Theory Overview

PyQED collects methods for molecular quantum dynamics, spectroscopy, light-matter interaction, open quantum systems, and quantum chemistry. The common starting point is a Hamiltonian model and a representation for the state, density matrix, or response function.

Hamiltonian Models

Most workflows start from a Hamiltonian split into a reference part and one or more interactions:

\[H(t) = H_0 + V(t).\]

For molecular problems, H_0 may be an electronic Hamiltonian, a vibronic Hamiltonian, a grid Hamiltonian, or an effective model Hamiltonian. The time-dependent perturbation V(t) often represents a laser pulse, a cavity mode, or a coupling to an external field.

The time-dependent Schrodinger equation is

\[i \frac{\partial}{\partial t} |\psi(t)\rangle = H(t)|\psi(t)\rangle,\]

and the corresponding density-matrix equation is

\[\frac{d\rho}{dt} = -i [H(t), \rho].\]

Atomic units are used in most quantum chemistry and molecular dynamics code paths unless a specific example states otherwise.

Quantum Chemistry

The nonrelativistic electronic Hamiltonian in a molecular orbital basis is

\[H = \sum_{pq} h_{pq} a_p^\dagger a_q + \frac{1}{2}\sum_{pqrs} (pq|rs) a_p^\dagger a_r^\dagger a_s a_q + E_\mathrm{nuc}.\]

Here h contains one-electron kinetic and nuclear-attraction integrals, (pq|rs) are electron-repulsion integrals, and E_nuc is the nuclear repulsion energy.

Hartree-Fock approximates the wavefunction by a single determinant and solves self-consistent orbital equations:

\[F C = S C \varepsilon.\]

Post-HF methods add correlation on top of this reference. PyQED includes CI, MP2, CASCI, and CASSCF-related workflows. In active-space methods, the orbital space is partitioned into inactive, active, and external orbitals. CASCI solves the active-space CI problem for fixed orbitals, while CASSCF optimizes both the CI coefficients and orbital rotations.

For larger calculations, electron-repulsion integrals may be represented by factors rather than a dense four-index tensor:

\[(pq|rs) \approx \sum_L L^L_{pq} L^L_{rs}.\]

This reduces memory pressure and enables direct factorized contractions in RHF, CASCI, and CASSCF workflows.

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Discrete Variable Representation

Discrete variable representation methods represent wavefunctions on a grid while keeping derivative operators as matrices. A typical one-dimensional grid Hamiltonian is

\[H_{ij} = T_{ij} + V(x_i)\delta_{ij}.\]

The potential is diagonal in the DVR grid, while the kinetic-energy matrix depends on the selected DVR basis. DVR methods are useful for wavepacket dynamics, model potentials, and grid-based quantum chemistry prototypes.

For tensor-product grids, multidimensional Hamiltonians are assembled from one-dimensional kinetic terms and a multidimensional potential:

\[H = \sum_\alpha I_1 \otimes \cdots \otimes T_\alpha \otimes \cdots \otimes I_d + V(q_1,\ldots,q_d).\]

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Nonlinear Spectroscopy

Spectroscopy observables are response functions of dipole or current operators. Linear absorption is determined by a two-point dipole correlation function:

\[C(t) = \langle \mu(t)\mu(0)\rangle.\]

The frequency-domain spectrum is obtained from a Fourier transform, with appropriate damping or windowing:

\[I(\omega) \propto \omega\, \mathrm{Im} \int_0^\infty dt\, e^{i\omega t} C(t).\]

Nonlinear spectroscopy uses higher-order response functions. For example, a third-order signal depends on nested commutators of dipole operators evaluated at multiple time delays. In practice, PyQED supports both sum-over-states and time-domain propagation viewpoints.

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Open Quantum Systems

Open-system dynamics describe a system coupled to an environment. A common Markovian model is the Lindblad master equation:

\[\frac{d\rho}{dt} = -i[H,\rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\} \right).\]

For structured baths, the environment is often characterized by a spectral density:

\[J(\omega) = \sum_k |g_k|^2 \delta(\omega - \omega_k).\]

The bath correlation function determines memory effects and dissipative rates. HEOM and related methods approximate the bath correlation function as a sum of exponentials and propagate auxiliary density operators.

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Floquet Theory

For a periodic Hamiltonian,

\[H(t + T) = H(t),\]

solutions can be written in Floquet form:

\[|\psi_\alpha(t)\rangle = e^{-i\epsilon_\alpha t} |\phi_\alpha(t)\rangle, \qquad |\phi_\alpha(t+T)\rangle = |\phi_\alpha(t)\rangle.\]

The quasienergies epsilon are eigenvalues of the Floquet operator in an extended time-periodic Hilbert space. This representation is useful for driven two-level systems, periodically driven lattices, and light-dressed matter.

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Light-Matter Coupling

Light-matter models couple electronic, vibrational, or model-system degrees of freedom to quantized or classical fields. A minimal cavity-QED Hamiltonian has the form

\[H = H_\mathrm{mol} + \omega_c a^\dagger a + g\,\mu\,(a^\dagger + a),\]

where omega_c is the cavity frequency, g is the coupling strength, and mu is the molecular dipole operator. Depending on gauge and approximation, additional self-polarization terms may be required.

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Nonadiabatic and Ehrenfest Dynamics

Nonadiabatic dynamics couple nuclear motion to multiple electronic states. In an adiabatic basis, nuclear motion feels state-dependent energies and derivative couplings:

\[d_{IJ}(R) = \langle \Phi_I(R) | \nabla_R \Phi_J(R) \rangle.\]

Ehrenfest dynamics propagates classical nuclei on the mean field generated by a coherent electronic state:

\[M_A \ddot{R}_A = -\nabla_{R_A} \langle \Psi_e(t;R) | H_e(R) | \Psi_e(t;R) \rangle.\]

In a locally diabatic overlap formulation, short-time electronic propagation can use state overlaps between neighboring geometries instead of explicit nonadiabatic coupling vectors. This is useful when analytic NACs are unavailable or expensive.

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Method Selection

Use these rough guidelines:

  • Use RHF as the starting point for closed-shell mean-field calculations.

  • Use MP2 for inexpensive dynamic correlation near a single-reference state.

  • Use CASCI when active orbitals are fixed and static correlation matters.

  • Use CASSCF when active orbitals need to relax with the CI wavefunction.

  • Use TDDFT or TDHF for linear-response excited states and spectra.

  • Use DVR methods for low-dimensional grid dynamics.

  • Use MPS/DMRG when an active space or lattice model is too large for exact diagonalization but has manageable entanglement.

  • Use HEOM or master equations when environmental dissipation is essential.

  • Use Floquet theory when the Hamiltonian is periodic in time.

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