FCI: Full Configuration Interaction#
Theory#
Forte contains a determinant-based implementation of the Full Configuration Interaction (FCI) method. The FCI method belongs to the category of active space solvers and can be used on its own or to produce reference states for subsequent multireference treatments of dynamical electron correlation. In FCI, the user specifies a set of active orbitals and all Slater determinants \(\Phi_I\) with the correct symmetry and value of \(M_S\) are generated. The FCI wave function \(\Psi\) is a linear combination of the determinants
where \(N_\mathrm{FCI}\) is the number of determinants in the FCI space and \(C_I\) are the expansion coefficients.
The FCI wave function is variationally optimized by solving the eigenvalue problem
where \(\hat{H}\) is the Hamiltonian operator.
In Forte, the FCI wave function is always assumed to be normalized, that is, the coefficients \(C_I\) are normalized such that
The FCI energy is given by the expectation value of the Hamiltonian operator \(\hat{H}\) with respect to the FCI wave function
Implementation details#
To solve the FCI equations, Forte’s implementation uses the Davidson–Liu solver, which requires computing only the action of the Hamiltonian onto a general FCI vector \(\sum_I b_I |\Phi_I\rangle\):
In this case, the final state can be written in terms of the coefficient vector \(\boldsymbol{\sigma} = \mathbf{H} \mathbf{b}\), where here we use matrix/vector notation.
The algorithm that computes \(\boldsymbol{\sigma} = \mathbf{H} \mathbf{b}\) is called the sigma algorithm. Forte implements the string-based sigma algorithm Bendazzoli and Evangelisti [Bendazzoli, G. L.; Evangelisti, S. J Chem Phys 98, 3141 (1993)].
In Forte’s string-based implementation, the FCI wave function is expressed in terms of separate occupation strings for the alpha (\(I_\alpha\)) and beta (\(I_\beta\)) electrons, so that a state can be written as
This means that internally, the FCI vector \(C_I\) is stored as a matrix indexed by the strings \(I_\alpha\) and \(I_\beta\): \(C_{[I_\alpha][I_\beta]}\).
A Few Practical Notes#
FCI computations in Forte run only in the active orbitals (defined by the
ACTIVEkeyword). See more information in the Tutorials to learn how to define the target active space.The default procedure in Forte solves the FCI eigenvalue equation in the Slater determinant basis. In this case Forte projects out contaminants of the wrong spin symmetry; however, this procedure can fail to yield a state with correct multiplicity. To guarantee that the final state is an eigenfunction of \(\hat{S}^2\), turn on spin adapation by setting the option
CI_SPIN_ADAPTtoTrue.Difficult cases of convergence can be resolved with a combination of using a spin-adapted FCI algorithm and by modifying the parameters of the Davidson–Liu solver used by the FCI code.
A First Example#
The following is an example of a FCI computation on the Li2 molecule using the cc-pVDZ basis.
# forte/tests/manual/fci-1/input.dat
import forte
molecule li2 {
0 1
Li
Li 1 1.6
}
set {
basis cc-pVDZ
reference rhf
}
set forte {
active_space_solver fci
}
# run a RHF computation
E_scf, scf_wfn = energy('scf', return_wfn=True)
# pass the RHF orbitals to Forte and run a FCI computation
energy('forte', ref_wfn=scf_wfn)
The output of this computation contains useful information about the number of determinants, symmetry, multiplicity of the root (\(2S + 1\)), number of roots required, etc.
==> FCI Solver <==
Number of determinants 1345608
Symmetry 0
Multiplicity 1
Number of roots 1
Target root 0
Trial vectors per root 10
Spin adapt false
Allocating memory for the Hamiltonian algorithm. Size: 2 x 435 x 435. Memory: 0.002820 GB
By default, to guess an intial solution, the FCI code identifies a small list of low-energy determinant that are spin complete and diagonalizes the Hamiltonian and \(\hat{S}^2\) operators. This procedure yields a list of root with their corresponding energy and expectation value of \(\hat{S}^2\). The roots projected out are listed at the bottom.
==> FCI Initial Guess <==
---------------------------------------------
Root Energy <S^2> Spin
---------------------------------------------
0 -14.821706304246 0.000 singlet
1 -14.701890096488 0.000 singlet
2 -14.697750811390 2.000 triplet
3 -14.688167598595 0.000 singlet
4 -14.626162130001 0.000 singlet
5 -14.623675382053 2.000 triplet
...
19 -14.374215745393 0.000 singlet
---------------------------------------------
Timing for initial guess = 0.002 s
Projecting out guess roots: [2,5,7,9,11,13,15]
The next block shows the convergence of the Davidson–Liu procedure
==> Diagonalizing Hamiltonian <==
Energy convergence: 1.00e-06
Residual convergence: 1.00e-06
-----------------------------------------------------
Iter. Avg. Energy Delta_E Res. Norm
-----------------------------------------------------
1 -14.821706304246 -1.482e+01 +2.474e-01
2 -14.834596416564 -1.289e-02 +3.363e-02
3 -14.835056965855 -4.605e-04 +1.352e-02
4 -14.835126439226 -6.947e-05 +4.432e-03
...
11 -14.835137265477 -1.047e-11 +1.575e-06
12 -14.835137265478 -9.344e-13 +4.629e-07
-----------------------------------------------------
The Davidson-Liu algorithm converged in 13 iterations.
For each target root(s), a list of the most important determinants is
shown next. In this printout, the orbitals are grouped by irrep and
their occupation is labeled by one character (0 = empty, a/b
= one alpha/beta electron, 2 doubly occupied):
==> Root No. 0 <==
2200000 0 000 000 0 2000000 000 000 -0.91351927
2000000 0 000 000 0 2000000 000 200 0.19711995
2000000 0 000 000 0 2000000 200 000 0.19711995
2000000 0 000 000 0 2000000 ab0 000 0.11601362
2000000 0 000 000 0 2000000 ba0 000 0.11601362
2000000 0 000 000 0 2000000 000 ab0 0.11601362
2000000 0 000 000 0 2000000 000 ba0 0.11601362
The energy of all the roots is summarized in a table:
==> Energy Summary <==
Multi.(2ms) Irrep. No. Energy <S^2>
--------------------------------------------------------
1 ( 0) Ag 0 -14.835137265478 0.000000
--------------------------------------------------------
The end of the output also shows the expectation value of the dipole and quadrupole operators and the occupation numer of natural orbitals (obtained from the FCI one-body reduced density matrix):
==> NATURAL ORBITALS <==
1Ag 1.998924 1B1u 1.998773 2Ag 1.679206
1B3u 0.143579 1B2u 0.143579 2B1u 0.024482
...
Initial guess and parameters of the Davidson-Liu algorithm#
The Davidson-Liu procedure requires an initial set of guess vectors. In Forte, the initial guesses are determined by diagonalizing the Hamiltonian in a small space of determinants \(\{ |\Phi'_I\rangle \}\). This space is spin complete. The procedure starts with the diagonalization of the matrix representation of the \(\hat{S}^2\) operator
which allows to group the solution according to the multiplicity. After this step, we transform the Hamiltonian to the basis of eigenstates of \(\hat{S}^2\) and diagonalize each block separately. The guess vectors are taken from the lowest energy solutions with the correct multiplicity. If a guess vector with wrong multiplicity happens to fall below the energy of guess vector with the correct multiplicity, we project the incorrect guess during the diagonalization procedure.
The number of guess vectors and the size of the determinant space used
to generate the initial guess is determined by options controlled by the
user. Once the user specifies the number of target states (roots) and
the multiplicity, then the option DL_GUESS_PER_ROOT (default = 1)
controls the number of guess states requested, that is:
Number of guess states = Number of roots \(\times\)
DL_GUESS_PER_ROOT
The subspace of determinants selected for the initial guess has size
controlled by the option DL_DETS_PER_GUESS (default = 50) , which
determines the number of determinants included in the following way
Number of determinants used to build guesses = Number of guess states
\(\times\) DL_DETS_PER_GUESS
Note that the number of determinants used in the initial guess procedure could be larger than the one determined by the equation above. This happens if the determinants selected do not form a spin-complete set. In this case, additional determinants are added to ensure that the determinant space is spin-complete.
Two other options control the Davidson-Liu procedure: -
DL_SUBSPACE_PER_ROOT: this option controls the maximum number of
subspace vectors stored by the DL algorithm:
Maximum number of subspace vectors = Number of roots $\times$ `DL_SUBSPACE_PER_ROOT`
DL_COLLAPSE_PER_ROOT: once the maximum number of subspace vectors is reached, the DL procedure forms the best solution vectors and collapses (resets) the subspace size. This option controls the number of vectors kept after collapes:Number of subspace vectors after collapse = Number of roots \(\times\)
DL_COLLAPSE_PER_ROOT
When should you modify these options? DL_GUESS_PER_ROOT and
DL_DETS_PER_GUESS should be modified if the DL procedure has
trouble finding the correct initial guess. For example, if the
determinant space is too small, the initial guess procedure may produce
a list of states with incorrect energetic ordering. See the test case
forte/tests/methods/detci-5 for an example.
DL_COLLAPSE_PER_ROOT and DL_SUBSPACE_PER_ROOT should be modified
if the DL procedure has trouble converging.
Note that these options need to be changed carefully. If you ask for
more guess vectors than those available in a given space the code will
fail. Additionally, DL_COLLAPSE_PER_ROOT should be greater or equal
to DL_GUESS_PER_ROOT, and DL_SUBSPACE_PER_ROOT should be greater
than DL_COLLAPSE_PER_ROOT.
Spin-adapted FCI#
In certain cases, convergence to a state with target multiplicity fails
due to either variational collapse to a root of lower energy and
different multiplicity or because no guess state can be found. Spin
adaptation can be turned on by setting the option CI_SPIN_ADAPT to
True.
Forte implements within the determinant-based FCI code a procedure to perform the Davidson–Liu procedure in a basis of configuration state funcions (CSFs). CSFs are spin-adapted linear combinations of Slater determinants with a given orbital occupation pattern (electron configuration).
When expressed in the CSF basis a FCI state is given by:
where the coefficients \(C'_{i}\) are different from the ones that express \(\Psi\) in the Slater determinant basis. Forte’s spin-adapted code computes the sigma vector in the determinant basis and, before feeding it to the Davidson–Liu solver, it converts it to the CSF basis. Spin-adapted FCI computation are more expensive than conventional ones, with the additional cost of the order of 10–15%.
To demonstrate the utility of spin adaptation, consider a computation of
the \(A_1\) quintet state of Li2. In a determinant code, a
straightforward modification of the previous example (fci-1) fails
because the algorithm that guesses the initial state cannot find a
quintet state.
In the following input we set ci_spin_adapt to True and specify
the multiplicity of the state (5).
# forte/tests/manual/fci-2/input.dat
import forte
molecule li2 {
0 1
Li
Li 1 2.0
}
set {
basis cc-pVDZ
reference rhf
e_convergence 9
}
set forte {
active_space_solver fci
ci_spin_adapt true
multiplicity 5
}
# run a RHF computation
E_scf, scf_wfn = energy('scf', return_wfn=True)
# pass the RHF orbitals to Forte and run a FCI computation
energy('forte', ref_wfn=scf_wfn)
The output file contains some extra/different sections. At the beginning of the computation we can read information about the number of CSF and CSF construction timing:
==> Spin Adapter <==
Number of CSFs: 295572
Number of couplings: 4570632
Timing for identifying configurations: 0.1099
Timing for finding the CSFs: 0.3369
The initial guess contains only CSFs with the correct value of spin:
==> FCI Initial Guess <==
Selected 2 CSF
---------------------------------------------
CSF Energy <S^2> Spin
---------------------------------------------
227224 -12.339361395518 6.000 quintet
110597 -12.339361395518 6.000 quintet
---------------------------------------------
Timing for initial guess = 0.002 s
The final state is a linear combination of many determinants
==> Root No. 0 <==
2a00000 0 0b0 000 0 a000000 000 b00 -0.13907742
2b00000 0 0a0 000 0 b000000 000 a00 -0.13907742
2b00000 0 0a0 000 0 a000000 000 b00 0.13907742
2a00000 0 0b0 000 0 b000000 000 a00 0.13907742
2a00000 0 0a0 000 0 b000000 000 b00 0.13907742
2b00000 0 0b0 000 0 a000000 000 a00 0.13907742
...
2a00000 0 000 a00 0 b000000 b00 000 -0.13247287
2b00000 0 000 b00 0 a000000 a00 000 -0.13247287
Since in this example the orbitals come from a RHF computation (same
number of alpha and beta electrons) Forte will assume that the target
state havs \(M_S = 0\). This can be seen from the determinant
composition and in the final energy summary that reports the value of
\(2 M_S\) (2ms)
==> Energy Summary <==
Multi.(2ms) Irrep. No. Energy <S^2>
--------------------------------------------------------
5 ( 0) Ag 0 -12.596862494551 6.000000
--------------------------------------------------------