“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

Derive the Fock Equations — The unitary freedom

Exercises

Quantum Chemistry I › Unit 7 · The Fock problem › Derive the Fock Equations › all problems

Practice Problem 5 of 9

The unitary freedom

Problem

Show that mixing two occupied orbitals,

ϕ_{1}^{'} = cos θ ϕ_{1} + sin θ ϕ_{2}

and

ϕ_{2}^{'} = - sin θ ϕ_{1} + cos θ ϕ_{2}

, leaves the Slater determinant (and hence

E

) unchanged. Why is this freedom essential to the worked example?

Solution

Predict before reading on. This justifies the canonicalization step — the rotation that turned $\sum_{j} ε_{jk} ϕ_{j}$ into $ε_{k} ϕ_{k}$ . What property of determinants does the proof reduce to?

A determinant is multiplied by the determinant of any matrix that mixes its columns; a rotation has determinant 1, so $Ψ^{'} = Ψ$ . Because $E$ only sees the determinant, the minimizer is a whole unitary orbit of orbital sets — and we may pick the member that diagonalizes $ε$ . Without this freedom there would be no orbital energies, only a coupled multiplier matrix.