# Symmetric and antisymmetric squares (crystod-group --product IR IR --symmetric /
# --antisymmetric) and Jahn-Teller modes (crystod-group --jahn-teller);
# testsuite section 9
# [IR x IR] is the symmetric square (characters (chi(g)^2 + chi(g^2))/2): quadratic
# forms of the components, Jahn-Teller couplings, two-electron singlets.
# {IR x IR} is the antisymmetric square (characters (chi(g)^2 - chi(g^2))/2):
# two-electron triplets. Jahn-Teller active modes = [IR x IR] minus the identity irrep;
# with two irreps, the symmetry-allowed pseudo-Jahn-Teller coupling modes of the plain
# product IR1 x IR2 (a necessary condition only: the energy gap and the coupling
# strength decide whether the mixing destabilizes the structure).

$ crystod-group --product T2g T2g --pg m-3m --symmetric --antisymmetric

* Point group *
m-3m

* Direct product *
T2g*T2g

* Result *
 1(A1g) + 1(Eg) + 1(T1g) + 1(T2g)

* Symmetric square *
[T2g x T2g] = A1g + Eg + T2g
dimension: 6 = 1 + 2 + 3

* Antisymmetric square *
{T2g x T2g} = T1g
dimension: 3 = 3


$ crystod-group --jahn-teller Eg --pg m-3m

* Point group *
m-3m

* Jahn-Teller active modes *
[Eg x Eg] = A1g + Eg
JT-active: Eg
  (linear vibronic coupling allowed by symmetry; the Jahn-Teller theorem)


$ crystod-group --jahn-teller T1u --pg m-3m

* Point group *
m-3m

* Jahn-Teller active modes *
[T1u x T1u] = A1g + Eg + T2g
JT-active: Eg + T2g
  (linear vibronic coupling allowed by symmetry; the Jahn-Teller theorem)


$ crystod-group --jahn-teller Eg T2g --pg m-3m

* Point group *
m-3m

* Pseudo-Jahn-Teller coupling *
Eg x T2g = T1g + T2g
symmetry-allowed coupling modes: T1g + T2g
  (necessary condition only: a mode Q can mix the two levels when Gamma_Q
   occurs in the product; whether the mixing destabilizes the high-symmetry
   structure depends on the energy gap and the coupling strength, which
   symmetry does not give)


$ crystod-group --jahn-teller T1u T1u --pg m-3m

* Point group *
m-3m

* Pseudo-Jahn-Teller coupling *
T1u x T1u = A1g + Eg + T1g + T2g
same degenerate level (first-order Jahn-Teller): [T1u x T1u] - A1g = Eg + T2g
two different levels of symmetry T1u, symmetry-allowed coupling modes: Eg + T1g + T2g
  (necessary condition only: a mode Q can mix the two levels when Gamma_Q
   occurs in the product; whether the mixing destabilizes the high-symmetry
   structure depends on the energy gap and the coupling strength, which
   symmetry does not give)


$ crystod-group --jahn-teller A1 E --pg 4mm

* Point group *
4mm

* Pseudo-Jahn-Teller coupling *
A1 x E = E
symmetry-allowed coupling modes: E
  (necessary condition only: a mode Q can mix the two levels when Gamma_Q
   occurs in the product; whether the mixing destabilizes the high-symmetry
   structure depends on the energy gap and the coupling strength, which
   symmetry does not give)


$ crystod-group --product R4+ R4+ --sg Pm-3m --symmetric --antisymmetric

* Space group *
Pm-3m (No. 221)

* K points (primitive basis) *
R: (1/2, 1/2, 1/2)   star of 1 arm(s)
GM: (0, 0, 0)   star of 1 arm(s)

* Direct product (full space-group irreps) *
R4+ x R4+ = GM1+ + GM3+ + GM4+ + GM5+

* Dimension check (star size x small dim) *
3 x 3 = 9 -> 1 + 2 + 3 + 3 = 9

* Symmetric square (full space-group irreps) *
[R4+ x R4+] = GM1+ + GM3+ + GM5+
dimension: 6 = 1 + 2 + 3

* Antisymmetric square (full space-group irreps) *
{R4+ x R4+} = GM4+
dimension: 3 = 3

Cross-validated against the Bilbao Crystallographic Server DIRPRO:
M. I. Aroyo et al., Acta Cryst. A62, 115-128 (2006); https://cryst.ehu.es/rep/dirpro.html



$ crystod-group --product X5+ X5+ --sg Pm-3m --symmetric --antisymmetric

* Space group *
Pm-3m (No. 221)

* K points (primitive basis) *
X: (0, 1/2, 0)   star of 3 arm(s)
GM: (0, 0, 0)   star of 1 arm(s)
M: (1/2, 1/2, 0)   star of 3 arm(s)

* Direct product (full space-group irreps) *
X5+ x X5+ = GM1+ + GM2+ + 2GM3+ + GM4+ + GM5+ + M1+ + M2+ + M3+ + M4+ + 2M5+

* Dimension check (star size x small dim) *
6 x 6 = 36 -> 1 + 1 + 2x2 + 3 + 3 + 3 + 3 + 3 + 3 + 2x6 = 36

* Symmetric square (full space-group irreps) *
[X5+ x X5+] = GM1+ + GM2+ + 2GM3+ + GM5+ + M1+ + M4+ + M5+
dimension: 21 = 1 + 1 + 2x2 + 3 + 3 + 3 + 6

* Antisymmetric square (full space-group irreps) *
{X5+ x X5+} = GM4+ + M2+ + M3+ + M5+
dimension: 15 = 3 + 3 + 3 + 6

Cross-validated against the Bilbao Crystallographic Server DIRPRO:
M. I. Aroyo et al., Acta Cryst. A62, 115-128 (2006); https://cryst.ehu.es/rep/dirpro.html



$ crystod-group --product P1 P1 --sg I4/mcm --symmetric

* Space group *
I4/mcm (No. 140)

* K points (primitive basis) *
P: (1/4, 1/4, 1/4)   star of 2 arm(s)
GM: (0, 0, 0)   star of 1 arm(s)
M: (1/2, 1/2, 1/2)   star of 1 arm(s)

* Direct product (full space-group irreps) *
P1 x P1 = GM4+ + GM3- + M2+ + M1-

* Dimension check (star size x small dim) *
2 x 2 = 4 -> 1 + 1 + 1 + 1 = 4

* Symmetric square (full space-group irreps) *
note: P1 is of complex type; the square is that of the physically irreducible real representation P1P3 (D + D*, dimension 4)
[P1P3 x P1P3] = GM1+ + 2GM4+ + GM2- + 2M2+ + M3+ + 2M1- + M4-
dimension: 10 = 1 + 2x1 + 1 + 2x1 + 1 + 2x1 + 1

Cross-validated against the Bilbao Crystallographic Server DIRPRO:
M. I. Aroyo et al., Acta Cryst. A62, 115-128 (2006); https://cryst.ehu.es/rep/dirpro.html


