Non-Hermitian operator

A comparison with equation (3.8) shows that if A is hermitian, then we have = A and A is said to be self-adjoint. The two terms, hermitian and self-adjoint, are synonymous. To find the adjoint of a non-hermitian operator, we apply equations (3.33). For example, we see from equation (3.10) that the adjoint of the operator d/dx is —d/dx. 

The hf is a non-Hermitian operator. Noting that the expectation value of a real non-Hermitian operator is equal to the expectation value of its Hermitian part, the equivalence of the two first-order energies is obvious. The difference in two Hamiltonians... 

G. K.-L. Chan and Troy Van Voorhis, Density-matrix renormahzation-group algorithms with nonorthogonal orbitals and non-Hermitian operators, and apphcations to polyenes. J. Chem. Phys. 122, 204101 (2005). 

Ht( y) = H(t ) = H (t ) / H( z) for complex values of the dilatation parameter jj, i.e., H( ) is non-Hermitian and as such, the variational theorem does not apply. However, there exists a bi-variational theorem /57,58/ for non-Hermitian operators. The bi-variational SCF equations for the dilated Hamiltonians are derived by extremizing the generalized functional... 

It is noteworthy that from the physical point of view the Hermitian conjugation means in this case simply the change of the order of monomer numeration along the chain (for a more detailed discussion of this point see20 ). Thus, the chains which are not symmetric with respect to this change, are characterized by the non-Hermitian operators g. 

The extension of the recursion method to non-Hermitian operators possessing real eigenvalues has been carried out by introducing an appropriate biorthogonal basis set in close analogy with the unsymmetric Lanczos procedure. Non-Hermitian operators with real eigenvalues are encountered, for instance, in the chemical pseudopotential theory. Notice that the two-sided recursion method in formulation (3.18) is also valid for relaxation operators, as previously discussed. 

The fact that 3 is a complex number implies that is a non-Hermitian operator. 

Now let us look at the paper. Eqn. (1) gives the form of the transcorrelated wave function C0, where C = li >jfiri,rj) is a Jastrow factor, and is a determinant. This compact wave function includes the effects of electron correlation through the introduction of r in C. The form C0 is taken as the trial wave function in quantum Monte Carlo (QMC) molecular computations today. Indeed the explicit form for/(r r,j is most often used by the QMC community. The transcorrelated wave function was obtained by solving (C //C - W) = 0, which Boys called the transcorrelation wave equation. Because C //C is a non-Hermitian operator, it was important to devise independent assessments of the accuracy of the wavefunction C0. 

In the classical theory of light and optical coherence the field is represented by complex vectorial amplitudes E(r, t) and E (r, t), which absolute values are complex numbers (c- numbers). In quantum theory of light the most important physical quantity is the electric field, which is represented by the field operator E(r, t). This Hermitian operator is usually expressed by the sum of two non-Hermitian operators as... 

To demonstrate results for an open-shell system. Table 20 reports results for 02- ° Intensities can also be obtained by considering the left eigenvector of the (non-Hermitian) operator H, along with R, and such EOM-CC examples are shown for electronic excited states later. 

The complex coordinate rotation (CCR) or complex scaling method (5,6,10,19) is directly based on the ABCS theory (1-3), therefore Reinhardt (5) also called it the direct approach. A complex rotated Hamiltonian, H 0), is obtained from the electron Hamiltonian of the atom, H, by replacing the radial coordinates r by re, where 0 is a real parameter. The eigenproblem of this non-Hermitian operator is solved variationally in a basis of square-integrable functions. The matrix representation of H ) is obtained by simple scaling of matrices T and V representing the kinetic and Coulomb potential part of the unrotated Hamiltonian H,... 

As a consequence of this Hermitization, the antisymmetrized products of the full Cl atomic solutions is not an eigenfunction of the sum of atomic operators Ha, as was the case for the sum of non-Hermitian operators Hj. However, considering H acting to right and H acting to left, one can easily see that the expectation value of that operator sum calculated with the antisymmetrized product of the atomic full Cl solutions will be equal to the sum of atomic full Cl energies. 

In order to obtain a consistent higher-order operator, the first idea is to refrain from expansion in 1/c. This approach leads in most cases to energy-dependent or non-hermitian operators, which prevent the formulation of a set of mutually orthogonal zero-order solutions. These could in turn be used as the basis for methods making use of configuration interaction or many-body perturbation theory. ... 

The matrix representing such an operator is thus Hermitian in the matrix sense (2.2.5), H = H. A useful consequence is that the adjoint of a non-Hermitian operator such as A + iB (A and B Hermitian) is... 

---