Atomic orbitals wave functions

With 4) containing a normalization factor and all permutations over the atomic orbital wave functions i (1 = 1,2,... 2n). Likewise, if all electron pairs were exchanged in a cyclic manner, the product wave function, 4>b, has the fonn ... 

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. 

Molecular orbital (MO) theory describes covalent bond formation as arising from a mathematical combination of atomic orbitals (wave functions) on different atoms to form molecular orbitals, so called because they belong to the entire molecule rather than to an individual atom. Just as an atomic orbital, whether un hybridized or hybridized, describes a region of space around an atom where an electron is likely to be found, so a molecular orbital describes a region of space in a molecule where electrons are most likely to be found. 

Molecular orbital (MO) theory (Section 1.11) A description of covalent bond formation as resulting from a mathematical combination of atomic orbitals (wave functions) to form molecular orbitals. 

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. 

All the atomic orbital wave functions contain the exponential term e p. where p = Zjr/a, which is zero only when r — oo. 

The assumption that the atomic orbital wave functions are separately normalized leads to the conclusion that the first and third integrals in equation 3.14 are both equal to 1. The second integral is known as the overlap integral symbolized by S. This allows equation 3.14 to be simplified to ... 

It has not proved mathematically feasible to calculate the electron-electron repulsion that causes this change in orbital-energies for many-electron molecules. It is even difficult to rationalize the qualitative changes in sequence on the basis of the shapes of the 11orbitals. Greater success has been achieved by an approximate method which begins with orbitals characteristic of the isolated atoms present in the molecule, and assumes that molecular orbital wave functions can be obtained by taking linear combinations of atomic orbital wave functions (abbreviated L.C.A.O.). For... 

We will consider first the combination of two s orbitals, as in H2. For convenience, we label the atoms of a diatomic molecule a and b, so the atomic orbital wave functions are i(i( Ir ) and il<( li ). We can visualize the two atoms moving closer to each other until the electron clouds overlap and merge into larger molecular electron clouds. The resulting molecular orbitals are linear combinations of the atomic orbitals, the sum of the two orbitals and the difference between them ... 

So far we have discussed the electron density for the ground state of the H atom. When the atom absorbs energy, it exists in an excited state and the region of space occupied by the electron is described by a different atomic orbital (wave function). As you ll see, each atomic orbital has a distinctive radial probability distribution and 90% probability contour. 

A FIGURE 9.A0 Molecular orbitals from atomic orbital wave functions. 

The following sketches show the atomic orbital wave functions (with phases) used to construct some of the MOs of a homonuclear diatomic molecule. For each sketch, determine the type of MO that will result from mixing the atomic orbital wave functions as drawn. Use the same labels for the MOs as in the Closer Look box on phases. 

Pipek, J., Mezey, P. G. (1989). A fast intrinsic localization procedure applicable for ab initio and semi-empirical linear combination of atomic orbital wave functions. Journal of Chemical Physics, 90,4916-4926. 

We know that molecular orbital wave functions must be normalized, just as atomic orbital wave functions must be. Therefore,... 

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-molecular orbital (LCAO-MO) approximation. The combination of atomic orbitals chosen is called the basis set. A minimal basis set for molecules containing C, H, O, and N would consist of Is-, 2s-, Ip -, 2py-, and 2p -orbitals for C, O, and N, and a Is-orbital for hydrogen, inclusion of additional orbitals in the basis set leads to an extended basis set. Economy dictates use of minimal basis sets whenever possible. 

Atomic orbital wave functions are commonly expressed in terms of polar coordinates, rather than Cartesian ones. Figure 2.1 shows the relationship between Cartesian ( y, v, ) and polar (r, 6, ([)) coordinates. The values of. y and r in terms of polar coordinates are given by ... 

A 2p orbital has a nodal surface (e.g. the xy plane in the case of the 2p. orbital), and in general orbitals have / such surfaces. Atomic orbital wave functions have n - / - 1 radial nodes (nodal spheres) plus / nodal surfaces which pass through the origin, making a total of n - 1 nodes. 

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