Functions linear combination

CALCULATION OF THE ELECTRONIC STRUCTURE OF ANTIFERROMAGNETIC CHROMIUM WITH A SINUSOIDAL SPIN DENSITY WAVE BY THE METHOD OF DIRAC FUNCTION LINEAR COMBINATION... 

As an example, to construct the character table for the Oh symmetry group we could apply the symmetry operations of the ABg center over a particularly suitable set of basis functions the orbital wavefunctions s, p, d,... of atom (ion) A. These orbitals are real functions (linear combinations of the imaginary atomic functions) and the electron density probability can be spatially represented. In such a way, it is easy to understand the effect of symmetry transformations over these atomic functions. 

Such a wave function is called an LCAO wave function ( linear combination of atomic orbitals ). 

We encountered basis sets in Sections 4.3.4,4.4.1.2, and 5.2.3.6.L A basis set is a set of mathematical functions (basis functions), linear combinations of which yield molecular orbitals, as shown in Eqs. 5.51 and 5.52. The functions are usually, but not invariably, centered on atomic nuclei (Fig. 5.7). Approximating molecular orbitals as linear combinations of basis functions is usually called the LCAO or linear combination of atomic orbitals approach, although the functions are not... 

Forming the basis of the AOs from contractions of Gaussian functions (linear combinations of a set of functions with constant coefficients) times angular functions followed by evaluating the overlap matrix S in this basis set. 

Basis functions In the context of solutions of the electronic Schrodinger s equation for hydrogen-bonded system, this term refers to Gaussian functions of the form exp[-ar ], where r is the position vector, multiplied by powers of the coordinates x, y, and z. The basis functions are usually located at the nuclear positions and near the midpoint of the hydrogen bond (midbond functions). Linear combinations of such basis functions form molecular orbitals. 

We considered only one-determinantal wave functions but the result of Eq. (4.27) holds for multideterminantal wave functions (linear combination of determinants) as well. 

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. 

The two diabatic nuclear wave functions xf and x can be expressed as linear combinations of auxiliary nuclear wave functions and, respec-... 

MCSCF methods describe a wave function by the linear combination of M configuration state functions (CSFs), with Cl coefficients, Ck,... 

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],... 

Adopting the view that any theory of aromaticity is also a theory of pericyclic reactions [19], we are now in a position to discuss pericyclic reactions in terms of phase change. Two reaction types are distinguished those that preserve the phase of the total electi onic wave-function - these are phase preserving reactions (p-type), and those in which the phase is inverted - these are phase inverting reactions (i-type). The fomier have an aromatic transition state, and the latter an antiaromatic one. The results of [28] may be applied to these systems. In distinction with the cyclic polyenes, the two basis wave functions need not be equivalent. The wave function of the reactants R) and the products P), respectively, can be used. The electronic wave function of the transition state may be represented by a linear combination of the electronic wave functions of the reactant and the product. Of the two possible combinations, the in-phase one [Eq. (11)] is phase preserving (p-type), while the out-of-phase one [Eq. (12)], is i-type (phase inverting), compare Eqs. (6) and (7). Normalization constants are assumed in both equations ... 

In this chapter, we resfiict the discussion to elementary chemical reactions, which we define as reactions having a single energy bamer in both dhections. As discussed in Section I, the wave function R) of any system undergoing an elementary reaction from a reactant A to a product B on the ground-state surface, is written as a linear combination of the wave functions of the reactant, A), and the product, B) [47,54] ... 

The task is now to calculate the structure and energy of the system in the transition state between A and B. Its wave function is assumed to be constmcted from a linear combination of the two. It is convenient to use VB terminology for this purpose. Let the wave function of A be denoted by a VB function A) and that of B by B). 

Now, consider a complex nucleai wave function given by a linear combination of the two real nuclear wave functions [42,53],... 

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,... 

These new wave functions are eigenfunctions of the z component of the angular momentum iij = —with eigenvalues = +2,0, —2 in units of h. Thus, Eqs. (D.l 1)-(D.13) represent states in which the vibrational angular momentum of the nuclei about the molecular axis has a definite value. When beating the vibrations as harmonic, there is no reason to prefer them to any other linear combinations that can be obtained from the original basis functions in... 

The solution to this problem is to use more than one basis function of each type some of them compact and others diffuse, Linear combinations of basis Functions of the same type can then produce MOs with spatial extents between the limits set by the most compact and the most diffuse basis functions. Such basis sets arc known as double is the usual symbol for the exponent of the basis function, which determines its spatial extent) if all orbitals arc split into two components, or split ualence if only the valence orbitals arc split. A typical early split valence basis set was known as 6-31G 124], This nomenclature means that the core (non-valence) orbitals are represented by six Gaussian functions and the valence AOs by two sets of three (compact) and one (more diffuse) Gaussian functions. 

To com pnte molecular orbitals, yon must give them mathematical Torm. The usual approach is to expand them as a linear combination ofkiiown function s, such as th e atom ic orbitals oT th e con stit-... 

HyperChem uses single detenu in am rather than spin-adapted wave fn n ction s to form a basis set for th e wave Fin ciion sin a con -figuration interaction expansion. That is, HyperChem expands a Cl wave function, m a linear combination of single Slater deterniinants P,... 

To this pom t, th e basic approxmi alien is th at th e total wave I lnic-tion IS a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ah miiio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ah mitio calculation. However, there are two main things to be considered m the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222). will require the few esl possible term s for an accurate representation of a molecular orbital. The second one is the speed of tW O-electron integral calculation. 

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