Bernoulli Expansion based Unitary Coupled-Cluster - #581
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Adds the bottom layer of the Bernoulli expansion of the unitary-CC similarity-transformed Hamiltonian: a Wick reduction that retains partial contractions, so a product of normal-ordered operators reduces to a sum of normal-ordered operators rather than collapsing to a scalar vacuum average, and the normal-ordered commutator built on it. WickTheorem::use_topology is disabled explicitly rather than left alone: it defaults to ON (wick.hpp), and its one-representative-times-multiplicity bookkeeping is only exercised by the fully-contracted path. On this partial-contraction path it rescales terms whose amplitude pairs are symmetric, which leaves vacuum averages correct while corrupting projections onto excited manifolds. wick_commutator reindexes B's summed indices to fresh temporaries before forming A*B, since A and B are independently constructed and may otherwise share labels, which would fuse two independent summations.
Adds the second layer: the split of an operator O into O_N, "the non-diagonal part containing all the excitation and de-excitation operators" (defined above Eq. (43) of 10.1063/1.5030344), and the rank-preserving remainder O_R = O - O_N. The Bernoulli expansion's inner commutators carry N/R subscripts, so every nesting level needs this classification. Classification needs definite index spaces, so expand_to_blocks first rewrites each general index of the residual NormalOperator as a sum over the base spaces it spans. Only the hole and particle spaces are expanded over: in the single-reference setting the remaining base spaces are empty, so restricting to those keeps the expansion 2-way per index instead of compounding across the nested commutators. That makes the routine single-reference only, which the header warns about. The rank cutoff mirrors pdaggerq (nt_bra > bernoulli_excitation_level -> R) rather than the paper's uncapped O_N, since that is the convention defining qUCCSD and the one the numbers are validated against; terms above the cutoff fall to R rather than being dropped.
Adds the top layer: hbar(N, rank, skip1) sums H̄⁰..H̄^rank of 10.1063/1.5030344 Eq. (45), each order transcribed from its equation with the published coefficients and per-level N/R subscripts. Bernoulli numbers B₁=-1/2, B₂=1/12, B₃=0, B₄=-1/720 (Eq. 40) enter as those coefficients; a subscript R/N on a commutator means "form the commutator, then keep only its R/N part before the next nesting", which is what the split from the previous commit provides. Two cancellations from the paper are relied on and noted in place: F enters H̄ only at first order (stated just below Eq. (50)), and the higher orders carry only R-subscripted inner commutators. Every term is a nested commutator whose prefix is shared with other terms, within a rank and across ranks, so nest() memoizes each prefix (keyed by the base operator plus the tags applied so far). The nine rank-4 terms have only 3 distinct level-1 and 6 distinct level-2 nodes. Reusing a memoized ExprPtr is safe because expression composition deep-copies its operands. Contributions accumulate through Sum::append rather than chained operator+, which deep-copies the whole accumulated Sum on every call and is quadratic in the term count at high rank.
Adds CC::Options::hbar_expansion (BCH by default, Bernoulli opt-in) and dispatches CC::hbar() to bernoulli::hbar() when it is selected. Two constructor assertions guard the combination: the Bernoulli expansion is defined for the unitary ansatz only, and it requires an explicit hbar_comm_rank, since CC::hbar() otherwise falls back to rank 4 and would silently select the most expensive and least exercised order. CC::energy() takes the plain reference expectation value under this expansion: the tensor-level H̄ is already fully expanded, so no operator connectivity remains to constrain. Its comm_rank argument defaults to the amplitude rank and is passed explicitly for the qUCCSD [2|3] split, where the energy is taken at H̄³ while the amplitudes stop at H̄².
Pins the derived equations at Bernoulli ranks 1-3 for the unitary ansatz: term counts for the energy and for the singles/doubles residuals, plus the guards on invalid configurations (Bernoulli with a non-unitary ansatz, and Bernoulli without an explicit hbar_comm_rank). The rank-3 numbers (46 energy, 32 singles, 38 doubles terms) are the ones cross-checked term-by-term against pdaggerq, so a change here means the derivation changed.
Corrections: - Eq. (45) is the assembly H̄ = Σ_k H̄^k; (46)-(50) are the rank-by-rank operators. The file header attributed (45)-(50) to the latter. - expand_to_blocks: the SR "o"/"g" base spaces are not empty, so the old justification for dropping them was wrong. They are droppable because the single-reference projection annihilates those terms. The header @warning said the same wrong thing. - R_part's result is not block-resolved; it stays in compact general-index form. The header claimed the opposite. - wick_reduce leaves at most one residual NormalOperator, not exactly one -- fully-contracted terms carry none, which find_nop already handled. - The memo shares level-1 nodes across ranks; it is not a prefix relation. - The use_topology rescaling set is {2, 1/2, 1/3, 8/3, 2/3}; the comment said 3 where it should have said 1/3.
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Adds the Bernoulli expansion of the UCC similarity-transformed Hamiltonian$\bar{H} = e^{-\sigma} H e^{\sigma}$ , $\sigma = T - T^\dagger$ , as an alternative to the BCH expansion in $\bar H^0$ – $\bar H^4$ , Eqs. (45)–(50) of Ref. 1.
mbpt::CC. RanksGround state energies are validated, excited states pending.
More details to be added.
References
self-consistent polarization propagator theory: A third-order formulation and
pilot applications, J. Chem. Phys. 148, 244110 (2018).
doi:10.1063/1.5030344 — Sec. III B, the
reference this module implements.