24 Table-CI Calculations

GAMESS-UK now contains two separate modules for performing Table-CI calculations, the original Conventional module that involves explicit storage of the CI hamiltonian on disk, and a new, semi-direct module that avoids explicit storage of the hamiltonian, and is capable of handling significantly larger secular problems. While the Conventional module will ultimately be phased out, our intention at this stage is to support both, so that the data requirements and file handling characteristics of both are described below.

Table-CI calculations are performed under control of the RUNTYPE CI specification, with data input characterising the nature of the CI introduced by a data line with the keyword MRDCI in the first data field. Termination of this data is accomplished by presenting a valid Class 2 directive, such as VECTORS or ENTER.

Before detailing example data files for performing both Conventional and Semi-direct Table-CI calculations on the X¹A₁ state of formaldehyde, we mention some general points on conducting such calculations.

1. The data requirements, computational strategy and overall philosophy of the Table-CI modules are quite distinct from the Direct-CI module described above.

2. The aim of both modules is to calculate one or more roots of a given symmetry from a Multi-Reference CI calculation. Both modules can also calculate transition moments (TM) between states of the same symmetry or states of different symmetry, in addition to CI-Dipole and Quadrupole moments. The modules are based on the Table-CI algorithm of R.J.Buenker [39], the main practical difference between this and the Direct-CI module being the use of configuration selection and energy extrapolation.

3. The final list of selected configurations is derived from an initial list of configurations generated by single plus double excitations from a user-specified list of reference functions. Note that there is effectively no limit on this number of initial configurations. The selection and extrapolation procedure may be applied to a number of roots of a given secular problem. The Direct-CI module, of course, considers explicitly all configurations which are single and double excitations of a given set of reference configurations: this module is typically limited to the lowest few roots of a given symmetry.

4. RUNTYPE CI is in fact a combination of tasks, requesting integral generation, SCF, and finally, the various sub-tasks associated with the Table-CI calculation itself. While in simple cases it may be feasible to perform all steps in a single calculation, it will often be necessary to break up the calculation into multiple jobs, driving through each of the tasks under control of the appropriate RUNTYPE directive, with use made of the BYPASS keyword on the data lines initiating each of the sub-modules. We illustrate this point in the sections below.

24.1 Table-CI and Molecular Symmetry

A crucial requirement in running the Table-CI modules is an understanding of the treatment of symmetry. Unlike the SCF and direct-CI modules, the molecular orbitals are automatically reordered at the outset of the CI into groups belonging to the same irreducible representation, with the ordering within each group dictated by the ordering encountered at orbital generation time (i.e. at SCF time). Note that each representation has an associated index number e.g. in a system of C_2v symmetry the four representations a₁, b₁, b₂ and a₂ have associated index numbers of 1,2 3 and 4 respectively. Groups of orbitals of common representation are ordered by virtue of increasing representation sequence number, so that in a C_2v system all molecular orbitals of a₁ symmetry would occur first in the list (with the occupied orbitals preceding the virtual orbitals in the subset), followed by the orbitals of b₁ symmetry (again with the DOMOS preceding the VMOS), followed by orbitals of b₂ symmetry (DOMOS before VMOS) and finally, orbitals of a₂ symmetry. Any subsequent reference to the orbitals, for example when specifying the reference functions, must be in this revised numbering scheme. Let us consider an example to try and clarify this point. Consider again the output from the closed shell SCF calculation on H₂CO, in particular the symmetry adapted basis set information,

            =============================
            IRREP  NO. OF SYMMETRY ADAPTED
                   BASIS FUNCTIONS
            =============================
              1          12
              2           4
              3           6
            =============================

and the list of MOs printed at convergence:

           ===============================================
           M.O.  IRREP  ORBITAL ENERGY   ORBITAL OCCUPANCY
           ===============================================
              1     1    -20.48275080           2.0000000
              2     1    -11.28286952           2.0000000
              3     1     -1.40833443           2.0000000
              4     1     -0.86648626           2.0000000
              5     3     -0.69818828           2.0000000
              6     1     -0.63034883           2.0000000
              7     2     -0.52027278           2.0000000
              8     3     -0.43433094           2.0000000
              9     2      0.14397469           0.0000000
             10     1      0.27419771           0.0000000
             11     3      0.36740523           0.0000000
             12     1      0.45123743           0.0000000
             13     2      0.93266602           0.0000000
             14     3      1.02032602           0.0000000
             15     1      1.02498516           0.0000000
             16     1      1.14613786           0.0000000
             17     3      1.27971217           0.0000000
             18     1      1.57176247           0.0000000
             19     2      1.86744709           0.0000000
             20     1      1.91087974           0.0000000
             21     3      1.98262324           0.0000000
             22     1      3.31460342           0.0000000
           ===============================================

Based on the reordering scheme outlined above, the table below outlines the sequence numbers of the MOs both prior to and after reordering. Note that a list of irreducible representations (IRreps) and their associated indices for each of the abelian point groups are given in Table 7. With the molecular orbitals reordered thus, the user must apply the revised numbering scheme in specification of, for example, the reference configurations. Thus consider the SCF configuration for H₂CO in terms of the doubly occupied SCF m.o.s:

m.o.	1a1	2a1	3a1	4a1	1b2	5a1	1b1	2b2
SCF ordering	1	2	3	4	5	6	7	8
Table ordering	1	2	3	4	17	5	13	18

Each reference function in the CI is defined in terms of the reordered MOs under control of the CONF directive, with the m.o.s in each representation presented in turn, in order of increasing representation number. Thus the following sequence:

          1   2   3   4   5   13   17   18

would define the SCF configuration for H₂CO. Note that an additional integer is required in specifying the number of open-shell orbitals (NONO, non-identically coupled orbitals) in each function. This value is specified first in the CONF data sequence, and would typically be followed by a sequence of NONO integers defining the orbitals in question. In the present case NONO is zero, as all m.o.s are doubly occupied, so that the full CONF data line would be:

       0   1   2   3   4   5   13   17   18

IRrep	IRrep	SCF Sequence	Table-CI	Occupation
	No.	No.	Sequence No.	No.
a1	1	1	1	2.0
		2	2	2.0
		3	3	2.0
		4	4	2.0
		6	5	2.0
		10	6	0.0
		12	7	0.0
		15	8	0.0
		16	9	0.0
		18	10	0.0
		20	11	0.0
		22	12	0.0
b1	2	7	13	2.0
		9	14	0.0
		13	15	0.0
		19	16	0.0
b2	3	5	17	2.0
		8	18	2.0
		11	19	0.0
		14	20	0.0
		17	21	0.0
		21	22	0.0

Let us consider the specification for the following configuration:

1a₁²2a₁²3a₁²4a₁²1b₂²(5a₁6a₁)(1b₁2b₁)2b₂² (1)

In this case there are two non-identically spin-coupled pairs i.e., 4 orbitals, which must be specified first in the CONF data line. This would then be:

   4  5  6  13  14    1   2   3   4   17   18

where the four orbitals, 5(5a₁), 6(6a₁), 13(1b₁) and 14(2b₁) precede the doubly occupied orbitals in the list.

Table: Irreducible Representations and Associated Indexing used in the Table-CI Module

Point Group	IRrep	Sequence No.
Cs	a'	1
	a''	2
C2	a	1
	b	2
Ci	ag	1
	au	2
C2v	a1	1
	b1	2
	b2	3
	a2	4
C2h	ag	1
	au	2
	bu	3
	bg	4
D2h	ag	1
	b3u	2
	b2u	3
	b1g	4
	b1u	5
	b2g	6
	b3g	7
	au	8

24.2 Conventional Table-CI Calculations

There is a formal limit of 200,000 selected configurations derived from an initial list of configurations generated by single plus double excitations from a user-specified list of reference functions. The selection and extrapolation procedure may be applied on up to twenty roots of a given secular problem.

1. The Conventional Table-CI module comprises a set of 9 sub-modules, which must be user-driven (either implicitly or explicitly, see below) through data input. These sub modules are as follows:
   • ADAPT: generation of a symmetry adapted list of integrals, derived by a pseudo-transformation from the list of `raw' integrals.
   • TRAN: integral transformation, using the list of adapted integrals generated above together with a molecular orbital coefficient array nominated by the user. Note that in contrast to the Direct-CI module, transformation is an integral part of the Conventional Table-CI module. required by both the SELECT and CI sub-modules (see below - this data base will usually be available on a given machine, but may be generated by the user).
   • SELECT: performs configuration generation and subsequent selection based on a user-specified set of reference configurations and appropriate thresholds.
   • CI: generates the CI-hamiltonian based on the set of selected configurations from SELECT and integrals from TRAN.
   • DIAG: calculates one or more CI eigenfunctions of the hamiltonian generated under CI.
   The remaining modules are optional, and may be used to analyse one or more of the CI eigenvectors:
   • NATORB: generate the spin-free natural orbitals for one or more of the calculated CI eigenvectors.
   • PROP: compute various 1-electron properties of the CI wavefunctions. Note that the natural orbitals generated above may be routed to the Dumpfile and examined by the other analysis modules of GAMESS-UK in a subsequent job.
   • TM: compute the transition moments between nominated CI eigenvectors.
   Note at this point that there may be additional data input associated with each of the sub-modules e.g., for defining the reference configurations and selection attributes in SELECT.

2. In the interests of efficiency the Table-CI module requires as input a `data-base' of pattern symbolic matrix elements for use in both the selection process and in construction of the final CI Hamiltonian over the selected configurations. These pattern elements are assumed to reside on a data set with LFN TABLE. The data base may be constructed in a given run of the Table-CI module by entering the TABLE sub-module prior to SELECT and CI. Thus the following data-driven loading of sub-modules:

               .
               .
              MRDCI
              ADAPT
              TRAN
              TABLE
              SELECT
              CI
              DIAG
               .
               .
   

   would be typical of that required when the user is explicitly constructing the TABLE data set in a given run of the program. Since TABLE generation is somewhat expensive, it will be more usual for the user to allocate a pre-generated version of the data set prior to executing the Table-CI modules. This allocation process and detailed locations of TABLE are, of course, machine specific, and will be outlined at the appropriate points in Parts 12-16 of the Manual. In this case the TABLE data line is simply omitted from the data sequence shown above, thus:

               .
               .
              MRDCI
              ADAPT
              TRAN
              SELECT
              CI
              DIAG
               .
               .
   

   Note that failure to correctly allocate TABLE when using the above sequence will lead to an error condition.
3. Several direct-access files will be generated under RUNTYPE CI processing. For Conventional Table-CI calculations, these include:
   • the Mainfile (ED2) and Dumpfile (ED3).
   • the Scratch file (ED7).
   • temporary files for sorting both transformed integrals and intermediate matrices in the CI calculation (the Sortfile).
   • in addition to the standard direct access files listed above, the Table-CI module makes extensive use of FORTRAN data sets (hereafter referred to as interfaces).
   Any restart jobs will require a subset of interfaces to be saved (see Table 8) in addition to the Dumpfile (ED3) and Mainfile (ED2). Extensive use is also made of scratch FORTRAN data sets, with LFNs FTN001, 002, 003, 004, 008, 009 and FTN010.
4. As mentioned above, generation of a valid Mainfile for subsequent use in the integral transformation routines requires the data line

              SUPER OFF NOSYM
   

   in the SCF run.

Table: FORTRAN Interfaces Used by the Conventional Table-CI Module

File	Contents	Generated by	Required by
		Sub-Module	Sub-Module
FTN022	Symmetry Adapted Integrals	ADAPT	TRAN
FTN031	Transformed Integrals	TRAN	SELECT, CI
FTN033	Partial Matrix Elements	SELECT	CI
FTN034	Partial Matrix Elements	SELECT	CI
FTN035	CI Hamiltonian	CI	DIAG
FTN036	CI Vectors	DIAG	NATORB
			PROP, TM

24.3 Conventional Table-CI - Single-reference CISD Calculations

A Conventional Table-CI calculation is to performed on the formaldehyde molecule, using the SCF configuration as the sole reference function. A valid data sequence for performing such a calculation is shown below.

          TITLE
          H2CO - 3-21G DEFAULT BASIS - MRDCI 1M/1R
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI
          ADAPT 
          TRAN
          TABLE
          SELECT
          SYMMETRY 1
          SPIN 1
          CNTRL 16
          SINGLES 1
          CONF
          0 1 2 3 4 5  13  17 18
          ROOTS 1
          THRESH 30 10
          CI
          DIAG
          EXTRAP 2
          ENTER

The following points should be noted:

1. The ADAPT data line specifies generation of a symmetry adapted list of 1- and 2-electron integrals.
2. Integral transformation, requested through the TRAN directive, will use the converged closed shell SCF vectors resident in section 1 (see Table 1), the default closed-shell section. If section specification is made on the TRAN directive, it should point to this section i.e. TRAN 1.
3. In this example we are generating the TABLE data base (as requested by the presence of the TABLE data line) rather than restoring from the library file.
4. The majority of the data input characterising the CI calculation is presented under the SELECT keyword. In the present case we are:
   • requesting CI wavefunctions of A₁ symmetry (SYMMETRY 1)
   • requesting singlet CI wavefunctions (SPIN 1)
   • defining the number of active electrons in the CI through the CNTRL directive
   • requesting the inclusion of all singly excited configurations with respect to the reference configuration (SINGLES 1) regardless of the computed energy lowerings.
   • defining the reference configuration under control of the CONF directive
   • controlling the selection process through the ROOTS and THRESH directives. In the Table-CI procedure this selection process involves construction of an explicit zero-order Hamiltonian H₀ (over the nominated reference functions) followed by perturbative selection of configurations with respect to the user-nominated roots of H₀. The ROOTS directive specifies these roots - typically the nominated roots will exhibit a strong overlap with the final CI wavefunctions, with the dominant configurations in the final CI wavefunction present in the vectors of the zero-order Hamiltonian (indeed the process of extrapolation based on selection assumes this to be true). In the present case this specification is trivial - the zero-order Hamiltonian is simply a unit matrix comprising the SCF configuration.
   • The thresholds to be used in selection are specified on the THRESH data line. The first integer specified is the minimum threshold to to be used (T_min, in units of micro-Hartree), the second integer the increment to be used in defining higher-threshold cases to be solved in the process of extrapolation [39].
5. The CI data line requests construction of the CI Hamiltonian over the set of selected configurations .
6. The DIAG directive requests diagonalisation of the CI Hamiltonian, with EXTRAP requesting two extrapolation passes to be performed in the process of extrapolation to the zero-threshold limit (T=0).

The sequence of data lines defining the Conventional Table-CI calculation is terminated by the VECTORS directive. Let us now consider a Conventional Table-CI calculation on the ²B₂ state of H₂CO⁺, again using the SCF configuration as the sole reference function. A valid data sequence for performing such a calculation is shown below, where we are still performing all the computation in a single job.

          TITLE
          H2CO+ - 2B2 - 3-21G  CISD TABLE-CI CALCULATION
          CHARGE 1
          MULT 2
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI
          ADAPT 
          TRAN 5 
          SELECT
          SYMMETRY 3
          SPIN 2
          CNTRL 15
          SINGLES 1
          CONF
          1 18   1 2 3 4 5  13  17 
          ROOTS 1
          THRESH 30 10
          CI
          DIAG
          EXTRAP 2
          ENTER

Considering the changes to the closed-shell run, the following points should be noted:

• The OPEN directive is not explicitly required since it we are performing a default high-spin RHF calculation. If presented, it should be specified prior to the Table-CI data.
• The set of vectors used in the Table-CI transformation will be restored from section 5 of the Dumpfile (specified by the TRAN 5 data line), having been placed in that Section by the SCF process. Note again that this corresponds to the default section number of the energy-ordered SCF eigenvectors generated by the open-shell SCF module (see Table 1).
• The majority of data changes appear within the SELECT data. Thus the SPIN directive now defines the spin multiplicity of the doublet CI wavefunction: SYMMETRY specifies the IRREP of the ²B₂ state (sequence number 3) while CNTRL defines the number of active electrons, now 15.
• The first integer of the CONF data line indicates a single open-shell orbital, the second the sequence number of that orbital (the 2b₂, no. 16 in the reordered sequence) with the remaining integers the sequence numbers of the doubly occupied MOs.
• The TABLE directive is omitted, for we assume the data-base generated in the closed-shell run has been retained and allocated to the open-shell calculation.

Now let us consider performing the closed-shell calculation above in a sequence of jobs, where the first job carries out the SCF, the second the Table-CI calculation. Valid data sequences for performing the calculation are shown below.

Run I: The SCF Job

          TITLE
          H2CO - 3-21G  SCF PRIOR TO TABLE-CI CALCULATION
          SUPER OFF NOSYM
          ZMATRIX ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          ENTER

The only obvious point to note is the use of the SUPER directive in requesting full integral list generation required in the subsequent symmetry adaption and integral transformation.

Run II: The Table-CI Job

          RESTART
          TITLE
          H2CO - 3-21G  TABLE-CI 1M/1R
          SUPER OFF NOSYM
          BYPASS SCF
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI
          ADAPT 
          TRAN 
          TABLE
          SELECT
          SYMMETRY 1
          SPIN 1
          CNTRL 16
          SINGLES 1
          CONF
          0 1 2 3 4 5  13  17 18
          ROOTS 1
          THRESH 30 10
          CI
          DIAG
          EXTRAP 2
          ENTER

Considering the changes to the complete run, the following points should be noted:

• The SCF computation is BYPASS'ed.
• Note that the default VECTORS and ENTER data will still apply in the RESTART job, with the default VECTORS section for closed-shell SCF orbitals that is written in the startup-job now being used as the source of eigenvectors. Use of this default section will be carried through into the Table-CI module, so that explicit specification on the TRAN directive (i.e. TRAN 1 in the above) is not in fact required.

The calculation may be further subdivided by splitting Run II above into separate integral transformation and CI runs using the BYPASS keyword on the data lines of the appropriate Table-CI sub-modules to deactivate the computation accordingly. Thus:

Run IIa: The Transformation Job

          RESTART
          TITLE
          H2CO - 3-21G  TABLE-CI 1M/1R -TRANSFORMATION
          SUPER OFF NOSYM
          BYPASS SCF
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI
          ADAPT 
          TRAN 
          SELECT BYPASS
          SYMMETRY 1
          SPIN 1
          CNTRL 16
          SINGLES 1
          CONF
          0 1 2 3 4 5  13  17 18
          ROOTS 1
          THRESH 30 10
          CI BYPASS
          DIAG BYPASS
          EXTRAP 2
          ENTER

Thus BYPASS is appended to the data lines requesting those Table-CI sub-modules (SELECT, CI and DIAG) to deactivate the associated processing.

Run IIb: The Table-CI Job

          RESTART
          TITLE
          H2CO - 3-21G  TABLE-CI 1M/1R - SELECTION AND CI
          SUPER OFF NOSYM
          BYPASS SCF
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI
          ADAPT BYPASS
          TRAN BYPASS
          TABLE 
          SELECT 
          SYMMETRY 1
          SPIN 1
          CNTRL 16
          SINGLES 1
          CONF
          0 1 2 3 4 5  13  17 18
          ROOTS 1
          THRESH 30 10
          CI 
          DIAG 
          EXTRAP 2
          ENTER

Now BYPASS is appended to both the ADAPT and TRAN data lines, since the associated processing has been completed in the previous job.

24.4 Conventional Table-CI - Freezing and Discarding Orbitals

In the examples above we have assumed that all MOs, typically generated at SCF time, are active in the subsequent CI calculation. In many instances however this will not be the case, for the user may wish to

• `freeze' inner-shell orbitals, performing a `valence-only' CI calculation.
• discard certain virtual orbitals from the CI calculation, typically the high-energy inner-shell complement orbitals.

The final subset of orbitals to be included in the CI is controlled by the specification of additional data for the TRAN sub-module. The freezing of core, or inner-shell, orbitals and the discarding of virtual orbitals is signaled by appropriate keywords on the TRAN directive (CORE and DISCARD respectively), with subsequent data lines nominating the number and sequence nos. of those orbitals within each IRrep to be frozen or discarded. Note that the sequence numbers to be specified refer to the Table reordered orbital set defined above. Consider the previous H₂CO calculation. Suppose we wish to freeze both the O1s and C1s orbitals (with SCF sequence numbers 1 and 2 respectively) and to discard the two highest-energy virtual orbitals (with SCF sequence numbers 21 and 22): The core orbitals are both of a₁ symmetry, and have sequence numbers 1 and 2. The virtual orbitals are of b₂ (SCF sequence no. 21) and a₁ (SCF sequence no. 22) symmetry, and as the highest orbital of each IRrep, correspond to the 6th orbital of b₂ symmetry and the 12th orbital of a₁ symmetry respectively. The TRAN data will then appear as follows

    
          TRAN CORE DISCARD
          2 0 0 0              ... core MOs
          1 2
          1 0 1 0              ... discarded MOs
          12 6

where two additional data lines are associated with each category, the first specifying the number of orbitals within each IRrep, the second the sequence number of the orbitals in question. Note again that the sequence numbers refer to the numbering within each IRrep. Thus if we were to also freeze the 1b₂ orbital, the revised TRAN data would appear as follows:

    
          TRAN CORE DISCARD
          2 0 1 0              ... core MOs
          1 2 1
          1 0 1 0              ... discarded MOs
          12 6

Before detailing the Table-CI data, we should mention that the revised numbering scheme used in the specification of, for example, the reference configurations is that in effect after the freezing and discarding of orbitals. Having effectively removed three orbitals of a₁ symmetry and one of b₂ from the subsequent CI, the table below presents the final orbital numbering to be used in CONF specification:

IRrep	IRrep	SCF Sequence	Table-CI	Occupation
	No.	No.	Sequence No.	No.
a1	1	3	1	2.0
		4	2	2.0
		6	3	2.0
		10	4	0.0
		12	5	0.0
		15	6	0.0
		16	7	0.0
		18	8	0.0
		20	9	0.0
b1	2	7	10	2.0
		9	11	0.0
		13	12	0.0
		19	13	0.0
b2	3	5	14	2.0
		8	15	2.0
		11	16	0.0
		14	17	0.0
		17	18	0.0

The data for performing the Table-CI calculation is shown below:

          TITLE
          H2CO - 3-21G DEFAULT BASIS - MRDCI 1M/1R
          SUPER OFF NOSYM
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI
          ADAPT 
          TRAN CORE DISCARD
          2 0 0 0  
          1 2
          1 0 1 0 
          12 6
          SELECT
          SYMMETRY 1
          SPIN 1
          CNTRL 12
          SINGLES 1
          CONF
          0 1 2 3  10  14 15
          ROOTS 1
          THRESH 30 10
          CI
          DIAG
          EXTRAP 2
          ENTER

The following points should be noted:

• the number of active electrons in the CI specified on the CNTRL data line is now 12.
• the integers specified on the CONF data line now label the six doubly-occupied orbitals in the revised numbering scheme outlined above.

24.5 Conventional Table-CI - Multi-reference CI Calculations

Specification of additional reference functions in the Table-CI input data is accomplished through the CONF directive, with each reference function characterised by an additional data line of integers defining

• the number, and sequence numbers, of any open shell orbitals, and
• the sequence numbers of the doubly-occupied orbitals.

Consider initially a 4-reference CI calculation for H₂CO, comprising the SCF configuration, that arising from the double excitation 1b₁ to 2b₁, that from the double excitation 2b₂ to 3b₂ and that from the excitation (1b₁2b₁) to (2b₂3b₂). This leads to the following occupation patterns for the 4 reference functions:

Reference	1a1	2a1	3a1	4a1	1b2	5a1	1b1	2b2	2b1	3b2
Function
1	2	2	2	2	2	2	2	2	0	0
2	2	2	2	2	2	2	0	2	2	0
3	2	2	2	2	2	2	2	0	0	2
4	2	2	2	2	2	2	1	1	1	1

Then each data line of the CONF directive will reflect the occupation patterns above:

        CONF
        0 1 2 3 4 5  13  17 18        ..   Ref.1
        0 1 2 3 4 5  14  17 18        ..   Ref.2
        0 1 2 3 4 5  13  17 19        ..   Ref.3
        4 13 14 18 19  1 2 3 4 5 17   ..   Ref.4

The full data input for the job would be as follows:

          RESTART
          TITLE
          H2CO - 3-21G DEFAULT BASIS - MRDCI 4M/1R
          SUPER OFF NOSYM
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI
          ADAPT 
          TRAN 
          SELECT
          SYMMETRY 1
          SPIN 1
          CNTRL 16
          SINGLES 1
          CONF
          0 1 2 3 4 5  13  17 18
          0 1 2 3 4 5  14  17 18
          0 1 2 3 4 5  13  17 19
          4 13 14 18 19  1 2 3 4 5 17
          ROOTS 1
          THRESH 30 10
          CI
          DIAG
          EXTRAP 2
          ENTER

24.6 Conventional Table-CI - Default Sub-module Attributes

To simplify the data-driven loading of sub-modules, the program assumes a default loading order so that, assuming no additional data input is required by a given sub-module i.e., the default attributes of that sub-module are in effect, the user may omit explicit specification of the module from the data input. The assumed default is shown below:

            .
            .
           MRDCI
           ADAPT
           TRAN
           SELECT
           CI
           DIAG
      .
      .

In practice the SELECT module will always require input, characterising for example the nature of the reference configurations, selection attributes etc, but in many instances the defaults of the other sub-modules will hold so that the associated data input may be omitted. Clearly this omission of data requires a firm understanding of the defaults in effect, which will only be apparent after the detailed description of directives presented in section 5. For the moment we illustrate this by considering the simplified data file for the multi-reference calculation on H₂CO above:

          RESTART
          TITLE
          H2CO - 3-21G DEFAULT BASIS - MRDCI 4M/1R
          SUPER OFF NOSYM
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI
          SELECT
          SINGLES 1
          CONF
          0 1 2 3 4 5  13  17 18
          0 1 2 3 4 5  14  17 18
          0 1 2 3 4 5  13  17 19
          4 13 14 18 19  1 2 3 4 5 17
          ENTER

The following points should be noted:

• It is assumed that the complete sequence of sub-tasks is to be carried out. If any task is to be 'BYPASS'ed, then the associated data line must be present.
• In the absence of the TRAN directive, the transformation module will use the default section number corresponding to use of "ENTER" (i.e. section 1) directive as the location of the molecular orbital coefficient array.
• The SYMMETRY, SPIN, CNTRL, ROOTS and THRESH directives of the SELECT sub-module may all be omitted, since the required specification corresponds in each case to the default values.
• Both CI and DIAG may be omitted, the EXTRAP specification of DIAG corresponding to the default value.

24.7 Conventional Table-CI - Restarting Calculations

In the examples considered above, we have assumed that the Table-CI job completes in the time allocated. This may not be the case and we need consider restarting the computation in a controlled fashion. Such a requirement may be met in RUNTYPE CI processing when:

• the associated integral evaluation or SCF has not completed, due either to lack of time or to convergence problems in the SCF;
• Table-CI processing itself has not completed.

In the present implementation it is not possible to restart Table-CI processing within a given sub-module in the event of job termination due to lack of time. It is possible however to fragment the calculation into separate sub-module runs, through the use of the BYPASS directive on the sub-module data lines. In such usage restarting the computation is achieved under control of the RESTART directive, which nominates the CI task for restarting. Consider the Table-CI job of §16.5; we show below the data files for fragmenting this CI into,

• symmetry adaptation and integral transformation;
• configuration selection and hamiltonian construction;
• Davidson diagonalisation.

The subset of interfaces to be saved between the various steps is given in Table 8.

Adaptation and Transformation

          RESTART CI
          TITLE
          H2CO - 3-21G DEFAULT BASIS - MRDCI 4M/1R
          SUPER OFF NOSYM
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI
          SELECT BYPASS
          SINGLES 1
          CONF
          0 1 2 3 4 5  13  17 18
          0 1 2 3 4 5  14  17 18
          0 1 2 3 4 5  13  17 19
          4 13 14 18 19  1 2 3 4 5 17
          CI BYPASS
          DIAG BYPASS
          ENTER

Selection and Hamiltonian Construction

          RESTART CI
          TITLE
          H2CO - 3-21G DEFAULT BASIS - MRDCI 4M/1R
          SUPER OFF NOSYM
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI
          ADAPT BYPASS
          TRAN BYPASS
          SELECT 
          SINGLES 1
          CONF
          0 1 2 3 4 5  13  17 18
          0 1 2 3 4 5  14  17 18
          0 1 2 3 4 5  13  17 19
          4 13 14 18 19  1 2 3 4 5 17
          DIAG BYPASS
          ENTER

Diagonalisation

          RESTART CI
          TITLE
          H2CO - 3-21G DEFAULT BASIS - MRDCI 4M/1R
          SUPER OFF NOSYM
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI
          ADAPT BYPASS
          TRAN BYPASS
          SELECT BYPASS
          SINGLES 1
          CONF
          0 1 2 3 4 5  13  17 18
          0 1 2 3 4 5  14  17 18
          0 1 2 3 4 5  13  17 19
          4 13 14 18 19  1 2 3 4 5 17
          CI BYPASS
          ENTER

24.8 Semi-direct Table-CI Calculations

The formal limits that apply to conventional calculations are significantly extended in the semi-direct module. There is now a limit of 800,000 selected configurations derived from an initial list of configurations generated by single plus double excitations from a user-specified list of reference functions, the number of which may not exceed 256. The selection and extrapolation procedure may now be applied on up to thirty roots of a given secular problem.

Semi-direct Table-CI calculations are performed under control of the RUNTYPE CI specification, with data input characterising the nature of the CI introduced by a data line with the keyword MRDCI in the first data field and the keyword DIRECT in the second field. Termination of this data is again accomplished by presenting a valid Class 2 directive, such as VECTORS or ENTER.

Note that while the Conventional and Semi-direct modules are based on the same Table-CI algorithm, there are significant differences in both file utilisation and data requirements. The most significant of these are highlighted below:

1. In contrast to the conventional module, the integral transformation is now performed under control of the conventional 4-index module of GAMESS, rather than the ADAPT and TRAN Table CI modules.
2. The memory requirements of the semi-direct module may be significantly greater than those associated with the conventional algorithm. While the default memory allocations will prove adequate for "small-medium" cases, the user should use the MEMORY pre-directive to request at least 8 MWords in calculations with, say, more than 20 active electrons.
3. The Semi-direct Table-CI module comprises a reduced set of 6 sub-modules, which may be user-driven (either implicitly or explicitly, see below) through data input. These sub modules are as follows:
   • TABLE: generates an input a `data-base' of pattern symbolic matrix elements for use in both the selection process and in solving the secular problem.
   • SELECT: performs configuration generation and subsequent selection based on a user-specified set of reference configurations and appropriate thresholds. Note that the semi-direct module requires at least two reference configurations.
   • CI: provides pre-processing prior to the semi-direct evaluation of the CI eigenfunctions, followed by calculation, in semi-direct fashion, of one or more CI eigenfunctions of the secular problem. In contrast to the conventional module, just two secular problems are solved as part of the extrapolation process, one at the lowest threshold (T_min) and one at the threshold (T_min + T_inc).
   • NATORB: generate the spin-free natural orbitals for one or more of the calculated CI eigenvectors. Note that this module is now executed in default.
   The remaining analysis modules remain optional, and may be used to further analyse one or more of the CI eigenvectors:
   • PROP: compute various 1-electron properties of the CI wavefunctions. Note that the natural orbitals generated above may be routed to the Dumpfile and examined by the other analysis modules of GAMESS-UK in a subsequent job.
   • TM: compute the transition moments between nominated CI eigenvectors.

   Note at this point that there may be additional data input associated with each of the sub-modules e.g., for defining the reference configurations and selection attributes in SELECT.

4. In the interests of efficiency the Table-CI module again requires as input a `data-base' of pattern symbolic matrix elements for use in both the selection process and in construction of the final CI Hamiltonian over the selected configurations. These pattern elements are assumed to reside on a data set with LFN "table-ci". The data base may be constructed in a given run of the Table-CI module by entering the TABLE sub-module prior to SELECT and CI. Thus the following data-driven loading of sub-modules:

               .
               .
              MRDCI DIRECT
              TABLE
              SELECT
              CI
              NATORB
               .
               .
   

   would be typical of that required when the user is explicitly constructing the TABLE data set in a given run of the program. This is now the recommended route in semi-direct calculations, rather than the user allocating a pre-generated version of the data set prior to executing the Table-CI modules. Note that failure to correctly allocate table-ci when using the above sequence will lead to an error condition.

5. Several direct-access files will be generated under RUNTYPE CI processing. For Semi-direct Table-CI calculations, these include:
   • the Mainfile (ED2) and Dumpfile (ED3).
   • the Transformed Integral file (ED6).
   • the Scratch file (ED7).
   • temporary files for sorting both transformed integrals and intermediate matrices in the CI calculation (the Sortfile).
   • in addition to the standard direct access files listed above, the Table-CI module again makes extensive use of FORTRAN data sets (hereafter referred to as interfaces).
   Any restart jobs will require a subset of interfaces to be saved (see Table 9) in addition to the Dumpfile (ED3), Mainfile (ED2) and Transformed Integral File (ED6). Extensive use is also made of scratch FORTRAN data sets, with LFNs FTN001, 002, 003, 004, 008, 009, 010, 011, 022, 041, 043, and FTN044.
6. As mentioned above, generation of a valid Mainfile for subsequent use in the integral transformation routines requires the data line

              SUPER OFF NOSYM
   

   in the SCF run.

Table: FORTRAN Interfaces Used by the Semi-direct Table-CI Module

File	Contents	Generated by	Required by
		Sub-Module	Sub-Module
FTN031	Transformed	Transformation	SELECT, CI
	Integrals	module
FTN033	Partial Matrix Elements	SELECT	CI
FTN034	Partial Matrix Elements	SELECT	CI
FTN042	Zero-order vectors	SELECT	CI
FTN012	Configuration data	SELECT	CI
FTN036	CI Vectors	CI	NATORB
			PROP, TM

24.9 Semi-direct Table-CI - Multi-reference CI Calculations

We would again point out that all semi-direct Table-CI calculations require at least two reference configurations i.e. CISD calculations based on a single reference configuration are not possible with this module. However we do not consider this to be a major disadvantage given that the process of configuration choice and specification has been simplified through the use of automated configuration generation (see below).

Note again that specification of additional reference functions in the Table-CI input data is again accomplished through the CONF directive; in contrast to CONF specification in the conventional module, however, the data lines specifying the configurations are now terminated by a single data line containing the character string END in the first data field. Each reference function is characterised by an additional data line of integers defining

• the number, and sequence numbers, of any open shell orbitals, and
• the sequence numbers of the doubly-occupied orbitals.

Consider initially a 4-reference CI calculation for H₂CO, comprising the SCF configuration, that arising from the double excitation 1b₁ to 2b₁, that from the double excitation 2b₂ to 3b₂ and that from the excitation (1b₁2b₁) to (2b₂3b₂). This leads to the following occupation patterns for the 4 reference functions:

Reference	1a1	2a1	3a1	4a1	1b2	5a1	1b1	2b2	2b1	3b2
Function
1	2	2	2	2	2	2	2	2	0	0
2	2	2	2	2	2	2	0	2	2	0
3	2	2	2	2	2	2	2	0	0	2
4	2	2	2	2	2	2	1	1	1	1

Then each data line of the CONF directive will reflect the occupation patterns above:

        CONF
        0 1 2 3 4 5  13  17 18        ..   Ref.1
        0 1 2 3 4 5  14  17 18        ..   Ref.2
        0 1 2 3 4 5  13  17 19        ..   Ref.3
        4 13 14 18 19  1 2 3 4 5 17   ..   Ref.4
        END                           ..   the directive terminator

The full data input for the job would be as follows:

          TITLE
          H2CO - 3-21G BASIS - semi-direct MRDCI 4M/1R
          SUPER OFF NOSYM
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI DIRECT
          TABLE
          SELECT
          SYMMETRY 1
          SPIN 1
          CNTRL 16
          SINGLES ALL
          CONF
          0 1 2 3 4 5  13  17 18
          0 1 2 3 4 5  14  17 18
          0 1 2 3 4 5  13  17 19
          4 13 14 18 19  1 2 3 4 5 17
          END
          ROOTS 1
          THRESH 2 2
          CI
          NATORB
          ENTER

Now let us consider performing the closed-shell calculation above in a sequence of jobs, where the first job carries out the SCF, the second the Table-CI calculation. Valid data sequences for performing the calculation are shown below.

Run I: The SCF Job

          TITLE
          H2CO - 3-21G SCF PRIOR TO TABLE-CI CALCULATION
          SUPER OFF NOSYM
          ZMATRIX ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          ENTER

The only obvious point to note is the use of the SUPER directive in requesting full integral list generation required in the subsequent symmetry adaption and integral transformation.

Run II: The Table-CI Job

          RESTART
          TITLE
          H2CO - 3-21G BASIS - semi-direct MRDCI 4M/1R
          SUPER OFF NOSYM
          BYPASS SCF
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI DIRECT
          TABLE
          SELECT
          SYMMETRY 1
          SPIN 1
          CNTRL 16
          SINGLES ALL
          CONF
          0 1 2 3 4 5  13  17 18
          0 1 2 3 4 5  14  17 18
          0 1 2 3 4 5  13  17 19
          4 13 14 18 19  1 2 3 4 5 17
          END
          ROOTS 1
          THRESH 2 2
          CI
          NATORB
          ENTER

Considering the changes to the complete run, the following points should be noted:

• The SCF computation is BYPASS'ed
• Note that the default VECTORS and ENTER data will still apply in the RESTART job, with the default VECTORS section for closed-shell SCF orbitals that is written in the startup-job now being used as the source of eigenvectors.

The calculation may be further subdivided by splitting Run II above into separate integral transformation and CI runs using the BYPASS keyword on the data lines of the appropriate Table-CI sub-modules to deactivate the computation accordingly. Thus:

Run IIa: The Transformation Job

          RESTART
          TITLE
          H2CO - 3-21G  TABLE-CI 4M/1R -TRANSFORMATION
          SUPER OFF NOSYM
          BYPASS SCF
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI DIRECT
          TABLE BYPASS
          SELECT BYPASS
          SYMMETRY 1
          SPIN 1
          CNTRL 16
          SINGLES ALL
          CONF
          0 1 2 3 4 5  13  17 18
          0 1 2 3 4 5  14  17 18
          0 1 2 3 4 5  13  17 19
          4 13 14 18 19  1 2 3 4 5 17
          END
          ROOTS 1
          THRESH 2 2
          CI BYPASS
          NATORB BYPASS
          ENTER

Thus BYPASS is appended to the data lines requesting those Table-CI sub-modules (SELECT, CI and NATORB) to deactivate the associated processing.

Run IIb: The Table-CI Job

          RESTART
          TITLE
          H2CO - 3-21G  TABLE-CI 4M/1R - SELECTION AND CI
          SUPER OFF NOSYM
          BYPASS SCF TRAN
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI DIRECT
          TABLE 
          SELECT 
          SYMMETRY 1
          SPIN 1
          CNTRL 16
          SINGLES ALL
          CONF
          0 1 2 3 4 5  13  17 18
          0 1 2 3 4 5  14  17 18
          0 1 2 3 4 5  13  17 19
          4 13 14 18 19  1 2 3 4 5 17
          END
          ROOTS 1
          THRESH 2 2
          CI 
          NATORB
          ENTER

Now that TRAN is now appended to the BYPASS directive since the associated processing has been completed in the previous job.

24.10 Semi-direct Table-CI - Default MRDCI Calculations

In order to simplify the process of configuration specification and data preparation, the semi-direct module now provides a set of default options that require little or no data input. While these defaults are not expected to cover most in-depth requirements, they do provide a starting point for users, and a route to subsequent, more extensive calculations. To illustrate this default working of the module, we consider below a number of example calculations.

A Semi-direct Table-CI calculation is to performed on the formaldehyde molecule. Given the following data sequence:

          TITLE
          H2CO - 3-21G DEFAULT TABLE-CI OPTIONS
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI DIRECT
          ENTER

then the calculation undertaken will be based on the following;

1. Integral transformation will use the set of orbitals from section 1, the default section for output of the closed-shell SCF eigenvectors.
2. The table-ci data base will be generated rather than restored from the library file.
3. A CI wavefunction of A₁ symmetry (i.e. SYMMETRY 1).
4. A singlet CI wavefunction (i.e. SPIN 1).
5. The number of active electrons in the CI will be set to be those involved in the SCF calculation (i.e. CNTRL 16).
6. Singly excited configurations with respect to each of the default reference configurations (SINGLES ALL) will be included, regardless of their computed energy lowerings.
7. The set of reference configurations to be employed will include the SCF configuration, plus those generated from this configuration by including (i) for each symmetry IRREP, the doubly excited configuration arising from excitation of the highest occupied DOMO of that symmetry to the lowest virtual orbital (VMO) of the same symmetry, and (ii) the lowest singly excited configuration, again arising from the highest occupied DOMO to the lowest VMO of the same symmetry. In the present example, this will correspond to the SCF configuration, the double and single excitation arising from the DOMO 5a₁ to VMO 6a₁, the double and single excitation arising from the DOMO 1b₁ to VMO 2b₁, and the double and single excitation arising from the DOMO 2b₂ to VMO 3b₂. No reference configurations will be included involving orbitals of a₂ symmetry given the absence of such orbitals involved in the occupied manifold. This results in a total reference set of 7 functions, as shown thus in the job output:

      numbers of open shells and corresponding main configurations
   
     0             1   2   3   4   5  13  17  18       ..   SCF configuration
     0             1   2   3   4   6  13  17  18       ..   5a1 -> 6a1 double
     2             5   6   1   2   3   4  13  17  18   ..   5a1 -> 6a1 single
     0             1   2   3   4   5  14  17  18       ..   1b1 -> 2b1 double
     2            13  14   1   2   3   4   5  17  18   ..   1b1 -> 2b1 single
     0             1   2   3   4   5  13  17  19       ..   2b2 -> 3b2 double
     2            18  19   1   2   3   4   5  13  17   ..   2b2 -> 3b2 single
   

8. The default selection process subsequently undertaken is equivalent to the following ROOTS and THRESH directives.

             THRESH 10 10
             ROOTS 1
   

   Thus this default selection process involves construction of an explicit zero-order Hamiltonian H₀ (over the reference functions described above) followed by perturbative selection of configurations with respect to the lowest root of H₀. The minimum threshold to be used in selection (T_min) is 10 micro-Hartree, with an increment of 10 uH to be used in defining the higher-threshold case to be solved in the process of extrapolation [39].

9. In default the module will, having solved the secular problem for the lowest root of the CI secular problem, generate the spinfree natural orbitals from the associated CI eigenfunction.

The sequence of data lines defining the Semi-direct Table-CI calculation is terminated by the VECTORS directive. Note at this stage that the full data specification corresponding to the defaults generated from the above data file is as follows:

          TITLE
          H2CO - 3-21G - EXPLICIT DATA FOR DEFAULT MRDCI SETTINGS  -113.43885803
          SUPER OFF NOSYM
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          ACTIVE
          1 TO 22 END
          MRDCI DIRECT
          TABLE
          SELECT
          CNTRL 16
          SPIN 1
          SYMM 1
          SINGLES ALL
          CONF
            0     1   2   3   4   5  13  17  18
            0     1   2   3   4   6  13  17  18
            2     5   6   1   2   3   4  13  17  18
            0     1   2   3   4   5  14  17  18
            2    13  14   1   2   3   4   5  17  18
            0     1   2   3   4   5  13  17  19
            2    18  19   1   2   3   4   5  13  17
          END
          THRESH 10 10
          ROOTS 1
          CI
          NATORB
          CIVEC 1
          ENTER

Let us now consider a Semi-direct Table-CI calculation on the ²B₂ state of H₂CO⁺, again using default options available within the module. A valid data sequence for performing such a calculation is shown below, where we are still performing all the computation in a single job.

          TITLE
          H2CO+ 2B2 3-21G - DEFAULT MRDCI SETTINGS  -113.06446075
          MULT 2
          CHARGE 1
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI DIRECT
          ENTER

Considering the changes to the closed-shell run, the following points should be noted:

• The set of vectors used in the Table-CI transformation will be the energy-ordered SCF orbitals from section 5 of the Dumpfile, the default section in the absence of section specification on the ENTER directive.
• The symmetry and spin of the CI wavefunction will be deduced from the preceding SCF calculation i.e. a CI wavefunction of B₂ symmetry (corresponding to SYMMETRY 3). and a doublet CI wavefunction (corresponding to SPIN 2).
• The number of active electrons in the CI will be set to be those involved in the SCF calculation (i.e. CNTRL 15).
• Singly excited configurations with respect to each of the default reference configurations (SINGLES ALL) will be included, regardless of their computed energy lowerings.
• The set of reference configurations to be employed will follow the same algorithm used in the closed shell case above i.e. the SCF configuration, plus those generated from this configuration by including (i) for each symmetry IRREP, the doubly excited configuration arising from excitation of the highest occupied DOMO of that symmetry to the lowest virtual orbital (VMO) of the same symmetry, and (ii) the lowest singly excited configuration, again arising from the highest occupied DOMO to the lowest VMO of the same symmetry. In the present example, this will correspond to the SCF configuration, the double and single excitation arising from the DOMO 5a₁ to VMO 6a₁, the double and single excitation arising from the DOMO 1b₁ to VMO 2b₁, and the double and single excitation arising from the DOMO 1b₂ to VMO 3b₂. Note that the DOMO involved in the latter configurations is now the 1b₂ given that the 2b₂ is now singly occupied, and again the absence of excitations involving a₂ MOs given the absence of such orbitals involved in the occupied manifold. This again results in a total reference set of 7 functions, as shown thus in the job output:

     numbers of open shells and corresponding main configurations
  
    1            18   1   2   3   4   5  13  17      ..   SCF configuration
    1            18   1   2   3   4   6  13  17      ..   5a1 -> 6a1 double
    3             5   6  18   1   2   3   4  13  17  ..   5a1 -> 6a1 single
    1            18   1   2   3   4   5  14  17      ..   1b1 -> 2b1 double
    3            13  14  18   1   2   3   4   5  17  ..   1b1 -> 2b1 single
    1            18   1   2   3   4   5  13  19      ..   1b2 -> 3b2 double
    3            17  18  19   1   2   3   4   5  13  ..   1b2 -> 3b2 single
  

The sequence of data lines defining the Semi-direct Table-CI calculation is again terminated by the ENTER directive. Note at this stage that the full data specification corresponding to the defaults generated from the above data file is as follows

          TITLE
          H2CO+ 2B2 3-21G - EXPLICIT DATA FOR DEFAULTS  -113.06446075
          MULT 2
          CHARGE 1
          SUPER OFF NOSYM
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          OPEN 1 1
          ACTIVE
          1 TO 22 END
          MRDCI DIRECT
          TABLE
          SELECT
          CNTRL 15
          SPIN 2
          SYMM 3
          SINGLES ALL
          CONF
           1    18           1   2   3   4   5  13  17
           1    18           1   2   3   4   6  13  17
           3     5   6  18   1   2   3   4  13  17
           1    18           1   2   3   4   5  14  17
           3    13  14  18   1   2   3   4   5  17
           1    18           1   2   3   4   5  13  19
           3    17  18  19   1   2   3   4   5  13
          END
          THRESH 10 10
          ROOTS 1
          CI
          NATORB
          CIVEC 1
          ENTER

24.11 Semi-direct Table-CI - Freezing and Discarding Orbitals

In the examples above we have assumed that all MOs, typically generated at SCF time, are active in the subsequent CI calculation. In many instances however this will not be the case, for the user may wish to

• `freeze' inner-shell orbitals, performing a `valence-only' CI calculation.
• discard certain virtual orbitals from the CI calculation, typically the high-energy inner-shell complement orbitals.

In contrast to the Conventional Table-CI module, the final subset of orbitals to be included in the Table-CI calculation is now controlled by the CORE and ACTIVE directives of the integral transformation module.

The freezing of core, or inner-shell, orbitals is achieved by nominating the sequence nos. of those orbitals to be frozen under control of the CORE directive. The discarding of orbitals is performed under control of the ACTIVE directive, which specifies the sequence nos. of the active set of orbitals to appear in the CI. Note that the sequence numbers to be specified refer to the input orbitals, typically those produced by the SCF code, and not the Table reordered orbital as in the conventional module.

Consider the previous H₂CO calculation. Suppose we wish to freeze both the O1s and C1s orbitals (with SCF sequence numbers 1 and 2 respectively) and to discard the two highest-energy virtual orbitals (with SCF sequence numbers 21 and 22): The CORE and ACTIVE data will then appear as follows

    
          CORE
          1 2 END
          ACTIVE
          3 TO 20 END

The core orbitals are both of a₁ symmetry, and have sequence numbers 1 and 2. The virtual orbitals are of b₂ (SCF sequence no. 21) and a₁ (SCF sequence no. 22) symmetry, and as the highest orbital of each IRrep, correspond to the 6th orbital of b₂ symmetry and the 12th orbital of a₁ symmetry respectively. Before detailing the Table-CI data, we should mention that the revised numbering scheme used in the specification of, for example, the reference configurations is, as in the conventional case, that in effect after the freezing and discarding of orbitals. Having effectively removed three orbitals of a₁ symmetry and one of b₂ from the subsequent CI, the table below presents the final orbital numbering to be used in CONF specification:

IRrep	IRrep	SCF Sequence	Table-CI	Occupation
	No.	No.	Sequence No.	No.
a1	1	3	1	2.0
		4	2	2.0
		6	3	2.0
		10	4	0.0
		12	5	0.0
		15	6	0.0
		16	7	0.0
		18	8	0.0
		20	9	0.0
b1	2	7	10	2.0
		9	11	0.0
		13	12	0.0
		19	13	0.0
b2	3	5	14	2.0
		8	15	2.0
		11	16	0.0
		14	17	0.0
		17	18	0.0

The data for performing the semi-direct Table-CI calculation is shown below:

          TITLE
          H2CO - 3-21G BASIS - valence direct-MRDCI 4M/1R
          SUPER OFF NOSYM
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          CORE
          1 2 END
          ACTIVE
          3 TO 20 END
          MRDCI DIRECT
          TABLE
          SELECT
          SYMMETRY 1
          SPIN 1
          CNTRL 12
          SINGLES ALL
          CONF
          0 1 2 3  10  14 15
          0 1 2 3  11  14 15
          0 1 2 3  10  14 16
          4 10 11 15 16 1 2 3 14
          END
          ROOTS 1
          THRESH 2 2
          CI
          NATORB
          ENTER

The following points should be noted:

• the number of active electrons in the CI specified on the CNTRL data line is now 12.
• the integers specified on the CONF data line now label the six doubly-occupied orbitals in the revised numbering scheme outlined above.

24.12 Semi-direct Table-CI - Default Sub-module Attributes

To simplify the data-driven loading of sub-modules, the program assumes a default loading order so that, assuming no additional data input is required by a given sub-module i.e., the default attributes of that sub-module are in effect, the user may omit explicit specification of the module from the data input. The assumed default is shown below:

            .
            .
           MRDCI DIRECT
           TABLE
           SELECT
           CI
           NATORB
            .
            .

In practice the SELECT module will require input (except in cases where the default configuration generation described above is used), characterising for example the nature of the reference configurations, selection attributes etc, but in many instances the defaults of the other sub-modules will hold so that the associated data input may be omitted. Clearly this omission of data requires a firm understanding of the defaults in effect, which will only be apparent after the detailed description of directives presented in Part 6. For the moment we illustrate this by considering the simplified data file for the multi-reference calculation on H₂CO above:

          RESTART
          TITLE
          H2CO - 3-21G DEFAULT BASIS - MRDCI 4M/1R
          SUPER OFF NOSYM
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI
          SELECT
          CNTRL 16
          CONF
          0 1 2 3 4 5  13  17 18
          0 1 2 3 4 5  14  17 18
          0 1 2 3 4 5  13  17 19
          4 13 14 18 19  1 2 3 4 5 17
          END
          THRESH 2 2
          ENTER

The following points should be noted:

• It is assumed that the complete sequence of sub-tasks is to be carried out. If any task is to be 'BYPASS'ed, then the associated data line must be present.
• The transformation module will again use the default section number corresponding to the closed-shell SCF vectors (section 1) as the location of the molecular orbital coefficient array.
• The SYMMETRY, SPIN, SINGLES and ROOTS directives of the SELECT sub-module may all be omitted, since the required specification corresponds in each case to the default values. Note however that the CNTRL directive must be specified in any SELECT data in which the CONF directive is also present (in contrast to the conventional module).
• To provide compatibility with the DIRECT-CI module, an alternative form of the CONF directive may also be used, where the configurations are defined in terms of occupation patterns rather than by the orbital numbering. In this case a second keyword, OCCUPATION, is specified on the CONF data line; thus in the present example, we may also use the following CONF data;

            CONF OCCUPATION
             2   2   2   2   2   2   2   2   0   0  0
             2   2   2   2   2   2   0   2   2   0  0
             2   2   2   2   2   2   2   0   0   0  2
             2   2   2   2   2   2   1   1   1   0  1
            END
  

  where the occupations specified correspond to the occupancies of the input SCF MOs. At this stage we leave it to the user to confirm that this data is equivalent to the CONF specification in the example above.
• Both CI and NATORB may be omitted, the default values being required.

24.13 Semi-direct Table-CI - Restarting Calculations

In the examples considered above, we have assumed that the Table-CI job completes in the time allocated. This may not be the case and we need consider restarting the computation in a controlled fashion. Such a requirement may be met in RUNTYPE CI processing when:

• the associated integral evaluation, SCF or integral transformation has not completed, due either to lack of time or to convergence problems in the SCF;
• Table-CI processing itself has not completed.

In the present implementation it is not possible to restart Table-CI processing within a given sub-module in the event of job termination due to lack of time. It is possible however to fragment the calculation into separate sub-module runs, through the use of the BYPASS directive on the sub-module data lines. In such usage restarting the computation is achieved under control of the RESTART directive, which nominates the CI task for restarting. Consider the Table-CI job of §16.5; we show below the data files for fragmenting this CI into,

• integral transformation;
• configuration selection;
• hamiltonian pre-processing and Davidson diagonalisation.

The subset of interfaces to be saved between the various steps is given in Table 7.

Integral Transformation

          RESTART CI
          TITLE
          H2CO - 3-21G DEFAULT BASIS - MRDCI 4M/1R
          SUPER OFF NOSYM
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI DIRECT
          TABLE BYPASS
          SELECT BYPASS
          SINGLES 1
          CONF
          0 1 2 3 4 5  13  17 18
          0 1 2 3 4 5  14  17 18
          0 1 2 3 4 5  13  17 19
          4 13 14 18 19  1 2 3 4 5 17
          END
          CI BYPASS
          NATORB BYPASS
          ENTER

Configuration Selection

          RESTART CI
          TITLE
          H2CO - 3-21G DEFAULT BASIS - MRDCI 4M/1R
          SUPER OFF NOSYM
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI DIRECT
          SELECT 
          SINGLES 1
          CONF
          0 1 2 3 4 5  13  17 18
          0 1 2 3 4 5  14  17 18
          0 1 2 3 4 5  13  17 19
          4 13 14 18 19  1 2 3 4 5 17
          END
          CI BYPASS
          NATORB BYPASS
          ENTER

Diagonalisation and Natural Orbital Generation

          RESTART CI
          TITLE
          H2CO - 3-21G DEFAULT BASIS - MRDCI 4M/1R
          SUPER OFF NOSYM
          ZMAT ANGSTROM
          C
          O 1 1.203
          H 1 1.099 2 121.8
          H 1 1.099 2 121.8 3 180.0
          END
          RUNTYPE CI
          MRDCI DIRECT
          SELECT BYPASS
          SINGLES 1
          CONF
          0 1 2 3 4 5  13  17 18
          0 1 2 3 4 5  14  17 18
          0 1 2 3 4 5  13  17 19
          4 13 14 18 19  1 2 3 4 5 17
          END
          CI
          NATORB 
          ENTER