3.23. Full Configuration Interaction¶
ORCA provides several exact and approximate approaches to tackle the full configuration interaction (FCI) problem. These methods are accessible via the CASSCF module (see Section Complete and Incomplete Active Space Self-Consistent Field (CASSCF and RAS/ORMAS)) or the ICE module (described in Section Approximate Full CI Calculations in Subspace: ICE-CI).
In the following, we compute the FCI energy of the lithium hydride
molecule using the CASSCF module, where a typical input requires the
declaration of an active space. The latter defines the number of active
electron and orbitals, which are evaluated with the FCI ansatz. In the
special case that all electrons and orbitals are treated with the FCI
ansatz, we can use the keyword DoFCI in the %CASSCF block and let
the program set the active space accordingly. In this example, we focus
on the singlet ground state. Note that excited states for arbitrary
multiplicities can be computed with the keywords Mult and NRoots.
The FCI approach is invariant to orbital rotations and thus orbital
optimization is skipped in the CASSCF module. Nevertheless, it is
important to employ a set of meaningful orbitals, e.g. from a converged
Hartree-Fock calculation, to reduce the number of FCI iterations.
# Hartree-Fock orbitals
!def2-tzvp RHF
*xyz 0 1
Li 0 0 0
H 0 0 1.597
*
The output of the Hartree-Fock calculation also reports on the total number of electrons and orbitals in your system (see snippet below).
Number of Electrons NEL .... 4
Basis Dimension Dim .... 20
In the given example, there are 4 electrons in 20 orbitals, which is a “CAS(4,20)”. Reading the converged RHF orbitals, we can start the FCI calculation.
!def2-tzvp extremescf
!moread
%moinp "RHF.gbw"
%maxcore 2000
%casscf
DoFCI true # sets NEL 4 and NORB 20 in this example.
end
*xyz 0 1
Li 0 0 0
H 0 0 1.597
*
The output reports on the detailed CI settings, the number of configuration state functions (CSFs) and the CI convergence thresholds.
CI-STEP:
CI strategy ... General CI
Number of multiplicity blocks ... 1
BLOCK 1 WEIGHT= 1.0000
Multiplicity ... 1
#(Configurations) ... 8455
#(CSFs) ... 13300
#(Roots) ... 1
ROOT=0 WEIGHT= 1.000000
PrintLevel ... 1
N(GuessMat) ... 512
MaxDim(CI) ... 10
MaxIter(CI) ... 64
Energy Tolerance CI ... 1.00e-13
Residual Tolerance CI ... 1.00e-13
Shift(CI) ... 1.00e-04
...
The program then prints the actual CI iterations, the final energy, and the composition of the wave function in terms of configurations (CFGs).
------------------
CAS-SCF ITERATIONS
------------------
MACRO-ITERATION 1:
--- Inactive Energy E0 = 0.99407115 Eh
--- All densities will be recomputed
CI-ITERATION 0:
-8.012799617 0.526896429727 ( 0.25)
CI-ITERATION 1:
-8.047996328 0.001601312242 ( 0.25)
CI-ITERATION 2:
-8.048134967 0.000022625293 ( 0.25)
CI-ITERATION 3:
-8.048137773 0.000000462227 ( 0.25)
CI-ITERATION 4:
-8.048137841 0.000000035496 ( 0.25)
CI-ITERATION 5:
-8.048137845 0.000000001357 ( 0.25)
CI-ITERATION 6:
-8.048137845 0.000000000254 ( 0.25)
CI-ITERATION 7:
-8.048137845 0.000000000006 ( 0.25)
CI-ITERATION 8:
-8.048137845 0.000000000001 ( 0.25)
CI-ITERATION 9:
-8.048137845 0.000000000000 ( 0.25)
CI-PROBLEM SOLVED
DENSITIES MADE
<<<<<<<<<<<<<<<<<<INITIAL CI STATE CHECK>>>>>>>>>>>>>>>>>>
BLOCK 1 MULT= 1 NROOTS= 1
ROOT 0: E= -8.0481378449 Eh
0.97242 [ 0]: 22000000000000000000
0.00296 [ 99]: 20000002000000000000
0.00258 [ 89]: 20000010001000000000
0.00252 [ 85]: 20000020000000000000
Aside from energies, the CASSCF module offers a number of properties (g-tensors, ZFS, …), which are described in Section CASSCF Properties.
The exact solution of the FCI problem has very steep scaling and is thus limited to smaller problems (at most active spaces of 16 electrons in 16 orbitals). Larger systems are accessible with approximate solutions, e.g. with the density matrix renormalization group approach (DMRG), described in Section Density Matrix Renormalization Group (DMRG), or the iterative configuration expansion (ICE) reported in Section Approximate Full CI Calculations in Subspace: ICE-CI. For fun, we repeat the calculation with the ICE-CI ansatz, which offers a more traditional approach to get an approximate full CI result.
!def2-TZVP ExtremeSCF
!moread
%moinp "RHF.gbw"
%maxcore 2000
%ice
Nel 4
Norb 20
end
*xyz 0 1
Li 0 0 0
H 0 0 1.597
*
The single most important parameter to control the accuracy is TGen.
It is printed with the more refined settings in the output. We note
passing that the wave function expansion and its truncation can be
carried out in the basis of CSFs, configurations, or determinants. The
different strategies are discussed in detail by Chilkuri et
al. [441, 442].
ICE-CI:
General Strategy ... CONFIGURATIONS (all CSFs to a given CFG, spin adapted)
Max. no of macroiterations ... 12
Variational selection threshold ... -1.000e-07
negative! => TVar will be set to 1.000e-07*Tgen=1.000e-11
Generator selection threshold ... 1.000e-04
Excitation level ... 2
Selection on initial CSF list ... YES
Selection on later CSFs lists ... YES
...
******************************
* ICECI MACROITERATION 3 *
******************************
# of active configurations = 2808
Initializing the CI ... (CI/Run=3,2 UseCC=0)done ( 0.0 sec)
Building coupling coefficients ... (CI/Run=3,2)Calling BuildCouplings_RI UseCCLib=0 DoRISX=0
CI_BuildCouplings NCFG= 2808 NORB=20 NEL=4 UseCCLib=0 MaxCore=2000
PASS 1 completed. NCFG= 2808 NCFGK= 8416 MaxNSOMOI=4 MaxNSOMOK=4
PASS 2 completed.
PASS 3 completed.
Memory used for RI tree = 2.99 MB (av. dim= 35)
Memory used for ONE tree = 1.32 MB (av. dim= 46)
Memory used for coupling coefficients= 0.01 MB
done ( 0 sec)
Now calling CI solver (4095 CSFs)
****Iteration 0****
Maximum residual norm : 0.000293130557
****Iteration 1****
Maximum residual norm : 0.000000565920
****Iteration 2****
Maximum residual norm : 0.000001755176
****Iteration 3****
Maximum residual norm : 0.000000435942
Rebuilding the expansion space
****Iteration 4****
*** CONVERGENCE OF ENERGIES REACHED ***
CI problem solved in 0.4 sec
CI SOLUTION :
STATE 0 MULT= 1: E= -8.0481340246 Eh W= 1.0000 DE= 0.000 eV 0.0 cm**-1
0.97249 : 22000000000000000000
Selecting new configurations ... (CI/Run=3,2)done ( 0.0 sec)
# of selected configurations ... 2747
# of generator configurations ... 43 (NEW=1 (CREF=43))
Performing single and double excitations relative to generators ... done ( 0.0 sec)
# of configurations after S+D ... 7038
Selecting from the generated configurations ... done ( 0.1 sec)
# of configurations after Selection ... 2808
Root 0: -8.048134025 -0.000000023 -8.048134048
==>>> CI space seems to have converged. No new configurations
maximum energy change ... 1.727e-05 Eh
********* ICECI IS CONVERGED *********
Initializing the CI ... (CI/Run=3,3 UseCC=0)done ( 0.0 sec)
Building coupling coefficients ... (CI/Run=3,3)Calling BuildCouplings_RI UseCCLib=0 DoRISX=
CI_BuildCouplings NCFG= 2808 NORB=20 NEL=4 UseCCLib=0 MaxCore=2000
PASS 1 completed. NCFG= 2808 NCFGK= 8416 MaxNSOMOI=4 MaxNSOMOK=4
PASS 2 completed.
PASS 3 completed.
Memory used for RI tree = 2.99 MB (av. dim= 35)
Memory used for ONE tree = 1.32 MB (av. dim= 46)
Memory used for coupling coefficients= 0.01 MB
done ( 0 sec)
Now calling CI solver (4095 CSFs)
****Iteration 0****
Maximum residual norm : 0.000000471011
****Iteration 1****
*** CONVERGENCE OF ENERGIES REACHED ***
CI problem solved in 0.1 sec
CI SOLUTION :
STATE 0 MULT= 1: E= -8.0481340245 Eh W= 1.0000 DE= 0.000 eV 0.0 cm**-1
0.97249 : 22000000000000000000
With Hartree-Fock orbitals and the default settings, the ICE converges in 3 macro iterations to an energy of \(-8.048134047513~E_\text{h}\). The deviation from the exact solution is just \(3.8 \times 10^{-6}~E_\text{h}\) in this example.