Hermitian symmetry

Here the first equality follows from the fact that f e C. The technical continuity condition on f and its first and second partial derivatives allows us to exchange the derivative and the integral sign (disguised as a complex scalar product). See, for example, [Bart, Theorem 31.7]. The third equality follows from the Hermitian symmetry of It follows that is an element... 

The DFT of any real valued signal will exhibit Hermitian symmetry (real spectral components are even, and imaginary components are odd). 

For two states, Demkov and Nikitin introduced analytic model matrix elements which became widely used because of their convenient applicability. The matrix elements do not imply translational factors, so that they are real. Analytical matrix elements for which Hermitian symmetry is adopted may be assumed to be based on orthogonal states. Nevertheless, the notation H will be used for the model matrix elements. With AH = Hn 22 and H21 = H,2 the results are ... 

As pointed out above, Hermitian symmetry in the potential coupling matrix is achieved when orthogonalized basis states are used. For two states a convenient (symmetric) orthogonalization is given by ... 

Orthogonalized basis states have been used in the calculations by Fritsch et and Stolterfoht. Moreover, Hermitian symmetric matrix elements have been achieved by Pfeiffer and Garcia using the arithmetic averaging procedure. Moreover, Hermitian symmetry is assumed for the analytic model matrix element. ... 

On the other hand, it is based on the Hermiticity of the two-electron operator l/ri2, which, in the case of real orbitals, results in the Hermitian symmetry ... 

Two properties of the first-order density matrix are worth mentioning. First, it has the Hermitian symmetry ... 

The symmetry properties of the second-order density matrix are easy to establish. First, its Hermitian symmetry follows from the definition of Eq. (7.17) and from the adjoint relation ... 

Since is an orthonormal basis, such matrices do provide a representation in the sense of (2.2.11) and (2.2.12) moreover, the matrices associated in this way with Hermitian operators possess Hermitian symmetry (tf = H) themselves, unlike the representation matrices defined through (2.2.15). 

The fact that molecular point groups, the existence of a Hermitian symmetry operator G having the properties... 

This matrix is the appropriate representation of an observable such as X. A Hermitian matrix is its own hermitian conjugate. The diagonal elements of a Hermitian matrix are real and each element is symmetry related to its complex conjugate across the main diagonal. 

Thus, symmetry projection need only be performed on the ket. Typically, projection operators are Hermitian and essentially idempotent cx in any... 

Recently, a unitarily invariant decomposition of Hermitian second-order matrices of arbitrary symmetry under permutation of the indices within the row or column subsets of indices has been reported by Alcoba [77]. This decomposition, which generalizes that of Coleman, also presents three components that are mutually orthogonal with respect to the trace scalar product [77] ... 

The coefficients A defined in Eq. 4.18 satisfy certain symmetry relationships [323, 391], From parity considerations, it follows that X +2.2 + L must be odd. Moreover, because the dipole operator is Hermitian, the expansion coefficients A are all real. The symmetry property... 

Since any operator can be written as the sum of Hermitian and anti-Hermitian operators, we can restrict our discussion to these two types only. Further, any operator can be written as a linear combination of irreducible symmetry operators, so we can restrict ourselves to irreducible tensor operators. An operator matrix 0(r, K) that transforms according to the symmetry (T, K) obeys the relationship... 

If the symmetry is different, then of course iL /, > can be nonzero. In this article we assume that 0t,..., VN have definite albeit different time reversal symmetries. The properties can be represented by vectors t/j >... t/jy>... in Hilbert space with scalar product defined above. It is a simple matter to demonstrate that L is Hermitian in this Hilbert Space. Define the time correlation function... 

Paraphrasing the words of Ceulemans the special configurational symmetry of the half-filled shell also affects the interaction matrix elements. In the case of a one-electron Hermitian operator Tim > one has the following [7] ... 

For systems with high symmetry, in particular for atoms, symmetry properties can be used to reduce the matrix of the //-electron Hamiltonian to separate noninteracting blocks characterized by global symmetry quantum numbers. A particular method will be outlined here [263], to complete the discussion of basis-set expansions. A symmetry-adapted function is defined by 0 = 04>, where O is an Hermitian projection operator (O2 = O) that characterizes a particular irreducible representation of the symmetry group of the electronic Hamiltonian. Thus H commutes with O. This implies the turnover rule (0 > II 0 >) = (), which removes the projection operator from one side of the matrix element. Since the expansion of OT may run to many individual terms, this can greatly simplify formulas and computing algorithms. Matrix elements (0/x H ) simplify to (4 H v) or... 

A few remarks are necessary, concerning the general properties of the density matrix. Keeping in mind Eq. (17) and that the density matrix is Hermitian (p = P a) we obtain the following symmetry relations ... 

---