Symmetric spin orbital basis

This problem can be avoided, however, if an appropriate open-shell perturbation theory is defined such that the zeroth-order Hamiltonian is diagonal in the truly spin-restricted molecular orbital basis. The Z-averaged perturbation theory (ZAPT) defined by Lee and Jayatilaka fulfills this requirement. ZAPT takes advantage of the symmetric spin orbital basis. For each doubly occupied spatial orbital and each unoccupied spatial orbital, the usual a and P spin functions are used, but for the singly occupied orbitals, new spin functions. 

The symmetric spin-orbital basis, which was also used to construct the spin-restricted (zT) correction, also provides a route to a spin-restricted open-shell B-CC theory (RB-CC). In this spin basis, the Tj amplitudes may be shown to have the symmetries... 

The structure of the parametric UA for the 4-RDM satisfies the fourth-order fermion relation (the expectation value of the commutator of four annihilator and four creator operators [26]) for any value of the parameter which is a basic and necessary A-representability condition. Also, the 4-RDM constructed in this way is symmetric for any value of On the other hand, the other A-representability conditions will be affected by this value. Hence it seems reasonable to optimize this parameter in such a way that at least one of these conditions is satisfied. Alcoba s working hypothesis [48] was the determination of the parameter value by imposing the trace condition to the 4-RDM. In order to test this working hypothesis, he constructed the 4-RDM for two states of the BeHa molecule in its linear form Dqo/,. The calculations were carried out with a minimal basis set formed by 14 Hartree-Fock spin orbitals belonging to three different symmetries. Thus orbitals 1, 2, and 3 are cr orbitals 4 and 5 are cr and orbitals 6 and 7 are degenerate % orbitals. The two states considered are the ground state, where... 

The model describes the simultaneous perturbation of the sixfold degenerate 2T2 basis by the effects of spin-orbit coupling and an axially symmetric ligand field distortion,... 

The spin-orbitals 17,) iGA + f form a basis set for the supersystem A —B. One possible procedure of realizing the transformation (218) as well as (219) is the Lowdin symmetric orthonormalization method137. ... 

In this subsection, we will briefly discuss how one may construct a basis

carrier space which is adapted not only to the treatment of the ground state of the Hamiltonian H but also to the study of the lowest excited states. In molecular and solid-state theory, it is often natural and convenient to start out from a set of n linearly independent wave functions = < > which are built up from atomic functions (spin orbitals, geminals, etc.) involved and which are hence usually of a nonorthogonal nature due to the overlap of the atomic elements. From this set O, one may then construct an orthonormal set tp = d>A by means of successive, symmetric, or canonical orthonormalization.27 For instance, using the symmetric procedure, one obtains... 

The total parity of a given class of levels (F fine structure component for E-states, upper versus lower A-doublet component for II-states) is found to alternate with 7. The second type of label, often loosely called the e// symmetry, factors out this (—l) 7 or (—l)-7-1/2 7-dependence (Brown et al., 1975) and becomes a rotation-independent label. (Note that e/f is not the parity of the symmetrized nonrotating molecule ASE) basis function. In fact, for half-integer S, it is not possible to construct eigenfunctions of crv in the form [ A, S, E) —A, S, — E)], because, for half-integer S, vice versa.) The third type of parity label arises when crv is allowed to operate only on the spatial coordinates of all electrons, resulting in a classification of A = 0 states according to their intrinsic E+ or E- symmetry. Only A = 0) basis functions have an intrinsic parity of this last type because, unlike A > 0) functions, they cannot be put into [ A) — A)] symmetrized form. The peculiarity of this E symmetry is underlined by the fact that the selection rule for spin-orbit perturbations (see Section 3.4.1) is E+ <-> E, whereas for all types of electronic states and all... 

Matrix elements of the total spin-orbit Hamiltonian between basis states differing by A A = 1, AE = +1, for a given signed value of fl, may be calculated using only the s) or l s part of Hso. Again, it is unnecessary to use symmetrized basis functions (except when one of the states involved is a Eq state) ... 

Once the system is parameterized, the many-electron matrix is set up by successively applying the operators ofEquation (1) to simple anti-symmetrized product functions (Slater determinants). The size of the basis set is given by the number of possible distributions of nj electrons over 10 spin orbitals, /> =, which is quite moderate Taking advantage of the electron/hole equivalence... 

While these phase factors are useful for the real groups, we would like to be able to make the Hamiltonian matrix real for any group, if possible. After all, for an even number of electrons, we can always construct a real basis, as demonstrated in section 9.3. In fact, we can transfer the principles from that section directly to the case of spin-orbit Cl and construct linear combinations of determinants that are symmetric under time reversal. 

The ground-state wavefunction will be antisymmetric in the spin coordinates of the two electrons and symmetric in their spatial coordinates. It will also have zero orbital angular momentum (an S state) the most general S state can be shown to depend only on the interparticle distances ri, r2, and ri2 [11]. We construct it from a basis of functions of the form... 

We say that z forms a basis for A,or that z belongs to Ai, or that z transforms according to the totally symmetric representation Ai. The s orbitals have spherical symmetry and so always belong to IY This is taken to be understood and is not stated explicitly in character tables. Rx, Ry, Rz tell us how rotations about x, y, and z transform (see Section 4.6). Table 4.5 is in fact only a partial character table, which includes only the vector representations. When we allow for the existence of electron spin, the state function ip(x y z) is replaced by f(x y z)x(ms), where x(ms) describes the electron spin. There are two ways of dealing with this complication. In the first one, the introduction of a new... 

These equations provide the basis for the RB-CC method, since they do not imply any loss of spin restriction on the molecular orbitals as the rotation is applied. Furthermore, the RB-CC method may be trivially implemented within existing ROHF-CCSD programs by a simple symmetrization of the standard (a, 3) Tj amplitudes into the new spin basis prior to the rotation. 

Jt is identified with the total angular momentum along the z axis, and St and Mt are the contributions from spin and orbital motion. The operator for the z component of the spin angular momentum St depends on the representation S, implicit in Zj, under which the function values are invariant. If the value of 0(ri, r, . .., t) is independent of the direction, ri, r2,. . . , and dependent only on their magnitudes, 0 is clearly a scalar function. In that event the function value space has all the symmetry of the sphere and s is the totally symmetric representation of the group. Zi V is then equal to zero, and J,0 = A/,0. The spin infinitesimal operator 1, only appears in the problem when the values of the function 0(ri, r, . .., 1) at various points of application form a basis for a representation other than the totally symmetric rep of the... 

In the previous discussion of the symmetry of the Dirac equation (chapter 6), it was shown that the Dirac equation was symmetric under time reversal, and that the fermion functions occur in Kramers pairs where the two members are related by time reversal. We will have to deal with a variety of operators, and in most cases the methodologies will be developed in the absence of an external magnetic field, or with the magnetic field considered as a perturbation. Consequently, we can make the developments in terms of a basis of Kramers pairs, which are the natural representation of the wave function in a system that is symmetric under time reversal. The development here is primarily the development of a second-quantized formalism. We will use the term Kramers-restricted to cover techniques and methods based on spinors that in some well-defined way appear as Kramers pairs. The analogous nonrelativistic situation is the spin-restricted formalism, which requires that orbitals appear as pairs with the same spatial part but with a and spins respectively. Spin restriction thus appears as a special case of Kramers restriction, because of the time-reversal connection between a and spin functions. 

The general problem is now clear the quantities i /,. p are tensor components, with respect to the group U(m), and we want to find linear combinations of these components that will display particular symmetries under electron permutations and hence under index permutations. Each set of symmetrized products, with a particular index symmetry, will provide a basis for constructing spin-free CFs (as in Section 7.6) for states of given spin multiplicity and in this way the full-CI secular equations will be reduced into the desired block form, each block corresponding to an irreducible representation of U(m). It is therefore necessary to study both groups U(m), which describes possible orbital transformations, and which provides a route (via the Young tableaux of Chapter 4) to the construction of rank-N tensors of particular symmetry type with reject to index permutations. 

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