Matrices and Cofactors

As a cofactor is itself a determinant, we may just consider the determinant of an overlap matrix. A determinant can be expressed as a sum of products of its matrix elements. The derivative of a product of matrix elements is obtained by taking the derivative of one matrix element and multiplying this by the product of the other matrix elements. This has to be done for all the matrix elements in the product, and the results have to be added. Another way to look at a determinant is by expanding it in its first order cofactors (cf. Eq. (11))  

Now the determinant is a linear combination of matrix elements of a row (or column) times the corresponding cofactors. The weight of a certain matrix element in the determinant is given by its first order cofactor. The derivative must be the sum of the derivatives of the matrix elements times their cofactors, like shown in the next equation  

Because cofactors are sub-determinants, one can immediately write down their derivatives. The first order cofactors of first order cofactors are second order cofactors, and first order cofactors of second order cofactors introduce third order cofactors. 

In these equations there is also a sign involved, which depends on the relative positions of the original indices ij,k and / with respect to r and s. The first indices refer to the original overlap matrix, while r and s should refer to the matrix where row i (and j) and column k (and /) have been removed. To keep the equations simple we will omit this sign in our equations  

we now need also a third order reduced density matrix, which involves the corresponding third order cofactors. 

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