Exam 4 Review. Exam Review: A exam review sheet for exam 4 will be posted on the course webpage. Additionally, a practice exam will also be posted.

Transcription

1 Chem 4502 Quantum Mechanics & Spectroscopy (Jason Goodpaster) Exam 4 Review Exam Review: A exam review sheet for exam 4 will be posted on the course webpage. Additionally, a practice exam will also be posted. The exam (like exam 1), will be made up of multiple choice, short and long answer. We discussed the contributions to the total energy of the H 2 molecule in terms of the virial theorem. According to this description (for internuclear separation R): A. The average potential energy decreases (becomes more negative) as the electrons become more delocalized B. The average kinetic energy decreases (becomes less positive) as the electrons become more delocalized. C. The average kinetic energy increases as R decreases for all values of R. D. The average kinetic energy decreases as R decreases for all values of R. E. The average potential energy increases as R decreases for all values of R. F. The average potential energy decreases as R decreases for all values of R. G. The equilibrium bond length is determined by potential energy contributions only. Last names beginning A - M Akerman 225; N Z Tate We discussed the contributions to the total energy of the H 2 molecule in terms of the virial theorem. According to this description (for internuclear separation R): A. The average potential energy decreases (becomes more negative) as the electrons become more delocalized B. The average kinetic energy decreases (becomes less positive) as the electrons become more delocalized. C. The average kinetic energy increases as R decreases for all values of R. D. The average kinetic energy decreases as R decreases for all values of R. E. The average potential energy increases as R decreases for all values of R. F. The average potential energy decreases as R decreases for all values of R. G. The equilibrium bond length is determined by potential energy contributions only. The following terms arise from a given electron configuration: 4 S, 2 P, 2 D Using Hund's Rules, order these terms in order of increasing energy (from lowest to highest). A. 2 P < 4 S < 2 D B. 2 P < 2 D < 4 S C. 4 S < 2 P < 2 D D. 4 S < 2 D < 2 P E. 2 D < 4 S < 2 P F. 2 D < 2 P < 4 S Answer: B. The average kinetic energy decreases (becomes less positive) as the electrons become more delocalized.

2 The following terms arise from a given electron configuration: 4 S, 2 P, 2 D Using Hund's Rules, order these terms in order of increasing energy (from lowest to highest). A. 2 P < 4 S < 2 D B. 2 P < 2 D < 4 S C. 4 S < 2 P < 2 D D. 4 S < 2 D < 2 P E. 2 D < 4 S < 2 P F. 2 D < 2 P < 4 S Answer: D. 4 S < 2 D < 2 P Hund's Rule #1 tells us that the state with the highest spin multiplicity will be lowest in energy ( 4 S). Then, Rule #2 predicts that for a given spin multiplicity (here, doublet) the state with the higher orbital angular momentum will be lower in energy (so 2 D < 2 P). Which statement is true concerning the Hartree-Fock method? Choose the best answer. A. Electron-electron repulsion is omitted. B. The instantaneous correlations of the electrons' motions are neglected. C. The total energy of a state is equal to the sum of the energies of its occupied orbitals. D. This is not a variational method. E. The total energy of the helium atom is calculated exactly. F. 2 of the above statements are correct. Which statement is true concerning the Hartree-Fock method? Choose the best answer. A. Electron-electron repulsion is omitted. B. The instantaneous correlations of the electrons' motions are neglected. C. The total energy of a state is equal to the sum of the energies of its occupied orbitals. D. This is not a variational method. E. The total energy of the helium atom is calculated exactly. F. 2 of the above statements are correct. If two rows of a determinant are interchanged, then the determinant ; this property of determinants helps to ensure that determinantal wave functions satisfy the. A. equals zero; Born-Oppenheimer Approximation B. changes sign; Born-Oppenheimer Approximation C. equals zero; Pauli Exclusion Principle D. changes sign; Pauli Exclusion Principle E. equals zero; Hartree-Fock Approximation F. changes sign; Hartree-Fock Approximation G. equals zero; Variational Theorem H. changes sign; Variational Theorem I. equals zero; Conservation of Angular Momentum J. changes sign; Conservation of Angular Momentum Answer: B. The instantaneous correlations of the electrons' motions are neglected.

3 If two rows of a determinant are interchanged, then the determinant ; this property of determinants helps to ensure that determinantal wave functions satisfy the. A. equals zero; Born-Oppenheimer Approximation B. changes sign; Born-Oppenheimer Approximation C. equals zero; Pauli Exclusion Principle D. changes sign; Pauli Exclusion Principle E. equals zero; Hartree-Fock Approximation F. changes sign; Hartree-Fock Approximation G. equals zero; Variational Theorem H. changes sign; Variational Theorem I. equals zero; Conservation of Angular Momentum J. changes sign; Conservation of Angular Momentum How many microstates are represented by the 1 D term of the carbon atom in its ground state electron configuration? A. 0 B. 1 C. 2 D. 3 E. 4 F. 5 G. 6 H. 7 I. 8 J. 9 For the carbon atom in its ground state electron configuration, describe each of the microstates associated with the 1 D term Answer: D. changes sign; Pauli Exclusion Principle How many microstates are represented by the 1 D term of the carbon atom in its ground state electron configuration? F. 5 Write a wave function for the lithium atom (atomic number 3) in the form of a Slater determinant. (You do not have to normalize it.) Answer: A 1 D ("singlet D") term corresponds to S = 0 (so M s must be 0) and L = 2 (so M L can have 5 possible values: 2, 1, 0, -1, -2).

4 Write a wave function for the lithium atom (atomic number 3) in the form of a Slater determinant. (You do not have to normalize it.) (4 points, 2 points each) Fill in the blanks. If two rows of a determinant are interchanged, then the determinant. This property helps to ensure that determinantal wave functions satisfy. (4 points, 2 points each) Fill in the blanks. If two rows of a determinant are interchanged, then the determinant. This property helps to ensure that determinantal wave functions satisfy. Sodium vapor lamps commonly used for street lighting emit yellow light at about 589 nm as Na atoms relax from an excited electronic state to the 2 S ground state. Using the selection rules for atomic spectroscopy, write a term symbol for the excited state. (You can omit J.) If two rows of a determinant are interchanged, then the determinant changes sign. This property helps to ensure that determinantal wave functions satisfy the Pauli Principle.

5 Sodium vapor lamps commonly used for street lighting emit yellow light at about 589 nm as Na atoms relax from an excited electronic state to the 2 S ground state. Using the selection rules for atomic spectroscopy, write a term symbol for the excited state. (You can omit J.) Answer: 2 P Must be a doublet state in view of the ΔS = 0 selection rule. Must be a P state (having L = 1) in view of the ΔL = ± 1 (that is, +1 or -1) selection rule. The total, normalized wave function for the ground state (1s) of the hydrogen atom is: For this state, calculate the most probable distance of the electron from the nucleus. (Note: this question asks for the most probable value, not the average value.) According to Koopmans' Theorem, what is the first ionization energy (ionization potential) of neon? That is, what is the minimum energy needed to produce Ne +?

6 According to Koopmans' Theorem, what is the first ionization energy (ionization potential) of neon? That is, what is the minimum energy needed to produce Ne +? What is the total electron-electron repulsion in this atom, according to these Hartree- Fock calculations? Answer: 0.85 E h = 23.1 ev What is the total electron-electron repulsion in this atom, according to these Hartree- Fock calculations? Consider the following trial function for the hydrogen atom: α r2 ϕ(r) = e Using the variational method to calculate the energy gives the following expression: E ϕ = 3!2 α 2µ e2 ε o α 2π 3 Starting with this expression for E φ, calculate the best value of the variational parameter α. (You can leave your answer in terms of fundamental constants such as ħ, etc.) We know that the ground state electron configuration of neon is: 1s 2 2s 2 2p 6. Putting two electrons in each orbital, the sum of the orbital energies of the 10 electrons is: 2 (-32.82) + 2 (-1.94) + 6(-0.85) = E h This differs from the total energy of the atom ( E h ) by: (-128.7) = 54.1 E h = 1472 ev = total electron-electron repulsion

7 \