Spin operator unitary transformation

The sum over eoulomb and exehange interaetions in the Foek operator runs only over those spin-orbitals that are oeeupied in the trial F. Beeause a unitary transformation among the orbitals that appear in F leaves the determinant unehanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to ehoose sueh a unitary transformation to make the 8i j matrix diagonal. Upon so doing, one is left with the so-ealled canonical Hartree-Fock equations ... 

We now consider how to eliminate the spin-orbit interaction, but not scalar relativistic effects, from the Dirac equation (25). The straightforward elimination of spin-dependent terms, taken to be terms involving the Pauli spin matrices, certainly does not work as it eliminates all kinetic energy as well. A minimum requirement for a correct procedure for the elimination of spin-orbit interaction is that the remaining operator should go to the correct non-relativistic limit. However, this check does not guarantee that some scalar relativistic effects are eliminated as well, as pointed out by Visscher and van Lenthe [44]. Dyall [12] suggested the elimination of the spin-orbit interaction by the non-unitary transformation... 

It is worth noting that the methods mentioned give rise to Hartree-Fock equations based on different Fock operators for the various orbitals of the same spin. This implies that off-diagonal Lagrange multipliers appear which cannot be eliminated by a suitable unitary transformation. Moreover, the complications arise when excited states above the first one are considered (see, e.g. [25] for further details). 

The analogy between spin and quasispin (q = s = 1/2) enabled Judd to introduce another unitary transformation - the operation R [12]... 

It should be noted that the SO operator is nondiagonal in the diabatic spin-orbital electronic basis which usually is employed to set up the E x E JT Hamiltonian, see (28, 33). The (usually ad hoc assumed) diagonal form of H o is obtained by the unitary transformation S which mixes spatial orbitals and spin functions of the electron. In this transformed basis, the electronic spin projection is thus no longer a good quantum number. 

If the wave function that one considers, is a single Slater determinant , the spin orbitals (p, from which , is constructed, are not uniquely determined, but rather there is an infinity of equivalent sets of qo, related by unitary transformations. To some extent one can make the qo, unique if one requires either that they are canonical (diagonalize the Fock operator) and are symmetry-adapted, or localized (e.g. according to the criteria of Edmiston and Ruedenberg or Foster and Boys [1-3]). The localized spin orbitals have some advantages both for the chemical interpretation and for the computation of correlation corrections. 

The operators V and S operate on the spatial and spin degrees of freedom respectively and transform like pseudovectors under the symmetry operations. Now, taking a general unitary transformation on a fixed set of spinors,... 

One has a separate operator for each spin orbital so the equation has to be solved several times and (controllable) problems of orthogonality have to be dealt with. Unlike the LSD energy, the LSD-SIC energy is not invariant to a unitary transformation among the occupied orbitals. For example, in a solid the SIC is zero for Bloch functions but not for Wannier functions. This clearly leads to arbitrariness in the application of LSD-SIC in situations involving wavefunctions which are delocalized by symmetry—a topic discussed further below. 

It is important to mention here that for any practical implementation of a UGA scheme, one need not demand that the n-electron CSF s be adapted to U n). Rather, it is expedient to demand that the CSF s be adapted to the subgroup of U n),u n) = U ric) <8> U tia) U nv), where Tic, na, and v are. respectively, the number of core, active, and virtual orbitals. This is simply due to the physical requirement that the maximum invariance of an approximate function that one may practically impose is the invariance of the function and the energy with respect to separate unitary transformation among core, active, and virtual orbitals. This was indeed done in the UGA CC papers by Paldus and others [33, 34], In particular, both these papers have used the h( )-adapted scalar tensor generators in their choice of excitation operators. We also point out two references [35, 36] in this context where useM discussions of other UGA-based approaches and their interrelation can be found. Reference [36] has also presented in considerable detail a number of approximately spin-adapted CC approaches and their relationship with UGA CC. We also point out that Li and Paldus have applied the UGA CC method to many problems (see e.g.. Refs. 39 8 in [37]). 

We can use the invariance of a single determinant to a unitary transformation of the spin orbitals to simplify Eq. (3.49) and put it in the form of an eigenvalue equation for a particular set of spin orbitals. First, however, we need to determine the effect of the above unitary transformation on the Fock operator / and the Lagrange multipliers The only parts of the Fock operator that depend on the spin orbitals are the coulomb and exchange terms. The transformed sum of the coulomb operators is... 

Thus the sum of coulomb operators is invariant to a unitary transformation of the spin orbitals. In an identical manner it is easy to show that the sum of exchange operators, and hence the Fock operator itself, is invariant to an... 

The canonical spin orbitals, which are a solution to this equation, will generally be delocalized and form a basis for an irreducible representation of the point group of the molecule, i.e., they will have certain symmetry properties characteristic of the symmetry of the molecule or, equivalently, of the Fock operator. Once the canonical spin orbitals have been obtained it would be possible to obtain an infinite number of equivalent sets by a unitary transformation of the canonical set. In particular, there are various criteria (see Further Reading) for choosing a unitary transformation so that the transformed set of spin orbitals is in some sense localized, more in line with our intuitive feeling for chemical bonds. 

Hence, the relativistic analog of the spin-restriction in nonrelativistic closed-shell Hartree-Fock theory is Kramers-restricted Dirac-Hartree-Fock theory. We should emphasize that our derivation of the Roothaan equation above is the pedestrian way chosen in order to produce this matrix-SCF equation step by step. The most sophisticated formulations are the Kramers-restricted quaternion Dirac-Hartree-Fock implementations [286,318,319]. A basis of Kramers pairs, i.e., one adapted to time-reversal s)mimetry, transforms into another basis under quatemionic unitary transformation [589]. This can be exploited not only for the optimization of Dirac-Hartree-Fock spinors, but also for MCSCF spinors. In a Kramers one-electron basis, an operator O invariant under time reversal possesses a specific block structure. 

The operator k in the form of equation (111) is needed for a general unitary transformation. Often, however, we are interested in restricted unitary operators with special properties. Thus, the following operator, where the Kp are real-valued parameters, generates real unitary (i.e., orthogonal) transformations that conserve the spin symmetry of the transformed electronic state... 

As already noted, there are many similarities between the exponential unitary transformations in configuration space and in orbital space. Comparing with the results for orbital transformations in Chapter 3, we note that the operators S and k are both anti-Hermitian, as are the matrices S and x. Moreover, whereas exp(—S)]P) represents a unitarily transformed configuration state, exp(—ic) P) represents a state where the spin orbitals have been unitarily transformed as exp(—x)ap exp(ic). 

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. 

Let Op and ap be the elementary creation and annihilation operators associated with the untransformed spin orbitals and let 0) be any state in Fock space expressed in terms of the untransformed elementary operators. We shall in this section demonstrate that the elementary operators Op and ap and state 0) generated by the unitary transformation (3.2.1) and (3.2.2) can be expressed in terms of the untransformed operators and states as... 

In this exercise, we consider the spin-flip operators - that is, unitary operators that transform alpha operators into beta operators and visa versa. Although we shall investigate only the effect of the spin-flip operators on the creation operators, we note that the corresponding results for the annihilation operating are readily obtained by taking the adjoints of the relations for the creation operators. 

To demonstrate (3E.4.3)-(3E.4.6), we shall evaluate the BCH expansions of the unitary transformations. For this purpose, we must first evaluate the commutators of the creation operators with Sx and Sy. Expressing Sx and Sy in terms of the spin-shift operators... 

Chapters 1-3 introduce second quantization, emphasizing those aspects of the theory that are useful for molecular electronic-structure theory. In Chapter 1, second quantization is introduced in the spin-orbital basis, and we show how first-quantization operators and states are represented in the language of second quantization. Next, in Chapter 2, we make spin adaptations of such operators and states, introducing spin tensor operators and configuration state functions. Finally, in Chapter 3, we discuss unitary transformations and, in particular, their nonredundant formulation in terms of exponentials of matrices and operators. Of particular importance is the exponential parametrization of unitary orbital transformations, used in the subsequent chapters of the book. 

Let us set up a 2D unitary matrix representation for the transformation of the spin functions a and (1 in Civ. So far, we have established only a relation between 0(3)+ and SU(2). The matrix representations of reflections or improper rotations do not belong to 0(3)+ because their determinants have a value of -1. To find out how a and p behave under reflections, we notice that any reflection in a plane can be thought of as a rotation through n about an axis perpendicular to that plane followed by the inversion operation. For instance, 6XZ may be constructed as xz = Cz(y) i. Herein, it is not necessarily required... 

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