2s orbital

The above mean-field potential is used to find the 2p orbital of the carbon atom, which is then used to define the mean-field potential experienced by, for example, an electron in the 2s orbital ... 

Sketch the probability of finding an electron in the 2s orbital of hydrogen at distance r from a hydrogen nucleus as a function of r as a contour map with heavy lines at high probability and light lines at low probability. How does this distribution differ from the Is orbital ... 

All of our orbitals have disappeared. How do we escape this terrible dilemma We insist that no two elections may have the same wave function. In the case of elections in spatially different orbitals, say. Is and 2s orbitals, there is no problem, but for the two elechons in the 1 s orbital of the helium atom, the space orbital is the same for both. Here we must recognize an extr a dimension of relativistic space-time... 

Plot of the orthogonalized 2s orbital probability density vs r note there is one node. 

To provide further insight why the SCF mean-field model in electronic structure theory is of limited accuracy, it can be noted that the average value of the kinetic energy plus the attraction to the Be nucleus plus the SCF interaction potential for one of the 2s orbitals of Be with the three remaining electrons in the s 2s configuration is ... 

This data elearly shows that eorreetions to the SCF model (see the above table) represent signifieant fraetions of the inter-eleetron interaetion energies (e.g., 1.234 eV eompared to 5.95- 1.234 = 4.72 eV for the two 2s eleetrons of Be), and that the inter-eleetron interaetion energies, in turn, eonstitute signifieant fraetions of the total energy of eaeh orbital (e.g., 5.95 -1.234 eV = 4.72 eV out of-15.4 eV for a 2s orbital of Be). 

The effeet of adding in the py orbitals is to polarize the 2s orbital along the y-axis. The amplitudes Cn are determined via the equations of perturbation theory developed below the ehange in the energy of the 2s orbital eaused by the applieation of the field is expressed in terms of the Cn eoeffieients and the (unperturbed) energies of the 2s and npy orbitals. 

To illustrate sueh symmetry adaptation, eonsider symmetry adapting the 2s orbital of N and the three Is orbitals of H. We begin by determining how these orbitals transform under the symmetry operations of the C3V point group. The aet of eaeh of the six symmetry operations on the four atomie orbitals ean be denoted as follows ... 

Orbitals are described by specifying their size shape and directional properties Spherically symmetrical ones such as shown m Figure 1 1 are called y orbitals The let ter s IS preceded by the principal quantum number n n = 2 3 etc ) which speci ties the shell and is related to the energy of the orbital An electron m a Is orbital is likely to be found closer to the nucleus is lower m energy and is more strongly held than an electron m a 2s orbital... 

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... 

Because each carbon m acetylene is bonded to two other atoms the orbital hybridization model requires each carbon to have two equivalent orbitals available for CT bonds as outlined m Figure 2 19 According to this model the carbon 2s orbital and one of Its 2p orbitals combine to generate two sp hybrid orbitals each of which has 50% s character and 50% p character These two sp orbitals share a common axis but their major lobes are oriented at an angle of 180° to each other Two of the original 2p orbitals remain unhybridized... 

Section 2 21 Carbon is sp hybridized m acetylene and the triple bond is of the ct + Tt + Tt type The 2s orbital and one of the 2p orbitals combine to give two equivalent sp orbitals that have their axes m a straight line A ct bond between the two carbons is supplemented by two tr bonds formed by overlap of the remaining half filled p orbitals... 

Take C—H bond distance and bond strength for example Recalling that an elec tron m a 2s orbital is on average closer to the nucleus and more strongly held than an... 

For all orbitals except s there are regions in space where 0, ) = 0 because either Yimt = 0 or = 0. In these regions the electron density is zero and we call them nodal surfaces or, simply, nodes. For example, the 2p orbital has a nodal plane, while each of the 3d orbitals has two nodal planes. In general, there are I such angular nodes where = 0. The 2s orbital has one spherical nodal plane, or radial node, as Figure 1.7 shows. In general, there are (n — 1) radial nodes for an ns orbital (or n if we count the one at infinity). 

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