Spin-free Hamiltonian

The complete Hamiltonian of the molecular system can be wrihen as H +H or H =H +H for the commutator being linear, where is the Hamiltonian corresponding to the spin contribution(s) such as, Fermi contact term, dipolar term, spin-orbit coupling, etc. (5). As a result, H ° would correspond to the spin free part of the Hamiltonian, which is usually employed in the electron propagator implementation. Accordingly, the k -th pole associated with the complete Hamiltonian H is , so that El is the A -th pole of the electron propagator for the spin free Hamiltonian H . 

The above operators apply only to primitive basis functions that have the spin degree of freedom included. In the current work we follow the work of Matsen and use a spin-free Hamiltonian and spin-free basis functions. This approach is valid for systems wherein spin-orbit type perturbations are not considered. In this case we must come up with a different way of obtaining the Young tableaux, and thus the correct projection operators. 

Those systems for which spin is conserved are those systems which are well described by a spin-free Hamiltonian. The spin-free Hamiltonian commutes with the symmetric group 5 F of permutations on electronic spatial indices. It follows that irreducible representations of this symmetric group are good quantum numbers. Certain irreducible representations of S F will be found to correspond to spin quantum numbers. 

Quantum numbers are in general associated with symmetry groups of effective Hamiltonians. Often the group theoretical nature of certain quantum numbers is not emphasized, or perhaps even realized. A case in point is provided by systems well described by a spin-free Hamiltonian in which case the symmetric group S%F yields the analogs of the usual spin quantum numbers. Often this problem is treated in a spin-oriented manner despite the fact that a spin-free Hamiltonian is used. A consequence of the use of a spin-oriented language is that many chemists implicitly assume... 

The zero-order, spin-free Hamiltonian HSF commutes with the symmetric group SnF of permutations on the N different spatial electronic indices,... 

The use of the conventional spin formulation in conjunction with a spin-free Hamiltonian HSF merely assures symmetry adaptation to a given spin-free permutational symmetry [Asp] without recourse to group theory. In fact, one may symmetry adapt to a given spin-free permutational symmetry without recourse to spin. This is the motivation behind the Spin-Free Quantum Chemistry series.107-116 In this spin-free formulation one uses a spatial electronic ket which is symmetry adapted to a given spin-free permutational symmetry by the application of an appropriate projector. The Pauli-allowed partitions are given by eq. (2-12) and the correspondence with spin by eqs. (2-14) and (2-15). Finally, since in this formulation [Asp] is the only type of permutational symmetry involved, we suppress the superscript SF on [Asp],... 

The total spin-free Hamiltonian including internal motion is taken to be... 

Here Q represents a vector of normal nuclear coordinates Qu Q2>- , Tn is the nuclear kinetic energy operator, and HSF(Q) is the electronic spin-free Hamiltonian... 

Prepared State. Here the Hamiltonian H is the time-independent molecular Hamiltonian. Both H0 and T are time independent. The initial prepared state is an eigenket to H0 and thus is nonstationary with respect to H = H0 + T. One example is provided by considering H0 as the spin-free Hamiltonian 77sp and the perturbation T as a spin interaction. A second example is provided by considering H0 as the spin-free Born-Oppenheimer Hamiltonian and T as a spin-free nonadiabatic perturbation. In the first example spin-free symmetry is not conserved but double-point group symmetry may be. In the second example point-group symmetry is not conserved, but spin-free symmetry is. The initial prepared state arises from some other time-dependent process as, for example, radiative absorption which occurs at a rate very much faster than the rate at which our prepared state evolves. Mechanisms for radiationless transitions in excited benzene may involve such prepared states, as is discussed in Section XI. 

Here MA, MB, MA , and MB are the z-components of the spins of A and of B. Such collisions are usually treated7,34,142,177 in an adiabatic approximation using spin-free Hamiltonian and spin-free potential curves. Thus, MA and Mb are only approximate local quantum numbers during a collision and so may change. The total M quantum number is, however, conserved... 

There are point-group selection rules in the presence of spin interactions.73,115117 172 We recall that a spin-free Hamiltonian //SF(Qeq) for a rigid nuclear framework Qeq has a point group SF which acts on electronic spatial coordinates, and that... 

Furthermore, there is a potential surface for each set of excited states for the N nuclei, i.e., for each set of singlets, triplets, etc. (assuming a spin-free Hamiltonian). Transitions from one surface to the next will, instantaneously, still be governed by the Franck-Condon principle and selection rules. Thus, the important question concerning purity of states in an electronic transition can be dismissed. 

Various approaches can be pursued to compute spin-orbit effects. Four-component ab initio methods automatically include scalar and magnetic relativistic corrections, but they put high demands on computer resources. (For reviews on this subject, see, e.g., Refs. 18,19,81,82.) The following discussion focuses on two-component methods treating SOC either perturbationally or variationally. Most of these procedures start off with orbitals optimized for a spin-free Hamiltonian. Spin-orbit coupling is added then at a later stage. The latter approaches can be divided again into so-called one-step or two-step procedures as explained below. 

When using a spin-free hamiltonian operator, the spin functions are introduced as mulplicative factors, yielding the spin-orbitals. There are two spin-orbitals per orbital. An electronic configuration is defined by the occupancies of the spin-orbitals. Open-shell configurations are those in which not all the orbitals are doubly occupied. 

Before continuing our discussion of gauge-origin dependence, we note that the substitution of Eq. 70 in the spin-free Hamiltonian Eq. 69 followed by expansion does not lead to the expression Eq. 52. To account for the missing Zeeman spin interaction, we must first replace the nonrelativistic spin-0 Hamiltonian Eq. 69 with a nonrelativistic spin- Hamiltonian, which for a one-electron system is given by... 

The configurational functions of all three components of the triplet state are listed on the three lines of Equation (1.17). From top to bottom, they correspond to the occupation of the MOs 0, and 0, with two electrons with an a spin, with one electron each with a and jS spin, and with two electrons with a p spin. The z component of the total spin is equal to M5 = 1,0, and — 1, respectively. The three triplet functions are degenerate (i.e., have the same energy) in the absence of external fields and ignoring relativistic effects (i.e., with a spin-free Hamiltonian). For our purposes, it is therefore sufficient to consider only one of the components, e.g., the one corresponding to Ms = 0. 

In this section I will outline the different methods that have been used and are currently used for the computation of parity violating effects in molecular systems. First one-component methods will be presented, then four-component schemes and finally two-component approaches. The term one-component shall imply herein that the orbitals employed for the zeroth-order description of the electronic wavefunction are either pure spin-up spin-orbitals or pure spin-down spin-orbitals and that the zeroth-order Hamiltonian does not cause couplings between the two different sets ( spin-free Hamiltonian). The two-component approaches use Pauli bispinors as basic objects for the description of the electronic wavefunction, while the four-component schemes employ Dirac four-spinors which contain an upper (or large) component and a lower (or small) component with each component being a Pauli bispinor. 

In the three mentioned spin-free Hamiltonians, the electron-electron interaction is described by the nonrelativistic Coulumb interaction,... 

Hyal [Eq. (20)] can be used in perturbation theory and in variational calculations, in particular in spin-orbit Cl calculations. In these, the lower symmetry of the spin-orbit Hamiltonian respect to the spin-free Hamiltonian imposes larger practical restrictions in the treatment of correlation effects, so that decoupling methods which allow the calculation of electron correlation and spin-orbit coupling in separated (although dependent) calculations are useful. These methods can be applied to any relativistic Hamiltonian that could be separated in a spin-free part and a spin-dependent part and we will comment on them in Section 2.2. 

Fig. 3.5/ and 6d manifolds in an octahedral field and their correlation with free ion one-electron levels. Note that the crystal field lowers the energy of the 6d levels (relative to 5J) prior to splitting. Bethe notation is used for the Kramers s doublets and Mulliken notation for the spin-free Hamiltonian states, as usual. 

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