Ground State Computations¶
Coupled Cluster parent class¶
-
class
coupled_cluster.cc.CoupledCluster(system, mixer=<class 'coupled_cluster.mix.DIIS'>, verbose=False)¶ Coupled Cluster Abstract class
Abstract base class defining the skeleton of a Coupled Cluster ground state solver class.
- Parameters
system (QuantumSystems) – Quantum systems class instance
mixer (AlphaMixer) – AlpaMixer object
verbose (bool) – Prints iterations for ground state computation if True
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compute_ground_state(t_args=[], t_kwargs={}, l_args=[], l_kwargs={})¶ Compute ground state energy
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compute_one_body_expectation_value(mat, make_hermitian=True)¶ Function computing the expectation value of a one-body operator \(\hat{A}\). This is done by evaluating
\[\langle A \rangle = \rho^{q}_{p} A^{p}_{q},\]where \(p, q\) are general single-particle indices and \(\rho^{q}_{p}\) is the one-body density matrix.
- Parameters
mat (np.ndarray) – The one-body operator to evaluate (\(\hat{A}\)), as a matrix. The dimensionality of the matrix must be the same as the one-body density matrix, i.e., \(\mathbb{C}^{l \times l}\), where
lis the number of basis functions.make_hermitian (bool) – Whether or not to make the one-body density matrix Hermitian. This is done by \(\tilde{\boldsymbol{\rho}} = \frac{1}{2}(\boldsymbol{\rho}^{\dagger} + \boldsymbol{\rho}), where :math:\)tilde{boldsymbol{rho}}` is the Hermitian one-body density matrix. Default is
make_hermitian=True.
- Returns
The expectation value of the one-body operator.
- Return type
complex
See also
CoupledCluster.compute_one_body_density_matrix
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compute_particle_density()¶ Computes one-particle density
- Returns
Particle density
- Return type
np.array
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compute_reference_energy()¶ Computes reference energy
- Returns
Reference energy
- Return type
np.array
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get_amplitudes(get_t_0=False)¶ Getter for amplitudes
- Parameters
get_t_0 (bool) – Returns amplitude at t=0 if True
- Returns
Amplitudes in AmplitudeContainer object
- Return type
AmplitudeContainer
CCS¶
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class
coupled_cluster.ccs.CCS(system, **kwargs)¶ Coupled cluster singles
Class for coupled cluster solver with singles excitations.
- Parameters
system (QuantumSystem) – QuantumSystem class describing the system
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compute_energy()¶ Compute ground state CCS energy.
- Returns
CCS ground state energy
- Return type
float
CCD¶
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class
coupled_cluster.ccd.CCD(system, **kwargs)¶ Coupled Cluster Doubles
Class for Coupled Cluster solver, including double excitations.
- Parameters
system (QuantumSystem) – QuantumSystem class describing the system
CCSD¶
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class
coupled_cluster.ccsd.CCSD(system, include_singles=True, **kwargs)¶ Coupled Cluster Singels Doubles
Coupled Cluster solver with single-, and double excitations.
- Parameters
system (QuantumSystems) – QuantumSystems class instance describing the system to be solved
include_singles (bool) – Include singles
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compute_energy()¶ Compute Energy
- Returns
Energy of current state
- Return type
float
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compute_one_body_density_matrix()¶ Computes one-body density matrix
- Returns
One-body density matrix
- Return type
np.array
RCCSD¶
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class
coupled_cluster.rccsd.RCCSD(system, include_singles=True, **kwargs)¶ Restricted Coupled Cluster Singels Doubles
Restricted coupled-cluster solver with single-, and double excitations. The excitation and de-exciation operators are parametrized according to chapter 13.7.5 in 1,
\[\begin{split}\hat{T}_1 &= \sum_{ai} \tau^a_i E_{ai} \\ \hat{T}_2 &= \frac{1}{2}\sum_{abij} \tau^{ab}_{ij} E_{ai} E_{bj} \\ \hat{\Lambda}_1 &= \frac{1}{2} \sum_{ai} \lambda^i_a E_{ia} \\ \hat{\Lambda}_2 &= \frac{1}{2} \sum_{abij} \lambda^{ij}_{ab} \left(\frac{1}{3} E_{ia} E_{jb} + \frac{1}{6} E_{ja} E_{ib} \right)\end{split}\]- Parameters
system (QuantumSystems) – QuantumSystems class instance describing the system to be solved
include_singles (bool) – Include singles
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t_1, t_2 \(\hat{T}\)-amplitudes \(\tau^a_i, \tau^{ab}_{ij}\)
- Type
np.ndarray
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l_1, l_2 \(\hat{\Lambda}\)-amplitudes \(\lambda^i_a, \lambda^{ij}_{ab}\)
- Type
np.ndarray
References
- 1
T. Helgaker, P. Jorgensen, J. Olsen “Molecular electronic-structure theory”, John Wiley & Sons, 2014.
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compute_energy()¶ Compute the total restricted coupled-cluster energy
\[E_{\text{RCCSD}} = E_{\text{ref}} + 2 f^i_a \tau^a_i + (2 \tau^{ab}_{ij} - \tau^{a}_i \tau^b_j) (u^{ij}_{ab} - u^{ij}_{ba} )\]- Returns
The total coupled-cluster energy of the current state.
- Return type
float
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compute_initial_guess()¶ Compute the initial guess for the coupled-cluster amplitudes. Currently the only option is the MP2 initial guess,
\[\begin{split}\tau^a_i &= \lambda^i_a = 0, \, \forall a,i \\ \tau^{ab}_{ij} &= \frac{u^{ab}_{ij}}{\epsilon_i+\epsilon_j - \epsilon_a - \epsilon_b} \\ \lambda^{ij}_{ab} &= \frac{u^{ij}_{ab}}{\epsilon_i+\epsilon_j - \epsilon_a - \epsilon_b}.\end{split}\]
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compute_one_body_density_matrix()¶ Computes the coupled-cluster one-body density matrix 2,
\[\rho^q_p \equiv \langle \tilde{\Psi} | a_p^\dagger a_q | \Psi \rangle.\]- Returns
One-body density matrix
- Return type
np.array
References
- 2
I. Shavitt, R. Bartlett “Many-body methods in chemistry and physics: MBPT and coupled-cluster theory”, Cambridge university press, 2009.
OACCD¶
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class
coupled_cluster.ccd.OACCD(system, **kwargs)¶ Orbital Adaptive Coupled Cluster Doubles
Implementation of the non-orthogonal coupled cluster method with double excitations. The code is based on a script written by Rolf H. Myhre and Simen Kvaal.
Requires orthonormal basis functions.
https://doi.org/10.1063/1.5006160
- Parameters
system (QuantumSystem) – QuantumSystem class instance description of system
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compute_ground_state(max_iterations=100, tol=0.0001, termination_tol=0.0001, tol_factor=0.1, change_system_basis=True, **mixer_kwargs)¶ Compute ground state
- Parameters
max_iterations (int) – Maximum number of iterations
tol (float) – Tolerance parameter, e.g. 1e-4
tol_factor (float) – Tolerance factor
change_system_basis (bool) – Whether or not to change the basis when the ground state is reached. Default is
True.
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compute_one_body_expectation_value(mat, make_hermitian=True)¶ Function computing the expectation value of a one-body operator \(\hat{A}\). This is done by evaluating
\[\langle A \rangle = \rho^{q}_{p} A^{p}_{q},\]where \(p, q\) are general single-particle indices and \(\rho^{q}_{p}\) is the one-body density matrix.
- Parameters
mat (np.ndarray) – The one-body operator to evaluate (\(\hat{A}\)), as a matrix. The dimensionality of the matrix must be the same as the one-body density matrix, i.e., \(\mathbb{C}^{l \times l}\), where
lis the number of basis functions.make_hermitian (bool) – Whether or not to make the one-body density matrix Hermitian. This is done by \(\tilde{\boldsymbol{\rho}} = \frac{1}{2}(\boldsymbol{\rho}^{\dagger} + \boldsymbol{\rho}), where :math:\)tilde{boldsymbol{rho}}` is the Hermitian one-body density matrix. Default is
make_hermitian=True.
- Returns
The expectation value of the one-body operator.
- Return type
complex
See also
CoupledCluster.compute_one_body_density_matrix
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get_amplitudes(get_t_0=False)¶ Getter for amplitudes, overwrites CC.get_amplitudes to also include coefficients.
- Parameters
get_t_0 (bool) – Returns amplitude at t=0 if True
- Returns
Amplitudes and coefficients in OACCVector object
- Return type
OACCVector