Atomic orbitals defined

Also plotted in Fig. 4.7 are the radial distribution Junctions (P) for the atomic orbitals, defined in terms of the radial wavefunction by... 

For some purposes, a basis set consisting of hybridized atomic orbitals is particularly suitable in LCAO—MO calculations. By taking hybrids directed along the chemical bonds instead of pure atomic orbitals defined in terms of arbitrary axes, one simultaneously retains the essential features of the bond orbital picture and the standard delocalized method. This method has been developed in a parametric form similar to the standard Hiickel method including or not including overlap integrals 41). 

Let us now describe the distributions of electrons in atoms. For each neutral atom, we must account for a number of electrons equal to the number of protons in the nucleus, that is, the atomic number of the atom. Each electron is said to occupy an atomic orbital defined by a set of quantum numbers n, (, and m. In any atom, each orbital can hold a maximum of two electrons. Within each atom, these atomic orbitals, taken together, can be represented as a diffuse cloud of electrons (Figure 5-19). 

By the rules of Section 5-15, each shell has an s subshell (defined by = 0) consisting of one s atomic orbital (defined >y m = 0). We distinguish among orbitals in different... 

In this section, we haven t yet discussed the fourth quantum number, the spin quantum number, m. Because has two possible values, and —j, each atomic orbital, defined... 

In the second equation, c are the coefficients of the linear combination of atomic orbitals defining each ith molecular orbital variational principle to a wave function in the form of Slater determinant leads to the set of the Hartree-Fock equations (HF) ... 

By the rules given in Section 4-16, each shell has an s subshell (defined by f = 0) consisting of one s atomic orbital (defined by m( = 0). We distinguish among orbitals in different principal shells (main energy levels) by using the principal quantum number as a coefficient Ir indicates the s orbital in the first shell, 2s is the s orbital in the second shell, 2p is a p orbital in the second shell, and so on (see Table 4-8). 

In this method we assume that the various Xj-Y, bonds separating the quantum part from the classical one are represented by localized bond orbitals and that the electrons of the quantum part are represented by canonical molecular orbitals, expanded on the basis of the atomic orbitals defined on the quantum atoms and their nearest neighbours Y,. One could obtain such a situation by performing a unitary transformation on the molecular orbitals of the whole molecule (assumed to be known) and setting equal to zero the coefficients of the atomic orbitals defined on the atoms of the now classical part If the X, -Y, bonds have been chosen according to the criteria above defined, this simplification may not have a major importance. We make some additional assumptions ... 

Let us denote by. .. p> v>... the basis set of N atomic orbitals defined on the quantum atoms and their nearest neighbours Yj and assumed to be linearly independent. The SLOs are expanded on a limited number of such AOs by ... 

The semi-empirical equivalent of the above scheme benefits strongly from the simplifications introduced by the basic assumptions of the semi-empirical quantum chemical methods. Owing to the assumed orthogonality of the atomic orbitals, the only requirement of the method is the orthogonalization of the atomic orbitals defined on atoms X to the hybrid orbital involved in the SLOs associated to the frontier bonds. Similarly, one can discard the atomic orbitals centered on the Y atoms and these atoms are considered as classical point charges like all the other classical atoms. This approach has been tested successfully at the NDDO level. 

The electrons in an atom occupy atomic orbitals, which are designated by the letters s, p, d, and f. Each orbital can contain a maximum of two electrons. An atomic orbital is a mathematical equation that describes the energy of an electron. The square of the equation for the atomic orbital defines the probability of finding an electron within a given region of space. 

The origins of the Finnis-Sinclair potential [Finnis and Sinclair 1984] lie in the density of states and the moments theorem. Recall that the density of states D(E) (see Section 3.8.5) describes the distribution of electronic states in the system. D(E) gives the number of states between E and E - - 8E. Such a distribution can be described in terms of its moments. The moments are usually defined relative to the energy of the atomic orbital from which the molecular orbitals are formed. The mth moment, fi", is given by ... 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2, but in somewhat different ways. Both assume that electron waves behave like more familiar waves, such as sound and light waves. One important property of waves is called interference in physics. Constructive interference occurs when two waves combine so as to reinforce each other (in phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2.2). Recall from Section 1.1 that electron waves in atoms are characterized by then- wave function, which is the same as an orbital. For an electron in the most stable state of a hydrogen atom, for example, this state is defined by the I5 wave function and is often called the I5 orbital. The valence bond model bases the connection between two atoms on the overlap between half-filled orbitals of the two atoms. The molecular orbital model assembles a set of molecular- orbitals by combining the atomic orbitals of all of the atoms in the molecule. 

The next approximation involves expressing the jiiolecular orhiiah as linear combinations of a pre-defined set of one-electron functions kjiown as basis functions. These basis functions are usually centered on the atomic nuclei and so bear some resemblance to atomic orbitals. However, the actual mathematical treatment is more general than this, and any set of appropriately defined functions may be u.sed. 

A convenient orbital method for describing eleetron motion in moleeules is the method of molecular orbitals. Molecular orbitals are defined and calculated in the same way as atomic orbitals and they display similar wave-like properties. The main difference between molecular and atomic orbitals is that molecular orbitals are not confined to a single atom. The crests and troughs in an atomic orbital are confined to a region close to the atomic nucleus (typieally within 1-2 A). The electrons in a molecule, on the other hand, do not stick to a single atom, and are free to move all around the molecule. Consequendy, the crests and troughs in a molecular orbital are usually spread over several atoms. 

In the PPP model, each first-row atom such as carbon and nitrogen contributes a single basis functiqn to the n system. Just as in Huckel theory, the orbitals x, m e not rigorously defined but we can visualize them as 2p j atomic orbitals. Each first-row atom contributes a certain number of ar-electrons—in the pyridine case, one electron per atom just as in Huckel 7r-electron theory. 

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 / . 

There is no reason for dividing the off-diagonal contributions equally between the two orbitals. It may be argued that the most electronegative (which then needs to be defined) atom (orbital) should receive most of the shared electrons. 

The concept of natural orbitals may be used for distributing electrons into atomic and molecular orbitals, and thereby for deriving atomic charges and molecular bonds. The idea in the Natural Atomic Orbital (NAO) and Natural Bond Orbital (NBO) analysis developed by F. Weinholt and co-workers " is to use the one-electron density matrix for defining the shape of the atomic orbitals in the molecular environment, and derive molecular bonds from electron density between atoms. 

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. 

As proposed in [131, 132], the general AFDF scheme can be given in terms of an atomic orbital membership function mk(i) defined as... 

Atomic orbital basis functions have several indices, each referring to a different listing of these basis functions. In order to facilitate the correct index assignment in each case, several auxiliary quantities are defined. 

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