Molecular Potential Energy, Intramolecular Forces, and Intermolecular Interactions Notes

Intramolecular Interactions & Molecular Degrees of Freedom

• Definition of Potential Energy Surface:

  • Intramolecular interactions dictate the chemical forces and energetic changes associated with covalent bonding within a single molecule.

  • For an $N$-atom molecule, the potential energy $U$ is fundamentally a function of the spatial position vectors of all nuclei:

$U = U(\vec{R})$

• Where the complete nuclear configuration vector is defined as:

$\vec{R} = \{\vec{r}_1, \vec{r}_2, \dots, \vec{r}_N\}$

• Generalized Internal Coordinates:

  • Potential energy depends exclusively on the relative positions of the atoms, making it invariant to rigid translational or rotational displacements of the entire molecule.

  • The potential is re-expressed in terms of internal generalized coordinates $S_j$ (e.g., bond lengths, bond angles, torsional dihedral angles):

  • For a non-linear molecule: $U = U(S_1, S_2, \dots, S_{3N-6})$

  • For a linear molecule: $U = U(S_1, S_2, \dots, S_{3N-5})$

• Decomposition of Total Degrees of Freedom (D.O.F.):

  • An $N$-atom system possesses $3N$ total degrees of freedom, partitioned into translational, rotational, and internal vibrational modes:

$\text{D.O.F.} = 3N = 3_{\text{translation}} + 3_{\text{rotation}} + (3N - 6)_{\text{internal/vibrational}}$

• For linear molecules, rotation occurs about only $2$ orthogonal axes perpendicular to the internuclear axis (rotation around the molecular axis produces no spatial displacement of nuclei):

$\text{D.O.F.}_{\text{linear}} = 3N = 3_{\text{translation}} + 2_{\text{rotation}} + (3N - 5)_{\text{internal/vibrational}}$

• Diatomic Molecule Example ($N = 2$, linear structure):

$\text{Internal D.O.F.} = 3(2) - 5 = 1$

• The potential energy of a diatomic molecule depends on exactly one generalized coordinate: the scalar internuclear distance $r$, yields $U = U(r)$.

Covalent Bonding & Diatomic Potential Energy Curves

• Reference State for Potential Energy:

  • Potential energy is defined relative to an arbitrary reference point. The standard convention sets $U = 0$ at infinite internuclear separation:

$U \rightarrow 0 \quad \text{as} \quad r \rightarrow \infty$

• Anatomy of the Diatomic Potential Energy Curve $U(r)$:

  • Attractive Branch ($r > r_e$): At long separations, atoms experience an attractive force ($F = -\frac{dU}{dr}$). As $r$ decreases toward equilibrium, potential energy decreases ($U < 0$).

  • Equilibrium Internuclear Separation ($r_e$): The potential energy reaches its global minimum value $-D_e$. At this point, net force is zero:

$\frac{dU}{dr}\Big|_{r = r_e} = 0$

• Repulsive Branch ($r < r_e$): At separations smaller than $r_e$, electron cloud overlap and core nuclear repulsions cause $U(r)$ to rise rapidly, creating a steep repulsive wall.

  • Classical vs. Quantum Dissociation Energies:

• Classical Well Depth ($D_e$): The energy difference between the asymptote $U(\infty) = 0$ and the minimum of the potential energy curve at $r_e$:

$D_e = -U(r_e)$

• Actual/Quantum Dissociation Energy ($D_0$): The actual physical energy required to dissociate the molecule from its ground vibrational state to separated atoms at rest:

$D_0 = D_e - \text{ZPE}$

• Where $\text{ZPE}$ represents the Zero-Point Energy of the ground vibrational state.

Harmonic Approximation & Quantum Vibrational Energy

• Taylor Series Expansion around Equilibrium:

  • For small displacements about the equilibrium distance $r_e$, the true electronic potential $U(r)$ is approximated by expanding in a Taylor series:

$U(r) \approx U(r_e) + \frac{dU}{dr}\Big|_{r_e} (r - r_e) + \frac{1}{2} \frac{d^2U}{dr^2}\Big|_{r_e} (r - r_e)^2$

• Because $r_e$ is defined at the potential minimum, the first derivative term vanishes identically:

$\frac{dU}{dr}\Big|_{r_e} = 0$

• Harmonic Potential Energy Expression:

$U(r) \approx U(r_e) + \frac{1}{2} f_r (r - r_e)^2$

• Where $f_r$ is the harmonic force constant (spring constant), defined mathematically as the local curvature of the potential energy surface at minimum:

$f_r = \left(\frac{d^2U}{dr^2}\right)\Big|_{r_e}$

• Quantum Harmonic Oscillator Mechanics:

  • Reduced Mass ($\mu$): For a diatomic system composed of atomic masses $m_1$ and $m_2$:

$\mu = \frac{m_1 m_2}{m_1 + m_2}$

• Fundamental Vibrational Frequency ($\nu_e$):

$\nu_e = \frac{1}{2\pi} \left(\frac{f_r}{\mu}\right)^{1/2}$

• Quantized Energy Levels:

$E_{\text{vib}}(n) = \left(n + \frac{1}{2}\right) h \nu_e \quad \text{for} \quad n = 0, 1, 2, \dots$

• Zero-Point Energy (ZPE): Due to Heisenberg's Uncertainty Principle, a quantum oscillator cannot rest motionless at the exact potential minimum $r_e$. The minimum attainable energy in the ground state ($n = 0$) is:

$\text{ZPE} = E_{\text{vib}}(0) = \frac{1}{2} h \nu_e$

• Energy Level Spacing: The spacing between adjacent vibrational levels is uniform:

$\Delta E = E_{\text{vib}}(n+1) - E_{\text{vib}}(n) = h \nu_e$

• Limitations of the Harmonic Approximation:

  • The parabolic harmonic potential curve $U_{\text{harmonic}}(r)$ diverges to $\infty$ as $r \rightarrow \infty$.

  • Real molecular potential curves flatten to $0$ at large internuclear distances, permitting bond dissociation. The harmonic approximation fails for large displacements $r \gg r_e$

Short-Range Repulsion & Electronic Cloud Overlap

• Pauli Repulsion Physical Origin:

  • When non-bonding electronic clouds are forced into close proximity, the Pauli Exclusion Principle dictates that no two electrons can occupy identical quantum states.

  • Bringing closed electron shells together forces electrons of identical spin into higher-energy spatial orbitals, generating strong, short-range repulsive forces.

• Empirical Formulations of Repulsive Potentials:

  • Because short-range Pauli repulsion arises from complex quantum exchange effects, empirical equations are used in molecular dynamics:

  • Inverse Power Law: $U(r) = C r^{-n}$ (where $n$ typically ranges between $9$ and $12$).

  • Exponential Decay: $U(r) = C e^{-n r}$ or $U(r) = C e^{-\alpha r}$.

Intermolecular Interactions & Electrostatic Potentials

• Overview & Thermal Energy Reference:

  • Intermolecular interactions govern non-covalent forces between separate molecules or non-bonded groups.

  • Thermal energy at room temperature ($T = 300\,\text{K}$):

$k_B T \approx 4.14 \times 10^{-21}\,\text{J} \implies R T \approx 2.5\,\text{kJ\,mol}^{-1}$

• Covalent bonds range from hundreds of kilojoules per mole (e.g., $C-H$ bond strength $\approx 400\,\text{kJ\,mol}^{-1}$; $O_2$ double bond $\approx 498\,\text{kJ\,mol}^{-1}$), whereas intermolecular forces range from strong ionic bonds down to fractions of $k_B T$.

  • 1. Point Charge – Point Charge (Coulombic Interaction):

• The interaction between two localized point charges $q_1 = z_1 e$ and $q_2 = z_2 e$ separated by distance $r$:

$U(r) = \frac{z_1 z_2 e^2}{(4\pi \epsilon_0 \epsilon) r}$

• Fundamental Parameters:

  • Fundamental charge unit: $e = 1.602 \times 10^{-19}\,\text{C}$.

  • Charge numbers: $z_1, z_2$ (e.g., $z = -1$ for an electron).

  • Permittivity of free space: $\epsilon_0 = 8.854 \times 10^{-12}\,\text{C}^2\,\text{J}^{-1}\,\text{m}^{-1}$.

  • Relative permittivity (dielectric constant) of medium ($\epsilon$):

    • Vacuum: $\epsilon = 1$

    • Water ($H_2O$): $\epsilon = 78.4$

    • Ideal Metal / Perfect Conductor: $\epsilon \rightarrow \infty$

• Characteristics:

  • Decays slowly as $\frac{1}{r}$, making bare Coulombic forces extremely long-range.

  • Negative values ($U < 0$) signify attraction; positive values ($U > 0$) signify repulsion.

• Quantitative Example ($Na^+$ and $Cl^-$ in Vacuum):

  • Contact distance: $r = 0.28\,\text{nm}$, charge numbers: $z_1 = +1$, $z_2 = -1$.

$U(0.28\,\text{nm}) = -8.4 \times 10^{-19}\,\text{J} = -506\,\text{kJ\,mol}^{-1} \approx -200\,k_B T$

* To reduce this unscreened potential to $\approx -4\,k_B T$ in a vacuum, the separation distance must increase to $r = 56\,\text{nm}$.

• Debye Screening in Electrolyte Solutions:

  • In solutions containing mobile ions, neighboring charges rearrange dynamically to form an ionic atmosphere, exponentially screening Coulombic interactions:

$U(r) \sim \frac{q_1 q_2}{(4\pi \epsilon_0 \epsilon) r} \exp(-\kappa r)$

• The inverse screening parameter $\kappa^{-1}$ is the Debye length:

$\kappa^{-1} \propto I^{-1/2}$

• Where $I$ is the solution ionic strength ($I = \frac{1}{2} \sum z_i^2 c_i$). Debye lengths typically range from $0.1\,\text{nm}$ to $100\,\text{nm}$.

• For a $1.0\,\text{M}\,NaCl$ aqueous solution, $\kappa^{-1} \approx 0.3\,\text{nm}$ ($3\,\text{\AA}$). At a separation $r = 3 \kappa^{-1} = 0.9\,\text{nm}$, the potential energy decays by a factor of $e^{-3} \approx 0.05$ (attenuated to $\approx 1\%$ of its unscreened value at $10\,\text{\AA}$).

  • 2. Point Charge – Permanent Dipole Interaction:

• Interaction between a charge $z_1 e$ and a permanent dipole moment $\mu_2 = q d$ at separation $r \gg d$:

$U(r, \theta) = -\frac{z_1 e \mu_2 \cos(\theta)}{(4\pi \epsilon_0 \epsilon) r^2}$

• Parameters & Units:

  • Angle $\theta$: Angle between the point charge separation axis and the dipole vector axis.

  • Dipole moment unit: Debye ($D$), where $1\,D = 3.336 \times 10^{-30}\,\text{C\,m}$.

  • Charges of $\pm e$ separated by $0.1\,\text{nm}$ produce a dipole moment:

$\mu = (1.602 \times 10^{-19}\,\text{C}) \times (10^{-10}\,\text{m}) = 1.602 \times 10^{-29}\,\text{C\,m} = 4.8\,D$

* Water molecule permanent dipole moment: $\mu_{H_2O} = 1.85\,D$

• Orientation Behavior:

  • For $\theta = 0$ (negative pole directed toward positive charge), $\cos(0) = 1$, yielding an attractive potential ($U < 0$).

  • 3. Permanent Dipole – Permanent Dipole Interaction:

• Interaction between two fixed dipoles $\mu_1$ and $\mu_2$ separated by distance $r$:

$U(r, \theta_1, \theta_2, \phi) = -\frac{\mu_1 \mu_2}{(4\pi \epsilon_0 \epsilon) r^3} \left(2 \cos(\theta_1) \cos(\theta_2) - \sin(\theta_1) \sin(\theta_2) \cos(\phi)\right)$

• Decays as $\frac{1}{r^3}$ for fixed spatial orientations.

• Maximum Attraction (Head-to-Tail Alignment):

  • When $\theta_1 = 0$, $\theta_2 = 0$, and $\phi = 0$:

$U_{\text{max}}(r) = -\frac{2 \mu_1 \mu_2}{(4\pi \epsilon_0 \epsilon) r^3}$

* For two dipoles with $\mu_1 = \mu_2 = 1\,D$ in head-to-tail orientation, $U(r) \approx k_B T$ at a separation of $r = 0.36\,\text{nm}$.

Orientational Averaging & Polarizability / Induction

• Thermally Averaged Rotating Dipoles (Keesom Interaction):

  • For freely rotating dipoles, an unweighted spatial average over all orientations gives zero net potential ($\langle U \rangle = 0$) because attractive and repulsive orientations cancel.

  • However, lower-energy attractive orientations are statistically favored according to a Boltzmann weighting factor $\exp\left(-\frac{U}{k_B T}\right)$:

$\langle U(r) \rangle = \frac{\int U(r, \Omega) \exp\left(-\frac{U(r, \Omega)}{k_B T}\right) d\Omega}{\int \exp\left(-\frac{U(r, \Omega)}{k_B T}\right) d\Omega}$

• Performing this orientation angle integration yields the Keesom (Orientation) Interaction:

$\langle U(r) \rangle = -\frac{\mu_1^2 \mu_2^2}{3 (4\pi \epsilon_0 \epsilon)^2 k_B T r^6}$

• Thermal averaging changes the spatial distance dependence from $\frac{1}{r^3}$ to $\frac{1}{r^6}$, and introduces an explicit inverse temperature dependence ($\propto T^{-1}$).

  • Polarization / Induction Interactions (Debye Interactions):

• Occur when a fixed charge or permanent dipole polarizes a neighboring nonpolar molecule, inducing a transient dipole moment.

• Electric Polarizability ($\alpha$):

  • Represents the ease with which an external electric field $\vec{E}$ distorts a molecule's electron cloud:

$\vec{\mu}_{\text{induced}} = \alpha \vec{E}$

* Polarizability scales generally with molecular volume; larger, diffuse electron clouds (e.g., argon, xenon) exhibit significantly higher polarizability than small, tightly bound atoms (e.g., helium).

• Point Charge – Induced Dipole Interaction:

• Electric field generated by point charge $z_1 e$: $E(r) \propto \frac{z_1 e}{r^2}$.

• The induced dipole moment is always aligned along the line of centers with the electric field vector, rendering the interaction strictly attractive:

$U(r) = -\frac{(z_1 e)^2 \alpha_2}{2 (4\pi \epsilon_0 \epsilon)^2 r^4}$

• The potential energy decays as $\frac{1}{r^4}$.

  • General Definition of Van der Waals Forces:

• The total attractive Van der Waals interaction combines three distinct physical interactions that share a characteristic $\frac{1}{r^6}$ distance dependence:

  1. Keesom Interaction: Thermally averaged permanent dipole – permanent dipole forces ($\propto r^{-6}$).

  2. Debye Interaction: Permanent dipole – induced dipole forces ($\propto r^{-6}$).

  3. London Dispersion Interaction: Instantaneous dipole – induced dipole quantum fluctuations ($\propto r^{-6}$).

• Combined with short-range Pauli repulsion ($\sim r^{-12}$ or $\sim e^{-\alpha r}$), these components constitute the complete Van der Waals intermolecular potential function.