Elementary Jacobi rotations

1) An Elementary Jacobi Rotation (EJR) over n-dimensional space can be constructed, in order to fulfill the needs of the present work, as an orthogonal matrix Jij(a)= Jij,pq(oc) with the following prescription  

2) The action of Jij over a n-dimensional column vector c= cp produces a new vector such as 

This is sufficient to ensure norm invariance for the involved 

4) It is well known that the product of a set of unitary matrices gives as a result a unitary matrix. Therefore, if S= i,j) is some set of active index pairs, and 91 = Jij and attached set 

Ramon Carbd, Joan Miro, Lloren Domingo, and Juan J. Novoa 

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Variation of Quantum Mechanical Objects in the MO framework via Elementary Jacobi Rotations... 

Some time ago, we started HI the development of the Elementary Jacobi Rotation (EJR) algorithms which were known from old times /2, 3, 4/. As a consequence of this development some intermediate work has been published /5,6/. The present paper corresponds, essentially, to the application of our EJR experience to multiconfigurational calculations. 

VARIATION OF QUANTUM MECHANICAL OBJECTS IN THE MO FRAMEWORK VIA ELEMENTARY JACOBI ROTATIONS... 

The Elementary Jacobi Rotations can be taken as an alternative form for the unitary matrix transformation of the multiconfiguration energy. The exponential and the EJR forms of the unitary matrix can be related through the expressions,already discussed in section 2.5, because the EJR are the minimal parts in which the exponential... 

Amat L, Carbo-Dorca R. Quantum similarity measures under atomic shell approximation first-order density fitting using elementary Jacobi rotations. J Comput Chem 1997 18 2023-2039. 

Convex sets [89] play a leading role in optimisation problems. They have been introduced to deal with some chemical problems [9a], Elementary Jacobi Rotations (EJR) [90] can be applied over a generating vector to obtain optimal coefficients, while preserving convex conditions [39e,41,49,9l]. 

The optimization is then no longer a linear problem and becomes a difficult optimization with the usual caveats such as the existence of local minima. Amat et al. used the elementary Jacobi rotation technique as an optimization technique to find the proper coefficients for the ASA s-type Gaussians. Other techniques can be applied as well, such as the Newton-Raphson technique. " Lists of s-type Gaussian coefficients and exponents may be fovmd on the Internet for different basis sets. ... 

Measures Under Atomic Shell Approximation First Order Density Fitting Using Elementary Jacobi Rotations. 

Amat L, Caibo-Dorca R (2000) Molecular electronic density fitting using elementary Jacobi rotations under atomic shell approximation. J Chem Inf Comput Sci 40 1188-1198... 

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