Probability distributions orbitals

Probability distribution (orbital) a representation indicating the probabilities of finding an electron at various points in space. 

Instead of probability distributions it is more common to represent orbitals by then-boundary surfaces, as shown m Figure 1 2 for the Is and 2s orbitals The boundary sur face encloses the region where the probability of finding an electron is high—on the order of 90-95% Like the probability distribution plot from which it is derived a pic ture of a boundary surface is usually described as a drawing of an orbital... 

A plot of radial probability distribution versus r/ao for a His orbital shows a maximum at 1.0 (that is, r = a0). The plot is shown below ... 

While some younger physicists, like Born s student Werner Heisenberg and Ralph Fowler s student P. A. M. Dirac, were not sanguine about Bom s interpretation, it pleased chemists like Lewis, who earlier had been willing to think about the "average" position of an electron in its orbit, so as to reconcile Bohr s dynamic atom with Lewis s static atom. For Pauling, it was a natural step to take Y2 to be the probability distribution function for an electron s position in space.32... 

In further studies of chemistry and physics, you will learn that the wave functions that are solutions to the Schrodinger equation have no direct, physical meaning. They are mathematical ideas. However, the square of a wave function does have a physical meaning. It is a quantity that describes the probability that an electron is at a particular point within the atom at a particular time. The square of each wave function (orbital) can be used to plot three-dimensional probability distribution graphs for that orbital. These plots help chemists visualize the space in which electrons are most likely to be found around atoms. These plots are... 

When y,mp (p = ) represent atomic orbitals, yfmp is a probability distribution, which should integrate to 1. The normalization condition is therefore... 

The Hartree-Fock or self-consistent-field approximation is a simplification useful in the treatment of systems containing more than one electron. It is motivated partly by the fact that the results of Hartree-Fock calculations are the most precise that still allow the notion of an orbital, or a state of a single electron. The results of a Hartree-Fock calculation are interpretable in terms of individual probability distributions for each electron, distinguished by characteristic sizes, shapes and symmetry properties. This pictorial analysis of atomic and molecular wave functions makes possible the understanding and prediction of structures, spectra and reactivities. 

It is an interesting Tact that just as the single s orbital is spherically symmetric, the summation or electron density of a set or three p orbitals, five d orbitals, or seven f orbitals is also spherical (UnsBld s theorem). Thus, although it might appear as though an atom such as neon with a filled set of sand p orbitals would have a lumpy electron cloud, the total probability distribution is perfectly spherical... 

The other important property of an electron that must be specified besides its spatial distribution is its spin. According to quantum-mechanical arguments, into which we need not go in detail, each electron has a spin which can take one of two values. It is convenient to include this description in the wave function by defining a and / so that at = 1 if the spin is in one direction and = 0 if it is in the other, ft is defined in a complementary manner. Thus the electron moving in an orbital ip(x, y, z) may be associated with two functions pix, y, z) . and fix, y, z)8 according to the direction of its spin. A function such as %p x, y, z)a whioh gives the probability distribution of the spin co-ordinate as well as that of its spatial co-ordinates is sometimes referred to as a spin orbital. 

The two descriptions are useful in rather different contexts. If we are interested in the relative positions of the two electrons, then the interpretation in terms of localised orbitals gives a clearer description of the qualitative features of the overall probability distribution. On the other hand, if we are interested in the removal of an electron, the first description is more appropriate, for the remaining electron must occupy an orbital which is a solution of the original Schrodinger equation. Thus the electron must be removed from y)t or y>2. 

Fig. 23.8 Probability distribution Nn x,y z) 2 for the intrashell wavefunction N = n = 6 in the x = 0 plane corresponding to the collinear arrangement rn = rj +r2. The axes have a quadratic scale to account for the wave propagation in coulombic systems, where nodal distances increase quadratically. The fundamental orbit (-----------) (as) as well as the symmetric stretch motion (------) (ss) along the Wannier ridge are overlayed on the figure (from...

Under the Hartree-Fock (i.e., HF) approximation, the function of in variables for the solutions of the electronic Hamiltonian is reduced to n functions, which are referenced as molecular orbitals (MOs), each dependent on only three variables. Each MO describes the probability distribution of a single electron moving in the average field of all other electrons. Because of the requirements of the Pauli principle or antisymmetry with respect to the interchange of any two electrons, and indistinguishability of electrons, the HF theory is to approximate the many-electron wavefunction by an antisymmetrized product of one-electron wavefunctions and to determine these wavefunctions by a variational condition applied to the expected value of the Hamiltonian in the resulting one-electron equations,... 

The electron clouds of the s orbital (or probability distribution of finding the electron) has spherical symmetry. The s orbital is the shape of a sphere, in which the electron cloud density becomes less dense from the center outward, and the nucleus is at the center. Look at the figure above right. 

---