17. Selected CI

The selected CI module is integrated into NWChem but as yet no input module has been written. The input thus consists of setting the appropriate variables in the database.

It is assumed that an initial SCF/MCSCF calculation has completed, and that MO vectors are available. These will be used to perform a four-index transformation, if this has not already been performed.

17.1 Background

This is a general spin-adapted, configuration-driven CI program which can perform arbitrary CI calculations, the only restriction being that all spin functions are present for each orbital occupation. CI wavefunctions may be specified using a simple configuration generation program, but the prime usage is intended to be in combination with perturbation correction and selection of new configurations. The second-order correction (Epstein-Nesbet) to the CI energy may be computed, and at the same time configurations that interact greater than a certain threshold with the current CI wavefunction may be chosen for inclusion in subsequent calculations. By repeating this process (typically twice is adequate) with the same threshold until no new configurations are added, the CI expansion may be made consistent with the selection threshold, enabling tentative extrapolation to the full-CI limit.

A typical sequence of calculations is as follows:

1. Pick as an initial CI reference the previously executed SCF/MCSCF.
2. Define an initial selection threshold.
3. Determine the roots of interest in the current reference space.
4. Compute the perturbation correction and select additional configurations that interact greater than the current threshold.
5. Repeat steps 3 and 4.
6. Lower the threshold (a factor of 10 is common) and repeat steps 3, 4, and 5. The first pass through step 4 will yield the approximately self-consistent CI and CI+PT energies from the previous selection threshold.

To illustrate this, below is some abbreviated output from a calculation on water in an augmented cc-PVDZ basis set with one frozen core orbital. The SCF was converged to high precision in $C_{2v}$ symmetry with the following input

  start h2o
  geometry; symmetry c2v
    O 0 0 0; H 0 1.43042809 -1.10715266
  end
  basis
    H library aug-cc-pvdz; O library aug-cc-pvdz
  end
  task scf
  scf; thresh 1d-8; end

The following input restarts from the SCF to perform a sequence of selected CI calculations with the specified tolerances, starting with the SCF reference.

  restart h2o
  set fourindex:occ_frozen 1
  set "selci:selection thresholds" \
      0.001 0.001 0.0001 0.0001 0.00001 0.00001 0.000001
  task selci

Table 17.1 summarizes the output from each of the major computational steps that were performed.

Table 17.1: Summary of steps performed in a selected CI calculation on water.

		CI
Step	Description	dimension	Energy
1	Four-index, one frozen-core
2	Config. generator, SCF default	1
3+4	CI diagonalization	1	$E_{CI} = -76.041983$
5	PT selection T=0.001	1	$E_{CI+PT} = -76.304797$
6+7	CI diagonalization	75	$E_{CI} = -76.110894$
8	PT selection T=0.001	75	$E_{CI+PT} = -76.277912$
9+10	CI diagonalization	75	$E_{CI}(T=0.001) = -76.110894$
11	PT selection T=0.0001	75	$E_{CI+PT}(T=0.001) = -76.277912$
12+13	CI diagonalization	823	$E_{CI} = -76.228419$
14	PT selection T=0.0001	823	$E_{CI+PT} = -76.273751$
15+16	CI diagonalization	841	$E_{CI}(T=0.0001) = -76.2300544$
17	PT selection T=0.00001	841	$E_{CI+PT}(T=0.0001) = -76.274073$
18+19	CI diagonalization	2180	$E_{CI} = -76.259285$
20	PT selection T=0.00001	2180	$E_{CI+PT} = -76.276418$
21+22	CI diagonalization	2235	$E_{CI}(T=0.00001) = -76.259818$
23	PT selection T=0.000001	2235	$E_{CI+PT}(T=0.00001) = -76.276478$
24	CI diagonalization	11489

17.2 Files

Currently, no direct control is provided over filenames. All files are prefixed with the standard file-prefix, and any files generated by all nodes are also postfixed with the processor number. Thus, for example the molecular integrals file, used only by process zero, might be called h2o.moints whereas the off-diagonal Hamiltonian matrix element file used by process number eight would be called h2o.hamil.8.

ciconf -- the CI configuration file, which holds information about the current CI expansion, indexing vectors, etc. This is the most important file and is required for all restarts. Note that the CI configuration generator is only run if this file does not exist. Referenced only by process zero.

moints -- the molecular integrals, generated by the four-index transformation. As noted above these must currently be manually deleted, or the database entry selci:moints:force set, to force regeneration. Referenced only by process zero.

civecs -- the CI vectors. Referenced only by process zero.

wmatrx -- temporary file used to hold coupling coefficients. Deleted at calculation end. Referenced only by process zero.

rtname, roname -- restart information for the PT selection. Should be automatically deleted if no restart is necessary. Referenced only by process zero.

hamdg -- diagonal elements of the Hamiltonian. Deleted at calculation end. Referenced only by process zero.

hamil -- off-diagonal Hamiltonian matrix elements. All processes generate a file containing a subset of these elements. These files can become very large. Deleted at calculation end.

17.3 Configuration Generation

If no configuration is explicitly specified then the previous SCF/MCSCF wavefunction is used, adjusting for any orbitals frozen in the four-index transformation. The four-index transformation must have completed successfully before this can execute. Orbital configurations for use as reference functions may also be explicitly specified.

Once the default/user-input reference configurations have been determined additional reference functions may be generated by applying multiple sets of creation-annihilation operators, permitting for instance, the ready specification of complete or restricted active spaces.

Finally, a uniform level of excitation from the current set of configurations into all orbitals may be applied, enabling, for instance, the simple creation of single or single+double excitation spaces from an MCSCF reference.

17.3.1 Specifying the reference occupation

A single orbital configuration or occupation is specified by

  ns  (socc(i),i=1,ns)  (docc(i),i=1,nd)

where ns specifies the number of singly occupied orbitals, socc() is the list of singly occupied orbitals, and docc() is the list of doubly occupied orbitals (the number of doubly occupied orbitals, nd, is inferred from ns and the total number of electrons). All occupations may be strung together and inserted into the database as a single integer array with name "selci:conf". For example, the input

  set "selci:conf" \
    0                1  2  3  4 \
    0                1  2  3 27 \
    0                1  3  4 19 \
    2   11 19        1  3  4 \
    2    8 27        1  2  3 \
    0                1  2  4 25 \
    4   3  4 25 27   1  2 \
    4   2  3 19 20   1 4 \
    4   2  4 20 23   1 3

specifies the following nine orbital configurations

   1(2)  2(2)  3(2)  4(2)
   1(2)  2(2)  3(2) 27(2)
   1(2)  3(2)  4(2) 19(2)
   1(2)  3(2)  4(2) 11(1) 19(1)
   1(2)  2(2)  3(2)  8(1) 27(1)
   1(2)  2(2)  4(2) 25(2)
   1(2)  2(2)  3(1)  4(1) 25(1) 27(1)
   1(2)  2(1)  3(1)  4(2) 19(1) 20(1)
   1(2)  2(1)  3(2)  4(1) 20(1) 23(1)

The optional formatting of the input is just to make this arcane notation easier to read. Relatively few configurations can be currently specified in this fashion because of the input line limit of 1024 characters.

17.3.2 Applying creation-annihilation operators

Up to 10 sets of creation-annihilation operator pairs may be specified, each set containing up to 255 pairs. This suffices to specify complete active spaces with up to ten electrons.

The number of sets is specified as follows,

  set selci:ngen 4

which indicates that there will be four sets. Each set is then specified as a separate integer array

  set "selci:refgen  1" 5 4    6 4   5 3   6 3  
  set "selci:refgen  2" 5 4    6 4   5 3   6 3  
  set "selci:refgen  3" 5 4    6 4   5 3   6 3  
  set "selci:refgen  4" 5 4    6 4   5 3   6 3

In the absence of friendly, input note that the names "selci:refgen n" must be formatted with n in I2 format. Each set specifies a list of creation-annihilation operator pairs (in that order). So for instance, in the above example each set is the same and causes the excitations

  4->5   4->6   3->5   3->6

If orbitals 3 and 4 were initially doubly occupied, and orbitals 5 and 6 initially unoccupied, then the application of this set of operators four times in succession is sufficient to generate the four electron in four orbital complete active space.

The precise sequence in which operators are applied is

1. loop through sets of operators
2. loop through reference configurations
3. loop through operators in the set
4. apply the operator to the configuration, if the result is new add it to the new list
5. end the loop over operators
6. end the loop over reference configurations
7. augment the configuration list with the new list
8. end the loop over sets of operators

17.3.3 Uniform excitation level

By default no excitation is applied to the reference configurations. If, for instance, you wanted to generate a single excitation CI space from the current configuration list, specify

set selci:exci 1

Any excitation level may be applied, but since the list of configurations is explicitly generated, as is the CI Hamiltonian matrix, you will run out of disk space if you attempt to use more than a few tens of thousands of configurations.

17.4 Number of roots

By default, only one root is generated in the CI diagonalization or perturbation selection. The following requests that 2 roots be generated

  set selci:nroot 2

There is no imposed upper limit. If many roots are required, then, to minimize root skipping problems, it helps to perform an initial approximate diagonalization with several more roots than required, and then resetting this parameter once satisfied that the desired states are obtained.

17.5 Accuracy of diagonalization

By default, the CI wavefunctions are converged to a residual norm of $10^{-6}$ which provides similar accuracy in the perturbation corrections to the energy, and much higher accuracy in the CI eigenvalues. This may be adjusted with

 set "selci:diag tol" 1d-3

the example setting much lower precision, appropriate for the approximate diagonalization discussed in the preceding section.

17.6 Selection thresholds

When running in the selected-CI mode the program will loop through a list of selection thresholds ($T$), performing the CI diagonalization, computing the perturbation correction, and augmenting the CI expansion with configurations that make an energy lowering to any root greater than $T$. The list of selection thresholds is specified as follows

  set "selci:selection thresholds" \
      0.001 0.001 0.0001 0.0001 0.00001 0.00001 0.000001

There is no default for this parameter.

17.7 Mode

By default the program runs in "ci+davids" mode and just determines the CI eigenvectors/values in the current configuration space. To perform a selected-CI with perturbation correction use the following

  set selci:mode select

and remember to define the selection thresholds.

17.8 Memory requirements

No global arrays are used inside the selected-CI, though the four-index transformation can be automatically invoked and it does use GAs. The selected CI replicates inside each process

• all unique two-electron integrals in the MO basis that are non-zero by symmetry, and
• all CI information, including the CI vectors.

These large data structures are allocated on the local stack. A fatal error will result if insufficient memory is available.

17.9 Forcing regeneration of the MO integrals

When scanning a potential energy surface or optimizing a geometry the MO integrals need to be regenerated each time. Specify

  set selci:moints:force logical .true.

to accomplish this.

17.10 Disabling update of the configuration list

When computing CI+PT energy the reference configuration list is normally updated to reflect all configurations that interact more than the specified threshold. This is usually desirable. But when scanning a potential energy surface or optimizing a geometry the reference list must be kept fixed to keep the potential energy surface continuous and well defined. To do this specify

  set selci:update logical .false.

17.11 Orbital locking in ci geometry optimization

The selected CI wavefunction is not invariant to orbital rotations or to swapping two or more orbitals. Orbitals could be swapped or rotated when the geometry is changed in a geometry optimization step. The keyword lock has to be set in the SCF/MCSCF (vectors) input block to keep the orbitals in the same order throughout the geometry optimization.