Shapes of f orbitals

Because of their importance to us, the object of the present section is to give the reader more familiarity with the shapes of f orbitals than that provided in Section 7.3. In that section the f orbitals were used to indicate the way that F terms (arising from d configurations, n = 2, 3, 7 or 8) split in an octahedral ligand field. The labels that were used to describe these f orbitals were abbreviated, just as the label is an abbreviation for d2z2 2 y2. The complete labels for the f orbitals accurately describe the lobes of the orbitals, their relative phases and their positions. Drawings of the cubic set of f orbitals are repeated in Fig. 11.2, together with their abbreviated and 

Atomic number Name Symbol Isolated atom electron configuration f electron configuration 

Note Promethium, effectively, does not occur in nature. It is a fission product of uranium and may be made, for example, by neutron bombardment of neodymium to give an isotope with a half-life of just under 4 years. 

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Example The electron configuration for Be is Is lsfi but we write [He]2s where [He] is equivalent to all the electron orbitals in the helium atom. The Letters, s, p, d, and f designate the shape of the orbitals and the superscript gives the number of electrons in that orbital. 

The shape of these orbitals does not change strongly between the lowest triplet state ( 2g) with the configuration. .. b2) 3e) (2b f and the lowest quintet state ( Aig) with the configuration. .. (Ifi2) 3e) (2fii) ... 

You should be able to calculate the number of f orbitals in the f subshell. Remember that each orbital can hold no more than two electrons. The f orbitals are even more complicated than the d orbitals, but you do not have to know or recognise their shapes. 

Even if the shapes of the orbitals for the first four subshells are given, the shapes of the d and f subshell orbitals are so detailed at this point that they are given to students who are especially interested in learning more about orbital geometry. Look at the figure to the left. 

To summarize at this point, it is reiterated that wavefunction tjr r,6,three-dimensional shape of each orbital can be represented by a contour surface, on which every point has the same value off. The three-dimensional shapes of nine hydrogenic orbitals (2s, 2p, and 3d) are displayed in Fig. 2.1.5. In these orbitals, the nodal surfaces are located at the intersections where f changes its sign. For instance, for the 2p orbital, the yz plane is a nodal plane. For the 3d y orbital, the xz and yz plane are the nodal planes. 

The second quantum number describes the shape of the orbital as s, p, d, f or g. These shapes do not describe the electron s path but rather are mathematical models showing the probability of the electron s location. The s and p orbital shapes are shown in Figure 8.9, but descriptions of the d and f orbitals are reserved for more advanced texts. 

The orbital angular momentum quantum number, , determines the shape of the orbital. Instead of expressing this as a number, letters are used to label the different shapes of orbitals, s orbitals have f = 0, and p orbitals have - 1. 

The orbital angular momentum quantum number, f, determines, as you might guess, the angular momentum of the electron as it moves in its orbital. This quantum number tells us the shape of the orbital, spherical or whatever. The values that f can take depend on the value of n i can have any value from 0 up to - 1 f = 0,... 

All this explains why the shape of an orbital depends on the orbital angular quantum number, f. All s orbitals (f= 0) are spherical, aU p orbitals (f = 1) are shaped like a hgure eight, and d orbitals (f = 2) are yet another different shape. The problem is that these probability density plots take a long time to draw—organic chemists need a simple easy way to represent orbitals. The contour diagrams were easier to draw but even they were a little tedious. Even simpler stiU is to draw just one contour within which there is, say, a 90% chance of finding the electron. This means that aU s orbitals can be represented by a circle, and aU p orbitals by a pair of lobes. 

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. 

The determinant (= total molecular wavefunction ) just described will lead to (remainder of section 5.2) n occupied, and a number of unoccupied, component spatial molecular orbitals yjr. These orbitals ir from the straightforward Slater determinant are called canonical (in mathematics the word means in simplest or standard form ) molecular orbitals. Since each occupied spatial ir can be thought of as a region of space which accommodates a pair of electrons, we might expect that when the shapes of these orbitals are displayed ( visualized section 5.5.6) each one would look like a bond or a lone pair. However, this is often not the case e.g. we do not find that one of the canonical MOs of water connects the O with one H, and another canonical MO connects the O with another H. Instead most of these MOs are spread over much of a molecule - delocalized (lone pairs, unlike conventional bonds, do tend to stand out). However, it is possible to combine the canonical MOs to get localized MOs which look like our conventional bonds and lone pairs. This is done by using the columns (or rows) of the Slater > to create a with modified columns (or rows) if a column/row of a determinant is multiplied by k and added to another column/row, the determinant is unchanged (section 4.3.3). We see that if this is applied to the Slater determinant, we will get a new determinant corresponding to exactly the same total wavefunction, i.e. to the same molecule, but built up from different component occupied MOs ifr. The new and the new i/f s are no less or more correct than the previous ones, but by appropriate manipulation of the columns/rows the s can be made to correspond to our ideas of bonds and lone pairs. These localized MOs are sometimes useful. 

F can be factored into a radial component and two angular components. The radial function R describes electron density at different distances from the nucleus the angular functions 0 and describe the shape of the orbital and its orientation in space. The two angnlar factors are sometimes combined into one factor, called F ... 

Atoms have a series of principal energy levels indexed by the letter n. The w = 1 level is closest to the nucleus, and the energies of the levels increase as the value of n (and distance horn the nucleus) increases. Each principal energy level is divided into sublevels (sets of orbitals) of different characteristic shapes designated by the letters s, p, d, and f. Each s subsheU consists of a single s orbital each p subsheU consists of a set of three p orbitals each d subsheU consists of a set of five d orbitals and so on. An orbital can be empty or it can contain one or two electrons, but never more than two electrons (if an orbital contains two electrons, then the electrons must have opposite spins). The shape of an orbital represents a probability map for finding electrons—it does not represent a trajectory or pathway for electron movements. 

The shapes of the orbitals for the other molecules of Fig. 2 are shown in Fig. 5. Molecules B and C of Fig. 2 have donors at the both ends. For B and C in Fig. 5, HOMO-1 spreads somewhat into extremities, and electron-hole symmetry does not hold either between HOMO and LUMO nor between HOMO-1 and LUMO-i-1. These shapes are beneficial for %(3). However, the orbital HOMO does not spread into the extremities. This means that transition from HOMO-1 to HOMO does not have so much moment as that of SBA. The orbital shape for molecules B and C is not so good as SBA. With respect to the orbitals of the molecules E and F, both HOMO and HOMO-1 spread, and the electron-hole symmetry does not hold. These are beneficial for Molecules D, E and F, which have amino at both ends and have nitrogen atoms at the X positions of Fig. 1, are expected to have large third order susceptibilities. These results correspond well with the experimental fact that their measured X is large and as good as conjugated polymers. The x values of SBA and SBAC, which is dichloro SBA, are 1.0 X 10 esu and 1.3 x lO" esu. The X( ) of PU-STAD is 1.5 X 10 esu, which is a derivative of molecule F in Fig. 2. 

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