Atomic orbitals nodal planes

Fig. 1.9 Boundary surfaces for the angular parts of the li and Ip atomic orbitals of the hydrogen atom. The nodal plane shown in grey for the Ip atomic orbital lies in the xy plane.

The optimum values of die oq and a coefficients are determined by the variational procedure. The HF wave function constrains both electrons to move in the same bonding orbital. By allowing the doubly excited state to enter the wave function, the electrons can better avoid each other, as the antibonding MO now is also available. The antibonding MO has a nodal plane (where opposite sides of this plane. This left-right correlation is a molecular equivalent of the atomic radial correlation discussed in Section 5.2. 

We encounter a different type of bond in a nitrogen molecule, N2. There is a single electron in each of the three 2p-orbitals on each atom (33). However, when we try to pair them and form three bonds, only one of the three orbitals on each atom can overlap end to end to form a (T-bond (Fig. 3.10). Two of the 2/7-orbitals on each atom (2px and 2py) are perpendicular to the internuclear axis, and each one contains an unpaired electron (Fig. 3.11, top). When the electrons in one of these p-orbitals on each N atom pair, the orbitals can overlap only in a side-by-side arrangement. This overlap results in a TT-bond, a bond in which the two electrons lie in two lobes, one on each side of the internuclear axis (Fig. 3.11, bottom). More formally, a 7T-bond has a single nodal plane containing the internuclear axis. Although a TT-bond has electron density on each side of the internuclear axis, it is only one bond, with the electron cloud in the form of two lobes, just as a p-orbital is one orbital with two lobes. In a molecule with two Tr-bonds, such as N2, the... 

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. 

The two atomic orbitals that contribute to the antibonding orbital in Eq. 2 are each proportional to e r a°, where r is the distance of the point from its parent nucleus. Confirm that there is a nodal plane lying halfway between the two nuclei. 

This expression indicates that there is a decreased probability (indicated by the term —2(/>A(/>B) of finding the electrons in the region between the two nuclei. In fact, there is a nodal plane between the positive and negative (with respect to algebraic sign) of the two regions of the molecular orbital. As a simple definition, we can describe a covalent bond as the increased probability of finding electrons between two nuclei or an increase in electron density between the two nuclei when compared to the probability or density that would exist simply because of the presence of two atoms. 

There is a nodal plane in ip2 at the middle carbon atom. The symmetry of the metal and ligands is appropriate for the interaction of the d orbital on the metal, so the interaction is as shown in Figure 21.13b. Combination of the dx/ orbital on the metal with the V3 allyl orbital is shown in Figure 21.13c. The bonding between other alkene molecules with 7r systems and metals is similar. 

However, the electron density between the atoms approaches zero at the nodal plane for the anti-bonding a orbital, which destabilizes the system. As an example, see below for the overlap of 2 Is orbitals (from Figure 9-2). Figure 9-3 shows the bonding and antibonding sigma orbitals formed by the combining head-on of two p orbitals. 

This simple treatment, formulated in a context of molecular bonding, was also what led Cox and Symons (1986) to propose the bond-center site as an explanation for anomalous muonium (Mu ). The location of the muon at the nodal plane of the nonbonding orbital explains the very small hyperfine coupling observed in pSR. Still, the muon is close to the electron, which occupies a nonbonding state on the neighboring semiconductor atoms. 

The molecular orbitals are labelled a and ir depending on whether they are symmetrical about the internuclear axis or have a nodal plane passing through the nuclei. The m.o. s are numbered in sequence of increasing energy, independent of the numbering of the atomic orbitals. This numbering serves to avoid any confusion in cases where atomic orbitals from different levels are combined, as in... 

If we imagine the nuclei to be forced together to = 0, the wave function Is A + Iss will approach, as a limit, a charge distribution around the united atom that has neither radial nor angular nodal planes. This limiting charge distribution has the same symmetry as the Is orbital on the united atom, Helium. On the other hand, the combination Isa Iss has a nodal plane perpendicular to the molecular axis at all intemuclear separations. Hence its limit in the united atom has the symmetry properties of a 2p orbital. A simple correlation diagram for this case is ... 

The molecular orbitals of lowest energy that can be formed from these basis orbitals correspond to sums of basis orbitals of the same symmetry species, containing no nodal planes between the atoms. These are a bonding orbital, symmetry aig... 

This molecular orbital, designated tts, has a positive value above carbon atoms 5 and 6, indicated by the solid contour negative value above carbon atoms 3 and 2, indicated by the dashed contour, and has a nodal plane including atom 4 and the nibogen atom. The contours are those that might be drawn in a plane parallel to the molecular plane, 1 A above it. 

The bond of ethylene (and other olefins) is a proper MO, highly localized to the two carbon atoms. It is the linear combination of the two 2p orbitals which is S with respect to reflection in the bisecting plane and A w.r.t. a 180° rotation about the C2 axis which contains that plane. All -type orbitals are A w.r.t. reflection in the nodal plane of the p orbitals themselves. 

For Eq. (9.35) to be useful the density matrix employed must be accurate. In particular, localization of excess spin must be well predicted. ROHF methods leave something to be desired in this regard. Since all doubly occupied orbitals at the ROHF level are spatially identical, they make no contribution to P only singly occupied orbitals contribute. As discussed in Section 6.3.3, this can lead to the incorrect prediction of a zero h.f.s. for all atoms in the nodal plane(s) of the singly occupied orbital(s), since their interaction with the unpaired spin(s) arises from spin polarization. In metal complexes as well, the importance of spin polarization compared to tire simple analysis of orbital amplitude for singly occupied molecular orbitals (SOMOs) has been emphasized (Braden and Tyler 1998). 

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