Explicit working equations for elements of F (Dˆ ) in case of Fermi-contact and
spin–dipolar operators D̂ in the second-order ADC polarization propagator
approximation
The expressions for the p-h and 2p-2h components of the F (Dˆ ) vector "of modified
transition moments" in the second-order ADC approximation for polarization propagator,
formulated in terms of spin-orbitals, have been derived by Schirmer [J. Schirmer, Phys. Rev.
A 26, 2395, 1982]. Using the intermediate-state representation (ISR) formalism [F. Mertins
and J. Schirmer, Phys. Rev. A 53, 2140, 1996], the F (Dˆ ) quantity can be explained as
~
FJ ( Dˆ )  J Dˆ 0
(1)
where D̂ is one-electron property operator, 0 is the exact electronic (closed-shell) ground
~
~
state, and J are intermediate states. From Eq. (1) it is clear, that the J states are either
of singlet or triplet type, depending on the spin properties of the operator D̂ . In particular, the
Fermi-contact (FC) and spin–dipolar (SD) operators given, respectively, by the expressions
ˆ FC    (riN ) sˆi and
H
N
(2)
i
T  r2 I
3riN riN
iN sˆ
i
5
r
i
iN
ˆ SD  
H
N
(3)
~
couple the (singlet) ground state 0 to the triplet intermediate states J . In Eqs. (2) and
(3), riN  ri  R N is the position vector of electron i with respect to the position of nucleus N,
ŝ i is the spin operator for electron i, and I is the 3×3 unit matrix; the summations run over all
electrons. In case of the FC and SD operators, explicit spin-free working equations
(formulated in terms of spatial orbital) for elements of F (Dˆ ) can be obtained using the
general rules of spin algebra [see, e.g., O. Vahtras, et al., J. Chem. Phys. 96, 6120, 1992]. The
ADC computations then can be carried out in spin-adapted manner, which reduces the
computational costs.
In the following we list the results of the transformation of the F (Dˆ ) matrix elements to
~
the basis of triplet-adapted states J (S = 1, Ms = 0) for the case when operators D̂ are of
~
~
the form suggested by Eqs. (2) and (3). The F (Dˆ ) components in p-h space ( J   jk )
are labeled by indices j and k, referring to unoccupied and occupied Hartree-Fock (HF)
1
orbitals, respectively. Similarly, F (Dˆ ) components in 2p-2h space are labeled by indices i, j,
~
~
k, and l ( J  ijkl ), where the pairs (i, j) and (k, l) refer to unoccupied and occupied
orbitals, respectively. For the 2p-2h components five distinct spin-coupling cases can be
distinguished: cases 1-3 ( i  j , k  l , giving rise to three different spin functions), case 4
( i  j , k  l ), and case 5 ( i  j , k  l ). Also, we make use of the following notations: the
indices r, s, t refer to unoccupied orbital and indices n, v, w refer to the occupied orbitals;
( pq | rs ) denote two-electron Coulomb integrals   p (1)q (1) | r (2)s (2)  involving
spatial parts of the HF orbitals |   ; ijkl  i   j   k  l and  ik   i   k are
combinations of the HF orbital energies  ; Drs  r Dˆ  s are orbital matrix elements of
the operator D̂ .
In the second-order ADC scheme, ADC(2), the p-h components of the F (Dˆ ) vector are
sums of zero-, first- and second-orders terms:
10
( 0)
(1)
( 2, A)
( 2, B )
( 2,C )
( 2, I )
F jk  F jk
 F jk
 F jk
 F jk
 F jk
  F jk
(4)
I 1
The corresponding triplet-adapted expressions read:
(0)
F jk

1
D jk
2
(5)
(1)
F jk

1
D ( jn | sk )
 ns
 jskn
2 ns
(6)
( 2, A)
F jk

1

Dsk
( jn | rv)( 2(ns | vr)  (nr | vs))
(7)
D jv
(rk | sn )( 2(vr | ns )  (vs | nr ))
(8)
2 2 rs  nvrj nvrs
vn
( 2, B )
F jk

1

2 2 rs  knrs nvrs
vn
( 2,C )
F jk

1

Dsv ( jn | rk )(vr | ns)
2 2 rs
 knjr vnsr
(9)
nv
( 2,1)
F jk

1

Dvk
2 rs  vj nvrs
(rn | sv)( 2(nr | js)  (ns | jr))
(10)
vn
2
1
( 2,2)
F jk


Dwk
2 nv  wj nvrj
(nr | vw)( 2(rn | jv)  (rv | jn))
(11)
wr
( 2,3)
F jk

1
D jr

2 rs  kr nvrs
(rn | sv )( 2(nk | vs)  (ns | vk))
(12)
vn
1
( 2, 4 )
F jk


D jt
2 rs  kt knrs
(rk | sn)( 2(tr | ns )  (ts | nr ))
(13)
tn
1
( 2,5)
F jk


Dvs ( jr | nk )( rv | sn)
(14)
Dvs (nv | sr )( jn | rk )
(15)
 kvjs nvrs
2 rs
vn
1
( 2,6)
F jk


 knjr kvjs
2 rs
vn
1
( 2,7)
F jk


Dnr
( jn | vs)(( sk | rv)  (sv | rk ))  (sv | rk )(( js | vn)  ( jn | vs))
(16)
Dvr
( jv | sn)(( nk | rs )  (ns | rk ))  (ns | rk )(( jn | sv)  ( jv | sn))
(17)
2 rs  knjr kvrs
vn
1
( 2,8)
F jk


2 rs  kvjr vnsj
vn
( 2,9)
F jk

1

Dnr ( js | rt )( sn | tk )
2 rs
 knjr knst
(18)
tn
( 2,10 )
F jk

1

Dnr ( jv | rw)(vn | wk )
2 nv
 knjr vwjr
(19)
wr
The 2p-2h components of the F (Dˆ ) vector include only first-orders terms. The
corresponding triplet-adapted expressions read:
Case 1 ( i  j , k  l ):

D

1
 D

(1)
Fijkl
    vk ((il | jv)  (iv | jl ))  k  l     is (( sk | jl )  ( sl | jk ))  i  j   (20)
2
 s   kljs
 
 v   vlij

Case 2 ( i  j , k  l ):


D

1
 D
(1)
Fijkl
    vk ((iv | jl )  (il | jv))  k  l     is (( sl | jk )  ( sk | jl ))  i  j   (21)
2
 s   kljs
 
 v   vlij

3
Case 3 ( i  j , k  l ):
(1)
Fijkl


D

1 
  Dvk

((il | jv)  (iv | jl ))  k  l     is (( sk | jl )  ( sl | jk ))  i  j   (22)
 
2
 s   kljs
 
 v   vlij

Case 4 ( i  j , k  l ):
(1)
Fiikl


1  Dvk
D
(iv | il )  is ( sl | ik )  k  l 

2  v  vlii

s  klis
(23)
Case 5 ( i  j , k  l ):
(1)
Fijkk

D js
1 
 Dvk (iv | jk )

( sk | ik )  i 



2
vkij
kkis
v
s



j


(24)
In Eqs. (20-24) notations i  j imply that the preceding part of the expression on the righthand side of the equation has to be repeated with the indices i and j interchanged.
4