ACI: Adaptive Configuration Interaction#
Theory#
The Adaptive Configuration Interaction Method (ACI) is an iterative selected CI method that optimizes a space of determinants such that the total error in the energy is controlled by a user-defined parameter, \(\sigma\),
The ACI algorithm grows a set of reference determinants (\(P\)) and screens its first-order interacting space using perturbative energy estimates. This screening is done in a cumulative fasion to produce an approximation to the total correlation energy ignored. The space of reference (\(P\)) and selected determinants (\(Q\)) define the ACI model space (\(M\)). The Hamiltonian is diagonalized in this space to produce the ACI energy and wave function,
The algorithm proceeds with a pruning step to get a new (\(P\)) space to start the next iteration. The iterations end when the ACI energy is satisfactorily converged, which produces a total error that matches \(sigma\) very closely. Additionally, the perturbative estimates of the determinants excluded from the model space can be used as a perturbative correction, which we denote as the ACI+PT2 energy.
A Few Practical Notes#
In Forte, ACI wave functions are defined only in the active orbitals.
The ACI wave function is defined is a set of Slater Determinants, so it is not guaranteed to be a pure spin eigenfunction. As a result, we augment ACI wave functions throughout the procedure to ensure each intermediate space of determinants is spin complete.
The initial \(P\) space is defined from a small CAS wave function so that there are fewer than 1000 determinants. This can be enlarged to improve convergence if needed using the ACTIVE_GUESS_SIZE option.
This portion of the manual will discuss ACI usage generally, but all content is transferrable to the case where ACI is used as a reference for DSRG computations. If that is the case, the option CAS_TYPE ACI needs to be set.
A First Example#
The simplest input for an ACI calculation involves specifying values for \(\sigma\).
import forte
molecule h2o {
0 1
O
H 1 0.96
H 1 0.96 2 104.5
}
set {
basis sto-3g
reference rhf
}
set forte {
job_type aci
sigma 0.001
}
E_scf, scf_wfn = energy('scf', return_wfn=True)
energy('forte', ref_wfn=scf_wfn)
Though not required, it is good practice to also specify the number of roots, multiplicity, symmetry, and charge. The output contains information about the sizes and energies of the \(P\) and \(M\) spaces at each step of the iteration, and provides a summary of the converged wave function:
==> ACI Summary <==
Iterations required: 3
Dimension of optimized determinant space: 24
* Adaptive-CI Energy Root 0 = -75.012317069484 Eh = 0.0000 eV
* Adaptive-CI Energy Root 0 + EPT2 = -75.013193884201 Eh = 0.0000 eV
==> Wavefunction Information <==
Most important contributions to root 0:
0 -0.987158 0.974480589 16 |2220220>
1 0.076700 0.005882905 15 |2220202>
2 -0.046105 0.002125685 13 |22-+2+->
3 -0.046105 0.002125685 14 |22+-2-+>
4 0.044825 0.002009273 12 |2202220>
5 0.043438 0.001886853 11 |2222200>
6 0.040971 0.001678638 10 |2200222>
7 0.033851 0.001145896 9 |22--2++>
8 0.033851 0.001145896 8 |22++2-->
9 0.032457 0.001053488 7 |2+-2220>
Spin state for root 0: S^2 = 0.000000, S = 0.000, singlet
==> Computing Coupling Lists <==
--------------------------------
α 0.000186 s
β 0.000186 s
αα 0.000333 s
ββ 0.000307 s
αβ 0.000866 s
--------------------------------
1-RDM took 0.000107 s (determinant)
==> NATURAL ORBITALS <==
1A1 2.000000 1B1 1.998476 2A1 1.998399
3A1 1.977478 1B2 1.974442 2B2 0.025891
4A1 0.025314
RDMS took 0.002290
Adaptive-CI ran in : 0.067389 s
For ground state computations, very few additional options are required unless very large determinants spaces are considered. In this case, memory efficient screening and diagonalization algorithms can be chosen.
Computing Excited States with ACI#
ACI Options#
Basic Options#
NROOT
Number of CI roots to find. If energy(‘aci’) is used, energy criteria will be computed for each root with respect to a trial wavefunction. The maximum value among each root will then be used for evaluation with \(\tau_{q}\).
Type: int
Default: 1
SELECT_TYPE
Specifies whether second order PT theory energy correction, or first order amplitude is used in selecting the \(Q\) space.
Type: string
Options: AMP, ENERGY, AIMED_AMP, AIMED_ENERGY
Default: AMP
TAUP
Threshold used to prune the \(P+Q\) space
Type: double
Default: 0.01
TAUQ
Threshold used to select the \(Q\) space
Type: double
Default: 0.000001
Expert Options#
DIAG_ALGORITHM
The algorithm used in all diagonalizations. This option is only needed for calculations with very large configuration spaces.
Type: string
Options: DAVIDSON, FULL, DAVIDSON_LIST
Default: DAVIDSON
SMOOTH
This option implements a smoothing function for the Hamiltonian that makes the energy an everywhere-differentiable function of a geometric coordinate by gradually gradually decoupling the determinant of least importance. This function is useful for correcting discontinuities in potential energy curves, but it can yeild non-physical curves if the discontinuities are large.
Type: bool
Default: False
SMOOTH_THRESHOLD
The threshold for smoothing the Hamiltonian
Type: double
Default: 0.01
EXCITED_ALGORITHM
This option determines the algorithm to compute excited states. Currently the only options implemented are “STATE_AVERAGE” which means that a function of the criteria among the excited states of interest are used to build the configuraiton space, and “ROOT_SELECT” where the determinant space is constructed with respect to a single root.
Type: string
Options: “STATE_AVERAGE”, “ROOT_SELECT”
Default: “STATE_AVERAGE”
PERTURB_SELECT
Option defines \(\tau_{q}\) as either MP2 estimate or estimate derived from 2D diagonalization. True uses the MP2 estimation.
Type: bool
Default: false
POST_DIAGONALIZE
Option to re-diagonalize Hamiltonian in final CI space. This can be is useful to compute more roots.
Type: bool
Default: False
POST_ROOT
Number of roots to compute on post-diagonalization. For this option to be used, post-diagonalize must be true.
Type: int
Default: 1
PQ_FUNCTION
Option that selects the function of energy estimates per root and the expansion coefficients per root. This option is only meaningful if more than one root is desired.
Type: string
Options: “MAX”, “AVERAGE”
Default: “MAX”
Q_REFERENCE
Reference state type to be used when computing estimates of the energy difference between two states. The estimation of the change in energy gap a determinant introduces can be done for all excited states with respect to the ground state (GS), or with respect to the nearest, lower state.
Type: string
Options: “GS”, “ADJACENT”
Default: “GS”
Q_REL
- Rather than using the absolute energy to define the importance of a determinant, an energy gap between
two states can be used. This allows the determinant space to be constructed such that the energy difference between to states is optimized.
Type: bool
Default: False
REF_ROOT
Option that selects the desired root that is used to build the determinant space. This option should only be used when the EXCITED_ALGORITHM is set to “ROOT_SELECT”.
Type: int
Default: 0
SPIN_TOL
For all of the algorithms in EX_ACI, roots are only used to build determinant spaces if their spin multiplicity is within a given tolerance of the input spin multiplicity. This option defines that spin tolerance. NOTE: the multiplicity must be defined within the EX_ACI scope. For poorly behaved systems, it may be useful to increase this to an arbitrarily large value such that the lowest-energy multiplicities can be confirmed
Type: double
Default: 1.0e-4