Correct sign of matrix elements for permuted configurations

[jagot]

For indistinguishable particles, the order of appearance in the configuration does not matter (hence the Slater determinant). At the moment, when generating matrix elements, EnergyExpressions.jl does not take this into account. In the examples below, the matrix elements between configuration a and each of the other configurations should be equal.

using EnergyExpressions

h = OneBodyHamiltonian() + CoulombInteraction()

a = SlaterDeterminant([:a, :b])
b = SlaterDeterminant([:b, :c])
c = SlaterDeterminant([:c, :b])

Matrix(Matrix(h, [a, b, c]))

3×3 Array{EnergyExpressions.NBodyMatrixElement,2}:
 (a|a) + (b|b) + F(b,a) - G(b,a)  - (a|c) + [b a|c b] - [b a|b c]    (a|c) + [b a|b c] - [b a|c b]
 - (c|a) + [c b|b a] - [c b|a b]  (b|b) + (c|c) + F(c,b) - G(c,b)    - (c|c) - (b|b) + G(c,b) - F(c,b)
 (c|a) + [b c|b a] - [b c|a b]    - (b|b) - (c|c) + G(b,c) - F(b,c)  (c|c) + (b|b) + F(b,c) - G(b,c)

a = SlaterDeterminant([:a, :b, :c])
b = SlaterDeterminant([:b, :c, :d])
c = SlaterDeterminant([:d, :b, :c])

Matrix(Matrix(h, [a, b, c]))

3×3 Array{EnergyExpressions.NBodyMatrixElement,2}:
 (a|a) + (b|b) + (c|c) + F(b,a) - G(b,a) + F(c,a) - G(c,a) + F(c,b) - G(c,b)  (a|d) - [b a|d b] + [b a|b d] - [c a|d c] + [c a|c d]                        (a|d) + [b a|b d] - [b a|d b] + [c a|c d] - [c a|d c]
 (d|a) - [d b|b a] + [d b|a b] - [d c|c a] + [d c|a c]                        (b|b) + (c|c) + (d|d) + F(c,b) - G(c,b) + F(d,b) - G(d,b) + F(d,c) - G(d,c)  (d|d) + (b|b) + (c|c) - G(d,b) + F(d,b) - G(d,c) + F(d,c) + F(c,b) - G(c,b)
 (d|a) + [b d|b a] - [b d|a b] + [c d|c a] - [c d|a c]                        (b|b) + (c|c) + (d|d) + F(c,b) - G(c,b) - G(b,d) + F(b,d) - G(c,d) + F(c,d)  (d|d) + (b|b) + (c|c) + F(b,d) - G(b,d) + F(c,d) - G(c,d) + F(c,b) - G(c,b)

a = SlaterDeterminant([:a, :b, :c, :d])
b = SlaterDeterminant([:c, :d, :e, :f])
c = SlaterDeterminant([:e, :f, :c, :d])
d = SlaterDeterminant([:f, :e, :c, :d])
e = SlaterDeterminant([:e, :f, :d, :c])
f = SlaterDeterminant([:f, :e, :d, :c])

Matrix(Matrix(h, [a, b, c, d, e, f]))

6×6 Array{EnergyExpressions.NBodyMatrixElement,2}:
 (a|a) + (b|b) + (c|c) + (d|d) + F(b,a) - G(b,a) + … + F(c,b) - G(c,b) + F(d,b) - G(d,b) + F(d,c) - G(d,c)  [b a|f e] - [b a|e f]                                                                                        …  - [b a|e f] + [b a|f e]
 [f e|b a] - [f e|a b]                                                                                      (c|c) + (d|d) + (e|e) + (f|f) + F(d,c) - G(d,c) + … + F(e,d) - G(e,d) + F(f,d) - G(f,d) + F(f,e) - G(f,e)       (f|f) + (e|e) + (d|d) + (c|c) - G(f,e) + F(f,e) + … - G(e,d) + F(e,d) - G(e,c) + F(e,c) - G(d,c) + F(d,c)
 [f e|b a] - [f e|a b]                                                                                      (c|c) + (d|d) + (e|e) + (f|f) + F(d,c) - G(d,c) + … - G(d,e) + F(d,e) - G(d,f) + F(d,f) + F(f,e) - G(f,e)       (f|f) + (e|e) + (d|d) + (c|c) - G(f,e) + F(f,e) + … + F(d,e) - G(d,e) + F(c,e) - G(c,e) - G(d,c) + F(d,c)
 [e f|b a] - [e f|a b]                                                                                      - (c|c) - (d|d) - (e|e) - (f|f) - F(d,c) + G(d,c) + … + G(d,e) - F(d,e) + G(d,f) - F(d,f) + G(e,f) - F(e,f)     - (f|f) - (e|e) - (d|d) - (c|c) - F(e,f) + G(e,f) + … - F(d,e) + G(d,e) - F(c,e) + G(c,e) + G(d,c) - F(d,c)
 - [f e|b a] + [f e|a b]                                                                                    - (c|c) - (d|d) - (e|e) - (f|f) + G(c,d) - F(c,d) + … + G(d,e) - F(d,e) + G(d,f) - F(d,f) - F(f,e) + G(f,e)     - (f|f) - (e|e) - (d|d) - (c|c) + G(f,e) - F(f,e) + … - F(d,e) + G(d,e) - F(c,e) + G(c,e) - F(c,d) + G(c,d)
 - [e f|b a] + [e f|a b]                                                                                    (c|c) + (d|d) + (e|e) + (f|f) - G(c,d) + F(c,d) + … - G(d,e) + F(d,e) - G(d,f) + F(d,f) - G(e,f) + F(e,f)    …  (f|f) + (e|e) + (d|d) + (c|c) + F(e,f) - G(e,f) + … + F(d,e) - G(d,e) + F(c,e) - G(c,e) + F(c,d) - G(c,d)