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qml.qchem.taylor_coeffs

taylor_coeffs(pes, max_deg=4, min_deg=3)[source]

Computes the coefficients of a Taylor vibrational Hamiltonian.

The coefficients are computed from a multi-dimensional polynomial fit over potential energy data computed along normal coordinates, with a polynomial specified by min_deg and max_deg.

Parameters:

  • pes (VibrationalPES) – the vibrational potential energy surface object

  • max_deg (int) – maximum degree of the polynomial used to compute the coefficients

  • min_deg (int) – minimum degree of the polynomial used to compute the coefficients

Returns:

the coefficients of the Taylor vibrational Hamiltonian

Return type:

List(TensorLike[float])

Example

>>> freqs = np.array([0.0249722])
>>> pes_onemode = np.array([[0.08477, 0.01437, 0.00000, 0.00937, 0.03414]])
>>> pes_object = qml.qchem.VibrationalPES(freqs=freqs, pes_data=[pes_onemode])
>>> coeffs = qml.qchem.taylor_coeffs(pes_object, 4, 2)
>>> print(coeffs)
[array([[-4.73959071e-05, -3.06785775e-03,  5.21798831e-04]])]

A molecular potential energy surface can be defined as [Eq. 7 of J. Chem. Phys. 135, 134108 (2011)]:

\[V = V_0 + \sum_{i} F_i q_i + \sum_{i,j} F_{ij} q_i q_j + \sum_{i,j,k} F_{ijk} q_i q_j q_k + \cdots,\]

where \(q\) is a normal coordinate and \(F\) represents the derivatives of the potential energy surface.

This function computes these derivatives via Taylor expansion of the potential energy data by performing a multi-dimensional polynomial fit.

The potential energy surface along the normal coordinate can be defined as

\[V(q_1,\cdots,q_M) = V_0 + \sum_{i=1}^M V_1^{(i)}(q_i) + \sum_{i>j} V_2^{(i,j)}(q_i,q_j) + \sum_{i<j<k} V_3^{(i,j,k)}(q_i,q_j,q_k) + \cdots,\]

where \(V_n\) represents the \(n\)-mode component of the potential energy surface computed along the normal coordinate. The \(V_n\) terms are defined as:

\[\begin{split}V_0 &\equiv V(q_1=0,\cdots,q_M=0) \\ V_1^{(i)}(q_i) &\equiv V(0,\cdots,0,q_i,0,\cdots,0) - V_0 \\ V_2^{(i,j)}(q_i,q_j) &\equiv V(0,\cdots,q_i,\cdots,q_j,\cdots,0) - V_1^{(i)}(q_i) - V_1^{(j)}(q_j) - V_0 \\ \nonumber \vdots\end{split}\]

Note that the terms \(V_n\) are represented here by an array of energy points computed along the normal coordinates. These energy data are then used in a multi-dimensional polynomial fit where each term \(V_n\) is expanded in terms of products of \(q\) with exponents specified by min_deg and max_deg.

The one-mode Taylor coefficients, \(\Phi\), computed here are related to the potential energy surface as:

\[V_1^{(j)}(q_j) \approx \Phi^{(2)}_j q_j^2 + \Phi^{(3)}_j q_j^3 + ... + \Phi^{(n)}_j q_j^n,\]

where the largest power \(n\) is determined by max_deg. Similarly, the two-mode and three-mode Taylor coefficients are computed if the two-mode and three-mode potential energy surface data, \(V_2^{(j, k)}(q_j, q_k)\) and \(V_3^{(j, k, l)}(q_j, q_k, q_l)\), are provided.

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