4. mqed: Molecular QED solvers

4.1. qedhf: QED Hartree-Fock

4.1.1. Theory and Background

4.1.1.1. Pauli-Fierz Hamiltonian

The light-matter Hamiltonian of molecular quantum electrodynamics (mQED) is the Pauli-Fierz (PF) Hamiltonian:

\[\begin{split}\hat{H}_{\tt{PF}} &= \hat{H}_{\tt{e}} + \hat{H}_{\tt{p}} + \hat{H}_{\tt{ep}} + \hat{H}_{\tt{DSE}} \\ &= \hat{H}_{\tt{e}} + \sum_\al \left[ \hat{H}_{\tt{p}}^{\al} + \hat{H}_{\tt{ep}}^{\al} + \hat{H}_{\tt{DSE}}^{\al} \right]\end{split}\]

where:

• \(\hat{H}_{\tt{p}}\) is the photonic Hamiltonian,

• \(\hat{H}_{\tt{ep}}\) is the bilinear electron-photon Hamiltonian,

• \(\hat{H}_{\tt{DSE}}\) is the dipole self-energy (DSE) Hamiltonian

which are dependent on the modes of the photon, \(\al\). The matter (electronic) Hamiltonian, \(\hat{H}_{\tt{e}}\), is:

\[\hat{H}_{\tt{e}} = \sum_{pq} \h{p}{q} \cf{p} \af{q} + \frac{1}{2} \sum_{pqrs} \v{pq}{rs} \cf{p} \cf{q} \af{s} \af{r}\]

where:

• \(\cf{}\) and \(\af{}\) are fermionic single-particle creation and annihilation operators, and

• \(\{pqrs\}\) are general electron orbital indices.

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Expanded, the terms of the PF Hamiltonian are:

\[\begin{split}\hat{H}_{\tt{PF}} &= \hat{H}_e + \sum_\al \left[ \om_\al \cb{\al} \ab{\al} + \sqrt{\frac{\om_\al}{2}}\bm{\la}_\al \cdot \hat{D} (\cb{\al} + \ab{\al}) + \frac{1}{2} (\bm{\la}_\al \cdot \hat{D})^2 \right] \\ &= \hat{H}_e + \sum_\al \left[ \om_\al \cb{\al} \ab{\al} + \sqrt{\frac{\om_\al}{2}} \bm{e}_{\al} \cdot \la_\al \cdot \hat{D} (\cb{\al} + \ab{\al}) + \frac{1}{2} (\bm{e}_{\al} \cdot \la_\al \cdot \hat{D})^2 \right]\end{split}\]

where:

• \(\cb{\al}\) and \(\ab{\al}\) are bosonic creation and annihilation operators of photon mode \(\al\),

• \(\om_\al\) is the frequency of the photon mode,

• \(\la_\al = \sqrt{\frac{1}{\epsilon V_\al}}\) is the amplitude/coupling strength of the photon mode,

• \(\bm{e}_\al\) is the unit vector of the photon mode,

• \(V\) is the cavity volume, and

• \(\hat{D}\) is the molecular dipole operator (electronic DOFs, can also include nuclear DOFs).

4.1.1.2. QED-HF Wavefunction

The QED-HF reference wavefunction ansatz is:

\[\ket{\Psi_{\tt{QED-HF}}} = \ket{\Psi_{\tt{HF}}} \otimes \ket{0_{\tt{p}}}\]

where \(\ket{\Psi_{\tt{HF}}}\) is the non-QED/HF wavefunction and \(\ket{0_{\tt{p}}}\) are zero photon states:

\[\begin{split}\ket{0_{\tt{p}}} &= \prod_\al \left[ \sum_{n} \ket{n} \right] \\ &= \prod_\al \left[ \bm{C}^n_{\al} \sum_{n} (\cb{\al})^n \ket{0} \right]\end{split}\]

in which each photon mode \(\al\) is expressed in terms of \(n\)-normalized photon number states, \(\ket{n}\), each of which are defined in terms of the photon vacuum state, \(\ket{0}\).

The QED-HF energy can be evaluated self-consistently after modifying the HF one- and two-electon integrals, as detailed in get_hcore() and get_jk(). These two functions make calls to functions get_dse_hcore() and get_dse_jk(), respectively.

4.1.1.3. Coherent-State Representation

The coherent-state (CS) representation is achieved by transforming the PF Hamiltonian above:

\[\begin{split}\ket{\Psi_{\tt{CS-QED-HF}}} &= \prod_\al e^{z_\al (\ab{\al} - \cb{\al})} \ket{\Psi_{\tt{QED-HF}}} \\ &= \prod_\al e^{z_\al \ab{\al} - {z^{*}_{\al}} \cb{\al}} \ket{\Psi_{\tt{QED-HF}}} \\ &= \bm{U_z} \ket{\Psi_{\tt{QED-HF}}}\end{split}\]

where \(z_\al\) denotes the displacement due to the coupling of mode \(\al\) with the electrons of the molecular system:

\[z_{\al} = \sum_\al \frac{\la_\al \cdot \mel*{\mu}{\hat{D}}{\nu}} {\sqrt{2 \om_\al}}\]

Consequently, \(\bm{U_z}\) also transforms the original PF Hamiltonian, \(\bm{U}_{\bm{z}} \hat{H}_{\tt{PF}} \bm{U}^\dagger_{\bm{z}}\), to form CS Hamiltonian, \(\hat{H}_{CS}\):

\[\begin{split}\hat{H}_{CS} = \hat{H}_{\tt{e}} &+ \sum_\al \om_\al \cb{\al} \ab{\al} \\ &- \sum_\al \sqrt{\frac{\om_\al}{2}} \mel*{\mu}{\bm{\la}_\al \cdot (\hat{D} - \ev*{\hat{D}}_{\mu\nu})}{\nu} (\cb{\al} + \ab{\al}) \\ &+ \sum_\al \frac{1}{2} \mel*{\mu}{[\bm{\la}_\al \cdot (\hat{D} - \ev*{\hat{D}}_{\mu\nu})]^2}{\nu}\end{split}\]

In the CS representation, the QED-HF energy is subject to an DSE-dependent energy difference:

\[E_{\tt{QEDHF}} = E_{HF} + \frac{1}{2} \sum_\al \mel*{\mu} {[ \bm{\la}_\al \cdot (\hat{D} - \ev*{\hat{D}}_{\mu\nu})]^2} {\nu}\]

which is can also be added by modifiying the one-electron integrals. This is explained in more detail here: get_dse_hcore().

4.1.2. RHF class definition

4.2. scqedhf: SC-QED-HF

4.2.1. Theory and Background

WIP

4.2.1.1. Polaron Transformation

WIP

4.2.2. RHF class definition

4.3. vtqedhf: VT-QED-HF

4.3.1. Theory and Background

WIP

4.3.1.1. Variational Transformation

WIP

4.3.2. RHF class definition

5. mqed: Multiscale mQED solvers

This is the multiscale QED module for polariton chemistry. WIP.

5.1. MSRHF: Multiscale QED-HF