7.8 Coupled-Cluster Excited-State and Open-Shell Methods

7.8.19 Active-Space EOM-CC(2,3): Tricks of the Trade (CCMAN only)

Active space calculations are less demanding with respect to the size of a hard drive. The main bottlenecks here are the memory usage and the CPU time. Both arise due to the increased number of orbital blocks in the active space calculations. In the current implementation, each block can contain from 0 up to 16 orbitals of the same symmetry irrep, occupancy, and spin-symmetry. For example, for a typical molecule of C2v symmetry, in a small/moderate basis set (e.g., TMM in 6-31G*), the number of blocks for each index is:

occupied: (α+β)×(a1+a2+b1+b2)=2×4=8
virtuals: (α+β)×(2a1+a2+b1+2b2)=2×6=12
(usually there are more than 16 a1 and b2 virtual orbitals).

In EOM-CCSD, the total number of blocks is O2V2=82×122=9216. In EOM-CC(2,3) the number of blocks in the EOM part is O3V3=83×123=884736. In active space EOM-CC(2,3), additional fragmentation of blocks occurs to distinguish between the restricted and active orbitals. For example, if the active space includes occupied and virtual orbitals of all symmetry irreps (this will be a very large active space), the number of occupied and virtual blocks for each index is 16 and 20, respectively, and the total number of blocks increases to 3.3×107. Not all of the blocks contain real information, some blocks are zero because of the spatial or spin-symmetry requirements. For the C2v symmetry group, the number of non-zero blocks is about 10–12 times less than the total number of blocks, i.e., 3×106. This is the number of non-zero blocks in one vector. Davidson diagonalization procedure requires (2*MAX_VECTORS + 2*NROOTS) vectors, where MAX_VECTORS is the maximum number of vectors in the subspace, and NROOTS is the number of the roots to solve for. Taking NROOTS = 2 and MAX_VECTORS = 20, we obtain 44 vectors with the total number of non-zero blocks being 1.3×108.

In CCMAN implementation, each block is a logical unit of information. Along with real data, which are kept on a hard drive at all the times except of their direct usage, each non-zero block contains an auxiliary information about its size, structure, relative position with respect to other blocks, location on a hard drive, and so on. The auxiliary information about blocks is always kept in memory. Currently, the approximate size of this auxiliary information is about 400 bytes per block. It means, that in order to keep information about one vector (3×106 blocks), 1.2 GB of memory is required! The information about 44 vectors amounts 53 GB. Moreover, the huge number of blocks significantly slows down the code.

To make the calculations of active space EOM-CC(2,3) feasible, we need to reduce the total number of blocks. One way to do this is to reduce the symmetry of the molecule to lower or C1 symmetry group (of course, this will result in more expensive calculation). For example, lowering the symmetry group from C2v to Cs would results in reducing the total number of blocks in active space EOM-CC(2,3) calculations in about 26=64 times, and the number of non-zero blocks in about 30 times (the relative portion of non-zero blocks in Cs symmetry group is smaller compared to that in C2v).

Alternatively, one may keep the MAX_VECTORS and NROOTS parameters of Davidson’s diagonalization procedure as small as possible (this mainly concerns the MAX_VECTORS parameter). For example, specifying MAX_VECTORS = 12 instead of 20 would require 30% less memory.

One more trick concerns specifying the active space. In a desperate situation of a severe lack of memory, should the two previous options fail, one can try to modify (increase) the active space in such a way that the fragmentation of active and restricted orbitals would be less. For example, if there is one restricted occupied b1 orbital and one active occupied B1 orbital, adding the restricted b1 to the active space will reduce the number of blocks, by the price of increasing the number of FLOPS. In principle, adding extra orbital to the active space should increase the accuracy of calculations, however, a special care should be taken about the (near) degenerate pairs of orbitals, which should be handled in the same way, i.e., both active or both restricted.