2 A Comment on the Question of Degeneracies in Quantum Mechanics
49
of quantum origin! Again, as already pointed out, all the aforementioned statements
are merely a consequence of the properties of the harmonic oscillator.
Let us now continue by applying the general transformations, (2.16), (2.18), to
the field Hamiltonian. Here we will only sketch the derivation, since it is, however,
very time-consuming. The final formula for the change of the ground state energy
has a surprisingly simple analytical form. Details of the derivation have been given
in previous work [15] as well as a more comprehensive discussion in [16].
For the correction of the ground state energy we finally get
ΔE 0 =
AI r
˜
ω r
c
r
AI
2 − ω r
˜
c
r
AI
2
(2.26)
where the summation refers to virtual spinorbitals A, occupied spinorbitals I , and
all hypervibrational modes r, r ∈ {V , R, T }. The coefficients c resp. ˜
c are related to
the adiabatic and the non-adiabatic transformation, respectively, and determined by
the set of equations
u
r
P Q +
ε
0
P − ε
0
Q
c
r
P Q +
AI
ν
0
P I QA − ν
0
P I AQ
c
r
AI −
ν
0
P AQI − ν
0
P AI Q
c
r
I A
− ω r ˜
c
r
P Q = ε
r
P δ P Q
(2.27)
ε
0
P − ε
0
Q
˜
c
r
P Q +
AI
ν
0
P I QA − ν
0
P I AQ
˜
c
r
AI −
ν
0
P AQI − ν
0
P AI Q
˜
c
r
I A
− ˜
ω r c
r
P Q = ˜
ε
r
P δ P Q
(2.28)
where u are the coefficients of the electron-hyperphonon interaction, ε 0 are oneelectron energies, and ν 0 two-electron potential energies.
For the derivation we stress the most interesting three limits of Eq. (2.26):
(a) The adiabatic limit, which means that all non-adiabatic coefficients ˜
c will be
equal to zero. Thus, we obtain the adiabatic correction
ΔE 0(ad) =
AI r
˜
ω r
c
r
AI
2 = 2
AI
r∈V
1
2
ω r +
r∈R
ρ r +
r∈T
τ r
c
r
AI
2 (2.29)
which we can directly compare with the Born-Handy ansatz. In the author’s
works [14, 15], the exact CPHF reformulation of the Born-Handy ansatz is displayed, leading to the identity between the field and the mechanical equations
at the adiabatic level
ΔE 0(ad) =
ψ(R)
T N
ψ(R)
R 0
= 2
AI
r∈V
1
2
ω r +
r∈R
ρ r +
r∈T
τ r
c
r
AI
2 .
(2.30)
Numerical verification was performed on the molecules H 2 , HD and D 2 [14].
It was surprising that the vibrational contribution only amounted about 20 %,
49
of quantum origin! Again, as already pointed out, all the aforementioned statements
are merely a consequence of the properties of the harmonic oscillator.
Let us now continue by applying the general transformations, (2.16), (2.18), to
the field Hamiltonian. Here we will only sketch the derivation, since it is, however,
very time-consuming. The final formula for the change of the ground state energy
has a surprisingly simple analytical form. Details of the derivation have been given
in previous work [15] as well as a more comprehensive discussion in [16].
For the correction of the ground state energy we finally get
ΔE 0 =
AI r
˜
ω r
c
r
AI
2 − ω r
˜
c
r
AI
2
(2.26)
where the summation refers to virtual spinorbitals A, occupied spinorbitals I , and
all hypervibrational modes r, r ∈ {V , R, T }. The coefficients c resp. ˜
c are related to
the adiabatic and the non-adiabatic transformation, respectively, and determined by
the set of equations
u
r
P Q +
ε
0
P − ε
0
Q
c
r
P Q +
AI
ν
0
P I QA − ν
0
P I AQ
c
r
AI −
ν
0
P AQI − ν
0
P AI Q
c
r
I A
− ω r ˜
c
r
P Q = ε
r
P δ P Q
(2.27)
ε
0
P − ε
0
Q
˜
c
r
P Q +
AI
ν
0
P I QA − ν
0
P I AQ
˜
c
r
AI −
ν
0
P AQI − ν
0
P AI Q
˜
c
r
I A
− ˜
ω r c
r
P Q = ˜
ε
r
P δ P Q
(2.28)
where u are the coefficients of the electron-hyperphonon interaction, ε 0 are oneelectron energies, and ν 0 two-electron potential energies.
For the derivation we stress the most interesting three limits of Eq. (2.26):
(a) The adiabatic limit, which means that all non-adiabatic coefficients ˜
c will be
equal to zero. Thus, we obtain the adiabatic correction
ΔE 0(ad) =
AI r
˜
ω r
c
r
AI
2 = 2
AI
r∈V
1
2
ω r +
r∈R
ρ r +
r∈T
τ r
c
r
AI
2 (2.29)
which we can directly compare with the Born-Handy ansatz. In the author’s
works [14, 15], the exact CPHF reformulation of the Born-Handy ansatz is displayed, leading to the identity between the field and the mechanical equations
at the adiabatic level
ΔE 0(ad) =
ψ(R)
T N
ψ(R)
R 0
= 2
AI
r∈V
1
2
ω r +
r∈R
ρ r +
r∈T
τ r
c
r
AI
2 .
(2.30)
Numerical verification was performed on the molecules H 2 , HD and D 2 [14].
It was surprising that the vibrational contribution only amounted about 20 %,