Doublet operators

We conclude that T - lvac) - provided that it does not vanish - represents a tenscH- state with spin eigenvalues 5 and M. Because of the close relationship between spin tensw operators and spin eigenfunctions, the terminology for spin functions is often used for spin tensor operators as well. Thus, a spin tensor operator with 5 = 0 is referred to as a singlet operator, S — gives a doublet operator, 5 = 1 a triplet, and so on. 

As an example of doublet operators, we first consider the creation operators From the... 

For strings containing two or more elementary operators, it is possible to constmct more than one tensor operator. We shall in Section 2.6.7 present a general method for the construction of tensor operators from strings of elementary operators. At present, we note that, by coupling the two doublet operators and it is possible to generate a singlet two-body creation... 

An example of a spectrum with a chemical shift is that of the tin 3d peaks in Eig. 2.8. A thin layer of oxide on the metallic tin surface enables photoelectrons from both the underlying metal and the oxide to appear together. Resolution of the doublet 3 ds/2, 3 dii2 into the components from the metal (Sn ) and from the oxide Sn " is shown in Eig. 2.8 B. The shift in this instance is 1.6-1.7 eV. Curve resolution is an operation that can be performed routinely by data processing systems associated with photoelectron spectrometers. 

The amount of spin contamination is given by the expectation value of die operator, (S ). The theoretical value for a pure spin state is S S + 1), i.e. 0 for a singlet (Sz = 0), 0.75 for a doublet (S = 1/2), 2.00 for a triplet (S = 1) etc. A UHF singlet wave function will contain some amounts of triplet, quintet etc. states, increasing the (S ) value from its theoretical value of zero for a pure spin state. Similarly, a UHF doublet wave function will contain some amounts of quartet, sextet etc. states. Usually the contribution from the next higher spin state from the desired is... 

P-Q-R triplets 225 P-R doublet 225, 249, 250 P-R exchange 135, 256 partial dipole moment operator 231 perturbation theory 5-6, 64-9 accuracy 78-9... 

The subsets of d orbitals in Fig. 3-4 may also be labelled according to their symmetry properties. The d ildxi y2 pair are labelled and the d yldxMyz trio as t2g. These are group-theoretical symbols describing how these functions transform under various symmetry operations. For our purposes, it is sufficient merely to recognize that the letters a ox b describe orbitally i.e. spatially) singly degenerate species, e refers to an orbital doublet and t to an orbital triplet. Lower case letters are used for one-electron wavefunctions (i.e. orbitals). The g subscript refers to the behaviour of... 

Any set of energetically well-isolated levels can be described by an effective spin Hamiltonian operator by choosing S to match the corresponding number of levels. This can be just one isolated Kramers doublet of a high-spin multiplet if the... 

When spin-orbit coupling is introduced the symmetry states in the double group CJ are found from the direct products of the orbital and spin components. Linear combinations of the C"V eigenfunctions are then taken which transform correctly in C when spin is explicitly included, and the space-spin combinations are formed according to Ballhausen (39) so as to be diagonal under the rotation operation Cf. For an odd-electron system the Kramers doublets transform as e ( /2)a, n =1, 3, 5,... whilst for even electron systems the degenerate levels transform as e na, n = 1, 2, 3,. For d1 systems the first term in H naturally vanishes and the orbital functions are at once invested with spin to construct the C functions. 

The studies on [LaTb], [TbLu], [LaEr] and [CeY] established that the individual ions ofthe molecules [Tb2(HL)2(H2L)Cl(py)(H20)] and [CeEr(HL)2(H2L)(N03) (py)(H20)] exhibit isolated, well-defined ground state doublets, thus leading to proper definitions of qubit states. The next step is to prove the existence of a weak coupling within each molecule conducive to the appropriate energy level spectrum for the realization of quantum gate operations. 

The effects of spin-orbit coupling on geometric phase may be illustrated by imagining the vibronic coupling between the two Kramers doublets arising from a 2E state, spin-orbit coupled to one of symmetry 2A. The formulation given below follows Stone [24]. The four 2E components are denoted by e, a), e a), e+ 3), c p), and those of 2A by coa), cop). The spin-orbit coupling operator has nonzero matrix elements... 

I" In actual fact, the three configurations corresponding to spin arrangements a/a/ft, a/ft/a and fl/a/a (or the corresponding ones with two ft and one a electron) are not eigenfunctions of the S2 operator. Upon proper combination, they give rise to three spectroscopic states, two doublets and one of the components of a quartet. 

At this point it is interesting to introduce a doublet notation that is more adequate to treat with infinite degrees of freedom. Considering then an arbitrary operator A we define... 

Spin-restricted procedures, signified by an R prefix (e.g. RHF, RMP), constrain the a and (3 orbitals to be the same. As such, the resulting wavefunctions are eigenfunctions of the spin-squared operator (S2) that correspond to pure spin states (doublets, triplets, etc). The disadvantage of this approach is that it restricts the flexibility in the... 

Sample calculations were carried out on H2, H, and H3. Geometry optimizations were carried out in internal coordinates. The projection operators used in the expansion (4) represented a singlet state for H2 and and a doublet state for H3. Starting geometries that were used are given in Table XVI. The initial wave functions were centered at the nuclei. For all of the initial functions the correlation parameters were set to zero (that is, the matrices A were... 

For the homonuclear (HON) species, the permutation-symmetry operator had the following form Y = 83) <8) Ye S2), where 83) is a Young operator for the third-order symmetric group which permutes the nuclear coordinates and 82) is a Young operator for the second-order symmetric group which permutes the electronic coordinates. For the fermionic nuclei (H and T, spin = 1/2) the Young operators corresponded to doublet-type representations, while for the bosonic D nuclei we use operators that correspond to the totally symmetric representation. In all cases the electronic operator corresponded to a singlet representation. 

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