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Contents 1 **HAMILTONIAN -- Specification of the Hamiltonian 1.1 .LVCORR 1.2 .GAUNT 1.3 .DOSSSS 1.4 .X2C 1.5 .X2Cmmf 1.6 .SPINFREE 1.7 .LEVY-LEBLOND 1.8 .NONREL 1.9 .DFT 1.10 .DFTAUTO 1.11 .LVNEW 1.12 .X2COLD 1.13 .X2C4 1.14 .NOSPIN 1.15 .HFXFAC 1.16 .ALDA 1.17 .OPERATOR 1.18 .INTFLG 1.19 .BSS 1.20 .ONESTEP 1.21 .NOAMFI 1.22 .DO2C4C 1.23 .DO4C2C 1.24 .USE_DF 1.25 .CONT2C 1.26 .ZORA 1.27 .SOLVENT 1.28 .FDE 1.29 .NOSFMU 1.30 .PRINT 1.31 .ONESYS 1.32 .URKBAL 1.33 .FREEPJ 1.34 .VEXTPJ 1.35 .NOSMLV 1.36 .SMLV1C 1.37 .ONECAP 1.38 .ONECNV 1.39 .CMPEIG 2 *DFT 3 *AMFI 4 *FDE 4.1 Sample inputs 4.2 References

**HAMILTONIAN -- Specification of the Hamiltonian

This section defines the electronic Hamiltonian that is to be used. Internally the program will always work with 4-component operators that are expanded using distinct large and small component basis sets. In the transformation to an orthogonal basis set one may, however, combine large and small component functions and/or functions of different symmetry in order to obtain a matrix expansion of e.g. the spinfree modified Dirac equation or the Lévy-Leblond equation ^[1].

In addition one can also modify the Hamiltonian by introduction of an additional operator, e.g. describing an external field. Any additional operator defined in this section must be totally symmetric under both the molecular point group and time reversal symmetry. The latter requirement precludes the introduction of external magnetic fields.

.LVCORR

This keyword gives Dirac-Coulomb Hamiltonian in which (SS|SS) integrals are neglected and replaced by an interatomic SS correction (calculated as a classical repulsion term of (tabulated) small component atomic charges) ^[2].

This is currently the most economical and accurate approximation to the full Dirac-Coulomb Hamiltonian and can certainly be used for the calculation of spectroscopic constants and valence properties; for core properties testing is recommended (see also .LVNEW). This is the default Hamiltonian choice in DIRAC11.

.GAUNT

Add the Gaunt interaction to the Hamiltonian. This will increase the computational time significantly but is important when studying spin-orbit splittings and/or performing accurate studies of light molecules. The current implementation is limited to including the Gaunt interaction in the construction of the Fock matrix and works for Hartree-Fock and DFT. For using Gaunt in combination with DFT see the *DFT section of the manual. Transformation of the Gaunt part of the two electron operator to the MO basis is not yet implemented, for this purpose we recommend the use of the molecular mean field approximation. This can be used for example in MP2/CC/FSCC/IHFSCC calculations with RELCCSD by means of the X2Cmmf Hamiltonian (see also our FAQ page: [1]).

For the relativistic two-component mode (see .X2C keyword) with AMFI contributions it uses both spin-same and spin-other orbit mean-field parts.

.DOSSSS

This keyword gives unmodified Dirac-Coulomb Hamiltonian which was default till the DIRAC11 release. Explicitly including (SS|SS) type Coulomb integrals does give the most accurate description of the system but does increase computational cost significantly. Use this option for high-accuracy calculations, preferably in conjunction with the .GAUNT keyword to also include the Gaunt correction to the two-electron interaction.

.X2C

Keyword referring to the new module for the Exact Two-Component relativistic Hamiltonian. ^{[3] [4]}.

DIRAC runs in 2-component mode. One should combine this option with an AMFI correction to the unscreened spin-orbit operator. This is the default in the development version unless .NOAMFI specified. See also the warning there.

.X2Cmmf

Keyword referring to the 2-component molecular-mean-field Hamiltonian apporach within the new module for the Exact Two-Component relativistic Hamiltonian. ^{[5] [4]}.

DIRAC starts with a 4c-SCF run and performs a transformation to 2-component mode (based on the converged Fock operator) prior to a post-HF correlation step. after the SCF . One can combine this option with the .GAUNT keyword which activates the inclusion of spin-other-orbit contributions in the Hamiltonian. The X2Cmmf-Hamiltonian can at present (DIRAC11 release) only be used for post-HF calculation within the RELCCSD module. Patches for other correlation modules in DIRAC are planned to be released soon. For further information and a detailed guideline please consult our FAQ page.

.SPINFREE

Use Dyall's spinfree Hamiltonian ^[6] to obtain results without spin-orbit coupling for the four-component Hamiltonian in the default restricted kinetic balance scheme. This keyword works also for two-component relativistic Hamiltonians where one can choose between two spin-free schemes - see the .BSS keyword.

Note that this option should not be used for response calculations with time-antisymmetric (magnetic) operators as it will eliminate important contributions. Add more...

.LEVY-LEBLOND

Use the nonrelativistic Lévy-Leblond Hamiltonian ^[7].

Use this option before any additional one-electron operators are specified, because it redefines the metric used in the calculation. Lévy-Leblond provides nonrelativistic mode in DIRAC.

.NONREL

Standard nonrelativistic calculation based on the Schrodinger equation. Should give identical energy results as with the .LEVY-LEBLOND keyword.

DIRAC runs in the 2-component mode.

.DFT

Perform a Kohn-Sham calculation. In the following line you must specify the desired DFT functional.

The functional can either be selected from a set of predefined combinations of exchange and correlation functionals, e.g.:

.DFT
 B3LYP

or composed by specifying GGAKEY followed by a list of the desired functionals together with their weights:

.DFT
 GGAKEY PW86X=1.0 P86C=1.0

Exchange functionals:

• LDA exchange ^[8]: SLATER
• Becke 1988 ^[9]: BECKE
• The correction term to LDA proposed by Becke ^[9]: B88CORR
• Perdew-Wang 1986 ^[10]: PW86X

Correlation functionals:

• The Vosko-Wilk-Nusair LDA correlation energy (VWN5) ^[11]: VWN
• The Lee-Yang-Parr functional ^[12]: LYP
• Perdew 1986 ^[13]: P86C

Some of the predefined combinations (see the Dalton manual for a more complete list):

• LDA
• CAMB3LYP
• BLYP
• B3LYP
• BP86
• BPW91
• PBE
• PBE0

.DFTAUTO

Perform a Kohn-Sham calculation using functionals provided by the XCFun library. In the following line you must specify the desired DFT functional.

Advanced options

.LVNEW

Modification of .LVCORR that obtains the atomic small component charge via a Mulliken analysis instead of the original table look-up. (The problem with the table look-up is that the electrostatics in the molecule will be wrong if you have specified a basis set which does not give the correct small-electron charge because of deficiencies in the core region.)

.X2COLD

One-step Exact (infinite order) 2-Component relativistic Hamiltonian ^[3].

note: in the previous release version Dirac2008 this keyword is called .X2C.: DIRAC runs in the (memory saving) 2-component mode. Note that one should combine this option with the SPINFREE option as X2C will only provide an unscreened (bare nucleus) spin-orbit operator that gives unphysically large spin-orbit contributions. To get a realistic screened spin-orbit operator AMFI is added in the development version unless .NOAMFI specified. See also the warning there.

.X2C4

One-step Exact (infinite order) 2-Component relativistic Hamiltonian ^[3].

DIRAC runs in the 4-component mode. This mode is useful if you wish to restart from a previous 4-component calculation and vice versa. AMFI is added in the development version unless .NOAMFI specified. See also the warning there.

.NOSPIN

Implies .SPINFREE, but also remove all spin-symmetry-breaking (quaternion "imaginary" or "triplet" terms) from property gradients in response calculations. Used for analyzing magnetic properties similarly to how it is done with non-relativistic methods.

.HFXFAC

Weight of exchange in Fock matrix construction.

Default:

.HFXFAC
 1.0

.ALDA

Invoke the adiabatic LDA approximation: anything beyond the Kohn-Sham functional (exchange-correlation kernel etc.) is approximated by LDA (exchange switched off in hybrid functionals). In the development version this keyword has been moved after *DFT.

.OPERATOR

Specification of an additional one-electron operator in the Hamiltonian. The operator must be totally symmetric both under the molecular point group and time reversal symmetry. The field strength of the operator is specified with COMFACTOR. The keyword can be repeated for addition of more than one operator.

See the specification of one-electron operators section for more information and explicit examples.

.INTFLG

Specify what two-electron integrals to include. All other modules use this as the default value.

Default: Include LL, SL, and SS integrals (1 = include; 0 = do not include):

.INTFLG
 1 1 0

.BSS

Use the 2-component relativistic Hamiltonian obtained after the Barysz-Sadlej-Snijders transformation of the Dirac Hamiltonian in the finite basis set.^[14].

Calculations using the 2-component BSS Hamiltonian are running only with large component basis functions.

The user can select order of scalar and spin-orbit terms in the .BSS Hamiltonian.

Both scalar and spin-orbit relativistic effects up to infinite order:

.BSS
 099

Scalar relativistic effects of type "from the beginning" up to the infinite order, no spin-orbit interaction:

.SPINFREE
.BSS
 109

Scalar relativistic effects "from the end" up to the infinite order:

.SPINFREE
.BSS
 009

Traditional scalar relativistic "from the beginning" second-order Douglas-Kroll-Hess Hamiltonian:

.SPINFREE
.BSS
 102

Scalar relativistic "from the end" second-order Douglas-Kroll-Hess Hamiltonian:

.SPINFREE
.BSS
 002

Douglas-Kroll-Hess Hamiltonian with first-order spin-orbit and second-order scalar relativistic effects:

.BSS
 012

Douglas-Kroll-Hess Hamiltonian with second-order spin-orbit and second-order scalar relativistic effects:

.BSS
 022

Only in the development version of DIRAC

In this two-component variational scheme is possible to combine external AMFI spin-orbit terms^[15] together with BSS integrals - see the *AMFI section. Note that AMFI provides only one-center integrals.

Infinite order scalar and spin-orbit terms, (one-electron) mean-field spin-same-orbit (MFSSO2) from AMFI

.BSS
2999

This is in fact the 'maximum' of two-component relativity, resembling Dirac-Coulomb Hamiltonian.

Second-order Douglas-Kroll-Hess spin-free from 'the beginning' with first order spin-orbit (SO1) term plus mean-field spin-same-orbit (MFSSO) from AMFI

.BSS
2112

SO1 may come either from BSS-transformation or from AMFI. In the latter case it is only for one-center and for point nucleus.

Infinite order scalar and spin-orbit terms, (one-electron) mean-field spin-same and spin-other-orbit (MFSO2) from AMFI

.BSS
3999

This mimics the Dirac-Coulomb-Gaunt Hamiltonian, as the spin-other-orbit comes from the Gaunt interaction term.

DIRAC enables switching-off AMFI spin-orbit contributions from various centers. See keyword .NOAMFIC.

Infinite order scalar terms "from the beginning", (one-electron) spin orbit (SO1) and the mean-field spin-same-orbit (MFSSO2) terms exclusively from AMFI

.BSS
4109

Second order (DKH) scalar terms "from the beginning", (one-electron) spin orbit (SO1) and the mean-field spin-same-orbit (MFSSO2) terms exclusively from AMFI

.BSS
4102

Infinite order scalar terms "from the end", (one-electron) spin orbit (SO1) and the mean-field spin-same-orbit and spin-other-orbit (MFSO2) terms from AMFI

.BSS
5009

Infinite order scalar terms "from the end", and the (one-electron) spin orbit term (SO1) from AMFI

.BSS
6009

AMFI one-electron spin-orbit terms - SO1 - are currently for the point nucleus.

For the BSS value 'axyz' of y=0, DIRAC employs spin-free picture change transformation of property operators, although the system is not in the boson (spin-free) symmetry for a>1.

Rough first order, DKH1 (not recommended for practical calculations)

.BSS
111

For comparison only between BSS-SO1 and AMFI-SO1 integrals (point nucleus only) use

.BSS
001

.ONESTEP

Together with the .BSS keyword - for the infinite order only - invokes the one-step infinite order method (which is otherwise called by .X2C, .X2C4 keywords).

.NOAMFI

Do not include the AMFI contribution in the development version where AMFI is the default. This holds also for keywords .X2C and .X2C4). In the DIRAC08 distribution version .NOAMFI is the default.

Warning: missing AMFI contribution (for .X2C, .X2C4 and .BSS keywords) may lead to overestimation of spin-orbit effects since these would be represented by one-electron terms only, without two-electron shielding.

.DO2C4C

Only in the development version of DIRAC

After iterations at the two-component level ascend to the four-component level.

.DO4C2C

Only in the development version of DIRAC

After iterations at the four-component level do the relativistic transformation to the two-component level.

.USE_DF

Only in the development version of DIRAC

After the four-component DC-SCF method do the infinite order transformation (either one- or two-step) upon the Fock-Dirac matrix. Otherwise it is transforming Dirac bare nucleus.

Used only with keyword .DO4C2C.

.CONT2C

Only in the development version of DIRAC

After four-component DC-SCF continue with two-component iterations.

Used only with keyword .DO4C2C.

Integer (3,4,5) should be specified in free format on the line following .CONT2C:

.CONT2C
 3

.ZORA

Use the zeroth-order regular approximation ^{[16] [17] [1]} of the Dirac Hamiltonian in the Hartree-Fock procedure. Works only for closed-shell systems. The implementation offers only little computational advantages and is intended chiefly for comparisons of methodology. Note that the combination .SPINFREE and .ZORA gives a spinfree formalism that differs from the conventional spinfree ZORA formulation. Two integers should be specified in free format on the line following .ZORA:

.ZORA
 1 1

The first number indicates whether the density is to be normalized over the 2-component (0; ZORA) or 4-component metric (1; ZORA4).

The second number specifies whether the orbital energies should be unmodified (0; normal ZORA) or scaled (1; scaled ZORA).

.SOLVENT

Model solvent effects by placing the molecule in a spherical cavity in a dielectric continuum.

.FDE

Activates the frozen density embedding (FDE) functionality. Options can be specified under the *FDE menu.

In order to use FDE, the user must generated an embedding potential with the ADF code (see the developer's website [[2]] for further information).

.NOSFMU

Only in the development version of DIRAC

In spinfree correlated calculations group multiplication tables are by default set up as direct products of spatial and spin symmetries. This flag turns off this, and so the spinfree case is treated similar to the spin-orbit case.

Programmers options

.PRINT

Print level.

Default:

.PRINT
 0

.ONESYS

Ignore two-particle interactions.

The keyword ensures the diagonalization of the bare nuclei Hamiltonian matrix without proceeding to the iterative SCF method.

.URKBAL

Unrestricted kinetic balance.

The default is restricted kinetic balance. This is imposed by deleting unphysical solutions from the free particle positronic spectrum. This leads to a 1:1 ratio of electronic and positronic solutions. This preprojection is sensitive to linear dependencies and should therefore preferably be used in conjunction with the spherical transformation of both large and small components.

.FREEPJ

Project out all free positronic solutions from the MO space.

.VEXTPJ

Project out all external field positronic solutions from the MO space.

.NOSMLV

Delete SS nuclear attraction integrals. This will take out contributions to the one-electron spin-orbit interaction and the Darwin interaction.

.SMLV1C

Neglect potential for multi-center SS blocks, i.e. multi-center SS nuclear attraction integrals and multi-center SS two-electron integrals.

.ONECAP

Consider taking the .SMLV1C model one step further. Only one-center contributions to the LS and SS two-electron integrals and SS nuclear attraction integrals are calculated explicitly. The electrostatic effects of the terms neglected this way are included by calculating the classical repulsion from small component charges based on a Mulliken population analysis. Note that we therefore only need to calculate the derivative of the LL integrals when calculating the molecular gradient.

.ONECNV

.ONECNV
 THRESHOLD

Employ the one-center model as given by .ONECAP until convergence to a specified THRESHOLD, whereafter the full set of two-electron integrals will be used. This threshold applies to whatever convergence criteria has been selected (.EVCCNV, .ERGCNV, or .FCKCNV).

.CMPEIG

When some two-component relativistic Hamiltonian is chosen, compare eigenvalues between the 'parent' four-component Dirac and derived two-component one-electron Hamiltonians.

For the infinite order (one- and two-step) two-component Hamiltonians eigenvalues are identical with four-component Dirac counterparts. For the second-order (and lower order) Douglas-Krol-Hess Hamiltonian they slightly differ.

*DFT

In this link the directives for modifying the DFT calculation are described (e.g. modifying grid, using ALDA and much more)

*AMFI

AMFI directives

*FDE

Frozen density embedding (FDE) directives.

Sample inputs

Some examples of the Hamiltonian input.

References

1. ↑ ^{1.0 1.1} L. Visscher and T. Saue, Approximate relativistic electronic structure methods based on the quaternion modified Dirac equation, J. Chem. Phys. 113, 3996 (2000) electronic version.
2. ↑ L. Visscher, Approximate molecular Dirac-Coulomb calculations using a simple Coulombic correction, Theor. Chem. Acc. 98, 68 (1997) electronic version.
3. ↑ ^{3.0 3.1 3.2} M. Iliaš and T. Saue, An infinite-order two-component relativistic Hamiltonian by a simple one-step transformation, J. Chem. Phys. 126, 064102 (2007) electronic version.
4. ↑ ^{4.0 4.1} S. Knecht and T. Saue, tba, xxx xxx, Universities of Strasbourg and Odense (2010-2011) .
5. ↑ J. Sikkema, L. Visscher, T. Saue, and M. Iliaš, The molecular mean-field approach for correlated relativistic calculations, J. Chem. Phys. 131, 124116 (2009) .
6. ↑ K. G. Dyall, An exact separation of the spin-free and spin-dependent terms of the Dirac-Coulomb-Breit hamiltonian, J. Chem. Phys. 100, 2118 (1994) electronic version.
7. ↑ J.-M. Lévy-Leblond, Nonrelativistic particles and wave equations, Commun. Math. Phys. 6, 286 (1967) electronic version.
8. ↑ P. A. M. Dirac, Note on Exchange Phenomena in the Thomas Atom, Proc. Roy. Soc. London 26, 376 (1930) .
9. ↑ ^{9.0 9.1} A. D. Becke, Density-Functional Exchange-Energy Approximation With Correct Asymptotic Behavior, Phys. Rev. A 38, 3098 (1988) electronic version.
10. ↑ J. P. Perdew and Y. Wang, Accurate and Simple Density Functional for the Electronic Exchange Energy: Generalized Gradient Approximation, Phys. Rev. B 33, 8800 (1986) electronic version.
11. ↑ S. J. Vosko, L. Wilk, and M. Nusair, Accurate Spin-Dependent Electron Liquid Correlation Energies for Local Spin Density Calculations: A Critical Analysis, Can. J. Phys. 58, 1200 (1980) .
12. ↑ C. Lee, W. Yang, and R. G. Parr, Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density, Phys. Rev. B 37, 785 (1988) electronic version.
13. ↑ J. P. Perdew, Density-Functional Approximation for the Correlation Energy of the Inhomogenous Electron Gas, Phys. Rev. B 33, 8822 (1986) electronic version.
14. ↑ M. Iliaš and H. J. Aa. Jensen and V. Kello and B. O. Roos and M. Urban, Theoretical study of PbO and the PbO anion, Chem.Phys.Lett. 408, 210 (2005) electronic version.
15. ↑ M. Iliaš and V. Kello and L. Visscher and B. Schimmelpfennig, Inclusion of mean-field spin–orbit effects based on all-electron two-component spinors: Pilot calculations on atomic and molecular properties, J.Chem.Phys. 115, 9667 (2001) electronic version.
16. ↑ E. van Lenthe, E. J. Baerends, and J. G. Snijders, Relativistic total energies using regular approximations, J. Chem. Phys. 101, 9783 (1994) electronic version.
17. ↑ E. van Lenthe, J. G. Snijders, and E. J. Baerends, The zero-order regular approximation for relativistic effects: The effect of spin-orbit coupling in closed shell molecules, J. Chem. Phys. 105, 6505 (1996) electronic version.