pp-RPA is a method developed by Weitao Yang et al. (van Aggelen, Yang, Yang, PRA 2013, 88, 030501; Yang, van Aggelen, Yang, JCP 2013, 139, 224105) that gives the ground state and excited state energies of a N-electron system starting from a (N-2)-electron reference. It is particularly suitable for systems where the desired N-electron system is strongly correlated but the corresponding (N-2)-electron system is weakly correlated. A prototypical example is the BH molecule, where two of the valence electrons form a weakly correlated B-H sigma bond, but the rest two valence electrons are strongly correlated.
We recommend the TDA variant of pp-RPA, i.e. pp-TDA, which decouples the particle-particle part from the hole-hole part (hh-TDA) so that the results are easier to interpret. The resulting error is small (~0.1 eV). The full pp-RPA code is still under debugging and is not recommended for use.
The following input performs a pp-TDA calculation on the BH molecule. Note that the SCF part is actually calculating [BH]2+, the (N-2)-electron system. Besides, the only difference from TD-DFT is to set imethod=4, itda=1, pprpa=1. The keyword pprpa separate the calculations of N-electron states (=1, pp-TDA) and (N-4)-electron states (=2, hh-TDA):
$COMPASS Title bh Basis cc-pvdz Geometry B 0. 0. 0. H 0. 0. 1.232 End geometry skeleton group c(2v) $END $xuanyuan direct schwarz $end $scf RKS dft b3lyp charge 2 spin 1 THRESHCONV 1.d-10 1.d-8 $end $tddft imethod 4 isf 0 # calculates singlet states; change to 1 for triplet states itda 1 iexit 10 pprpa 1 $end
The resulting "excitation energies" are actually E(N)-E(N-2) (where the (N-2)-electron system is at the ground state but the N-electron system is not necessarily at the ground state), so expect that most of them be negative:
No. Pair ExSym ExEnergies Wavelengths f D<S^2> Dominant Excitations IPA Ova En-E1
1 A1 2 A1 -38.4575 eV -32.24 nm 0.0000 0.0000 87.1% VV(0): A1( 3 )-> A1( 3 ) -51.538 0.000 0.0000
2 B2 1 B2 -35.2970 eV -35.13 nm 0.0000 0.0000 90.8% VV(0): B2( 1 )-> A1( 3 ) -48.452 0.000 3.1605
3 B1 1 B1 -35.2970 eV -35.13 nm 0.0000 0.0000 90.8% VV(0): B1( 1 )-> A1( 3 ) -48.452 0.000 3.1605
4 A1 3 A1 -32.4585 eV -38.20 nm 0.0000 0.0000 44.7% VV(0): B1( 1 )-> B1( 1 ) -45.366 0.000 5.9989
5 A2 1 A2 -32.4585 eV -38.20 nm 0.0000 0.0000 89.5% VV(0): B2( 1 )-> B1( 1 ) -45.366 0.000 5.9989
6 A1 4 A1 -31.2166 eV -39.72 nm 0.0000 0.0000 41.5% VV(0): B2( 1 )-> B2( 1 ) -45.366 0.000 7.2408
...Herein the first state, dominated by VV(0): A1( 3 )-> A1( 3 ), is the ground state of the N-electron system (the arrow is somewhat misleading here, it should better be interpreted as "adding two electrons onto the A1(3) orbital of the reference"; analogously, state 2 is dominated by the addition of two electrons onto the B2(1) and A1(3) orbitals of the reference, respectively). The rest are excited states of the N-electron system, whose excitation energies can be read off from the En-E1 column (unit: eV). The absolute energy of the ground state can be obtained by adding -38.4575 eV to the SCF energy of the (N-2)-electron system, -24.11146566 a.u., or alternatively it can be directly read off from earlier sections of the output file:
No. 1 w= -38.4575 eV -25.5247507904 a.u. f= 0.0000 D<Pab>= 0.0000 Ova= 0.0000
VV(0): A1( 3 )-> A1( 3 ) c_i: -0.9335 Per: 87.1% IPA: -51.538 eV Oai: 0.0000
VV(0): A1( 4 )-> A1( 3 ) c_i: -0.2075 Per: 4.3% IPA: -38.814 eV Oai: 0.0000
VV(0): A1( 5 )-> A1( 3 ) c_i: -0.1371 Per: 1.9% IPA: -33.656 eV Oai: 0.0000
VV(0): A1( 6 )-> A1( 3 ) c_i: 0.1328 Per: 1.8% IPA: -32.570 eV Oai: 0.0000
VV(0): B1( 1 )-> B1( 1 ) c_i: 0.1326 Per: 1.8% IPA: -45.366 eV Oai: 0.0000
VV(0): B2( 1 )-> B2( 1 ) c_i: 0.1326 Per: 1.8% IPA: -45.366 eV Oai: 0.0000Note that, as for now, the code only supports RHF/RKS references and Abelian point group symmetry.
The gradients of single states, and the transition dipole moments and NAC between two states can be computed similar to those for TD-DFT using the resp module.