Hamiltonian hermitian

In order to obtain the theoretical expressions for the g- and /-tensor elements, we have to compare the theoretical Hamiltonian given in Eq. (IV.59) with the phenomenological Hamiltonian in Eq. (1.4). For this comparison we have to keep in mind that in the phenomenological expression corresponds to Pyjk in Eq. (IV.59) and that in order to make the phenomenological Hamiltonian Hermitian, products such as cos yZ Jy> must be symmetrisized to (cos yZJy- +Jy cos yZ)j2, etc. 

Molecular aspects of geometric phase are associated with conical intersections between electronic energy surfaces, W(Q), where Q denotes the set of say k vibrational coordinates. In the simplest two-state case, the W Q) are eigen-surfaces of the nuclear coordinate dependent Hermitian electronic Hamiltonian... 

For the kind of potentials that arise in atomic and molecular structure, the Hamiltonian H is a Hermitian operator that is bounded from below (i.e., it has a lowest eigenvalue). Because it is Hermitian, it possesses a complete set of orthonormal eigenfunctions ( /j Any function spin variables on which H operates and obeys the same boundary conditions that the ( /j obey can be expanded in this complete set... 

The full Hamiltonian H thus eontains differential operators over the 3N eleetronie eoordinates (denoted r as a shorthand) and the 3M nuelear eoordinates (denoted R as a shorthand). In eontrast, the eleetronie Hamiltonian Hg is a Hermitian differential operator in... 

Let us, therefore, assume that the amplitude >ft x) describing a relativistic spin particle is an -component object. We are then looking for a hermitian operator H, the hamiltonian or energy operator, which is. linear in p and has the property that H2 = c2p2 + m2c4 = — 2c2V2 + m2c4. We also require H to be the infinitesimal operator for time translations, i.e., that... 

The matrix Hfj would be the transpose of Hf, if it were Hermitian. The Hermiticity of the superoperator Hamiltonian has been a concern since the beginnings of the electron propagator theory (46,129). For a Hermitian spin ftee Hamiltonian (// ) the following relation can be written describing the Hermiticity problem,... 

This raises a dilemma in treating second- and higher-order properties in coupled-cluster theory. In the EOM-CC approach, which is basically a Cl calculation for a non-Hermitian Hamiltonian H= that incorporates... 

Here, /j and rj are the l" left- and the J right-hand eigenvectors of the non-Hermitian Hamiltonian H. The operator is represented on the space spanned by the manifold created by the excitations out of a Hartree-Fock reference determinant, including the null excitation (the reference function). When we calculate the transition probability between a ground state g) and an excited state ]e), we need to evaluate and The reference function is a right-... 

This contribution considers systems which can be described with just the Hamiltonian, and do not need a dissipative term so that TZd = 0- This would be the case for an isolated system, or in phenomena where the dissipation effects can be represented by an additional operator to form a new effective non-Hermitian Hamiltonian. These will be called here Hamiltonian systems. For isolated systems with a Hermitian Hamiltonian, the normalization is constant over time and the density operator may be constructed in a simpler way. In effect, the initial operator may be expanded in its orthonormal eigenstates (density amplitudes) and eigenvalues Wn (positive populations), where n labels the states, in the form... 

The Hermitian Hamiltonian matrix H, the diagonal matrix E, and the unitary matrix... 

Since the Hamiltonian operator is hermitian, the energy eigenvalues E are real. 

A theory for nonequilibrium quantum statistical mechanics can be developed using a time-dependent, Hermitian, Hamiltonian operator Hit). In the quantum case it is the wave functions [/ that are the microstates analogous to a point in phase space. The complex conjugate / plays the role of the conjugate point in phase space, since, according to Schrodinger, it has equal and opposite time derivative to v /. 

Note that because the effective Hamiltonian matrix is not Hermitian, the eigenvectors are not orthogonal. However, when ac is small, the orthogonality properties are satisfactorily verified. 

The equations (17) are a set of N coupled equations for the N components of ll. Since the Hamiltonian is required to be Hermitian, the matrices must also be Hermitian, such that a = ad, f3 = ft. 

There is no proper perturbative basis for the mnemonic diagram in Fig. 3.58, because the non-orthogonal unperturbed orbitals cannot correspond to any physical (Hermitian) unperturbed Hamiltonian operator,79 as illustrated in Examples 3.17 and 3.18 below. The PMO interpretation of Fig. 3.58 therefore rests on an nnphysical starting point. Removal of orbital overlap (to restore Hermiticity) eliminates the supposed effect. 80... 

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. 

However, this Hamiltonian is manifestly non-Hermitian (unless S = 0) and therefore cannot correspond to a physical unperturbed system. Neither of the operators H/(0) and H"(°) can serve as a proper unperturbed Hamiltonian for the PMO rationalization (unless S = 0, when both are equivalent to a standard H(0) such as that underlying, e.g., Example 1.1 or Fig. 3.13). 

Note that, in contrast to other forms of intermolecular perturbation theory to be considered below, the NBO-based decomposition (5.8) is based on a full matrix representation of the supermolecule Hamiltonian H. All terms in (5.8) are therefore fully consistent with the Pauli principle, and both /7units(0, and Vunits(mt) are properly Hermitian (and thus, physically interpretable) at all separations. 

The aforementioned applications of recursive methods in reaction dynamics do not involve diagonalization explicitly. In some quantum mechanical formulations of reactive scattering problems, however, diagonalization of sub-Hamiltonian matrices is needed. Recursive diagonalizers for Hermitian and real-symmetric matrices described earlier in this chapter have been used by several authors.73,81... 

Boundary Conditions via a Symmetrically Damped, Hermitian Hamiltonian Operator. 

As seen from equation (50), the ESC Hamiltonian is energy dependent and Hermitian. For a fixed value of E, the ESC Hamiltonian can be diagonalized and the resulting solutions, in principle, form a complete orthonormal set. The eigenfunctions of are identical to the large component of the Dirac spinor. When Z — 0, equations (38) and (44) give us the similarity transformed Hamiltonian... 

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