The CC2 ground-state energy is—similarly to other coupled-cluster energies—obtained from the expression

Ω_{μ1} | = ⟨μ_{1}|Ĥ + [Ĥ,T_{2}]|HF⟩ = 0 , | (10.5) |

Ω_{μ2} | = ⟨μ_{2}|Ĥ + [F,T_{2}]|HF⟩ = 0 , | (10.6) |

Ĥ = exp(-T_{1})H exp(T_{1}), |

The residual of the cluster equations Ω(t_{ai},t_{aibj}) is the so-called vector function. The
recommended reference for the CC2 model is ref. [133], the implementation with the
resolution-of-the-identity approximation, RI-CC2, was first described in ref. [8].

Advantages of the RI approximation:
For RI-CC2 calculations, the operation count and thereby the CPU and the wall time
increases—as for RI-MP2 calculations—approximately with (O^{2}V ^{2}N_{x}), where O is the number
of occupied and V the number of virtual orbitals and N_{x} the dimension of the auxiliary basis set
for the resolution of the identity. Since RI-CC2 calculations require the (iterative) solution of the
cluster equations (10.5) and (10.6), they are about 10–20 times more expensive than MP2
calculations. The disk space requirements are approximately O(2V + N)N_{x} + N_{x}^{2} double
precision words. The details of the algorithms are described in ref. [8], for the error introduced by
the RI approximation see refs. [117,135].

Required input data: In addition to the above mentioned prerequisites ground-state energy calculations with the ricc2 module require only the data group $ricc2 (see Section 21.2.17), which defines the methods, convergence thresholds and limits for the number of iterations etc. If this data group is not set, the program will carry out a CC2 calculation. With the input

$ricc2

mp2

cc2

conv=6

mp2

cc2

conv=6

the ricc2 program will calculate the MP2 and CC2 ground-state energies, the latter converged to
approximately 10^{-6} a.u. The solution for the single-substitution cluster amplitudes is saved in the
file CCR0--1--1---0, which can be kept for a later restart.

Ground-State calculations for other methods than CC2:
The MP2 equations and the energy are obtained by restricting in the CC2 equations the
single-substitution amplitudes t_{ai} to zero. In this sense MP2 can be derived as a simplification of
CC2. But it should be noted that CC2 energies and geometries are usually not more accurate than
MP2.

For CCS and CIS the double-substitution amplitudes are excluded from the cluster expansion and the single-substitution amplitudes for the ground state wavefunction are zero for closed–shell RHF and open–shell UHF reference wavefunctions and thus energy is identical to the SCF energy.

For the Methods CIS(D), CIS(D_{∞}) and ADC(2) the ground state is identified with the MP2
ground state to define is total energy of the excited state, which is needed for the definition of
gradients and (relaxed) first-order properties which are obtained as (analytic) derivatives the total
energy.

Diagnostics:
Together with the MP2 and/or CC2 ground state energy the program evaluates the D_{1} diagnostic
proposed by Janssen and Nielsen [118], which is defined as:

| (10.7) |

where λ_{max}[M] is the largest eigenvalue of a positive definite matrix M. (For CC2 the D_{1}
diagnostic will be computed automatically. For MP2 is must explictly be requested with the
d1diag option in the $ricc2 data group, since for RI-MP2 the calculation of D_{1} will contribute
significantly to the computational costs.) Large values of D_{1} indicate a multireference character of
the ground-state introduced by strong orbital relaxation effects. In difference to the T_{1} and S_{2}
diagnostics proposed earlier by Lee and coworkers, the D_{1} diagnostic is strictly size-intensive and
can thus be used also for large systems and to compare results for molecules of different size. MP2
and CC2 results for geometries and vibrational frequencies are, in general, in excellent agreement
with those of higher-order correlation methods if, respectively, D_{1}(MP2) ≤ 0.015 and
D_{1}(CC2) ≤ 0.030 [11,118]. For D_{1}(MP2) ≤ 0.040 and D_{1}(CC2) ≤ 0.050 MP2 and/or CC2
usually still perform well, but results should be carefully checked. Larger values of D_{1}
indicate that MP2 and CC2 are inadequate to describe the ground state of the system
correctly!

The D_{2} diagnostic proposed by Nielsen and Janssen [119] can also be evaluated. This analysis can
be triggered, whenever a response property is calculated, e.g. dipole moment, with the keyword
$D2-diagnostic. Note that the calculation of D_{2} requires an additional O(N^{5}) step!
D_{2}(MP2/CC2) ≤ 0.15 are in excellent agreement with those of higher-order correlation methods,
for D_{2}(MP2/CC2) ≥ 0.18 the results should be carefully checked.