Megascopic Quantum Phenomena
329
(3) the Fröhlich’s effective two-electron interaction [102]
(4) the Lee-Low-Pines polarons and their self-energy [106].
During the numerical tests in 1998 the above mentioned theory totally failed in the
calculations of the first correction beyond the B-O approximation, i.e. the adiabatic
correction known as the Born-Huang ansatz [7]:
E 0(ad) = ψ 0 (R)|T N |ψ 0 (R) R 0
(11.19)
This was a great disappointment for me. In order to directly compare the field formula
for the adiabatic correction with the mechanical Born-Huang ansatz, I did rewrite
the latter into its field form. In the adiabatic case the N-electron function ψ 0 (R) can
be expanded as a single Slater determinant though the one-electron functions ϕ I (R):
ψ 0 (R) =
1
√
N !
N
I
ϕ I (R)
(11.20)
The function ϕ P (R) can be expanded in terms of the coefficients c P Q (R), dependent
on the nuclear coordinates, and the orthonormal set of one-electron wavefunctions
defined in the equilibrium position R 0 :
ϕ P (R) =
Q
c P Q (R) ϕ Q (R 0 )
(11.21)
In the following work we will use the following notation for the spin-orbitals: I, J,
K, L—occupied; A, B, C, D—virtual (unoccupied); P, Q, R, S—the arbitrary ones.
One obtains then a very simple expression for the Born-Huang ansatz (11.19)
E 0(ad) =
AI iα
2
2M i
c
iα
AI
2
(11.22)
where M i stands for the nuclear mass and α for the Cartesian coordinates.
It is now useful to proceed from the Cartesian to the normal coordinate system
using the following transformation
c
r
P Q =
iα
c
iα
P Q α
r
iα
(11.23)
One wants to substitute for c
iα
P Q in (11.22), so we need to know the inverse matrix
β
r
ia . From
α
+
β = I
(11.24)
Equation (11.22) can be expressed in normal coordinates as
329
(3) the Fröhlich’s effective two-electron interaction [102]
(4) the Lee-Low-Pines polarons and their self-energy [106].
During the numerical tests in 1998 the above mentioned theory totally failed in the
calculations of the first correction beyond the B-O approximation, i.e. the adiabatic
correction known as the Born-Huang ansatz [7]:
E 0(ad) = ψ 0 (R)|T N |ψ 0 (R) R 0
(11.19)
This was a great disappointment for me. In order to directly compare the field formula
for the adiabatic correction with the mechanical Born-Huang ansatz, I did rewrite
the latter into its field form. In the adiabatic case the N-electron function ψ 0 (R) can
be expanded as a single Slater determinant though the one-electron functions ϕ I (R):
ψ 0 (R) =
1
√
N !
N
I
ϕ I (R)
(11.20)
The function ϕ P (R) can be expanded in terms of the coefficients c P Q (R), dependent
on the nuclear coordinates, and the orthonormal set of one-electron wavefunctions
defined in the equilibrium position R 0 :
ϕ P (R) =
Q
c P Q (R) ϕ Q (R 0 )
(11.21)
In the following work we will use the following notation for the spin-orbitals: I, J,
K, L—occupied; A, B, C, D—virtual (unoccupied); P, Q, R, S—the arbitrary ones.
One obtains then a very simple expression for the Born-Huang ansatz (11.19)
E 0(ad) =
AI iα
2
2M i
c
iα
AI
2
(11.22)
where M i stands for the nuclear mass and α for the Cartesian coordinates.
It is now useful to proceed from the Cartesian to the normal coordinate system
using the following transformation
c
r
P Q =
iα
c
iα
P Q α
r
iα
(11.23)
One wants to substitute for c
iα
P Q in (11.22), so we need to know the inverse matrix
β
r
ia . From
α
+
β = I
(11.24)
Equation (11.22) can be expressed in normal coordinates as