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<TITLE>9. Relativistic All-electron Approximations</TITLE>
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<BR>
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<B> Next:</B> <A NAME="tex2html1137"
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HREF="node12.html">10. Hartree-Fock or Self-consistent</A>
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<B> Up:</B> <A NAME="tex2html1133"
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HREF="user.html">user</A>
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<B> Previous:</B> <A NAME="tex2html1127"
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HREF="node10.html">8. Effective Core Potentials</A>
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  <B> <A NAME="tex2html1135"
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HREF="node2.html">Contents</A></B>
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<BR>
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<BR>
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<!--End of Navigation Panel-->
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<UL>
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<LI><A NAME="tex2html1138"
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HREF="node11.html#SECTION001110000000000000000">9.1 Douglas-Kroll approximation</A>
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<LI><A NAME="tex2html1139"
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HREF="node11.html#SECTION001120000000000000000">9.2 Dyall's Modified Dirac Hamitonian approximation</A>
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</UL>
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<!--End of Table of Child-Links-->
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<HR>
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<H1><A NAME="SECTION001100000000000000000">
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9. Relativistic All-electron Approximations</A>
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</H1>
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<A NAME="sec:rel"></A>All methods which include treatment of relativistic effects are ultimately
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based on the Dirac equation, which has a four component wave function. The
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solutions to the Dirac equation describe both positrons (the ``negative
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energy'' states) and electrons (the ``positive energy'' states), as well as
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both spin orientations, hence the four components. The wave function may be
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broken down into two-component functions traditionally known as the large
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and small components; these may further be broken down into the spin
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components.
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<P>
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The implementation of approximate all-electron relativistic methods in
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quantum chemical codes requires the removal of the negative energy states
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and the factoring out of the spin-free terms. Both of these may be achieved
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using a transformation of the Dirac Hamiltonian known in general as a
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Foldy-Wouthuysen transformation. Unfortunately this transformation cannot be
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represented in closed form for a general potential, and must be
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approximated. One popular approach is that originally formulated by Douglas
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and Kroll<A NAME="tex2html16"
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HREF="footnode.html#foot2244"><SUP>9.1</SUP></A> and developed by Hess<A NAME="tex2html17"
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HREF="footnode.html#foot2245"><SUP>9.2</SUP></A>. This approach decouples the positive and
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negative energy parts to second order in the external potential (and also
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fourth order in the fine structure constant, <IMG
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WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img42.gif"
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ALT="$\alpha$">). Another approach is
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based on a modification of the Dirac equation by Dyall<A NAME="tex2html18"
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HREF="footnode.html#foot2246"><SUP>9.3</SUP></A>, and involves an exact FW
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transformation on the atomic basis set level<A NAME="tex2html19"
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HREF="footnode.html#foot2247"><SUP>9.4</SUP></A>.
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<P>
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Since these approximations only modify the integrals, they can in principle
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be used at all levels of theory. At present the Douglas-Kroll implementation
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can be used at all levels of theory whereas Dyall's approach is currently
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available at the Hartree-Fock level.
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The derivatives have been implemented, allowing both methods to be used in
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geometry optimizations and frequency calculations.
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<P>
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The <code>RELATIVISTIC</code> directive provides input for the implemented relativistic
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approximations and is a compound directive that encloses additional directives
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specific to the approximations:
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<PRE>
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RELATIVISTIC
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[DOUGLAS-KROLL [<string (ON||OFF) default ON> \
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<string (FPP||DKH||DKFULL||DK3||DK3FULL) default DKH>] ||
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DYALL-MOD-DIRAC [ (ON || OFF) default ON ]
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[ (NESC1E || NESC2E) default NESC1E ] ]
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[CLIGHT <real clight default 137.0359895>]
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END
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</PRE>
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<P>
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Only one of the methods may be chosen at a time. If both methods are found
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to be on in the input block, NWChem will stop and print an error message.
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There is one general option for both methods, the definition of the speed
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of light in atomic units:
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<P>
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<PRE>
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CLIGHT <real clight default 137.0359895>
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</PRE>
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<P>
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The following sections describe the optional sub-directives that
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can be specified within the <code>RELATIVISTIC</code> block.
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<P>
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<H1><A NAME="SECTION001110000000000000000"></A>
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<A NAME="sec:douglas-kroll"></A>
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<BR>
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9.1 Douglas-Kroll approximation
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</H1>
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<P>
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The (spin-free) one-electron Douglas-Kroll approximation has been
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implemented. The use of relativistic effects from this Douglas-Kroll
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approximation can be invoked by specifying:
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<P>
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<PRE>
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DOUGLAS-KROLL [<string (ON||OFF) default ON> \
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<string (FPP||DKH||DKFULL|DK3|DK3FULL) default DKH>]
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</PRE>
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<P>
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The <code>ON|OFF</code> string is used to turn on or off the
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Douglas-Kroll approximation. By default, if the <code>DOUGLAS-KROLL</code>
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keyword is found, the approximation will be used in the calculation.
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If the user wishes to calculate a non-relativistic quantity after turning
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on Douglas-Kroll, the user will need to define a new <code>RELATIVISTIC</code>
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block and turn the approximation <code>OFF</code>. The user could also simply
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put a blank <code>RELATIVISTIC</code> block in the input file and all options
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will be turned off.
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<P>
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The <code>FPP</code> is the approximation based on free-particle projection
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operators<A NAME="tex2html20"
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HREF="footnode.html#foot2248"><SUP>9.5</SUP></A> whereas the
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<code>DKH</code> and <code>DKFULL</code> approximations are based on external-field
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projection operators<A NAME="tex2html21"
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HREF="footnode.html#foot2249"><SUP>9.6</SUP></A>.
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The latter two are considerably better approximations than the former. <code>DKH</code>
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is the Douglas-Kroll-Hess approach and is the approach that is generally
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implemented in quantum chemistry codes. <code>DKFULL</code> includes certain
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cross-product integral terms ignored in the <code>DKH</code> approach (see for example
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Häberlen and Rösch<A NAME="tex2html22"
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HREF="footnode.html#foot2250"><SUP>9.7</SUP></A>). The third-order Douglas-Kroll
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approximation has been implemented by T. Nakajima and K. Hirao<A NAME="tex2html23"
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HREF="footnode.html#foot2251"><SUP>9.8</SUP></A>. This approximation can be called using
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<code>DK3</code> (DK3 without cross-product integral terms) or <code>DK3FULL</code> (DK3 with
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cross-product integral terms).
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<P>
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The contracted basis sets used in the calculations should reflect the relativistic
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effects, i.e. one should use contracted basis sets which were generated using the
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Douglas-Kroll Hamiltonian. Basis sets that were contracted using the
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non-relativistic (Schödinger) Hamiltonian WILL PRODUCE ERRONEOUS RESULTS for
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elements beyond the first row. See appendix <A HREF="node41.html#sec:knownbasis">A</A> for available
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basis sets and their naming convention.
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<P>
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NOTE: we suggest that spherical basis sets are used in the calculation. The use of
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high quality cartesian basis sets can lead to numerical inaccuracies.
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<P>
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In order to compute the integrals needed for the Douglas-Kroll approximation
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the implementation makes use of a fitting basis set (see literature given
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above for details). The current code will create this fitting basis set
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based on the given <TT>"ao basis"</TT> by simply uncontracting that basis. This
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again is what is commonly implemented in quantum chemistry codes that
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include the Douglas-Kroll method. Additional flexibility is available to
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the user by explicitly specifying a Douglas-Kroll fitting basis
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set. This basis set must be named <TT>"D-K basis"</TT> (see Chapter
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<A HREF="node9.html#sec:basis">7</A>).
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<P>
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<H1><A NAME="SECTION001120000000000000000"></A>
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<A NAME="sec:dyall-mod-dir"></A>
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<BR>
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9.2 Dyall's Modified Dirac Hamitonian approximation
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</H1>
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<P>
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The approximate methods described in this section are all based on Dyall's
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modified Dirac Hamiltonian. This Hamiltonian is entirely equivalent to the
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original Dirac Hamiltonian, and its solutions have the same properties.
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The modification is achieved by a transformation on the small component,
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extracting out <!-- MATH
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$\sigma\cdot{\bf p}/2mc$
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-->
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<IMG
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WIDTH="72" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
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SRC="img85.gif"
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ALT="$\sigma\cdot{\bf p}/2mc$">. This gives the modified small
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component the same symmetry as the large component, and in fact it differs
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from the large component only at order <IMG
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WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img86.gif"
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ALT="$\alpha^2$">. The advantage of the
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modification is that the operators now resemble the operators of the
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Breit-Pauli Hamiltonian, and can be classified in a similar fashion into
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spin-free, spin-orbit and spin-spin terms. It is the spin-free terms which
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have been implemented in NWChem, with a number of further approximations.
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<P>
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The first is that the negative energy states are removed by a normalized
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elimination of the small component (NESC), which is equivalent to an exact
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Foldy-Wouthuysen (EFW) transformation. The number of components in the wave
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function is thereby effectively reduced from 4 to 2. NESC on its own does
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not provide any advantages, and in fact complicates things because the
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transformation is energy-dependent. The second approximation therefore
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performs the elimination on an atom-by-atom basis, which is equivalent to
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neglecting blocks which couple different atoms in the EFW transformation.
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The advantage of this approximation is that all the energy dependence can be
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included in the contraction coefficients of the basis set. The tests which
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have been done show that this approximation gives results well within
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chemical accuracy. The third approximation neglects the commutator of the
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EFW transformation with the two-electron Coulomb interaction, so that the
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only corrections that need to be made are in the one-electron integrals.
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This is the equivalent of the Douglas-Kroll(-Hess) approximation as it is
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usually applied.
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<P>
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The use of these approximations can be invoked with the use of the
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<code>DYALL-MOD-DIRAC</code> directive in the <code>RELATIVISTIC</code> directive block.
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The syntax is as follows.
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<P>
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<PRE>
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DYALL-MOD-DIRAC [ (ON || OFF) default ON ]
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[ (NESC1E || NESC2E) default NESC1E ]
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</PRE>
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<P>
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The <code>ON|OFF</code> string is used to turn on or off the
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Dyall's modified Dirac approximation. By default, if the <code>DYALL-MOD-DIRAC</code>
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keyword is found, the approximation will be used in the calculation.
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If the user wishes to calculate a non-relativistic quantity after turning
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on Dyall's modified Dirac, the user will need to define a new
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<code>RELATIVISTIC</code>
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block and turn the approximation <code>OFF</code>. The user could also simply
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put a blank <code>RELATIVISTIC</code> block in the input file and all options
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will be turned off.
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<P>
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Both one- and two-electron approximations are available
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<code>NESC1E || NESC2E</code>, and both have
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analytic gradients. The one-electron approximation is the default.
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The two-electron approximation specified by <code>NESC2E</code> has some sub
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options which are placed on the same logical line as the
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<code>DYALL-MOD-DIRAC</code> directive, with the following syntax:
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<P>
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<PRE>
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NESC2E [ (SS1CENT [ (ON || OFF) default ON ] || SSALL) default SSALL ]
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[ (SSSS [ (ON || OFF) default ON ] || NOSSSS) default SSSS ]
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</PRE>
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<P>
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The first sub-option gives the capability to limit the two-electron
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corrections to those in which the small components in any density must be on
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the same center. This reduces the <IMG
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WIDTH="64" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
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SRC="img87.gif"
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ALT="$(LL\vert SS)$"> contributions to at most
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three-center integrals and the <IMG
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WIDTH="64" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
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SRC="img88.gif"
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ALT="$(SS\vert SS)$"> contributions to two centers. For a
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case with only one relativistic atom this option is redundant. The second
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controls the inclusion of the <IMG
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WIDTH="64" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
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SRC="img88.gif"
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ALT="$(SS\vert SS)$"> integrals which are of order
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<IMG
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WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img89.gif"
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ALT="$\alpha^4$">. For light atoms they may safely be neglected, but for heavy
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atoms they should be included.
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<P>
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In addition to the selection of this keyword in the <code>RELATIVISTIC</code>
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directive block, it is necessary to supply basis sets in addition to the
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<code>ao basis</code>. For the one-electron approximation, three basis sets are
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needed: the atomic FW basis set, the large component basis set and the small
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component basis set. The atomic FW basis set should be included in the
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<code>ao basis</code>.
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The large and small components should similarly be incorporated
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in basis sets named <code>large component</code> and <code>small component</code>,
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respectively. For the two-electron approximation, only two basis sets are
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needed. These are the large component and the small component. The large component
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should be included in the <code>ao basis</code> and the small component
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is specified separately as <code>small component</code>, as for the one-electron
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approximation. This means that the two approximations can <I>not</I> be run
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correctly without changing the <code>ao basis</code>, and it is up to the user to
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ensure that the basis sets are correctly specified.
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<P>
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There is one further requirement in the specification of the basis sets. In
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the <code>ao basis</code>, it is necessary to add the <code>rel</code> keyword either to the
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<code>basis</code> directive or the library tag line (See below for examples).
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The former marks the basis
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functions specified by the tag as relativistic, the latter marks the whole
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basis as relativistic. The marking is actually done at the unique shell
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level, so that it is possible not only to have relativistic and
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nonrelativistic atoms, it is also possible to have relativistic and
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nonrelativistic shells on a given atom. This would be useful, for example,
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for diffuse functions or for high angular momentum correlating functions,
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where the influence of relativity was small. The marking of shells as
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relativistic is necessary to set up a mapping between the ao basis and the
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large and/or small component basis sets. For the one-electron approximation
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the large and small component basis sets MUST be of the same size and
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construction, i.e. differing only in the contraction coefficients.
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<P>
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It should also be noted that the relativistic code will NOT work with basis
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sets that contain sp shells, nor will it work with ECPs. Both of these are
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tested and flagged as an error.
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<P>
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Some examples follow. The first example sets up the data for relativistic
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calculations on water with the one-electron approximation and the
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two-electron approximation, using the library basis sets.
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<P>
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<PRE>
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start h2o-dmd
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geometry units bohr
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symmetry c2v
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O 0.000000000 0.000000000 -0.009000000
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H 1.515260000 0.000000000 -1.058900000
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H -1.515260000 0.000000000 -1.058900000
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end
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basis "fw" rel
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oxygen library cc-pvdz_pt_sf_fw
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hydrogen library cc-pvdz_pt_sf_fw
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end
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basis "large"
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oxygen library cc-pvdz_pt_sf_lc
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hydrogen library cc-pvdz_pt_sf_lc
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end
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basis "large2" rel
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oxygen library cc-pvdz_pt_sf_lc
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hydrogen library cc-pvdz_pt_sf_lc
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end
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basis "small"
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oxygen library cc-pvdz_pt_sf_sc
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hydrogen library cc-pvdz_pt_sf_sc
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end
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set "ao basis" fw
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set "large component" large
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set "small component" small
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relativistic
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dyall-mod-dirac
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end
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task scf
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set "ao basis" large2
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unset "large component"
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set "small component" small
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relativistic
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dyall-mod-dirac nesc2e
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end
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task scf
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</PRE>
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<P>
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The second example has oxygen as a relativistic atom and hydrogen nonrelativistic.
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<P>
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<PRE>
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start h2o-dmd2
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geometry units bohr
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symmetry c2v
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O 0.000000000 0.000000000 -0.009000000
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H 1.515260000 0.000000000 -1.058900000
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H -1.515260000 0.000000000 -1.058900000
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end
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basis "ao basis"
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oxygen library cc-pvdz_pt_sf_fw rel
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hydrogen library cc-pvdz
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end
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basis "large component"
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oxygen library cc-pvdz_pt_sf_lc
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end
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basis "small component"
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oxygen library cc-pvdz_pt_sf_sc
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end
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relativistic
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dyall-mod-dirac
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end
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task scf
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</PRE>
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Edoardo Apra
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2004-05-25
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