1% 2% $Id$ 3% 4\label{sec:rel} 5All methods which include treatment of relativistic effects are ultimately 6based on the Dirac equation, which has a four component wave function. The 7solutions to the Dirac equation describe both positrons (the ``negative 8energy'' states) and electrons (the ``positive energy'' states), as well as 9both spin orientations, hence the four components. The wave function may be 10broken down into two-component functions traditionally known as the large 11and small components; these may further be broken down into the spin 12components. 13 14The implementation of approximate all-electron relativistic methods in 15quantum chemical codes requires the removal of the negative energy states 16and the factoring out of the spin-free terms. Both of these may be achieved 17using a transformation of the Dirac Hamiltonian known in general as a 18Foldy-Wouthuysen transformation. Unfortunately this transformation cannot be 19represented in closed form for a general potential, and must be 20approximated. One popular approach is that originally formulated by Douglas 21and Kroll\footnote{M.~Douglas and N.~M.~Kroll, Ann. Phys. (N.Y.) {\bf 82}, 2289 (1974)} and developed by Hess\footnote{B.A.~Hess, Phys.~Rev.~A~{\bf 32}, 23756 (1985); {\bf 33}, 3742 (1986)}. This approach decouples the positive and 24negative energy parts to second order in the external potential (and also 25fourth order in the fine structure constant, $\alpha$). Other approaches include 26the Zeroth Order Regular Approximation (ZORA)\footnote{C.~Chang, M.~Pelissier, 27M.~Durand, Physica Scripta ~{\bf 34}, 294 (1986); E.~van Lenthe, ~{\it The ZORA Equation}, 28doctoral thesis, Vrije Universiteit, Amsterdam (1996); S.~Faas, J.G.~Snijders, 29J.H.~van Lenthe, E.~van Lenthe, and E.J.~Baerends, Chem.~Phys.~ Lett.~{\bf 246}, 632 (1995).} 30and modification of the Dirac equation by Dyall\footnote{K.~G.~Dyall, 31J.~Chem.~Phys.~{\bf 100}, 2118 (1994)}, and involves an exact FW 32transformation on the atomic basis set level\footnote{K.~G.~Dyall, 33J.~Chem.~Phys.~{\bf 106}, 9618 (1997); K.~G.~Dyall and T.~Enevoldsen, 34J.~Chem.~Phys.~{\bf 111}, 10000 (1999).}. 35 36Since these approximations only modify the integrals, they can in principle 37be used at all levels of theory. At present the Douglas-Kroll and ZORA 38implementations can be used at all levels of theory whereas 39Dyall's approach is currently available at the Hartree-Fock level. 40The derivatives have been implemented, allowing both methods to be used in 41geometry optimizations and frequency calculations. 42 43The \verb+RELATIVISTIC+ directive provides input for the implemented relativistic 44approximations and is a compound directive that encloses additional directives 45specific to the approximations: 46\begin{verbatim} 47 RELATIVISTIC 48 [DOUGLAS-KROLL [<string (ON||OFF) default ON> \ 49 <string (FPP||DKH||DKFULL||DK3||DK3FULL) default DKH>] || 50 ZORA [ (ON || OFF) default ON ] || 51 DYALL-MOD-DIRAC [ (ON || OFF) default ON ] 52 [ (NESC1E || NESC2E) default NESC1E ] ] 53 [CLIGHT <real clight default 137.0359895>] 54 END 55\end{verbatim} 56 57Only one of the methods may be chosen at a time. If both methods are found 58to be on in the input block, NWChem will stop and print an error message. 59There is one general option for both methods, the definition of the speed 60of light in atomic units: 61 62\begin{verbatim} 63 CLIGHT <real clight default 137.0359895> 64\end{verbatim} 65 66The following sections describe the optional sub-directives that 67can be specified within the \verb+RELATIVISTIC+ block. 68 69\section{Douglas-Kroll approximation} 70\label{sec:douglas-kroll} 71 72The spin-free and spin-orbit one-electron Douglas-Kroll 73approximation have been implemented. The use of relativistic effects 74from this Douglas-Kroll approximation can be invoked by specifying: 75 76\begin{verbatim} 77 DOUGLAS-KROLL [<string (ON||OFF) default ON> \ 78 <string (FPP||DKH||DKFULL|DK3|DK3FULL) default DKH>] 79\end{verbatim} 80 81The \verb+ON|OFF+ string is used to turn on or off the 82Douglas-Kroll approximation. By default, if the \verb+DOUGLAS-KROLL+ 83keyword is found, the approximation will be used in the calculation. 84If the user wishes to calculate a non-relativistic quantity after turning 85on Douglas-Kroll, the user will need to define a new \verb+RELATIVISTIC+ 86block and turn the approximation \verb+OFF+. The user could also simply 87put a blank \verb+RELATIVISTIC+ block in the input file and all options 88will be turned off. 89 90The \verb+FPP+ is the approximation based on free-particle projection 91operators\footnote{B.A.~Hess, Phys.~Rev.~A~{\bf 32}, 756 (1985)} whereas the 92\verb+DKH+ and \verb+DKFULL+ approximations are based on external-field 93projection operators\footnote{B.A.~Hess, Phys.~Rev.~A~{\bf 33}, 3742 (1986)}. 94The latter two are considerably better approximations than the former. \verb+DKH+ 95is the Douglas-Kroll-Hess approach and is the approach that is generally 96implemented in quantum chemistry codes. \verb+DKFULL+ includes certain 97cross-product integral terms ignored in the \verb+DKH+ approach (see for example 98H\"{a}berlen and R\"{o}sch\footnote{O.D.~H\"{a}berlen, N.~R\"{o}sch, 99Chem.~Phys.~Lett.~{\bf 199}, 491 (1992)}). The third-order Douglas-Kroll 100approximation has been implemented by T. Nakajima and K. Hirao\footnote{T. Nakajima 101and K. Hirao, Chem.~Phys.~Lett.~{\bf 329}, 5111 (2000); T. Nakajima and K. Hirao, 102J.~Chem.~Phys.~{\bf 113}, 7786 (2000)}. This approximation can be called using 103\verb+DK3+ (DK3 without cross-product integral terms) or \verb+DK3FULL+ (DK3 with 104cross-product integral terms). 105 106The contracted basis sets used in the calculations should reflect the relativistic 107effects, i.e. one should use contracted basis sets which were generated using the 108Douglas-Kroll Hamiltonian. Basis sets that were contracted using the 109non-relativistic (Sch\"{o}dinger) Hamiltonian WILL PRODUCE ERRONEOUS RESULTS for 110elements beyond the first row. See appendix \ref{sec:knownbasis} for available 111basis sets and their naming convention. 112 113NOTE: we suggest that spherical basis sets are used in the calculation. The use of 114high quality cartesian basis sets can lead to numerical inaccuracies. 115 116In order to compute the integrals needed for the Douglas-Kroll approximation 117the implementation makes use of a fitting basis set (see literature given 118above for details). The current code will create this fitting basis set 119based on the given {\tt "ao basis"} by simply uncontracting that basis. This 120again is what is commonly implemented in quantum chemistry codes that 121include the Douglas-Kroll method. Additional flexibility is available to 122the user by explicitly specifying a Douglas-Kroll fitting basis 123set. This basis set must be named {\tt "D-K basis"} (see Chapter 124\ref{sec:basis}). 125 126\section{Zeroth Order regular approximation (ZORA)} 127\label{sec:zora} 128 129The spin-free and spin-orbit one-electron zeroth-order regular approximation (ZORA) 130have been implemented. The use of relativistic effects with ZORA 131can be invoked by specifying: 132 133\begin{verbatim} 134 ZORA [<string (ON||OFF) default ON> 135\end{verbatim} 136 137The \verb+ON|OFF+ string is used to turn on or off ZORA. 138By default, if the \verb+ZORA+ keyword is found, the approximation 139will be used in the calculation. If the user wishes to calculate 140a non-relativistic quantity after turning on ZORA, the user 141will need to define a new \verb+RELATIVISTIC+ block and turn 142the approximation \verb+OFF+. The user can also simply put 143a blank \verb+RELATIVISTIC+ block in the input file and all options 144will be turned off. 145 146\section{Dyall's Modified Dirac Hamitonian approximation} 147\label{sec:dyall-mod-dir} 148 149The approximate methods described in this section are all based on Dyall's 150modified Dirac Hamiltonian. This Hamiltonian is entirely equivalent to the 151original Dirac Hamiltonian, and its solutions have the same properties. 152The modification is achieved by a transformation on the small component, 153extracting out \hbox{$\sigma\cdot{\bf p}/2mc$}. This gives the modified small 154component the same symmetry as the large component, and in fact it differs 155from the large component only at order $\alpha^2$. The advantage of the 156modification is that the operators now resemble the operators of the 157Breit-Pauli Hamiltonian, and can be classified in a similar fashion into 158spin-free, spin-orbit and spin-spin terms. It is the spin-free terms which 159have been implemented in NWChem, with a number of further approximations. 160 161The first is that the negative energy states are removed by a normalized 162elimination of the small component (NESC), which is equivalent to an exact 163Foldy-Wouthuysen (EFW) transformation. The number of components in the wave 164function is thereby effectively reduced from 4 to 2. NESC on its own does 165not provide any advantages, and in fact complicates things because the 166transformation is energy-dependent. The second approximation therefore 167performs the elimination on an atom-by-atom basis, which is equivalent to 168neglecting blocks which couple different atoms in the EFW transformation. 169The advantage of this approximation is that all the energy dependence can be 170included in the contraction coefficients of the basis set. The tests which 171have been done show that this approximation gives results well within 172chemical accuracy. The third approximation neglects the commutator of the 173EFW transformation with the two-electron Coulomb interaction, so that the 174only corrections that need to be made are in the one-electron integrals. 175This is the equivalent of the Douglas-Kroll(-Hess) approximation as it is 176usually applied. 177 178The use of these approximations can be invoked with the use of the 179\verb+DYALL-MOD-DIRAC+ directive in the \verb+RELATIVISTIC+ directive block. 180The syntax is as follows. 181 182\begin{verbatim} 183 DYALL-MOD-DIRAC [ (ON || OFF) default ON ] 184 [ (NESC1E || NESC2E) default NESC1E ] 185\end{verbatim} 186 187The \verb+ON|OFF+ string is used to turn on or off the 188Dyall's modified Dirac approximation. By default, if the \verb+DYALL-MOD-DIRAC+ 189keyword is found, the approximation will be used in the calculation. 190If the user wishes to calculate a non-relativistic quantity after turning 191on Dyall's modified Dirac, the user will need to define a new 192\verb+RELATIVISTIC+ 193block and turn the approximation \verb+OFF+. The user could also simply 194put a blank \verb+RELATIVISTIC+ block in the input file and all options 195will be turned off. 196 197Both one- and two-electron approximations are available 198\verb+NESC1E || NESC2E+, and both have 199analytic gradients. The one-electron approximation is the default. 200The two-electron approximation specified by \verb+NESC2E+ has some sub 201options which are placed on the same logical line as the 202\verb+DYALL-MOD-DIRAC+ directive, with the following syntax: 203 204\begin{verbatim} 205 NESC2E [ (SS1CENT [ (ON || OFF) default ON ] || SSALL) default SSALL ] 206 [ (SSSS [ (ON || OFF) default ON ] || NOSSSS) default SSSS ] 207\end{verbatim} 208 209The first sub-option gives the capability to limit the two-electron 210corrections to those in which the small components in any density must be on 211the same center. This reduces the $(LL|SS)$ contributions to at most 212three-center integrals and the $(SS|SS)$ contributions to two centers. For a 213case with only one relativistic atom this option is redundant. The second 214controls the inclusion of the $(SS|SS)$ integrals which are of order 215$\alpha^4$. For light atoms they may safely be neglected, but for heavy 216atoms they should be included. 217 218In addition to the selection of this keyword in the \verb+RELATIVISTIC+ 219directive block, it is necessary to supply basis sets in addition to the 220\verb+ao basis+. For the one-electron approximation, three basis sets are 221needed: the atomic FW basis set, the large component basis set and the small 222component basis set. The atomic FW basis set should be included in the 223\verb+ao basis+. 224The large and small components should similarly be incorporated 225in basis sets named \verb+large component+ and \verb+small component+, 226respectively. For the two-electron approximation, only two basis sets are 227needed. These are the large component and the small component. The large component 228should be included in the \verb+ao basis+ and the small component 229is specified separately as \verb+small component+, as for the one-electron 230approximation. This means that the two approximations can {\it not} be run 231correctly without changing the \verb+ao basis+, and it is up to the user to 232ensure that the basis sets are correctly specified. 233 234There is one further requirement in the specification of the basis sets. In 235the \verb+ao basis+, it is necessary to add the \verb+rel+ keyword either to the 236\verb+basis+ directive or the library tag line (See below for examples). 237The former marks the basis 238functions specified by the tag as relativistic, the latter marks the whole 239basis as relativistic. The marking is actually done at the unique shell 240level, so that it is possible not only to have relativistic and 241nonrelativistic atoms, it is also possible to have relativistic and 242nonrelativistic shells on a given atom. This would be useful, for example, 243for diffuse functions or for high angular momentum correlating functions, 244where the influence of relativity was small. The marking of shells as 245relativistic is necessary to set up a mapping between the ao basis and the 246large and/or small component basis sets. For the one-electron approximation 247the large and small component basis sets MUST be of the same size and 248construction, i.e. differing only in the contraction coefficients. 249 250It should also be noted that the relativistic code will NOT work with basis 251sets that contain sp shells, nor will it work with ECPs. Both of these are 252tested and flagged as an error. 253 254Some examples follow. The first example sets up the data for relativistic 255calculations on water with the one-electron approximation and the 256two-electron approximation, using the library basis sets. 257 258\begin{verbatim} 259 start h2o-dmd 260 261 geometry units bohr 262 symmetry c2v 263 O 0.000000000 0.000000000 -0.009000000 264 H 1.515260000 0.000000000 -1.058900000 265 H -1.515260000 0.000000000 -1.058900000 266 end 267 268 basis "fw" rel 269 oxygen library cc-pvdz_pt_sf_fw 270 hydrogen library cc-pvdz_pt_sf_fw 271 end 272 273 basis "large" 274 oxygen library cc-pvdz_pt_sf_lc 275 hydrogen library cc-pvdz_pt_sf_lc 276 end 277 278 basis "large2" rel 279 oxygen library cc-pvdz_pt_sf_lc 280 hydrogen library cc-pvdz_pt_sf_lc 281 end 282 283 basis "small" 284 oxygen library cc-pvdz_pt_sf_sc 285 hydrogen library cc-pvdz_pt_sf_sc 286 end 287 288 set "ao basis" fw 289 set "large component" large 290 set "small component" small 291 292 relativistic 293 dyall-mod-dirac 294 end 295 296 task scf 297 298 set "ao basis" large2 299 unset "large component" 300 set "small component" small 301 302 relativistic 303 dyall-mod-dirac nesc2e 304 end 305 306 task scf 307\end{verbatim} 308 309The second example has oxygen as a relativistic atom and hydrogen nonrelativistic. 310 311\begin{verbatim} 312 start h2o-dmd2 313 314 geometry units bohr 315 symmetry c2v 316 O 0.000000000 0.000000000 -0.009000000 317 H 1.515260000 0.000000000 -1.058900000 318 H -1.515260000 0.000000000 -1.058900000 319 end 320 321 basis "ao basis" 322 oxygen library cc-pvdz_pt_sf_fw rel 323 hydrogen library cc-pvdz 324 end 325 326 basis "large component" 327 oxygen library cc-pvdz_pt_sf_lc 328 end 329 330 basis "small component" 331 oxygen library cc-pvdz_pt_sf_sc 332 end 333 334 relativistic 335 dyall-mod-dirac 336 end 337 338 task scf 339\end{verbatim} 340