PHZ3400-09 Problem Set 1a: Difference between revisions
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Consider a diatomic molecule formed of two nonequivalent atoms A and B. Assume that the known ground state energies <math>\varepsilon_A</math> and <math>\varepsilon_B</math> of the two atoms are different, and that <math>\varepsilon_A = \varepsilon_B + \Delta</math>, with <math>\Delta > 0</math>; in other words, atom A has a higher ground state energy. Assume that the hopping element <math><\psi_o^A |H|\psi_o^B>=-t</math> is also known. | Consider a diatomic molecule formed of two nonequivalent atoms A and B. Assume that the known ground state energies <math>\varepsilon_A</math> and <math>\varepsilon_B</math> of the two atoms are different, and that <math>\varepsilon_A = \varepsilon_B + \Delta</math>, with <math>\Delta > 0</math>; in other words, atom A has a higher ground state energy. Assume that the hopping element <math><\psi_o^A |H|\psi_o^B> =-t</math> 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. | 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. | ||
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c) Determine the ground state molecular orbital wavefunction for this molecule, expressing it as a linear combination of the two ground state atomic orbitals. | c) Determine the ground state molecular orbital wavefunction for this molecule, expressing it as a linear combination of the two ground state atomic orbitals. | ||
<math>|\psi_o^{mol.}>= |c_A \psi_o^A > + |c_B \psi_o^B ></math> | <math>|\psi_o^{mol.}>= |c_A \psi_o^A> + |c_B \psi_o^B></math> | ||
Determine the weight factors <math>c_A</math> and <math>c_B</math> as a function of the matrix elements. | Determine the weight factors <math>c_A</math> and <math>c_B</math> as a function of the matrix elements. | ||
Revision as of 22:41, 18 January 2011
Consider a diatomic molecule formed of two nonequivalent atoms A and B. Assume that the known ground state energies 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 \varepsilon_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 \varepsilon_B} of the two atoms are different, and 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 \varepsilon_A = \varepsilon_B + \Delta} , with 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} ; 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 and 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 . If the two atoms are identical, i.e. , then it is clear from symmetry that . Determine how the probabilities start to differ as 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} increases. Explain why 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} the electron spends most of its time on atom B.
e) Explain how 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} corresponds to a "pure" covalent bond, while 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} 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.