Biorthogonality condition

Biorthogonality conditions (2.186-2.189) and completeness relation (2.190) are equivalent to the biorthogonality and completeness relations (2.5) and (2.6) obeyed by partial derivatives in the full space, if we identify... 

We now restrict ourselves to periodic functions V x) of x, so that we may impose periodic boundary conditions. Thus directly proportional to the equilibrium distribution. The probability density P may now be written in terms of the so-called biorthogonal expansion given by Morse and Feschbach [61]... 

In this paper we use biorthogonal wavelets on the interval constructed by (Cema Finek 2008 Cerna Finek 2009) which outperforms similar construction in the sense of better conditioning of base functions as well as in better conditioning of wavelet transform. The condition number seems to be nearly optimal most especially in the case of cubic spline wavelets. From the viewpoint of numerical stabihty, ideal wavelet bases are orthogonal wavelet bases. However, they are... 

We note that the incoming wave boundary conditions in the bra state in Eqs. (2.2) and (2.4) can be enforced by not complex conjugating radial functions in bra states of Eq. (2.7). This inner product is called the biorthogonal inner product [29], and is formally related to the use of complex scaled coordinates and absorbing boundary conditions. 

In spin-adapted theory, it is also convenient to employ a biorthogonal basis for (mI and m) see Section 13.7.5. In such cases, condition (13.6.10) still holds and no extra complications are introduced in the theory as presented here. 

Tensor notation may be applied to quantum chemical entities such as basis functions and matrix elements. For example, 1%, ) is a covariant tensor of rank one. Like before, superscripts, e.g., x ), denote contravariant tensors. Co- and contravariant basis functions are defined to be biorthogonal that is, they obey the conditions of... 

T, and (4) Tf. Alternative (1) is said to be fully covariant, (2) is fully cowtravariant, and the other two are mixed representations. In principle, one is free to formulate physical laws and quantum chemical equations in any of these alternative representations, because the results are independent of the choice of representation. Furthermore, by applying the metric tensors, one may convert between all of these alternatives. It turns out, however, that it is convenient to use representations (3) or (4), which are sometimes called the natural representation. In this notation, every ket is considered to be a covariant tensor, and every bra is contravariant, which is advantageous as a result of the condition of biorthogonality in the natural representation, one obtains equations that are formally identical to those in an orthogonal basis, and operator equations may be translated directly into tensor equations in this natural representation. On the contrary, in fully co- or contravariant equations, one has to take the metric into account in many places, leading to formally more difficult equations. 

---