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266 lines
11 KiB
TeX
266 lines
11 KiB
TeX
%
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% $Id$
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%
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\label{sec:ecp}
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\def\ell{l}
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Effective core potentials (ECPs) are a useful means of replacing the core
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electrons in a calculation with an effective potential, thereby eliminating
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the need for the core basis functions, which usually require a large set of
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Gaussians to describe them. In addition to replacing the core, they may be
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used to represent relativistic effects, which are largely confined to the
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core. In this context, both the scalar (spin-free) relativistic effects and
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spin-orbit (spin-dependent) relativistic effects may be included in
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effective potentials. NWChem has the facility to use both, and these are
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described in the next two sections.
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A brief recapitulation of the development of RECPs is given here, following
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Pacios and Christiansen\footnote{l.~F.~Pacios and P.~A.~Christiansen,
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J.~Chem.~Phys.~{\bf 82}, 2664 (1985)}. The process can be viewed as starting
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from an atomic Dirac-Hartree-Fock calculation, done in {\it jj} coupling,
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and producing relativistic effective potentials (REPs) for each $\ell$ and
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$j$ value, $U^{\rm REP}_{\ell j}$. From these, a local potential is
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extracted, which for example contains the Coulomb potential of the core
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electrons balanced by the part of the nuclear attraction which cancels the
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core electron charge. The residue is expressed in a semi-local form,
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\begin{equation}
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U^{\rm REP} = U^{\rm REP}_{LJ}(r) + \sum_{\ell=0}^{L-1}
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\sum_{j=|\ell-1/2}^{\ell+1/2} \left[ U^{\rm REP}_{\ell j}(r) -
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U^{\rm REP}_{LJ}(r) \right] \sum_m | \ell j m \rangle \langle \ell j m |
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\end{equation}
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where $L$ is one larger than the maximum angular momentum in the atom.
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The scalar potential is obtained by averaging the REPs for each $j$ for a
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given $\ell$ to give an averaged relativistic effective potential, or AREP,
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\begin{equation}
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U^{\rm AREP}_\ell(r) = \frac{1}{2\ell+1} \left[ \ell U^{\rm REP}_{\ell-1/2}(r)
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+ (\ell+1) U^{\rm REP}_{\ell+1/2}(r) \right].
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\end{equation}
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These are summed into the full potential.
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%\begin{equation}
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%U^{\rm AREP} = U^{\rm AREP}_{L}(r) + \sum_{\ell=0}^{L-1}
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%\left[ U^{\rm AREP}_{\ell}(r) - U^{\rm AREP}_{L}(r)
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%\sum_m | \ell m \rangle \langle \ell m |.
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%\end{equation}
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The spin-orbit potential is obtained from the difference between the REPs
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for the two $j$ values for a given \ell, and may be
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represented in terms of an effective spin-orbit operator,
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\begin{equation}
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H^{\rm SO} = {\bf s} \cdot \sum_{\ell=1}^{L-1} \frac{2}{2\ell+1}
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\Delta U^{\rm REP}_{\ell} \sum_{mm'}
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| \ell m \rangle \langle \ell m | \hat\ell | \ell m' \rangle \langle \ell m' |.
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\end{equation}
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where
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\begin{equation}
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\Delta U^{\rm REP}_{\ell} = U^{\rm REP}_{\ell+1/2}(r)
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- U^{\rm REP}_{\ell-1/2}(r).
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\end{equation}
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The spin-orbit integrals generated by NWChem are the integrals over the sum,
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including the factor of $2/(2\ell+1)$, so that they may be treated as an
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effective spin-orbit operator without further factors introduced.
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The effective potentials, both scalar and spin-orbit, are fitted to
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Gaussians with the form
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\[
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r^2U_l(r) = \sum_{k} A_{lk} r^{n_{lk}} e^{-B_{lk}r^{2}}
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\]
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where $A_{lk}$ is the contraction coefficient, $n_{lk}$ is the
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exponent of the ``r'' term (r-exponent), and $B_{lk}$ is the Gaussian
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exponent. The $n_{lk}$ is shifted by 2, in accordance with most of the ECP
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literature and implementations, i.e., an $n_{lk} = 0$ implies
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$r^{-2}$. The current implementation allows $n_{lk}$ values
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of only 0, 1, or 2.
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\section{Scalar ECPs}
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\label{sec:scalar_ecp}
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The optional directive \verb+ECP+ allows the user to describe an effective core
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potential (ECP) in terms of contracted Gaussian functions as given above.
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Potentials using these functions must be specified explicitly by user input
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in the \verb+ECP+ directive. This directive has essentially the same form
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and properties as the standard \verb+BASIS+ directive, except for essential
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differences required for ECPs. Because of this, the ECP is treated
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internally as a basis set. The form of the input for the
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\verb+ECP+ directive is as follows:
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% [spherical || cartesian default cartesian]
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% [segment || nosegment default segment]
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\begin{verbatim}
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ECP [<string name default "ecp basis">] \
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[print || noprint default print]
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<string tag> library [<string tag_in_lib>] \
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<string standard_set> [file <filename>] \
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[except <string tag list>]
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<string tag> [nelec] <integer number_of_electrons_replaced>
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...
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<string tag> <string shell_type>
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<real r-exponent> <real Gaussian-exponent> <real list_of_coefficients>
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...
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END
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\end{verbatim}
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ECPs are automatically segmented, even if general contractions are input.
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The projection operators defined in an ECP are spherical by default, so
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there is no need to include the \verb+CARTESIAN+ or \verb+SPHERICAL+ keyword
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as there is for a standard basis set. ECPs are associated with centers in
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geometries through tags or names of centers. These tags must match in the
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same manner as for basis sets the tags in a \verb+GEOMETRY+ and
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\verb+ECP+ directives, and are limited to sixteen (16) characters.
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Each center with the same tag will have the same ECP. By default, the
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input module prints each ECP that it encounters. The \verb+NOPRINT+
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option can be used to disable printing. There can be only one active
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ECP, even though several may exist in the input deck. The ECP modules
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load ``ecp basis'' inputs along with any ``ao basis'' inputs present.
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ECPs may be used in both energy and gradient calculations.
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ECPs are named in the same fashion as geometries or regular basis
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sets, with the default name being \verb+"ecp basis"+. It should be
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clear from the above discussion on geometries and database entries how
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indirection is supported. All directives that are in common with the
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standard Gaussian basis set input have the same function and syntax.
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As for regular basis sets, ECPs may be obtained from the standard library.
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The names of the sets of ECPs available in the standard
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library (their coverage is described in Appendix \ref{sec:knownbasis}) are
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\begin{itemize}
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\item \verb,"Hay-Wadt MB (n+1) ECP",
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\item \verb,"Hay-Wadt VDZ (n+1) ECP",
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\item \verb+"LANL2DZ ECP"+
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\item \verb+"SBKJC VDZ ECP"+
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\item \verb+"Stuttgart RLC ECP"+
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\item \verb+"Stuttgart RSC ECP"+
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\item \verb+"CRENBL ECP"+
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\item \verb+"CRENBS ECP"+
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\end{itemize}
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The keyword \verb+nelec+ allows the user to specify the number of core
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electrons replaced by the ECP. Additional input lines define the
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specific coefficients and exponents. The variable \verb+<shell_type>+
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is used to specify the components of the ECP. The keyword \verb+ul+
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entered for \verb+<shell_type>+ denotes the local part of the ECP.
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This is equivalent to the highest angular momentum functions specified
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in the literature for most ECPs. The standard entries (\verb+s, p, d+,
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etc.) for \verb+shell_type+ specify the angular momentum projector
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onto the local function. The shell type label of \verb+s+ indicates
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the \verb+ul-s+ projector input, \verb+p+ indicates the \verb+ul-p+,
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etc.
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For example, the Christiansen, Ross and Ermler ARECPs are available in
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the standard basis set libary named \verb+{crenbl_ecp}+. To perform a
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calculation on uranyl (UO$_2^{2+}$) with all-electron oxygen
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(aug-cc-pvdz basis), and uranium with an ARECP and using the
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corresponding basis the following input can be used
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\begin{verbatim}
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geometry
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U 0 0 0
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O 0 0 1.65
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O 0 0 -1.65
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end
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basis
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U library crenbl_ecp
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O library aug-cc-pvdz
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end
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ecp
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U library crenbl_ecp
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end
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\end{verbatim}
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The following is an example of explicit input of an ECP for H$_2$CO.
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It defines an ECP for the carbon and oxygen atoms in the molecule.
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% \centerline{{\bf H$_2$CO }}
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\begin{verbatim}
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ecp
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C nelec 2 # ecp replaces 2 electrons on C
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C ul # d
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1 80.0000000 -1.60000000
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1 30.0000000 -0.40000000
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2 0.5498205 -0.03990210
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C s # s - d
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0 0.7374760 0.63810832
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0 135.2354832 11.00916230
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2 8.5605569 20.13797020
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C p # p - d
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2 10.6863587 -3.24684280
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2 23.4979897 0.78505765
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O nelec 2 # ecp replaces 2 electrons on O
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O ul # d
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1 80.0000000 -1.60000000
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1 30.0000000 -0.40000000
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2 1.0953760 -0.06623814
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O s # s - d
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0 0.9212952 0.39552179
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0 28.6481971 2.51654843
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2 9.3033500 17.04478500
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O p # p - s
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2 52.3427019 27.97790770
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2 30.7220233 -16.49630500
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end
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\end{verbatim}
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Various ECPs without a local function are available, including those of
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the Stuttgart group. For those, no "ul" part needs to be defined. To
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define the absence of the local potential, simply specify one contraction
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with a zero coefficient:
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\begin{verbatim}
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<string tag> ul
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2 1.00000 0.00000
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\end{verbatim}
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\section{Spin-orbit ECPs}
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\label{sec:spinorb_ecp}
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The Spin-orbit ECPs can be used with the Density Functional Approach, but
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one has to run the calculations without symmetry. Note: when a Hartree-Fock
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method is specified the spin-orbit input will be ignored.
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Spin-orbit ECPs are fitted in precisely the same functional form as the
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scalar RECPs and have the same properties, with the exception that there is
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no local potential ul, no $s$ potential and no effective charge has to be
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defined. Spin-orbit potentials are
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specified in the same way as ECPs except that the directive \verb+SO+ is
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used instead of \verb+ECP+. Note that there currently are no spin-orbit
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ECPs defined in the standard NWChem library. The \verb+SO+
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directive is as follows:
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\begin{verbatim}
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SO [<string name default "so basis">] \
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[print || noprint default print]
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<string tag> library [<string tag_in_lib>] \
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<string standard_set> [file <filename>]
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[except <string tag list>]
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...
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<string tag> <string shell_type>
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<real r-exponent> <real Gaussian-exponent> <real list_of_coefficients>
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...
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END
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\end{verbatim}
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Note: in the literature the coefficients of the spin-orbit potentials are NOT
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always defined in the same manner. The NWChem code assumes that the spin-orbit
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potential defined in the input is of the form:
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\begin{equation}
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\Delta U^{\rm NWChem}_{\ell} = \frac{2}{2\ell+1} \Delta U_{\ell}
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\end{equation}
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For example, in the literature the Stuttgart potentials are defined as
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$\Delta U_{\ell}$ and, hence, have to be multiplied by $2/(2{\ell}+1)$. On the
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other hand, the CRENBL potentials in the published papers are defined as
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$\frac{\ell}{2\ell+1} \Delta U_{\ell}$ and, hence, have to be multiplied by
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$2/{\ell}$ (Warning: on the CRENBL website the spin-orbit potentials already
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have been corrected with the $2/{\ell}$ factor).
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%RJH: Align six comment symbols (#) with the other two.--fmr.
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%Generally contracted ECP basis sets are not in wide use, but the
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%functionality has been included in NWChem for applications where they
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%might be useful.
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