LCAO Approximation

Note that the sums of the squares of the coefficients in a given MO must equal 1 (e.g., 0.3717 + 0.6015 + 0.3717 + 0.6015 = 1.0 for Pi) because each of the AOs represents a probability distribution of finding the electron at a given point in space. The total probability of finding an electron in all space for an MO must be unity, exactly as for its constituent AOs. We now can see that the LCAO approximation is only one of many possibilities to describe the electron density (= probability) for MOs. We do not have to express the electron density as a linear combination of the electron densities of AOs centered at the atoms. We could also... 

Population anaiysis methods of assigning charges rely on the LCAO approximation and express the numbers of electrons assigned to an atom as the sum of the populations of the AOs centered at its nucleus. The simplest of these methods is the Coulson analysis usually used in semi-empirical MO theory. This analysis assumes that the orbitals are orthogonal, which leads to the very simple expression for the electronic population of atom i that is given by Eq. (53), where Natomic orbitals centered... 

One of the things illustrated by this calculation is that a surprisingly good approximation to the eigenvalue can often be obtained from a combination of approximate functions that does not represent the exact eigenfunction very closely. Eigenvalues are not vei y sensitive to the eigenfunctions. This is one reason why the LCAO approximation and Huckel theory in particular work as well as they do. 

The LCAO approximation for the wave functions in the Hartree-Fock equations... 

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). 

Any set of one-electron functions can be a basis set in the LCAO approximation. However, a well-defined basis set will predict electronic properties using fewer terms than a poorly-defined basis set. So, choosing a proper basis set in ab initio calculations is critical to the reliability and accuracy of the calculated results. 

LCAO Approximation. Linear Combination of Atomic Orbitals approximation. Expresses the Molecular Orbitals by linear combinations of atom-centered functions (Atomic Orbitals). 

In order to test the accuracy of the LCAO approximations, we use the variation principle if V lcao is an approximate solution then the variational integral... 

Here the a, are the LCAO coefficients, which have to be determined. The formulation of HF theory where we use the LCAO approximation is usually attributed to Roothaan (1951a). His formulation applies only to electronic configurations of the type 1/ 3,...,. Following the discussion of Chapter 5, the charge... 

The sums run over the occupied orbitals note that we have not made any reference to the LCAO approximation. The energy expression is correct for a determinantal wavefunction irrespective of whether the orbitals are of LCAO form or not. 

Brillouin s theorem (Brillouin, 1933) tells us that the singly excited states do not interact with the HF ground state. This theorem is true for all HF wavefunc-tions, and does not depend on the ZDO or LCAO approximations. This means that... 

A special aspect of this description appears if one starts the orbital optimisation process with orbitals obtained by linear combinations ofRHF orbitals of the isolated atoms (LCAO approximation s.str.). Let Pn.opt and be the starting and final orbitals of such a calculation. Then the difference between c n.opi and Papt in the vicinity of each atom merely consists in a distortion of the atomic orbitals of each atom. This distortion just compensates the contribution of the orbitals of the other atoms to Pn.ctpt in order to restore the proportionality between the partial waves of ipopi and the appropriate atomic orbital. 

Because of the terms Ir-RAl and Ir-rT explicit solutions to Eq. 3 carmot be obtained in position space. In such cases approximate solutions are usually expressed as truncated linear combinations of basis functions (LCAO expressions). In spite of its successes, the LCAO approximation experiences various difficulties (truncation limits, nature of the basis functions, etc.) hard to estimate and which are not entirely controllable [51]. 

Show that the variational energies of a homonuclear diatomic molecule are given in the LCAO approximation by Eq. (137) and that the corresponding wavefiine-tions are as indicated in Eqs. (141) and (142). 

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. 

Catlow and Stoneham (1983) have shown that ionic term corresponds to the difference of the diagonal matrix elements of the Hamiltonian of an AB molecule in a simple LCAO approximation (Haa cf. section 1.18.1), whereas the covalent energy gap corresponds to the double of the off-diagonal term Hab—i e.,... 

Figure 8.IB shows an experimental contour map of electron density for the H2O molecule in plane y-z, after Bader and Jones (1963). The electron density is higher around the nuclei and along the bond directrix. The experimental electron density map conforms quite well to the hybrid orbital model of Duncan and Pople (1953) with the LCAO approximation.

These one-electron basis functions, 4>, constitute the basis set. When the basis functions represent the atomic orbitals for the atoms in the molecule, eq. 3.4 corresponds to a linear combination of atomic orbitals (LCAO) approximation. 

The 5-position of the nonprotonated 1,2,4-thiadiazole system was calculated to be the most reactive in nucleophilic substitution reactions using a simple molecular orbital method with LCAO approximation (84CHEC-I(6)463>. 

Therefore, with the LCAO approximation, Eq. (1.8) transforms to a system of N i equations with Nat unknown parameters cj. The resoluhon of Eq. (1.18) imphes that the determinant of the — E Stj matrix has to be zero, otherwise we would obtain the trivial soluhon cj = 0 Vy, which obviously has no physical meaiung. Therefore,... 

The Linear Combination of Atomic Orbitals (LCAO) approximation... 

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