Questions: Electron Correlation and Computational Approximations

Question 1

You compute the energy of two non-interacting molecules A and B using CISD: E(A+B) calculated together gives a result that differs from E(A) + E(B) computed separately. What fundamental flaw in CISD does this reveal?

• A. CISD includes too many excited determinants, causing double-counting of correlation
• B. CISD is not size-consistent — its energy does not scale correctly with system size
• C. CISD uses an incorrect basis set for multi-molecule calculations
• D. CISD ignores triple excitations, which become important when two molecules are present

Answer: (B)

Size-consistency means the energy of two non-interacting systems computed together equals the sum of their individual energies. Truncated CI methods like CISD fail this property because when you include only singles and doubles in the full system, you are implicitly treating the two-molecule problem differently than two separate one-molecule problems. Coupled cluster theory solves this by using an exponential ansatz that automatically generates higher-order terms (like products of double excitations) even when the cluster operator is truncated.

Question 2

Which statement best explains why correlation energy matters for chemical predictions even though it is 'small' on an absolute scale?

• A. Correlation energy changes sign near transition states, reversing the predicted reaction direction
• B. Correlation energy (~1 eV per electron pair) is often comparable to the reaction barriers and bond strength differences being predicted
• C. Correlation energy only matters for heavy elements with many electrons, not for organic molecules
• D. Correlation energy affects the molecular geometry but not the electronic energy

Answer: (B)

The correlation energy for a single electron pair is roughly 1 eV — a small fraction of total electronic energy. But the chemical quantities of interest (reaction barriers, bond dissociation energies, relative conformational stabilities) are often measured in fractions of an eV or tens of kJ/mol. Missing 1 eV per electron pair can therefore completely change whether a reaction is predicted to proceed, or which isomer is more stable. This is why the 'small' absolute error has large practical consequences.

Question 3

The Hartree-Fock method ignores electron-electron repulsion largely, which is why it fails for most molecular systems.

• T. True
• F. False

Answer: False

Hartree-Fock does account for electron-electron repulsion — but only in an averaged, mean-field sense. Each electron moves in the static field of all other electrons treated as a smeared-out charge cloud. What HF misses is the instantaneous, dynamic correlation: the fact that electrons actually avoid each other moment-to-moment, reducing their repulsion energy below the mean-field prediction. The difference between the exact (non-relativistic) energy and the HF energy is the correlation energy, which arises from this neglected instantaneous avoidance.

Question 4

CCSD(T) recovers more correlation energy than MP2 for most single-reference molecular systems, at the cost of higher computational scaling.

• T. True
• F. False

Answer: True

MP2 scales as N⁵ and captures a significant fraction of correlation energy through doubly-excited determinants, but misses higher-order effects. CCSD(T) scales as N⁷ and systematically includes singles, doubles, and perturbative triples, recovering roughly 99% of correlation energy for well-behaved single-reference systems. This is why CCSD(T) is called the 'gold standard' — it is the most accurate routine method before switching to multireference approaches. The N⁷ scaling limits it to systems with ~20–50 heavy atoms.

Question 5

What is size-consistency, and why does it matter for quantum chemical calculations?

Model answer: Size-consistency means that the energy of two non-interacting fragments A and B calculated together equals the sum of their energies calculated separately: E(A···B) = E(A) + E(B). It matters because if a method is not size-consistent, energies calculated for small model systems cannot be meaningfully transferred to larger ones, and errors grow non-systematically with system size. Truncated CI methods (CISD) fail size-consistency; coupled cluster methods satisfy it by using an exponential wavefunction ansatz that automatically generates disconnected higher excitations.

Size-consistency is a basic requirement for a method to give chemically meaningful results — otherwise, dissociation energies, intermolecular interaction energies, and any property that involves comparing systems of different sizes will be artificially biased. This is one of the key practical advantages of coupled cluster over truncated CI, and why CC methods are preferred for accurate thermochemical benchmarks despite their higher cost.