List of top Chemistry Questions on Quantum Chemistry

Wavefunctions and energies for a particle confined in a cubic box are \( \psi_{n_x,n_y,n_z} \) and \( E_{n_x,n_y,n_z} \), respectively. The functions \( \phi_1, \phi_2, \phi_3 \), and \( \phi_4 \) are written as linear combinations of \( \psi_{n_x,n_y,n_z} \). Among these functions, the eigenfunction(s) of the Hamiltonian operator for this particle is/are \[ \phi_1 = \frac{1}{\sqrt{2}} \psi_{1,4,1} - \frac{1}{\sqrt{2}} \psi_{2,2,3} \] \[ \phi_2 = \frac{1}{\sqrt{2}} \psi_{1,5,1} + \frac{1}{\sqrt{2}} \psi_{3,3,3} \] \[ \phi_3 = \frac{1}{\sqrt{2}} \psi_{1,3,8} + \frac{1}{\sqrt{2}} \psi_{3,8,1} \] \[ \phi_4 = \frac{1}{2} \psi_{3,3,1} + \frac{\sqrt{3}}{2} \psi_{2,4,1} \]

The correct option with regard to the following statements is 
(a) Time-independent Schrödinger equation can be exactly solved for Be\(^{2+}\). 
(b) For a particle confined in a one-dimensional box of length \( l \) with infinite potential barriers, the trial variation function \( \phi = \left[ \left( \frac{3}{l^3} \right)^{1/2} x \right] \) is not an acceptable trial wavefunction for \( 0 \le x \le l \). 
(c) Wavefunctions for system of Fermions must be anti-symmetric with respect to exchange of any two Fermions in the system. 
(d) Born-Oppenheimer approximation can be used to separate the vibrational and rotational motion of a molecule.

Consider two non-interacting particles confined to a one-dimensional box with infinite potential barriers. Their wavefunctions are \( \psi_1 \) and \( \psi_2 \), and energies are \( E_1 \) and \( E_2 \), respectively. The INCORRECT statement(s) about this system is/are:
Rigid rotor wavefunctions are given by \( Y_{l,m}(\theta, \phi) \). The wavefunctions \( Y_{1,0}(\theta, \phi) \) and \( Y_{2,0}(\theta, \phi) \) are given below: \[ Y_{1,0}(\theta, \phi) = \sqrt{\frac{3}{4\pi}} \cos \theta \qquad Y_{2,0}(\theta, \phi) = \sqrt{\frac{5}{16\pi}} (3\cos^2 \theta - 1) \] For a non-polar diatomic molecule, the value of transition dipole moment integral for the transition between \( Y_{1,0}(\theta, \phi) \) and \( Y_{2,0}(\theta, \phi) \) is equal to:
An electron is confined to move in a one-dimensional box of length 1˚ A. Its energy in the first excited state is approximately
Calculate the wavelength (in nanometer) associated with a proton moving at \(1.0 \times 10^3 \, \text{m/s}\)
Choose the correct answer from the options given below:

Match the following 

Column IColumn II
(P) Associated Legendre polynomials(III) Angular part of H atom
(Q) Hermite polynomials(I) Harmonic oscillator
(R) Associated Laguerre polynomials(IV) Radial part of H atom
(S) Trigonometric functions(II) Particle in a box model
The wavefunction for Be\(^{3+}\) in a certain state is given by \(\psi = N e^{-\left(\tfrac{4}{a_0}\right)r}\), where \(N\) is the normalization constant, \(r\) is the distance of the electron from the nucleus and \(a_0\) is the Bohr radius. The most probable distance of the electron from the nucleus in this state is
In the scheme \( \mathrm{P_2} \xrightleftharpoons[\;k_1\;]{\;I_a\;} 2\mathrm{Q} \xrightarrow{k_2} \mathrm{R} \), \(I_a\) represents the intensity of light absorbed. Assuming the quantum yield of the first step is one, the steady-state concentration of \(\mathrm{Q}\) is given by
The INCORRECT statement is
The maximum number of electrons present in an orbital with \(n = 4,\ l = 3\) is
If \( \Delta x \) is the uncertainty in position and \( \Delta v \) is the uncertainty in velocity of a particle and both are equal, then the correct expression for uncertainty in momentum for the same particle is:
A light from Paschen series of hydrogen atom is able to eject photoelectrons from a metal. Then the work function of the metal is