LCAO-MO approximation

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 ... 

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... 

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. 

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... 

In the conventional MO-LCAO theory, the function u is approximated by a Is orbital, but better approximations may be obtained by including higher orbitals. The total wave function is such that, for separated atoms, there is a fifty per cent chance that the mole-... 

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 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... 

These properties of the d-shell chromophore (group) prove the necessity of the localized description of d-electrons of transition metal atom in TMCs with explicit account for effects of electron correlations in it. Incidentally, during the time of QC development (more than three quarters of century) there was a period when two directions based on two different approximate descriptions of electronic structure of molecular systems coexisted. This reproduced division of chemistry itself to organic and inorganic and took into account specificity of the molecules related to these classical fields. The organic QC was then limited by the Hiickel method, the elementary version of the HFR MO LCAO method. The description of inorganic compounds — mainly TMCs,— within the QC of that time was based on the crystal field... 

Several valence-bond (VB) treatments of heterocyclic compounds were reported in the thirties and forties.1, 2 The known difficulty in applying the VB method to complicated molecules has made an overwhelming majority of authors use the molecular orbital (MO) method. In most cases its simplest version, the naive MO LCAO method, has been used. This approximation differs from the well-known Hiickel... 

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 ... 

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... 

A further simple application of the present table , is to the computation of overlap integrals in. which one or both atoms are replaced by groups of atoms. S s of this sort frequently occur when one is working with iron-localized MO s (molecular orbitals) in LCAO approximation. The non-localized MO structure of H20 furnishes a convenient example. Neglecting, p hybridization, the electron conjuration rosy be written ... 

The electronic charge density in an MO extends over the whole molecule, or at least over a volume containing two or more atoms, and therefore the MOs must form bases for the symmetry point group of the molecule. Useful deductions about bonding can often be made without doing any quantum chemical calculations at all by finding these symmetry-adapted MOs expressed as linear combinations of AOs (the LCAO approximation). So we seek the LCAO MOs... 

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