Restricted spin orbitals

Section 2.S discusses electron spin and spin operators in many-electron systems and contains a description of restricted and unrestricted spin orbitals and spin-adapted configurations. Spin-adapted configurations, unlike many single determinants derived from restricted spin orbitals, are correct eigenfunctions of the total electron spin operator. Singlet, doublet, and triplet spin-adapted configurations as well as unrestricted wave functions, which are not eigenfunctions of the total electron spin operator, are described. 

Such spin orbitals are called restricted spin orbitals, and determinants formed from them are restricted determinants. In such a determinant a given spatial... 

With restricted spin orbitals and restricted determinants, the spatial orbitals are constrained to be identical for a and fi spins. For example, the restricted Hartree-Fock (RHF) ground state of the Li atom is... 

So fiir in this chapter we have discussed the Hartree-Fock equations from a formal point of view in terms of a general set of spin orbitals xj. We are now in a position to consider the actual calculation of Hartree-Fock wave functions, and we must be more specific about the form of the spin orbitals. In the last chapter we briefly discussed two types of spin orbitals restricted spin orbitals, which are constrained to have the same spatial function for a (spin up) and jS (spin down) spin functions and unrestricted spin orbitals, which have different spatial functions for a and P spins. Later in this chapter we will discuss the unrestricted Hartree-Fock formalism and unrestricted calculations. In this section we are concerned with procedures for calculating restricted Hartree-Fock wave functions and, specifically, we consider here... 

Closed-Shell Hartree-Fock Restricted Spin Orbitals... 

At the beginning of this chapter we derived and discussed formal properties of the Hartree-Fock equations independent of any particular form for the spin orbitals. We then introduced a set of restricted spin orbitals and have since been concerned solely with restricted closed-shell calculations of the type... 

Analogous to Eq. (3.110) for restricted spin orbitals, an unrestricted set of spin orbitals has the following form... 

There are two types of spin orbitals restricted spin orbitals, which are constrained to have the same spatial function for a and spin functions and unrestricted spin orbitals, which have different spatial functions for a and spins. A restricted set of spin orbitals has the form X ( ) = whereas an unrestricted set has the form ... 

The exponential parametrization of a unitary operator is independent in the sense that there are no restrictions on the allowed values of the numerical parameters in the operator - any choice of numerical parameters gives rise to a bona fide unitary operator. In many situations, however, we would like to carry out restricted spin-orbital and orbital rotations in order to preserve, for example, the spin symmetries of the electronic state. Such constrained transformations are also considered in this chapter, which contains an analysis of the symmetry properties of unitary orbital-rotation operators in second quantization. We begin, however, our exposition of spin-orbital and orbital rotations in second quantization with a discussion of unitary matrices and matrix exponentials. 

Having seen the need for restricted spin-orbital rotations, let us first see how the antisymmetric k operator (3.2.6)... 

In this section, we briefly discuss spectroscopic consequences of the R-T coupling in tiiatomic molecules. We shall restrict ourselves to an analysis of the vibronic and spin-orbit structure, detennined by the bending vibrational quantum number o (in the usual spectroscopic notation 02) and the vibronic quantum numbers K, P. 

In the ordinary Hartree-Fock scheme, the total wave function is approximated by a single Slater determinant and, if the system possesses certain symmetry properties, they may impose rather severe restrictions on the occupied spin orbitals see, e.g., Eq. 11.61. These restrictions may be removed and the total energy correspondingly decreased, if instead we approximate the total wave function by means of the first term in the symmetry adapted set, i.e., by the projection of a single determinant. Since in both cases,... 

The existing SCF procedures are of two types in restricted methods, the MO s, except for the hipest (singly) occupied MO, are filled by two electrons with antiparallel spin, while in unrestricted methods, the variation procedure is performed with individual spin orbitals. In the latter, a total wave function is not an eigenvalue of the spin operator S, which is disadvantageous in many applications because of a necessary annihilation of higher multiplets by the projection operator. Since in practical applications the unrestricted methods have not proved to be remarkably superior, we shall call our attention in this review mainly to the restricted methods. 

The expressions (4.22)-(4.23) found in chap. 4 for the isomer shift 5 in nonrelativ-istic form may be applied to lighter elements up to iron without causing too much of an error. In heavier elements, however, the wave function j/ is subject to considerable modification by relativistic effects, particularly near the nucleus (remember that the spin-orbit coupling coefficient increases with Z ). Therefore, the electron density at the nucleus l /(o)P will be modified as well and the aforementioned equations for the isomer shift require relativistic correction. This has been considered [1] in a somewhat restricted approach by using Dirac wave functions and first-order perturbation theory in this approximation the relativistic correction simply consists of a dimensionless factor S (Z), which is introduced in the above equations for S,... 

The AO composition of the SOMO can often be deduced from the dipolar hyperfine matrix, particularly when the radical has enough symmetry to restrict possible hybridization. Thus an axial hyperfine matrix can usually be interpreted in terms of coupling to a SOMO composed of a single p- or d-orbital. A departure from axial symmetry may be due to spin orbit coupling effects, if (for example) /) Az and Ax AyxP(gx gy). If the departure from axial symmetry is larger, it is usually caused by d-orbital hybridization. The procedure is best illustrated by examples. 

In the 5 d series however it is possible to derive additional information bearing upon the problem of the relative extent of central field and symmetry restricted covalency. For many 5 d complexes reasonable estimates of the effective spin-orbit coupling constant can be derived from the spectra, and thence the relativistic ratio, / (= complex/ gas). When both f) and / are known for a given system, Jorgensen (74) has suggested how estimates of both covalencv contributions may be made. 

The Restricted Active Space (RAS) State Interaction Approach With Spin-Orbit Coupling. 

Sebastian has emphasized that (17a) implies Pi <0.5 (since 0restricted form the wavefunction (11) has when the a and / -spin orbitals are constrained to be equal. It can be circumvented by removing this constraint and using different spatial orbitals for electrons with different spin, which is accomplished by making different choices for the coupling functions. 

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

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