QuantumSpectra

QuantumSpectra is a computational quantum chemistry and molecular vibrational normal mode analysis library written in Julia. It diagonalizes mass-weighted Hessian tensors, projects out infinitesimal translations and rotations via Eckart-Sayvetz conditions, computes harmonic vibrational frequencies in $\text{cm}^{-1}$, evaluates zero-point vibrational energy (ZPE) and statistical thermodynamic functions ($Q_{\text{vib}}, U_{\text{vib}}, C_{v,\text{vib}}, S_{\text{vib}}$), and generates synthetic infrared (IR) absorption spectra.

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Theoretical Foundations

1. Mass-Weighted Hessian Transformation

For an $N$-atom molecule with Cartesian force constant matrix (Hessian) $H_{ij} = \frac{\partial^2 V}{\partial x_i \partial x_j}$, the mass-weighted Hessian $H_{\text{MW}}$ is defined by scaling by atomic masses $m_i$:

$$ H_{\text{MW}, ij} = \frac{H_{ij}}{\sqrt{m_i m_j}} $$

Where the mass matrix $M = \operatorname{diag}(m_1, m_1, m_1, \dots, m_N, m_N, m_N)$.

2. Eckart Translational & Rotational Projection

To isolate the $3N - 6$ (or $3N - 5$ for linear molecules) internal vibrational degrees of freedom from zero-frequency external motions, an orthonormal basis $U \in \mathbb{R}^{3N \times 6}$ is constructed from center-of-mass translations and infinitesimal rigid rotations:

$$ \mathbf{T}_\alpha = \sqrt{m_i} \hat{\mathbf{e}}_\alpha, \quad \mathbf{R}_\alpha = \sqrt{m_i} (\hat{\mathbf{e}}_\alpha \times \mathbf{r}_i) $$

The idempotent Eckart projection operator $P$ satisfies:

$$ P = I_{3N} - \sum_{k=1}^{6} \mathbf{u}_k \mathbf{u}_k^T, \quad P^2 = P, \quad P^T = P $$

Projecting the mass-weighted Hessian:

$$ H_{\text{proj}} = P H_{\text{MW}} P $$

3. Harmonic Normal Mode Frequencies

Eigen-decomposition of the projected mass-weighted Hessian yields eigenvalues $\lambda_k$ (in $\text{s}^{-2}$):

$$ H_{\text{proj}} \mathbf{L}_k = \lambda_k \mathbf{L}_k $$

The vibrational wavenumber $\tilde{\nu}_k$ in $\text{cm}^{-1}$ is:

$$ \tilde{\nu}_k = \frac{1}{2\pi c} \sqrt{\lambda_k} $$

4. Zero-Point Energy & Vibrational Thermodynamics

Zero-Point Vibrational Energy (ZPE):

$$ \text{ZPE} = \frac{1}{2} \sum_{k=1}^{3N-6} h \nu_k = \frac{1}{2} h c \sum_{k=1}^{3N-6} \tilde{\nu}_k $$

Vibrational partition function $Q_{\text{vib}}(T)$ at temperature $T$:

$$ Q_{\text{vib}}(T) = \prod_{k=1}^{3N-6} \frac{1}{1 - e^{-\Theta_{v, k} / T}}, \quad \Theta_{v, k} = \frac{h \nu_k}{k_B} $$

Internal energy $U_{\text{vib}}$ and constant-volume heat capacity $C_{v, \text{vib}}$:

$$ U_{\text{vib}}(T) = N_A \sum_{k=1}^{3N-6} h \nu_k \left( \frac{1}{2} + \frac{1}{e^{\Theta_{v, k} / T} - 1} \right) $$

$$ C_{v, \text{vib}}(T) = R \sum_{k=1}^{3N-6} \left(\frac{\Theta_{v, k}}{T}\right)^2 \frac{e^{\Theta_{v, k} / T}}{(e^{\Theta_{v, k} / T} - 1)^2} $$

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Architectural Flow

flowchart TD
    subgraph Geometry ["Molecular Specification"]
        ATOMS[Atomic Coordinates & Masses] --> COM[Center of Mass Translation]
        COM --> INERTIA[Inertia Tensor & Linearity Test]
    end

    subgraph Projector ["Eckart Projection Engine"]
        COM --> TRANS[3 Translations Tx, Ty, Tz]
        COM --> ROT[3 Rotations Rx, Ry, Rz]
        TRANS & ROT --> GS[Gram-Schmidt Orthonormalization]
        GS --> P_MAT[Projector Matrix P = I - UU^T]
    end

    subgraph Dynamics ["Hessian & Normal Mode Diagonalization"]
        H_CART[Cartesian Hessian SI] --> H_MW[Mass-Weighted Hessian H_mw = M^-1/2 H M^-1/2]
        H_MW & P_MAT --> H_PROJ[Projected Hessian H_proj = P H_mw P]
        H_PROJ --> EIGEN[Eigen-decomposition L, Lambda]
        EIGEN --> FREQ[Frequencies in cm^-1]
        EIGEN --> DISP[Cartesian Displacements]
    end

    subgraph Output ["Spectroscopy & Thermodynamics"]
        FREQ --> ZPE[Zero-Point Energy ZPE]
        FREQ --> THERMO[Q_vib, U_vib, Cv, S_vib]
        FREQ --> IR[Lorentzian/Gaussian Convoluted IR Spectrum]
    end

Loading

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Module Layout

src/
├── QuantumSpectra.jl   # Main module re-exports
├── constants.jl        # Physical constants (h, c, N_A, k_B, amu to kg)
├── molecule.jl         # Atom and Molecule types, COM, moment of inertia
├── projection.jl       # Eckart translational and rotational projection operator
├── normal_modes.jl     # Mass-weighted Hessian transformation and diagonalization
├── thermodynamics.jl   # ZPE, partition function, enthalpy, entropy, heat capacity
└── spectrum.jl         # Lorentzian and Gaussian line-broadened IR spectrum synthesis
test/
└── runtests.jl         # Verification against analytical diatomic oscillator and water

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Getting Started

using Pkg
Pkg.activate(".")
Pkg.instantiate()

using QuantumSpectra

# Define Water Molecule (H2O)
h2o = Molecule([
    Atom("O", 15.9994, [0.0, 0.0, 0.0]),
    Atom("H", 1.00784, [0.0, 0.757, 0.586]),
    Atom("H", 1.00784, [0.0, -0.757, 0.586]),
])

# Compute normal modes and thermodynamics
# analysis = calculate_normal_modes(h2o, H_cart)
# thermo = calculate_vibrational_thermodynamics(analysis, 298.15)

Run test suite:

julia --project=. test/runtests.jl

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License

This project is licensed under the MIT License.