Complex scaling technique

One of the most powerful tools to study resonances is complex scaling techniques (see Ref. 157 and references therein). In complex scaling the coordinate x of the Hamiltonian was rotated into the complex plane that is, H(x) > ll(xe 2). For resonances that have 0 .v tan-1 [Im(/i <,s )/Re ( (" ,v))] < < ) the wave functions of both the bound and resonance states are represented by square-integrable functions and can be expanded in standard L2 basis functions. 

One recent development in DFT is the advent of linear scaling algorithms. These algorithms replace the Coulomb terms for distant regions of the molecule with multipole expansions. This results in a method with a time complexity of N for sufficiently large molecules. The most common linear scaling techniques are the fast multipole method (FMM) and the continuous fast multipole method (CFMM). 

In some diseases a simple ordinal scale or a VAS scale cannot describe the full spectrum of the disease. There are many examples of this including depression and erectile dysfunction. Measurement in such circumstances involves the use of multiple ordinal rating scales, often termed items. A patient is scored on each item and the summation of the scores on the individual items represents an overall assessment of the severity of the patient s disease status at the time of measurement. Considerable amoimts of work have to be done to ensure the vahdity of these complex scales, including investigations of their reprodu-cibihty and sensitivity to measuring treatment effects. It may also be important in international trials to assess to what extent there is cross-cultural imiformity in the use and imderstand-ing of the scales. Complex statistical techniques such as principal components analysis and factor analysis are used as part of this process and one of the issues that need to be addressed is whether the individual items should be given equal weighting. 

THE SCIENCE OF ECOLOGY emerged at the turn of the last century and brought with it the experimental approaches that were already central to the study of physiology (1-3). Manipulations of whole aquatic ecosystems— excluding aquaculture, which dates back 2500 years (4)—developed more slowly, mainly because of difficulties associated with increased biotic complexity and physical scale in larger systems. One technique initially used to overcome the problems of complexity, scale, and replicability was creation of controlled microcosms that embodied a more or less natural representation of the whole system (5, 6). 

In this tribute and memorial to Per-Olov Lowdin we discuss and review the extension of Quantum Mechanics to so-called open dissipative systems via complex deformation techniques of both Hamiltonian and Liouvillian dynamics. The review also covers briefly the emergence of time scales, the definition of the quasibosonic pair entropy as well as the precise quantization relation between the temperature and the phenomenological relaxation time. The issue of microscopic selforganization is approached through the formation of certain units identified as classical Jordan blocks appearing naturally in the generalised dynamical picture. 

We give now two examples showing, either in a one-channel case or in a multichannel case, that very high accuracy can be achieved in this way. The propagation technique itself has not been commented upon. We leave this for the next sections where more general paths than those implied by ordinary complex scaling are considered. 

Recently Yao and Chu have developed complex-scaling grid techniques which can be used to discretize the time-independent Floquet Hamiltonian... 

Corresponding to the above idea are methods that implement the regularization technique of "exterior complex scaling" (ECS), which was introduced to atomic physics in Ref. [92b]—see also Refs. [95,96], and which is discussed in Section 4.3. 

Most parts of ADF have been efficiently parallelized. Because of the exponential spatial decay of the STO basis functions, linear scaling techniques reduce the computational complexity from 0 NI to O(Vat) for the most time-consuming parts of the calculation. " A density-fit procedure and the possibility of making a frozen core approximation " further reduce the cost of the calculations. 

Tsuchida and Tsukada have developed an adaptive FE algorithm for large-scale electronic structure calculations. " " They utilize a three-dimensional cubic Hermite basis to represent the electron orbitals and have extended their method to Hnear-scaling complexity.MG techniques were employed in the solution of the Poisson problem, and accurate forces suitable for molecular dynamics simulations were computed. The method was tested on diamond and cubic BN lattices and the C o molecule. The FE code has also been applied to simulate Hquid formamide at the ab initio level.Those simulations were used to assess the accmacy of previous simulations using empirical force fields. The current version of the Tsuchida-Tsukada code is called FEMTECK this code incorporates their recently developed linear-scaling technology. 

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