Polariton

TeraChem can compute energies, gradients, and non-adiabatic couplings for a molecule strongly coupled to a single quantized cavity mode. This is the regime of polariton chemistry, in which the molecular and photonic degrees of freedom hybridize into polaritonic states. Strong-coupling calculations are layered on top of an underlying excited-state method (typically CASCI/FOMO-CASCI or CASSCF) and produce polaritonic adiabatic states from which gradients and couplings are derived in the usual way.

The cavity is treated as a single harmonic mode with frequency \(\omega\), a fixed polarization direction \(\hat{\mathbf{g}}\), and coupling strength \(\lambda\). The electronic-photonic Hamiltonian is constructed in a basis consisting of the lowest \(N\) electronic states multiplied by photon-number states \(\lvert n\rangle\). The resulting polaritonic Hamiltonian is diagonalized at each geometry; gradients are obtained by propagating derivatives of the underlying electronic states through this diagonalization.

Cavity Hamiltonians

The form of the light-matter Hamiltonian is selected with sc_model:

jc — Jaynes-Cummings. Couples molecular dipoles to the cavity mode but drops the dipole self-energy term. State-specific (ground- and excited-state) dipole moments and excited-state-to-excited-state dipolar couplings are included by default.
pf — Pauli-Fierz. Like jc but additionally includes the dipole self-energy term, which is required for gauge consistency in the long-wavelength limit. Recommended for quantitative work.

The individual dipole contributions can be overridden after the model is chosen using the sc_ss_dipoles, sc_ex_dipoles, and sc_dipole_self keywords; this is mostly useful for diagnostic comparisons.

Specifying the cavity

A cavity calculation needs the photon energy, the polarization direction of the cavity mode (the "g vector"), and the coupling strength:

strongcoupling   yes
sc_model         pf
sc_omega         0.3768          # photon energy in atomic units (Hartree)
sc_gvec_x        0.0
sc_gvec_y        0.0
sc_gvec_z        1.5418          # polarization direction (renormalized internally)
sc_strength      0.007           # coupling strength in atomic units

The components of sc_gvec_{x,y,z} give the polarization direction and are renormalized to unit length by the parser; only the direction matters.

sc_states controls how many molecular electronic states are included before constructing the polaritonic Hamiltonian (default 2). sc_target selects which polaritonic state is being followed when a single state is required (e.g. for a gradient or coupling calculation; default 0, the lower polariton).

Non-adiabatic couplings between polaritonic states are computed when sc_nacs yes is set; this is needed for dynamics or for conical-intersection optimization in the polaritonic manifold.

Example: gradient on the lower polariton of ethylene (Pauli-Fierz)

This input computes the gradient of the first polaritonic excited state of ethylene at the FOMO-CASCI(2,2)/6-31G* level, with a cavity tuned to 0.3768 a.u. (\(\approx 10.25\) eV) polarized along z:

ethylene_PF_grad.in

coordinates       geom.xyz
basis             6-31gs
method            hf
run               gradient
threall           1.0e-14
convthre          1.0e-8
threspdp          1.0e-8
precision         mixed
charge            0
spinmult          1

fon               yes
fon_temperature   0.25
casci             yes
closed            7
active            2
cassinglets       2

cphfiter          500
cphftol           1.0e-5
cphfalgorithm     inc_diis

strongcoupling    yes
sc_model          pf
sc_states         2
sc_target         1
sc_omega          0.3768
sc_gvec_x         0.000
sc_gvec_y         0.000
sc_gvec_z         1.5418
sc_strength       0.007

end

Example: same calculation with the Jaynes-Cummings Hamiltonian

Switching sc_model pf to sc_model jc performs the same calculation without the dipole self-energy term. This is useful for assessing the magnitude of dipole self-energy effects but is not recommended for production work.

ethylene_JC_grad.in

coordinates       geom.xyz
basis             6-31gs
method            hf
run               gradient
charge            0
spinmult          1

fon               yes
fon_temperature   0.25
casci             yes
closed            7
active            2
cassinglets       2

cphfiter          500
cphftol           1.0e-5

strongcoupling    yes
sc_model          jc
sc_states         2
sc_target         1
sc_omega          0.3768
sc_gvec_x         0.000
sc_gvec_y         0.000
sc_gvec_z         1.5418
sc_strength       0.007

end

Summary of keywords

Keyword	Type	Default	Description
`strongcoupling`	bool	no	Master switch; must be `yes` to enable any other `sc_*` keyword
`sc_model`	string	none	`jc` (Jaynes-Cummings) or `pf` (Pauli-Fierz). Sets defaults for the dipole flags below.
`sc_states`	int	2	Number of molecular electronic states included before polaritonic diagonalization
`sc_target`	int	0	Index of the polaritonic state followed for gradients and properties
`sc_omega`	float	0.0	Photon energy in atomic units (Hartree)
`sc_gvec_x`	float	0.0	x-component of the cavity polarization vector
`sc_gvec_y`	float	0.0	y-component of the cavity polarization vector
`sc_gvec_z`	float	0.0	z-component of the cavity polarization vector
`sc_strength`	float	0.0	Light-matter coupling strength in atomic units
`sc_ss_dipoles`	bool	model-dependent	Include state-specific (permanent) dipole moments in the polaritonic Hamiltonian. Defaults to `yes` for both `jc` and `pf`.
`sc_ex_dipoles`	bool	model-dependent	Include excited-state-to-excited-state dipolar couplings. Defaults to `yes` for both `jc` and `pf`.
`sc_dipole_self`	bool	model-dependent	Include the dipole self-energy term. Defaults to `no` for `jc` and `yes` for `pf`.
`sc_nacs`	bool	no	Compute non-adiabatic couplings between polaritonic states