Jacobi rotation

There are several ways to diagonalize a real-symmetric matrix. The common strategy is to use orthogonal matrices to gradually reduce the magnitude of off-diagonal elements.18,19 The most rudimental and well-known approach is the Jacobi rotation, which zeros one such element at a time. The alternative rotation scheme devised by Givens can be used to make the process more efficient. Perhaps the most efficient and most commonly used approach is that... 

Note that since SVD is based on eigenvector decompositions of cross-product matrices, this algorithm gives equivalent results as the Jacobi rotation when the sample covariance matrix C is used. This means that SVD will not allow a robust PCA solution however, for Jacobi rotation a robust estimation of the covariance matrix can be used. 

Calculation of eigenvectors requires an iterative procedure. The traditional method for the calculation of eigenvectors is Jacobi rotation (Section 3.6.2). Another method—easy to program—is the NIPALS algorithm (Section 3.6.4). In most software products, singular value decomposition (SVD), see Sections A.2.7 and 3.6.3, is applied. The example in Figure A.2.7 can be performed in R as follows ... 

The final step is the orbital optimization for the truncated SDTQ-CI expansion. We used the Jacobi-rotation-based MCSCF method of Ivanic and Ruedenberg (55) for that purpose. Table 1 contains the results for the FORS 1 and FORS2 wavefiinctions of HNO and NCCN, obtained using cc-pVTZ basis sets (51). In all cases, the configurations were based on split-localized orbitals. For each case, four energies are listed corresponding to (i) whether the full or the truncated SDTQ-CI expansion was used and (ii) whether the split-localized orbitals were those deduced from the SD naturals orbitals or were eventually MCSCF optimized. It is seen that... 

The MOs in eq 5 are typically optimized using a reorthogonalization technique that has been described by Gianinetti et al.,(30) though they can also be obtained using a Jacobi rotation method that sequentially and iteratively optimizes each individual orbital.(28,37)... 

JACOBI ROTATIONS A GENERAL PROCEDURE FOR ELECTRONIC ENERGY OPTIMIZATION ... 

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. 

The current work presented here must be necessarily considered as a new step towards an easy, comprehensive and cheap way to obtain atomic and molecular wavefunctions. In the same path we are attempting here to show that the Jacobi Rotation techniques may be considered viable alternative procedures to the usual SCF and Unitary Transformation algorithms. 

After this, perhaps, the reader will be indulgent with the structure of this paper. First we will present a broad survey of the ideas which are necessary to build up the Jacobi Rotation techniques. A second part will study various energy... 

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 ... 

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

Any Jacobi rotation between two MO s will simply modify both density matrices in the way described in section 4.1. Once these density variation expressions, see formula (4.3), are applied on the Roothaan-Bagus energy and after collecting terms of similar order the usual four-parameter expression for the energy increment (5.2) is obtained. The parameters are now defined by the formulae ... 

The variation of the integrals under a single Jacobi rotation over an active pair of orbitals i,j has already been defined in section 4.3, and noting that both P and Q hypermatrices transform in the same way, these expressions alone will suffice, through substitution in the energy expression, to obtain the formulae for the definition of the energy variation parameters ... 

This expression toghether with the ones that perform the rotation of the h, P and Q integral sets, provide the algorithm to implement the Jacobi rotation method on the MO basis using the Roothaan-Bagus energy expression. 

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... 

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