10.1 CC2 Ground-State Energy Calculations

The CC2 ground-state energy is—similarly to other coupled-cluster energies—obtained from the expression \[\begin{align} E_{\text{CC}} & = \langle\text{HF}| H |\text{CC}\rangle = \langle\text{HF}| H \exp(T) |\text{HF}\rangle ~~, \\ & = E_{\text{SCF}} + \sum_{iajb} \Big[{t}_{ab}^{ij} + {t}_{a}^{i} {t}_{b}^{j} \Big] \; \Big[ 2(ia|jb) - (ja|ib) \Big] , \end{align}\](10.1–10.2) where the cluster operator $T$ is expanded as $T= T_1 + T_2$ with \[\begin{align} T_1 & = \sum_{ai} t_{a}^{i} \tau_{ai} \\ T_2 & = \frac{1}{2}\sum_{aibj} t_{ab}^{ij} \tau_{aibj} \end{align}\](10.3–10.4) (for a closed-shell case; in an open-shell case an additional spin summation has to be included). The cluster amplitudes $t_{a}^{i}$ and $t_{ab}^{ij}$ are obtained as solution of the CC2 cluster equations [260]: \[\begin{align} \Omega_{\mu_1} & = \langle\mu_{1} | \hat{H} + [\hat{H},T_2] |\text{HF}\rangle = 0 ~~, \\ \Omega_{\mu_2} & = \langle\mu_{2} | \hat{H} + [F,T_2] |\text{HF}\rangle = 0 ~~, \end{align}\](10.5–10.6) with \[\begin{align*} \hat{H} = \exp(-T_1) H \exp(T_1), \end{align*}\] where $\mu_{1}$ and $\mu_{2}$ denote, respectively, the sets of all singly and doubly excited determinants.

The residual of the cluster equations $\Omega(t_{ai},t_{aibj})$ is the so-called vector function. The recommended reference for the CC2 model is ref. [260], the implementation with the resolution-of-the-identity approximation, RI-CC2, was first described in ref. [12].

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 ${\cal O}(O^2V^2N_x)$, 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. [12], for the error introduced by the RI approximation see refs. [243, 267].

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 25.2.23), 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

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$_\infty$) 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 [244], which is defined as: \[\begin{equation} D_1 = \sqrt{\max\Big(\lambda_{\mathrm{max}}\Big[\sum_i t_{ai} t_{bi}\Big], \lambda_{\mathrm{max}}\Big[\sum_a t_{ai} t_{aj}\Big]\Big)} \end{equation}\](10.7) where $\lambda_{\mathrm{max}}[{\mathbf{M}}]$ is the largest eigenvalue of a positive definite matrix $\mathbf{M}$. (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, $D_1(\text{MP2}) \le 0.015$ and $D_1(\text{CC2}) \le 0.030$ [244, 15]. For $D_1(\text{MP2}) \le 0.040$ and $D_1(\text{CC2}) \le 0.050$ 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 [245] 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! $D_2(\text{MP2/CC2}) \le 0.15$ are in excellent agreement with those of higher-order correlation methods, for $D_2(\text{MP2/CC2}) \ge 0.18$ the results should be carefully checked.