# *KRCI¶

or the (deprecated) *KRCICALC.

**2- and 4-component relativistic GASCI module**¶

Written by Timo Fleig, based on LUCIA by Jeppe Olsen, parallelization by Stefan Knecht and Hans Joergen Aagaard Jensen.

The KRCI module is a string-based Hamiltonian-direct configuration interaction (CI) program [Fleig2003], [Fleig2006] , based on the LUCIA code [Olsen1990]. It is capable of doing efficient CI computations at arbitrary excitation level, e.g. FCI, SDCI, RASCI, and MRCI using general active spaces. The code is interfaced to molecular integrals obtained in the relativistic 2c- and 4-component framework and uses double-point group symmetry. It is implemented as a full parallel version [Knecht2010a] .

A central feature of the program is the Generalized Active Space (GAS) concept, in which the underlying total orbital space is subdivided into a basically arbitrary number (save for an upper limit) of subspaces with arbitrary occupation constraints. This is the most general approach to orbital space subdivisions. The program uses DIRAC Kramers pairs from either a closed- or an open-shell calculation in a relativistic two- and four-component formalism (see **HAMILTONIAN).

The technical limitations are roughly set by several 100 million determinants in the CI expansion on PCs and common computing clusters and several billions of determinants on supercomputers with ample memory.

If desired, the program also computes 1-particle densities from optimized CI wave functions from which the natural orbital occupations are printed.

The program can also be used for the computation of various one-electron properties at the GASCI level [Knecht2009] .

**Mandatory keywords**¶

### .CI PROGRAM¶

specifies the CI module behind KRCI. Two choices: LUCIAREL and GASCIP

```
.CI PROGRAM
LUCIAREL
```

### .CIROOTS¶

specifies the number of states (roots) *Y* in irrep *X* to optimize on. This keyword may be
repeated several times for multi-root multi-symmetry calculations
step-by-step. *default:* none.

Example for calculating *Y* = 5 roots in irrep *X* = 1:

```
.CIROOTS
1 5
```

The KRCI module exploits the character tables for the Abelian double groups handled by DIRAC as detailed here,
Symmetry-handling at the correlated level. In contrast to other correlation
modules, in KRCI the order is reversed that is boson irreps come **before** fermion irreps.

If you run the calculation in **linear symmetry** you have to specify the **2 x OMEGA** value
of the state to optimize on, and the .CI PROGRAM must be LUCIAREL. The doubling (*2 x*) stems from the fact
that we want to avoid non-integer input, e.g. in case of an odd number
of electrons we might have \(\Omega\) = 1/2, 3/2, 5/2, etc. values and the
corresponding input for one root with the \(\Omega\) =1/2 would then read as

```
.CIROOTS
1 1
```

If we have a system with an even number of electrons and inversion symmetry the input for two \(\Omega\) =2g states read as

```
.CIROOTS
4g 2
```

### .GAS SHELLS¶

Specification of CI calculation with electron distribution in orbital (GAS) spaces.
The first line contains the number of GA spaces to be used (1-7),
followed by one line per GAS with a separation by a “/” of the min/max number of electrons in each GAS and the
number of orbitals per fermion corep (either one (no inversion symmetry) or two (inversion
symmetry: *gerade* *ungerade*) entries per line).

The first entry before the “/” gives the minimal number of accumulated (!) electrons after consideration of this GAS, the second the corresponding maximum number, separated by blanks. The minimum and maximum accumulated occupations allow for a very flexible parameterization of the wave function. All determinants fulfilling the occupation constraints will be constructed. The second entry after the “/” gives the number of orbitals per fermion corep. See also the open-shell input in *SCF which is similar to the syntax used here. The design of a GAS scheme is non-trivial and should be motivated by the electronic structure of the system (e.g. inner core, outer core, valence, virtual space). Sometimes it is useful to subdivide the valence space, for scientific reasons, or/and the virtual space, for technical reasons (save core memory). See reference [Fleig2006a], pp. 27 for more details. example for 4 GAS spaces with an advanced excitation pattern:

```
.GAS SHELLS
4
1 2 / 1 0
6 8 / 1 2
14 16 / 7 7
16 16 / 30 24
```

if all remaining orbitals (fullCI) should be included in the last GAS space, use:

```
.GAS SHELLS
4
1 2 / 1 0
6 8 / 1 2
14 16 / 7 7
16 16 / all
```

### .INACTIVE¶

Inactive orbitals per fermion corep. *default*: all orbitals active. example for 4 *gerade* and 2 *ungerade* orbitals frozen in a molecule with inversion center:

```
.INACTIVE
4 2
```

**Optional keywords**¶

### .NOOCCN¶

compute natural orbital occupation numbers for each electronic state. *default*: do not compute natural orbital occupation numbers.

### .ANALYZ¶

analyze the final CI wave function printing the coefficients for each determinant above a given threshold
\(10^{-2}\). *default*: do not analyze the final CI wave function.

### .THRGCI¶

set CI convergence threshold based on the norm of the CI gradient (aka CI residual). *default*:

```
.THRGCI
1.0e-4
```

### .MAX CI¶

maximum number of CI iterations. *default*:

```
.MAX CI
5
```

### .MXCIVE¶

maximum size of Davidson subspace. *default*: 3 times the number of eigenstates (see .CIROOTS) to optimize on.
example:

```
.MXCIVE
24
```

### .RSTRCI¶

Restart CI from vector(s) on file KRCI_CVECS.x where **x** is determined by the symmetry of the wave
function.

```
.RSTRCI
1
```

*default*: no restart.

```
.RSTRCI
0
```

*further infomation*: the convention in linear symmetry for KRCI_CVECS.x is the following where the offset/range for systems
with a gerade number of electrons is for x: offset –> 1; range: x =[1-64] and with an ungerade number of electrons x: offset –>
65; range: x =[65-128]. In more detail (with MJ == \(\Omega\)):

for systems with a gerade number of e- and no inversion center:

```
MJ = 0: KRCI_CVECS.1
MJ = +1: KRCI_CVECS.2
MJ = -1: KRCI_CVECS.3
MJ = +2: KRCI_CVECS.4
MJ = -2: KRCI_CVECS.5
...
up to
MJ = +32: KRCI_CVECS.64
```

for systems with a gerade number of e- and an inversion center:

```
MJ = 0g: KRCI_CVECS.1
MJ = +1g: KRCI_CVECS.2
MJ = -1g: KRCI_CVECS.3
MJ = +2g: KRCI_CVECS.4
MJ = -2g: KRCI_CVECS.5
...
up to
MJ = +16g: KRCI_CVECS.32
and
MJ = 0u: KRCI_CVECS.33
MJ = +1u: KRCI_CVECS.34
MJ = -1u: KRCI_CVECS.35
MJ = +2u: KRCI_CVECS.36
MJ = -2u: KRCI_CVECS.37
...
up to
MJ = +16u: KRCI_CVECS.64
```

for systems with an ungerade number of e- and no inversion center:

```
MJ = +1/2: KRCI_CVECS.65
MJ = -1/2: KRCI_CVECS.66
MJ = +3/2: KRCI_CVECS.67
MJ = -3/2: KRCI_CVECS.68
...
up to
MJ = +63/2: KRCI_CVECS.128
```

for systems with an ungerade number of e- and an inversion center:

```
MJ = +1/2g: KRCI_CVECS.65
MJ = -1/2g: KRCI_CVECS.66
MJ = +3/2g: KRCI_CVECS.67
MJ = -3/2g: KRCI_CVECS.68
...
up to
MJ = -31/2g: KRCI_CVECS.96
and
MJ = +1/2u: KRCI_CVECS.97
MJ = -1/2u: KRCI_CVECS.98
MJ = +3/2u: KRCI_CVECS.99
MJ = -3/2u: KRCI_CVECS.100
...
up to
MJ = -31/u: KRCI_CVECS.128
```

### .CHECKP¶

enables a check point write of the current solution vectors to the file
KRCI_CVECS.x (see above in .CIROOTS for an explanation of how **x** is
supposed to be replaced) during the Davidson iterations. A checkpoint
file will be written roughly every 6th iteration. *default*: do not write check points.

**Advanced options**¶

### .IJKLRO¶

enables the storage of the resorted two-electron integrals on file
IJKL_REOD which can then be read-in in a restart. This avoids a
multiple reading from the 4IND* files and subsequent resorting. The
keyword has to be present also in the restart to enable the potential
read-in procedure. Hint: This keyword may be combined with .MAX CI == 0
in a precedent step in order to save memory for the actual production
run. This is due to the fact that the resorting step itself requires the
in-core storage of the unsorted and sorted integrals. *default*: do not write the resorted integrals to file IJKL_REOD.

### .IJKLSP¶

enables the splitting of the resorted integrals among the co-workers
according to their needs in the computation of the sigma vector. It can
only be used in combination with .IJKLRO.
*default*: do no split the resorted integrals among the co-workers.

**KRCI properties**¶

This part referes to the *2- and 4-component relativistic KR-CI property module* written by
Stefan Knecht and Hans Joergen Aa. Jensen, parallelization by Stefan Knecht

The KR-CI property module [Knecht2009] takes advantage of the 2- and 4-component KRCI module. It can be used for the computation of:

permanent dipole moments in ground and excited states,

transition dipole moments

computation of \(\hat{l}_z\), \(\hat{s}_z\) and \(\hat{j}_z (\Omega)\) expectation values (only for linear molecules).

Upon request the analysis of other one-electron properties may also be implemented. The module could in principle be used for each one-electron operator that is specified in One-electron operators.

### .OPERATOR¶

Further modification by Malaya K. Nayak for the proper functioning of .OPERATOR keyword

Here is some specific examples of One-electron operators using .OPERATOR keyword.

For the electron electric dipole moment (eEDM) effective electric field one can use

```
.OPERATOR
'A2-EDM'
iBETAGAM
EDM
```

The nucleus-electron scalar-pseudoscalar (Ne-SPS) interaction in BeH can be defined as

```
.OPERATOR
'A1-SPS'
iBETAGAM
'PVCBe 01' for the first atom (as specied in .mol file) Be in BeH molecule
.OPERATOR
'A2-SPS'
iBETAGAM
'PVCH 02' for the second atom (as specied in .mol file) H in BeH molecule
```

Similarly, the magnetic hyperfine-structure constants in BeH can be defined as follows

```
.OPERATOR
'X1-HYP' (X1-HYP, Y1-HYP and Z1-HYP are for the first atom as specied in .mol file)
XAVECTOR
'NEF 001'
'NEF 005'
.OPERATOR
'Y1-HYP'
YAVECTOR
'NEF 003'
'NEF 001'
.OPERATOR
'Z1-HYP'
ZAVECTOR
'NEF 005'
'NEF 003'
.OPERATOR
'X2-HYP' (X2-HYP, Y2-HYP and Z2-HYP are for second atom as specied in the .mol file)
XAVECTOR
'NEF 002'
'NEF 006'
.OPERATOR
'Y2-HYP'
YAVECTOR
'NEF 004'
'NEF 002'
.OPERATOR
'Z2-HYP'
ZAVECTOR
'NEF 006'
'NEF 004'
```

The nuclear Magnetic-Quadrupole-Moment (MQM) constants in BeH can be defined by defination as

```
.OPERATOR
'Z1-MQM' (Z1-MQM, for the first atom as specified in the .mol file)
ZAVECTOR
YZEFG011
XZEFG011
COMFACTOR
-0.333333333D0
.OPERATOR
'Z2-MQM' (Z2-MQM, for the second atom as specfied in the .mol file)
ZAVECTOR
YZEFG021
XZEFG021
COMFACTOR
-0.333333333D0
```

One can define many more One-electron operators as per their requirements.

### .DIPMOM¶

compute the permanent dipole moments in electronic ground and excited states.

### .TRDM¶

compute the transition dipole moments between electronic states.

### .OMEGAQ¶

*Note*: this keyword may only be used for linear molecules.

compute the expectation values of the spin- and angular momentum operator in z-direction and print the total \(\hat{j}_z\) = \(\hat{s}_z\) + \(\hat{l}_z\) expectation value (\(\Omega\) value) for each electronic state.

### .MHYP¶

compute magnetic hyperfine interaction constants as expectation values over Fermi’s four-component operator (Z. Phys. 60 (1930) 332). Example for the hydrogen atom (proton nucleus):

.MHYP 0.5 nuclear spin quantum number I 2.793 nuclear magnetic moment \mu

In the molecular case add a column with I and \(\mu\) for every atom in the order given by the basis file. Reference: [Fleig2014]

### .EEDM¶

compute electron electric dipole moment effective electric field as an expectation value over the effective one-electron momentum-form operator (J. Phys. B: At. Mol. Opt. Phys. 22 (1989) 559, stratagem II). Reference: [Fleig2013]

### .ENSPS¶

compute nucleon-electron scalar-pseudoscalar interaction constant as an expectation value over the corresponding effective four-fermion operator in the limit for an infinitely heavy nucleon (Sov. Phys. JETP 62 (1985) 872). Reference: [Fleig2015]

### .NMQM¶

compute nuclear Magnetic-Quadrupole-Moment (MQM) Interaction constatn as an expectation value over the corresponding effective four-component operator (Sov. Phys. JETP 60 (1984) 873), Reference: [Fleig2016]