# Output chemical properties#

You can specify quantities you want to calculate as a list, and for each quantity, you can specify the target electronic state in `parameters`. You can specify the electronic states with indices: 0: ground state, 1: first excited state, 2: second excited state and so on. Please note that the states of output chemical properties are confined by `num_excited_states`. For Hartree-Fock, only `gradient` and `hessian` of the electronic ground state in molecular system are available.

• `dipole_moment`: Permanent electric dipole moment. For post-HF calculation, CASCI is supported.

• `parameters`:

• `state`: The target electronic state

• `transition_dipole_moment`: Electronic transition dipole moment. For post-HF calculation, CASCI is supported.

• `parameters`:

• `state_pair`: The target pair of two electronic states

• `oscillator_strength`: The oscillator strength. For post-HF calculation, CASCI is supported.

• `parameters`:

• `state_pair`: The target pair of two electronic states

• `gradient`: First-order derivative with respect to nuclear coordinates. For post-HF calculation, this property will be calculated analytically.

• `type`: Choice of the method for calculation. Please See below.

• `dx`: Interval for numerical differentiation (Å). Default is 0.001.

• `parameters`:

• `state`: The target electronic state

• `hessian`: Second-order derivative with respect to nuclear coordinates. For post-HF calculation, this property will be calculated using numerically derivative with analytical gradients.

• `type`: Choice of the method for calculation. Please See below.

• `dx`: Interval for numerical differentiation (Å). Default is 0.001.

• `parameters`:

• `state`: The target electronic state

• `vibrational_analysis`: Vibrational analysis. For post-HF calculation, this property will be calculated using numerically derivative with analytical gradients.

• `type`: Choice of the method for calculation. Please See below.

• `dx`: Interval for numerical differentiation (Å). Default is 0.001.

• `parameters`:

• `state`: the target electronic state

• `non_adiabatic_coupling`: Non-adiabatic coupling between two states. For post-HF calculation, CASCI is supported.

• `type`: Choice of the method for calculation. Please See below.

• `dx`: Interval for numerical differentiation (Å). Default is 0.001.

• `parameters`:

• `state_pair`: The target pair of two electronic states

• `band_structure`: Band structure calculation. For post-HF calculation, this property will not be calculated.

• `algorithm`: The algorithm for finding excited states. You can specify the algorithm from the following options.

• QSE [McC17][Details] (default)

• QEOM[Oll20]

Choice of `type` in `gradient`, `hessian`, `vibrational_analysis` and `non_adiabatic_coupling`

• `HAMILTONIAN_NUMERICAL`: using the method reported in [Mit20] with the numerical derivative of Hamiltonian. For orbital optimization case, this option does not work. (default)

• `HAMILTONIAN_ANALYTICAL`: using the method reported in [Mit20] with the analytical derivative of Hamiltonian. For non-singlet `hf_multiplicity` or orbital optimization cases, this option does not work. In addition, when degenerate molecular orbitals exist, this option does not work.

The following options are not available for `non_adiabatic_coupling`.

• `ANALYTICAL_PYSCF`: using analytical gradient method implemented in PySCF. Note this option does not support yet in the cases of open shell systems, computing electronic integrals with density fitting. In addition, for orbital optimization case, this option works only for `STATE_SPECIFIC`.

• `ANALYTICAL_QAMUY`: using analytical gradient method reported in [Omi22]. Note this option works only for SA-OO-SSVQE (`SSVQE` solver with `STATE_AVERAGED` type orbital optimization) and SA-OO-VQD (`VQD` solver with `STATE_AVERAGED` type orbital optimization).

type

HAMILTONI AN_NUMERICAL

HAMILTONIA N_ANALYTICAL

ANAL YTICAL_PYSCF

ANAL YTICAL_QAMUY

OK

OK

OK (Note 3)

Hessian

OK

OK (Note 5)

OK

OK

OK (Note 3)

SS (Note 1) Hessian

SS (Note 1) Non- adiabatic coupling

OK (Note 4)

SA (Note 2) Hessian

SA (Note 2) Non- adiabatic coupling

(Note 1): State-specific orbital optimization

(Note 2): State-averaged orbital optimization

(Note 3): Not available for the cases of open shell systems or computing electronic integrals with density fitting

(Note 4): Only available for SA-OO-SSVQE and SA-OO-VQD

(Note 5): Only available for singlet Hartree-Fock level

## References#

[Mit20] “Theory of analytical energy derivatives for the variational quantum eigensolver”, Kosuke Mitarai, Yuya O. Nakagawa, Wataru Mizukami, Phys. Rev. Research 2, 013129 (2020).

[Omi22] “Analytical energy gradient for state-averaged orbital-optimized variational quantum eigensolvers and its application to a photochemical reaction”, Keita Omiya, Yuya O. Nakagawa, Sho Koh, Wataru Mizukami, Qi Gao, Takao Kobayashi, https://arxiv.org/abs/2107.12705

[McC17] “Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states”, J. R. McClean et al., Phys. Rev. A 95, 042308 (2017).

[Oll20] “Quantum equation of motion for computing molecular excitation energies on a noisy quantum processor”, P. J. Ollitrault et al., Phys. Rev. Research 2, 043140 (2020).

## Notes#

• MCVQE does not support `hessian`, `vibrational_analysis`, or `non_adiabatic_coupling`.

• VQD does not support `non_adiabatic_coupling`.

• For `transition_dipole_moment`, VQD calculates the absolute values only.

## Input example#

```"output_chemical_properties": [
{
"dipole_moment": {
"parameters": [{"state": 0}, {"state": 1}]
}
},
{
"transition_dipole_moment": {
"parameters": [{"state_pair": [0, 1]}]
}
},
{
"oscillator_strength": {
"parameters": [{"state_pair": [0, 1]}]
}
},
{
"dx": 1e-6,
"parameters": [{"state": 0}, {"state": 1}],
"type": "HAMILTONIAN_NUMERICAL"
}
},
{
"hessian": {
"dx": 1e-6,
"parameters": [{"state": 0}, {"state": 1}]
}
},
{
"vibrational_analysis": {
"dx": 1e-6,
"parameters": [{"state": 0}, {"state": 1}],
"type": "HAMILTONIAN_NUMERICAL"
}
},
{