4. SymmetriesΒΆ
CheMPS2 exploits the \(\mathsf{SU(2)}\) spin symmetry, \(\mathsf{U(1)}\) particle number symmetry, and abelian point group symmetries \(\{ \mathsf{C_1}, \mathsf{C_i}, \mathsf{C_2}, \mathsf{C_s}, \mathsf{D_2}, \mathsf{C_{2v}}, \mathsf{C_{2h}}, \mathsf{D_{2h}} \}\) of ab initio quantum chemistry Hamiltonians. Thereto the orbital occupation and virtual indices have to be represented by states which transform according to a particular row of one of the irreps of the symmetry group of the Hamiltonian. For example for orbital \(k\):
Then the MPS tensors factorize into ClebschGordan coefficients and reduced tensors due to the WignerEckart theorem:
This has three important consequences:
 There is blocksparsity due to the ClebschGordan coefficients. Remember that the ClebschGordan coefficients of abelian groups are Kronecker \(\delta\)‘s. The blocksparsity results in both memory and CPU time savings.
 There is information compression for spin symmetry sectors other than singlets, as the tensor \(\mathbf{T[i]}\) does not contain spin projection indices. The virtual dimension associated with \(\mathbf{T[i]}\) is called the reduced virtual dimension \(D_{\mathsf{SU(2)}}\). This also results in both memory and CPU time savings.
 Excited states in different symmetry sectors can be obtained by groundstate calculations.
The operators
of orbital \(c\) transform according to row \((s = \frac{1}{2}; s^z=\sigma; N=\pm 1; I_c)\) of irrep \((s = \frac{1}{2}; N=\pm 1; I_c)\). \(\hat{b}^{\dagger}\) and \(\hat{b}\) are hence both doublet irreducible tensor operators, and the WignerEckart theorem allows to factorize corresponding matrix elements into ClebschGordan coefficients and reduced matrix elements. Together with the WignerEckart theorem for the MPS tensors, this allows to work with reduced quantities only in CheMPS2. Only Wigner 6j and 9j symbols are needed, but never Wigner 3j symbols or ClebschGordan coefficients.
For more information on the exploitation of symmetry in the DMRG method, please read Ref. [SYMM1].
[SYMM1] 
