Higher-order equation-of-motion coupled-cluster methods.

The equation-of-motion coupled-cluster (EOM-CC) methods truncated after double, triple, or quadruple cluster and linear excitation operators (EOM-CCSD, EOM-CCSDT, and EOM-CCSDTQ) have been derived and implemented into parallel execution programs. They compute excitation energies, excited-state dipole moments, and transition moments of closed- and open-shell systems, taking advantage of spin, spatial (real Abelian), and permutation symmetries simultaneously and fully (within the spin-orbital formalisms). The related Lambda equation solvers for coupled-cluster (CC) methods through and up to connected quadruple excitation (CCSD, CCSDT, and CCSDTQ) have also been developed. These developments have been achieved, by virtue of the algebraic and symbolic manipulation program that automated the formula derivation and implementation altogether. The EOM-CC methods and CC Lambda equations introduce a class of second quantized ansatz with a de-excitation operator (Y), a number of excitation operators (X), and a physical (e.g., the Hamiltonian) operator (A), leading to the tensor contraction expressions that can be performed in the order of ((...((yx)x)...)x)a or ((...((ax)x)...)x)y at the minimal peak operation cost, where x, y, and a are basis-set representations (i.e., tensors) of the respective operators X, Y, and A. Any intermediate tensor resulting from either contraction order is shown to have at most six groups of permutable indices, and this knowledge is used to guide the computer-synthesized programs to fully exploit the permutation symmetry of any tensor to minimize the arithmetic and memory costs.