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.

For more information on the exploitation of symmetry in the DMRG method, please read Ref. [SYMM1].

SYMM1

  1. Wouters and D. Van Neck, European Physical Journal D 68, 272 (2014), doi: 10.1140/epjd/e2014-50500-1