Operator spin, formal

In the usual formalism of quantum mechanics, the first quantization formalism, observables are represented by operators and the wave functions are normal functions. In the method of second quantization, the wave functions are also expressed in terms of operators. The formalism starts with the introduction of an abstract vector space, the Fock space. The basis vectors of the Fock space are occupation number vectors, with each vector defined by a set of occupation numbers (0 or 1 for fermions). An occupation number vector represents a Slater determinant with each occupation number giving the occupation of given spin orbital. Creation and annihilation operators that respectively adds and removes electrons are then introduced. Representations of usual operators are expressed in terms of the very same operators. 

Computation of the spin-orbit contribution to the electronic g-tensor shift can in principle be carried out using linear density functional response theory, however, one needs to introduce an efficient approximation of the two-electron spin-orbit operator, which formally can not be described in density functional theory. One way to solve this problem is to introduce the atomic mean-field (AMEI) approximation of the spin-orbit operator, which is well known for its accurate description of the spin-orbit interaction in molecules containing heavy atoms. Another two-electron operator appears in the first order diamagnetic two-electron contribution to the g-tensor shift, but in most molecules the contribution of this operator is negligible and can be safely omitted from actual calculations. These approximations have effectively resolved the DET dilemma of dealing with two-electron operators and have so allowed to take a practical approach to evaluate electronic g-tensors in DET. Conventionally, DET calculations of this kind are based on the unrestricted... 

The key feature of our approach is the concept of a formal reference determinant, 0). This determinant is used to generate the complete MRCC wave function by acting on it with an appropriate CC excitation operator. The formal reference determinant, in the case of the ground electronic state, can be the Hartree-Fock determinant. In most cases this determinant has the largest weight in the wave function of the considered state. Moreover, 0) defines the partition of spin-orbital space into occupied and unoccupied orbital subspaces. These orbitals are often referred to as holes and particles, respectively. Hence, the holes correspond to the spin-orbitals occupied in 0) and the particles correspond to the unoccupied orbitals. The formal reference determinant is used to generate all necessary electronically excited configurations in the wave function in the Oliphant-Adamowicz coupled cluster model. The simplest case discussed in the early works of Oliphant and Adamowicz [14-16] is the case of two determinantal reference space ... 

The orbital angular-momentum operators are of great importance in any central-field system other operators with formally similar properties are the spin operators (used extensively in Chapters 4 and 11), and it is therefore useful to collect the properties that characterize any kind of angular momentum. If the operators associated with the components are denoted by HKy and then... 

The last term is the intrinsic change in the operator P, which is zero when P does not formally depend on t, as is the case for momentum, angular momentum, and spin operators. In deriving Eq. (7-74), no use was made of the adiabatic property of the w-functions. Therefore, it holds for all time-dependent bases. In the moving representation, w = , D = H by virtue of Eq. (7-49), and (7-74) reverts to Eq. (7-59). 

We have thus far only considered the relativistic quantum mechanical description of a single spin 0, mass m particle. We next turn to the problem of describing a system of n such noninteracting spin 0, mass m, particles. The most concise description of a system of such identical particles is in terms of an operator formalism known as second quantization. It is described in Chapter 8, The Mathematical Formalism of Quantum Statistics, and Hie reader is referred to that chapter for detailed exposition of the formalism. We here shall assume familiarity with it. 

Quantization of the Electromagnetic Field.—Instead of proceeding as in the previous discussion of spin 0 and spin particles, we shall here adopt essentially the opposite point of view. Namely, instead of formulating the quantum theory of a system of many photons in terms of operators and showing the equivalence of this formalism to the imposition of quantum rules on classical electrodynamics, we shall take as our point of departure certain commutation rules which we assume the field operators to satisfy. We shall then show that a... 

In this section analytical expressions for ENDOR transition frequencies and intensities will be given, which allow an adequate description of ENDOR spectra of transition metal complexes. The formalism is based on operator transforms of the spin Hamiltonian under the most general symmetry conditions. The transparent first and second order formulae are expressed as compact quadratic and bilinear forms of simple equations. Second order contributions, and in particular cross-terms between hf interactions of different nuclei, will be discussed for spin systems possessing different symmetries. Finally, methods to determine relative and absolute signs of hf and quadrupole coupling constants will be summarized. 

There are a variety of formalisms that allow for a mapping of a discrete quantum system onto a continuous analog (for reviews see Refs. 218 and 219). The most prominent examples are Schwinger s theory of angular momentum [98] and the Holstein-Primakoff transformation [97], both of which allow a continuous representation of spin degrees of freedom. To discuss these two theories, we consider a spin DoF that is described by the spin operators Si,S2,Si with commutation relations... 

As discussed in Section VI.A for the case of spin systems, the formalism described above is not the only way to construct a mapping of a A -level system. First of all, it is clear that one may again eliminate one boson DoF by exploiting the operator (which corresponds to the identity operator in the physical... 

After the energy is expressed as a functional of the 2-RDM, a systematic hierarchy of V-representabihty constraints, known as p-positivity conditions, is derived [17]. We develop the details of the 2-positivity, 3-positivity, and partial 3-positivity conditions [21, 27, 34, 33]. In Section II.E the formal solution of V-representability for the 2-RDM is presented through a convex set of two-particle reduced Hamiltonian matrices [7, 21]. It is shown that the positivity conditions correspond to certain classes of reduced Hamiltonian matrices, and consequently, they are exact for certain classes of Hamiltonian operators at any interaction strength. In Section II.F the size of the 2-RDM is reduced through the use of spin and spatial symmetries [32, 34], and in Section II.G the variational 2-RDM method is extended to open-shell molecules [35]. 

These operators satisfy the formal definition for spin-adapted operators in Eqs. (79) and (78). Inserting these four operators into Eq. (8), we can generate four... 

In our formalism [5-9] excitation operators play a central role. Let an orthonormal basis p of spin orbitals be given. This basis has usually a finite dimension d, but it should be chosen such that in the limit —> cxd it becomes complete (in the so-called first Sobolev space [10]). We start from creation and annihilation operators for the ij/p in the usual way, but we use a tensor notation, in which subscripts refer to annihilation and superscripts to creation ... 

Superexchange describes interaction between localized moments of ions in insulators that are too far apart to interact by direct exchange. It operates through the intermediary of a nonmagnetic ion. Superexchange arises from the fact that localized-electron states as described by the formal valences are stabilized by an admixture of excited states involving electron transfer between the cation and the anion. A typical example is the 180° cation-anion-cation interaction in oxides of rocksalt structure, where antiparallel orientation of spins on neighbouring cations is favoured by covalent... 

---