The LCAO MO approximation

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)  

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Formula for the Probability of Electron Excitation in the MO LCAO Approximation... 

In order to perform a qualitative analysis of the /1-decay-induced redistribution of electron density it is sufficient to calculate the molecular electron states in the MO LCAO approximation, i.e., not taking into account the correlation of electrons. Below we present the calculation data for a number of molecules, which we have obtained using the Gaussian-70 program with the basis of s and p functions. We have used the extended atomic basis 4-31G, which contains about twice as many atomic functions as the minimal one (Ditchfield et al, 1971). 

The electron density distribution was calculated according to Mulliken (1955 see also Herzberg, 1966). In the MO LCAO approximation the ith molecular orbital is... 

Molecule In the MO LCAO approximation With allowance for electron correlation ... 

We have calculated the data presented in the table in collaboration with G. V. Smeloy (Kaplan et al., 1983, 1985). In the MO LCAO approximation we have used the same bases of atomic functions as in calculations of the excitation probabilities of the corresponding molecules (see Section III,B,1). Allowing for electron correlation, calculations of the number of Cl configurations and the atomic bases were the same as those given in Section III,B,2. 

In view of the large size of the valine molecule, our calculations (Kaplan et al., 1983) were carried out on a minimal basis of 51 Slater orbitals in the MO LCAO approximation using the method described in Sections II, C, and III, B, 1. We have taken into account 608 singly excited states of the ion (valine-He)+. The results of calculations for valine on a minimal basis were corrected with regard to the results of calculation of the influence of the basis length on the excitation probabilities of the fragments that are shown enclosed in boxes in Fig. 9 (the rest of the molecule was replaced by a hydrogen atom). 

In the original meaning of the MO-LCAO approximation, the molecular orbitals are constructed from the combinations of orbitals of atoms constituting the molecule. For molecular orbitals so formed it may be assumed that their quality will reflect the quality of atomic orbitals used. But the AO description of atoms is accurate only for the hydrogen atom and the so-called hydrogen-like atoms, i.e, for... 

In the MO-LCAO approximation introduced above, the first three contributions to the interaction energy of eq.(1.22) (the last term, C/jvN> is a constant dependent only on position and charge of solute nuclei) can be expressed in terms of matrices defined on the AOs. Actually, the two mixed e N interactions being formally equivalent can be cast in a single one-electron term,... 

In the MO LCAO approximation RHF method (4.21) transform to the matrix equations... 

When approximate MOs are used, each approximate MO should, of course, belong to the same irreducible representation as the MO it is approximating. In the LCAO-MO method, the MOs are approximated as linear combinations of some set of AOs ... 

Here the 7r-system is treated with a very simple, but still quantum mechanical method e.g. by the Hiickel Hamiltonian and MO LCAO approximation (which in the particular case of the Hiickel Hamiltonian gives the exact answer). No explicit interaction, i.e. junction, between the subsystems was assumed at that time however, the effects of the geometry of the classically moving nuclei were very naturally reproduced by a linear dependence of the one-electron hopping matrix elements of the bond length ... 

The 7r-electron structures and energies of the singlet tt-tt transitions for a number of l//-pyrrolo[l,2-a jimidazoles (39), l/f-pyrrolo[l,2-f>]-s-triazoles (40) and l//-pyrrolo[2,l-c]-s-triazoles (41) were calculated by the MO LCAO method within the semiempirical self-consistent field (SCF) approximation. A comparison of the data shows that the maximum... 

There is some preliminary data <74CHE230> based on the MO LCAO method within the semi-empirical self-consistent field (SCF) approximation, on l/f-pyrrolo[l,2-a]imidazoles the maximum TT-electron densities are localized on C-5 and C-7 <84CHEC-I(6)979>. No other theoretical methods have been used. 

Simple HUckel molecular orbital (HMO) calculations on the pyrrolo[2,l-c][l,2,4]triazole (28) suggest that electrophilic attack would occur most readily at C-10, and this prediction was borne out by observations that acid-catalyzed deuteration and bromination by NBS in the dark both occur at this position <85JCR(S)363>. Various reactivity indices have been calculated for a number of pyrrolo[ 1,2-b][, 2,4]triazoles (29) and pyrrolo[2,1 -c][ 1,2,4]triazoles (30) using the MO LCAO method within the semiempirical SCF approximation. These indicate that the 5-position is most susceptible to electrophilic attack, followed by the 7-position <74CHE230>. 

The individual bonds, or if we prefer a more realistic MO description, also the molecular orbitals, will be expressed in the form of the usual LCAO approximation in the basis of atomic orbitals. In our case, if we confine ourselves to a simpler localized description, the conesponding orbitals (bonds) are given by eq. (12). 

In Sect. 4.1.5 the Hartree-Fock LCAO approximation for periodic qrstems was considered. The main difference of the CO LCAO method (crystalline orbitals as linear combination of atomic orbitals) from that used in molecular quantum chemistry, the MO LCAO (molecular orbitals as hnear combination of atomic orbitals) method was explained. In the CO LCAO approximation the one-electron wavefunction of a crystal (CO - ifih R)) is expanded in Bloch sums Xt kiR) of AOs ... 

The possibility of consideration of atoms as elementary subunits of the molecular systems is a consequence of Born-Oppenheimer or adiabatic approximation ( separation of electron and nuclear movements) aU quantum chemistry approaches start from this assumption. Additivity (or linear combination) is a common approach to construction of complex functions for physical description of the systems of various levels of complexity (cf orbital approximation, MO LCAO approximation, basis sets of wave functions, and some other approximations in quantum mechanics). The final justification of the method is good correlation of the results of its applications with the available experimental data and the potential to predict the characteristics of molecular systems before experimental data become available. It can be achieved after careful parameter adjustment and proper use of the force field in the area of its validity. The contributions not considered explicitly in the force field formulae are included implicitly into parameter values of the energy terms considered. 

Starting within the one-electron approximation, the MO-LCAO wavefunction for a diatomic metal-ligand (M-L) model system is conveniently set up as... 

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... 

To this point, the basic approximation is that the total wave function 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 ab initio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ab initio calculation. However, there are two main th ings to be con sidered in 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 fewest possible terms for an accurate representation of a molecular orbital. The second one is the speed of two-electron integral calculation. 

In molecular applications the calculation of the HF energy is a still more difficult problem. It should be observed that, in the SCF-MO-LCAO now commonly in use, one does not determine the exact HF functions but only the best approximation to these functions obtainable within the framework given by the ordinarily occupied AO s. Since the set of these atomic orbitals is usually very far from being complete, the approximation may come out rather poor, and the correlation energy estimated from such a calculation may then turn out to be much too large in absolute order of magnitude. The best calculation so far is perhaps Coulson s treatment of... 

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