Atomic Energies, Electrostatic Forces, and Intermolecular Interactions

Charged Particle Attraction and Repulsion in Atomic Structure

Electrostatic potential energy is the fundamental potential energy that governs atomic structure, chemical bonding, and molecular interactions. At the atomic scale, subatomic particles carry specific electric charges:

• Protons carry a positive charge ($q = +e$).
• Electrons carry a negative charge ($q = -e$).
• Neutrons carry a neutral charge ($q = 0$).

The elementary charge constant is defined as:

$e = 1.602 \times 10^{-19}\,\text{C}$

In an atom, the positively charged nucleus contains protons and neutrons, while negatively charged electrons occupy the space surrounding the nucleus.

For example, a neutral Carbon atom (${}^{12}_{6}\text{C}$) possesses an atomic number of $6$ and an atomic weight of $12.011\,\text{g\,mol}^{-1}$:

• Nucleus: Contains $6$ protons ($q_{\text{nucleus}} = +6e = +9.612 \times 10^{-19}\,\text{C}$) and $6$ neutrons.
• Electron cloud: Contains $6$ electrons orbiting the nucleus ($q_{\text{total\,electrons}} = -6e = -9.612 \times 10^{-19}\,\text{C}$).

Two primary electrostatic forces act simultaneously within an atom:

1. Attraction: Operates between particles of opposite charges (e.g., between the positively charged nucleus and negatively charged electrons). This attractive electrostatic potential energy holds the electrons to the nucleus.
2. Repulsion: Operates between particles of like charges (e.g., between negatively charged electrons). This repulsive interaction causes electrons to push away from one another within the atomic orbitals.

Calculating Electrostatic Potential Energy (Coulomb Potential)

The quantitative interaction between point charges is modeled using Coulomb's law. The Coulomb potential ($V$) calculates the electrostatic potential energy between two charged particles ($i$ and $j$):

$V(r_{ij}) = \frac{q_i q_j}{4\pi \epsilon_0 r_{ij}}$

Where:

• $V(r_{ij})$: Electrostatic potential energy (in Joules, \text{J}).
• $q_i$: Charge of particle $i$ (in Coulombs, \text{C}).
• $q_j$: Charge of particle $j$ (in Coulombs, \text{C}).
• $r_{ij}$: Distance separating particle $i$ and particle $j$ (in meters, \text{m}).
• $\epsilon_0$: Permittivity of free space, a fundamental constant equal to $8.854 \times 10^{-12}\,\text{C}^2\,\text{J}^{-1}\,\text{m}^{-1}$.

In chemical systems, interatomic and subatomic distances are frequently expressed in Angstroms ($\text{\AA}$):

$1\,\text{\AA} = 0.1\,\text{nm} = 10^{-10}\,\text{m}$

Typical single covalent bond lengths involving hydrogen atoms approximate $1\,\text{\AA}$, including:

• $\text{O-H}$ bond length: $0.96\,\text{\AA} = 0.096\,\text{nm} = 0.96 \times 10^{-10}\,\text{m}$
• $\text{N-H}$ bond length: $1.01\,\text{\AA} = 0.101\,\text{nm} = 1.01 \times 10^{-10}\,\text{m}$
• $\text{C-H}$ bond length: $1.09\,\text{\AA} = 0.109\,\text{nm} = 1.09 \times 10^{-10}\,\text{m}$

Quantifying Electrostatic Interactions in Media and Biomolecules

To compute the electrostatic potential energy between a pair of elementary positive and negative charges ($+e$ and $-e$) separated by a distance of $3.000\,\text{\AA}$ ($3.000 \times 10^{-10}\,\text{m}$) in a vacuum:

$V(r) = \frac{(1.602 \times 10^{-19}\,\text{C})(-1.602 \times 10^{-19}\,\text{C})}{4\pi (8.854 \times 10^{-12}\,\text{C}^2\,\text{J}^{-1}\,\text{m}^{-1})(3.000 \times 10^{-10}\,\text{m})}$

$V(r) = -7.689 \times 10^{-19}\,\text{J\,per\,particle\,pair}$

Converting this single-pair interaction energy to molar quantities ($\text{kJ\,mol}^{-1}$) using Avogadro's number ($N_A = 6.022 \times 10^{23}\,\text{mol}^{-1}$):

$V_{\text{molar}} = (-7.689 \times 10^{-19}\,\text{J}) \times \left(\frac{6.022 \times 10^{23}\,\text{pairs}}{1\,\text{mol}}\right) \times \left(\frac{1\,\text{kJ}}{1000\,\text{J}}\right) = -463.0\,\text{kJ\,mol}^{-1}$

The local dielectric environment significantly attenuates the strength of electrostatic interactions:

• Vacuum: Charges separated by $3\,\text{\AA}$ yield an interaction energy of $\sim -500\,\text{kJ\,mol}^{-1}$ ($-463.0\,\text{kJ\,mol}^{-1}$ calculated).
• Bulk Water: Due to the high dielectric constant of water ($\epsilon_r \approx 80$), the charges are screened, reducing interaction energy to $\sim -6\,\text{kJ\,mol}^{-1}$.
• Protein Interior: Hydrophobic packing lowers dielectric screening relative to water, resulting in an interaction energy of $\sim -250\,\text{kJ\,mol}^{-1}$.
• Protein Surface: Partial exposure to aqueous solvent results in intermediate dielectric screening, yielding an interaction energy of $\sim -20\,\text{kJ\,mol}^{-1}$.

Mathematical Relationship Between Potential Energy and Force

The electrostatic force ($F(r)$) acting on charged particles is equal to the negative first spatial derivative of the electrostatic potential energy ($V(r)$):

$F(r) = -\frac{d}{dr}V(r)$

Substituting the expression for the Coulomb potential:

$V(r_{ij}) = \frac{q_i q_j}{4\pi \epsilon_0 r_{ij}}$

$F(r_{ij}) = -\frac{d}{dr}\left(\frac{q_i q_j}{4\pi \epsilon_0 r_{ij}}\right) = \frac{q_i q_j}{4\pi \epsilon_0 r_{ij}^2}$

This expression represents Coulomb's Law for force.

Key derivative properties between energy and force:

• Slopes on Potential Energy Curves: The magnitude and direction of the electrostatic force at any interatomic distance $r$ correspond directly to the negative slope ($-\frac{dV}{dr}$) of the potential energy curve.
• Spatial Scaling: Force scales inversely with the square of the distance ($F \propto \frac{1}{r^2}$), whereas potential energy scales inversely with distance ($V \propto \frac{1}{r}$). Consequently, repulsive forces between like charges grow dramatically faster than potential energy as distance $r$ decreases.

Chemical Bonds and Covalent Interaction Potential

Chemical bonds are formed through a balance of electrostatic attractions and repulsions between multiple nuclei and electrons. In a diatomic molecule such as molecular Hydrogen ($\text{H}_2$), four distinct pairwise forces occur between two nuclei ($\text{H}_A, \text{H}_B$) and two electrons ($e_1, e_2$):

1. Nucleus-Electron Attractions: Between $\text{H}_A - e_1$, $\text{H}_A - e_2$, $\text{H}_B - e_1$, and $\text{H}_B - e_2$ (distances $r_{1\text{A}}, r_{2\text{A}}, r_{1\text{B}}, r_{2\text{B}}$). These attractive forces pull the nuclei toward the shared electron density cloud, acting as electrostatic glue.
2. Nucleus-Nucleus Repulsion: Between $\text{H}_A - \text{H}_B$ (distance $R_{\text{AB}}$).
3. Electron-Electron Repulsion: Between $e_1 - e_2$ (distance $r_{12}$).

The net effective potential energy ($V_{\text{eff}}(R_{\text{AB}})$) as a function of internuclear separation $R_{\text{AB}}$ establishes three main regions:

• Large Separation ($R_{\text{AB}} \to \infty$): Interactions approach zero ($V_{\text{eff}} \approx 0$).
• Ideal Bond Length ($R_{\text{AB}} = R_e$): Net attractive forces equal net repulsive forces ($\frac{dV_{\text{eff}}}{dR_{\text{AB}}} = 0$). Potential energy reaches a minimum, defining the equilibrium bond length ($R_e$) and the bond dissociation energy ($\Delta E_d$).
• Short Separation ($R_{\text{AB}} < R_e$): Strong nuclear and electronic repulsions dominate, causing potential energy to rise rapidly.

Non-Bonded Interactions and van der Waals Forces

Non-bonded neutral atoms and molecules exert attractive forces on one another through subtle fluctuations in their electron distributions, giving rise to van der Waals forces. For nonpolar molecules, London dispersion forces represent the primary van der Waals attraction.

Mechanism of London Dispersion Interactions:

1. An isolated neutral atom possesses a time-averaged spherical, unpolarized electron cloud.
2. Transient, instantaneous quantum-mechanical fluctuations in electron position break spherical symmetry, creating a temporary instant dipole.
3. This instantaneous dipole induces a complementary, polarized charge distribution in adjacent atomic electron clouds.
4. Both polarized states exist simultaneously in a correlated manner, establishing an attractive electrostatic potential between non-bonded neutral species.

Balance of Repulsive and Dispersion Energies

When two non-bonded atoms approach one another, their interaction is governed by two opposing distance-dependent energy terms:

1. London Dispersion Energy (Attractive): Long-range attraction scaling as:    $V_{\text{dispersion}} \propto -\frac{1}{r^6}$
2. Electronic Repulsion Energy (Repulsive): Short-range Pauli exclusion repulsion arising from overlapping electron clouds, scaling as:    $V_{\text{repulsion}} \propto +\frac{1}{r^{12}}$

The sum of these terms produces the net interaction potential energy curve:

• Ideal Contact Distance: The minimum of the net interaction curve occurs where attractive dispersion and short-range repulsion balance.
• van der Waals Radius ($r_w$): Defined as half of the ideal interatomic contact distance between two non-bonded atoms of the same element:

$r_w = \frac{\text{Ideal Interatomic Distance}}{2}$

In space-filling molecular representations, atomic surfaces are drawn using their van der Waals radii ($r_w$).

Energy Units, Scales, and Conversions

Energy values across molecular dynamics, thermodynamics, and physical chemistry use several common units:

1. Joule (\text{J}): The SI base unit of energy defined as:    $1\,\text{J} = 1\,\text{kg\,m}^2\,\text{s}^{-2} = 1\,\text{N\,m}$
2. Kilojoules per Mole ($\text{kJ\,mol}^{-1}$): The standard energy metric in chemical applications.
3. Thermal Energy ($RT$): Thermal kinetic energy per mole at room temperature ($25^\circ\text{C} = 298.15\,\text{K}$):    $RT = (8.314\,\text{J\,K}^{-1}\,\text{mol}^{-1})(298.15\,\text{K}) = 2.479\,\text{kJ\,mol}^{-1} \approx 2.48\,\text{kJ\,mol}^{-1}$
4. Kilocalories per Mole ($\text{kcal\,mol}^{-1}$):    $1\,\text{kcal\,mol}^{-1} = 4.184\,\text{kJ\,mol}^{-1}$Note on Food Calories: In nutritional labeling, $1\,\text{Calorie (Cal)} = 1000\,\text{calories (cal)} = 1\,\text{kcal} = 4.184\,\text{kJ}$.

5. Electron-Volt per Molecule ($\text{eV/molecule}$):    $1\,\text{eV/molecule} = 96.485\,\text{kJ\,mol}^{-1} \approx 96.5\,\text{kJ\,mol}^{-1}$
6. Nuclear Binding Energy Scale: Nuclear binding energies range from $1\,\text{to}\,9\,\text{MeV}$ ($10^6\,\text{eV}$) per nucleon—roughly $1$ to $9$ million times larger than typical chemical bond and thermal interaction energies.

Comparison of Relative Energy Magnitudes ($\text{kJ\,mol}^{-1}$):

Key Quantitative Capabilities and Theoretical Concepts

• Convert arbitrary electrical charges expressed as integer multiples of elementary charge $e$ into Coulombs ($\text{C}$) via $q = z \times (1.602 \times 10^{-19}\,\text{C})$.
• Calculate explicit electrostatic potential energies ($V$) and force magnitudes ($F$) for arbitrary pairs of charges across defined interatomic distances ($r$) using Coulomb's law and dielectric constants.
• Describe the origin of van der Waals and London dispersion forces through transient fluctuations in electron density and induced dipole polarization.
• Differentiate between covalent bond potential energy profiles, short-range Pauli repulsions ($r^{-12}$), and long-range dispersion attractions ($-r^{-6}$).