Complex orthonormal basis

In the approximate treatment based on the bivariational principle and the use of a finite truncated (real or complex) orthonormal basis [Pg.124]

In EucUdean space, orthonormal bases help both to simplify calculations and to prove theorems. Unitary bases, also called complex orthonormal bases, play the same role in complex scalar product spaces. To define a unitary basis for arbitrary (including infinite-dimensional) complex scalar product spaces, we first define spanning. 

We start by constructing an orthonormal basis a, b, c (where c is a unit vector along the wavevector k, with a and b in the two-dimensional vector space orthogonal to k). The significance of this ansatz is that any vector function F(x,y,z) is divergence free if and only if its Fourier coefficients F(k) are orthogonal to k, that is if k -F(k) =0. Thus, F(k) is a linear combination of a(k) and b(k). Lesieur defines the complex helical waves as... 

In this situation, one can achieve a further simplification by observing that it is not necessary to choose the linearly independent set 0 complex instead one can start from a real set = %, (p2,..., complete orthonormal basis for the L2 Hilbert space. The existence of such bases is well known. This means that in the approximate treatment of the eigenvalue problem for H, the relation, Eq. (2.73), is now replaced by the simpler relation... 

We note finally that, in the numerical studies of the complex symmetric operators Tu and Tueff, it is usually convenient to use orthonormal basis sets which are real, since the associated matrices will then automatically be symmetric with complex elements. In the case of a truncated basis of order m, most of the eigenvalues will usually turn out to be complex, but we observe that one has to study their behaviour when m goes to infinity and the set becomes complete, before one can make any definite conclusions as to the existence of true complex eigenvalues to Tu and Tuefj, respectively. The connection between the results of approximate numerical treatments and the exact theory is still a very interesting but mostly unsolved problem. 

Each microphase has its own symmetry and unit cell, and a full solution to the NSCF problem must be consistent with them. For example, the unit cell for the cylindrical phase is hexagonal, that for bcc is the appropriate Wigner-Seitz cell, and so on. The early NSCFT calculations approximated the unit cells for the latter two phases by cylinders and spheres, respectively. Matsen and Schick eliminated this unit cell approximation (UCA), by expanding each function of position in a series of orthonormal basis functions, each of which has the full symmetry of the phase under consideration. This provides a more accurate description of the C and S phases and, most importantly, permits the treatment of the more complex, nonclassical phases. 

In this case the most probable mode shape cannot be obtained directly by solving a standard eigenvalue problem. The mathematical structure of the optimization problem is more complex because the mode shapes are not necessarily orthogonal to each other. However, it is possible to reduce the complexity by representing the mode shape via a set of orthonormal basis and noting that the dimension of the subspace spanned by such basis does not exceed the number of modes. The... 

The orthonormality of the (271) 2 e 1(0t functions will be demonstrated explicitly later. Before doing so however, it is useful to review both complex numbers and basis sets. 

The general theory of electronic structure of complex systems and their PES are based on the tacit assumption that the basis orbitals are well defined orthonormal functions, which can be conveniently divided into two (or more if necessary) classes. The reality is much more tough and results in serious conceptual problems in all the existing packages offering hybrid modelization techniques in their respective menus. These have been addressed in the previous section. Now we address the meaning of the results obtained so far. In fact, up to this point, we obtained the description suitable for any hybrid QM/QM method. Within this context, the distribution of orbitals... 

The following restrictions are hereafter imposed on the two-electron wavefunction variations. The CSF expansion coefficients C are assumed to be real and any transformation applied to these coeffieients must be real. The orbital basis is allowed to be complex but any transformation applied to the orbitals must be real. These restrictions have no effect on the expectation values of real Hamiltonian operators. Finally, an orthonormal orbital basis is assumed and only orthogonal orbital transformations are allowed. This of course does place restrictions on some of the present discussions but is considered crucial for the extension of these results to the general N-electron case. 

The method is an approximate self-consistent-field (SCF) ab initio method, as it contains no empirical parameters. All of the SCF matrix elements depend entirely on the geometry and basis set, which must be orthonormal atomic orbitals. Originally, the impetus for its development was to mimic Hartree-Fock-Roothaan [5] (HFR) calculations especially for large transition metal complexes where full HFR calculations were still impossible (40 years ago). However, as we will show here, the method may be better described as an approximate Kohn-Sham (KS) density functional theory (DFT)... 

The overlap of the F function with itself is the dot product of the coefficient vector and its complex conjugate transpose if, as we assume here, the basis functions are orthonormal. 

The Hartree-Fock method can be applied to truly large molecules containing several hundred atoms. For such systems, it becomes impossible to construct a set of orthonormal orbitals (5.1.5), much less a set of canonical orbitals. However, as we shall see in Chapter 10, all information about the Hartree-Fock wave function is contained in the one-electron density matrix, which may be expressed directly in the basis of AOs. For large molecules, the density-matrix elements can be optimized by an algorithm whose complexity scales linearly with the size of the system. 

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