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351 lines
14 KiB
TeX
351 lines
14 KiB
TeX
%
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% $Id$
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%
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\label{sec:selci}
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The selected CI module is integrated into NWChem but as yet no
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input module has been written. The input thus consists of setting the
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appropriate variables in the database.
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It is assumed that an initial SCF/MCSCF calculation has completed, and
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that MO vectors are available. These will be used to perform a
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four-index transformation, if this has not already been performed.
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\section{Background}
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This is a general spin-adapted, configuration-driven CI program
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which can perform arbitrary CI calculations, the only restriction
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being that all spin functions are present for each orbital occupation.
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CI wavefunctions may be specified using a simple configuration
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generation program, but the prime usage is intended to be in
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combination with perturbation correction and selection of new
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configurations. The second-order correction (Epstein-Nesbet) to the
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CI energy may be computed, and at the same time configurations that
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interact greater than a certain threshold with the current CI
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wavefunction may be chosen for inclusion in subsequent calculations.
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By repeating this process (typically twice is adequate) with the same
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threshold until no new configurations are added, the CI expansion may
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be made consistent with the selection threshold, enabling tentative
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extrapolation to the full-CI limit.
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A typical sequence of calculations is as follows:
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\begin{enumerate}
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\item Pick as an initial CI reference the previously executed
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SCF/MCSCF.
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\item Define an initial selection threshold.
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\item Determine the roots of interest in the current reference space.
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\item Compute the perturbation correction and select additional
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configurations that interact greater than the current threshold.
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\item Repeat steps 3 and 4.
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\item Lower the threshold (a factor of 10 is common) and repeat steps
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3, 4, and 5. The {\em first} pass through step 4 will yield the
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approximately self-consistent CI and CI+PT energies from the {\em
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previous} selection threshold.
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\end{enumerate}
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To illustrate this, below is some abbreviated output from a
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calculation on water in an augmented cc-PVDZ basis set with one frozen
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core orbital. The SCF was converged to high precision in $C_{2v}$
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symmetry with the following input
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\begin{verbatim}
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start h2o
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geometry; symmetry c2v
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O 0 0 0; H 0 1.43042809 -1.10715266
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end
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basis
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H library aug-cc-pvdz; O library aug-cc-pvdz
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end
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task scf
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scf; thresh 1d-8; end
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\end{verbatim}
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The following input restarts from the SCF to perform a sequence of
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selected CI calculations with the specified tolerances, starting with
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the SCF reference.
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\begin{verbatim}
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restart h2o
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set fourindex:occ_frozen 1
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set selci:mode select
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set "selci:selection thresholds" \
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0.001 0.001 0.0001 0.0001 0.00001 0.00001 0.000001
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task selci
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\end{verbatim}
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Table \ref{selcitab} summarizes the output from each of the major
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computational steps that were performed.
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\begin{table}[htbp]
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\begin{tabular}{c|l|r|l}
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& & CI & \\
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Step & Description & dimension & Energy \\ \hline
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& & & \\
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1 & Four-index, one frozen-core & & \\
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2 & Config. generator, SCF default & 1 & \\
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3+4& CI diagonalization & 1 & $E_{CI} = -76.041983$ \\
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5 & PT selection T=0.001 & 1 & $E_{CI+PT} = -76.304797$ \\
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6+7 & CI diagonalization & 75 & $E_{CI} = -76.110894$ \\
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8 & PT selection T=0.001& 75 & $E_{CI+PT} = -76.277912$ \\
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9+10& CI diagonalization & 75 & $E_{CI}(T=0.001) = -76.110894$ \\
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11 & PT selection T=0.0001 & 75 & $E_{CI+PT}(T=0.001) = -76.277912$ \\
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12+13 & CI diagonalization & 823 & $E_{CI} = -76.228419$ \\
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14 & PT selection T=0.0001 & 823 & $E_{CI+PT} = -76.273751$ \\
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15+16 & CI diagonalization & 841 & $E_{CI}(T=0.0001) = -76.2300544$ \\
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17 & PT selection T=0.00001& 841 & $E_{CI+PT}(T=0.0001) = -76.274073$ \\
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18+19 & CI diagonalization & 2180 & $E_{CI} = -76.259285$ \\
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20 & PT selection T=0.00001& 2180 & $E_{CI+PT} = -76.276418$ \\
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21+22 & CI diagonalization & 2235 & $E_{CI}(T=0.00001) = -76.259818$ \\
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23 & PT selection T=0.000001 & 2235 & $E_{CI+PT}(T=0.00001) = -76.276478$\\
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24 & CI diagonalization & 11489 & \\ \hline
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\end{tabular}
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\caption{\label{selcitab} Summary of steps performed in a selected CI
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calculation on water.}
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\end{table}
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\section{Files}
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Currently, no direct control is provided over filenames. All files
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are prefixed with the standard file-prefix, and any files generated by
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all nodes are also postfixed with the processor number. Thus, for
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example the molecular integrals file, used only by process zero, might
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be called {\tt h2o.moints} whereas the off-diagonal Hamiltonian matrix
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element file used by process number eight would be called {\tt
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h2o.hamil.8}.
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\sloppy
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\begin{description}
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\item{\tt ciconf} --- the CI configuration file, which holds
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information about the current CI expansion, indexing vectors, etc.
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This is the most important file and is required for all restarts.
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Note that the CI configuration generator is only run if this file
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does not exist. Referenced only by process zero.
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\item{\tt moints} --- the molecular integrals, generated by the four-index
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transformation. As noted above these must currently be manually
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deleted, or the database entry \verb+selci:moints:force+ set, to
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force regeneration. Referenced only by process zero.
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\item{\tt civecs} --- the CI vectors. Referenced only by process zero.
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\item{\tt wmatrx} --- temporary file used to hold coupling coefficients.
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Deleted at calculation end. Referenced only by process zero.
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\item{\tt rtname, roname} --- restart information for the PT
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selection. Should be automatically deleted if no restart is
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necessary. Referenced only by process zero.
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\item{\tt hamdg} --- diagonal elements of the Hamiltonian.
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Deleted at calculation end. Referenced only by process zero.
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\item{\tt hamil} --- off-diagonal Hamiltonian matrix elements. All
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processes generate a file containing a subset of these elements.
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These files can become very large. Deleted at calculation end.
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\end{description}
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\fussy
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\section{Configuration Generation}
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If no configuration is explicitly specified then the previous
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SCF/MCSCF wavefunction is used, adjusting for any orbitals frozen in
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the four-index transformation. The four-index transformation must
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have completed successfully before this can execute. Orbital
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configurations for use as reference functions may also be explicitly
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specified.
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Once the default/user-input reference configurations have been
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determined additional reference functions may be generated by applying
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multiple sets of creation-annihilation operators, permitting for
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instance, the ready specification of complete or restricted active
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spaces.
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Finally, a uniform level of excitation from the current set of
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configurations into all orbitals may be applied, enabling, for
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instance, the simple creation of single or single+double excitation
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spaces from an MCSCF reference.
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\subsection{Specifying the reference occupation}
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A single orbital configuration or occupation is specified by
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\begin{verbatim}
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ns (socc(i),i=1,ns) (docc(i),i=1,nd)
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\end{verbatim}
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where \verb+ns+ specifies the number of singly occupied orbitals,
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\verb+socc()+ is the list of singly occupied orbitals, and
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\verb+docc()+ is the list of doubly occupied orbitals (the
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number of doubly occupied orbitals, \verb+nd+, is inferred from
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\verb+ns+ and the total number of electrons). All occupations may be
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strung together and inserted into the database as a single integer
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array with name \verb+"selci:conf"+. For example, the input
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\begin{verbatim}
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set "selci:conf" \
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0 1 2 3 4 \
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0 1 2 3 27 \
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0 1 3 4 19 \
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2 11 19 1 3 4 \
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2 8 27 1 2 3 \
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0 1 2 4 25 \
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4 3 4 25 27 1 2 \
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4 2 3 19 20 1 4 \
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4 2 4 20 23 1 3
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\end{verbatim}
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specifies the following nine orbital configurations
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\begin{verbatim}
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1(2) 2(2) 3(2) 4(2)
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1(2) 2(2) 3(2) 27(2)
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1(2) 3(2) 4(2) 19(2)
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1(2) 3(2) 4(2) 11(1) 19(1)
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1(2) 2(2) 3(2) 8(1) 27(1)
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1(2) 2(2) 4(2) 25(2)
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1(2) 2(2) 3(1) 4(1) 25(1) 27(1)
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1(2) 2(1) 3(1) 4(2) 19(1) 20(1)
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1(2) 2(1) 3(2) 4(1) 20(1) 23(1)
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\end{verbatim}
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The optional formatting of the input is just to make this arcane
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notation easier to read. Relatively few configurations can be
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currently specified in this fashion because of the input line limit of
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1024 characters.
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\subsection{Applying creation-annihilation operators}
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Up to 10 sets of creation-annihilation operator pairs may be
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specified, each set containing up to 255 pairs. This suffices to
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specify complete active spaces with up to ten electrons.
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The number of sets is specified as follows,
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\begin{verbatim}
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set selci:ngen 4
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\end{verbatim}
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which indicates that there will be four sets. Each set is then
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specified as a separate integer array
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\begin{verbatim}
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set "selci:refgen 1" 5 4 6 4 5 3 6 3
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set "selci:refgen 2" 5 4 6 4 5 3 6 3
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set "selci:refgen 3" 5 4 6 4 5 3 6 3
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set "selci:refgen 4" 5 4 6 4 5 3 6 3
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\end{verbatim}
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In the absence of friendly, input note that the names
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\verb+"selci:refgen n"+ must be formatted with n in \verb+I2+
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format. Each set specifies a list of creation-annihilation operator
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pairs (in that order). So for instance, in the above example each set
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is the same and causes the excitations
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\begin{verbatim}
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4->5 4->6 3->5 3->6
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\end{verbatim}
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If orbitals 3 and 4 were initially doubly occupied, and orbitals 5 and
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6 initially unoccupied, then the application of this set of operators
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four times in succession is sufficient to generate the four electron
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in four orbital complete active space.
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The precise sequence in which operators are applied is
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\begin{enumerate}
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\item loop through sets of operators
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\item loop through reference configurations
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\item loop through operators in the set
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\item apply the operator to the configuration, if the result is new add it
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to the new list
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\item end the loop over operators
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\item end the loop over reference configurations
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\item augment the configuration list with the new list
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\item end the loop over sets of operators
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\end{enumerate}
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\subsection{Uniform excitation level}
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By default no excitation is applied to the reference configurations.
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If, for instance, you wanted to generate a single excitation CI space
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from the current configuration list, specify
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\begin{verbatim}
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set selci:exci 1
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\end{verbatim}
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Any excitation level may be applied, but since the list of
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configurations is explicitly generated, as is the CI Hamiltonian
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matrix, you will run out of disk space if you attempt to use more than
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a few tens of thousands of configurations.
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\section{Number of roots}
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By default, only one root is generated in the CI diagonalization or
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perturbation selection. The following requests that 2 roots be
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generated
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\begin{verbatim}
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set selci:nroot 2
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\end{verbatim}
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There is no imposed upper limit. If many roots are required, then, to
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minimize root skipping problems, it helps to perform an initial
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approximate diagonalization with several more roots than required, and
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then resetting this parameter once satisfied that the desired
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states are obtained.
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\section{Accuracy of diagonalization}
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By default, the CI wavefunctions are converged to a residual norm of
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$10^{-6}$ which provides similar accuracy in the perturbation
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corrections to the energy, and much higher accuracy in the CI
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eigenvalues. This may be adjusted with
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\begin{verbatim}
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set "selci:diag tol" 1d-3
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\end{verbatim}
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the example setting much lower precision, appropriate for the
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approximate diagonalization discussed in the preceding section.
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\section{Selection thresholds}
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When running in the selected-CI mode the program will loop
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through a list of selection thresholds ($T$), performing the CI
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diagonalization, computing the perturbation correction, and augmenting
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the CI expansion with configurations that make an energy lowering to
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any root greater than $T$. The list of selection thresholds is
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specified as follows
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\begin{verbatim}
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set "selci:selection thresholds" \
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0.001 0.001 0.0001 0.0001 0.00001 0.00001 0.000001
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\end{verbatim}
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There is no default for this parameter.
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\section{Mode}
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By default the program runs in \verb="ci+davids"= mode and just
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determines the CI eigenvectors/values in the current configuration
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space. To perform a selected-CI with perturbation correction use the
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following
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\begin{verbatim}
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set selci:mode select
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\end{verbatim}
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and remember to define the selection thresholds.
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\section{Memory requirements}
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No global arrays are used inside the selected-CI, though the
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four-index transformation can be automatically invoked and it does use
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GAs. The selected CI replicates inside each process
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\begin{itemize}
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\item all unique two-electron integrals in the MO basis that are
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non-zero by symmetry, and
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\item all CI information, including the CI vectors.
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\end{itemize}
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These large data structures are allocated on the local stack. A fatal
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error will result if insufficient memory is available.
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\section{Forcing regeneration of the MO integrals}
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When scanning a potential energy surface or optimizing a geometry the
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MO integrals need to be regenerated each time. Specify
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\begin{verbatim}
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set selci:moints:force logical .true.
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\end{verbatim}
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to accomplish this.
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\section{Disabling update of the configuration list}
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When computing CI+PT energy the reference configuration list is
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normally updated to reflect all configurations that interact more than
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the specified threshold. This is usually desirable. But when
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scanning a potential energy surface or optimizing a geometry the
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reference list must be kept fixed to keep the potential energy surface
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continuous and well defined. To do this specify
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\begin{verbatim}
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set selci:update logical .false.
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\end{verbatim}
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\section{Orbital locking in CI geometry optimization}
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The selected CI wavefunction is not invariant to orbital rotations or
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to swapping two or more orbitals. Orbitals could be swapped or rotated
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when the geometry is changed in a geometry optimization step. The keyword
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\verb#lock# has to be set in the SCF/MCSCF (vectors) input block to keep the
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orbitals in the same order throughout the geometry optimization.
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