PHZ3400-09 Problem Set 1a

Consider a diatomic molecule formed of two nonequivalent atoms A and B. Assume that the known ground state energies $\varepsilon _{A}$ and $\varepsilon _{B}$ of the two atoms are different, and that $\varepsilon _{A}=\varepsilon _{B}+\Delta$, with $\Delta >0$; in other words, atom A has a higher ground state energy. Assume that the hopping element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <\psi_o^A |H|\psi_o^B>=-t} is also known.

a) Using the tight-binding approximation, determine the energies of the bonding and the anti-bonding orbital for this molecule. Express the energy gap between the ground state and the first excited state in terms of the above parameters.

b) How does the energy gap between the bonding and the anti-bonding orbital behave for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta \gg t} ?

c) Determine the ground state molecular orbital wavefunction for this molecule, expressing it as a linear combination of the two ground state atomic orbitals. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi_o^{mol.}>= |c_A \psi_o^A > + |c_B \psi_o^B >}

Determine the weight factors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_A} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_B} as a function of the matrix elements.

d) Show that the probability for an electron to be found on atom A is given by the expression Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_A = |c_A |^2} . If the two atoms are identical, i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta = 0} , then it is clear from symmetry that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_A = P_B} . Determine how the probabilities start to differ as $\Delta$ increases. Explain why for $\Delta \gg t$ the electron spends most of its time on atom B.

e) Explain how $\Delta =0$ corresponds to a "pure" covalent bond, while $\Delta >0$ leads to partial charge transfer from atom A to atom B. If this happens, each atom acquires a net charge, and the corresponding Coulomb attraction brings in "ionic" character to the chemical bond. The standard ionic bond, such as found in NaCl corresponds to almost complete charge transfer, while the covalent character of the chemical bond is almost gone.