Combinations, linear orthogonal

Neither of these wavefunctions generates the uncontaminated spectroscopic ground-states of the separated reactants. These states are obtained via the linear combination P = /i - V2/2 which ensures that each of the the three degenerate Ms = 1 components of both 02 and R contributes to the ground-state resonance scheme for the reactants. This linear combination is orthogonal to /2, which generates 5 = 0 excited states for the reactants. 

LCOAO linear combination of orthogonalized atomic orbitals... 

The AO of the two boron atoms, f (2 ) and f (B2) which are approximately sp3 and the orbital of hydrogen atom f (H) which is an s orbital have similar energy and the appreciable spatial overlap, but it is only the combination f (B ) + f (B2) which has the correct symmetry to combine linearly with f (.H). The three normalized and orthogonal MOs have the approximate form. 

The derivation of the product operator formalism from the density matrix is relatively straightforward. Starting with the density matrix of an arbitrary defined spin system, the density matrix is expanded into a linear combination of orthogonal matrices, the so-called product operators which specify an orthogonal coherence component... 

Equation (23.12) shows that when S is not zero (R is not infinite), i/ j and ij/n are not orthogonal. Consequently, they are no longer proper wave functions f or the system when R is not infinite. Two orthogonal wave functions can be constructed from and J/n by taking them in linear combination. These orthogonal wave functions serve as the first approximate wave functions of the system. 

The above four linear combinations are orthogonal to each other. Therefore, following the same method as developed above, we get... 

By a linear orthogonal transformation of axes we shall mean a change from one Cartesian system to another with the same origin. Let ei, C2 and es be the base vectors and x, X2 and jcs the coordinate axes of the first system and e j, and Cj the base vectors and and the coordinate axes of the second one. The vectors of the second base can be expressed as linear combination of the vectors of the first one... 

The second valence-only model starts from a molecular orbital viewpoint and was derived in the mid 1970s by Hay, Thibeault and Hoffmann (HTH) [2], approximately at the same time as the Kahn-Briat model. The magnetic orbitals are defined as linear combinations of orthogonal atomic-like orbitals... 

The Cl formalism in semiempirical methods is in principal the same as in ab initio methods. The MOs xj/-, are obtained as linear combinations of orthogonalized atomic orbitals (OAOs) A. (equation 1). 

According to the Karhunen-Loeve (K-L) theorem, a stochastic process on a bounded interval can be represented as an infinite linear combination of orthogonal functions, the coefficients of which constitute uncorrelated random variables. The basis functions in K-L expansions are obtained by eigendecomposition of the autocovariance function of the stochastic process and are shown to be its most optimal series representation. The deterministic basis functions, which are orthonormal, are the eigenfunctions of the autocovariance function and their magnitudes are the eigenvalues. The Karhunen-Loeve expansion converges in the mean-square sense for any distribution of the stochastic process (Papoulis and Pillai 2002). A K-L representation of a zero-mean stochastic process f(t, 6) can be represented in the form... 

A set of eigenfunctions of a Hermitian operator wifh fhe same eigenvalues can be transformed (linearly combined) into orthogonal functions while remaining eigenfunctions of the operator. 

In orthogonal collocation, the trial function given in Elquation 2.23 is written in terms of linear combinations of orthogonal pol3momials Pm x) of the order 1 to m - - 1, with Pq as the starting point. 

Another approach is spin-coupled valence bond theory, which divides the electrons into two sets core electrons, which are described by doubly occupied orthogonal orbitals, and active electrons, which occupy singly occupied non-orthogonal orbitals. Both types of orbital are expressed in the usual way as a linear combination of basis functions. The overall wavefunction is completed by two spin fimctions one that describes the coupling of the spins of the core electrons and one that deals with the active electrons. The choice of spin function for these active electrons is a key component of the theory [Gerratt ef al. 1997]. One of the distinctive features of this theory is that a considerable amount of chemically significant electronic correlation is incorporated into the wavefunction, giving an accuracy comparable to CASSCF. An additional benefit is that the orbitals tend to be... 

If a 33 = 0, we have a i3 = 0, and the function 03 is then a linear combination of the functions 0X and 0 2 and should be omitted in the orthogonalization process, which is here simply accomplished by means of the Gaussian elimination technique developed for solving equation systems. The connection between the matrices a and a may be written in the form ... 

The classical method is known as Schmidt orthogonalization. In the general step, the im column of A has added to it a linear combination of... 

The form of the functions may be closely similar to that of the molecular orbitals used in the simple theory of metals. If there are M interatomic positions in the crystal which might be occupied by any one of the N electron-pair bonds, then the M functions linear aggregates that approximate the solutions of the wave equation with inclusion of the interaction terms representing resonance. This combination can be effected with use of Bloch factors ... 

One attitude would consist in restoring symmetry by a symmetric superposition of the degenerate and linearly independent but non orthogonal symmetry-broken solutions, considering the gerade and ungerade combinations of the A+A and AA solutions in the... 

To build up vin the cluster function (1) we use the functions (PvA vA2---fvBi 9vs2 -- all of which satisfy the strong orthogonality condition in the sense of to (2), but do not satisfy the strong orthogonality needed for (1) We therefore consider the linear combination... 

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