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LESSON NOTES · 04

Beyond one determinant: MP2, CI, and CC

Prerequisites: All molecular lessons; perturbation expansions

Prerequisite lessons: Hartree–Fock from constrained variation · Roothaan equations and a reproducible SCF cycle

1. Define correlation relative to the same problem

At fixed geometry, Hamiltonian and finite basis, correlation recovery relative to optimized HF is \(E_{FCI}-E_{HF}\leq0\). Only the complete-basis limit approaches exact nonrelativistic correlation for that Hamiltonian. MP2 is perturbative, CI variational in a determinant subspace, and standard CC projected with an exponential ansatz. These are not interchangeable accuracy labels.

2. Derive second-order perturbation

Use electronic \(H\) only here, adding \(V_{NN}\) at the end. Expand the state and energy in \(\lambda\), project the first-order equation onto an excited determinant, then project back onto the reference to obtain \(E^{(2)}\). Canonical stationary HF eliminates singles (Brillouin’s theorem); the two-body Hamiltonian connects directly to at most doubles. Ordered spin-orbital sums count each double four times, explaining the factor \(1/4\).

\[ \begin{aligned}H(\lambda)&=H_0+\lambda V,\quad H_0=\sum_k f(k),\quad V=H-H_0,\\|\Psi\rangle&=|0\rangle+\lambda|\Psi^{(1)}\rangle+\cdots,\\\langle I|\Psi^{(1)}\rangle&=\frac{\langle I|V|0\rangle}{E_0^{(0)}-E_I^{(0)}},\\E^{(2)}&=\sum_{I\ne0}\frac{|\langle I|V|0\rangle|^2}{E_0^{(0)}-E_I^{(0)}},\\E_{MP2}&=E_{HF}+\tfrac14\sum_{ij}^{occ}\sum_{ab}^{virt}\frac{|\langle ij\Vert ab\rangle|^2}{\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b}.\end{aligned} \]

3. CI: linear variation and size consistency

Vary a linear combination of orthonormal determinants. FCI includes all determinants in the chosen one-particle basis, while CISD retains reference, singles and doubles. For two separated fragments, a double excitation on each gives a global disconnected quadruple. Global CISD omits it, so fragment correlation energies need not add. Truncated CI remains variational but is generally not size-consistent.

\[ |\Psi_{CI}\rangle=\sum_I c_I|\Phi_I\rangle,\quad\delta[c^\dagger Hc-E(c^\dagger c-1)]=0\Rightarrow Hc=Ec. \]

4. CC: exponential ansatz and projection

CCSD retains \(T_1+T_2\), yet the exponential generates higher disconnected excitations such as \(T_2^2/2\). For separated fragments with commuting cluster operators and factorizing references/equations, the exponential factorizes and explains size extensivity. Standard CC is not variational; strong multireference character can cause failure. CCSD(T) has perturbative triples, not full CCSDT.

\[ \begin{aligned}|\Psi_{CC}\rangle&=e^T|\Phi_0\rangle,\quad T=T_1+T_2+\cdots,\quad\bar H=e^{-T}He^T,\\E&=\langle\Phi_0|\bar H|\Phi_0\rangle,\quad0=\langle\Phi_\mu|\bar H|\Phi_0\rangle,\\T&=T_A+T_B,\quad[T_A,T_B]=0\Rightarrow e^T=e^{T_A}e^{T_B}.\end{aligned} \]

Beyond one determinant: MP2, CI, and CC — schematic under the stated model assumptions

Teaching schematic drawn from the equations or algorithm steps in this lesson; not measured data.

Worked small-gap breakdown

Use a dimensionless two-determinant teaching Hamiltonian, not molecular MP2 results. For \(\Delta=1,v=0.1\), exact \(E_-=-0.00990195\) while second order is \(-0.01\). For \(\Delta=0.02,v=0.1\), exact \(E_-=-0.0904988\) but second order gives \(-0.5\), badly wrong. At \(\Delta\to0\) the exact solution stays finite at \(-|v|\) while the perturbation diverges. Lower MP2 energy is not proof of reliability.

\[ H=\begin{pmatrix}0&v\\v&\Delta\end{pmatrix},\quad E_-=\frac{\Delta-\sqrt{\Delta^2+4v^2}}2,\quad E^{(2)}=-v^2/\Delta. \]

Check yourself

Is finite-basis FCI exact for nature?

Answer and reasoning

No, basis and Hamiltonian approximations remain.

Does CCSD contain only doubly excited determinants?

Answer and reasoning

No, exponentiation generates higher disconnected excitations.

Why can MP2 lie below exact energy without violating the variational theorem?

Answer and reasoning

Its truncated perturbation energy is not a trial-state Rayleigh quotient.

References and further reading

Derivations and numerical examples here are original teaching constructions, not copied passages or reported research data.

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