Orthogonal basis set

It is convenient to describe the polarization vector in terms of an orthogonal basis set of unit vectors, (a1 a2), such that aj a2 = 0. Two choices are commonly used, one for... 

The above approaches estimate the excitation rate by using either second-order perturbation theory [6] or a re-summation to all orders in perturbation theory [20,21]. In order to be able to sum the infinite series of perturbation theory references [20,21], we use an orthogonal basis-set of the model Hamiltonian (2) (the creation and destruction operators need to... 

The eigenvectors from the analysis of Z(mxm) are sometimes referred to as the row-mode eigenvectors because they form an orthogonal basis set that spans the row space of A. The eigenvectors from the analysis of Z(nxn) are sometimes referred to as the column-mode eigenvectors because they form an orthogonal basis set that spans the column space of A. 

We now return to the task of formulating the solvent perturbation operator in Eq. (9-1). To formulate ourselves in terms of matrix elements, we introduce a real and orthogonal basis set for the quantum chemical region, =i, When the discussion turns to the quantum chemical method, the details of this basis will be dealt with, but for the moment the discussion is kept general. The permanent electrostatic contribution to Vsoiv., called Vperm., comes from the interaction between the quantum chemical charge distribution and the point-charges of the solvent. In other words,... 

An adequate basis set may be obtained from Bowden et al.73 which was developed for multiple-quantum experiments. In this formalism, the irreducible spin operators Tqn,n=0,..., 21 q= — n, n, of rank n and order q form an orthogonal basis set. The orthonormal operators, 7, derived from these form a complete basis set. These are defined as... 

The orbital exponent a, and the location of the orbital center Ri are the variational parameters. These Gaussians are positioned in the molecule in the same way as in Lewis s concept of valence theory. A single Slater determinant wave function for the non-orthogonal basis set of doubly occupied orbitals, Gj is constructed, and the total electronic energy is given by the well-known formula... 

Here, hi is the operator for kinetic and electron nuclear attraction energies, and / 2 is the two electron operator (l/r,). Since the Gaussian orbitals in eqn (1-A-l) form a non-orthogonal basis set, the following transformation can be used in order to convert it to an orthogonal basis set ... 

Ivanciuc, O., Taraviras, S.L. and Cabrol-Bass, D. (2000c). Quasi-Orthogonal Basis Sets of Molecular Graph Descriptors as a Chemically Diversity Measure. J.Chem.lnf.Comput.ScL, 40, 126-134. 

The set <5) = (pi) i=i,n forms a carrier space which is in one to one correspondence with the elements of the orbit Q, H c G). An orthogonal basis set for ) may then always be defined by forming the h — 1 traceless combinations of these n components. As an example in the case of a tetrahedron an arbitrary function space, transforming as T2, will have exactly one component which is totally symmetric under a Csv subgroup, and which we will label as a)- Four such components can be formed, one for each trigonal site. The T2 basis may then be expressed (up to... 

For extending the preceding definitions to the case of non-orthogonal basis set, two procedures have been devised, which can be reconciled on the basis of a more physical definition of charges based on dipole moments (vide infra) ... 

Principle components analysis (PCA), a form of factor analysis (FA), is one of the most common unsupervised methods used in the analysis of NMR data. Also known as Eigenanalysis or principal factor analysis (PEA), this method involves the transformation of data matrix D into an orthogonal basis set which describes the variance within the data set. The data matrix D can be described as the product of a scores matrix T, and a loading matrix P,... 

The charge most often calculated within a ZDO calculation stems from the orthogonalized basis set. This quantity is similar to that obtained by... 

Other Functional Series with Orthogonal Basis Sets... 

D. Cabrol-Bass, Quasi-orthogonal basis sets of molecular graph descriptors as a chemical diversity measure, J. Chem. Inf. Comput. Sci. 2000, 40, 126-134. 

A feature common to the semi-empirical methods is that the overlap matrix, S (in Equation (2.225)), is set equal to the identity matrix I. Thus all diagonal elements of the overlap matrix are equal to 1 and all off-diagonal elements are zero. Some of the off-diagonal elements would naturally be zero due to the use of orthogonal basis sets on each atom, but in addition the elements that correspond to the overlap between two atomic orbitals on different atoms are also set to zero. The main implication of this is that the Roothaan-Hall equations are simplified FC = SCE becomes FC = CE and so is immediately in standard matrix form. It is important to note that setting S equal to the identity matrix does not mean that aU overlap integrals are set to zero in the calculation of Fock matrix elements. Indeed, it is important specifically to include some of the overlaps in even the simplest of the semi-empirical models. 

The principal computational steps are first to orthogonalize the Xijk basis set, and then to diagonalize the Hamiltonian matrix H in the orthogonalized basis set so as to satisfy the Rayleigh-Schrbdinger variational principle... 

The Lowdin method is equivalent to a population analysis of the density matrix in the orthogonalized basis set (Section 13.2) formed by transforming the original set of functions by... 

Orthonormal Basis An orthogonal basis set with all the basis vectors having length 11, 11 = 1. Thus, for the orthonormal basis set we have = 5,y-. 

Using all the above, Cizek presents the explicit, spin orbital and spin-adapted CC doubles equations (CCD) i.e., T = T2 (then called coupled-pair many-electron theory) in terms of one- and two-electron integrals over an orthogonal basis set. Assisted by Joe Paldus with some computations, he also reports some CCD results for N2, which though limited to only ti to Ug excitations, uses ab initio integrals in an Slater-type-orbital basis. He also does the full Cl calculation to assess convergence, a tool widely used in Cizek s and Paldus work and by most of us, today. He also reports results for the minimum-basis TT-electron approximation to benzene. 

The effective Hamiltonian and the metric (or overlap) operator J are Hermitian. The equation looks like those frequently used for solving the Schrodinger equation by means of a non-orthogonal basis set, as for instance in valence-bond (VB) theory. The operators Jf and y seem to have interesting transferability potentialities but up to now they do not seem to have been used for practical investigations. A more general approach has recently been given by Suzuki. 

The choice of the basis set. One may perform a new S transform for the H conformation and identify the new orthogonalized AOs with that of the bicentric problem. This is the most direct solution but the orthogonaliz-ation tails will be different and the use in this new basis of the effective bielectronic Hamiltonian given in Eq. (136) for instance may result in uncontrolled effects. One may also express the bielectronic operator of Eq. (136) in the non-orthogonal basis set and calculate the Hamiltonian matrix of H in the basis of non-orthogonal determinants, antisymmetrized products of Is AOs. The problem to solve is then of H-ES) type and it faces a typical non-orthogon ity problem of VB methods, which has been a major drawback of these approaches. 

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