# **HAMILTONIAN¶

## Introduction¶

This section defines the electronic Hamiltonian that is to be used. Within the Born-Oppenheimer approximation the generic form of the electronic Hamiltonian is

$H = \sum_{i}\ h\left(i\right) + \frac{1}{2}\sum_{i\ne j}\ g\left(i,j\right) + V_{NN}; \quad V_{NN} = \frac{1}{2}\sum_{A\ne B} \frac{Z_A Z_B}{R_{AB}}$

where $$V_{NN}$$ is the operator of the repulsion of classical nuclei. The one-electron hamiltonian $$h\left(i\right)$$ splits into the free-electron Hamiltonian $$h_0$$ and the electron-nucleus interaction $$V_{eN}$$. In the non-relativistic case the free-electron Hamiltonian is simply the kinetic energy operator, whereas a rest mass term is added in the relativistic case, e.g. for the Dirac (bare-nucleus) Hamiltonian

$h_D = \beta mc^2 + c\left(\alpha\cdot\mathbf{p}\right) + V_{eN}$

In the non-relativistic case the two-electron operator $$g\left(i,j\right)$$ is the instantaneous Coulomb interaction.

$g^{Coulomb}\left(1,2\right)=\frac{1}{r_{12}}$

In the relativistic case, the two-electron interaction is vastly more complex, including magnetic interactions as well as retardation effects. In the relativistic framework the instantenous Coulomb-interaction is the zeroth-order term in an expansion in $$c^{-2}$$ of the full Lorentz invariant two-electron interaction. Note, however, that although the mathematical form of the Coulomb term is the same as in the non-relativistic domain, the physical content is different. For instance, in the relativistic domain the Coulomb term contains the spin-same orbit (SSO) interaction. The first-order term is the Breit interaction

$g^{Breit}\left(1,2\right)=-\frac{c{\alpha}_{1}\cdot c{\alpha}_{2}}{2c^{2}r_{12}}-\frac{\left(c{\alpha}_{1}\cdot\mathbf{r_{12}}\right)\left(c{\alpha}_{2}\cdot\mathbf{r}_{12}\right)}{2c^{2}r_{12}^{3}}$

which can be rearranged to

$g^{Breit}\left(1,2\right)=g^{Gaunt}\left(1,2\right)+g^{gauge}\left(1,2\right)=-\frac{c{\alpha}_{1}\cdot c{\alpha}_{2}}{c^{2}r_{12}}-\frac{\left(c{\alpha}_{1}\cdot{\nabla}_{1}\right)\left(c{\alpha}_{2}\cdot{\nabla}_{2}\right)r_{12}}{2c^{2}}$

The Gaunt term, which contains the spin-other orbit interaction, is implemented at the SCF level in DIRAC.

The Dirac Hamiltonian (or effective one-electron Hamiltonians such as the Fock or Kohn-Sham operators) give electronic solutions of both positive and negative energy. 2-component relativistic Hamiltonians can be generated by a unitary decoupling transformation. The exact decoupling gives the eXact 2-Component Hamiltonian (X2C) (we use [Ilias2007]), whereas the Zeroth-Order Regular Approximation (ZORA), Douglas-Kroll-Hess (DKH) and Barysz-Sadlej-Snijders (BSS) Hamiltonians are generated by approximate decouplings. For more a detailed discussion of relativistic Hamiltonians, see [Saue2011].

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 spin-free modified Dirac equation or the Lévy-Leblond equation [Visscher2000].

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.

## General¶

### .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.

## 4-component Hamiltonians¶

The default Hamiltonian of DIRAC is the Dirac-Coulomb Hamiltonian using the Simple Coulombic Correction (see .LVCORR). The Dirac-Coulomb Hamiltonian formally has no bound solution and is therefore embedded in projection operators. By default, these are the projection operators obtained iteratively in the SCF process.

### .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 a 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/Tutorial pages).

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

### .DOSSSS¶

This keyword gives unmodified Dirac-Coulomb Hamiltonian which was the default untill 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 .GAUNT to also include the Gaunt correction to the two-electron interaction.

### .LVCORR¶

This keyword activates the 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) [Visscher1997a] .

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 since DIRAC11.

### .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.)

### .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.

### .INTFLG¶

Specify what two-electron integrals to include. All other modules use this as the default value. By default include LL and SL, exclude SS integrals (1 = include; 0 = do not include):

.INTFLG
1 1 0

## Turning off spin dependence¶

### .SPINFREE¶

Use Dyall’s spin-free Hamiltonian, Ref. [Dyall1994] , 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.

### .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.

### .NOSFMU¶

In spin-free 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 spin-free case is treated similar to the spin-orbit case.

Warning

only in development version

### .SPINF2¶

Warning

documentation missing

## Advanced: Other projection operators¶

### .FREEPJ¶

Project out all negative-energy solutions of the free-particle one-electron Dirac Hamiltonian from the MO space. This corresponds to the Feynmann (1948) basis for QED and no-pair Hamiltonians.

### .VEXTPJ¶

Project out all negative-energy solutions of the bare-nucleus one-electron Dirac Hamiltonian from the MO space. This corresponds to the Furry (1951) basis for QED and no-pair Hamiltonians.

## Exact 2-component (X2C) Hamiltonians¶

### .X2C¶

This keyword activates the Exact 2-Component one-electron Hamiltonian [Ilias2007] based on its implementation in the module X2Cmod, Refs. [Knecht2010] and [Knecht2014]. It yields exactly the same energies as the 4-component one-electron Hamiltonian in the same large component basis set with restricted kinetic balance, and it yields the same division in spinfree (scalar relativistic) and spin-orbit like contributions as the 4-component one-electron Hamiltonian. The two-electron part of the Hamiltonian is treated on a non-relativistic level.

One should therefore combine X2C with a correction to the unscreened one-electron spin-orbit operator. It is default with .X2C to include the atomic mean field two-electron spin-orbit correction “AMFI”, unless .NOAMFI is specified. To use the spinfree version of X2C with only scalar relativistic terms one needs to add the keyword .SPINFREE (.SPINFREE implies .NOAMFI). The spinfree X2C is equivalent to and numerically yields the same numbers as X2C in the quantum chemistry packages CFour, Turbomole, and Molcas.

An overview of the (local) X2C approach is given in the corresponding tutorial section X2C and local X2C.

### .X2Cmmf¶

This keyword activates the 2-component molecular-mean-field (X2C) Hamiltonian approach [Sikkema2009] within the module X2Cmod, Ref. [Knecht2014].

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 .GAUNT which activates the inclusion of spin-other-orbit contributions in the Hamiltonian. The X2Cmmf-Hamiltonian can at present only be used for post-HF calculation within the RELCCSD module. Patches for other correlation modules in DIRAC will be part of the Dirac2014 release. See also the Molecular mean-field X2C tutorial, section Molecular mean-field X2C, for further information.

### .BSS¶

Use the 2-component relativistic Hamiltonian obtained after the Barysz–Sadlej–Snijders transformation of the Dirac Hamiltonian in the finite basis set, see Ref. [Ilias2005]. Calculations using the 2-component BSS Hamiltonian are running only with large component basis functions.

## Approximate 2-component Hamiltonians¶

Please note that these are only tested for energies and generally do not work for properties !

### .ZORA¶

Use the zeroth-order regular approximation ( [vanLenthe1994], [vanLenthe1996], [Visscher2000] ) 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 approximate Hamiltonians methodologies. Note that the combination .SPINFREE and .ZORA gives a spin-free formalism that differs from the conventional spin-free 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).

### .DKH1¶

First-order Douglas-Kroll Hamiltonian

### .DKH2¶

Second-order Douglas-Kroll Hamiltonian

## Non-relativistic Hamiltonians¶

### .LEVY-LEBLOND¶

Use the nonrelativistic Levy-Leblond Hamiltonian [Levy1967].

Use this option before any additional one-electron operators are specified, because it redefines the metric used in the calculation.

### .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 spin-free mode, which in fact represents the traditional one-component mode.

## Effective core potentials¶

### .ECP¶

Perform an effective core potential calculation. The ECP parameters should be set in the MOL file. Both spin-orbit and spin-free calculations are available by specifying spin-orbit (SO) parameters in the MOL file. With SO parameters, the 2-component spin-orbit calculation is conducted, whereas the spin-free (1-component) calculation is performed by omitting the SO part in the ECP parameter. (See How to specify ECP parameters in mol files)

## External fields/Environment¶

### .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 One-electron operators section for more information and explicit examples.

### .PCM¶

Model solvent effects by placing the molecule in a cavity in a dielectric continuum. The cavity is shaped on the actual geometry of the solute, the full molecular electrostatic potential is used. DIRAC makes use of the external PCMSolver module.

### .SOLVENT¶

Model solvent effects by placing the molecule in a spherical cavity in a dielectric continuum. The solute electrostatic potential is represented in terms of a truncated multipolar expansion.

### .FDE¶

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

In order to use FDE the user must have generated an embedding potential and/or frozen densities for the environment, either directly with the ADF code (see the developer’s website http://www.scm.com for further information) or via the PyADF scripting framework (see [Jacob2011] or visit the developer’s website http://pyadf.org for further information).

### .PEQM¶

Activates the polarizable embedding model. Options can be specified under the *PEQM menu.

### .CAP¶

Complex Absorption Potential (only in the development version).

## Kohn-Sham Hamiltonian¶

### .DFT¶

Perform a Kohn–Sham density functional theory 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

Alternatively, it can be composed by specifying GGAKEY followed by a list of the desired functionals together with their weights:

.DFT
GGAKEY PW86X=1.0 P86C=1.0

GGAKEY also allows the definition of (global) hybrid functionals, for instance B3LYP can be specified as:

.DFT
GGAKEY Slater=0.8 Becke=0.72 HF=0.2 VWN=0.19 LYP=0.81

where HF indicates weight of Hartree-Fock exchange 20%. It is also possible to specify long-range corrected or Coulomb-attenuated functionals using the CAM keyword, e.g. CAMB3LYP is predefined but can also be specified as:

.DFT
CAM p:alpha=0.19 p:beta=0.46 p:mu=0.33 x:slater=1 x:becke=1 c:lyp=0.81 c:vwn5=0.19

Coulomb-attenuated functionals are based on a separation of the two-electron interaction into a long- and short-range part

$\frac{1}{r_{12}}=\frac{\left[\alpha+\beta\ erf\left(\mu r_{12}\right)\right]}{r_{12}} - \frac{1-\left[\alpha+\beta\ erf\left(\mu r_{12}\right)\right]}{r_{12}}$

For CAM or long-range corrected functionals this separation is only invoked in the evaluation of exchange. The above CAM input first reads the three parameters(p) $$\alpha$$ (alpha), $$\beta$$ (beta) and $$\mu$$ (mu), followed by exchange (x) and correlation (c) functionals with their respective weights. Standard exchange functionals are automatically short-range corrected following the approach of [IIkura2001]. The activation of this separation for correlation as well leads to long-range WFT/short-range DFT methods, as represented by MP2-srDFT in DIRAC.

### .DFTAUTO¶

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

### .HFXFAC¶

Weight of exchange in Fock matrix construction. Default:

.HFXFAC
1.0

### .HFXATT¶

Warning

documentation missing

### .HFXMU¶

Warning

documentation missing

## Advanced modification of the 4-component Hamiltonian¶

### .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.

### .JZOUT¶

Print out Jz MO matrices for the diagonalization into own formatted files (suitable for testing). Only for the linear symmetry. Programmer’s option.

### .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¶

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:

.ONECNV
THRESHOLD

This threshold applies to whatever convergence criteria has been selected (.EVCCNV, .ERGCNV or .FCKCNV).

## Advanced BSS keywords/Experimental¶

### .X2COLD¶

One-step Exact (infinite order) 2-Component relativistic Hamiltonian [Ilias2007]. This keyword was called .X2C prior to DIRAC10. DIRAC runs in the (memory saving) 2-component mode. Note that one should combine this option with the spin-free 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.

### .X2C4¶

One-step Exact (infinite order) 2-Component relativistic Hamiltonian [Ilias2007] .

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.

### .IOTC4¶

Warning

documentation missing

### .BSS4¶

Warning

documentation missing

### .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–Kroll–Hess Hamiltonian they slightly differ.

### .BEG_2C¶

Warning

documentation missing

### .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 where AMFI is the default. This holds also for keywords .X2C and .X2C4). In the DIRAC08 distribution version .NOAMFI was 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 the two-electron shielding.

### .DO2C4C¶

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

Warning

only in development version

### .DO4C2C¶

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

Warning

only in development version

### .USE_DF¶

Warning

only in development version

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¶

Warning

only in development version

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

Warning

and what does this integer mean?

### .MO4C2C¶

Warning

documentation missing

## Various¶

### .SCQSET¶

Warning

documentation missing

### .YREQ1¶

Warning

documentation missing

### .BLOCKD¶

Warning

documentation missing

### .VACPOL¶

Warning

documentation missing

### .MIXVAC¶

Warning

documentation missing

### .NOVREF¶

Warning

documentation missing

### .POSVAC¶

Warning

documentation missing

### .QDOTS¶

Warning

documentation missing

### .MMF¶

Warning

documentation missing

### .xB¶

Warning

documentation missing

### .xC¶

Warning

documentation missing