Defects:  F-center in LiF,  a paramagnetic defect G. Mallia

Overview and goals

A simple case: F-center in LiF

• The perfect system: LiF bulk
  • Defining the bulk structure
  • Supercell size and shape
  • Size-consistency
  • Properties:
    • Band structure,
    • Density of states,
    • Charge density
• Defective system
  • F-center modeling
  • F-center basis set
  • Relaxation geometry
  • Properties:
    • Band structure,
    • Density of states,
    • Charge and spin density
    • Hyperfine coupling
• Inputs

Calculation of hyperfine coupling constants with CRYSTAL

References

Overview and goals

Defects are created in crystals when the periodic structure is altered, for instance, by displacing, removing, adding or substituting one or more atoms. Such a local perturbation in the structure causes a modification of the properties of the system, which is often relevant.

Defects may display ability to trap and/or release electrons into states which are very localized. These states can fall inside the band gap of the material: when electrons are trapped or released from this states, energy is emitted or absorbed, and can be seen as sharp peaks in the optical spectra of the material. The trapping of charge and the optical activity have very high impact in the design of laser systems, microelectronic planar devices and optical communication systems.

Defects can also alter the density of the material, specially if structural relaxation of the defects takes place. This influences both the mechanical reliability of the material and its optical properties. A clear and concise book [1] about the point defects is a good starting point for the beginner: definitions and classification, thermodynamics, formation of defects and experimental techniques are explained in a simple way.

Understanding the mechanisms relating defects and change of properties in crystals is a major challenge where computational models are useful tools. In particular, electronic structure calculations can help to evaluate the energies implied in  formation,  relaxation and migration of defects. The detailed description of the electron density provided by the calculations allows understanding how the chemical bonding is modified around the defect.

When a point defect is created, the translational symmetry of the system is lost. Periodic models can nevertheless be applied to simulate a local defect by constructing a supercell, that is, a set of several unit cells inside which the defect is inserted.

Generation of a supercell: the new lattice vectors length is 5 times the original unit cell ones. Translational symmetry is maintained, the area of the new cell is 25 times the original one.

A defect is inserted at the centre of the cell (it could be anywhere, due to translational invariance). The distance between two point defects determines the entity of their interaction.

Such models should be used with some caution in order to obtain reliable results. The cost of the computation of a model supercell large enough to obtain negligible interaction between defects may be unaffordable.

There are several other methods of modelling localized defects, using clusters or embedded cluster techniques, reviewed in chapter 13 of C. Pisani, editor: Quantum-Mechanical Ab-initio calculation of the Properties of Crystalline Materials, Lecture Notes in Chemistry, Vol. 67, Springer Verlag, Heidelberg, 1996

This tutorial presents the procedure followed to study a simple defect, the F-centre in LiF, with the CRYSTAL package.
The study is published in: G. Mallia, R. Orlando, C. Roetti, P. Ugliengo, R. Dovesi, Phys. Rev. B 63, 235102 (2001).

A simple case: F-centre in LiF - Bulk

F-centre

Electrons bound to a negative ion vacancy are known as  F-centres. Because of their optical activity, F centres are also called colour centres. References [4,5] present experimental investigation of F-centre in Alkali Halides and in LiF in particular. 

F-center in LiF is  an electron bound to a fluorine vacancy.

The interaction between the unpaired electron at the anion vacancy and the nuclear spins of surrounding atoms is studied up to the seventh nearest neighbours by mean of the electron spin resonance (ESR).

CRYSTAL is a suitable tool to model this kind of defect for the following reasons:

• The host compound is an ionic crystal, well described by the Hartree-Fock method.
• The crystal structure is fcc: the high number of symmetry operators reduces the computational costs.
• The F centre, where an electron is trapped into the anion vacancy, is a neutral defect.

Modelling of the system consists of two steps:

• definition of the perfect system, and choice of a supercell of suitable size;
• creation of the defect.

Perfect system

Defining the bulk structure

The space group of LiF is Fm-3m, the asymmetric unit includes two atoms in special positions of multiplicity 1:

   F (0,0,0)     Li (1/2,1/2,1/2)

or exchanging the coordinates

   Li (0,0,0)     F (1/2,1/2,1/2)

From the point of view of physics the two possibilities are equivalent, but not from the computational point of view for this study. The defect is an anion vacancy (a Fluorine atom missing): if the Fluorine atom is at the origin, when a supercell is created and the F atom at the origin is removed to generate the defect, the number of symmetry operators is not reduced. See "F-centre formation".

See "Geometry input", "Basis set input" , "Hamiltonian and SCF for input instructions.

Li and F basis sets as in the site www.crystal.unito.it

Default values for all computational parameters. Restricted closed shell Hartree-Fock Hamiltonian.

Mixing of Fock matrices is chosen to speed up SCF convergence.

Exercise 1: Run CRYSTAL with file LiF.d12 as an input (command: runcry LiF - output in LiF.out)

The default SCF guess is a superposition of atomic densities. The number of electrons attributed to each atom is the sum of shell charges. In this case we have chosen a neutral atomic configuration for the SCF atomic guess:

   9 electrons for F         3 electrons for Li

Mulliken analysis of the wave function at the end of SCF gives:

 9.977  electrons for F      2.023  electrons for Li

The crystal is fully ionic: the choice on an atomic (not ionic, closer to the real one) configuration  maintains neutrality of the cell when the defect is created by removing a Fluorine atom.

CRYSTAL input and output

Supercell size and shape

A supercell is defined by an expansion matrix, given in input, to multiply the lattice parameters matrix (see keyword SUPERCEL in CRYSTAL User's Manual). The expansion matrix to obtain a supercell of given volume must be chosen so as not to reduce the number of symmetry operators.
crystal checks the compatibility of the symmetry operators of the original cell with the new cell, and removes the symmetry operators not compatible with the structure. The number of symmetry operators can be reduced, not increased.

Hints to define an expansion matrix of LiF primitive cell:

• multiply the translational vectors by an integer i; the supercell maintains the original shape, in this case rhombohedral. If i=2, a super-cell with a volume equal to 2x2x2xV=8V is generated, where V is the volume of the LiF primitive cell with 2 atoms. Since the considered super-cell is composed of 16 atoms, it is named S₁₆.

  Expansion matrix: SUPERCEL i 0 0 0 i 0 0 0 i

  i volume LiF - atoms/cell name 2 8 x V 16 S16 3 27 x V 54 S54 4 64 x V 128 S128

• exploit the transformation matrix primitive => conventional cell (See Appendix A.4 in CRYSTAL User's Manual). In the case of LiF, the supercell obtained has a volume 4 times the original one (the volume of fcc crystallographic cell is 4 times the volume of the primitive cell). The shape of the cell is cubic.

  Expansion matrix: from primitive S2 to crystallographic S8: SUPERCEL -1 1 1 1 -1 1 1 1 -1

  x 2

  Expansion matrix: from primitive S2 to S64: SUPERCEL -2 2 2 2 -2 2 2 2 -2

  A larger supercell is obtained by multiplying the translational vectors of the conventional cell by an integer i; if i=2, the supercell contains 64 atoms (S_8² = S₆₄); the shape of the cell is cubic.

• exploit the matrix T that transforms the conventional body-centred cubic cell (with 4 atoms) into the primitive body-centred cubic cell (consisting of 2 atoms): (see Appendix A.4 in CRYSTAL User's Manual). When multiplying the matrix T with the expansion matrix E64 that defines S₆₄, we obtain a supercell with 32 atoms (S₃₂). We calculate the expansion matrix E32 that transforms the primitive face-centred cubic cell into the S₃₂ as follows:

  T	x	E64	=	E32
  -1/2 1/2 1/2 1/2 -1/2 1/2 1/2 1/2 -1/2	x	-2 2 2 2 -2 2 2 2 -2	=	3 -1 -1 -1 3 -1 -1 -1 3

  If we multiply the translational vectors of S₃₂ by an integer i=2, S₂₅₆ is obtained. The expansion matrix to obtain S₃₂ and S₂₅₆ are as follows:

  Expansion matrix: from primitive S2 to S32: SUPERCEL 3 -1 -1 -1 3 -1 -1 -1 3

  x 2

  Expansion matrix: from primitive S2 to S256: SUPERCEL 6 -2 -2 -2 6 -2 -2 -2 6

  Similarly, this expansion matrix can be obtained by multiplying the matrix T with the expansion matrix for S₅₁₂, which is a simple cubic cell like S₆₄. The generic term representing supercells like S₃₂ and S₂₅₆ is S_4(2i)³ with i=1,2,... They are rhombohedral with all the three angles \(\alpha\), \(\beta\) and \(\gamma\) nearly equal to 109.5�.

• S_2i³ (the shape is like primitive face-centred cell)
• S_8i³ (the shape is like simple cubic cell)
• S_4(2i)³ (the shape is like primitive body-centred cell)

Supercells up to 32 atoms will be studied. The exercises will use a 16 atoms supercell, to obtain results quickly.
Cells bigger than S₃₂ are not required because (see reference  [6]:

1. the interaction between neighbouring defects is negligible;
2. the hyperfine coupling between the magnetic momentum of the trapped electron and the first and second neighbour nuclei is evaluated correctly.

Exercise 2:

• Insert into the file LiF.d12 the SUPERCEL directive to define a supercell with 8, 16, 32 atoms

     volume x 4         volume x 8         volume x 16
      8 atoms            16 atoms           32 atoms
      SUPERCEL           SUPERCEL           SUPERCEL
      -1  1  1            2  0  0            3 -1 -1
       1 -1  1            0  2  0           -1  3 -1
       1  1 -1            0  0  2           -1 -1  3  

• Insert the keyword TESTGEOM in geometry input, to stop execution after definition of the crystal structure.
• Run CRYSTAL Verify that the number of symmetry operators is kept, when the cubic supercells, with 8, 16 and 32 atoms are built.

CRYSTAL input and output for S₈
CRYSTAL input and output for S₁₆

Exercise 3:
The size of the reciprocal lattice cell reduces with increasing size of the direct lattice cell. The sampling in k space requires less and less points, at the limit 1 k point for very large supercell.
The Pack-Monkhorst net shrinking factor (see "SCF") ranges from 8 to 2. The value of IS must be checked to guarantee stability of the energy value within the SCF convergence tolerance.

Run CRYSTAL for the supercell S₁₆ (remove TESTGEOM) with IS = 8, 4, 2 and check the energy value with respect to the SCF convergence threshold

Size-consistency

The first essential requirement of supercell method is the size-consistency. Each extensive property for a n-cells system must be equal to n times the corresponding value obtained for 1 cell system.

Exercise 4: Verify that the energy per LiF pair is equal up to the SCF convergence threshold  for the supercells S_8, S₁₆

Band structure and density of states

Exercise 5: Use the wave function obtained in Exercise 2 and run a properties calculation (runprop script) in order to compute the band structure of the perfect crystal for S₁₆ along the path identified by the k-points \(\Gamma\)-X-W.

The coordinates of the k-points are:

\(\Gamma\)  (0    0    0)
X  (1/2   0   1/2)
W  (1/2  1/4  3/4)

In the properties input the coordinates must be inserted in integer format, in units of the shrinking factor (second input  datum):

\(\Gamma\)  (0  0  0)
X   (2  0  2)
W   (2  1  3)

The band range chosen includes the valence bands and the first 4 virtual bands (96 electrons per cell, 32 core electrons, RHF hamiltonian: 48 occupied bands, 16 core bands)

The input file (with extension d3, if the script runprop is used) is:

Visualize the band structure by using the band script and the appropriate control file.

Density of states in the energy range of the bands previously considered is computed executing properties with the following input:

To identify the "label" of the atoms see the output:

 *******************************************************************************
    1    9 F     4     0.000     0.000     0.000   1.370E-01       9.977
    2    9 F     4     2.010     2.010     0.000   1.370E-01       9.977
    3    9 F     4     2.010     0.000     2.010   1.370E-01       9.977
    4    9 F     4     4.020     2.010     2.010   1.370E-01       9.977
    5    9 F     4     0.000     2.010     2.010   1.370E-01       9.977
    6    9 F     4     2.010     4.020     2.010   1.370E-01       9.977
    7    9 F     4     2.010     2.010     4.020   1.370E-01       9.977
    8    9 F     4     4.020     4.020     4.020   1.370E-01       9.977
    9    3 LI    2     0.000     0.000    -2.010   5.250E-01       2.023
   10    3 LI    2    -2.010    -2.010    -2.010   5.250E-01       2.023
   11    3 LI    2     2.010     0.000     0.000   5.250E-01       2.023
   12    3 LI    2     0.000    -2.010     0.000   5.250E-01       2.023
   13    3 LI    2     0.000     2.010     0.000   5.250E-01       2.023
   14    3 LI    2    -2.010     0.000     0.000   5.250E-01       2.023
   15    3 LI    2     2.010     2.010     2.010   5.250E-01       2.023
   16    3 LI    2     0.000     0.000     2.010   5.250E-01       2.023
 *******************************************************************************

In the perfect crystal at site 1 and 2 there are equivalent atoms, fluorine.
At the end of calculation the following band structure and density of states should be obtained.

The system is insulator, with band gap of 0.801 hartree (21.795 eV).
The band width (BWIDTH keyword, see CRYSTAL User's Manual) of all valence bands is very narrow, indicating that there is little overlap of the ionic charge distributions. The valence bands are mainly Fluorine.

 BAND LIMITS FROM BAND  17 TO BAND  52

 BAND  17 EMIN -1.466479E+00 EMAX -1.450493E+00
 BAND  18 EMIN -1.456663E+00 EMAX -1.445848E+00
 BAND  19 EMIN -1.451432E+00 EMAX -1.440662E+00
 BAND  20 EMIN -1.450493E+00 EMAX -1.440662E+00
 BAND  21 EMIN -1.446320E+00 EMAX -1.438974E+00
 BAND  22 EMIN -1.445502E+00 EMAX -1.434563E+00
 BAND  23 EMIN -1.440662E+00 EMAX -1.434563E+00
 BAND  24 EMIN -1.440662E+00 EMAX -1.434563E+00
 BAND  25 EMIN -6.318031E-01 EMAX -5.887851E-01
 BAND  26 EMIN -6.318031E-01 EMAX -5.887851E-01
 BAND  27 EMIN -6.318031E-01 EMAX -5.878500E-01
 BAND  28 EMIN -6.151087E-01 EMAX -5.869528E-01
 BAND  29 EMIN -6.151087E-01 EMAX -5.804994E-01
 BAND  30 EMIN -6.151087E-01 EMAX -5.786599E-01
 BAND  31 EMIN -6.151087E-01 EMAX -5.588019E-01
 BAND  32 EMIN -5.829380E-01 EMAX -5.442011E-01
 BAND  33 EMIN -5.746830E-01 EMAX -5.442011E-01
 BAND  34 EMIN -5.703980E-01 EMAX -5.394351E-01
 BAND  35 EMIN -5.703980E-01 EMAX -5.362066E-01
 BAND  36 EMIN -5.703980E-01 EMAX -5.349269E-01
 BAND  37 EMIN -5.559182E-01 EMAX -5.303390E-01
 BAND  38 EMIN -5.559182E-01 EMAX -5.080550E-01
 BAND  39 EMIN -5.360308E-01 EMAX -5.080550E-01
 BAND  40 EMIN -5.346110E-01 EMAX -5.080550E-01
 BAND  41 EMIN -5.330866E-01 EMAX -5.080550E-01
 BAND  42 EMIN -5.288004E-01 EMAX -5.079541E-01
 BAND  43 EMIN -5.261544E-01 EMAX -5.077259E-01
 BAND  44 EMIN -5.261544E-01 EMAX -5.074979E-01
 BAND  45 EMIN -5.196737E-01 EMAX -5.013619E-01
 BAND  46 EMIN -5.151926E-01 EMAX -4.963703E-01
 BAND  47 EMIN -5.151926E-01 EMAX -4.963703E-01
 BAND  48 EMIN -5.088824E-01 EMAX -4.960240E-01
 BAND  49 EMIN  3.043997E-01 EMAX  4.128116E-01
 BAND  50 EMIN  3.680616E-01 EMAX  4.423662E-01
 BAND  51 EMIN  4.069737E-01 EMAX  4.832150E-01
 BAND  52 EMIN  4.127770E-01 EMAX  4.832150E-01

CRYSTAL input and output

Exercise 6: Compare the band structure of supercell S₁₆ with the band structure of LiF bulk, crystallographic cell.

Note - The same path \(\Gamma\)-X-W of S₁₆ for BAND properties can be used, taking into account that there are 2 atoms, 12 electrons, 4 core electrons in the cell: 6 bands are occupied (2 core bands).
Tha band range must be modified:  17 -> 3, 52 -> 10:

Charge density

A window in the plane  (001), which contains the vacancy site, is used to compute the charge density in a grid of points. Looking at the output obtained, the ionicity of the system is confirmed by Mulliken population analysis:

ALPHA+BETA ELECTRONS MULLIKEN POPULATION ANALYSIS - NO. OF ELECTRONS 96.000000 ATOM Z CHARGE A.O. POPULATION 1 F 9 9.977 1.999 0.809 1.028 1.028 1.028 1.027 0.728 0.728 0.728 0.155 0.239 0.239 0.239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 LI 3 2.023 1.615 0.372 0.012 0.012 0.012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OVERLAP POPULATION CONDENSED TO ATOMS FOR FIRST 6 NEIGHBORS ATOM A 1 F ATOM B CELL R(AB)/AU R(AB)/ANG OVPOP(AB) 9 LI ( 0 0 0) 3.798 2.010 -0.001 2 F ( 0 0 0) 5.372 2.843 -0.027 . . . . . . . . . . . . . . . . . . . . . . . . . ATOM A 2 F ATOM B CELL R(AB)/AU R(AB)/ANG OVPOP(AB) 10 LI ( 0 0 1) 3.798 2.010 -0.001 1 F ( 0 0 0) 5.372 2.843 -0.027 . . . . . . . . . . . . . . . . . . . . . . . . .

Li^+0.977 F^-0.977, very close to the simple model Li⁺¹ F^-1.  The overlap population between first neighbours is zero.

The charge density of the perfect crystal shows almost spherical distributions, not overlapping.

Comparison between the density in the crystal and the density as sum of the atomic densities confirms that there is no build up of charge between neighbouring atoms.

Defective system

F-center formation

An F-centre is an electron bound to a negative ion vacancy. Modelling a F-centre requires definition of a crystal geometry with an "isolated" anion vacancy, and of a basis set allowing build up of electron charge in the vacancy during the SCF process.

To model the isolated defect with CRYSTAL, a periodic program, the supercell technique is adopted: a supercell large enough to avoid defects interaction is generated, and then the defect is created.

The keyword GHOSTS of CRYSTAL package (second input block, basis set definition) allows generation of a vacancy, leaving at the site the variational basis set necessary to allow build up of charge. 

The keyword GHOSTS removes electrons and nuclear charge from the selected atomic site, leaving the basis set centred at the atomic position.

A model obtained with removal of an atom (ATOMREMO) is inadequate, as the basis set associated with that centre is removed as well. There is no variational freedom to allow build up of electron charge at that site.

In the supercell S₁₆ there are 8 Lithium (3 electrons in the neutral atom) and 8 Fluorine atoms (9 electrons in the neutral atom), for a total of 96 electrons, corresponding to 48 occupied bands (RHF hamiltonian).  When a Fluorine atom is removed, in the supercell there are 8 Li, 7 F, for a total of 87 electrons. The odd number of electrons requires independent treatment of \(\alpha\) and \(\beta\) electrons, allowed by Unrestricted Hartree Fock method, chosen by the keyword UHF (third input block, "Hamiltonian &c")

When the charge distribution of \(\alpha\) and \(\beta\) electrons is different, the resulting spin density  is responsible for the magnetic properties of the system. To drive the SCF process towards a given state, it is possible to force the program to start SCF with a defined  difference between the number of \(\alpha\) and the number of \(\beta\) electrons. This trick should be used only at the beginning of the calculation, in order to allow the electron density to relax properly.

The keyword SPINLOCK (input block 3, "Hamiltonian & SCF")  allows definition of the difference (n_\(\alpha\)-n_\(\beta\)) for a given number of SCF cycles.

Exercise 7: In the previous input for S₁₆ insert the keyword GHOSTS, selecting the fluorine in the origin and run CRYSTAL. Remember to insert the keyword UHF  in the method input and the SPINLOCK directives in SCF part, because the unpaired electron causes a different occupancy of \(\alpha\) and \(\beta\) levels . 
The last part of the input deck follows and changes with respects to the previous input for S₁₆ (exercise 2) are highlighted:

CRYSTAL input and output

The removal of a Fluorine atom located at (0,0,0) did not modify the number of symmetry operators, there is 1 symmetry allowed degree of freedom for geometry optimization (see below).

Exercise 8: Repeat  exercise 7 by modifying in basis set input the electronic configuration of atoms to obtain ions Li⁺ and F^-. The program stops with the error message "UNIT CELL NOT NEUTRAL": GHOSTS removed the Fluorine centre, corresponding to 10 electrons and 9 nuclear charges.

Exercise 9: The 9 Fluorine atoms in the supercell are equivalent, but as the translational symmetry within the supercell is not recognized, the symmetry of the 9 sites may be different. The keyword ATOMSYMM (input block 1, Geometry input ) allows symmetry analysis at the atomic sites.

Repeat exercise 7: insert the keyword ATOMSYMM in geometry input (and TESTGEOM to stop the program after geometry step) and run the two cases:

1)    F (0,0,0)     2)    F (-1/2,-1/2,0)

**** ATOMS BELONGING TO THE SUPERCELL
 LABEL AT.NO.      COORDINATES (ANGSTROM AND FRACTIONARY)
   1     9     0.0000     0.0000     0.0000     0.0000     0.0000     0.0000
   2     9    -2.0100    -2.0100     0.0000     0.0000     0.0000    -0.5000
   3     9    -2.0100     0.0000    -2.0100     0.0000    -0.5000     0.0000
   4     9    -4.0200    -2.0100    -2.0100     0.0000    -0.5000    -0.5000
   5     9     0.0000    -2.0100    -2.0100    -0.5000     0.0000     0.0000
   6     9    -2.0100    -4.0200    -2.0100    -0.5000     0.0000    -0.5000
   7     9    -2.0100    -2.0100    -4.0200    -0.5000    -0.5000     0.0000 
   8     9    -4.0200    -4.0200    -4.0200    -0.5000    -0.5000    -0.5000
   9     3     0.0000     0.0000    -2.0100    -0.2500    -0.2500     0.2500
  10     3    -2.0100    -2.0100    -2.0100    -0.2500    -0.2500    -0.2500
  11     3     2.0100     0.0000     0.0000    -0.2500     0.2500     0.2500
  12     3     0.0000    -2.0100     0.0000    -0.2500     0.2500    -0.2500
  13     3     0.0000     2.0100     0.0000     0.2500    -0.2500     0.2500
  14     3    -2.0100     0.0000     0.0000     0.2500    -0.2500    -0.2500
  15     3     2.0100     2.0100     2.0100     0.2500     0.2500     0.2500
  16     3     0.0000     0.0000     2.0100     0.2500     0.2500    -0.2500

 **************************** SUPERCELL GENERATED ****************************
 *********************************************************************
                        ATOMIC SITE SYMMETRIES
 *********************************************************************

 ATOM   1 ATOMIC NUMBER   9 - NUMBER OF SYMMOPS WHICH DO NOT MOVE THE ATOM  48
 SEQUENCE NUMBERS OF THE SYMMOPS
    1    2    3    4    5    6    7    8    9   10   11   12   13   14   15
   16   17   18   19   20   21   22   23   24   25   26   27   28   29   30
   31   32   33   34   35   36   37   38   39   40   41   42   43   44   45
   46   47   48
 NO EQUIVALENT ATOMS


 ATOM   7 ATOMIC NUMBER   9 - NUMBER OF SYMMOPS WHICH DO NOT MOVE THE ATOM   8
 SEQUENCE NUMBERS OF THE SYMMOPS
    1    2   13   14   25   26   37   38
 NUMBER OF ATOMS EQUIVALENT BY SYMMETRY    5
 SEQUENCE NUMBERS OF THESE ATOMS
    2    4    6    5    3
 SEQUENCE NUMBERS OF THE CORRESPONDING SYMMOPS
    3    5    6    7   10

F-center basis set

Creation of the F-centre using the keyword GHOSTS leaves at the vacancy site the basis set of the atom removed. This basis set may be oversize to describe a single electron charge distribution.

The keyword ATOMSUBS (input block 1, Geometry input ) offers an alternative way to generate a vacancy, with a basis set associated to the vacancy site.

ATOMSUBS substitutes with different atoms selected atoms. The basis set of the new atoms must be defined.
If the conventional atomic number of the new atom is 0 (symbol XX), no nuclear charge is attributed to that site, and no electrons should be attributed through the basis set input

Example - F-centre creation from perfect bulk supercell using ATOMSUBS

1) Geometry - input block 1

2) Basis set - input block 2 - insert the following data:

Exercise 10: Run CRYSTAL for F-centre in LiF specifying the ATOMSUBS directive and using only the two most diffuse sp shells of fluorine basis set.

Exercise 11: Test the effect of a 1 shell (s and sp type) basis set at the vacancy site.

The characteristics of the wave function describing the unpaired electron clearly emerge from the data reported in the following table, where the effect of the modification of the basis set centred at the anion vacancy is explored. See table from reference  [6]

			HF
case	basis set type		\(\Delta\)E	(electrons)	(102 e/bohr3)	(102 e/bohr3)	(102 e/bohr3)
1	7-311(s,sp,sp,sp)		0.00	0.91	2.19	2.25	1.88
2	11(sp,sp)	0.437  0.137	+0.10	0.90	1.94	2.26	1.88
3	1(sp)	0.093	-0.50	1.05	2.08	2.25	1.84
4	1(s)	0.093	-0.09	1.09	2.08	2.24	1.77

Effect of the basis set adopted for describing the F-center on total energy, on the Mulliken charge and on the value of the function , spin density, computed at the site of the vacancy XX and at the position of nearest and next nearest neighbours. \(\Delta\)E (in mhartree) is the energy difference with respect to CASE 1, which refers to a calculation with the fluorine anion basis set centred at XX. In CASE 2 only the two most diffuse sp shells of the F- basis set are kept. In CASE 3 only the most diffuse sp shell is kept and its exponent is reoptimized. In CASE 4 the defect basis set is simplified to a single s function.

Relaxation geometry

As a defect is created, the local perturbation breaks the equilibrium between the different forces inside the crystal. A relaxation takes place, involving several atoms surrounding the defect, to find a new balance between repulsive and attractive forces. 

Structural relaxation modifies geometry, and as consequence total energy,  and electronic properties.
The magnitude of these changes depends on the nature of the material.

Relaxation around F-centre must be taken into account.  The neighbours analysis gives:

The F-center (XX) is surrounded by 6 Li ions (the nearest neighbours) and by 12 F ions (the next nearest neighbours).

The keyword SYMMDIR (input block 1, Geometry input ) allows analysis of the symmetrized directions allowed in geometry optimization. Only one direction is allowed, corresponding to the relaxation of the position of the stars of neighbours around the defect.

 SYMMETRY ALLOWED DIRECTION   1 ATOM    9 (Z=  3)
   0.0000000   0.0000000   0.0000000       0.0000000   0.0000000   0.0000000
   0.0000000   0.0000000   0.0000000       0.0000000   0.0000000   0.0000000
   0.0000000   0.0000000   0.0000000       0.0000000   0.0000000   0.0000000
   0.0000000   0.0000000   0.0000000       0.0000000   0.0000000   0.0000000
   0.0000000   0.0000000   0.4082483       0.0000000   0.0000000   0.0000000
  -0.4082483   0.0000000   0.0000000       0.0000000   0.4082483   0.0000000
   0.0000000  -0.4082483   0.0000000       0.4082483   0.0000000   0.0000000
   0.0000000   0.0000000   0.0000000       0.0000000   0.0000000  -0.4082483

 SYMMETRY ALLOWED INTERNAL DEGREE(S) OF FREEDOM:   1

Displacement of atoms  9, 11, 12, 13,14, 16, the first stars of defect neighbours, corresponds to the unique degree of freedom in a 16 atoms supercell.
In a 32 atoms supercell the first 2 stars of neighbours are allowed to be displaced in geometry optimization, without loss of symmetry.

The keyword OPTGEOM ("Geometry optimization") allows geometry optimization, according to the degrees of freedom symmetry allowed. After optimization, the neighbours analysis gives:

 NEIGHBORS OF THE NON-EQUIVALENT ATOMS

 N = NUMBER OF NEIGHBORS AT DISTANCE R
    ATOM  N     R/ANG      R/AU   NEIGHBORS (ATOM LABELS AND CELL INDICES)
   1 XX   6     2.0329     3.8416   9 LI   0 0 0  11 LI   0 0 0  12 LI   0 0 0
                                   13 LI   0 0 0  14 LI   0 0 0  16 LI   0 0 0
   1 XX  12     2.8426     5.3717   2 F    0 0 0   2 F    0 0-1   3 F    0 0 0
                                    3 F    0-1 0   4 F    0 0-1   4 F    0-1 0
                                    5 F    0 0 0   5 F   -1 0 0   6 F    0 0-1
                                    6 F   -1 0 0   7 F   -1 0 0   7 F    0-1 0
   1 XX   8     3.4814     6.5789  10 LI   0 0 0  10 LI   1 0 0  10 LI   0 1 0
                                   10 LI   0 0 1  15 LI   0 0 0  15 LI  -1 0 0
                                   15 LI   0-1 0  15 LI   0 0-1

Comparison between the given outputs shows that the six Li⁺ first neighbours relax by 0.02 Angstrom away from the center of the defect in agreement with Ref [6]. In this paper, the total energy of the defective system has been optimised with respect to the position of the first and second sphere of neighbours in a S₃₂. In the table below the first line refers to the case in which ions have not been displaced, the second one to the relaxation of the six Li⁺ first neighbours and the last one to the further relaxation of the second sphere of ions. The six Li⁺ first neighbours relax far away from the centre of defect, because they feel the reciprocal electrostatic repulsion. The six F^- second neighbours move also far away along radial direction with respect to the defect, but in this case the displacement of the anions is very small since the perturbation caused by the defect almost does not propagate to the second shell of neighbours. The optimization of Li⁺ position determines an energy gain of nearly 1 mhartree, instead the F^- relaxation is negligible (0.2 mhartree) with respects to the non relaxed energy.

Given the small magnitude of the energies involved in the relaxation of the F-centre, we conclude that the perturbation caused by this defect is small. This is mainly due to the ionic character of LiF, since the main interaction in this type of materials is determined by an equilibrium between the attractive and repulsive long-range interactions between anions and cations. Both types of ions have charges several times higher than that of the trapped electron, and therefore the effect of the presence of the defect is relatively small.

Data from reference [6].

Defect formation energy

The formation energy E_n ^f  of the F-centre defect is defined as:

   E_n^f = E_n-1^d + E_F - E_n^p

where
           E_n-1^d   =  energy of the defective system
           E_n^p      =  energy of perfect system
           E_F       =  energy of the isolated fluorine atom.

Charge and mass balance must be preserved in the equation.

Exercise 12: Calculate the formation energy of F-centre using the results obtained with   supercells of 16 atoms (output of exercises 2 and 7).  The energy of the isolated atom should be computed with an atomic basis set, with appropriate diffuse functions. The keyword ATOMHF (third input block) controls a crystal run with no periodic calculation: the wave functions of the irreducible atoms in the cell are computed by the atomic program inserted in crystal; the program then stops.

Electronic properties

To compute properties of the defective system edit the input (BAND, DOSS) used for the perfect crystal , to take into account that there are 87 electrons, 44 occupied alpha bands and 43 occupied beta bands, 15 core bands for each spin..

Band structure and density of states

One of the most relevant effects caused by the presence of a defect is the appearance of localized states in the band gap of the material. Trapping or releasing electrons to and from these states requires less energy than to excite electrons from the top of the valence band to the bottom of the conduction band.

\(\alpha\) and \(\beta\) electrons are described by different set of orbitals. Two band structures (drawn by continuous and dashed lines) are obtained, for the majority and minority spin states. The shape of the bands is similar to the bands of the perfect system, but a new band appear in the band gap; this is the state associated to the F-centre. Projected density of states confirm the attribution of the band in the gap to the defect.

BWIDTH prints the band width of selected bands.

ANBD allows analysis of the principal AO's components of selected bands. Submit the following input:

    ALPHA      ELECTRONS

 BAND LIMITS FROM BAND  16 TO BAND  48

 BAND  16 EMIN -1.469412E+00 EMAX -1.457189E+00
 BAND  17 EMIN -1.459120E+00 EMAX -1.452635E+00
 BAND  18 EMIN -1.456492E+00 EMAX -1.447512E+00
 BAND  19 EMIN -1.453093E+00 EMAX -1.447512E+00
 BAND  20 EMIN -1.451948E+00 EMAX -1.444787E+00
 BAND  21 EMIN -1.447213E+00 EMAX -1.441488E+00
 BAND  22 EMIN -1.446108E+00 EMAX -1.441488E+00
 BAND  23 EMIN -6.323549E-01 EMAX -5.968586E-01
 BAND  24 EMIN -6.323549E-01 EMAX -5.936570E-01
 BAND  25 EMIN -6.323549E-01 EMAX -5.936570E-01
 BAND  26 EMIN -6.207675E-01 EMAX -5.876354E-01
 BAND  27 EMIN -6.039257E-01 EMAX -5.838262E-01
 BAND  28 EMIN -6.039257E-01 EMAX -5.829773E-01
 BAND  29 EMIN -6.039257E-01 EMAX -5.689758E-01
 BAND  30 EMIN -5.808023E-01 EMAX -5.496637E-01
 BAND  31 EMIN -5.808023E-01 EMAX -5.496637E-01
 BAND  32 EMIN -5.632076E-01 EMAX -5.456283E-01
 BAND  33 EMIN -5.616351E-01 EMAX -5.359277E-01
 BAND  34 EMIN -5.596968E-01 EMAX -5.350725E-01
 BAND  35 EMIN -5.596968E-01 EMAX -5.350725E-01
 BAND  36 EMIN -5.460655E-01 EMAX -5.138960E-01
 BAND  37 EMIN -5.422544E-01 EMAX -5.138960E-01
 BAND  38 EMIN -5.323224E-01 EMAX -5.134964E-01
 BAND  39 EMIN -5.317481E-01 EMAX -5.134964E-01
 BAND  40 EMIN -5.263183E-01 EMAX -5.134964E-01
 BAND  41 EMIN -5.200227E-01 EMAX -5.046798E-01
 BAND  42 EMIN -5.177353E-01 EMAX -5.046798E-01
 BAND  43 EMIN -5.154251E-01 EMAX -5.046798E-01
 BAND  44 EMIN -1.810962E-01 EMAX -1.334811E-01
 BAND  45 EMIN  2.957130E-01 EMAX  3.367300E-01
 BAND  46 EMIN  3.212653E-01 EMAX  4.057058E-01
 BAND  47 EMIN  3.485757E-01 EMAX  4.057058E-01
 BAND  48 EMIN  3.813605E-01 EMAX  4.223566E-01

    BETA       ELECTRONS

 BAND LIMITS FROM BAND  16 TO BAND  48

 BAND  16 EMIN -1.468045E+00 EMAX -1.455173E+00
 BAND  17 EMIN -1.457007E+00 EMAX -1.450095E+00
 BAND  18 EMIN -1.454712E+00 EMAX -1.445496E+00
 BAND  19 EMIN -1.450561E+00 EMAX -1.444908E+00
 BAND  20 EMIN -1.449560E+00 EMAX -1.443599E+00
 BAND  21 EMIN -1.444863E+00 EMAX -1.438800E+00
 BAND  22 EMIN -1.444863E+00 EMAX -1.438800E+00
 BAND  23 EMIN -6.314273E-01 EMAX -5.940887E-01
 BAND  24 EMIN -6.314273E-01 EMAX -5.925618E-01
 BAND  25 EMIN -6.314273E-01 EMAX -5.919625E-01
 BAND  26 EMIN -6.200823E-01 EMAX -5.855450E-01
 BAND  27 EMIN -5.985977E-01 EMAX -5.787253E-01
 BAND  28 EMIN -5.985977E-01 EMAX -5.787253E-01
 BAND  29 EMIN -5.985977E-01 EMAX -5.635124E-01
 BAND  30 EMIN -5.798698E-01 EMAX -5.473294E-01
 BAND  31 EMIN -5.760465E-01 EMAX -5.473294E-01
 BAND  32 EMIN -5.590414E-01 EMAX -5.426076E-01
 BAND  33 EMIN -5.590414E-01 EMAX -5.342512E-01
 BAND  34 EMIN -5.541856E-01 EMAX -5.332294E-01
 BAND  35 EMIN -5.541856E-01 EMAX -5.332294E-01
 BAND  36 EMIN -5.454173E-01 EMAX -5.131146E-01
 BAND  37 EMIN -5.378406E-01 EMAX -5.131146E-01
 BAND  38 EMIN -5.317572E-01 EMAX -5.107263E-01
 BAND  39 EMIN -5.268313E-01 EMAX -5.107263E-01
 BAND  40 EMIN -5.249740E-01 EMAX -5.107263E-01
 BAND  41 EMIN -5.181217E-01 EMAX -5.039255E-01
 BAND  42 EMIN -5.168710E-01 EMAX -5.039255E-01
 BAND  43 EMIN -5.133662E-01 EMAX -5.039255E-01
 BAND  44 EMIN  1.974520E-01 EMAX  2.481626E-01
 BAND  45 EMIN  3.380283E-01 EMAX  3.674521E-01
 BAND  46 EMIN  3.674521E-01 EMAX  4.187525E-01
 BAND  47 EMIN  3.920745E-01 EMAX  4.232495E-01
 BAND  48 EMIN  4.114568E-01 EMAX  4.390525E-01

\(\alpha\) band 44, at -0.181096 hartree (occupied), and \(\beta\) band 44, at 0.197452  (not occupied) are analysed:

     BAND   44    E(A.U.)=      -0.181096
 AO    2  S XX    1   RE=     0.01652
 AO    6  S XX    1   RE=    -0.13232
 AO   10  S XX    1   RE=     1.06897
 . . . . . . . . . . . . . . . .

     BAND   44    E(A.U.)=       0.197452
 AO    2  S XX    1   RE=     0.03423
 AO    6  S XX    1   RE=    -0.23607
 AO   10  S XX    1   RE=     0.98791
 . . . . . . . . . . . . . . . .

The bands in the gap are due to the defect.

Charge and spin density maps

Mulliken population analysis and charge and spin density maps  can be computed submitting the same (no reference to the number of electrons) input deck used for the perfect crystal, using the defective systems wave function. See "How to run crystal".

Data are computed with reference to total and spin density.

 MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM
     ALPHA+BETA ELECTRONS
 MULLIKEN POPULATION ANALYSIS - NO. OF ELECTRONS   87.000000

  ATOM    Z CHARGE  SHELL POPULATION

   1 XX   0  0.911  0.000  0.004  0.000  0.000  0.000 -0.082 -0.001 -0.001
                   -0.001  0.997 -0.002 -0.002 -0.002
   2 F    9  9.988  1.999  0.810  1.027  1.027  1.029  1.027  0.728  0.728
                    0.727  0.161  0.243  0.243  0.240
   3 F    9  9.988  1.999  0.810  1.027  1.029  1.027  1.027  0.728  0.727
                    0.728  0.161  0.243  0.240  0.243
   4 F    9  9.988  1.999  0.810  1.029  1.027  1.027  1.027  0.727  0.728
                    0.728  0.161  0.240  0.243  0.243
   5 F    9  9.988  1.999  0.810  1.029  1.027  1.027  1.027  0.727  0.728
                    0.728  0.161  0.240  0.243  0.243
   6 F    9  9.988  1.999  0.810  1.027  1.029  1.027  1.027  0.728  0.727
                    0.728  0.161  0.243  0.240  0.243
   7 F    9  9.988  1.999  0.810  1.027  1.027  1.029  1.027  0.728  0.728
                    0.727  0.161  0.243  0.243  0.240
   8 F    9  9.973  1.999  0.809  1.028  1.028  1.028  1.027  0.728  0.728
                    0.728  0.155  0.239  0.239  0.239
   9 LI   3  2.023  1.615  0.371  0.012  0.012  0.013
  10 LI   3  2.023  1.615  0.372  0.012  0.012  0.012
  11 LI   3  2.023  1.615  0.371  0.013  0.012  0.012
  12 LI   3  2.023  1.615  0.371  0.012  0.013  0.012
  13 LI   3  2.023  1.615  0.371  0.012  0.013  0.012
  14 LI   3  2.023  1.615  0.371  0.013  0.012  0.012
  15 LI   3  2.023  1.615  0.372  0.012  0.012  0.012
  16 LI   3  2.023  1.615  0.371  0.012  0.012  0.013

  ATOM    Z CHARGE  SHELL POPULATION

   1 XX   0  0.911  0.000  0.004 -0.084  0.990
   2 F    9  9.988  1.999  3.893  3.209  0.887
   3 F    9  9.988  1.999  3.893  3.209  0.887
   4 F    9  9.988  1.999  3.893  3.209  0.887
   5 F    9  9.988  1.999  3.893  3.209  0.887
   6 F    9  9.988  1.999  3.893  3.209  0.887
   7 F    9  9.988  1.999  3.893  3.209  0.887
   8 F    9  9.973  1.999  3.894  3.210  0.871
   9 LI   3  2.023  1.615  0.408
  10 LI   3  2.023  1.615  0.408
  11 LI   3  2.023  1.615  0.408
  12 LI   3  2.023  1.615  0.408
  13 LI   3  2.023  1.615  0.408
  14 LI   3  2.023  1.615  0.408
  15 LI   3  2.023  1.615  0.408
  16 LI   3  2.023  1.615  0.408

 OVERLAP POPULATION CONDENSED TO ATOMS FOR FIRST   6 NEIGHBORS

 ATOM A   1 XX   ATOM B     CELL      R(AB)/AU R(AB)/ANG    OVPOP(AB)
                  9 LI   (  0  0  0)     3.798     2.010    -0.003
                  2 F    (  0  0  0)     5.372     2.843    -0.007
 . . . . . . . . . 
 ATOM A   2 F    ATOM B     CELL      R(AB)/AU R(AB)/ANG    OVPOP(AB)
                 10 LI   (  0  0  1)     3.798     2.010    -0.001
                  1 XX   (  0  0  0)     5.372     2.843    -0.007
                  3 F    (  0  0  0)     5.372     2.843    -0.028
 . . . . . . . . . 

 ATOM A   9 LI   ATOM B     CELL      R(AB)/AU R(AB)/ANG    OVPOP(AB)
                  1 XX   (  0  0  0)     3.798     2.010    -0.003
                  3 F    (  0 -1  0)     3.798     2.010    -0.001
                 10 LI   (  0  0  0)     5.372     2.843     0.000
 
 MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM
     ALPHA-BETA ELECTRONS
 MULLIKEN POPULATION ANALYSIS - NO. OF ELECTRONS    1.000000

  ATOM    Z CHARGE  SHELL POPULATION

   1 XX   0  0.969  0.000  0.005  0.000  0.000  0.000 -0.084  0.000  0.000
                    0.000  1.015  0.011  0.011  0.011
   2 F    9  0.001  0.000  0.002  0.004  0.004  0.000  0.000  0.000  0.000
                    0.000  0.002 -0.005 -0.005  0.001
   3 F    9  0.001  0.000  0.002  0.004  0.000  0.004  0.000  0.000  0.000
                    0.000  0.002 -0.005  0.001 -0.005
   4 F    9  0.001  0.000  0.002  0.000  0.004  0.004  0.000  0.000  0.000
                    0.000  0.002  0.001 -0.005 -0.005
   5 F    9  0.001  0.000  0.002  0.000  0.004  0.004  0.000  0.000  0.000
                    0.000  0.002  0.001 -0.005 -0.005
   6 F    9  0.001  0.000  0.002  0.004  0.000  0.004  0.000  0.000  0.000
                    0.000  0.002 -0.005  0.001 -0.005
   7 F    9  0.001  0.000  0.002  0.004  0.004  0.000  0.000  0.000  0.000
                    0.000  0.002 -0.005 -0.005  0.001
   8 F    9 -0.001  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000
                    0.000  0.000  0.000  0.000  0.000
   9 LI   3  0.004  0.001 -0.003  0.000  0.000  0.006
  10 LI   3  0.000  0.000  0.000  0.000  0.000  0.000
  11 LI   3  0.004  0.001 -0.003  0.006  0.000  0.000
  12 LI   3  0.004  0.001 -0.003  0.000  0.006  0.000
  13 LI   3  0.004  0.001 -0.003  0.000  0.006  0.000
  14 LI   3  0.004  0.001 -0.003  0.006  0.000  0.000
  15 LI   3  0.000  0.000  0.000  0.000  0.000  0.000
  16 LI   3  0.004  0.001 -0.003  0.000  0.000  0.006

According to Mulliken population analysis one electro is localized at the vacancy site.

The presence of the unpaired electron can be also revealed by charge and spin density maps.
Same input as for perfect crystal.

The maps above correspond to a charge density (left) and a spin density (right).

The lowest level in the map to the left is the highest level in the map to the right.
There is spin density at the vacancy site and at the neighbouring atoms.

A better understanding of the spin density is given by plot of the spin density along a path crossing vacancy and Lithium atoms, and a path crossing vacancy and Fluorine atoms.

In ECHG input, when the coordinates of point B and C coincide, a x-y file is written, with the name RHOLINE.DAT. A blank line separate total density and spin density data. It is saved by the script runprop as filename.RHOLINE 

Hyperfine coupling

A hyperfine structure of the EPR spectra can be detected, due to the interaction between the unpaired spin and the spin of neighbouring nuclei. This information gives better insight into the structure of the defect. The isotropic hyperfine interaction is caused by the non-zero probability of an electron being in the position of a given nuclei. This is only true for the s type orbitals (they are the only ones which do not become zero at the origin). The spherical symmetry of this type of orbitals causes the contribution to be isotropic. The anisotropic contribution is due to the presence of higher order poles, and it is indicative of the deformation of the electronic density with respect to the spherical distribution.

The data obtained from the analysis of the hyperfine interaction can also be simulated in an electronic structure calculation, once the electron density distribution is known. In CRYSTAL, the following directives should be provided:

The spin of the adjacent nuclei is assumed to be known, and is kept in a database inside the code.

Exercise 13: Run a properties calculation with the input above using  S₁₆ and S₃₂ wave function of the defective system 
Compare the isotropic and anisotropic hyperfine coupling constants for S₁₆ and S₃₂.

CRYSTAL input and output

S₁₆ wave function gives the following results:

 *******************************************************************************
 SPIN DENSITY AT THE NUCLEAR POSITIONS
 *******************************************************************************
 POINT  ATOM     X(AU)     Y(AU)     Z(AU)       BOHR**(-3)
   1    1 XX    0.0000    0.0000    0.0000         0.021452
   2    2 F     3.7983    3.7983    0.0000         0.036803
   3    8 F     7.5967    7.5967    7.5967         0.000836
   4    9 LI    0.0000    0.0000   -3.7983         0.022533
   5   10 LI   -3.7983   -3.7983   -3.7983         0.000450

 *******************************************************************************
 HYPERFINE COUPLING CONSTANTS (AN)
 *******************************************************************************
 POINT  ATOM  MASS N0    NUCLEAR      AN (MT)       AN (MHZ)    AN (CM**(-1))
                        G FACTOR
   2    2 F      19    5.2577710    0.55253E+01   0.15485E+03   0.51651E-02
   3    8 F      19    5.2577710    0.12550E+00   0.35172E+01   0.11732E-03
   4    9 LI      6    0.8220575    0.52893E+00   0.14823E+02   0.49445E-03
   4    9 LI      7    2.1709770    0.13969E+01   0.39147E+02   0.13058E-02
   5   10 LI      6    0.8220575    0.10557E-01   0.29587E+00   0.98693E-05
   5   10 LI      7    2.1709770    0.27881E-01   0.78137E+00   0.26064E-04

  CONVERSION FACTORS
     AN (MILLI T)        28.554470 * GN * P(SPIN) (BOHR**(-3))
     AN (MHZ)            28.024940 * AN (MILLI T)
     AN (CM**(-1))   0.3335641E-04 * AN (MHZ)
 TTTTTTTTTTTTTTTTTTTTTTTTTTTTTT ISOTROPIC   TELAPSE        1.08 TCPU        0.06

The atom labeled as ( 9 LI ) is one of the nearest neighbors of the F-centre; the experimental value of Fermi contact, determined for the ⁷Li, is equal to 39.06 MHz [4,5], the calculated one is 39.147 MHz (0.39147E+02).

The anisotropy of hyperfine interaction is described by T tensor evaluated at nuclear position, 
T is diagonalized and its eigenvalues are printed:

 *******************************************************************************
 HYPERFINE ELECTRON-NUCLEAR SPIN INTERACTION TENSOR
 ANISOTROPIC COMPONENTS - THE TENSOR IS TRACELESS
 *******************************************************************************


 POINT   ATOM  MASS NO  NUCLEAR G FACTOR


 TENSOR IN PRINCIPAL AXES SYSTEM
 AA  1.528725E-17 BB  4.716279E-18 CC -2.000353E-17


 POINT   ATOM  MASS NO  NUCLEAR G FACTOR
    2    2 F      19       5.2577710

 TENSOR IN PRINCIPAL AXES SYSTEM
 AA  8.725216E-02 BB -4.296771E-02 CC -4.428445E-02


 POINT   ATOM  MASS NO  NUCLEAR G FACTOR
    3    8 F      19       5.2577710

 TENSOR IN PRINCIPAL AXES SYSTEM
 AA -3.252607E-19 BB -3.089976E-18 CC  3.415237E-18


 POINT   ATOM  MASS NO  NUCLEAR G FACTOR
    4    9 LI      6       0.8220575
    4    9 LI      7       2.1709770

 TENSOR IN PRINCIPAL AXES SYSTEM
 AA -1.431286E-02 BB -1.431286E-02 CC  2.862571E-02


 POINT   ATOM  MASS NO  NUCLEAR G FACTOR
    5   10 LI      6       0.8220575
    5   10 LI      7       2.1709770

 TENSOR IN PRINCIPAL AXES SYSTEM
 AA  3.794708E-18 BB -1.084202E-19 CC -3.686287E-18

 *******************************************************************************
 THE COMPONENTS OF THE TENSOR IN MILLITESLA ARE OBTAINED BY MULTIPLYING
 THE COMPUTED ONES BY 3.4066697*NUCLEAR G FACTOR
 *******************************************************************************
  OTHER CONVERSION FACTORS
     T (MHZ)      =      28.024940 * T (MILLI T)
     T (CM**(-1)) =  0.3335641E-04 * T (MHZ)
 TTTTTTTTTTTTTTTTTTTTTTTTTTTTTT ANISOTROPI  TELAPSE        1.16 TCPU        0.14

References

[ 1] B. Henderson, Defect in crystalline solids, Edward Arnold, Ltd., London (1972).
[ 2] H. Seidel, H.C. Wolf, The Physics of Colour Centers., Ed. Fowler, New York (1968).
[3] John A. Weil, James R. Bolton and John E. Wertz, Electron Paramagnetic Resonance, JOHN WILEY SONS, INC, 605 Third Avenue, New York (1994).
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