRACTThe general equation for the adiabatic second-order derivative of the energy En of an eigenstate with respect to parameters λ and λ′ occurring in the Hamiltonian ℋ isThe applications of this equation to molecules (λ, λ′ = nuclear position coordinates) and to enclosed assemblies of interacting particles (λ = λ′ = volume) are discussed, and the classical analogue of the equation for a micro-canonical ensemble is derived.

scholarly journals A Vector Operation to Extract Second-Order Terrain Derivatives from Digital Elevation Models

2020 ◽  
Vol 12 (19) ◽  
pp. 3134 ◽  
Author(s):  
Guanghui Hu ◽  
Wen Dai ◽  
Sijin Li ◽  
Liyang Xiong ◽  
Guoan Tang
Keyword(s):  
Digital Elevation Models ◽  
Original Data ◽  
Second Order ◽  
Comparison Analysis ◽  
Initial Direction ◽  
Vector Method ◽  
Vector Operation ◽  
Digital Elevation ◽  
Gaussian Surface ◽  
Derivatives Of

Terrain derivatives exhibit surface morphology in various aspects. However, existing spatial change calculation methods for terrain derivatives are based on a mathematical scalar operating system, which may disregard the directional property of the original data to a certain extent. This situation is particularly true in second-order terrain derivatives, in which original data can be terrain derivatives with clear directional properties, such as slope or aspect. Thus, this study proposes a mathematical vector operation method for the calculation of second-order terrain derivatives. Given the examples of the first-order terrain derivatives of slope and aspect, their second-order terrain derivatives are calculated using the proposed vector method. Directional properties are considered and vectorized using the following steps: rotation-type judgment, standardization of initial direction, and vector representation. The proposed vector method is applied to one mathematical Gaussian surface and three different ground landform areas using digital elevation models (DEMs) with 5 and 1 m resolutions. Comparison analysis results between the vector and scalar methods show that the former achieves more reasonable and accurate second-order terrain derivatives than the latter. Moreover, the vector method avoids overexpression or even exaggeration errors. This vector operation concept and its expanded methods can be applied in calculating other terrain derivatives in geomorphometry.

Analytic energy derivatives in many‐body methods. I. First derivatives

1989 ◽  
Vol 90 (3) ◽  
pp. 1752-1766 ◽  
Author(s):  
E. A. Salter ◽  
Gary W. Trucks ◽  
Rodney J. Bartlett
Keyword(s):  
Many Body ◽  
Energy Derivatives ◽  
Analytic Energy Derivatives

scholarly journals Analytic Gradients for Restricted Active Space Second-order Perturbation Theory (RASPT2)

2021 ◽  
Author(s):  
Yoshio Nishimoto
Keyword(s):  
Perturbation Theory ◽  
Computational Cost ◽  
Second Order ◽  
Order Perturbation ◽  
Order Perturbation Theory ◽  
Local Version ◽  
Field Calculation ◽  
Vector Method ◽  
Active Space ◽  
Second Order Perturbation Theory

The computational cost of analytic derivatives in multireference perturbation theory is strongly affected by the size of the active space employed in the reference self-consistent field calculation. To overcome previous limits on active space size, the analytic gradients of single-state restricted active space second-order perturbation theory (RASPT2) and its complete active space variant (CASPT2) have been developed and implemented in a local version of OpenMolcas. Similar to previous implementations of CASPT2, the RASPT2 implementation employs the Lagrangian or Z-vector method.<br>The numerical results show that restricted active spaces with up to 20 electrons in 20 orbitals can now be employed for geometry optimizations.<br>

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