GATE Chemistry Physical Chemistry Flashcards

24 sample cards from the Physical Chemistry deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. State the Hermite polynomial recurrence relation.

   $$H_{v+1}(y) = 2y\,H_v(y) - 2v\,H_{v-1}(y)$$

2. How does an anharmonic (Morse) potential differ from the harmonic potential, and what are the spectral consequences?

   The Morse potential $V=D_e(1-e^{-a(r-r_e)})^{2}$ is asymmetric and allows bond dissociation. Energy levels converge (spacing decreases at high $v$), there are a finite number of levels, and overtone transitions ($\Delta v=\pm2,\pm3$) become allowed, unlike the equally-spaced harmonic levels.

3. Write the anharmonic oscillator energy expression including the anharmonicity constant.

   $$E_v = \left(v+\tfrac{1}{2}\right)\hbar\omega - \left(v+\tfrac{1}{2}\right)^{2}\hbar\omega\,x_e + \cdots$$ where $x_e$ is the anharmonicity constant; the negative term causes level convergence.

4. Write the operators for the square of total angular momentum $\hat{L}^{2}$ and its $z$-component $\hat{L}_z$ in spherical coordinates.

   $$\hat{L}_z = -i\hbar\frac{\partial}{\partial \phi}, \qquad \hat{L}^{2} = -\hbar^{2}\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\!\left(\sin\theta\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}\right]$$

5. State the eigenvalues of $\hat{L}^{2}$ and $\hat{L}_z$ acting on spherical harmonics $Y_l^{m}$.

   $$\hat{L}^{2}Y_l^{m} = l(l+1)\hbar^{2}\,Y_l^{m}, \qquad \hat{L}_z Y_l^{m} = m\hbar\,Y_l^{m}$$ with $l=0,1,2,\dots$ and $m=-l,\dots,+l$.

6. State the commutation relations among the angular momentum components.

   $$[\hat{L}_x,\hat{L}_y]=i\hbar\hat{L}_z,\;[\hat{L}_y,\hat{L}_z]=i\hbar\hat{L}_x,\;[\hat{L}_z,\hat{L}_x]=i\hbar\hat{L}_y$$ and $[\hat{L}^{2},\hat{L}_i]=0$. So $L^2$ and one component (e.g. $L_z$) can be known simultaneously, but not two components.

7. Define the angular momentum raising and lowering (ladder) operators and their action.

   $$\hat{L}_{\pm} = \hat{L}_x \pm i\hat{L}_y$$ They raise/lower $m$: $\hat{L}_{\pm}Y_l^{m} \propto Y_l^{m\pm1}$, while leaving $l$ unchanged.

8. Give the energy levels and degeneracy of a rigid rotor (rotational energy levels).

   $$E_J = \frac{\hbar^{2}}{2I}J(J+1) = hcB\,J(J+1), \quad J=0,1,2,\dots$$ Each level has degeneracy $g_J=2J+1$; $I=\mu r^{2}$ and $B=\dfrac{h}{8\pi^{2}cI}$.

9. What is the rotational constant $B$ and how does it relate to bond length?

   $B=\dfrac{h}{8\pi^{2}cI}$ (in cm$^{-1}$), with $I=\mu r^{2}$. A larger moment of inertia (longer bond or heavier atoms) gives a smaller $B$, so $B$ lets one determine bond lengths from rotational spectra.

10. What is the form of a hydrogenic atomic orbital wavefunction?

    $$\psi_{n,l,m}(r,\theta,\phi) = R_{n,l}(r)\,Y_l^{m}(\theta,\phi)$$ It factors into a radial part $R_{n,l}(r)$ and an angular part (spherical harmonic) $Y_l^{m}$.

11. Give the allowed values and meaning of the quantum numbers $n$, $l$, and $m_l$.

    $n=1,2,3,\dots$ (principal, size/energy); $l=0,1,\dots,n-1$ (azimuthal, shape, gives s,p,d,f); $m_l=-l,\dots,+l$ (magnetic, orientation), giving $2l+1$ orbitals per subshell.

12. State the energy of a hydrogen-like atom and how many nodes an orbital has.

    $$E_n = -\frac{Z^{2}me^{4}}{8\varepsilon_0^{2}h^{2}n^{2}} = -13.6\,\frac{Z^{2}}{n^{2}}\;\text{eV}$$ Total nodes $=n-1$: angular nodes $=l$, radial nodes $=n-l-1$.

13. Define the radial distribution function (RDF) and write its expression.

    The RDF gives the probability of finding the electron in a thin shell at radius $r$: $$P(r) = r^{2}R_{n,l}^{2}(r) \quad (\text{or } 4\pi r^{2}\psi^{2}\text{ for s})$$ The $r^{2}$ factor accounts for the increasing shell volume with $r$.

14. For the hydrogen 1s orbital, where does the radial distribution function peak?

    The RDF $P(r)=4r^{2}a_0^{-3}e^{-2r/a_0}$ peaks at $r=a_0$ (the Bohr radius, $\approx 52.9$ pm)—the most probable radius despite $\psi^{2}$ being maximum at the nucleus.

15. What is the orbital approximation in many-electron atoms?

    It approximates the total wavefunction as a product of one-electron orbitals: $\Psi(1,2,\dots,n)\approx \psi_1(1)\psi_2(2)\cdots\psi_n(n)$. Each electron is assigned its own hydrogen-like orbital, ignoring instantaneous electron-electron correlation.

16. What is electron spin, and what are the spin quantum number values for an electron?

    Spin is an intrinsic angular momentum of the electron with spin quantum number $s=\tfrac{1}{2}$. Its $z$-projection $m_s=\pm\tfrac{1}{2}$, giving spin states $\alpha$ (spin up) and $\beta$ (spin down).

17. Give the eigenvalue equations for the spin operators $\hat{S}^{2}$ and $\hat{S}_z$.

    $$\hat{S}^{2}\chi = s(s+1)\hbar^{2}\chi = \tfrac{3}{4}\hbar^{2}\chi, \qquad \hat{S}_z\chi = m_s\hbar\chi = \pm\tfrac{1}{2}\hbar\chi$$

18. State the Pauli exclusion principle in both its forms.

    No two electrons in an atom can have the same set of all four quantum numbers $(n,l,m_l,m_s)$. Equivalently (antisymmetry principle): the total wavefunction of fermions must be antisymmetric under exchange of any two electrons.

19. What does the antisymmetry requirement mean physically, and what does it forbid?

    Exchanging two electrons must change the sign of the total wavefunction: $\Psi(1,2)=-\Psi(2,1)$. It forbids two electrons from occupying the same spin-orbital, since $\Psi$ would then vanish—the basis of the exclusion principle.

20. For two electrons in helium, write the spatial-spin wavefunction respecting antisymmetry for the ground state.

    Ground state: symmetric spatial $\times$ antisymmetric (singlet) spin: $$\Psi = 1s(1)1s(2)\cdot\frac{1}{\sqrt{2}}\big[\alpha(1)\beta(2)-\beta(1)\alpha(2)\big]$$

21. What is a Slater determinant and why is it used?

    It is an antisymmetrized product of spin-orbitals written as a determinant. Because swapping two rows (electrons) or columns changes its sign, it automatically satisfies the Pauli antisymmetry principle, and it vanishes if two spin-orbitals are identical.

22. Write the Slater determinant for a two-electron system with spin-orbitals $\chi_a$ and $\chi_b$.

    $$\Psi(1,2) = \frac{1}{\sqrt{2}}\begin{vmatrix} \chi_a(1) & \chi_b(1) \\ \chi_a(2) & \chi_b(2) \end{vmatrix} = \frac{1}{\sqrt{2}}\big[\chi_a(1)\chi_b(2)-\chi_b(1)\chi_a(2)\big]$$

23. What is the normalization factor of an $N$-electron Slater determinant, and what do its rows and columns represent?

    The prefactor is $\dfrac{1}{\sqrt{N!}}$. Each column corresponds to one spin-orbital and each row to one electron (or vice versa); interchanging two electrons swaps two rows, flipping the sign as required by antisymmetry.

24. Compare the energy-level spacing of the particle in a box, harmonic oscillator, and rigid rotor.

    Particle in a box: spacing increases with $n$ ($E\propto n^{2}$). Harmonic oscillator: equally spaced ($\Delta E=\hbar\omega$). Rigid rotor: spacing increases linearly with $J$ ($\Delta E = 2B(J+1)$, since $E\propto J(J+1)$).

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