Linear combination of atomic orbitals coefficients

Note n is the position of the carbon in naphthalene or anthracene (see structure 29 or 30, respectively) c2in is the HMO (Hiickel molecular orbital) spin density, which is the square of the LCAO (linear combination of atomic orbitals) coefficient of the single-occupied HMO i /< at the center n (49) and pn is the McLachlan spin density (50). SOURCE Reproduced with permission from reference 30. Copyright 1986 Verlag der Zeitschrift fur Naturforschung. 

Mathematically, the molecular orbitals are treated as linear combinations of atomic orbitals, so that the wave function, is expressed as a sum of individual atomic orbitals multiplied by appropriate weighting factors (atomic coefficients) ... 

The coefficients indicate the contribution of each atomic orbital to the molecular orbital. This method of representing the molecular orbital wave function in terms of combinations of atomic orbital wave functions is known as the linear combination of atomic orbitals approximation (LCAO). The combination of atomic orbitals chosen is called the basis set. 

The raw output of a molecular structure calculation is a list of the coefficients of the atomic orbitals in each LCAO (linear combination of atomic orbitals) molecular orbital and the energies of the orbitals. The software commonly calculates dipole moments too. Various graphical representations are used to simplify the interpretation of the coefficients. Thus, a typical graphical representation of a molecular orbital uses stylized shapes (spheres for s-orbitals, for instance) to represent the basis set and then scales their size to indicate the value of the coefficient in the LCAO. Different signs of the wavefunctions are typically represented by different colors. The total electron density at any point (the sum of the squares of the occupied wavefunctions evaluated at that point) is commonly represented by an isodensity surface, a surface of constant total electron density. 

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

In analogy to using a linear combination of atomic orbitals to form MOs, a variational procedure is used to construct many-electron wavefunctions from a set of N Slater determinants y, i.e. one sets up a N x. N matrix of elements flij = (d>, H d>y) which, upon diagonalization, yields state energies and associated vectors of coefficients a used to define (fi as a linear combination of A,s ... 

Mulliken s population analysis is rooted in the LCAO (linear combination of atomic orbitals) formulation it is not directly applicable to other types of wavefunctions. With Cr i representing the coefficient of the rth type of atomic orbital (li, 2s, etc.) of atom k in the ith molecular orbital, we describe the latter by... 

The present article is an attempt to review those studies of pyridinelike heterocycles (mono-azines) and, to a lesser extent, their analogues and derivatives that have interpreted the behavior and estimated various physico-chemical properties of the compounds by the use of data calculated by the simplest version of the MO LCAO (molecular orbital, linear combination of atomic orbitals) method (both molecular orbital energies and expansion coefficients). In this review, attention is focused upon the use of the simple method because it has been applied to quite extensive sets of compounds and to the calculation of the most diverse properties. On the other hand, many fewer compounds and physico-chemical properties have been investigated by the more sophisticated methods. Such studies are referred to without being discussed in detail. In a couple of years, we believe, the extent of the applications of such methods will also be wide enough to warrant a detailed review. 

These O, are called Linear Combination of Atomic Orbitals Molecular Orbitals (LCAO MOs) and if they are introduced into the Hartree-Fock equations (eqns (10-2.5)), a simple set of equations (the Hartree-Fock-Roothaan equations) is obtained which can be used to determine the optimum coefficients Cti. For those systems where the space part of each MO is doubly occupied, i.e. there are two electrons in each 0, with spin a and spin respectively so that the complete MOs including spin are different, the total wavefunction is... 

Note that, upon examination of the AMI linear combination of atomic orbitals (LCAO) coefficients for the two aluminum on poly(/>-phenylenevinylene) systems, the highest occupied molecular orbitals are localized in character. The HOMO and HOMO-1 levels are almost totally localized to the aluminum atoms and to the carbon atoms within the moieties to which the aluminum atoms are attached. The conjugation within the frontier orbitals is thus totally lost... 

As introduced in Chapter 3, for AX systems, the hybrid orbitals are the linear combinations of atomic orbitals on central atom A that point toward the X atoms. In addition, the construction of the sp" hybrids was demonstrated. In this section, we will consider hybrids that have d-orbital contributions, as well as the relationship between the hybrid orbital coefficient matrix and that of the molecular orbitals, all from the viewpoint of group theory. 

Molecular orbitals are characterized by energies and amplitudes expressing the distribution of electron density over the nuclear framework (1-3). In the linear combination of atomic orbital (LCAO) approximation, the latter are expressed in terms of AO coefficients which in turn can be processed using the Mulliken approach into atomic and overlap populations. These in turn are related to relative charge distribution and atom-atom bonding interactions. Although in principle all occupied MOs are required to describe an observable molecular property, in fact certain aspects of structure and reactivity correlate rather well with the nature of selected filled and unfilled MOs. In particular, the properties of the highest occupied MO (HOMO) and lowest unoccupied MO (LUMO) permit the rationalization of trends in structural and reaction properties (28). A qualitative predictor of stability or, alternatively, a predictor of electron... 

We now compose a wave function for each tz electron in a conjugated system which describes its behaviour in the whole system, thus beforehand without restriction of the electron to particular bonds (molecular orbital, M.O.). It is now customary to compose these M.O. according first to Lennard-Jones as a linear combination of atomic orbitals (A.O.) provided with coefficients (L.C.A.O. approximation). These atomic orbitals in a molecule with double bonds are the p2 functions of the carbon atoms. The square of a particular coefficient indicates the contribution of the electron in question to the charge around this particular atom. 

There is no difficulty in defining an effective charge in terms of the LCAO electronic structure even when the softening is large. Ultimately the band slates can be written as linear combinations of atomic orbitals, with coefficients, as... 

A molecular orbital is assumed to be represented by a linear combination of atomic orbitals. This assumption is called the linear combination of atomic orbitals (LCAO) approximation. When the atomic orbitals and developing coefficients are denoted by xrand Crj, respectively, the molecular orbital (,) can be written by... 

As the chemist has gained his sea legs in the use of wave mechanics he has attempted to define covalent and ionic character in terms of wave functions. The new point of view is presented, for example, by Coulson, who maintains, there are two distinct definitions of a covalent bond (447, p. 145). He proceeds to describe the bond wave function, first by a molecular orbital type approximation, and then by a valence bond type approximation. In either case the approximate wave functions consist of linear combinations of atomic orbitals. Partial ionic bonding is revealed by the magnitudes of coefficients which imply asymmetry of electron distribution. 

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