Finite Basis Representation

The explicit construction to which Cioslowski refers is that provided by the density-driven approach, advanced in 1988. But, already in 1986, an alternative way for carrying out this explicit construction had been set forward by Kryachko, Petkov and Stoitsov [28]. This new approach - based on localscaling transformations - was further developed by these same authors [29, 30, 32, 34], by Kryachko and Ludena [1, 20, 31, 33, 35-37], and by Koga [51]. In this Section we show that Cioslowski s density-driven method corresponds to a finite basis representation of the local-scaling transformation version of density functional theory [38]. 

Here, both the expansion coefficients and the phase (j) are energy-dependent, and Ai7 is the so-called spectral range of the Hamiltonian, A/f = (iJmax iJmin) /2, where Tfmin and i7max are the minimum and maximum eigenvalues of the Hamiltonian in a finite basis representation. The Chebyshev vectors = Tj H)xo can be iteratively generated from the recursion relation, designed by Mandelshtam and Taylor [221],... 

Czako, G., Szalay, V, Csaszar, A.G. Finite basis representations with nondirect product basis functions having structure similar to that of spherical harmonics, J. Chem. Phys. 2006,124, 014110. 

In general, theoretical studies of triatomic and tetra-atomic molecules employ analytical PESs carefully fitted to large grids of ab initio data points, and curvilinear vibrational coordinates, to take into account large-amplitude motions. On the other hand, larger polyatomic molecules are investigated with simple polynomial PES, whose parameters are obtained from ah initio data, and with normal coordinates, possibly considering only the active ones. Finite basis representations (FBR),... 

The DVR is related to, but distinct from pseudo-spectral and collocation methods of solving differential equations. For the DVR there is an orthogonal transformation which defines die relation of die DVR to the finite basis representation (FBR). > Thus, for example, the Hermidan character of operators remains obvious in the DVR. Both pseudo-spect and collocation methods, however, use a "mixed" representation operators and, as such, do not display the Hermitian character of operators such as H. Thus the advantages of the DVR are that the accuracy is that of a Gaussian quadrature and it is a true representation, while the collocation methods permit more freedom in the choice of points, a distinct advantage in some multidimensional problems. 

This last equation would hold exactly for an infinite local basis, as in this case, the DVR would be equivalent to the continuous coordinate representation. We stress here that, because the local basis q.), a = 1,..., V) and the global basis ( ), = 1,..., V are related by a unitary transformation, the operators obtained in the global basis contain the same approximation. Therefore, to distinguish it from the VBR, the representation in terms of the global basis Xn). n = 1, - -, A is called finite basis representation (FBR). 

For each partial wave J and parity e, the Hamiltonian and wave packet are discretized in the BF frame in mixed representation [21, 64, 80, 89,160] discrete variable representation (DVR) is employed for the two radial degrees of freedom and finite basis representation (FBR) of normalized associated Legendre function i jK(O) for the angular degree of freedom. Thus the wave packet in the BF frame is written as... 

The finite basis representation (FBR) of the -f Coriolis term using a symmetry adapted AL basis set with symmetry p gives... 

DVR = discrete variable representation FBR = finite basis representation PO = potential optimized. 

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