LCAO-MO linear combination of atomic

Mulliken, R.S., Electronic population analysis on LCAO-MO [linear combination of atomic orbital-molecular orbital] molecular wave functions. I, J. Chem. Phys., 23, 1833-1840, 1955. 

R. S. Mulliken.y. Chew. Phys., 23, 1833 (1955). Electronic Population Analysis on LCAO-MO (Linear Combination of Atomic Orbital-Molecular Orbital) Molecular Wave Functions. 

LCAO-MO (Linear Combination of Atomic Orbitals-Molecular Orbital) Molecular Wave Functions. 

LCAO MO Linear Combination of Atomic Orbitals Molecular Orbitals The expression of molecular orbitals (MO) as a linear eombination of atomic orbitals (LCAO). 

The energy spectrum of the dodecaborides (YB,2, YbB,2, LuBj2> computed by the MO-linear combination of atomic orbitals (LCAO) method shows that the direction of the electron transfer is uncertain". ... 

There are a variety of methods for the computation of the MOs that interact in the transition states of [4+2]-cycloadditions. The LCAO method (linear combination of atomic orbitals) is often employed, and the basic idea is as follows. The MOs of the -systems of alkenes, conjugated polyenes, or conjugated polyenyl cations, radicals, or anions all are built by so-called linear combinations of 2p AOs. In a somewhat casual formulation, one might say that the MOs of these -systems are constructed with the help of the 2pz AOs. These AOs are centered at the positions of the n C atoms that are part of the -system. LCAO computations describe a conjugated -electron system that extends over n s/ 2-hybridized C atoms by way of n Ji-type MOs. 

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. 

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... 

You can interpret results, including dipole moments and atomic charges, using the simple concepts and familiar vocabulary of the Linear Combination of Atomic Orbitals (LCAO)-molecular orbital (MO) theory. 

To compute molecular orbitals, you must give them mathematical form. The usual approach is to expand them as a linear combination of known functions, such as the atomic orbitals of the constituent atoms of the molecular system. If the atomic orbitals, (Is, 2s, 2px, 2py, 2pz, etc.) are denoted as then this equation describes the molecular orbitals as linear combination of atomic orbitals (MO-LCAO) ... 

The basis of constructing the MOs is the linear combination of atomic orbitals (LCAO) method. This takes account of the fact that, in the region close to a nucleus, the MO wave function resembles an AO wave function for the atom of which the nucleus is a part. It is reasonable, then, to express an MO wave function 1/ as a linear combination of AO wave functions Xi on both nuclei ... 

Each MO is expanded in terms of the basis functions, conventionally called atomic orbitals (MO=LCAO, Linear Combination of Atomic Orbitals), although they are generally not solutions to the atomic HF problem. 

In MO calculations, a wave function is formulated that is a linear combination of the atomic orbitals that have overlapped (this method is often called the linear combination of atomic orbitals, or LCAO). Addition of the atomic orbitals gives the bonding MO ... 

The VB and the MO methods are rooted in very different philosophies of describing molecules. Although at the outset each method leads to different approximate wave functions, when successive improvements are made the two ultimately converge to the same wave function. In both the VB and MO methods, an approximate molecular wave function is obtained by combining appropriate hydrogen-like orbitals on each of the atoms in the molecule. This is called the linear combination of atomic orbitals (LCAO) approximation. 

In Eq. (2.30), F is the Fock operator and Hcore is the Hamiltonian describing the motion of an electron in the field of the spatially fixed atomic nuclei. The operators and K. are operators that introduce the effects of electrons in the other occupied MOs. Hence, when i = j, J( (= K.) is the potential from the other electron that occupies the same MO, i ff IC is termed the exchange potential and does not have a simple functional form as it describes the effect of wavefunction asymmetry on the correlation of electrons with identical spin. Although simple in form, Eq. (2.29) (which is obtained after relatively complex mathematical analysis) represents a system of differential equations that are impractical to solve for systems of any interest to biochemists. Furthermore, the orbital solutions do not allow a simple association of molecular properties with individual atoms, which is the model most useful to experimental chemists and biochemists. A series of soluble linear equations, however, can be derived by assuming that the MOs can be expressed as a linear combination of atomic orbitals (LCAO)44 ... 

The various MO calculations use different basis sets and have different ways of calculating multicenter coulomb and exchange integrals. The current trend in MO is to expand as a linear combination of atomic orbitals (LCAO). The atomic orbitals are represented by Slater functions with expansion in gaussian functions, taking advantage of the additive rule. When the calculation is performed in this... 

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... 

Most of the commonly used electronic-structure methods are based upon Hartree-Fock theory, with electron correlation sometimes included in various ways (Slater, 1974). Typically one begins with a many-electron wave function comprised of one or several Slater determinants and takes the one-electron wave functions to be molecular orbitals (MO s) in the form of linear combinations of atomic orbitals (LCAO s) (An alternative approach, the generalized valence-bond method (see, for example, Schultz and Messmer, 1986), has been used in a few cases but has not been widely applied to defect problems.)... 

The MO theory differs greatly from the VB approach and the basic MO theory is an extension of the atomic structure theory to molecular regime. MOs are delocalized over the nuclear framework and have led to equations, which are computationally tractable. At the heart of the MO approach lies the linear combination of atomic orbitals (LCAO) formahsm... 

In the MO approach molecular orbitals are expressed as a linear combination of atomic orbitals (LCAO) atomic orbitals (AO), in return, are determined from the approximate numerical solution of the electronic Schrodinger equation for each of the parent atoms in the molecule. This is the reason why hydrogen-atom-like wavefunctions continue to be so important in quantum mechanics. Mathematically, MO-LCAO means that the wave-functions of the molecule containing N atoms can be expressed as... 

Equation 1.132 has the property of never giving an energy that is lower than the true energy resonance principle cf Pauling, 1960). This property allows us to assign to the MO wave functions that are obtained by linear combination of the AO functions of the separate atoms (Linear Combination of Atomic Orbitals, or LCAO, method), by progressive adjustment of the combinatory parameters, up to achievement of the lowest energy. 

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