Spin-free approach

Other multiconfiguration VB methods have also been devised, like the biorthogonal valence bond method of McDouall (35,36) or the spin-free approach of McWeeny (37). For an overview of these methods, the reader is advised to consult a recent review (1). 

A spin-free approach for valence bond theory and its applications... 

A spin-free approach for valence bond (VB) theory, based on symmetric group techniques, is introduced in this chapter. Bonded tableaux (BT) are adopted to represent VB structures, and a paired-permanent-determinant algorithm is developed to solve the so-called IV problem in the nonorthogonal VB method, followed by the introduction of our ab initio VB program, Xiamen-99. Furthermore, applications of ab initio VB method to the resonance effect, chemical reactions, and excited states are carried out by the Xiamen package. 

It is evident that the matrix elements of the Hamiltonian and overlap are independent of the index r of BT in Eq. (13) and only the first diagonal element of the irreducible representation matrix, D P), is required, which has been well discussed [31,33,42,43], and is easily determined. It is worthwhile to emphasize that Eqs. (21) and (22) are the unique formulas of the matrix elements in the spin-free approach, even though one can take some other forms of VB functions. For example, it is possible to construct VB functions by Young operator [2], but the forms of the matrix elements are identical to Eqs. (21) and (22) [44],... 

Fully ab initio variational calculations, using a wavefunction consisting of VB structures and with optimization of both orbitals and structure coefficients, may be carried out by a variety of methods these range from the direct spin-free approach, in which only the permutation symmetry of the wavefunction is used (as in the pre-Slater era), to methods in which the structures (with spin factors included) cure expanded over determinants of spin-orbitals. 

The VBCI method can be viewed as a MRCI extension of the VBSCF approach. This method, which has been developed as a spin-free approach, starts with the calculation of a VBSCF wavefunction. The orbitals used to construct the initial wavefunction are formed as linear combinations of AOs from different subsets (or blocks ) as in eqn (3.6). The virtual orbitals needed for the additional VBCI configurations come from the orthogonal complements to the occupied orbitals for each subset from the original VBSCF wavefunction. The most convenient way of finding these virtual orbitals to diagonalise the representation of the projection onto the occupied space operator for each subset. 

In a similar way as in (13) the procedure is iterative. The main differences with the spin-free approach are ... 

W. Hiberty, Y. Mo, Z. Cao, Q. Zhang, A Spin-Free Approach for Valence Bond Theory and its Applications. In Theoretical and Computational Chemistry, volume 10, p. 143, D. L. Cooper, ed., Elsevier, Amsterdam (2002)... 

Amsterdam, The Netherlands, 2002, pp. 143-186. A Spin Free Approach for Valence Bond Theory and Its Application. 

The second method, which we now develop, is due mainly to Gallup and coworkers (for a review see Gallup et al., 1982), and may be regarded as another application (cf. p. 226) of the spin-free approach (Matsen, 1964) it leads to rather efficient procedures, and also involves ideas needed in Chapter 10 that are conveniently introduced at this point. 

Since we wish to adopt a spin-free approach, (10.4.6) cannot be used as it stands but to pass to the spin-free formalism only a few details need be changed. First, we are using orbitals not spin-orbitals, and so, writing al as the creation operator for spin-orbital ipr = we ought to be using the analogue of (10.4.6) after spin integrations. This is easily found to be... 

In the absence of a correlation between the local dynamics and the overall rotational diffusion of the protein, as assumed in the model-free approach, the total correlation function that determines the 15N spin-relaxation properties (Eqs. (1-5)) can be deconvolved (Tfast, Tslow < Tc) ... 

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. 

A model-free approach to analysis of DEER data in the absence of orientation selection was proposed based on shell factorization.22 The decay curves are simulated as the products of orientationally averaged thin shells of interacting electrons. The dipolar time-evolution data can be separated into a linear contribution and a non-linear contribution from background. The linear contribution can be converted to a radial distribution function for spin-spin interaction. 

The local permutational symmetry [Aa ] [AB ] is restricted such that the total permutational symmetry [A] is contained in T a 1 [V . When [Aa] [Ab] and [Aa ] [AB ] are not equal the corresponding separated molecule energies are different. Then for [Aa] [AB] / [Aa ] [AB ], the [Aa] [Ab] and [Aa ] [AB ] states are on different potential surfaces, and the process (5-9) is nonadiabatic. Thus the nonadiabatic reaction (5-9) might be expected to be most probably when the spin-free adiabatic potentials approach close to one another, since this is just the condition for the breakdown of the adiabatic approximation (see Sect. IV). 

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... 

---