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# Complete Basis Set

Code author: Lori A. Burns

Section author: Lori A. Burns

`complete_basis_set`(name[, scf_basis, scf_scheme, corl_wfn, corl_basis, corl_scheme, delta_wfn, delta_wfn_lesser, delta_basis, delta_scheme, delta2_wfn, delta2_wfn_lesser, delta2_basis, delta2_scheme, delta3_wfn, delta3_wfn_lesser, delta3_basis, delta3_scheme, delta4_wfn, delta4_wfn_lesser, delta4_basis, delta4_scheme, delta5_wfn, delta5_wfn_lesser, delta5_basis, delta5_scheme])

  Function to define a multistage energy method from combinations of basis set extrapolations and delta corrections and condense the components into a minimum number of calculations.

  Aliases: | cbs()
  Returns: | (float) – Total electronic energy in Hartrees
  PSI variables: |

  - `CBS TOTAL ENERGY`
  - `CBS REFERENCE ENERGY`
  - `CBS CORRELATION ENERGY`
  - `CURRENT ENERGY`
  - `CURRENT REFERENCE ENERGY`
  - `CURRENT CORRELATION ENERGY`

  Caution

  Some features are not yet implemented. Buy a developer a coffee.

  - Not all methods hooked in through PSI variables, configuration interaction and arbitrary order MP in particular.
  - No scheme defaults for given basis zeta number, so scheme must be specified explicitly.
  - No way to tell function to boost fitting basis size for all calculations.
  - No way to extrapolate def2 family basis sets
  - Need to add more extrapolation schemes

  As represented in the equation below, a CBS energy method is defined in several sequential stages (scf, corl, delta, delta2, delta3, delta4, delta5) covering treatment of the reference total energy, the correlation energy, a delta correction to the correlation energy, and a second delta correction, etc.. Each is activated by its stage_wfn keyword and is only allowed if all preceding stages are active.

  Here, is an energy or energy extrapolation scheme, and the following also hold.

  A translation of this ungainly equation to example [5] below is as follows. In words, this is a double- and triple-zeta 2-point Helgaker-extrapolated CCSD(T) coupled-cluster correlation correction appended to a triple- and quadruple-zeta 2-point Helgaker-extrapolated MP2 correlation energy appended to a SCF/aug-cc-pVQZ reference energy.

  - Energy Methods

      The presence of a stage_wfn keyword is the indicator to incorporate (and check for stage_basis and stage_scheme keywords) and compute that stage in defining the CBS energy.

      The cbs() function requires, at a minimum, `name='scf'` and `scf_basis` keywords to be specified for reference-step only jobs and `name` and `corl_basis` keywords for correlated jobs.

      The following energy methods have been set up for cbs().

      - scf
      - mp2
      - mp2.5
      - mp3
      - mp4(sdq)
      - mp4
      - omp2

      - omp3
      - ocepa
      - cepa0
      - cepa(0)
      - cepa(1)
      - cepa(3)
      - acpf

      - aqcc
      - qcisd
      - cc2
      - ccsd
      - fno-df-ccsd
      - bccd
      - cc3

      - qcisd(t)
      - ccsd(t)
      - fno-df-ccsd(t)
      - bccd(t)
      - cisd
      - cisdt
      - cisdtq

      - cin
      - fci
      - mrccsd
      - mrccsd(t)
      - mrccsdt
      - mrccsdt(q)

  Parameters:

  - name (string) –

    `'scf'` || `'ccsd'` || etc.

    First argument, usually unlabeled. Indicates the computational method for the correlation energy, unless only reference step to be performed, in which case should be `'scf'`. Overruled if stage_wfn keywords supplied.

  - corl_wfn (string) –

    `'mp2'` || `'ccsd(t)'` || etc.

    Indicates the energy method for which the correlation energy is to be obtained. Can also be specified with `name` or as the unlabeled first argument to the function.

  - delta_wfn (string) –

    `'ccsd'` || `'ccsd(t)'` || etc.

    Indicates the (superior) energy method for which a delta correction to the correlation energy is to be obtained.

  - delta_wfn_lesser (string) –

    `'mp2'` || `'ccsd'` || etc.

    Indicates the inferior energy method for which a delta correction to the correlation energy is to be obtained.

  - delta2_wfn (string) –

    `'ccsd'` || `'ccsd(t)'` || etc.

    Indicates the (superior) energy method for which a second delta correction to the correlation energy is to be obtained.

  - delta2_wfn_lesser (string) –

    `'mp2'` || `'ccsd(t)'` || etc.

    Indicates the inferior energy method for which a second delta correction to the correlation energy is to be obtained.

  - delta3_wfn (string) –

    `'ccsd'` || `'ccsd(t)'` || etc.

    Indicates the (superior) energy method for which a third delta correction to the correlation energy is to be obtained.

  - delta3_wfn_lesser (string) –

    `'mp2'` || `'ccsd(t)'` || etc.

    Indicates the inferior energy method for which a third delta correction to the correlation energy is to be obtained.

  - delta4_wfn (string) –

    `'ccsd'` || `'ccsd(t)'` || etc.

    Indicates the (superior) energy method for which a fourth delta correction to the correlation energy is to be obtained.

  - delta4_wfn_lesser (string) –

    `'mp2'` || `'ccsd(t)'` || etc.

    Indicates the inferior energy method for which a fourth delta correction to the correlation energy is to be obtained.

  - delta5_wfn (string) –

    `'ccsd'` || `'ccsd(t)'` || etc.

    Indicates the (superior) energy method for which a fifth delta correction to the correlation energy is to be obtained.

  - delta5_wfn_lesser (string) –

    `'mp2'` || `'ccsd(t)'` || etc.

    Indicates the inferior energy method for which a fifth delta correction to the correlation energy is to be obtained.

  - Basis Sets

      Currently, the basis set set through `set` commands have no influence on a cbs calculation.

  Parameters:

  - scf_basis (basis string) –

    `corl_basis` || `'cc-pV[TQ]Z'` || `'jun-cc-pv[tq5]z'` || `'6-31G*'` || etc.

    Indicates the sequence of basis sets employed for the reference energy. If any correlation method is specified, `scf_basis` can default to `corl_basis`.

  - corl_basis (basis string) –

    `'cc-pV[TQ]Z'` || `'jun-cc-pv[tq5]z'` || `'6-31G*'` || etc.

    Indicates the sequence of basis sets employed for the correlation energy.

  - delta_basis (basis string) –

    `'cc-pV[TQ]Z'` || `'jun-cc-pv[tq5]z'` || `'6-31G*'` || etc.

    Indicates the sequence of basis sets employed for the delta correction to the correlation energy.

  - delta2_basis (basis string) –

    `'cc-pV[TQ]Z'` || `'jun-cc-pv[tq5]z'` || `'6-31G*'` || etc.

    Indicates the sequence of basis sets employed for the second delta correction to the correlation energy.

  - delta3_basis (basis string) –

    `'cc-pV[TQ]Z'` || `'jun-cc-pv[tq5]z'` || `'6-31G*'` || etc.

    Indicates the sequence of basis sets employed for the third delta correction to the correlation energy.

  - delta4_basis (basis string) –

    `'cc-pV[TQ]Z'` || `'jun-cc-pv[tq5]z'` || `'6-31G*'` || etc.

    Indicates the sequence of basis sets employed for the fourth delta correction to the correlation energy.

  - delta5_basis (basis string) –

    `'cc-pV[TQ]Z'` || `'jun-cc-pv[tq5]z'` || `'6-31G*'` || etc.

    Indicates the sequence of basis sets employed for the fifth delta correction to the correlation energy.

  - Schemes

      Transformations of the energy through basis set extrapolation for each stage of the CBS definition. A complaint is generated if number of basis sets in stage_basis does not exactly satisfy requirements of stage_scheme. An exception is the default, `'highest_1'`, which uses the best basis set available. See Extrapolation Schemes for all available schemes.

  Parameters:

  - scf_scheme (function) –

    `highest_1` || `scf_xtpl_helgaker_3` || etc.

    Indicates the basis set extrapolation scheme to be applied to the reference energy.

  - corl_scheme (function) –

    `highest_1` || `corl_xtpl_helgaker_2` || etc.

    Indicates the basis set extrapolation scheme to be applied to the correlation energy.

  - delta_scheme (function) –

    `highest_1` || `corl_xtpl_helgaker_2` || etc.

    Indicates the basis set extrapolation scheme to be applied to the delta correction to the correlation energy.

  - delta2_scheme (function) –

    `highest_1` || `corl_xtpl_helgaker_2` || etc.

    Indicates the basis set extrapolation scheme to be applied to the second delta correction to the correlation energy.

  - delta3_scheme (function) –

    `highest_1` || `corl_xtpl_helgaker_2` || etc.

    Indicates the basis set extrapolation scheme to be applied to the third delta correction to the correlation energy.

  - delta4_scheme (function) –

    `highest_1` || `corl_xtpl_helgaker_2` || etc.

    Indicates the basis set extrapolation scheme to be applied to the fourth delta correction to the correlation energy.

  - delta5_scheme (function) –

    `highest_1` || `corl_xtpl_helgaker_2` || etc.

    Indicates the basis set extrapolation scheme to be applied to the fifth delta correction to the correlation energy.

  - molecule (molecule) –

    `h2o` || etc.

    The target molecule, if not the last molecule defined.

  Examples: |

      >>> # [1] replicates with cbs() the simple model chemistry scf/cc-pVDZ: set basis cc-pVDZ energy('scf')
      >>> cbs('scf', scf_basis='cc-pVDZ')

      >>> # [2] replicates with cbs() the simple model chemistry mp2/jun-cc-pVDZ: set basis jun-cc-pVDZ energy('mp2')
      >>> cbs('mp2', corl_basis='jun-cc-pVDZ')

      >>> # [3] DTQ-zeta extrapolated scf reference energy
      >>> cbs('scf', scf_basis='cc-pV[DTQ]Z', scf_scheme=scf_xtpl_helgaker_3)

      >>> # [4] DT-zeta extrapolated mp2 correlation energy atop a T-zeta reference
      >>> cbs('mp2', corl_basis='cc-pv[dt]z', corl_scheme=corl_xtpl_helgaker_2)

      >>> # [5] a DT-zeta extrapolated coupled-cluster correction atop a TQ-zeta extrapolated mp2 correlation energy atop a Q-zeta reference
      >>> cbs('mp2', corl_basis='aug-cc-pv[tq]z', corl_scheme=corl_xtpl_helgaker_2, delta_wfn='ccsd(t)', delta_basis='aug-cc-pv[dt]z', delta_scheme=corl_xtpl_helgaker_2)

      >>> # [6] a D-zeta ccsd(t) correction atop a DT-zeta extrapolated ccsd cluster correction atop a TQ-zeta extrapolated mp2 correlation energy atop a Q-zeta reference
      >>> cbs('mp2', corl_basis='aug-cc-pv[tq]z', corl_scheme=corl_xtpl_helgaker_2, delta_wfn='ccsd', delta_basis='aug-cc-pv[dt]z', delta_scheme=corl_xtpl_helgaker_2, delta2_wfn='ccsd(t)', delta2_wfn_lesser='ccsd', delta2_basis='aug-cc-pvdz')

      >>> # [7] cbs() coupled with database()
      >>> database('mp2', 'BASIC', subset=['h2o','nh3'], symm='on', func=cbs, corl_basis='cc-pV[tq]z', corl_scheme=corl_xtpl_helgaker_2, delta_wfn='ccsd(t)', delta_basis='sto-3g')

      >>> # [8] cbs() coupled with optimize()
      >>> optimize('mp2', corl_basis='cc-pV[DT]Z', corl_scheme=corl_xtpl_helgaker_2, func=cbs)

## Output

At the beginning of a cbs() job is printed a listing of the individual energy calculations which will be performed. The output snippet below is from the example job [2] above. It shows first each model chemistry needed to compute the aggregate model chemistry requested through cbs(). Then, since, for example, an `energy('ccsd(t)')` yields CCSD(T), CCSD, MP2, and SCF energy values, the wrapper condenses this task list into the second list of minimum number of calculations which will actually be run.

    Naive listing of computations required.
            scf / aug-cc-pvqz              for  SCF TOTAL ENERGY
            mp2 / aug-cc-pvtz              for  MP2 CORRELATION ENERGY
            mp2 / aug-cc-pvqz              for  MP2 CORRELATION ENERGY
        ccsd(t) / aug-cc-pvdz              for  CCSD(T) CORRELATION ENERGY
        ccsd(t) / aug-cc-pvtz              for  CCSD(T) CORRELATION ENERGY
            mp2 / aug-cc-pvdz              for  MP2 CORRELATION ENERGY
            mp2 / aug-cc-pvtz              for  MP2 CORRELATION ENERGY

    Enlightened listing of computations required.
            mp2 / aug-cc-pvqz              for  MP2 CORRELATION ENERGY
        ccsd(t) / aug-cc-pvdz              for  CCSD(T) CORRELATION ENERGY
        ccsd(t) / aug-cc-pvtz              for  CCSD(T) CORRELATION ENERGY

At the end of a cbs() job is printed a summary section like the one below. First, in the components section, are listed the results for each model chemistry available, whether required for the cbs job (*) or not. Next, in the stages section, are listed the results for each extrapolation. The energies of this section must be dotted with the weightings in column Wt to get the total cbs energy. Finally, in the CBS section, are listed the results for each stage of the cbs procedure. The stage energies of this section sum outright to the total cbs energy.

    ==> Components <==

    ----------------------------------------------------------------------------------
                   Method / Basis            Rqd   Energy [H]   Variable
    ----------------------------------------------------------------------------------
                      scf / aug-cc-pvqz        *  -1.11916375   SCF TOTAL ENERGY
                      mp2 / aug-cc-pvqz        *  -0.03407997   MP2 CORRELATION ENERGY
                      scf / aug-cc-pvdz           -1.11662884   SCF TOTAL ENERGY
                      mp2 / aug-cc-pvdz        *  -0.02881480   MP2 CORRELATION ENERGY
                  ccsd(t) / aug-cc-pvdz        *  -0.03893812   CCSD(T) CORRELATION ENERGY
                     ccsd / aug-cc-pvdz           -0.03893812   CCSD CORRELATION ENERGY
                      scf / aug-cc-pvtz           -1.11881134   SCF TOTAL ENERGY
                      mp2 / aug-cc-pvtz        *  -0.03288936   MP2 CORRELATION ENERGY
                  ccsd(t) / aug-cc-pvtz        *  -0.04201004   CCSD(T) CORRELATION ENERGY
                     ccsd / aug-cc-pvtz           -0.04201004   CCSD CORRELATION ENERGY
    ----------------------------------------------------------------------------------

    ==> Stages <==

    ----------------------------------------------------------------------------------
     Stage         Method / Basis             Wt   Energy [H]   Scheme
    ----------------------------------------------------------------------------------
       scf            scf / aug-cc-pvqz        1  -1.11916375   highest_1
      corl            mp2 / aug-cc-pv[tq]z     1  -0.03494879   corl_xtpl_helgaker_2
     delta        ccsd(t) / aug-cc-pv[dt]z     1  -0.04330347   corl_xtpl_helgaker_2
     delta            mp2 / aug-cc-pv[dt]z    -1  -0.03460497   corl_xtpl_helgaker_2
    ----------------------------------------------------------------------------------

    ==> CBS <==

    ----------------------------------------------------------------------------------
     Stage         Method / Basis                  Energy [H]   Scheme
    ----------------------------------------------------------------------------------
       scf            scf / aug-cc-pvqz           -1.11916375   highest_1
      corl            mp2 / aug-cc-pv[tq]z        -0.03494879   corl_xtpl_helgaker_2
     delta  ccsd(t) - mp2 / aug-cc-pv[dt]z        -0.00869851   corl_xtpl_helgaker_2
     total            CBS                         -1.16281105
    ----------------------------------------------------------------------------------

## Extrapolation Schemes

`highest_1`(**largs)

  Scheme for total or correlation energies with a single basis or the highest zeta-level among an array of bases. Used by `complete_basis_set()`.

`scf_xtpl_helgaker_2`(**largs)

  Extrapolation scheme for reference energies with two adjacent zeta-level bases. Used by `complete_basis_set()`.

`scf_xtpl_helgaker_3`(**largs)

  Extrapolation scheme for reference energies with three adjacent zeta-level bases. Used by `complete_basis_set()`.

`corl_xtpl_helgaker_2`(**largs)

  Extrapolation scheme for correlation energies with two adjacent zeta-level bases. Used by `complete_basis_set()`.

## Aliases

When a particular composite method or its functional form is going to be reused often, it is convenient to define an alias to it. A convenient place for such Python code to reside is in psi4/lib/python/aliases.py (source location) or `psi4/share/psi/python/aliases.py` (installed location). No recompilation is necessary after defining an alias. Some existing examples are below.

`sherrill_gold_standard`(name='mp2', **kwargs)

  Function to call the quantum chemical method known as ‘Gold Standard’ in the Sherrill group. Uses `complete_basis_set()` to evaluate the following expression. Two-point extrapolation of the correlation energy performed according to `corl_xtpl_helgaker_2()`.

      >>> # [1] single-point energy by this composite method
      >>> energy('sherrill_gold_standard')

      >>> # [2] finite-difference geometry optimization
      >>> optimize('sherrill_gold_standard')

      >>> # [3] finite-difference geometry optimization, overwriting some pre-defined sherrill_gold_standard options
      >>> optimize('sherrill_gold_standard', corl_basis='cc-pV[DT]Z', delta_basis='3-21g')

`allen_focal_point`(name='mp2', **kwargs)

  Function to call Wes Allen-style Focal Point Analysis. JCP 127 014306. Uses `complete_basis_set()` to evaluate the following expression. SCF employs a three-point extrapolation according to `scf_xtpl_helgaker_3()`. MP2, CCSD, and CCSD(T) employ two-point extrapolation performed according to `corl_xtpl_helgaker_2()`. CCSDT and CCSDT(Q) are plain deltas. This wrapper requires Kallay’s MRCC code.

      >>> # [1] single-point energy by this composite method
      >>> energy('allen_focal_point')

      >>> # [2] finite-difference geometry optimization embarrasingly parallel
      >>> optimize('allen_focal_point', mode='sow')

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