**Energy of polarisable H**_{2}O The energy of H

_{2}O in which the H's are not polarisable and where O is polarisable (with polarisability α) can be constructed from the electrostatic energy and the induction energy. The H's have a partial charge of δ+ and at O there is a partial charge of δ2-.

E = U

_{electrostatic} + U

_{induction} (1)

U

_{induction} is the energy that is gained when polarisation is allowed at O. It costs energy to polarise the electrons at O, to make the induced dipoles. The energy gain is that in interaction between the induced moments at O:

U

_{induction} = U

_{polarisation} + U

_{interaction} (2)

where:

U

_{polarisation} = +½αF² (3)

(This follows from the Virial Theorem)

U

_{polarisation} is the energy that it costs to make induced moments at O which has a polarisability α and which is located in the field F generated by the two protons (δ+).

U

_{interaction} = - μF. (4)

This is the general interaction energy of any dipole μ in any field F.

The dipole μ is the induced dipole at O due to the field generated by the H's. It is equal to the field F generated by the two H's times the polarisability of O:

μ = μ

_{induced} = αF

Using this in (4) we get:

U

_{interaction} = - μF = - αFF = -αF² (5)

Now we can write the induction energy U

_{induction} (2) as:

U

_{induction} = U

_{polarisation} + U

_{interaction} = +½αF² -αF² = -½αF² (6)

What remains to complete formula (1) is the electrostatic energy.

U

_{electrostatic} = U

_{H-H} + 2U

_{H-O} (6)

Using this in formula (1) gives the final result:

E = U

_{electrostatic} + U

_{induction} = U

_{H-H} + 2U

_{H-O} -½αF² (7)

If Θ is the H-O-H angle we can show the angle dependency in (7) as:

E(Θ) = U

_{H-H}(Θ) + 2U

_{H-O} -½αF(Θ)² (8)

__See also:__ Virial theorem

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