Boundary surface, orbitals

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... 

FIGURE 1 2 Boundary surfaces of a Is orbital and a 2s orbital The boundary surfaces enclose the volume where there is a 90-95% probability of finding an electron... 

FIGURE 1 3 Boundary surfaces of the 2p orbitals The wave function changes sign at the nucleus The two halves of each orbital are indicated by different colors The yz plane is a nodal surface for the Ip orbital The probability of finding a electron in the yz plane is zero Anal ogously the xz plane is a nodal surface for the 2py orbital and the xy plane is a nodal surface for the 2pz orbital You may examine different presentations of a 2p orbital on Learning By Modeling... 

Optically pure (Section 7 4) Descnbing a chiral substance in which only a single enantiomer is present Orbital (Section 1 1) Strictly speaking a wave function i i It is convenient however to think of an orbital in terms of the probability i i of finding an electron at some point relative to the nucleus as the volume inside the boundary surface of an atom or the region in space where the probability of finding an electron is high... 

Tj FIGURE 1.33 The three s-orbitals of 5 lowest energy. The simplest way of drawing an atomic orbital is as a g boundary surface, a surface within which there is a high probability (typically 90%) of finding the electron. We shall use blue to denote s-orbitals, but that color is only an aid to their identification. The shading Jp within the boundary surfaces is an 9 approximate indication of the electron density at each point. 

FIGURE 1.35 The boundary surface and the radial variation of a 2p-orbital along the (vertical) z-axis. 

All p-orbitals have boundary surfaces with similar shapes, including one nodal plane. Note that the orbital has opposite signs (as depicted by the depth of color) on each side of the nodal plane. 

FIGURE 3.8 When electrons with opposite spins (depicted as t and 1) in two hydrogen 1s-orbitals pair and thes-orbitals overlap, they lorm a boundary surface of the electron cloud. The cloud has cylindrical symmetry around the internuclear axis and spreads over both nuclei. In the illustrations in this book, cr-bonds are usually colored blue... 

FIGURE 3.9 A o-bond can also be formed when electrons in 1a- and 2p,-orbitals pair (where z is the direction along the internuclear axis). The two electrons in the bond are spread over the entire region of space enclosed by the boundary surface. 

FIGURE 3.10 A (T-bond is formed by the pairing ol electron spins in two 2p7-orbitals on neighboring atoms. At this stage, we are ignoring the interactions of any 2p,-(and 2p -) orbitals that also contain unpaired electrons, because they cannot form electron pair may be found anywhere within the boundary surface shown in the bottom diagram. Notice that the nodal plane of each p7-orbital survives in the tr-bond. 

FIGURE 3.11 A iT-bond is formed when electrons in two 2p-orbitals pair and overlap side by side. The middle diagram shows the extent of the overlap, and the bottom diagram shows the corresponding boundary surface. Even though the bond has a complicated shape, with two lobes, it is occupied by one pair of electrons and counts as one bond. In this text, ir-bonds are usually colored yellow. 

The square of the wavefunction, T2, relates to the probability of finding the electron at a particular location in space, with atomic orbitals being conveniently pictured as boundary surfaces (regions of space where there is a 90% probability of finding the electron within the enclosed volume). 

The orbital angular-momentum quantum number, , defines the shape of the atomic orbital (for example, s-orbitals have a spherical boundary surface, while p-orbitals are represented by a two-lobed shaped boundary surface). can have integral values from 0 to (n - 1) for each value of n. The value of for a particular orbital is designated by the letters s, p, d and f, corresponding to values of 0, 1, 2 and 3 respectively (Table 1.2). 

Figure 1.5 Boundary surfaces of a (bonding) and a (antibonding) molecular orbitals...

Figure 1.9 Molecular orbital diagram for methanal, showing the n —> tt and n —> n transitions. The boundary surfaces of the n, n and Jt molecular orbitals...

Boundary surfaces and atomic orbital envelope diagrams... 

The solutions of the Schrodinger equation show how j/ is distributed in the space around the nucleus of the hydrogen atom. The solutions for v / are characterized by the values of three quantum numbers and every allowed set of values for the quantum numbers, together with the associated wave function, strictly defines that space which is termed an atomic orbital. Other representations are used for atomic orbitals, such as the boundary surface and orbital envelopes described later in the chapter. 

Although the formal method of describing orbitals is to use mathematical expressions, much understanding of orbital properties may be gained by the use of pictorial representations. The most useful pictorial representations of atomic orbitals are similar to boundary surfaces (which are based on V /2), but are based upon the distribution of jf values, with the sign of / being indicated in the various parts of the diagram. The shapes of these distributions are based upon the contours of jf within... 

FIGURE 1.3 Boundary surfaces of the 2p orbitals. The wave function changes sign at the... 

Boundary surface of a carbon 2s orbital Boundary surface of a carbon 2p orbital... 

This is the most stable orbital of a hydrogen-like atom—that is, the orbital with the lowest energy. Since a Is orbital has no angular dependency, the probability density 2 is spherically symmetrical. Furthermore, this is true for all s orbitals. We depict the boundary surface for an electron in an s orbital as a sphere (Figure 1-2). The radial function ensures that the probability for finding the particle goes to zero for r — °°. 

Figure 1-2 The boundary surface of an s orbital an s orbital has no angular dependency.

The subscripts x, y, and z indicate the angular dependencies. As already mentioned, the three p orbitals are orthogonal to each other, and it is obvious that they are not spherically symmetrical about the nucleus. A boundary surface for each of the three p orbitals is given in Figure 1-3. The radial function is of course the same for all three p orbitals, and the first radial function is... 

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