Destructive interference atomic orbitals

We can arrive at this result in a different way. Consider what happens when two hydrogen atoms approach so that their Is AOs overlap (Fig. 1.10b). From the analogy between waves and wave functions, one might expect an interference effect analogous to that of the experiment indicated in Fig. 1.2. In other words, the orbitals should combine. When they do so, we may get constructive interference, the orbitals adding [c equation (1.13)], or we may get destructive interference, the orbitals subtracting [ct equation (1.14)]. The first situation is indicated in Fig. 1.10(c, d) the electron density is given by... 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

In molecular hydrogen, where we are building LCAO-MOs from two atomic orbitals, we expect two molecular orbitals. In the second molecular orbital, the two atomic orbitals interfere destructively where they overlap. This orbital has the form... 

Molecular orbitals are formed by combining atomic orbitals when atomic orbitals interfere constructively, they give rise to bonding orbitals when they interfere destructively, they give rise to antibonding orbitals. N atomic orbitals combine to give N molecular orbitals. 

De Broglie s work clearly shows that a moving electron can be considered as a wave. If it behaves in that way, a stable orbit in a hydrogen atom must contain a whole number of wavelengths, or otherwise there would be interference that would lead to cancellation (destructive interference). This condition can be expressed as... 

In order to reveal the mechanism of this molecular half-adder, the T(E) spectra of the molecule are presented in Fig. 26b. When perpendicular to the plane of the molecule, each NO2 contributes a very sharp resonance which does not participate in the overall conductance. When rotated by 90°, an NO2 introduces a supplementary resonance in the gap of the molecule. Due to its asymmetrical delocalization over the atomic orbitals, this resonance increase the conduction between the drive and the XOR electrode, but not between the drive and the AND electrode. This insures a 1 output for the former and a 0 for the latter. When the two NO2S are rotated, the two resonances they introduce create a deep interference between the drive and the XOR electrode. Located on the Fermi energy of the molecule, this interference leads to a low conductance state and a 0 logical output for the XOR gate. In contrast, the two resonances do not interfere destructively between the drive and the AND electrode, leading to a high conductance state and a 1 logical output. 

The hydrogen electron visualized as a standing wave around the nucleus. The circumference of a particular circular orbit has to correspond to a whole number of wavelengths, as shown in (a) and (b), or else destructive interference occurs, as shown in (c). This model is consistent with the fact that only certain electron energies are allowed the atom is quantized. (Although this idea has encouraged scientists to use a wave theory, it does not mean that the electron really travels in circular orbits.)... 

The combination of hydrogen Is atomic orbitals to form MOs. The phases of the orbitals are shown by signs inside the boundary surfaces. When the orbitals are added, the matching phases produce constructive interference, which give enhanced electron probability between the nuclei. This results in a bonding molecular orbital. When one orbital is subtracted from the other, destructive interference occurs between the opposite phases, leading to a node between the nuclei. This is an antibonding MO. 

Bond formation during reactions is entirely analogous to the process whereby we form molecular orbitals from atomic orbitals in individual molecules. Interaction between orbitals of like sign results in constructive interference (i.e., bonding), whereas interaction between orbitals of opposing sign results in destructive interference. 

The formation and the shapes of the sp hybrid orbitals and their participation in chemical bonds are shown in Figure 6.41. The first column shows the non-hybridized orbitals on the Be atom, and the second column shows the hybrid orbitals. The amplitude for each hybrid at any point r from the beryllium nucleus is easily visualized as the result of constructive and destructive interference of the 2s and 2p wave functions at that point. Because the sign of the 2s orbital is always positive, whereas that of the 2p orbital is different in the -I- and -z directions, the amplitude of Xi is greatest along -l-z, and that of 2 is greatest along —z. Because the probabilities are the squares of the amplitudes, an electron in xi is much more likely to be found on the left side of the nucleus than on the right the opposite is... 

Multiplying one of the two H atom orbitals by —1 is required for constructive interference combination with two + signs leads to destruction interference. 

Increased amplitude or electron density between atoms Antibonding orbital (destructive interference) ... 

The valence bond and molecular orbital theories differ in how they use the orbitals of two hydrogen atoms to describe the orbital that contains the electron pair in H2. Both theories assume that electron waves behave much like more familiar waves, such as sound and light waves. One property of waves that is important here is called interference in physics. Constructive interference occurs when two waves combine so as to reinforce each other ( in phase ) destructive interference occurs when they oppose each other ( out of phase ) (Figure 1.15). In the valence bond model constructive interference between two electron waves is seen as the basis for the shared electron-pair bond. In the molecular orbital model, the wave functions of molecules are derived by combining wave functions of atoms. 

In the bonding molecular orbital the electron density is greatest between the nuclei of the bonding atoms. In the antibonding molecular orbital, on the other hand, the electron density decreases to zero between the nuclei. We can understand this distinction if we recall that electrons in orbitals have wave characteristics. A property unique to waves allows waves of the same type to interact in such a way that the resultant wave has either an enhanced amplitude or a diminished amphtude. In the former case, we call the interaction constructive interference in the latter case, it is destructive interference (Figure 10.21). 

An approximate description of the MOs in H2 can be obtained by considering them as linear combinations of atomic orbitals (LCAOs). Each of the H atoms has one 1 atomic orbital let the two associated wavefunctions be ipi and In Section 1.6, we mentioned the importance of the signs of the wavefunctions with respect to their overlap during bond formation. The sign of the wavefunction associated with the I5 atomic orbital may be either + or —. Just as transverse waves interfere in a constructive (in-phase) or destructive (out-of-phase) manner, so too do orbitals. Mathematically, we represent the possible combinations of the two I5 atomic orbitals by equations 1.27 and 1.28, where N and N are the normalization factors. Whereas V mo is nn in-phase (bonding) interaction, ipuQ is an out-of-phase (antibonding) interaction. 

When calculations and plots are produced for the R = 10 case, constructive and destructive interference is seen to be much weaker because of the weak atomic orbital overlap. 

The mismatched phases produce destructive interference leading to a node of electron probability between the two nuclei. With electron distribution pushed to the outside, the energy of an antibonding orbital is higher than the energy of the atomic orbitals from which it is composed. 31. a. H2 has two valence electrons to put in the MO diagram whereas He2 has 4 valence electrons. 

At the equilibrium intemuclear separation of 106 pm, the overlap integral is equal to 0.59. Hence, = 0.56 and N = 1.10. A larger normalization constant is needed for the antibonding orbital because there is destructive interference of the two atomic orbitals between the two nuclei. 

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