Hermitian symmetry operator

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

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... 

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... 

Equations for other operators can be obtained from the symmetry and by Hermitian conjugation. Operators of Langevin forces L r ensure the validity of... 

Let us symbolize the identity operator by 1= n >< n + /IE > < E dE. I is a Hermitian projection operator defined in terms of the complete set of discrete, n>, and scattering, E>, stationary states of the system for each symmetry. Insertion of 7 into the amplitude leads to the... 

The operator A is Hermitian. Since P represents a (permutational) symmetry operator, it therefore conserves the scalar product. This means that for the two vectors i/fi and i/ 2 of the Hilbert space we obtain d... 

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 ... 

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). 

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

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... 

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] ... 

Factorization of the Cl space is difficult in the relativistic case. The first problem is the increase in number of possible interactions due to the spin-orbit coupling. The second problem is the rather arbitrary distinction in barred and unbarred spinors that should be used to mimic alpha and beta-spinorbitals. Unlike the non-relativistic case the spinors can not be made eigenfunctions of a generally applicable hermitian operator that commutes with the Hamiltonian. If the system under consideration possesses spatial symmetry the functions may be constrained to transform according to the representations of the appropriate double group but even in this case the precise distinction may depend on arbitrary criteria like the choice of the main rotation axis. 

The second quantized forms of quantum mechanical operators should also be Hermitian. This can be proven explicitly by using the above symmetry properties of the integral list and the Hermitian conjugate properties of the creation and annihilation operators discussed in Sect. 2.6. 

Here the general mathematical rules are utilized for taking the Hermitian conjugate, and the fact that the creation and annihilation operators are the Hermitian conjugates of each other [cf. Eq. (2.45)]. Using the symmetry rule of Eq. (4.42) one gets ... 

The FDP defined in (12.1.7) exhibits certain symmetry properties. For any pair of operators A and B, not necessarily Hermitian, it is easily verified that... 

To introduce orbital approximations, it is only necessary to write the operators in each FDP explicitly as in (14.4.5) the integrations then lead to sums of ordinary 2-electron integrals multiplied by FDPs for pairs of the E operators. In the most commonly occurring case, where the orbitals are real, the permutation symmetry of the integrals may be exploited by introducing the Hermitian combinations... 

In view of the fact that the operators V and A do not commute, the order of operators chosen in the definition of the symmetry-forcing technique (Eqs. 10 and 11) is not the only possible one and a different theory is obtained if one replaces the operator VA hy its Hermitian conjugate, the operator AV. This conjugate formulation of SAPT allows one to define the Amos-Musher perturbation theory [62]. It was also employed in one of the original formulations of the ELHAV method [59]. 

We consider the expectation value of a Hermitian operator of singlet spin symmetry... 

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