General Hermitian

To determine the form of M it is assumed to have a general Hermitian 2x2 matrix structure,... 

In summary, Erdahl s treatment is more general and allows a more concise formulation because he works in Fock space, conserving only the parity of the number of particles however, he finds it necessary to restrict the coefficients to be real. We work at fixed particle number and have no reason for the restriction to real coefficients. If the Hamiltonian should be general Hermitian, in which case the RDM must likewise be assumed to be general Hermitian, then our approach leads to Hermitian semidefinite conditions. 

In this equation stands for the supermatrix of operators with row and column n missing, and similarly e " represents a diagonal matrix of all the orbital energies except All the operators are constructed from (N — 1) orbitals, and consequently the effective operator may be viewed as determining orbital in the field of the (A — 1) other electrons. Since is in general Hermitian, solution of (30) gives rise to a whole set of orthonormal functions... 

The property operators are generally Hermitian, that is /j = -1-1, which gives the relation = g. It is therefore the hermiticity of the property operator that... 

The commutator of two Hermitian operators is an anti-Hermitian operator. From (2.3.2), we can therefore conclude that spin tensor operators are not in general Hermitian. Indeed, the only possible exception to this rule are the operators where M = 0, which may or may not be Hermitian. It is therefore of some interest to examine the Hermitian adjoints of the spin tensor operators. Taking the conjugate of the relations (2.3.1) and (2.3.2), we obtain ... 

For general matrices the reduction by triangular matrices requires less computation and is probably to be preferred. But if A is hermi-tian, observe that the use of the unitary reduction produces a matrix H that is again hermitian, hence, that is tridiagonal in form, having zeros everywhere except along, just above, and just below the main diagonal. 

The components of the operator P are hermitian.2 In general, any differential operator Q has a hermitian adjoint Qf, defined by the integral relation... 

The density operator r(<) is a Hermitian and positive function of time, and satisfies the generalized Liouville-von Neumann (LvN) equation(47, 45)... 

By setting B equal to A in the product AB in equation (3.11), we see that the square of a hermitian operator is hermitian. This result can be generalized to... 

The orthogonality theorem can also be extended to cover a somewhat more general form of the eigenvalue equation. For the sake of convenience, we present in detail the case of a single variable, although the treatment can be generalized to any number of variables. Suppose that instead of the eigenvalue equation (3.5), we have for a hermitian operator 4 of one variable... 

Both matrices are L x L dimensional and, as long as we are dealing with real basis functions, are symmetric, i. e., Mllv = Mvll (in the general case, they are self-adjoint or hermitian, i. e., = M 1(1). Using S and F and introducing the L x L dimensional matri-... 

Since has been already constrained to be hermitian, it is legitimate to assume, withoutany loss of generality that is always diagonalizable into, say, , by a unitary transformation of the basis elements [10], The diagonal elements of , then called its eigenvalues, are real. The rank constraint on P (which is basis independent) further reduces the number of non-zero eigenvalues toN. Let % (i = l,. .., N), be these non-zero eigenvalues. 

The matrix W1 K is in general skew-Hermitian due to Eq. (10), and hence its diagonal elements w P(R J are pure imaginary quantities. If we require that the /f,ad be real, then the matrix W ad becomes real and skew-symmetric with the diagonal elements equal to zero and the off-diagonal elements satisfying the relation... 

These coupling matrix elements are scalars due to the presence of the scalar Laplacian in Eq. (25). These elements are, in general, complex but if we require the /)L id to be real they become real. The matrix Wl-2 ad(R-/.), unlike its first-derivative counterpart, is neither skew-Hermitian nor skew-symmetric. 

Since H is Hermitian, the eigenvectors Vj of H form a complete orthonormal set and the vector representing a general state at t = 0 may be expressed as a linear superposition of these eigenvectors, (0) = CjVj, ... 

However, it is easy to verify that neither %a nor %b is an eigenvector of this unperturbed Hamiltonian, and neither are ea and eb its eigenvalues (see Example 3.18). More generally, since H/(0) is clearly Hermitian, it cannot have any non-orthogonal eigenvectors, by virtue of the theorem (note 79) quoted above. 

Without losing generality, this review will concentrate on real-symmetric matrices, whereas their Hermitian counterparts can be handled in a similar way. In some special cases, solutions of complex-symmetric matrices are required. This situation will be discussed separately. 

We have reproduced the c terms by means of the analysis of the different terms of the Breit hamiltonian with the exception of Darwin ones. These terms, denoted by Hp and given by Eq. (24) are not hermitian in general. The adjoint is given by... 

A Hermitian operator p is a von Neumann density if it is nonnegative and has unit trace. In more concrete terms, if is the finite-dimensional Fock space for a quantum model where electrons are distributed over a finite number of states, then p is a von Neumann density if (i) v,pv) > 0 for all operators v on and (ii) l)g = 1. By the formula v,pv) we mean the trace scalar product of the operators v and pv, that is, v,pv) = traceg(u pu) since (p, l)g = tracegp = 1 we have used this scalar product to express the trace condition. More generally. 

The unitary decomposition may be applied to any Hermitian, antisymmetric two-particle matrix including the 2-RDM, the two-hole RDM, and the two-particle reduced Hamiltonian. The decomposition is also readily generalized to treat p-particle matrices [80-82]. The trial 2-RDM to be purified may be written... 

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