Atomic Orbital, Linear Combination LCAO

In the present approach, the KS orbitals are expanded in a set of functions related to atomic orbitals (Linear Combination of Atomic Orbitals, LCAO). These functions usually are optimized in atomic calculations. In our implementation a basis set of contracted Gaussians VF/ is used. The basis set is in general a truncated (finite) basis set reasonably selected . 

Configuration Interaction (or electron correlation) adds to the single determinant of the Hartree-Fock wave function a linear combination of determinants that play the role of atomic orbitals. This is similar to constructing a molecular orbital as a linear combination of atomic orbitals. Like the LCAO approximation. Cl calculations determine the weighting of each determinant to produce the lowest energy ground state (see SCFTechnique on page 43). 

The raw output of a molecular structure calculation is a list of the coefficients of the atomic orbitals in each LCAO (linear combination of atomic orbitals) molecular orbital and the energies of the orbitals. The software commonly calculates dipole moments too. Various graphical representations are used to simplify the interpretation of the coefficients. Thus, a typical graphical representation of a molecular orbital uses stylized shapes (spheres for s-orbitals, for instance) to represent the basis set and then scales their size to indicate the value of the coefficient in the LCAO. Different signs of the wavefunctions are typically represented by different colors. The total electron density at any point (the sum of the squares of the occupied wavefunctions evaluated at that point) is commonly represented by an isodensity surface, a surface of constant total electron density. 

In order to describe the hydrogen molecule by quantum mechanical methods, it is necessary to make use of the principles given in Chapter 2. It was shown that a wave function provided the starting point for application of the methods that permitted the calculation of values for the dynamical variables. It is with a wave function that we must again begin our treatment of the H2 molecule by the molecular orbital method. But what wave function do we need The answer is that we need a wave function for the H2 molecule, and that wave function is constructed from the atomic wave functions. The technique used to construct molecular wave functions is known as the linear combination of atomic orbitals (abbreviated as LCAO-MO). The linear combination of atomic orbitals can be written mathematically as... 

In molecular orbital calculations, a wave function is formulated which is a linear combination of the atomic orbitals (also called LCAO) and is written as ... 

The present article is an attempt to review those studies of pyridinelike heterocycles (mono-azines) and, to a lesser extent, their analogues and derivatives that have interpreted the behavior and estimated various physico-chemical properties of the compounds by the use of data calculated by the simplest version of the MO LCAO (molecular orbital, linear combination of atomic orbitals) method (both molecular orbital energies and expansion coefficients). In this review, attention is focused upon the use of the simple method because it has been applied to quite extensive sets of compounds and to the calculation of the most diverse properties. On the other hand, many fewer compounds and physico-chemical properties have been investigated by the more sophisticated methods. Such studies are referred to without being discussed in detail. In a couple of years, we believe, the extent of the applications of such methods will also be wide enough to warrant a detailed review. 

In the molecular-orbital theory, each successive valence electron is considered as entering a field of positive electric charge furnished by the nuclei. One mathematical approach is quite extensively used as an aid in setting up a description of electron probability densities in the vicinity of more than one nucleus this approach is called the method of linear combinations of atomic orbitals (coveniently abbreviated LCAO). 

Atomic orbitals can combine and overlap to give more complex standing waves. We can add and subtract their wave functions to give the wave functions of new orbitals. This process is called the linear combination of atomic orbitals (LCAO). The number of new orbitals generated always equals the number of starting orbitals. 

Hartree-Fock-Roothaan SCF theory, using molecular orbitals expanded as linear combinations of atomic orbital basis functions (LCAO-MO theory)... 

In order to solve the Kohn-Sham equations (Eqn. (2)) we used the molecular orbital-linear combination of atomic orbitals (MO-LCAO) approach. The molecular wave functions 0j are expanded the symmetry adapted orbitals Xj) which are also expanded in terms of the atomic orbitals... 

What we have just described has its counterpart in a mathematical treatment called the LCAO (linear combination of atomic orbitals) method. In the LCAO treatment, wave functions for the atomic orbitals are combined in a linear fashion (by addition or subtraction) in order to obtain new wave functions for the molecular orbitals. 

To get approximations to higher MOs, we can use the linear-variation-function method. We saw that it was natural to take variation functions for Hj as linear combinations of hydrogenlike atomic-orbital functions, giving LCAO-MOs. To get approximate MOs for higher states, we add in more AOs to the linear combination. Thus, to get approximate wave functions for the six lowest linear combination of the three lowest m = 0 hydrogenlike functions on each atom ... 

The large number of two-electron integrals (p.v X(r) in the linear-combination-of-atomic-orbitals Hartree-Fock (LCAO-HF) approximation scales formally with N, where N is the number of basis functions. The use of direct meth-... 

In order to further describe the molecular wavefunctions or the molecular orbitals. Linear Combinations of Atomic Orbitals (LCAO) are normally used (LCAO method). Such a method of solution is possible since the directional dependence of the spherical-harmonic functions for the atomic orbitals can be used. The Pauli principle can be applied to the single-electron molecular orbitals and by filling the states with the available electrons the molecular electron configurations are attained. Coupling of the angular momenta of the open shell then gives rise to molecular terms. 

The key to most covalent models is the Linear Combination of Atomic Orbitals treatment or LCAO. Its principles, as carefully detailed by Roothaan (1951) are closely connected to Hartree-Fock calculations described by Slater (1960) and Cowan (1981). Let us recall the basic formulae and hypothesis ... 

SCF-MO- Self-consistent field-molecular orbital-linear combination of atomic LCAO orbitals... 

However, many atoms, especially carbon, form bonds with like atoms e.g., Oj, Fj, Nj, H3C-CH3, and ethane) clearly, these do not involve ions and electron transfer. In these molecules, the atomic orbitals are combined to give molecular orbitals. This is described as linear combination of atomic orbitals (LCAO). At a crude and simplistic level, we could imagine pushing together two Is orbitals in hydrogen, in phase, to form what we describe as a o-bond (Figure 1.15). 

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. 

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

ITyperChem uses the Linear Combination of Atomic Orhilals-Molecular Orbital (LCAO-MO) approximation for all of itsah initio sem i-empirical melh ods. If rg, represen is a molecular orbital and... 

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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