Combinations, linear

A high degree of correlation may be beneficial. When the parameters are strongly related, some linear combination of the two parameters may represent the data as well as do the individual parameters. In that case a method similar to that of Bruin and Praus-... 

LCAO method A method of calculation of molecular orbitals based upon the concept that the molecular orbital can be expressed as a linear combination of the atomic orbitals. 

An important application of this type of analysis is in the determination of the calculated cetane index. The procedure is as follows the cetane number is measured using the standard CFR engine method for a large number of gas oil samples covering a wide range of chemical compositions. It was shown that this measured number is a linear combination of chemical family concentrations as determined by the D 2425 method. An example of the correlation obtained is given in Figure 3.3. 

Since indistinguishability is a necessary property of exact wavefiinctions, it is reasonable to impose the same constraint on the approximate wavefiinctions ( ) fonned from products of single-particle solutions. Flowever, if two or more of the Xj the product are different, it is necessary to fonn linear combinations if the condition P. i = vj/ is to be met. An additional consequence of indistinguishability is that the h. operators corresponding to identical particles must also be identical and therefore have precisely the same eigenfiinctions. It should be noted that there is nothing mysterious about this perfectly reasonable restriction placed on the mathematical fonn of wavefiinctions. 

While the first and fourth of these are synnnetric with respect to particle iuterchauge and thereby satisfy the iudistiuguishability criterion, the other two are not and appropriate linear combinations must be fonned. Doing so, one finds the following four wavefimctions ... 

For many-electron systems such as atoms and molecules, it is obviously important that approximate wavefiinctions obey the same boundary conditions and symmetry properties as the exact solutions. Therefore, they should be antisynnnetric with respect to interchange of each pair of electrons. Such states can always be constmcted as linear combinations of products such as... 

The representation of trial fiinctions as linear combinations of fixed basis fiinctions is perhaps the most connnon approach used in variational calculations optimization of the coefficients is often said to be an application of tire linear variational principle. Altliough some very accurate work on small atoms (notably helium and lithium) has been based on complicated trial functions with several nonlinear parameters, attempts to extend tliese calculations to larger atoms and molecules quickly runs into fonnidable difficulties (not the least of which is how to choose the fomi of the trial fiinction). Basis set expansions like that given by equation (A1.1.113) are much simpler to design, and the procedures required to obtain the coefficients that minimize are all easily carried out by computers. 

These fiiuctions are linear combinations of products, where the fiiuction 4/ is an... 

Other linear combinations of simple potentials are also widely used to mimic the interactions in real systems. An example is the following. 

Taking the natural logaritlnn of (A3.1.54), we see that In+ In has to be conserved for an equilibrium solution of the Boltzmaim equation. Therefore, Incan generally be expressed as a linear combination with constant coefficients... 

In WKB theory, one generates a wavefunction that is valid in the D—> 0 limit using a linear combination of exponentials of the fomi... 

Alternatively, the electron can exchange parallel momentum with the lattice, but only in well defined amounts given by vectors that belong to the reciprocal lattice of the surface. That is, the vector is a linear combination of two reciprocal lattice vectors a and b, with integer coefficients. Thus, g = ha + kb, with arbitrary integers h and k (note that all the vectors a,b, a, b and g are parallel to the surface). The reciprocal lattice vectors a and are related to tire direct-space lattice vectors a and b through the following non-transparent definitions, which also use a vector n that is perpendicular to the surface plane, as well as vectorial dot and cross products ... 

The asymptotic kr —> oo) fomi of may then be written in temis of the following linear combinations ... 

In most of the connnonly used ab initio quantum chemical methods [26], one fonns a set of configurations by placing N electrons into spin orbitals in a maimer that produces the spatial, spin and angular momentum syimnetry of the electronic state of interest. The correct wavefimction T is then written as a linear combination of tire mean-field configuration fimctions qj = example, to describe the... 

B3.1.5.2 THE LINEAR COMBINATIONS OF ATOMIC ORBITALS TO FORM MOLECULAR ORBITALS EXPANSION OF THE SPIN ORBITALS... 

Let us express the displacement coordinates as linear combinations of a set of new coordinates y >q= Uy then AE = y U HUy. U can be an arbitrary non-singular matrix, and thus can be chosen to diagonalize the synmietric matrix H U HU = A, where the diagonal matrix A contains the (real) eigenvalues of H. In this fomi, the energy change from the stationary point is simply AF. = t Uj A 7- h is clear now that a sufBcient... 

To derive the DIIS equations, let us consider a linear combination of coordmate vectors q ... 

By combining the Lagrange multiplier method with the highly efficient delocalized internal coordinates, a very powerfiil algoritlun for constrained optimization has been developed [ ]. Given that delocalized internal coordinates are potentially linear combinations of all possible primitive stretches, bends and torsions in the system, cf Z-matrix coordinates which are individual primitives, it would seem very difficult to impose any constraints at all however, as... 

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