Contraction algorithm

Artificial Intelligence Contraction Algorithms in Quantum Dynamics Application to CD3H and... 

Table 1 Energies Associated with the Eigenvectors Obtained from the Initial State nv,)( Either by Performing Four Iterations of the Wave Operator Contraction Algorithm or by Performing an Exact Calculation (based upon filtered Lanczos calculations).

The value of d is frequently >106, so the wave operator contraction algorithm, described earlier in Section II.B.2, was used to select functions from the primitive space to build the active space. 

The neighbourhood contraction algorithm is conceptually very simple starting with a large value of k, the tangent space is constructed and the criteria for neigh-... 

Fig. 5.4 The three basic moves permitted to the simplex algorithm (reflection, and its close relation reflect-and-expmd contract in one dimension and contract around the lowest point). (Figure adapted from Press W H, B P Flannery,...

The first summation requires electron repulsion integrals with four virtuaJ indices. Efficient algorithms that avoid the storage of these integrals have been discussed in detail [20]. For every orbital index, p, this OV contraction must be repeated for each energy considered in the pole search it is usually the computational bottleneck. 

AuH and Au2 serve as benchmark molecules to test the performance of various relativistic approximations. Figure 4.7 shows predictions for relativistic bond contractions of Au2 from various quantum chemical calculations over more than a decade. In the early years of relativistic quantum chemistry these predictions varied significantly (between 0.2 and 0.3 A), but as the methods and algorithms became more refined, and the computers more powerful, the relativistic bond contraction for Au2 converged and is now at 0.26 A. 

Nelder and Mead (1965) described a more efficient (but more complex) version of the simplex method that permitted the geometric figures to expand and contract continuously during the search. Their method minimized a function of n variables using (n + 1) vertices of a flexible polyhedron. Details of the method together with a computer code to execute the algorithm can be found in Avriel (1976). 

Iterative Solution of the Contracted Schrodinger Equation The Reduced Density Matrices Construction Algorithms A. The 2-RDM Construction Algorithm... 

In order to avoid keeping in computer memory aU the 4-RDM elements, the contraction of the 4-RDM in order to get a consistent 3-RDM is simulated by another algorithm. Thus the only 4-RDM elements to be stored are the diagonal elements. All the elements are only calculated once and entered in all the places where they appear. 

As mentioned previously, the off-diagonal elements of the 3-RDM are determined by an algorithm obtained by contracting the 4-RDM. For simplicity sake, the expression given here for the contraction of the 4-RDM (Eq. (71)) corresponds to the spin block, ... 

Here we synthesize the concepts of the last four sections, (i) CSE, (ii) reconstruction, (iii) purification, and (iv) a contracted power method, to obtain an iterative algorithm for the direct calculation of the 2-RDM. 

As can be seen, the 2-CSE depends not only on the 2-RDM but also on the 3- and 4-RDMs. This fact lies at the root of the indeterminacy of this equation [63, 107]. As already mentioned, in the method proposed by Colmenero and Valdemoro [46 8] and in those further proposed by Nakatsuji and Yasuda [49, 51] and by Mazziotti [52, 111], a set of algorithms for approximating the higher-order ROMs in terms of the lower-order ones [46, 47, 108] allows this equation to be solved iteratively until converging to a self-consistent solution. In the approach considered in this work, the spin-adapted 2-CSE has been used. This equation is obtained by coupling the 2-CSE with the second-order contracted spin equation [50]. 

Data analysis. We use a combination of two movement detection algorithms (23) written in Madab environment (The MathWorks, Inc.) to accurately track movement of the heart edges see Note 10). Measurements of diastolic and systolic diameters as well as diastolic and systolic intervals, % fractional shortening (% PS), arrhythmicity index, and contraction direction and velocity are obtained as output from this analysis program. Heart rate is calculated as the inverse of the heart period where one period corresponds to a single diastolic interval plus subsequent systolic interval. % FS is quantified based on diameter measurements and is calculated as ((diastolic diameter-systolic diameter)/diastolic diameter) x 100 (%). Heart beat rhythmicity is quantified based on the standard deviation of the mean heart period normalized to the median heart period for each fly providing a dimensionless arrhythmicity index. 

When the HF basis set is minimal, this is fairly simple (there is a one-to-one correspondence in basis functions) but when it is larger, some algorithmic choices are made about how to carry out the mapping (e.g., always map to the tightest function or map based on overlap between the semiempirical STO and the large-basis contracted GTO). Thus, it is... 

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