10. Allyl radical (CH2-CH-CH2) is a planar molecular radical in which the three a electrons are
delocalized. Hückel theory is a simplified molecular orbital (MO) theory describing the MOs of
the n-electron system as linear combinations of the 2p, atomic orbitals at each carbon atom
involved in the delocalized system. The basic assumptions of Hückel theory are: All overlap
integrals are set equal to zero; All resonance integrals between non-neighboring atoms are set
equal to zero; All coulomb integrals are set equal to a; All resonance integrals between
neighboring atoms are set equal to ẞ. What is the total energy of the three electrons predicted
by Hückel MO theory?

(A) 3α +2√2B
(B) 2α +2√2B
(C) 2a - √2B
(D) 2a +4B
(E) 4α +4B

1.Given that the ionization energy of hydrogen-atom ground state is 13.6 eV, how much energy in eV is required to remove an electron from the 3d2 orbital in a Be3+ ion? (A) 6.8 eV (B) 18.1 eV (C) 24.2 eV (D) 54.4 eV (E) 61.2 eV

2. Consider a quantum particle of mass m trapped in a 3-dimensional cubic box. The potential energy is zero inside the box and infinity outside the box. The length on each side of the box is a. What is the degeneracy of the energy level of E=14h ? (h = Planck constant) 8ma² (A) 1 (B) 2 (C) 4 (D) 6 (E) 8

3. In quantum mechanics, the measurements of two physical quantities, a and b, are represented by operators à and &, respectively. If the two operators commute, which of the following statements is NOT true? (A) It is possible to measure a and b precisely and simultaneously. (B) The two operators possess a common set of eigenfunctions. (C) The two operators possess a common set of eigenvalues. (D) ÂBf =BAf, where f is a well-behaved function. (E) [A, B] =Ô.

4. Consider a quantum particle confined in a 1-D symmetric finite potential well defined as U(x)=U(0,0≤xsa [U,x＜0 or x ＞ a Which of the following schematic diagrams of the bound eigenstate energy distribution and probability density functions is most qualitatively correct in all aspects? (A) (B) E E U Uo E (C) (D) yCrf () tot Uo E3 -E3 (E) ( Lo -E E (x) ( -E -E -E E ( -E E 이 -E E E=0 0 0 0 注

5. a A quantum harmonic oscillator (QHO) with a fundamental vibrational frequency v is in superposition state described by a wave function (x)=(x) √3 %(x) +c,(x)+ (x) + √3 (x), where 4(x)'s are normalized eigenfunctions of the QHO (the subscripts "v" denote the vibrational quantum numbers) and c₁ is the expansion coefficient for 4₁(x). What is the average energy when a very large number of identically prepared QHO's in (x) state are measured?7, (A) 2 (B) 2hv (C) 12"3 hv (D) hv 2 (E) 4hv

6. The first excited state of helium (He) atom is described by electron configuration 1s¹2s1. Which of the followings is the correct form of spin-orbital functions for He atom first excited-state wave function satisfying the Pauli Principle? (Note: 1S(i) and 2S(1) are the spatial orbitals; α(i) and B(i) are the spin functions; i=1, 2 denotes the electron coordinates; N is the normalization constant) (A) N[IS(1)2S(2) -1S(2)2S(1)][a(1)B(2)] (B) N[IS(1)2S(2)+1S(2)2S(1)][a(1)a(2)] (C) N[1S(1)a(1)][2S(2)B(2)] (D) N[IS(1)2S(2)+1S(2)2S(1)][a(1)B(2) +a(2)B(1([ (E) N[IS(1)2S(2)-1S(2)2S(1)][a(1)f(2)+a(2)B(1)]

7. Consider the variational method for the ground state of a two-electron atom. The simplest trial function is the Hartree product of the form (r,2) =h()4(r), where3/2 (1) =√x a 0 isthe hydrogen-like satomic orbitalwith anefective nuclear changes , as the variational parameter. The variational integral can be evaluated analytically to give By = E[-5²+25(6-Z)+5], where Z is the true nuclear charge and E,-27.2 eV is the Hartree energy.What is the ground state energy of Li* ion at this level of approximation? (A) -77.5 eV (B) -196.5 eV (C) -54.4 eV (D) -108.8 eV (E) -98.2 eV

8. The hydrogen atomic wave functions can be expressed as nm (r,0,4)= R (r) Y" (0,4), where2- IIe-r12a0 a √24 a) a T x =(2) ; 1 = (2) C080 ; } = (2) (sinee (12/2 yt1=34π An electron is in the 2p orbital of a H atom. What is the most probable radius (the radius where the radial distribution function is at maximum) of finding that electron? (A) 5ao(B) 4a (C) 3do (D) 2do (Ε) α

9. The complex form of H-atom wave functions for n=3, 1=2, m=±1 states are Pu(r,0,4)= Ra()},,where R, (r) is the radial function and X,"=(}) (sinOcos0)e are spherical harmonic functions with 1=2 and m=±1. A real form function (3dxz orbital) can be constructed प 1 3dxz √2 correct? with the linear combination of the two complex wave functions as '3d +3d1). Which of the following statements about the 3d,,orbital is NOT (A) 3d wave function is an eigenfunction ofH-atom H (Hamiltonian operator). (B) 3d wave function is an eigenfunction of H-atom E (operator for square of orbital angular momentum). (C) 3d wave function is an eigenfunction of H-atom £. (operator for angular momentum along the z axis). (D) The magnitude of orbital angular momentum is not zero. (E) the average values of Lz is zero.

11. The first excited electron configuration of HF is (10)2(20)2²(3o)2(1π)3(40*)1. What is (are) molecular term(s) that can arise from this excited configuration? (A) Σ and 3Σ (B) 'Π and 3Π (C) 3Π and Σ (D) 3H (Ε) 3Δ

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