Canonical diagonalization

The deviation of the diagonal Lagrangian-multipliers (see Eq. 6, 7) obtained for the orbitals after the given transformation from the canonical diagonal Fock-matrix elements. 

The ACSE has important connections to other approaches to electronic structure including (i) variational methods that calculate the 2-RDM directly [36-39] and (ii) wavefunction methods that employ a two-body unitary transformation including canonical diagonalization [22, 29, 30], the effective valence Hamiltonian method [31, 32], and unitary coupled cluster [33-35]. A 2-RDM that is representable by an ensemble of V-particle states is said to be ensemble V-representable, while a 2-RDM that is representable by a single V-particle state is said to be pure V-representable. The variational method, within the accuracy of the V-representabihty conditions, constrains the 2-RDM to be ensemble N-representable while the ACSE, within the accuracy of 3-RDM reconstruction, constrains the 2-RDM to be pure V-representable. The ACSE and variational methods, therefore, may be viewed as complementary methods that provide approximate solutions to, respectively, the pure and ensemble V-representabihty problems. 

Unitary coupled-cluster theory [33, 50]. Canonical diagonalization [22]. 

First consider a Hartree-Fock reference function and transform to the Fermi vacuum (aU occupied orbitals are in the vacuum). Then all particle density matrices are zero and the cumulant decomposition, Eq. (23), based on this reference corresponds to simply neglecting aU three and higher particle-rank operators generated by commutators. This type of operator truncation is used in the canonical diagonalization theory of White [22]. 

If the wave function that one considers, is a single Slater determinant , the spin orbitals (p, from which , is constructed, are not uniquely determined, but rather there is an infinity of equivalent sets of qo, related by unitary transformations. To some extent one can make the qo, unique if one requires either that they are canonical (diagonalize the Fock operator) and are symmetry-adapted, or localized (e.g. according to the criteria of Edmiston and Ruedenberg or Foster and Boys [1-3]). The localized spin orbitals have some advantages both for the chemical interpretation and for the computation of correlation corrections. 

Also the canonical diagonal AIM hardnesses, tih = ya (electron repulsion integrals), can be interpolated, rja = ftiiCffi). to account for the actual AIM charges, again using the known final difference estimates [18],... 

As has already been mentioned, the Riemannian metric g j can always be reduced at one point to a canonical diagonal form Sij by a choice of appropriate local coordinates on the manifold. Precisely in the same manner, the symplectic... 

The sum over eoulomb and exehange interaetions in the Foek operator runs only over those spin-orbitals that are oeeupied in the trial F. Beeause a unitary transformation among the orbitals that appear in F leaves the determinant unehanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to ehoose sueh a unitary transformation to make the 8i j matrix diagonal. Upon so doing, one is left with the so-ealled canonical Hartree-Fock equations ... 

The equations may be simplified by choosing a unitary transformation (Chapter 13) which makes the matrix of Lagrange multipliers diagonal, i.e. Ay 0 and A This special set of molecular orbitals (f> ) are called canonical MOs, and they transform eq. (3.40) mto a set of pseudo-eigenvalue equations. 

This is an occupied-virtual off-diagonal element of the Fock matrix in the MO basis, and is identical to the gradient of the energy with respect to an occupied-virtual mixing parameter (except for a factor of 4), see eq. (3.67). If the determinants are constructed from optimized canonical HF MOs, the gradient is zero, and the matrix element is zero. This may also be realized by noting that the MOs are eigenfunctions of the Fock operator, eq. (3.41). 

We may again chose a unitary transfonnation which makes tlie matrix of the Lagrange multiplier diagonal, producing a set of canonical Kohn-Sham (KS) orbitals. The resulting pseudo-eigenvalue equations are known as the Kohn-Sham equations. 

For computational purposes it is convenient to work with canonical MOs, i.e. those which make the matrix of Lagrange multipliers diagonal, and which are eigenfunctions of the Fock operator at convergence (eq, (3.41)). This corresponds to a specific choice of a unitary transformation of the occupied MOs. Once the SCF procedure has converged, however, we may chose other sets of orbitals by forming linear combinations of the canonical MOs. The total wave function, and thus all observable properties, are independent of such a rotation of the MOs. 

The canonical molecular orbitals of any molecule can by obtained by computer calculations. All MO methods involve the diagonalization of a secular matrix. It can be said that by moving from AOs to FOs to BOs basis sets one proceeds through the various stages of this diagonalization process, as the number of non-zero off-diagonal overlap matrix elements decreases. 

The canon () function by default will return the diagonalized system, and in thise case, in the system object sd. For example, we should find sd. a to be identical to the matrix l that we obtained a few steps back. 

In coordinate space, the diagonal elements of the canonical density matrix in the Fourier path integral representation are given by [20]... 

Diagonal matrix elements of the P3 self-energy approximation may be expressed in terms of canonical Hartree-Fock orbital energies and electron repulsion integrals in this basis. For ionization energies, where the index p pertains to an occupied spinorbital in the Hartree-Fock determinant,... 

For many ionization energies and electron affinities, diagonal selfenergy approximations are inappropriate. Methods with nondiagonal self-energies allow Dyson orbitals to be written as linear combinations of reference-state orbitals. In most of these approximations, combinations of canonical, Hartree-Fock orbitals are used for this purpose, i.e. 

There are several types of transformations that can be applied to matrices. One of the most useful is the canonical transformation. To transform a matrix, you pre-multiply by a matrix of constants and postmultiply by another matrix of constants. The canonical transformation is one which converts a matrix into another matrix that is diagonal and has the eigenvalues of the original matrix as its diagonal elements. 

Let us canonically transform the Q matrix into a diagonal matrix which has as its diagonal elements the eigenvalues of Q. 

It is somewhat similar to canonical transformation. But it is different in that the diagonal 2 matrix contains as its diagonal elements, not the eigenvalues of the Kj, matrix, but its singular values. 

To describe the EMs, Ugi uses the so-called BE-matrices (from Bond and Electron), the diagonal entries of which are the number of free valence electrones and the off-diagonal are the formal covalent bond orders. The sum of all the elements of a row (or a column, since all BE-matrices are symmetrical, i.e., they have the same number of rows than columns) give the total number of electrons surrounding the atom associated to this row. In fact, the n atoms of an EM can be enumerated in n different ways, which would lead to n distinguishable but equivalent BE matrices. However, by appropiate rules one of these numberings can be considered canonical. 

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