Hamiltonians spin-free

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 spin free electronic Hamiltonian of the stem,, is partitioned according to the usual Moller-Plesset form (129),... 

The spin free many-body Hamiltonian Operator can be written in compact form by employing the 2-RO... 

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. 

But this is not the full story. The Hamiltonian operator employed is a spin-free operator and does not work on the spin functions a and p. H commutes therefore with the spin operators Sz and S ... 

The second term on the right-hand side of the equation gives for point nuclei directly the one-electron spin-orhit operator (2) of the Breit-Pauli Hamiltonian and can he eliminated to give a spin-free equation that becomes equivalent to the Schrddinger equation in the non-relativistic limit. In a quaternion formulation of the Dirac equation the elimination becomes particularly simple. The algebra of the quaternion units is that of the Pauli spin matrices... 

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. 

Table 4. The isotropic indirect spin-spin coupling constant of calculated at various levels of theory. LL refers to the Levy-Leblond Hamiltonian, std refers to a full relativistic calculation using restricted (RKB) or unrestricted (UKB) kinetic balance, spf refers to calculations based on a spin-free relativistic Hamiltonian. Columns F, G and whether quaternion imaginary parts are deleted (0) or not (1) from the regular Fock matrix F prior to one-index transformation, from the two-electron Fock matrix G...

Table 5. The NMR shielding constant and shielding polarizabilities of the xenon atom calculated at the Hartree-Fock level using the Drrac-Coulomb Hamiltonian (SR + SO), its spin-free version (SR) as well as the non-relativistic Levy-Leblond Hamiltonian. The shielding constant is given in ppm and shielding polarizabilities in ppm/(au field2) (1 a.u. field = 5.14220642X 10" V...

Most Hamiltonians of physical interest are spin-free. Then the matrix elements in Eq. (9) depend only on the space part of the spin orbitals and vanish for different spin by integration over the spin part. Then it is recommended to eliminate the spin and to deal with spin-free operators only. We start with a basis of spin-free orbitals cpp, from which we construct the spin orbitals excitation operators carry orbital labels (capital letters) and spin labels... 

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,... 

Thus the eigenstates of HSF are labeled by the partitions [ASP] associated with the irreducible representations of S%v. These [Asp] labels are called permutation quantum numbers. The spin-free Hilbert space Fsp of the Hamiltonian may be decomposed into subspaces FSF([ASF]) invariant to 5 p... 

The decomposition eq. (2-6) of the spin-free space FSP induces a decomposition on the Pauli-allowed portion of the Hilbert space of the Hamiltonian H of eq. (2-1). The Hamiltonian H which includes spin interactions may operate on any ket of the space Fsp V", where V is the electronic spin space. Here the symbol indicates a tensor product, so that Fsp Va consists of all spatial-spin kets which are composed of linear combinations of a simple product of a spatial ket of FSP and a spin ket of Va. The Pauli-allowed portion of the total A-electron Hilbert space of is... 

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... 

Such localized states as under discussion here may arise in a system with local permutational symmetries [Aa] and [AB], If [Aa] + [S] and [Ab] = [5], the outer direct product [Aa] 0 [AB] gives rise to a number of different Pauli-allowed [A], If the A and B subsystems interact only weakly, these different spin-free [A] levels will be closely spaced in energy. The extent of mixing of these closely spaced spin-free states under the full Hamiltonian, H = HSF + f2, may then be large. Thus, systems which admit a description in terms of local permutational symmetries may in some cases readily undergo spin-forbidden processes, such as intersystem crossing. 

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. 

The primitive VB model is defined in terms of overlap and Hamiltonian matrix elements over the basis states of eqn. (2.1.3). For fixed there are 2N possible spin-product functions so that this gives the dimension of the model s space. Indeed (though not originally formulated in this manner) the model may be mathematically represented entirely in spin space, despite the fundamental spin-free nature of the interactions. One may introduce a spin-space overlap operator by integrating out the spin-free coordinates... 

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. 

As in all perturbational approaches, the Hamiltonian is divided into an unperturbed part and a perturbation V. The operator is a spin-free, one-component Hamiltonian and the spin-orbit coupling operator takes the role of the perturbation. There is no natural perturbation parameter X in this particular case. Instead, J4 so is assumed to represent a first-order perturbation The perturbational treatment of fine structure is an inherent two-step approach. It starts with the computation of correlated wave functions and energies for pure spin states—mostly at the Cl level. In a second step, spin-orbit perturbed energies and wavefunctions are determined. 

Hess et al.119 utilized a Hamiltonian matrix approach to determine the spin-orbit coupling between a spin-free correlated wave function and the configuration state functions (CSFs) of the perturbing symmetries. Havriliak and Yarkony120 proposed to solve the matrix equation... 

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