Molecular orbitals LCAO method linear combination

The quantum mechanics methods in HyperChem differ in how they approximate the Schrodinger equation and how they compute potential energy. The ab initio method expands molecular orbitals into a linear combination of atomic orbitals (LCAO) and does not introduce any further approximation. 

To apply the variational method, we first need to have a trial function. When the system under study is a molecule composed of n atoms, the trial function, which is to become a molecular orbital, is a linear combination of atomic orbitals (LCAO) ... 

The second comment which it is appropriate to make here is that we can already see the reason for o-k separation by this method. In the LCAO-scheme we represent a molecular orbital as a linear combination of atomic orbitals, as per equation (2-1). We are going now to ask what kind of conclusion would be reached if, in the summation of equation (2-1), we supposed that some of the were u-orbitals, in addition to the orbitals of Ji-symmetry... 

Any CO will be a linear combination of such Bloch functions, each corresponding to a given /. This is equivalent to the LCAO expansion for molecular orbitals, the only difference is that we have cleverly preorganized the atomic orbitals (of one type) into symmetry orbitals (Bloch functions). Hence, it is indeed appropriate to call this approach as the LCAO CO method (Linear Combination of Atomic Orbitals — Crystal Orbitals), analogous to the LCAO MO (cf. p. 429). There is, however, a problem. Each CO should be a linear combination of the for various types of x and for various k. Only then would we have the full analogy a molecular orbital is a linear combination of all the atomic orhitals belonging to the atomic basis seL ... 

Of the various methods of approximating the correct molecular orbitals, only the linear combination of atomic orbitals (LCAO) is discussed. Consider two atoms A and B whose atomic orbitals are described by wave functions and titg. When these orbitals approach each other, the electron clouds of the two orbitals overlap and the wave function of the molecular orbital is given by a linear combination of atomic orbitals (LCAO). 

In general, semiempirical methods use molecular orbitals composed of linear combination of atomic orbitals (the LCAO approximation). 

The molecular orbital approach to chemical bonding rests on the notion that as elec trons m atoms occupy atomic orbitals electrons m molecules occupy molecular orbitals Just as our first task m writing the electron configuration of an atom is to identify the atomic orbitals that are available to it so too must we first describe the orbitals avail able to a molecule In the molecular orbital method this is done by representing molec ular orbitals as combinations of atomic orbitals the linear combination of atomic orbitals molecular orbital (LCAO MO) method... 

HyperChem uses the Linear Combination of Atomic Orbitals-Molecular Orbital (LCAO-MO) approximation for all ofitsnl) initio semi-empirical methods. If /j represents a molecular orbital and... 

The coefficients indicate the contribution of each atomic orbital to the molecular orbital. This method of representing the molecular orbital wave function in terms of combinations of atomic orbital wave functions is known as the linear combination of atomic orbitals approximation (LCAO). The combination of atomic orbitals chosen is called the basis set. 

The molecular orbitals used in these calculations are generated using the linear combination of atomic orbitals (LCAO) method, in which an orbital is... 

Various theoretical methods and approaches have been used to model properties and reactivities of metalloporphyrins. They range from the early use of qualitative molecular orbital diagrams (24,25), linear combination of atomic orbitals to yield molecular orbitals (LCAO-MO) calculations (26-30), molecular mechanics (31,32) and semi-empirical methods (33-35), and self-consistent field method (SCF) calculations (36-43) to the methods commonly used nowadays (molecular dynamic simulations (31,44,45), density functional theory (DFT) (35,46-49), Moller-Plesset perturbation theory ( ) (50-53), configuration interaction (Cl) (35,42,54-56), coupled cluster (CC) (57,58), and CASSCF/CASPT2 (59-63)). 

For explanation of experimental results and for correlation of charge densities with NMR data, semiempirical quantum-chemical calculations of benzo[c]pyrylium cation have been employed. Interestingly, the first calculation of 1,3-dimethyl-benzo[r]pyrylium cation by the simple linear combination of atomic orbitals/molecular orbital (LCAO/MO) method (70KGS1308) revealed a preference for the resonance from a in which the value of the charge density at C was three times as much as at C3. 

This is the classical and general LCAO approximation (linear combination of atomic orbitals) of the molecular orbital method. 

A crystalline solid can be considered as a huge, single molecule subsequently, the electronic wave functions of this giant molecule can be constructed with the help of the molecular orbital (MO) methodology [19]. That is, the electrons are introduced into crystal orbitals, which are extended along the entire crystal, where each crystal orbital can accommodate two electrons with opposite spins. A good approximation for the construction of a crystal MO is the linear combination of atomic orbitals (LCAO) method, where the MOs are constructed as a LCAO of the atoms composing the crystal [19]. 

As was emphasized before (cf. Chapter 3), a molecule is not simply a collection of its constituting atoms. Rather, it is a system of atomic nuclei and a common electron distribution. Nevertheless, in describing the electronic structure of a molecule, the most convenient way is to approximate the molecular electron distribution by the sum of atomic electron distributions. This approach is called the linear combination of atomic orbitals (LCAO) method. The orbitals produced by the LCAO procedure are called molecular orbitals (MOs). An important common property of the atomic and molecular orbitals is that both are one-electron wave functions. Combining a certain number of one-electron atomic orbitals yields the same number of one-electron molecular orbitals. Finally, the total molecular wave function is the... 

The electron shell of molecules is constructed by linear combination of atomic orbitals (LCAO method) of the participating atoms to form bonding ct and n molecular orbitals (MOs), non-bonding n (lone pairs of electrons) and anti-bonding... 

Molecular orbital theory originated from the theoretical work of German physicist Friederich Hund (1896-1997) and its apphcation to the interpretation of the spectra of diatomic molecules by American physical chemist Robert S. MuUiken (1896-1986) (Hund, 1926, 1927a, b Mulliken, 1926, 1928a, b, 1932). Inspired by the success of Heitler and London s approach, Finklestein and Horowitz introduced the linear combination of atomic orbitals (LCAO) method for approximating the MOs (Finkelstein and Horowitz, 1928). The British physicist John Edward Lennard-Jones (1894-1954) later suggested that only valence electrons need be treated as delocalized inner electrons could be considered as remaining in atomic orbitals (Lennard-Jones, 1929). 

It is well known that one of the standard approaches to Eq. (2.1) is the linear combination of atomic orbitals (LCAO) method it consists in expanding the states of the solid in linear combination of atomic (or molecular) orbitals of the composing atoms (or molecules). This method, when not applied in oversimplified form, provides an accurate description of core and valence bands in any type of crystal (metals, semiconductors, and insulators). Applied with some caution, the method also provides precious information on lowest lying conduction States, replacing whenever necessary atomic orbitals with appropriate localized orbitals. ... 

Of the various methods of approximaliitg the correct moiecular orbitals, we shall discuss only one- the linear combinaben of atomic orbitals (LCAO) method. We assume that we can approximate the correct molecular orbitals by combining the atomic orbitals of the atoms that form the molecule. The rationale is that most of the time the electrons will be nearer and hence controlled by one or the other of the two nuclei, and when this is so, the molecular orbital should be very nearly the same as the atomic orbital for that atom. The basic process is the same as the one wc employed in construaing hybrid atomic orbitals except that now we are coniiining orbitals on different atoms to form new orbitals that are associated with the entire molecule. We... 

This highly successful qualitative model parallels the most convenient quantum mechanical approach to molecular orbitals the method of linear combination of atomic orbitals (LCAO). We have assumed that the shapes and dispositions of bond orbitals are related, in a simple way to the shapes and dispositions of atomic orbitals. The LCAO method makes the same assumption mathematically to... 

Normally these molecular orbitals are obtained as expansions in a set of atom-centred basis functions (the linear combination of atomic orbitals (LCAO) method), m being the number of such functions. Recently, two-dimensional numerical integration methods have been developed to solve the MCSCF equations for linear molecules. The dimension m is then, in principle, infinite (practice, it is determined by the size of the grid used in the numerical integration). The molecular-orbital space is further divided into three subspaces the inactive, the active and the external orbitals. The inactive and active subspaces constitute the internal (occupied) orbital subspace, while the external orbitals are unoccupied. The CASSCF wavefunction is formed as a linear combination of configuration state functions (CSFs) generated from these orbitals in the following way. 

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