HarmonicBasis#

class HarmonicBasis(num_modals, *, threshold=1e-06)[source]#

Bases: VibrationalBasis

The Harmonic basis.

This class uses the Hermite polynomials (eigenstates of the harmonic oscillator) as a modal basis for the expression of the Watson Hamiltonian or any bosonic operator.

References

[1] Pauline J. Ollitrault et al. “Hardware efficient quantum algorithms for vibrational

structure calculations” Chem. Sci., 2020, 11, 6842-6855. https://doi.org/10.1039/D0SC01908A

Parameters:

  • num_modals (list[int]) – the number of modals to be used for each mode.

  • threshold (float) – the threshold value below which an integral coefficient gets neglected.

Methods

eval_integral(mode, modal_1, modal_2, power, kinetic_term=False)[source]#

The integral evaluation method of this basis.

Parameters:

  • mode (int) – the index of the mode.

  • modal_1 (int) – the index of the first modal.

  • modal_2 (int) – the index of the second modal.

  • power (int) – the exponent of the coordinate.

  • kinetic_term (bool) – if this is True, the method should compute the integral of the kinetic term of the vibrational Hamiltonian, :math:d^2/dQ^2.

Returns:

The evaluated integral for the specified coordinate or None if this integral value falls below the threshold.

Raises:

ValueError – if the power exceeds 4.

Return type:

complex | None

References

[1] J. Chem. Phys. 135, 134108 (2011)

https://doi.org/10.1063/1.3644895 (Table 1)

map(coefficient, modes)#

Maps the provided coefficient and mode index to this second-quantization basis.

This applies the actual basis and expands each mode into the number of modals with which the basis instance was initialized.

Parameters:

  • coefficient (complex) – the initial coefficient associated with the mode indices.

  • modes (tuple[int, ...]) – the mode indices. If all of these are negative, the coefficient is treated as belonging to a kinetic term.

Yields:

Pairs of integral values and indices. The indices are now three times as long as the initially provided modes index. The reason for that is that each mode index gets expanded into three indices, denoting the (mode, modal_1, modal_2) indices.

Return type:

Generator[tuple[complex, tuple[int, …]], None, None]