Atomic orbitals electron density surfaces

Unsaturated organic molecules, such as ethylene, can be chemisorbed on transition metal surfaces in two ways, namely in -coordination or di-o coordination. As shown in Fig. 2.24, the n type of bonding of ethylene involves donation of electron density from the doubly occupied n orbital (which is o-symmetric with respect to the normal to the surface) to the metal ds-hybrid orbitals. Electron density is also backdonated from the px and dM metal orbitals into the lowest unoccupied molecular orbital (LUMO) of the ethylene molecule, which is the empty asymmetric 71 orbital. The corresponding overall interaction is relatively weak, thus the sp2 hybridization of the carbon atoms involved in the ethylene double bond is retained. 

When the catalyst is illuminated, the ZnO bonds are excited and loosened. In the linear combination of the ZnO bonds (Eq. 25) the nonbonding form increases at the expense of the bonding forms. This means that there is a shift of the bond electrons from the oxygen to the zinc atom. The electron density in the orbitals of the Zn atoms is increased, while the electron density in the orbitals of the oxygen atoms is decreased as compared with that of the ground state of the bond. A similar state is generated by the oxygen desorption of the surface where electron-enriched zinc atoms remain. ZnO, as an n-conductor, even in the... 

FIGURE 2-8 Constant Electron Density Surfaces for Selected Atomic Orbitals. (a) (d) The cross-sectional plane is any plane containing the z axis, (e) The cross section is taken through the xz or yz plane, (f) The cross section is taken through the xy plane. (Figures (b)-(f) reproduced with permission from E. A. Orgyzlo and G. B. Porter, J. Chem. Educ., 1963, 40, 258.)... 

The uneven orbital extension is observed by plotting the value of the LUMO coellicicnt at a given distance from the atom onto an electron density surface, as supported by SPARTAN. 

FIGURE 2.8 Constant Electron Density Surfaces for Selected Atomic Orbitals, (a)-(d) The cross-sectional plane is any plane containing the z axis. 

Figure 10.2 Schematic Illustration of overlap population density of states (OPDOS) of the electronic structure of the same adsorption model as in Figure 10.1. In the tight-binding adsorption model, one s-type adsorbate atomic orbital interacts with surface atom of a cubic s-atomic orbital metal lattice, fi is overlap energy of metal atomic orbitals / the overlap energy between adsorbate atomic orbital and surface atom atomic orbital. Ns...

Figure 2.3 Contour plots of several low-lying H atom orbitals. Curves are surfaces on which the wavefunction exhibits constant values solid and dashed curves correspond to positive and negative values, respectively. The outermost contour in all cases defines a surface containing 90% of the electron probability density. The incremental change in wavefunction value between adjacent contours is 0.04, 0.008, 0.015, 0.003, 0.005, and 0.003 bohr respectively for the Is, 2s, 2p, 3p, and 3d orbitals. Boxes exhibit side lengths of 20 bohrs (Is, 2s, 2p) and 40 bohrs (3s, 3p, 3c0, so that orbital sizes can be compared. Straight dashed lines in 3p and 3d 2 plots show locations of nodes.

The calculated lowest unoccupied molecular orbital (LUMO) for BF3 is shown by solid red and blue lobes. Most of the volume represented by the LUMO corresponds to the empty p orbital in the sp -hybridized state of BF3 (located perpendicular to the plane of the atoms). This orbital is where electron density fills (bonding occurs) when BF3 is attacked by NH3. The van der Waals surface electron density of BF3 is indicated by the mesh. As the structure shows, the LUMO extends beyond the electron density surface, and hence it is easily accessible for reaction. 

Let s take a moment to add just a little more detail to this argument. Figure 14.4 shows a schematic representation of electron density (or electron probability ) versus atomic radius for the Af,Sd, and 6s orbitals. (Electron density here is defined as the function ATTr tf/, sometimes known as the radial distribution function. It gives the probability of finding the electron on the surface of a series of concentric spheres of radius r. You may have studied this in previous chemistry courses but you need not know the mathematical details to understand the following argument.) Note that most of the time the 6s electron, as expected, is found farther away from the nucleus (located at r = 0) than the 4f and 5d electrons. We say the 6s electron is shielded from the nucleus by the intervening 4f and 5d electrons. 

The usefulness of spin density surfaces can be seen in the following models of methyl radical, CH3, and allyl radical, CH2=CHCH2. In each case, the surface is shaped somewhat like a 2p atomic orbital on carbon. There are some interesting differences between the two radicals, however. While the unpaired electron is confined to the carbon atom in methyl radical, it is delocalized over the two terminal carbons in allyl radical. 

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. 

The raw output of a molecular structure calculation is a list of the coefficients of the atomic orbitals in each LCAO (linear combination of atomic orbitals) molecular orbital and the energies of the orbitals. The software commonly calculates dipole moments too. Various graphical representations are used to simplify the interpretation of the coefficients. Thus, a typical graphical representation of a molecular orbital uses stylized shapes (spheres for s-orbitals, for instance) to represent the basis set and then scales their size to indicate the value of the coefficient in the LCAO. Different signs of the wavefunctions are typically represented by different colors. The total electron density at any point (the sum of the squares of the occupied wavefunctions evaluated at that point) is commonly represented by an isodensity surface, a surface of constant total electron density. 

The mechanism involves a metal atom insertion into the O—H bond, thus resulting in the formation of an adsorbed metal—OH species (at the same or similar binding site) and a new metal—H bond. This is a classic bond activation process, which involves a significant stretch of the O—H bond in order to lower the antibonding ooh orbital to enable it to accept electron density from the metal. The reaction has been calculated by DFT to be endothermic by +90 kJ/mol over Pt(lll) surfaces with an activation barrier of +130 kJ/mol [Desai et al., 2003b]. 

Figure 3.10 Representations of the electron density ip2 of the Is orbital and the 2p orbital of the hydrogen atom. (b,e) Contour maps for the xe plane. (c,f) Surfaces of constant electron density. (a,d) Dot density diagrams the density of dots is proportional to the electron density. (Reproduced with permission from the Journal of Chemical Education 40, 256, 1963 and M. J. Winter, Chemical Bonding, 1994, Oxford University Press, Fig. 1.10 and Fig. 1.11.)...

Figure 3.15 An sp hybrid orbital, (a) left, radial functions for the 2s and 2p atomic orbitals right, radial function for the sp hybrid orbital (b) left, the shapes of the 2s and 2p atomic orbitals as indicated by a single contour value right, the shape of the sp hybrid orbital as indicated by the same contour, (c) The shape of a surface of constant electron density for the sp hybrid orbital, (d) Simplified representation of (c). (Reproduced with permission from R. J. Gillespie, D. A. Humphreys, N. C. Baird, and E. A. Robinson, Chemistry, 2nd Ed., 1989, Allyn and Bacon, Boston.)...

An approximate quantum mechanical expressions- that allows one to calculate the electrostatic surface potential around atoms, radicals, ions, and molecules by assuming that the ground-state electron density uniquely specifies the Hamiltonian of the system and thereby all the properties of the ground state. This approach greatly facilitates computational schemes for exact calculation of the ground-state energy and electron density of orbitals. 

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