Linearly independent functions

In addition to the growth of the small component p space, this also leads to another possible problem. If the large component s and d functions are optimized independently, they may well have exponents that are quite close in value. This is no problem for the large component, where symmetry makes these functions linearly independent. However, for the small component, these end up in the same symmetry, and one may in fact encounter serious linear dependencies. Again, this may be removed by projection or other methods, but from the point of view of basis sets size it is very unfortunate to fill the primitive space with functions that overlap strongly. 

The electronic wave functions of the different spin-paired systems are not necessarily linearly independent. Writing out the VB wave function shows that one of them may be expressed as a linear combination of the other two. Nevertheless, each of them is obviously a separate chemical entity, that can he clearly distinguished from the other two. [This is readily checked by considering a hypothetical system containing four isotopic H atoms (H, D, T, and U). The anchors will be HD - - TU, HT - - DU, and HU -I- DT],... 

Note that only the polynomial factors have been given, since the exponential parts are identical for all wave functions. Of course, any linear combination of the wave functions in Eqs. (D.5)-(D.7) will still be an eigenfunction of the vibrational Hamiltonian, and hence a possible state. There are three such linearly independent combinations which assume special importance, namely,... 

Any linearly independent set of simultaneous homogeneous equations we can construct has only the zero vector as its solution set. This is not acceptable, for it means that the wave function vanishes, which is contrai y to hypothesis (the electron has to be somewhere). We are driven to the conclusion that the normal equations (6-38) must be linearly dependent. 

According to the Floquet theorem [Arnold 1978], this equation has a pair of linearly-independent solutions of the form x(z,t) = u(z, t)e p( 2nizt/p), where the function u is -periodic. The solution becomes periodic at integer z = +n, so that the eigenvalues e we need are = ( + n). To find the infinite product of the we employ the analytical properties of the function e z). It has two simple zeros in the complex plane such that... 

The matrices F and M can be found from straightforward integration of (5.9) with the initial conditions being N linearly independent vectors. Then the quasienergy partition function equals... 

Thus, one can be far from the ideal world often assumed by statisticians tidy models, theoretical distribution functions, and independent, essentially uncorrupted measured values with just a bit of measurement noise superimposed. Furthermore, because of the costs associated with obtaining and analyzing samples, small sample numbers are the rule. On the other hand, linear ranges upwards of 1 100 and relative standard deviations of usually 2% and less compensate for the lack of data points. 

In general, the function cp obtained by the application of the operator A on an arbitrary function ip, as expressed in equation (3.1), is linearly independent of Ip. However, for some particular function 0i, it is possible that... 

Wave functions for the orbitals of molecules are calculated by linear combinations of all wave functions of all atoms involved. The total number of orbitals remains unaltered, i.e. the total number of contributing atomic orbitals must be equal to the number of molecular orbitals. Furthermore, certain conditions have to be obeyed in the calculation these include linear independence of the molecular orbital functions and normalization. In the following we will designate wave functions of atoms by % and wave functions of molecules by y/. We obtain the wave functions of an H2 molecule by linear combination of the Is functions X and of the two hydrogen atoms ... 

It should be noted that the functions Xn need not necessarily form an orthonormal set The linearly independent coefficients c can be considered to be variable parameters that are determined by minimization of the variational energy, W. If the functions Xn are not orthonormal, Eq. (105) can be rewritten in the form... 

The electronic wave functions of the different spin-paired systems are not necessarily linearly independent. Writing out the VB wave function shows that one of them may be expressed as a linear combination of the other two. 

This result can be generalized into the statement that any arbitrary vector in n dimensions can always be expressed as a linear combination of re basic vectors, provided these are linearly independent. It will be shown that the latent solutions of a singular matrix provide an acceptable set of basis vectors, just like the eigen-solutions of certain differential equations provide an acceptable set of basis functions. 

Here functions Qnt X), Qj(X), and QP(X) can be determined experimentally using calibration samples. If these functions are linear independent then the parameters Ank, A, and Ap can be uniquely determined from the variation of P /1, , n2,. .. /( . /. / considered as a function of X. In particular, the side effects, i.e., the temperature and pressure dependences, can be eliminated from the transmission spectrum. The sensing method based on this simple idea was applied in Ref. 69 for determination of microfluidic refractive index changes in two microcapillaries coupled to a single MNF illustrated in Fig. 13.26c. The developed approach allowed to compensate the side temperature and pressure variation effects. 

However, care must be taken to avoid the singularity that occurs when C is not full rank. In general, the rank of C will be equal to the number of random variables needed to define the joint PDF. Likewise, its rank deficiency will be equal to the number of random variables that can be expressed as linear functions of other random variables. Thus, the covariance matrix can be used to decompose the composition vector into its linearly independent and linearly dependent components. The joint PDF of the linearly independent components can then be approximated by (5.332). 

That a is nonracemic means that it does not identically annihilate all chirality functions, i.e., that at least one of the coefficients , with j different from zero. Qualitative completeness for % means, therefore, that it is not annihilated by any of the ety with j nor by any linear combination of them. This means that % must possess zr components belonging to each 71linearly independent, but also all functions ey must be linearly independent. For example, if zr= 2, it will not do to have xV —etx since in this case the chirality function % = x l + y(2 would be annihilated by the chiral ensemble operator a — e( l + —... 

In view of the preceding considerations it should be emphasized that it is incorrect to talk about the self-consistent-field molecular orbitals of a molecular system in the Hartree-Fock approximation. The correct point of view is to associate the molecular orbital wavefunction of Eq. (1) with the N-dimen-sional linear Hilbert space spanned by the orbitals t/2,... uN any set of N linearly independent functions in this space can be used as molecular orbitals for forming the antisymmetrized product. 

The different r " terms in the modified function are not linearly independent and, therefore, inclusion of additional terms does not guarantee the improvement in the fit. Buckingham function also suffers from the fact that as r —> 0,... 

Ordinary Points of a Linear Differential Equation. We shall have occasion to discuss ordinary linear differential equations of the second order with variable coefficients whose solutions cannot he obtained in terms of Lhe elementary functions of mathematical analysis, la such cases one of the standard procedures is to derive n pair of linearly independent solutions in the form ofinfinite series and from these series to compute tables of standard solutions. With the aid of such tables the solution appropriate to any given initial conditions may then he readily found. The object of this note is to outline briefly the procedure to he followed in these instances for proofs of the theorems... 

Within the Matlab s numerical precision X is singular, i.e. the two rows (and columns) are identical, and this represents the simplest form of linear dependence. In this context, it is convenient to introduce the rank of a matrix as the number of linearly independent rows (and columns). If the rank of a square matrix is less than its dimensions then the matrix is call rank-deficient and singular. In the latter example, rank(X)=l, and less than the dimensions of X. Thus, matrix inversion is impossible due to singularity, while, in the former example, matrix X must have had full rank. Matlab provides the function rank in order to test for the rank of a matrix. For more information on this topic see Chapter 2.2, Solving Systems of Linear Equations, the Matlab manuals or any textbook on linear algebra. 

On the other hand, the Lie bracket of the functions h x),Lfh x), U j h x) is linearly independent. Therefore, it can be proved that necessarily r < n and that the linearly independent functions qualify as a set of new coordinate functions around the point x°. 

The singlet spin function 0 q for the valence electrons (where the two subscripts indicate the eigenvalues of and 2 for the active space, S = M=Q)is expressed as a linear combination of all five linearly-independent spin-coupling modes for a singlet system of six electrons ... 

When expanded out, the determinant is a polynomial of degree n in the variable and it has n real roots if ff and S are both Hermitian matrices, and S is positive definite. Indeed, if S were not positive definite, this would signal that the basis functions were not all linearly independent, and that the basis was defective. If takes on one of the roots of Eq. (1.16) the matrix ff — is of rank... 

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