Intemuclear axis

For example, the Carbon-atom 3P(Ml=1, Ms=0) = [ p ppQ(x + p apoP ] and 3P(Ml=0, Ms=0) = 2-C2 [Ip Pp. aj + piap-iP ] states interact quite differently in a collision with a closed-shell Ne atom. The Ml = 1 state s two determinants both have an electron in an orbital directed toward the Ne atom (the 2po orbital) as well as an electron in an orbital directed perpendicular to the C-Ne intemuclear axis (the 2pi orbital) the Ml = 0 state s two determinants have both electrons in orbitals directed perpendicular to the C-Ne axis. Because Ne is a closed-shell species, any electron density directed toward it will produce a "repulsive" antibonding interaction. As a result, we expect the Ml = 1 state to undergo a more repulsive interaction with the Ne atom than the Ml = 0 state. 

The vector L is so strongly coupled to the electrostatic field and the consequent frequency of precession about the intemuclear axis is so high that the magnitude of L is not defined in other words L is not a good quantum number. Only the component H of the orbital angular momentum along the intemuclear axis is defined, where the quantum number A can take the values... 

The coupling of S to the intemuclear axis is caused not by the electrostatic field, which has no effect on it, but by the magnetic field along the axis due to the orbital motion of the electrons. Figure 7.16(a) shows that the component of S along the intemuclear axis is Ffi. The quantum number F is analogous to Mg in an atom and can take the values... 

The component of the total (orbital plus electron spin) angular momentum along the intemuclear axis is Qfi, shown in Figure 7.16(a), where the quantum number Q is given by... 

For 5 states there is no orbital angular momentum and therefore no resulting magnetic field to couple S to the intemuclear axis. The result is that a 5 state has only one component, whatever the multiplicity. 

The second symmetry property applies to all diatomics and concerns the symmetry of with respect to reflection across any (n ) plane containing the intemuclear axis. If is symmetric to (i.e. unchanged by) this reflection the state is labelled -I- and if it is antisymmetric to (i.e. changed in sign by) this reflection the state is labelled —as in or Ig. This symbolism is normally used only for I states. Although U, A, doubly degenerate state is... 

The + or — label indicates whether the wave function is symmetric or antisymmetric, respectively, to reflection across any plane containing the intemuclear axis. Whether the + component is below or above the — component for, say, J = 1 depends on the sign of q in Equation (7.94). The selection rules ... 

Organic chemists usually think of a double bond as the combination of a a bond and a 7i bond. An alternative is to consider a double bond as made up of two equivalent bent bonds (these bonds point above and below the intemuclear axis). 

Potential energy surfaces are also central to our quantum-mechanical studies, and we are going to meet them again and again in subsequent chapters. Let s start then with Figure 3.1, which shows H2+. We are not going to be concerned with the overall translational motion of the molecule. For simphcity, I have drawn a local axis system with the centre of mass as the origin. By convention, we label the intemuclear axis the z-axis. 

Figure 3.4 LCAO wavefunction along intemuclear axis...

Many graphics packages allow for contour diagrams and surface plots. These are given above for the square of the LCAO plus combination, for any plane containing the intemuclear axis. 

In the bond framework in Figure 10-18. all the bonds form from end-on overlap of orbitals directed toward each other. As illustrated by the three examples in Figure 10-20. this type of overlap gives high electron density distributed symmetrically along the intemuclear axis. A bond of this type is called a sigma (cr) bond, and a bonding orbital that describes a cr bond is a (7 orbital. 

All a bonds have high electron density concentrated along the intemuclear axis and axial symmetry, so their end-on profiles are circles. ... 

Trans-polyenes H-(-HC=CH-),, -H, trans-polyenynes H-(HC=CH-C=C) -H, cumulenes H2C=(C=C) =CH2 and polyynes H-(C=C) -H have been studied (M=N-1). For eentrosymmetrie molecules, the first order hyperpolarizability p is equal to zero so that non linear effects are of second order nature. Furthermore, (the x axis goes through the middle of the C-C bonds of the polyenes, or is the intemuclear axis in the case of linear molecules) is the most important component of the second order y hyperpolarizability tensor, the other components being negligible. Both y and the mean hyperpolarizability... 

Rather than giving the general expression for the Hellmann-Feynman theorem, we focus on the equation for a general diatomic molecule, because from it we can leam how p influences the stability of a bond. We take the intemuclear axis as the z axis. By symmetry, the x and y components of the forces on the two nuclei in a diatomic are zero. The force on a nucleus a therefore reduces to the z component only, Fz A, which is given by... 

Figure 6.1 Binding and antibinding regions for a heteronuclear diatomic molecule consisting of two nuclei A and B with ZA = ZB. The coordinate system is superimposed. The distance from a point with coordinates (x,y,z) to nucleus A is rA and to nucleus B is rB. the distance between the nuclei is RAb To obtain the 3D binding and antibinding regions rotate the figure about the intemuclear axis.

The basic principles dealing with the molecular orbital description of the bonding in diatomic molecules have been presented in the previous section. However, somewhat different considerations are involved when second-row elements are involved in the bonding because of the differences between s and p orbitals. When the orbitals being combined are p orbitals, the lobes can combine in such a way that the overlap is symmetric around the intemuclear axis. Overlap in this way gives rise to a a bond. This type of overlap involves p orbitals for which the overlap is essentially "end on" as shown in Figure 3.5. For reasons that will become clear later, it will be assumed that the pz orbital is the one used in this type of combination. 

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