VbvNeuE-001

Chapter 5
The one-electron bond
between two protons
Let us consider a quantum system consisting of two protons and one electron.
If protons are separated by a large distance, this system consists of a hydrogen
atom and the proton. If the hydrogen atom is at the origin, then the operator
of energy and wave function of the ground state have the form:
H (1)
0 = − 2
2m ∇2 r − e2
r , ϕ 1 = 1 √
πa
3 e− r a (5.1)
If hydrogen is at point R, then respectively
H (2)
0 = − 2 e 2
2m ∇2 r −
| −→ R − −→ r | , ϕ 2 = √ 1 |−→ R − −→ r |
πa
3 e− a (5.2)
In the assumption of fixed protons, the Hamiltonian of the total system has
the form:
H = − 2
2m ∇2 r − e2
r − e 2
| −→ R − −→ r | + e2
(5.3)
R
At that if one proton removed on infinity, then the energy of the system is
equal to the energy of the ground state E 0 , and the wave function satisfies the
stationary Schrodinger equation:
H (1,2)
0 ϕ 1,2 = E 0 ϕ 1,2 (5.4)
We seek a zero-approximation solution in the form of a linear combination of
basis functions:
ψ = a 1 (t)ϕ 1 + a 2 (t)ϕ 2 (5.5)
where coefficients a 1 (t) and a 2 (t) are functions of time, and the desired function
satisfies to the energy-dependent Schrodinger equation:
i dψ
dt = (H(1,2) 0 + V 1,2 )ψ, (5.6)
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