Introduction

This section provides the theoretical foundation for cavity-coupled molecular dynamics simulations in Cavity HOOMD.

Overview of Cavity QED in Molecular Dynamics

Cavity Quantum Electrodynamics (Cavity QED) studies the interaction between light confined in an optical cavity and matter. When molecular vibrations couple strongly to cavity photon modes, hybrid light-matter states called polaritons form.

Classical Treatment

Cavity HOOMD uses a classical description of both nuclear and photonic degrees of freedom:

  • Nuclear motion: Classical trajectories governed by Newton’s equations

  • Photonic modes: Harmonic oscillators with classical coordinates and momenta

  • Coupling: Bilinear dipole-field interaction

This approach is valid when:

  1. Many photons populate the cavity mode (large n)

  2. Thermal energy \(k_B T\) exceeds photon energy \(\hbar\omega\)

  3. Nuclear motion is classical (heavy nuclei)

Quantum vs Classical:

Aspect

Quantum

Classical (Cavity HOOMD)

Nuclear dynamics

Wave packets

Trajectories

Photonic modes

Fock states

Harmonic oscillators

Coupling

\(\hat{H}\)

Classical Hamiltonian

Validity

T → 0, few photons

T > 0, many photons

Theoretical Foundation

Hamiltonian Derivation Overview

The cavity-molecule Hamiltonian in Cavity HOOMD is derived from quantum electrodynamics and reduced to a classical form through several approximations. This section provides a high-level overview; see Cavity Forces for the complete derivation.

Starting Point: Minimal Coupling Hamiltonian

The quantum description begins with charged particles (electrons and nuclei) coupled to the electromagnetic field through the vector potential:

\[\hat{H} = \hat{H}_\mathrm{M} + \hat{H}_\mathrm{EM}\]

where the matter Hamiltonian includes kinetic energy with minimal coupling:

\[\hat{H}_\mathrm{M} = \sum_i \frac{|\hat{\mathbf{P}}_i - Z_i e \hat{\mathbf{A}}(\mathbf{R}_i)|^2}{2M_i} + \sum_j \frac{|\hat{\mathbf{p}}_j + e \hat{\mathbf{A}}(\mathbf{r}_j)|^2}{2m_j} + \hat{V}\]

and the electromagnetic field energy is:

\[\hat{H}_\mathrm{EM} = \frac{\epsilon_0}{2} \int d^3\mathbf{r} \left( |\hat{\mathbf{E}}|^2 + c^2|\hat{\mathbf{B}}|^2 \right)\]

Key Approximations:

  1. Coulomb Gauge: \(\nabla \cdot \hat{\mathbf{A}} = 0\), electromagnetic fields are purely transverse

  2. Mode Expansion: The vector potential is expanded in cavity normal modes with frequencies \(\omega_\ell = \ell \pi c / L\)

  3. Uniform Field Approximation: Since molecular length scales \(\ll\) cavity length, \(\hat{\mathbf{A}}(\mathbf{r}) \approx \hat{\mathbf{A}}(0)\) is spatially uniform

  4. Gauge Transformations: Two unitary transformations are applied:

    • Remove vector potential from kinetic energy (length gauge)

    • 90° phase rotation (coordinate displacement form)

  5. Pauli-Fierz Form: After transformations, the Hamiltonian becomes:

\[\hat{H}_\mathrm{PF} = \hat{H}_\mathrm{M} + \frac{1}{2}\sum_{\ell,\alpha} \left[ \hat{p}_{\ell,\alpha}^2 + \omega_\ell^2 \left( \hat{q}_{\ell,\alpha} + \frac{\lambda \hat{\mu}_\alpha}{\omega_\ell} \right)^2 \right]\]

where \(\lambda = 1/\sqrt{\epsilon_0 V}\) is the light-matter coupling strength and \(\hat{\mu}_\alpha\) is the molecular dipole moment operator.

Cavity Born-Oppenheimer Approximation

Timescale Separation:

Electrons move much faster than nuclei, but electromagnetic modes typically match nuclear vibrational timescales. We therefore:

  1. Solve electronic structure separately for nuclear positions \(\{\mathbf{R}_i\}\) and cavity coordinates \(\{q_{\ell,\alpha}\}\):

\[\hat{H}_\mathrm{e} |\psi_g\rangle = V_\mathrm{BO}(\{\mathbf{R}_i, q_{\ell,\alpha}\}) |\psi_g\rangle\]

  1. Invoke adiabatic approximation: Since cavity modes are typically far off-resonant from electronic transitions:

\[V_\mathrm{BO}(\{\mathbf{R}_i, q_{\ell,\alpha}\}) \approx V_\mathrm{BO}(\{\mathbf{R}_i\})\]

This avoids treating polaritonic potential energy surfaces while retaining cavity effects on nuclear dynamics.

Classical Limit and Mean-Field Approximation

From Quantum to Classical:

Treating slower degrees of freedom (nuclei and photons) classically, the potential energy becomes:

\[V(\{\mathbf{R}_i, q_{\ell,\alpha}\}) = V_\mathrm{BO}(\{\mathbf{R}_i\}) + \sum_{\ell,\alpha} \left[ \frac{\omega_\ell^2 q_{\ell,\alpha}^2}{2} + \lambda \omega_\ell \langle\psi_g|\hat{\mu}_\alpha|\psi_g\rangle q_{\ell,\alpha} + \frac{\lambda^2}{2}\langle\psi_g|\hat{\mu}_\alpha^2|\psi_g\rangle \right]\]

Mean-Field Treatment:

The quantum fluctuation term \(\langle\psi_g|\hat{\mu}_\alpha^2|\psi_g\rangle\) is approximated by its classical mean-field value:

\[\langle\psi_g|\hat{\mu}_\alpha^2|\psi_g\rangle \approx \left(\langle\psi_g|\hat{\mu}_\alpha|\psi_g\rangle\right)^2 = \mu_\alpha^2\]

where \(\mu_\alpha(\{\mathbf{R}_i\})\) is the classical dipole moment.

Final Classical Hamiltonian:

\[H = \sum_i \frac{\mathbf{P}_i^2}{2M_i} + V_\mathrm{BO}(\{\mathbf{R}_i\}) + \sum_{\ell,\alpha} \left[ \frac{p_{\ell,\alpha}^2}{2} + \frac{\omega_\ell^2}{2}\left(q_{\ell,\alpha} + \frac{\lambda\mu_\alpha}{\omega_\ell}\right)^2 \right]\]

This is the starting point for cavity molecular dynamics simulations.

Single-Mode vs Multi-Mode Treatment

Single-Mode Approximation:

In Cavity HOOMD, we typically retain only the lowest-frequency electromagnetic mode:

\[\omega_{\ell=1} = \frac{2\pi c}{L} = \omega_\mathrm{c}\]

Justification:

  • Energy scale separation: Higher modes have \(\omega_\ell > \omega_c\) and are off-resonant with molecular vibrations

  • Coupling strength: Higher modes couple more weakly due to larger frequency denominators

  • Computational efficiency: Single mode reduces degrees of freedom from \(\mathcal{O}(N_\text{modes})\) to \(\mathcal{O}(1)\)

Multi-Mode Considerations:

Multi-mode treatment is necessary when:

  • Multiple cavity modes are near-resonant with molecular transitions

  • Studying mode-selective effects

  • Investigating multimode interference phenomena

The formalism in Cavity Forces naturally extends to multiple modes.

Scope and Applicability

What Cavity HOOMD Can Simulate

Strong Coupling Regime:

  • Collective vibrational strong coupling

  • Polariton formation and dynamics

  • Cavity-modified reaction rates

  • Energy transfer in cavity-coupled systems

  • Non-equilibrium switching experiments

System Types:

  • Molecular liquids and solids

  • Organic semiconductors

  • Biomolecular systems

  • Any polarizable molecular system

Temperature Range:

  • 10 K to 500 K (typical)

  • Limited by MD force field validity

  • Best suited for T > 50 K where classical treatment is valid

Limitations

1. Classical Approximation:

Cannot capture:

  • Zero-point energy effects

  • Quantum coherence

  • Tunneling

  • Photon number quantization

2. Single-Mode Approximation:

  • Only one or few cavity modes

  • No multimode interference effects

  • No cavity dispersion

3. No Electronic Excitations:

  • Ground state electronic structure only

  • No photochemistry or excited state dynamics

  • Coupling through nuclear dipole moments only

4. Periodic Boundary Conditions:

  • Infinite periodic system

  • No cavity boundaries or mirrors

  • Uniform field approximation (q=0) or single standing wave (finite-q)

Units and Conventions

Atomic Units

Cavity HOOMD uses HOOMD reduced units internally, which are related to atomic units:

Fundamental Constants:

  • Electron mass: \(m_e = 1\)

  • Elementary charge: \(e = 1\)

  • Reduced Planck constant: \(\hbar = 1\)

  • Coulomb constant: \(k_e = 1\)

Derived Units:

Quantity

Atomic Unit

SI Equivalent

Length

Bohr radius \(a_0\)

0.529 Å

Energy

Hartree \(E_h\)

27.2 eV = 4.36×10⁻¹⁸ J

Time

\(\hbar/E_h\)

2.42×10⁻¹⁷ s = 0.024 fs

Mass

Electron mass \(m_e\)

9.11×10⁻³¹ kg

Temperature

\(E_h/k_B\)

315,775 K

Frequency

\(E_h/\hbar\)

4.13×10¹⁶ Hz

Practical Conversions:

For typical molecular simulations:

# Energy
1 eV = 0.0367 Hartree
1 kcal/mol = 0.00159 Hartree

# Length
1 Å = 1.889 Bohr
1 nm = 18.89 Bohr

# Time
1 fs = 41.34 atomic time units
1 ps = 41,341 atomic time units

# Temperature
300 K = 0.00095 Hartree/k_B
100 K = 0.00032 Hartree/k_B

# Frequency
2000 cm⁻¹ = 0.00916 Hartree/ℏ

User Interface Units

Input Parameters (User-Friendly):

  • Temperature: Kelvin (K)

  • Cavity frequency: wavenumbers (cm⁻¹)

  • Time: picoseconds (ps)

  • Length: Angstroms (Å)

  • Mass: atomic mass units (amu)

Internal Computation:

All converted to reduced/atomic units automatically.

Output:

Can be in either reduced units or converted back to SI/conventional units.

Physical Constants Table

Useful Physical Constants:

Constant

Symbol

Value

Boltzmann constant

\(k_B\)

1.381×10⁻²³ J/K = 8.617×10⁻⁵ eV/K

Planck constant

\(h\)

6.626×10⁻³⁴ J·s

Reduced Planck constant

\(\hbar\)

1.055×10⁻³⁴ J·s

Speed of light

\(c\)

2.998×10⁸ m/s

Elementary charge

\(e\)

1.602×10⁻¹⁹ C

Avogadro constant

\(N_A\)

6.022×10²³ mol⁻¹

Vacuum permittivity

\(\varepsilon_0\)

8.854×10⁻¹² F/m

Molecular Constants:

Molecule

Vibrational Frequency

Dipole Moment

O₂

1580 cm⁻¹

0.0 D (non-polar)

CO

2143 cm⁻¹

0.11 D

H₂O

3650 cm⁻¹ (stretch)

1.85 D

CO₂

1333 cm⁻¹ (asymmetric)

0.0 D (linear)

Energy Scales and Typical Values

Molecular Energy Scales

At room temperature (300 K):

\[k_B T \approx 0.026 \text{ eV} \approx 200 \text{ cm}^{-1}\]

Vibrational energies:

  • O-H stretch: ~3600 cm⁻¹ ≈ 0.45 eV

  • C=O stretch: ~1700 cm⁻¹ ≈ 0.21 eV

  • C-H stretch: ~3000 cm⁻¹ ≈ 0.37 eV

  • Bending modes: ~1000-1500 cm⁻¹

Bonding energies:

  • Covalent bonds: 1-5 eV

  • Hydrogen bonds: 0.1-0.4 eV

  • van der Waals: 0.01-0.1 eV

Cavity Energy Scales

Typical cavity frequencies:

  • Mid-IR: 1000-4000 cm⁻¹ (0.12-0.50 eV)

  • Near-IR: 4000-13,000 cm⁻¹ (0.50-1.6 eV)

  • Visible: 13,000-25,000 cm⁻¹ (1.6-3.1 eV)

Single-molecule coupling:

\[g_{\text{single}} \sim 10^{-5} \text{ to } 10^{-3} \text{ (atomic units)}\]

Collective coupling (N molecules):

\[g_{\text{collective}} = g_{\text{single}} \sqrt{N}\]

For N=512 molecules:

\[g_{\text{collective}} \approx 22.6 \times g_{\text{single}}\]

Strong Coupling Criterion

Condition for strong coupling:

\[\Omega_R > \sqrt{\gamma_{\text{mol}} \kappa_{\text{cav}}}\]

where:

  • \(\Omega_R = 2g_{\text{collective}}\): Rabi splitting

  • \(\gamma_{\text{mol}}\): Molecular dephasing rate

  • \(\kappa_{\text{cav}}\): Cavity loss rate

Typical values:

  • \(\Omega_R \sim 50-200\) cm⁻¹ (experimental)

  • \(\gamma_{\text{mol}} \sim 10-100\) cm⁻¹

  • \(\kappa_{\text{cav}} \sim 1-50\) cm⁻¹

Strong coupling achieved when: \(\Omega_R > 50\) cm⁻¹

Parameter Guidelines

Parameter Relationships

Cavity Quality Factor:

\[Q = \frac{\omega_{\text{cav}}}{\kappa}\]

where \(\kappa\) is the cavity loss rate.

Cavity Lifetime:

\[\tau_{\text{cav}} = \frac{1}{\kappa}\]

Molecular Dephasing Time:

\[T_2 = \frac{1}{\gamma_{\text{mol}}}\]

Rabi Period:

\[T_{\text{Rabi}} = \frac{2\pi}{\Omega_R}\]

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