Moment expansion Cauchy

This is often called the Cauchy moment expansion of the frequency-dependent polarizability and the sums S k) for even but negative values of k are called Cauchy moments. S (—2) in particular turns out to be proportional to the static polarizability... 

These moments are related to many physical properties. The Thomas-Kulm-Reiche sum rule says that. S (0) equals the number of electrons in the molecule. Other sum rules [36] relate S(2),, S (1) and. S (-l) to ground state expectation values. The mean static dipole polarizability is md = e-S(-2)/m,.J Q Cauchy expansion... 

The calculation of frequency-dependent linear-response properties may be an expensive task, since first-order response equations have to be solved for each considered frequency [1]. The cost may be reduced by introducing the Cauchy expansion in even powers of the frequency for the linear-response function [2], The expansion coefficients, or Cauchy moments [3], are frequency independent and need to be calculated only once for a given property. The Cauchy expansion is valid only for the frequencies below the first pole of the linear-response function. 

The even Cauchy moments arise in the expansion of a frequency-dependent polarizability a((o)... 

This truncation is a well-defined procedure, if the higher moments become progressively smaller. If the jump density w z) is even, then we obtain the standard diffusion equation. However, this naive Taylor series expansion is not valid for heavy-tailed probability density functions, such as a Cauchy PDF,... 

Transitioning from the stress state of a particle to the stress field of the continuum, the interaction of the Cauchy stress tensor components of neighboring points needs to be investigated. They have to satisfy the conditions of local equilibrium to be established with the aid of an arbitrary infinitesimal volume element. Such an element with faces in parallel to the planes of the Cartesian coordinate system is subjected to the volume force and on the faces to the components of the Cauchy stress tensor with additional increments in the form of the first element of Taylor expansions on one of the respective opposing faces. The balance of moments proves the symmetry of the stress tensor,... 

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