Linear combination of atomic orbitals LCAO-MO

The division of the ( + 1) bonding AO s into two types, a unique radial orbital and n surface orbitals, incidentally is also a feature of a free-electron MO treatment (198) which has been applied to borane polyhedra, as distinct from the linear combination of atomic orbitals (LCAO) MO treatments mentioned above. 

The molecular orbitals (MOs) are formed by the linear combination of atomic orbitals (LCAO-MO method). For diatomic molecules, the component of the angular momentum (A) in the direction of the bond axis is now important. The energy states are expressed by the symbol... 

Let us consider the construction of molecular orbitals from linear combinations of atomic orbitals (LCAO MOs) for an octahedral molecule... 

Formed by superposition (linear combination) of atomic orbitals (LCAO-MO)... 

Spin properties are notoriously difficult to calculate accurately57. Here, we are actually calculating spin populations, with their intrinsic uncertainties, and not the directly observed hyperfine interactions. On the other hand, analyses of the hyperfine interactions in the ESR spectra to give experimental atomic orbital occupancies for the radical electron are based on a simplistic, rigid linear combination of atomic orbitals (LCAO)-MO model with the reference electron-nuclear coupling parameters taken from the free atom. No allowance is made for radial or angular polarization of the atomic orbitals in the molecular environment. Thus agreement at these levels between calculated and experimental values can only be qualitative, at best. 

Let us consider angular momentum eigenstates of it electrons in an aromatic molecule of Djvk symmetry. The z axis is set to the Cn axis. Complex MOs ytm) of the molecule are given as linear combinations of atomic orbitals (LCAO-MOs) in the form [20]... 

As we saw briefly in the previous chapter, perhaps the simplest way to represent a molecular orbital is to base it on the atomic orbitals of the atoms involved in the bonding, generally as a linear combination of atomic orbitals (LCAO-MO). Because... 

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. 

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 ... 

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". ... 

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... 

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... 

What guidance for improving the scattering formalism can be obtained from theory In the linear combination of atomic orbitals (LCAO) formalism, a molecular orbital (MO) is described as a combination of atomic basis function ... 

Hiickel MO calculations of the 7r-electron density for pyrazolo[3,4-d]-pyrimidine 290 reveal N-3 to be the most electron rich (69CJC1129). The same conclusion was reached with simple linear combination of atomic orbitals (LCAO) calculations (291). LCAO data for electron densities on pyrazolo[4,3-LCAO calculations exaggerate electronegativities of nitrogen atoms (see 293)... 

Although MO s encompass the entire molecule, it is best to visualize most of them as being localized between pairs of bonding atoms. This description of bonding is called linear combination of atomic orbitals (LCAO). 

The molecular Schrodinger equation can be solved exacdy for the case of Hj when VAB is simply the sum of two hydrogen ion potentials. In general, however, an exact solution is not possible. Following the well-worn tracks of MO theory we look instead for an approximate solution that is given by some linear combination of atomic orbitals (LCAO). Considering the AB dimer illustrated in Fig. 3.1 we write... 

The wave functions for the molecular systems are described in terms of the atomic orbitals of the constituent atoms. The molecular orbitals or MOs are obtained as algebraic summation or linear combination of atomic orbitals (LCAO) with suitable weighting factors (LCAO—MQ method)... 

By far the commonest approximation employed to reduce the notion of an MO to an explicit, practical form is the linear combination of atomic orbitals (LCAO) approximation. Each MO is written as a linear combination of atomic orbitals on the various atoms. Denoting the /th atomic orbital , and the A th molecular orbital y/k, we write... 

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