Chemistry Reference
In-Depth Information
is the Rayleigh variational principle for the ground state, E
0
being the true
ground state energy;
«½w
E
1
provided
hc
0
jwi¼
0
ð
4
:
3
Þ
is the Rayleigh variational principle for the first excited state, provided the
trial function
w
is taken orthogonal to the true ground state function
c
0
.
The proofs of these statements are given in Magnasco (2007).
Therefore, evaluation of the integrals in (4.1) under the suitable con-
straints of normalization and orthogonality gives the upper bounds to
the energy of the ground and excited states depicted in Figure 4.1. This is
of fundamental importance in applications, since the variational energy
must always lie above the true energy.
It is easily seen that:
.
The equality sign holds for the exact functions.
.
If the variational function is affected by a first-order error, then the
error in the variational energy is second order. Therefore, energy is
Energy
(< 0)
ε
[
ϕ
]
E
1
ε
[
ϕ
]
E
0
E
0
Figure 4.1
Energy upper bounds to ground (left) and first excited state (right)
