Formal linear combination

X(n) parametrizes effective zero-cycles of degree n on X, i.e. formal linear combinations n,-[zj] of points X in X with coefficients n, IN fulfilling ni =n. X has a natural stratification into locally closed subschemes ... 

It seems that here the AOM offers a distinct advantage over the LFT parameterization scheme. While using the same number of parameters as the LFT scheme, indeed formal linear combinations of those parameters, it seems to account fairly well for the -orbital energies not only in octahedral but also in environments of less than cubic symmetry. The LFT approach failed to do this, as discussed in Section 6.2.2.4. It is this ability to deal with low symmetry systems by the summation of the parameters for the individual ligands that is the strength of the AOM. 

In the present section it is assumed that the components have an atomic structure, i.e. they can be conceived of as formal linear combinations of the atoms (which may in fact be atoms or ions or other particles as electrons etc.) (1), (2),... 

Nevertheless, the formal A/ scaling has spawned approaches which reduce the dependence to A/. This may be achieved by fitting the electron density to a linear combination of functions, and using the fitted density in evaluating the J integrals in the Coulomb term. 

The projection operator formalism also gives interesting aspects on the correlation problem. Previously one mainly used the secular equation (Eq. III.21) for investigating the symmetry properties of the solutions, and one was often satisfied with those approximate wave functions which were the simplest linear combinations of the basic functions having the correct symmetry. In our opinion, this problem is now better solved by means of the projection operators, and the use of the secular equations can be reserved for handling actual correlation effects. This implies also that, in place of the ordinary Slater determinants (Eq. III.17), we will essentially consider the projections of these functions as our basis. 

The problem of evaluating the effect of the perturbation created by the ligands thus reduces to the solution of the secular determinant with matrix elements of the type rp[ lICT (pk, where rpj) and cpk) identify the eigenfunctions of the free ion. Since cpt) and cpk) are spherically symmetric, and can be expressed in terms of spherical harmonics, the potential is expanded in terms of spherical harmonics to fully exploit the symmetry of the system in evaluating these matrix elements. In detail, two different formalisms have been developed in the past to deal with the calculation of matrix elements of Equation 1.13 [2, 3]. Since t/CF is the sum of one-electron operators, while cpi) and cpk) are many-electron functions, both the formalisms require decomposition of free ion terms in linear combinations of monoelectronic functions. 

Formally, we are free to choose any linear combination of the three chemical species as the reacting scalar under the condition that the combination is linearly independent of the rows of A.12 Arbitrarily choosing C3, a new scalar vector can be defined by the linear transformation13... 

In Appendix A2, we have formally applied the perturbation method to find the energy levels of a d ion in an octahedral environment, considering the ligand ions as point charges. However, in order to understand the effect of the crystalline field over d ions, it is very illustrative to consider another set of basis functions, the d orbitals displayed in Figure 5.2. These orbitals are real functions that are derived from the following linear combinations of the spherical harmonics ... 

What guidance for improving the scattering formalism can be obtained from theory In the linear combination of atomic orbitals (LCAO) formalism, a molecular orbital (MO) is described as a combination of atomic basis function ... 

Diamond (1966) has applied a filtering procedure in the refinement of protein structures, in which poorly determined linear combinations are not varied. In charge density analysis, the principal component analysis has been tested in a refinement of theoretical structure factors on diborane, B2H6, with a formalism including both one-center and two-center overlap terms (Jones et al. 1972). Not unexpectedly, it was found that the sum of the populations of the 2s and spherically averaged 2p shells on the boron atoms constitutes a well-determined eigenparameter, while the difference is very poorly determined. Correlation between one- and two-center terms was also evident in the analysis. 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

In general, then, an MCSCF calculation involves a specification of what MOs may be occupied in the CSFs appearing in the expansion of Eq. (7.1). Given that specification, the formalism finds a variational optimum for the shape of each MO (as a linear combination of basis functions) and for the weight of each CSF in the MCSCF wave function. 

This particular example illustrates what can be shown more formally to be true in general the energy of the wave function is invariant to expressing the wave function using any normalized linear combination of the occupied HF orbitals, as are the expectation values of all other quantum mechanical operators. Since all such choices of hnear combinations of orbitals satisfy the variational criterion, one may legitimately ask why the HF orbitals should be assigned any privileged status of their own as chemical entities. 

The central idea for solving Eq. 1 can be explained without the details of the mathematical formalism. With an appropriate linear combination of the variables ya we can define new variables za (called eigenfunctions) such that the resulting differential equation is ... 

An intrinsic feature of the thermodynamic formalism is the freedom to consider general combinations of extensive or intensive variables [cf. (8.70), (8.75)] as alternatives to standard choices. This freedom is used, for example, in considering the Gibbs free energy G = U — (T)S + (P)V as a linear combination of standard (U, S, V) extensities, or the phase-coexistence coordinate a [cf. (7.27), (7.28)] as a linear combination of standard (T, P) intensities. 

The occupation number vectors are basis vectors in an abstract linear vector space and specify thus only the occupation of the spin orbitals. The occupation number vectors contain no reference to the basis set. The reference to the basis set is built into the operators in the second quantization formalism. Observables are described by expectation values of operators and must be independent of the representation given to the operators and states. The matrix elements of a first quantization operator between two Slater determinants must therefore equal its counterpart of the second quantization formulation. For a given basis set the operators in the Fock space can thus be determined by requiring that the matrix elements between two occupation number vectors of the second quantization operator, must equal the matrix elements between the corresponding two Slater determinants of the corresponding first quantization operators. Operators that are considered in first quantization like the kinetic energy and the coulomb repulsion conserve the number of electrons. In the Fock space these operators must be represented as linear combinations of multipla of the ajaj... 

These results express the fact that any linear combination of conserved densities (a generalized moment density) is itself a conserved density in thermodynamics. We have shown, therefore, that if the free energy of the system depends only on K moment densities p,... pK, we can view these as the densities of K quasi-species of particles and can construct the phase diagram via the usual construction of tangencies and the lever rule. Formally this has reduced the problem to finite dimensionality, although this is trivial... 

We now turn to the second general question, regarding the choice of weight functions for the extra moment densities. Comparing Eq. (60) with the formally exact solution (59) of the coexistence problem tells us at least in principle what is required The log ratio ln/f(cr)/p(0)(cr) of the effective prior and the parent needs to be well approximated by a linear combination of the weight functions of the extra moment densities. However, the effective prior is unknown (otherwise we would already have the exact solution of the phase coexistence problem), and so this criterion is of little use [56]. 

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