# 4. Symmetries¶

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 $$k$$:

$\begin{split}\left|-\right\rangle & \rightarrow & \left|s = 0;s^z=0;N=0; I=I_0\right\rangle\\ \left|\uparrow\right\rangle & \rightarrow & \left|s = \frac{1}{2};s^z=\frac{1}{2};N=1; I=I_k\right\rangle\\ \left|\downarrow\right\rangle & \rightarrow & \left|s = \frac{1}{2};s^z=-\frac{1}{2};N=1; I=I_k\right\rangle\\ \left|\uparrow\downarrow\right\rangle & \rightarrow & \left|s = 0;s^z=0;N=2; I=I_k \otimes I_k = I_0\right\rangle.\end{split}$

Then the MPS tensors factorize into Clebsch-Gordan coefficients and reduced tensors due to the Wigner-Eckart theorem:

$A[i]^{(ss^zNI)}_{(j_L j_L^z N_L I_L \alpha_L);(j_R j_R^z N_R I_R \alpha_R)} = \left\langle j_L j_L^z s s^z \mid j_R j_R^z \right\rangle \delta_{N_L+N,N_R} \delta_{I_L\otimes I, I_R} T[i]^{(sNI)}_{(j_L N_L I_L \alpha_L);(j_R N_R I_R \alpha_R)}.$

This has three important consequences:

1. There is block-sparsity due to the Clebsch-Gordan coefficients. Remember that the Clebsch-Gordan coefficients of abelian groups are Kronecker $$\delta$$‘s. The block-sparsity results in both memory and CPU time savings.
2. 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.
3. Excited states in different symmetry sectors can be obtained by ground-state calculations.

The operators

$\begin{split}\hat{b}^{\dagger}_{k\sigma} & = & \hat{a}^{\dagger}_{k\sigma}\\ \hat{b}_{k\sigma} & = & (-1)^{\frac{1}{2}-\sigma}\hat{a}_{k-\sigma}\end{split}$

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 Wigner-Eckart theorem allows to factorize corresponding matrix elements into Clebsch-Gordan coefficients and reduced matrix elements. Together with the Wigner-Eckart theorem for the MPS tensors, this allows to work with reduced quantities only in CheMPS2. Only Wigner 6-j and 9-j symbols are needed, but never Wigner 3-j symbols or Clebsch-Gordan coefficients.