= Alternative of TD-DFT: particle-particle TDA (pp-TDA) based properties =

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: 0.0000
}}}

Note 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.