where M = Am
i
, in which A is the number of nucleons and E
is the total relativistic energy which has the form
E = Mc
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1+
p
2
M
2
c
2
r
: ð4Þ
The kinetic energy can be decomposed into two contribu-
tions: the first one is the relative space contribution
T
r
, and
the other is the center of mass contribution
T
cm
[34, 35].
T = 〠
i
T
r
−
T
cm
=
∑
A
i
p
i
2
2mA
−
∑
A
i
p
2
i
2mA
: ð5Þ
The second part of Equation (5) can be neglected. This
neglects the center of mass term ð∑
A
i
ðp
2
i
/2mAÞÞ according
to [36]. Applying the binomial theorem for E and substitut-
ing into Equation (3), the relativistic kinetic energy
T takes
the form
T =
∑
A
i
p
i
2
2mA
=
1
2m
〠
A
i=1
p
2
i
−
2
mA
〠
A
i<j
p
2
ij
, ð6Þ
where p
ij
= 1/2ðp
i
− p
j
Þ is the relative momentum of the two
nucleon systems. By substituting Equation (6) into Equation
(2), this leads to the effective nuclear Hamiltonian operator.
H = 〠
A
i
cα
!
i
: p
!
i
+ β
i
− I
ðÞ
m
i
c
2
−
1
2m
p
2
+ 〠
A
i<j
V
ij
+
2
mA
〠
A
i<j
p
2
ij
:
ð7Þ
In Hartree-Fock theory, we seek the best single state
given by the lowest energy expectation value of this
Hamiltonian.
2.1. Variational and Modified Hartree-Fock Wave Function.
One is able to ensure the antisymmetry of the fermions’ wave
functions with the aid of a Slater determinant and Hartree
product to have the convenient form in calculating the
ground-state energy as the following wave function which
is suitable for fermions [33]. So, the wave function of nucleus
ΨðrÞ becomes
Ψ r
ðÞ
=
1
ffiffiffiffiffi
A!
p
det ψ
i
r
!
i
, ð8Þ
where ψ
i
is the nucleon wave function which can be
expanded as
ψ
i
r
!
i
= 〠
α
C
iα
F
α
r
!
i
ð9Þ
where C
iα
is the oscillator constant and F
α
ðr
!
i
Þ is the oscilla-
tor wave function which has two components, radial compo-
nent Φ
α
and spin component χ
α
.
F
a
ji
=
Φ
a
X
a
+
: ð10Þ
The two components have the following relation between
them [17, 37], as
χ =1−
ε − v
2Mc
2
σ
!
· p
!
2mc
ϕ: ð11Þ
The principle of antisymmetry of the wave function was
not clarified by the Hartree method only, but the accurate pic-
ture of the ground-state energy calculations should have the
Hartree-Fock approximation besides the Slater determinant
as in Equation (8) for the wave function. To facilitate the cal-
culation of the wave function, we will use Equation (11) which
enables replacing between the two parts of the oscillator wave
function with the quantities of 1, 2, and 3. Here, we are dealing
with the ground state, so ðε − vÞ/c
2
makes the value of the sec-
ond term very small and can be neglected.
χ ≅
σ
!
· p
!
2mc
ϕ ð12Þ
The wave functions for two nucleons i and j have the for-
mula for bra part hΦ
a
ðr
i
Þϕ
γ
ðr
j
Þj and ket part jϕ
β
ðr
i
Þϕ
δ
ðr
j
Þi
as the bracket needs two wave functions in each side of the
bracket:
ϕ
α
r
i
ðÞ
ϕ
γ
r
j
D
= 〠
m
l
α
m
s
α
〠
m
l
γ
m
s
γ
l
α
s
α
m
l
α
m
s
α
j
α
M
α
j
l
γ
s
γ
m
l
γ
m
s
γ
j
γ
M
γ
ϕ
n
α
l
α
m
l
α
r
i
ðÞ
ϕ
n
γ
l
γ
m
l
γ
r
j
D
χ
1/2
m
s
α
χ
1/2
m
s
γ
D
P
T
α
P
T
γ
D
,
ð13Þ
where ðl
α
s
α
m
l
α
m
s
α
jj
α
M
α
Þ is the Clebsch-Gordon coefficient,
χ
1/2
m
s
α
is the spin function, and
P
T
α
is the function of isotopic
spin. The two wave functions depend on r
i
and r
j
which can
be merged to one wave by changing the special coordinates
foritthatconvertstotherelativeandcenterofmasscoordinates
(see Appendix A for more details). Then, we have the formula
ϕ
α
r
i
ðÞ
ϕ
γ
r
j
D
= 〠
m
l
α
m
s
α
〠
m
l
γ
m
s
γ
〠
lS
〠
λμ
〠
nlNL
〠
mM
〠
sm
s
〠
T
l
α
s
α
m
l
α
m
s
α
j
α
M
α
j
l
γ
s
γ
m
l
γ
m
s
γ
j
γ
M
γ
l
α
l
γ
m
l
α
m
l
γ
λμ
j
n
α
l
α
n
γ
l
γ
NLnl
j
lSm
l
m
S
JM
j
ðÞ
LlMm λμ
j
ðÞ
s
α
s
γ
m
s
α
m
s
γ
SM
s
j
s
α
s
γ
T
α
T
γ
TM
T
j
ϕ
nlm
r
ðÞ
ϕ
NLM
R
ðÞ
hj
χ
S
m
s
i, j
ðÞ
D
P
T
i, j
ðÞ
,
ð14Þ
3Advances in High Energy Physics