Atomic orbitals functions

The deformation functions, however, must also describe density accumulation in the bond regions, which in the one-center formalism is represented by the atom-centered terms. They must be more diffuse, with a different radial dependence. Since the electron density is a sum over the products of atomic orbitals, an argument can be made for using a radial dependence derived from the atomic orbital functions. The radial dependence is based on that of hydrogenic orbitals, which are valid for the one-electron atom. They have Slater-type radial functions, equal to exponentials multiplied by r1 times a polynomial of degree n — l — 1 in the radial coordinate r. As an example, the 2s and 2p hydrogenic orbitals are given by... 

In molecular orbital calculations, the molecular orbitals are represented as a linear combination of atomic orbital functions (Xi). A variety of different mathematical functions may be used to represent these atomic orbital functions. If a very sophisticated mathematical function is used, then the resulting answer is higher in quality, providing very accurate energies and geometries for the drug molecule being studied however, such calculations may be extremely expensive in terms of computer time required. If a... 

The radial part of the 3s atomic orbital function for the hydrogen atom is a good chemical example of a product of a polynomial with an exponential function, and takes the form ... 

Figure 1.1 Hydrogen atomic orbital functions, (a) Is (b) 2p (c) 3d. The edges drawn are artificial, because orbitals have no edges but merely decrease in magnitude as distance from the nucleus increases. The important features of the orbitals are the nodal planes indicated, and the algebraic signs of the orbital functions, positive in the shaded regions and negative in the unshaded regions.

We know that an electron in a hydrogen atom in a stationary state will be described by one of the atomic orbital functions [Pg.13]

The symbol MO means that the new function is a molecular orbital a molecular orbital is any orbital function that extends over more than one atom. jjince the technical term for a sum of functions of the type 1.4 is a linear combination, the prOr ccdure of adding up atomic orbital functions is called linear combination oj alomic orbitals, or ICAO. 

We assume from here on that the reader is familiar with the number and shape of each type of atomic orbital function. This information may be found in standard introductory college chemistry texts. 

The next approximation is the expression of each molecular orbital p as a linear combination of atomic orbitals (LCAO) (Equation A2.4), where atomic orbital to the molecular orbital. The

basis functions discussed in Section 1.2. Valence atomic orbitals are ordinarily chosen for the basis. 

I n In Molecular orbtital functions comprised of y Atomic orbital functions that form the /n... 

It is worth considering at this stage what the overall n electron distribution will be in this conjugated system. The electron population in any molecular orbital is derived from the square of the atomic orbital functions, so that the sine waves describing the coefficients in Fig. 1.34a are squared to describe the electron distribution in Fig. 1,34b. The n electron population in the molecule as a whole is then obtained by adding up the electron populations, allowing for the number of electrons in each orbital, for all the filled n molecular orbitals. Looking only at the n system, we can see that the overall % electron distribution for the cation is... 

It is appropriate, too, to test the utility of the spreadsheet integration procedures for the determination of the orthonormality relationships among the angular parts of the atomic orbital functions. These are the integrations of equation 1.8, over the unit sphere, of the form... 

Here are the molecular orbital expansion coefficients for the basis functions < ). The problem of determining the molecular orbital has been reduced from finding a complete description of the three-dimensional function i , to finding only a finite set of linear coefficients for each orbital. If the basis functions 4) are atomic orbital functions, then this linear expansion is known as a linear combination of atomic orbitals (LCAO). 

The discrete variational (DV) method numerically calculates the basis atomic orbitals using the following wave equation for the radial atomic orbital function Rja r) in spherical coordinates... 

At this point it is useful to remember that mathematical functions have signs. For example, a simple sine wave (see Fig. 12.1) oscillates from positive to negative and repeats this pattern. Atomic orbital functions also have signs. The functions for s orbitals are positive everywhere in three-dimensional space. That is, when the s orbital function is evaluated at any point in space, it results in a positive number. In contrast, the p orbital functions have different signs in different regions of space. For example, the orbital has a positive... 

We begin with the ground state of He2. The separated helium atoms each have the ground-state configuration Is. This closed-subshell configuration does not have any unpaired electrons to form valence bonds, and the VB wave function is simply the antisymmetrized product of the atomic-orbital functions ... 

For each of the degenerate orbitals the oijerator Or will give two answers deiiending on which atomic orbital function is o()erated on. If a third function... 

In a lattice, the set of atomic orbitals are ideal for the formation of Bloch functions. A band can be obtained by combining the atomic orbital functions. This model can be said to be composed of the orbital functions of the solid. Eq. (2.15) describes the combination for the one-dimensional case... 

Turning the formula 180" around the z axis (the c operation) changes the position of the numbers of the atomic orbital functions.centered on each atom ... 

To get approximations to higher MOs, we can use the linear-variation-function method. We saw that it was natural to take variation functions for Hj as linear combinations of hydrogenlike atomic-orbital functions, giving LCAO-MOs. To get approximate MOs for higher states, we add in more AOs to the linear combination. Thus, to get approximate wave functions for the six lowest linear combination of the three lowest m = 0 hydrogenlike functions on each atom ... 

In practice, one would not specify the values (since these do not determine the symmetry of the atomic orbital), but would simply choose as many basis functions as practical to adequately model the shape of the radial probability density. The maximum 1 value, on the other hand, is normally specified for each atom, and does not need to be limited to the atom s ground state 1 value. For example, the atomic orbital functions on a hydrogen nucleus typically include up to 1 = 1 functions, while a carbon atom in the same calculation might be given basis functions up through 1=2. 

If we express this function in terms of the original atomic orbital functions, we get... 

Our atomic orbital functions are products of radial and angular terms, and so this type of multiple integral can be treated as a product of integrals over each of the coordinates. For the Is orbital, for example ... 

For example, (rioo>, the average radial position for an electron in an H-like atom Is orbital, can be found using the simple operator r and the appropriate atomic orbital functions. To calculate the expectation value we would write... 

In 1933, James and Coolidge offered the first purely variational treatment of the hydrogen molecule not developed directly from atomic orbital functions (LCAO). The mathematical complexity of this treatment made it both theoretically demanding and practically time-consuming to apply. But James and Coolidge argue forcefully that despite its computational complexity the procedure has important theoretical virtues absent in its predecessors ... 

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