Probability functions, orbitals

Figure 10-1. Radial probability functions for typical 4/ and 6s orbitals.

Clearly, however, electrons exist. And they must exist somewhere. To describe where that somewhere is, scientists used an idea from a branch of mathematics called statistics. Although you cannot talk about electrons in terms of certainties, you can talk about them in terms of probabilities. Schrodinger used a type of equation called a wave equation to define the probability of finding an atom s electrons at a particular point within the atom. There are many solutions to this wave equation, and each solution represents a particular wave function. Each wave function gives information about an electron s energy and location witbin an atom. Chemists call these wave functions orbitals. 

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... 

Consider the radial portion of the wave function for the Is orbital as plotted in Fig. 21. When it is squared and multiplied by 4nr2, we obtain the probability function... 

Similar probability functions (including the factor 4nr2) for the 2s, 2p, 3s, 3p, and 3d orbitals are also shown in Fig. 2.4. Note that although the radial function for the 2s orbital is both positive (r < 2a0/Z) and negative (r > 2u0/Z), the probability function is everywhere positive (as or course it must be to have any physical meaning) as a result or the squaring operation. 

Fig. JL5 (a) V-a and ifo for mfividual hydrogen atoms (cf. Fig. 2.1). (bj = if>A + V b-(c) Probability function for the bending orbital.. (d) 4> - tfiA 4>b fe> Probability function for the antibonding orbital.

The radial probability function Dni or rRni 2, drawn to the same scale, of the first six hydrogenic orbitals. 

A plausible estimate of the spatial extension of a hydrogenic orbital is the radius of a spherical boundary surface within which there is a high probability of finding the electron. In order to develop this criterion on a quantitative basis, it is useful to define a cumulative probability function FW(p) which gives the probability of finding the electron at a distance less than or equal to p from the nucleus ... 

Table 2.1.6. Cumulative probability functions for hydrogenic orbitals...

The cumulative probability functions listed in Table 2.1.6 can be used to give some measure of correlation between the various criteria for orbital size. Once a particular value for P is chosen, it can be equated to each of the expresssions in turn and the resulting transcendental equation solved numerically. The p values thus obtained for P = 0.50, 0.90, 0.95, and 0.99 are tabulated in Table 2.1.7. The significant quantites are the size ratios, with the Is p value as standard, which provide a rational scale of relative orbital size based on any adopted probability criterion. As the prescribed P value approaches unity, the size ratios gradually decrease in magnitude and orbitals in the same shell tend to converge to a similar size. 

Orbitals. Atomic orbitals represent the angular distribution of electron density about a nucleus. The shapes and energies of these amplitude probability functions are obtained as solutions to the Schrodinger wave equation. Corresponding to a given principal quantum number, for example n = 3, there are one 3s, three 3p and five 3d orbitals. The s orbitals are spherical, the p orbitals are dumb-bell shaped and the d orbitals crossed dumb-bell shaped. Each orbital can accomodate two electrons spinning in opposite directions, so that the d orbitals may contain up to ten electrons. 

This result for P2(ri,ra) can only be true in the limit that the motion of electron 1 and hence its probability of being at a certain point in space is totally independent of the motions of electron 2, i.e. there are no forces acting between the two electrons and their motions are totally uncorrelated. The best that can be done in a model which employs an expression of the form of equation (19) for the pair probability function is to determine the m (which in turn determine the one-particle probability function) in such a way that the electron it describes experiences the average field of the other electrons, a situation which is attained in the Hartree-Pock limit of the orbital model. 

Electrons that are bound to nuclei are found in orbitals. Orbitals are mathematical descriptions that chemists use to explain and predict the properties of atoms and molecules. The Heisenberg uncertainty principle states that we can never determine exactly where the electron is nevertheless, we can determine the electron density, the probability of finding the electron in a particular part of the orbital. An orbital, then, is an allowed energy state for an electron, with an associated probability function that defines the distribution of electron density in space. 

How do we depict a probability function One way would be to draw contours connecting regions where there is an equal probability of finding the electron. If F2 for a Is orbital is plotted, a three-dimensional plot emerges. Of course, this is a two-dimensional representation of a three-dimensional plot—the contours are really spherical like the different layers of an onion. These circles are rather like the contour lines on a map except that they represent areas of equal probability of finding the electron instead of areas of equal altitude. 

Hybrid orbitals are formed from constructive and destructive combinations of 2p and 2s atomic wave functions, as illustrated by the line plots below. The solid figures depict the corresponding probability functions y2 which describe the electron density in the various directions around the bonded atm. 

In all the radial probability plots, the electron density, or probability of finding the electron, falls off rapidly as the distance from the nucleus increases. It falls olf most quickly for the 1 orbital by r = Sa, the probability is approaching zero. By contrast, the 3d orbital has a maximum at r = 9ao and does not approach zero until approximately r = 20ao- All the orbitals, including the s orbitals, have zero probability at the center of the nucleus, because Anir R = 0 at r = 0. The radial probability ftinctions are a combination of which increases rapidly with r, and R, which may have maxima and minima, but generally decreases exponentially with r. The product of these two factors gives the characteristic probabilities seen in the plots. Because chemical reactions depend on the shape and extent of orbitals at large distances from the nucleus, the radial probability functions help show which orbitals are most likely to be involved in reactions. 

Fig. 2.6 Angular probability function for hydrogcn-likc p orbitals. Only two dimensions of the three-dimensional function have been shown.

The second quantum number is the azimuthal quantum number, t The azimuthal quantum number designates the subshell. These are the orbital shapes with which we are familiar such as s, p, d, and f. If C = 0, we are in the s subshell if = 1, we are in the p subshell and so on. For each new shell, there exists an additional subshell with the azimuthal quantum number f = it -1. Each subshell has a peculiar shape to its orbitals. The shapes are based on probability functions of the position of the electron. There is a 90% chance of finding the electron somewhere inside a given shape. You should recognize the shapes of the orbitals in the s and p subshells. 

Figure 1.1. Representation of the geometry and energy of the bonding and antibonding ( a) molecular orbitals formed by covalent bonding between the Is atomic orbitals of two H atoms, A and B. The probability function, indicates electron density along the internuclear A-B axis.

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