272
Appendix
With the help of Eq. (A.1.10), the second-order ground-state energy is given by
E
(2)
0 = = 0 | ˆ
H I |
(1)
0 = −
a
2
a + b − k − l
(A.1.18)
The second-order ground state consists of two terms,
|
(2)
0 =
ˆ
Q 0
E
(0)
0 − ˆ
H 0
ˆ
H I
ˆ
Q 0
E
(0)
0 − ˆ
H 0
ˆ
H I | 0 − E
(1)
0
ˆ
Q 0
(E
(0)
0 − ˆ
H 0 ) 2
ˆ
H I | 0 (A.1.19)
where E
(1)
0 = = 0 | ˆ
H I | 0 . Applying again the resolution of the identity (twice in
the first term) and using the first-order result, |
(2)
0 may be written as
|
(2)
0 =
J =0
c
E
(0)
0 − E
(0)
J
| J J | ˆ
H I | cdi j
V cd[i j]
c + d − i − j
+ E
(1)
0
a
V ab[kl]
( a + b − k − l ) 2
(A.1.20)
Here, the sum over the states | J in the first term comprises single, double, triple,
and quadruple excitations (ν p-νh, ν = 1, 2, 3, 4).
Let us consider the | abkl component in |
(2)
0 . The corresponding amplitude
x
(2)
abkl can be written as
x
(2)
abkl = = abkl |
(2)
0 = −
c
abkl | ˆ
H I | cdi j V cd[i j]
( a + b − k − l )( c + d − i − j )
−
abkl | ˆ
H I − E
(1)
0 | abkl V ab[kl]
( a + b − k − l ) 2
(A.1.21)
Here, we have separated the diagonal contribution in the sum over the 2 p-2h states
which can be combined with the second term of Eq. (A.1.20), effecting the modified
diagonal matrix element abkl | ˆ
H I − E
(1)
0 | abkl in the numerator.
As another example, the 1 p-1h excitations in |
(2)
0 contribute with the amplitudes
x
(2)
ak = = ak |
(2)
0 = −
1
a − k
c
H I | cdi j
V cd[i j]
c + d − i − j
(A.1.22)
Appendix
With the help of Eq. (A.1.10), the second-order ground-state energy is given by
E
(2)
0 = = 0 | ˆ
H I |
(1)
0 = −
a
2
a + b − k − l
(A.1.18)
The second-order ground state consists of two terms,
|
(2)
0 =
ˆ
Q 0
E
(0)
0 − ˆ
H 0
ˆ
H I
ˆ
Q 0
E
(0)
0 − ˆ
H 0
ˆ
H I | 0 − E
(1)
0
ˆ
Q 0
(E
(0)
0 − ˆ
H 0 ) 2
ˆ
H I | 0 (A.1.19)
where E
(1)
0 = = 0 | ˆ
H I | 0 . Applying again the resolution of the identity (twice in
the first term) and using the first-order result, |
(2)
0 may be written as
|
(2)
0 =
J =0
c
E
(0)
0 − E
(0)
J
| J J | ˆ
H I | cdi j
V cd[i j]
c + d − i − j
+ E
(1)
0
a
V ab[kl]
( a + b − k − l ) 2
(A.1.20)
Here, the sum over the states | J in the first term comprises single, double, triple,
and quadruple excitations (ν p-νh, ν = 1, 2, 3, 4).
Let us consider the | abkl component in |
(2)
0 . The corresponding amplitude
x
(2)
abkl can be written as
x
(2)
abkl = = abkl |
(2)
0 = −
c
abkl | ˆ
H I | cdi j V cd[i j]
( a + b − k − l )( c + d − i − j )
−
abkl | ˆ
H I − E
(1)
0 | abkl V ab[kl]
( a + b − k − l ) 2
(A.1.21)
Here, we have separated the diagonal contribution in the sum over the 2 p-2h states
which can be combined with the second term of Eq. (A.1.20), effecting the modified
diagonal matrix element abkl | ˆ
H I − E
(1)
0 | abkl in the numerator.
As another example, the 1 p-1h excitations in |
(2)
0 contribute with the amplitudes
x
(2)
ak = = ak |
(2)
0 = −
1
a − k
c
H I | cdi j
V cd[i j]
c + d − i − j
(A.1.22)