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Ground State Properties, Gradients and Geometries

For CC2, one distinguishes between orbital-relaxed and unrelaxed properties. Both are calculated as first derivatives of the respective energy with respect to an external field corresponding to the calculated property. They differ in the treatment of the SCF orbitals. In the orbital-relaxed case the external field is (formally) already included at the SCF stage and the orbitals are allowed to relax in the external field; in the orbital-unrelaxed case the external field is first applied after the SCF calculation and the orbitals do not respond to the external field. Orbital-unrelaxed CC2 properties are calculated as first derivatives of the real part of the unrelaxed Lagrangian [101]
Lur CC2(t,$\displaystyle \bar{{t}}$, β) = 〈HF| H| CC〉 + $\displaystyle \sum_{{\mu_1}}^{}$$\displaystyle \bar{{t}}_{{\mu_1}}^{}$μ1|$\displaystyle \hat{{H}}$ + [$\displaystyle \hat{{H}}$, T2]| HF〉 (9.12)
    + $\displaystyle \sum_{{\mu_2}}^{}$$\displaystyle \bar{{t}}_{{\mu_2}}^{}$μ2|$\displaystyle \hat{{H}}$ + [F0 + β$\displaystyle \hat{{V}}$, T2]| HF〉  

with H = H0 + βV--where V is the (one-electron) operator describing the external field, β the field strength, and H0 and F0 are the Hamiltonian and Fock operators of the unperturbed system--by the expression:
Vur CC2 = $\displaystyle \left(\vphantom{\frac{ \partial L^{\rm ur~CC2}(t,\bar{t},\beta) }{\partial \beta}}\right.$$\displaystyle {\frac{{ \partial L^{\rm ur~CC2}(t,\bar{t},\beta) }}{{\partial \beta}}}$$\displaystyle \left.\vphantom{\frac{ \partial L^{\rm ur~CC2}(t,\bar{t},\beta) }{\partial \beta}}\right)_{0}^{}$  = $\displaystyle \sum_{{pq}}^{}$Durpq Vpq  , (9.13)
  = $\displaystyle \bigg($〈HF|$\displaystyle \hat{{V}}$| HF〉 + $\displaystyle \sum_{{\mu_1}}^{}$$\displaystyle \bar{{t}}_{{\mu_1}}^{}$μ1|$\displaystyle \hat{{V}}$ + [V, T2]| HF〉 (9.14)
    + $\displaystyle \sum_{{\mu_2}}^{}$$\displaystyle \bar{{t}}_{{\mu_2}}^{}$μ2|[$\displaystyle \hat{{V}}$, T2]| HF〉$\displaystyle \bigg)$  ,  

where ℜ indicates that the real part is taken. Relaxed CC2 properties (and gradients) are calculated from the the full variational density including the contributions from the orbital response to the external perturbation, which are derived from the Lagrangian [105,13]
Lrel CC2(t,$\displaystyle \bar{{t}}$) = 〈HF| H| CC〉 + $\displaystyle \sum_{{\mu_1}}^{}$$\displaystyle \bar{{t}}_{{\mu_1}}^{}$μ1|$\displaystyle \hat{{H}}$ + [$\displaystyle \hat{{H}}$, T2]| HF〉 (9.15)
    + $\displaystyle \sum_{{\mu_2}}^{}$$\displaystyle \bar{{t}}_{{\mu_2}}^{}$μ2|$\displaystyle \hat{{H}}$ + [F, T2]| HF〉 + $\displaystyle \sum_{{\mu_0}}^{}$$\displaystyle \bar{{\kappa}}_{{\mu_0}}^{}$Fμ0  ,  

where F is the Fock operator corresponding to the Hamiltonian of the perturbed system H = H0 + βV. One-electron properties are then obtained as:
Vrel CC2 = $\displaystyle \bigg($〈HF|$\displaystyle \hat{{V}}$| HF〉 + $\displaystyle \sum_{{\mu_1}}^{}$$\displaystyle \bar{{t}}_{{\mu_1}}^{}$μ1|$\displaystyle \hat{{V}}$ + [V, T2]| HF〉 (9.16)
    + $\displaystyle \sum_{{\mu_2}}^{}$$\displaystyle \bar{{t}}_{{\mu_2}}^{}$μ2|[V, T2]| HF〉 + $\displaystyle \sum_{{\mu_0}}^{}$$\displaystyle \bar{{\kappa}}_{{\mu_0}}^{}$Vμ0$\displaystyle \bigg)$  ,  
  = $\displaystyle \sum_{{pq}}^{}$Drelpq Vpq  . (9.17)

The calculation of one-electron first-order properties requires that in addition to the cluster equations also the linear equations for the Lagrangian multipliers $ \bar{{t}}_{\mu}^{}$ are solved, which requires similar resources (CPU, disk space, and memory) as the calculation of a single excitation energy. For orbital-relaxed properties also a CPHF-like linear equation for the Lagrangian multipliers $ \bar{{\kappa}}_{{\mu_0}}^{}$ needs to be solved and the two-electron density has to be build, since it is needed to set up the inhomogeneity (right-hand side). The calculation of relaxed properties is therefore somewhat more expensive--the operation count for solving the so-called Z-vector equations is similar to what is needed for an SCF calculation--and requires also more disk space to keep intermediates for the two-electron density--about O(2V + 2N)Nx + Nx2 in addition to what is needed for the solution of the cluster equations. For ground states, orbital-relaxed first-order properties are standard in the literature.

The calculation of the gradient implies the calculation of the same variational densities as needed for relaxed one-electron properties and the solution of the same equations. The construction of the gradient contributions from the densities and derivative integrals takes about the same CPU time as 3-4 SCF iterations and only minor extra disk space. For details of the implementation of CC2 relaxed first-order properties and gradients and a discussion of applicability and trends of CC2 ground-state equilibrium geometries see ref. [13]. The following is in example input for a MP2 and CC2 single point calculation of first-order properties and gradients:

  static relaxed operators=diplen,qudlen
A different input is required for geometry optimizations: in this case the model for which the geometry should be optimized must be specified in the data group $ricc2 by the keyword geoopt:
  geoopt model=cc2

For CC2 calculations, the single-substitution part of the Lagrangian multipliers $ \bar{{t}}_{\mu}^{}$ are saved in the file CCL0--1--1---0 and can be kept for a restart (for MP2 and CCS, the single-substitution part $ \bar{{t}}_{\mu}^{}$ vanishes).

For MP2 only relaxed first-order properties and gradients are implemented (unrelaxed MP2 properties are defined differently than in CC response theory and are not implemented). For MP2, only the CPHF-like Z-vector equations for $ \bar{{\kappa}}_{{\mu_0}}^{}$ need to be solved, no equations have to be solved for the Lagrangian multipliers $ \bar{{t}}_{\mu}^{}$. CPU time and disk space requirements are thus somewhat smaller than for CC2 properties or gradients.

For SCF/CIS/CCS it is recommended to use the modules grad and rdgrad for the calculation of, ground state gradients and first-order properties.

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