Density matrix energy weighted

Energy-weighted density matrix element in AO basis... 

The expression of the energy-weighted density matrix is more easily given in the MO basis [4, 26]... 

Here, W is an energy-weighted density matrix element as in the Hartree-Fock gradient formula (23). Equation 5 contains two "difficult" terms (the last two), the derivative of density matrix elements and the fitting coefficients e. It turns out that these terms can be eliminated by using the relationship... 

It is, of course, not possible to include a statistical weighting function exp — WJkT) directly in the proton wave function, since this must actually be inserted in the density matrix. The total jump rate, however, amounts essentially to a summation of quantum contributions like (9.68) from energy levels below the barrier top together with a classical contribution like (9.59) frorh higher levels. 

The density-matrix method of McWeeny was found to converge in all cases, but too slowly for practical purposes. The energy-weighted steepest descent (EWSD) method developed by Hillier and Saunders, is related to the McWeeny method it is claimed that the modifications adopted bring about improved convergence. 

The energy spectrum and the spectral weights are directly useful for the determination of atomic properties. For instance, the one-electron reduced density matrix becomes... 

Here the energy-weighted density matrix W has been introduced. 

This formulation requires one to compute the energy-weighted density matrix using the molecular orbital coefficient matrix C, which must be avoided to... 

In this way the final density matrix-based energy gradient expression results, which avoids the use of the conventional energy-weighted density matrix (which is conventionally constructed via the delocalized, canonical MO coefficients) ... 

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