Dual basis

These expressions show that a non-ON basis is more difficult to handle there are repeated inserts of the inverse S matrix, and double as many summation indices. However, there is a simple way to improve things Define the dual basis c,) as... 

The integrals over the one- and two-electron operators have the dual basis to the left of the operator and the primary basis to the right. The one- and two-... 

These quantities may be evaluated by a variety of techniques [15-18]. In the dual basis the one-electron integrals have no exploitable symmetry and the two-electron integrals possess only a two-fold symmetry... 

Since one has the general property W = H, it is convenient to introduce a dual basis 4 and having the property = iff. For this purpose, we will now introduce a linearly independent set 0 = l5 2,..., m consisting of m complex functions, which have the additional property that the overlap matrix A = <0 10) is nonsingular, and make the dual choice of bases ... 

Even if the conditions for persistent, lost, and new eigenvalues are completely clear for the exact eigenvalue problems to the operators T and Tt, it is considerably more difficult to translate them to the approximate eigenvalue problems associated with the application of the bi-variational principle for the operators T and Tt to truncated basis sets. In this connection, the relations (A. 1.40-1.49) may turn out to be useful in formulating the problem. Some of the computational aspects, particularly the choice of the dual basis sets, are further discussed in reference A. 

In addition to the chosen set of orbitals , we introduce the dual basis ... 

Let C be a non-hyperelliptic curve of genus g. Petri starts by choosing g points X, ..., xg on C in a reasonably general position (we won t worry about this). Let (fi.,(pg be a dual basis of differential forms, i.e.,... 

R. Jurgens-Lutovsky and J. Almidf, Chem. Phys. Lett., 178, 451 (1991). Dual Basis Sets in Calculation of Electron Correlation. 

The dynamic interaction between two bodies may be described by a resultant force and moment. Thus, when bodies (i - 1) and i interact, whether through a joint or contact, a generalized force, f<, is exerted on body i by body (i - 1) and with a negative sign on body (i - 1). Its form is given in Equation 2.3. This force vector may be resolved in the dual basis defined above ... 

Note that if it is necessary to model f using the dual basis composed of and V , the same elimination may be carried out using instead. 

This remark demands great care and consideration. Through the signal-wavelet inversion formula, derived later on, we can represent the (physical) wavefunction as a superposition of dual basis functions and wavelet transform coefficients. We symbolically denote this, for the dyadic representation (Sec. 1.3.2), by 9(b) = i J2j,i Thus at a given point b, the... 

In the last section we indicated that in order to implement a succesfull TPQ-scalet quantization procedure, we must regard the scalet solutions as the generators of the wavelet coefficients that contribute to the signal( f)-wavelet reconstruction formula (i.e. either CWT or DCWT). The TPQ conditions can then be applied to this wavefunction representation. The basic reason this approcich works is that the CWT/DCWT -reconstruction formula implicitly works with a (dual) basis that is both bounded and infinitely differentiable (i.e. for the Mexican hat wavelet and dual representation). Any truncated expansion of this type will clearly emulate the physical solution, as well as serve to filter out unphysical solutions (generated through the unphysical scalet configurations), since these will not be bounded or infinitely differentiable. 

The basis a a, constitutes a dual basis on the tangent plane, with as the dual metric. 

R. P. Steele, R. A. DiStasio, Jr., Y. Shao, J. Kong, and M. Head-Gordon, J. Chem. Phys., 125(074108), 1-11 (2006). Dual-Basis Second-Order Moller-Plesset Perturbation Theory A Reduced-Cost Reference for Correlation Calculations. 

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