12.2 Gradients Theory

All details on the theory and results are published in [153]. The RI-RPA energy is a function of the MO coefficients C and the Lagrange multipliers ϵ and depends parametrically (i) on the interacting Hamiltonian Ĥ, (ii) on the AO basis functions and the auxiliary basis functions. All parameters may be gathered in a supervector X and thus

ERIRPA ≡ ERIRPA (C, ϵ|X ).
(12.9)

C and ϵ in turn depend parametrically on X, the exchange-correlation matrix VXC, and the overlap matrix S through the KS equations and the orbital orthonormality constraint. First-order properties may be defined in a rigorous and general fashion as total derivatives of the energy with respect to a “perturbation” parameter ξ. However, the RI-RPA energy is not directly differentiated in our method. Instead, we define the RI-RPA energy Lagrangian

LRIRPA(C,E,DΔ,W|X,VXC,S)
= ERIRPA(C,E|X) + σ(⟨D Δσ (CTσF σCσ - Eσ)⟩- ⟨W σ(CTσSCσ - 1 )⟩). (12.10)
C, E, DΔ, and W are independent variables. LRIRPA is required to be stationary with respect to C, E, DΔ, and W. DΔ and W act as Lagrange multipliers enforcing that C and E satisfy the KS equations and the orbital orthonormality constraint,
(   RIRPA)
  ∂L------
   ∂D Δσstat = CσTFσCσ -Eσ = 0, (12.11)
( ∂LRIRPA)
   ∂W σstat = CσTSCσ - 1 = 0. (12.12)
DΔ and W are determined by the remaining stationarity conditions,
(        )
  ∂LRIRPA-    = 0,
    ∂E     stat
(12.13)

and

(   RIRPA)
  ∂L------    = 0.
    ∂C     stat
(12.14)

It turns out from eqs (12.13) and (12.14) that the determination of DΔ and W requires the solution of a single Coupled-Perturbed KS equation. Complete expressions for DΔ and W are given in [153]. At the stationary point “stat = (C = C,E = ϵ,DΔ = DΔ,W = W)”, first-order RI-RPA properties are thus efficiently obtained from

dERIRPA-(C,-ϵ|X-)
      dξ = ⟨ (        )      ⟩
   ∂LRIRPA-    dX-
      ∂X    stat dξ + ⟨ (        )   (      )   ⟩
   ∂LRIRPA-      ∂VXC--
     ∂VXC   stat   ∂ ξ   stat
+ ⟨( ∂LRIRPA )   dS ⟩
   --------    ---
      ∂S    statdξ. (12.15)
Finally, the RPA energy gradients may be explicitly expanded as follows:
dERIRPA(C,ϵ|X)-
      dξ = ⟨          ⟩
   RIRPAdh-
  D      dξ + ⟨         ⟩
   (4)dΠ-(4)
 Γ    dξ + ⟨    (         )   ⟩
   Δ   ∂VXC-[D-]
  D      ∂ ξ    stat
+ ⟨       (3)⟩
 Γ (3)dΠ---
      dξ + ⟨       (2)⟩
  Γ (2)dΠ--
       dξ-⟨      ⟩
  W dS-
     dξ. (12.16)
where DRIRPA is the KS ground state one-particle density matrix D plus the RI-RPA difference density matrix DΔ which corrects for correlation and orbital relaxation effects. h is the one-electron Hamiltonian; Π(234) are 2-, 3-, and 4-centre electron repulsion integrals and the Γ(234) are the corresponding 2-, 3-, and 4-index relaxed 2-particle density matrices; W may be interpreted as the energy-weighted total spin one-particle density matrix.

This result illustrates the key advantage of the Lagrangian method: Total RI-RPA energy derivatives featuring a complicated implicit dependence on the parameter X through the variables C and ϵ are replaced by partial derivatives of the RI-RPA Lagrangian, whose computation is straightforward once the stationary point of the Lagrangian has been fully determined.

Gradients Prerequisites

Geometry optimizations and first order molecular property calculations can be executed by adding the keyword rpagrad to the $rirpa section in the control file. RPA gradients also require

The following gradient-specific options may be further added to the $rirpa section in the control file

In order to run a geometry optimization, jobex must be invoked with the level set to rirpa, and the -ri option (E.g. jobex -ri -level rirpa).

In order to run a numerical frequency calculation, NumForce must be invoked with the level set to rirpa, e.g., NumForce -d 0.02 -central -ri -level rirpa.