Water: MP2 and CCSD(T) Energies¶

Hartree–Fock treats each electron in the average field of the others. The difference from the exact energy is the correlation energy. MP2 recovers most of it at low cost, and CCSD(T), the usual reference method for small molecules, recovers more. This example computes both for water in the cc-pVDZ basis set.

Copy and run¶

water-mp2.oqp

mp2/cc-pvdz
geom="""
O      0.000000     0.000000     0.122745
H      0.000000     0.759541    -0.471922
H      0.000000    -0.759541    -0.471922
"""

water-ccsdt.oqp

ccsd_t/cc-pvdz
geom="""
O      0.000000     0.000000     0.122745
H      0.000000     0.759541    -0.471922
H      0.000000    -0.759541    -0.471922
"""

Download water-mp2.oqp · Download water-ccsdt.oqp

openqp water-mp2.oqp
openqp water-ccsdt.oqp

Both start from an RHF reference. All electrons are correlated, including the oxygen 1s pair; add cc(nfzc=1) to the CCSD(T) route to freeze it. cc-pVDZ uses spherical d functions, so water has 24 basis functions. Each run takes 2–3 seconds.

What you get¶

The MP2 block of water-mp2.log separates same-spin and opposite-spin pairs:

  E(reference, SCF)      =       -76.0263514813
  E(MP2, same-spin total)=        -0.0516187358
  E(MP2, opp-spin  ab)   =        -0.1529199712
  E(MP2, correlation)    =        -0.2045387069
  E(MP2, total)          =       -76.2308901882

The CCSD(T) run starts its iterations from the MP2 amplitudes and reports:

  E(reference, SCF)      =       -76.0263514813
  E(CCSD, correlation)   =        -0.2138605036
  E(CCSD, total)         =       -76.2402119848
  E((T), correction)     =        -0.0030969638
  E(CCSD(T), correlation)=        -0.2169574674
  E(CCSD(T), total)      =       -76.2433089486

Method	Total energy (hartree)	Correlation energy (hartree)
RHF	−76.0263515	—
MP2	−76.2308902	−0.2045387
CCSD	−76.2402120	−0.2138605
CCSD(T)	−76.2433089	−0.2169575

MP2 recovers 94% of the CCSD(T) correlation energy here, and the perturbative triples (T) add 3.1 millihartree (1.9 kcal/mol) to CCSD. Differences of this size matter for reaction energies, which is why CCSD(T) is used to benchmark cheaper methods.

Cross-check¶

PySCF 2.13 at the same geometry and basis set, with all electrons correlated, gives RHF −76.02635148, MP2 −76.23089019, CCSD −76.24021200, and CCSD(T) −76.24330897 hartree. OpenQP agrees to 10⁻⁸ hartree.

Try next¶

Freeze the oxygen core with ccsd_t/cc-pvdz cc(nfzc=1): the CCSD(T) correlation energy becomes −0.2148543 hartree, 2.1 millihartree smaller in magnitude.
Repeat with cc-pvtz: the CCSD(T) correlation energy grows to −0.2891606 hartree, one third larger than with cc-pVDZ. Correlation energies converge slowly with the basis set, which is why correlated calculations need larger basis sets than HF or DFT.