Orthogonal sets

Another approach involves starting with an initial wavefimction Iq, represented on a grid, then generating // /q, and consider that tiiis, after orthogonalization to Jq, defines a new state vector. Successive applications //can now be used to define an orthogonal set of vectors which defines as a Krylov space via the iteration (n = 0,.. ., A)... 

In the Lowdin approach to population analysis [Ldwdin 1970 Cusachs and Politzer 1968] the atomic orbitals are transformed to an orthogonal set, along with the molecular orbital coefficients. The transformed orbitals in the orthogonal set are given by ... 

Tie first consideration is that the total wavefunction and the molecular properties calculated rom it should be the same when a transformed basis set is used. We have already encoun-ered this requirement in our discussion of the transformation of the Roothaan-Hall quations to an orthogonal set. To reiterate suppose a molecular orbital is written as a inear combination of atomic orbitals ... 

If we can find A, we shall have found an orthogonal set of eigenvectors. It is interesting and significant to note at this point that A is only one of many equally valid orthogonal sets of eigenvectors. 

The Lowdin population analysis scheme was created to circumvent some of the unreasonable orbital populations predicted by the Mulliken scheme, which it does. It is different in that the atomic orbitals are first transformed into an orthogonal set, and the molecular orbital coefficients are transformed to give the representation of the wave function in this new basis. This is less often used since it requires more computational work to complete the orthogonalization and has been incorporated into fewer software packages. The results are still basis-set-dependent. 

The first approximation we ll consider comes from the interpretation of as a probability density for the electrons within the system. Molecular orbital theory decomposes t(/ into a combination of molecular orbitals <()j, (jij,. To fulfill some of the conditions on we discussed previously, we choose a normalized, orthogonal set of molecular orbitals ... 

This group is stable to strong acids and bases, TMSI, Pd-C/H2, DDQ, TBAF, and LAH at low temperatures and thus has the potential to participate in a large number of orthogonal sets/... 

The complementary ABO ester (2,7,8-trioxabicyclo[3.2.1]octyl ester) is prepared from the epoxy ester by rearrangement with Cp2ZrCl2/AgC104. The OBO ester is more easily cleaved by Brpnsted acids than is the ABO ester, but the ABO ester is cleaved more easily by Lewis acids, thus forming an orthogonal set. The ABO ester can be cleaved with PPTS (MeOH, H2O, 22°, 2 h LiOH) the OBO ester is cleaved at 0° in 2 min. ... 

Power Series Expansions and Formal Solutions (a) Helium Atom. If the method of superposition of configurations is based on the use of expansions in orthogonal sets, the method of correlated wave functions has so far been founded on power series expansions. The classical example is, of course, Hyl-leraas expansion (Eq. III.4) for the ground state of the He atom, which is a power series in the three variables... 

The expansion coefficients, affl, are functions of space and time, and will be determined from the Boltzmann equation. The product Vi/af1 Yf in the expansion forms a complete orthogonal set in... 

This procedure of multiplication by one of the members of the orthogonal set of functions and integration can be continued. The equations developed above will be sufficient to obtain the Navier-Stokes... 

The next step in our process is to expand b / /bz and u each into a series of suitable functions. For this purpose it seems natural to use the set of solutions of the equation of the unperturbed atom, i.e., the set w, tt), as was done by Schrodinger. Unfortunately, this set is not a complete orthogonal set unless a continuous range of complicated functions corresponding to imaginary values of / are included. To avoid this complication we follow a procedme analogous to that used by Epstein for a similar purpose, i.e., we use for our expansion another set of functions, T V, w, )> defined as follows... 

It can be shown that r(/, w, n, a) thus defined, for any constant real value of a, form a complete orthogonal set with respect to a function decreasing rapidly with increasing r, such as rd l/oldz. We shall assume a to be negative. We may, therefore, write... 

So far we have considered an orthonormal basis set x In actnal calcnlations, employing non orthogonal sets of Gaussian fnnctions with overlap matrix... 

This construction is known as the Schmidt orthogonalization procedure. Since the initial selection for 0i can be any of the original functions ip, or any linear combination of them, an infinite number of orthogonal sets 0, can be obtained by the Schmidt procedure. 

Because both and Lz are hermitian, the spherical harmonics Yim(0, q>) form an orthogonal set, so that... 

Although the functions Rniir) according to equation (6.20) form an orthogonal set with w r) = r, the orthogonal relationships do not apply to the set of functions Sxiip) with w p) = p. Since the variable p introduced in equation (6.22) depends not only on r, but also on the eigenvalue E, or equivalently on X, the situation is more complex. To determine the proper orthogonal relationships for Sxiip), we express equation (6.24) in the form... 

The Hermite polynomials Hn ) form an orthogonal set over the range —oo oo with a weighting factor e . If we equate coefficients of stY on each side of equation (D.12), we obtain... 

Thus, the associated Laguerre polynomials form an orthogonal set over the range 0 p ss 00 with a weighting factor p er< . For the case where s and t on the left-hand side have the same exponent, we pick out the term fi = am the summation over (i, giving... 

Here 0 is the Heaviside function. The projection operator formalism must be carried out in matrix from and in this connection it is useful to define the orthogonal set of variables, k,uk,5k > where the entropy density is sk = ek — CvTrik with Cv the specific heat. In terms of these variables the linearized hydrodynamic equations take the form... 

They are assumed to be known. Furthermore, it should be recalled that the eigenfunctions VrJ form a complete orthogonal set. The equation of interest is... 

Here are the basic rules of the game For a system with electron spin S, the known complete orthogonal set of 2,S + I wavefunctions is associated with the values ms and is written as... 

Any three-dimensional orthogonal coordinate system may be specified in terms of the three coordinates q, q2 and q3. Because of the orthogonality of the coordinate surfaces, it is possible to set up, at any point, an orthogonal set of three unit vectors ex, e2, e3, in the directions of increasing qx, q2, q3, respectively. It is important to select the qt such that the unit vectors define a right-handed system of axes. The set of three unit vectors defines a Cartesian coordinate system that coincides with the curvilinear system in... 

The fact that quantum observables are represented by matrices immediately suggests problems of non-commutation. For instance, the observables can be measured at the same time only if they have a complete orthogonal set of eigenvectors in common. This happens only when they commute, i.e. XY = YX, or the commutator [X, Y] = XY — YX = 0. This is a central feature of the matrix formulation of quantum theory discovered by Heisenberg, Born and Jordan while trying to explain the observed spectral transitions of the hydrogen atom in a more fundamental way than the quantization... 

In practice it is most common and convenient to side-step one of the problems associated with strong orthogonality. We can work with an orbital basis which satisfies (11) independently of the choice of the physical structure of the groups of electrons a basis for which (11) is guaranteed for all R, S. Any orbital basis which forms an orthogonal set will fulfill this condition all overlap integrals are then zero.6)... 

Just as a vector is projected as components on orthogonal axes, a given function defined on a given domain can be projected onto an orthogonal set of functions. The Fourier series decomposition of a function /(x) defined over the interval [ —X, X] is a convenient example... 

Given a finite number of measurements at a given latitude (90° — 6) and longitude on the surface of the Earth, we look for a smooth function that could be fitted to the data and represent their variations to within any desired precision. Spherical harmonics are suitable because they make an orthogonal set of functions which can... 

The determination of the projections can be regarded as a linear least-squares fit only now we have an orthogonal set of vectors V =, as in Figure 5-28, rather than a general set of non-orthogonal vectors in F in the equivalent Figure 4-12. The projected test vector tproj is a linear combination of the vectors V. 

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