Hermitian-Lagrange multipliers

Associate the Lagrange multiplier ji (chemical potential) with the normalization condition in Eq. (6), the set of Hermitian-Lagrange multipliers X[ with orthonormality constraints in Eq. (4), and define the auxiliary functional Q, by the formula... 

From (3.40), 8 is a Hermitian matrix. It is always possible, therefore, to find a unitary matrix U such that the transformation (3.67) diagonalizes 8. We are not concerned with how to obtain such a matrix, only that such a matrix exists and is unique. There must exist, then, a set of spin orbitals for which the matrix of Lagrange multipliers is diagonal. 

The matrix of the Lagrange multipliers e is hermitian, therefore there exists a unitary transformation U that diagonalizes e. Applying this unitary transformation to the set of MOs j> ... 

If we interchange v and u in the above derivation and require that the matrix of Lagrange multipliers be Hermitian, we get the equation... 

It is also easily established (Problem 12.10) that the matrix of Lagrange multipliers is Hermitian, and (12.6.7) is thus exactly like the usual stationary-state HF equation (6.1.18) except for the presence of the time derivative. 

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