Coupled-Cluster Computation

Coupled-cluster (CC) computation represents a correlated state as an exponential of excitation operators acting on a reference determinant. Standard projected CC gives nonlinear amplitude equations rather than a Hermitian variational eigenproblem. Its residuals can be converged tightly while the chosen excitation truncation remains inaccurate.

Helpful background. Configuration interaction supplies determinants and excitation ranks; Hartree–Fock computation supplies a common reference.

Exponential ansatz and projected equations

Let $|\Phi_0\rangle$ be a reference determinant and write

$$|\Psi\rangle=e^T|\Phi_0\rangle,\qquad T=T_1+T_2+\cdots.$$

In spin-orbital notation, $T_1=\sum_{ia}t_i^a a_a^\dagger a_i$ and $T_2=\frac14\sum_{ijab}t_{ij}^{ab}a_a^\dagger a_b^\dagger a_j a_i$, where $i,j$ are occupied and $a,b$ virtual indices. CCSD retains $T_1+T_2$. Its exponential still produces disconnected triples, quadruples, and higher excitation products; it does not contain independent connected amplitudes of those ranks.

Introduce the similarity-transformed Hamiltonian

$$\bar H=e^{-T}He^T =H+[H,T]+\frac12[[H,T],T]+\cdots.$$

For an ordinary two-body electronic Hamiltonian and pure excitation operators, the connected expansion terminates after the fourth nested commutator. Evaluate connected tensor contractions rather than constructing a dense exponential in determinant space.

The energy and amplitude equations are

$$E=\langle\Phi_0|\bar H|\Phi_0\rangle,\qquad R_\mu=\langle\Phi_\mu|\bar H|\Phi_0\rangle=0,$$

for the chosen excited projections. $\bar H$ is generally non-Hermitian. The projected energy is not the Rayleigh quotient of $e^T|\Phi_0\rangle$, so the usual variational upper bound does not apply.

A solved truncation with a nonzero physical residual

Consider two two-state fragments, each with basis $|0\rangle,|1\rangle$, and the Hermitian model

$$H=h\otimes I+I\otimes h+U n\otimes n,\qquad h=\begin{pmatrix}0&g\\g&\Delta\end{pmatrix},\qquad n=|1\rangle\langle1|,$$

where $\Delta>0$, real $g\ne0$, and $U\geq0$ have energy units. This finite model isolates the projected-equation issue; its two-state fragments are not electronic orbitals and the truncation below is not molecular CCSD.

Use reference $|00\rangle$ and retain only the single-fragment excitations $T=t_A D_A+t_B D_B$, where $D_A=|1\rangle\langle0|\otimes I$ and $D_B=I\otimes|1\rangle\langle0|$. Since $D_A^2=D_B^2=0$ and $[D_A,D_B]=0$,

$$e^T|00\rangle=|00\rangle+t_B|01\rangle+t_A|10\rangle+t_A t_B|11\rangle.$$

The double excitation is present, but its coefficient cannot vary independently. Multiplying these four-dimensional matrices gives

$$E_{\rm CC}=g(t_A+t_B),\qquad R_A=g+\Delta t_A-g t_A^2,\qquad R_B=g+\Delta t_B-g t_B^2.$$

The branch continuous from $|00\rangle$ as $g\to0$ has $t_A=t_B=t=-2g/(\Delta+\sqrt{\Delta^2+4g^2})$. Neither retained equation senses $U$. At this exact root,

$$E_{\rm CC}=\Delta-\sqrt{\Delta^2+4g^2},\qquad \langle11|\bar H|00\rangle=Ut^2,\qquad (H-E_{\rm CC})e^T|00\rangle=Ut^2|11\rangle.$$

Thus tightening the two retained residuals cannot remove the omitted double-excitation residual. At $U=0$ the state is the exact product ground state and the energy is additive. For $U>0$, the normalized right-state Rayleigh quotient is

$$E_{\rm R} =\frac{\langle00|e^{T^\dagger}He^T|00\rangle} {\langle00|e^{T^\dagger}e^T|00\rangle} =E_{\rm CC}+\frac{Ut^4}{(1+t^2)^2}.$$

In a common energy unit, take $\Delta=1$, $g=1/2$, and $U=2$. Direct Hermitian diagonalization gives $E_0\simeq-0.391382381$, between $E_{\rm CC}\simeq-0.414213562$ and $E_{\rm R}\simeq-0.371320344$. The projected energy lies below the exact ground energy even though both retained amplitude residuals vanish. The right-state quotient still obeys the variational bound. This comparison separates iterative convergence, excitation truncation, and the definition of the reported energy.

Iterative solution and computational cost

A common initial guess uses perturbative amplitudes with orbital-energy denominators. Residual-based iterations, DIIS, or Newton-like methods then update the amplitudes. Near-zero denominators signal strong mixing and can destabilize such guesses; level shifting or damping may improve iteration without fixing the inadequacy of a single reference.

Conventional dense CCSD has leading work scaling $O(n_o^2n_v^4)$ and doubles storage $O(n_o^2n_v^2)$, with implementation-dependent intermediates. Integral factorization and local approximations alter cost and introduce further controls.

Require the amplitude residual, energy change, and relevant properties to converge. Multiple nonlinear roots can exist; continuation from weak coupling and comparison against small full-CI cases help identify the intended branch. A tiny residual confirms a root of the truncated equations, not its physical accuracy.

Size extensivity and a two-fragment check

For noninteracting fragments with a product reference and compatible excitation truncation, let $T=T_A+T_B$. The fragment operators commute, hence

$$e^{T_A+T_B}=e^{T_A}e^{T_B}.$$

The exponential automatically produces products such as $T_{2,A}T_{2,B}$ that a combined CISD expansion omits. The projected CC equations factorize and the energy is additive under these separability assumptions. Verify this numerically with two disconnected copies of a small Hamiltonian.

For two electrons in a complete finite orbital space, singles and doubles span all possible excitation ranks. If the exact state has nonzero reference overlap and the correct nonlinear root is reached, CCSD can reproduce full CI in that space. This provides a stringent test of signs and contractions without asserting that CCSD is generally exact.

Properties and strong-correlation limits

For a property defined as a derivative of the projected CC energy, standard response theory introduces left amplitudes through $\langle\Phi_0|(1+\Lambda)e^{-T}$ and a stationary Lagrangian. This accounts for the amplitude response. A normalized expectation value in the right state $e^T|\Phi_0\rangle$ is also mathematically well defined, but at finite truncation it generally differs from this energy derivative. Finite differences of converged energies can independently check response properties.

Large amplitudes, stretched bonds, near degeneracy, and reference sensitivity warn that a single-reference truncation may be inadequate. Common diagnostics are indicators, not universal error bars. Perturbative triples corrections can improve favorable cases but may become unreliable in the same small-denominator regime.

Exercises

Show why $e^{T_2}$ contains quadruple excitations even when $T_4$ is omitted.

Solution

The expansion includes $\tfrac12T_2^2$. Two doubles acting on disjoint occupied and virtual sets create a quadruple excitation. Its coefficient is constrained by products of doubles amplitudes; an independent connected quadruple amplitude would require $T_4$.

For the two-fragment model, restore an independent connected double amplitude by writing $T=t_A D_A+t_B D_B+sD_AD_B$. If an exact eigenvector has nonzero reference overlap and is rescaled to $|\psi\rangle=|00\rangle+c_B|01\rangle+c_A|10\rangle+c_{AB}|11\rangle$, determine $t_A,t_B,s$. What happens to the omitted residual?

Solution

The exponential gives singles $t_A,t_B$ and double coefficient $s+t_A t_B$. Set $t_A=c_A$, $t_B=c_B$, and $s=c_{AB}-c_Ac_B$. Then $e^T|00\rangle=|\psi\rangle$, so $e^{-T}He^T|00\rangle=E|00\rangle$: all three excited projections vanish. The enlarged ansatz spans this complete four-state space whenever the reference overlap is nonzero. It removes the truncation in this model; convergence of a nonlinear solver to the intended root remains a separate issue.

References

• Bartlett, R. J., and M. Musiał, “Coupled-cluster theory in quantum chemistry,” Reviews of Modern Physics 79, 291–352 (2007), doi:10.1103/RevModPhys.79.291 — projected equations, connected structure, properties, and limitations.
• Shavitt, I., and R. J. Bartlett, Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory, Cambridge University Press, 2009 — diagrammatic and algebraic implementations.