Operator anti-Hermitian

Thus unitary operators for the group are associated with anti-Hermitian operators for the Lie algebra. Replacing P — iP, gives P = P ... 

If 5(2) is restricted to be an anti-Hermitian operator with no more than two-particle interactions, the variational degrees of freedom of 5(2) can be... 

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... 

The coupling to the continuum is implicitly contained in the second term in Eq. (376). The decay of the population in the bound state subspace is due to the imaginary part of P E — PHPY P. The contribution from the real part of P E — PHP) P, the so-called level shift due to the coupling to the continuum, can be neglected, if the energy dependence of the coupling term QHP is weak. In this case the second term in Eq. (376) can be regarded as a purely anti-Hermitian operator, and the effective Hamiltonian reduces to... 

Here 0) is tire unperturbed Kohn-Sham determinant, and /<( ) the anti-Hermitian operator... 

The variation of the MC function is described by a unitary operator, exp ( ), where is an anti-Hermitian operator... 

In the above expression 0) is the reference determinant, whereas exp[—ic(t)] is the (unitary) orbital-rotation operator, being the exponential of the anti-Hermitian operator... 

The unitarily transformed Hamiltonian may now be expanded in the following power series in the anti-Hermitian operator k ... 

For any anti-hermitian operator Wi with = — W, it is easily seen that U i is unitary. This transformation is applied to the first-order result (for details see Ref. 75), and the operator Wi is determined such that odd terms (i.e., tho.se which couple upper and lower components in the Dirac spinor) are annihilated to second order. 

Here, a and a are creation and annihilation operators, respectively. therefore has the effect of substituting an electron in state i for one in state j. As was done for the Cl coefficients, we now define an anti-hermitian operator, T in terms of the variational parameters Tif. 

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 ... 

The Fock space as introduced in Chapter I is defined in terms of a set of orthonormal spin orbitals. In many situations - for example, during the optimization of an electronic state or in the calculation of the response of an electronic state to an external perturbation - it becomes necessary to carry out transformations between different sets of orthonormal spin orbitals. In this chapter, we consider the unitary transformations of creation and annihilation operators and of Fock-space states that are generated by such transformations of the underlying spin-orbital basis. In particular, we shall see how, in second quantization, the unitary transformations can be conveniently carried out by the exponential of an anti-Hermitian operator, written as a linear combination of excitation operators. 

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