Nodal representation

Fig. 46. The [(U02)3(Se04)5]" sheets in the crystal structures of a- (a) and 3-Mg2[(U02)3(Se04)5](H20)i6 (b) and their nodal representations (c and d, respectively). Legend [UO7] bipyramids = black circles [Se04] tetrahedra = white circles.

Fig. 59. The structure of (NH4)4[(U02)5(Mo04)7](H20) projected along the c axis (a), nodal representation of its [(1102)5(1 004)7] framework (b), nodal representation of its fundamental chain (c), and graphs isomorphous to nodal representations of fundamental chains of chiral uranyl molybdate frameworks with the U Mo ratio of 5 7, 4 5 and 6 7 (d, e and f, respectively).

Fig. 10. The [(U02)(Se04)2] sheet in the structure of (H30MCi2Ft3oN2]3[(U02)4(Se04)8] (Ft20)5 (a), its nodal representation (U and Se polyhedra are symbolized by black and white circles, respectively) (b), and organization of 1,12-dodecanediamine molecules into a micelle (c)...

Figure 7.3 Nodal representation of HEN corresponding to the process shown in Figure 7.2. In this HEN, E-1, E-2 and E-3 are heat exchangers, and heaters/coolers are not shown. HEN structure is represented by matrices SM, NOD, NODE pcl SPU on the right side. See Section 7.2.1 for more details.

The quantum number / — 1 corresponds to a p orbital. A p electron can have any of three values for Jitt/, so for each value of tt there are three different p orbitals. The p orbitals, which are not spherical, can be shown in various ways. The most convenient representation shows the three orbitals with identical shapes but pointing in three different directions. Figure 7-22 shows electron contour drawings of the 2p orbitals. Each p orbital has high electron density in one particular direction, perpendicular to the other two orbitals, with the nucleus at the center of the system. The three different orbitals can be represented so that each has its electron density concentrated on both sides of the nucleus along a preferred axis. We can write subscripts on the orbitals to distinguish the three distinct orientations Px, Py, and Pz Each p orbital also has a nodal plane that passes through the nucleus. The nodal plane for the p orbital is the J z plane, for the Py orbital the nodal plane is the X Z plane, and for the Pz orbital it is the Jt plane. 

Figure 4.52 The leading donor-acceptor (nN->-szn ) interaction between the donor ammine lone pair and the acceptor 4s metal orbital in 22e [Zn(NH3)6]2+ (of. Fig. 4.51). (Note that the inner nodal structure of the Zn 4s orbital is absent in the effective-core-potential representation of the metal atom.)...

Fig. 2.7 (a) Pictorial representation or the electron density in a hydrogen-like 2p orbital compared with lb) the electron density contours Tor the hydrogen-like 2pr orbital of carbon. Contour values are relative to the electron density maximum The xy plane is a nodal surface. The signs (+ and —) refer to those of the original wave function. [The contour diagram is from Ogryzlo, E. A4 Porter G. B J. Client. Ethic. 1963.40. 258. Reproduced with permission. ... 

How many nodal surfaces does a 4s orbital have Draw a cutaway representation of a 4s orbital showing the nodes and the regions of maximum electron probability. 

The quadratic functions x + y% x2 — y2, z2 (more correctly 37 — 1), xy, xz, and yz. The x2 + y2, function cannot represent a d orbital because it does not have a node. The remaining five are taken as pictorial representations of the five d orbitals of central atom A in the AH2 molecule. Note that xy, xz, yz, and x2 — y2 each has two nodal planes, whereas die nodes of z2 are the curved surfaces of two co-axial cones sharing a common vertex. 

What does a H 2 for a 2p orbital look like The probability density plot is no longer spherically symmetrical. This time the shape is completely different—the orbital now has an orientation in space and it has two lobes. Notice also that there is a region where there is no electron density between the two lobes—another nodal surface. This time the node is a plane in between the two lobes and so it is known as a nodal plane. One representation of the 2p orbitals is a three-dimensional plot, which gives a clear idea of the true shape of the orbital. 

Hybrids constructed from hydrogenic eigenfunctions are examined in their momentum-space representation. It is shown that the absence of certain cross-terms that cause the breaking of symmetry in position space, cause inversion symmetry in the complementary momentum representation. Analytical expressions for some simple hybrids in the momentum representation are given, and their nodal and extremal structure is examined. Some rather unusual features are demonstrated by graphical representations. Finally, special attention is paid to the topology at the momentum-space origin and to the explicit form of the moments of the electron density in both spaces. 

The two concepts have on occasion been brought together Coulson and Duncanson[4] gave an explicit formula for sp-orbitals based on Slater type orbitals (STO s). Rozendaal and Baerends used hybrids to describe chemical bonding in a momentum representation [5], and more recently, Cooper considered the shape of sp hybrids in momentum space, and their impact on momentum densities [6], We would like to have a closer look at them, in terms of their functional behavior, their nodal structure and their topology. We will do... 

Figure 5. Nodal surfaces of a sp d2z2 hybrid orbital with Z = 1 in the momentum-space representation. The left-hand plot contains two surfaces. One is the spherical node of the imaginary part. The second more complex surface consists of two closed and flattened spheres. These are the nodal surfaces belonging to the real part of the hybrid and are aligned along the -axis. The intersection of the two types of nodes are two circles around the -axis. The right-hand plot displays a cut through the a -plane. Note that the (polar) -axis is the horizontal axis in this plot. To avoid confusion, the nodal planes of the imaginary part are not displayed in either graph.

Figure 8. Plots of the nodal surfaces in a spxdx2 y2 hybrid orbital (Z = 1) in the momentum representation. The left plot shows the surface due to the real part (i.e., s and d contributions) only, whereas the right one combines it with the planar and spherical nodal surfaces characteristic of the imaginary (i.e., p-) component.

Figure 7. Simple two-state coupling scheme in HCN e-> CNH, according to Eq. (20). (Left) Wavefunction of the adiabatically delocalized state 241 (E = 16,612 cm ), which is assigned as (0,56,0),. This is state x ) of Eq. (20). (Right) Wavefunction of the resulting nonadiabatically delocalized state 242 (E = 16,623 cm ), which can be assigned as (2,16,0)jj( but displays the nodal structure of (0,26,0)f,j jj on the CNH side. This is state of Eq. (20). The various assignments refer to the adiabatic description (upper) and to the nodal structures in the isolated wells (lower). The value of the overlap integral Sk, is indicated along the line connecting the two states. The wavefunctions are shown in the same representation as in Eig. 6.

We have seen that the meaning of an orbital is illustrated most clearly by a probability distribution. Each orbital in the hydrogen atom has a unique probability distribution. We also have seen that another means of representing an orbital is by the surface that surrounds 90% of the total electron probability. These three types of representations for the hydrogen Is, 2s, and 3s orbitals are shown in Fig. 12.18. Note the characteristic spherical shape of each of the s orbitals. Note also that the 2s and 3s orbitals contain areas of high probability separated by areas of zero probability. These latter areas are called nodal surfaces, or simply nodes. The number of nodes increases as n increases. For s orbitals the number of nodes is given by n — 1. For our purposes, however, we will think of s orbitals only in terms of their overall spherical shape, which becomes larger as the value of n increases. 

Figure 2.12. Nodal properties of the transition densities of the first four transitions in benzene, a) Representation of the complex LCAO coefficients of HOMOs 01 and 0, as well as LUMOs nd by means of a phase polygon. Each coefficient has the absolute magnitude n and the complex phase shown by a dot in the complex plane of which the real and imaginary axes are abscissa and ordinate, b) Representation of the overlap densities evaluated from the complex coefficients, and c) values of the overlap densities at the vertices of the perimeter and the resulting nodal properties.

Figure 4.10. Two-step procedure for the derivation of the orbital correlation diagram of the iii<] otatory ring opening of cyclobutene a) intended orbital correlation that results ii>lthe absence of interaction between orbitals of the a bond that is being broken and orfcitals of the ji bond. The schematic representation of the orbitals shows that due to different nodal properties only o-Ji and MO interactions are pos-...

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