Atomic orbitals for

Thus for H and He, the basis set consists of one orbital, a Is atomic orbital. For atomsLi to Ne the 2 inner-shell electrons are combined with the nucleus and the basis set consists of 4 orbitals, the 2s, 2p, ... 

In die HMO approximation, the n-electron wave function is expressed as a linear combination of the atomic orbitals (for the case in which the plane of the molecule coincides with the x-y plane). Minimizing the total rt-electron energy with respect to the coefficients leads to a series of equations from which the atomic coefficients can be extracted. Although the mathematical operations involved in solving the equation are not... 

If I write possible atomic orbitals for hydrogen atom A as Xa possible atomic orbitals for hydrogen atom B as the molecular electronic wavefunction will be... 

So, for some match point / to infinity, the atomic pseudo-orbital is identical to the valence HF atomic orbital. For radial distances less than / the pseudoorbital is defined by a polynomial expansion that goes to zero. The values of the polynomial are found by matching the value and first three derivatives of the HF orbital at / . 

The Natural Atomic Orbitals for atom A in the molecular environment may be defined as those which diagonalize the block, NAOs for atom B as those which diagonalize the D block etc. These NAOs will in general not be orthogonal, and the orbital oecupation numbers will therefore not sum to the total number of electrons. To achieve a well-defined division of the electrons, the orbitals should be orthogonalized. 

Pi (77 ) bond (Section 1.8) The covalent bond formed by sideways overlap of atomic orbitals. For example, carbon-carbon double bonds contain a 7r bond formed by sideways overlap of two / orbitals. 

Constmct the a bonding framework using hybrid orbitals for inner atoms and atomic orbitals for outer atoms, as described in Section 10-1. Any hybrid orbitals not used to form a bonds contain lone pairs of electrons. 

Charge and configurational variations of atomic orbitals Although oversimplified treatments commonly speak of the atomic orbitals for a given atom, it is clear that AOs must depend on the charge state and specific configuration of the atom. Thus, the (N)AOs of an anionic atom A- will tend to be markedly more diffuse (i.e., spread out to greater distances from the nucleus) than those of a cationic A+. 

We dispose of the simplest problems first. Any orbital-basis theory of molecular electronic structure which purports to be interpretable as a theory of valence is committed to the use of atom-centred functions (or, at least, functions which go over into atomic orbitals for some values of their parameters).7) We therefore wish to stay as... 

The orbital angular-momentum quantum number, , defines the shape of the atomic orbital (for example, s-orbitals have a spherical boundary surface, while p-orbitals are represented by a two-lobed shaped boundary surface). can have integral values from 0 to (n - 1) for each value of n. The value of for a particular orbital is designated by the letters s, p, d and f, corresponding to values of 0, 1, 2 and 3 respectively (Table 1.2). 

Atomic orbitals are actually graphical representations for mathematical solutions to the Schrodinger wave equation. The equation provides not one, but a series of solutions termed wave functions t[ . The square of the wave function, is proportional to the electron density and thus provides us with the probability of finding an electron within a given space. Calculations have allowed us to appreciate the shape of atomic orbitals for the simplest atom, i.e. hydrogen, and we make the assumption that these shapes also apply for the heavier atoms, like carbon. 

As is well known, conventional hydrogenoid spherical orbitals are strictly linked to tetradimensional harmonics when the atomic orbitals for the tridimensional hydrogen atom are considered in momentum space. We have therefore studied an alternative representation, providing the Stark and Zeeman basis sets, related to the spherical one by orthogonal transformation, see eqs. (12) and (15). The latter can also be interpreted as suitable timber coefficients relating different tree structures of hyperspherical harmonics for R (Fig. 1). 

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 electronic states of a diatomic molecule such as H2 are approximately equal to linear combinations of atomic orbitals. For example, the groimd state is approximately proportional to the sum of Is orbitals from the two atoms IsA + Isb- An excited state is approximately proportional to the difference IsA — Isb- Although these LCAO (Linear Combination of Atomic Orbitals) wave functions are not quantitatively correct representations of the true wave functions, their shape, and hence their symmetry, is correct. 

FIGURE 4.8 Energy bands formed from ns and np atomic orbitals for (a) a body-centred cubic crystal and (b) a crystal of diamond structure, depicting filled levels for 4iVelectrons. 

To evaluate both the coupling and the magnitude of the moments, we consider first an array of one-electron atoms in non-degenerate states, and let 0 denote an atomic orbital for an electron on atom (i). Then the electron on atom (i) can overlap on to neighbouring atoms (/), so we may write its wave function in the form... 

Fig. 2. The epA orbital does not interact with other atomic orbitals for the sake of numerical calculations within the HMO framework, the 9f and 9, orbitals may simply be replaced by carbon 2pz orbitals. The model of thiophene is presented in Fig. 3 the x and y axes are shown, the z-axis is perpendicular to the plane of the molecule.

Of the various methods of approximating the correct molecular orbitals, we shall discuss only one- the linear combination of atomic orbitals (LCAO) method. We assume that we can approximate the correct molecular orbitals by combining the atomic orbitals of the atoms that form the molecule. The rationale is that most of the time the electrons will be nearer and hence controlled by oneor the other of the two nuclei, and when this is so, the molecular orbital should be very nearly the same as the atomic orbital for that atom. The basic process is the same as the one wc employed in constructing hybrid atomic orbitals except that now we are combining orbitals on different atoms to form new orbitals that are associated with the entire molecule. We... 

The radial function of the 3s atomic orbital for the hydrogen atom has the form given in equation (2.44). 

---