Symmetry adapted formulation

Using the variables introduced in the previous sections, the symmetry analysis of [41] can be reformulated as follows. The deformation of the molecule of a CC )q) is a vector with the components referring to the individual nuclear shifts  

For a symmetric (say, octahedral) molecule, it may be rewritten using the symmetry adapted nuclear shifts  

The meaning of the notation for the individual nuclear shifts is that 7l 7 ) represents a unit shift in the positive direction along the 7 axis of the ligand located at the 7 semiaxis of the coordinate frame. 

A remarkable feature is that the derivative of the one-electron part of the Fock operator with respect to the symmetry adapted nuclear shift Sqr (an operator acting on the one-electron states in the CLS carrier space) itself transforms according to the irreducible representation T and its row 7. That means that applying the deformation T7) to a complex results in a perturbation of the Fock operator having the same symmetry 1 7. This allows us to write the vibronic operator in a symmetry-adapted form  

Finally, the substitution operator can be expanded as a sum of symmetry-adapted components. For example, in the octahedral complex, single substitution MLf, — ML5X results in the substitution operator  

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In the symmetry-adapted formulation, the 43- term no longer occurs because the d-orbital density contains a vertical mirror plane even if such a plane is absent in the point group. This is illustrated as follows. Point groups without vertical mirror planes differ from those with vertical mirror planes by the occurrence of both dlm+ and d(m functions, with m being restricted to n, the order of the rotation axis. But the coordinate system can be rotated around the main symmetry axis such that P4 becomes zero. As proof, we write the (p dependence as... 

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],... 

In this section we will review the symmetry-adapted perturbation theory of pairwise nonadditive interactions in trimers. This theory was formulated in Ref. (302). We will show that pure three-body polarization and exchange components can be explicitly separated out and that the three-body polarization contributions through the third-order of perturbation theory naturally separate into terms describing the pure induction, mixed induction-dispersion, and pure dispersion interactions. 

In the earlier sections of this chapter we reviewed the many-electron formulation of the symmetry-adapted perturbation theory of two-body interactions. As we saw, all physically important contributions to the potential could be identified and computed separately. We follow the same program for the three-body forces and discuss a triple perturbation theory for interactions in trimers. We show how the pure three-body effects can be separated out and give working equations for the components in terms of molecular integrals and linear and quadratic response functions. These formulas have a clear, partly classical, partly quantum mechanical interpretation. The exchange terms are also classified for the explicit orbital formulas we refer to Ref. (302). 

This relation shows how the action of the antisymmetrizer can mix different orders in perturbation theory. Secondly, the projected functions AglO ) 0 > do not form an orthogonal set in the antisymmetric subspace of the Hilbert space L2(r3N) if we take all excited states a > and b > in order to obtain a complete set a > b >, the projections As a > b > form a linearly dependent set. Expanding a given (antisymmetric) function in this overcomplete set is always possible, but the expansion coefficients are not uniquely defined. How the different symmetry adapted perturbation theories that have been formulated since the original treatment by Eisenschitz and London in 1930 , actually deal with these two problems can be read in the following reviews Usually, the first order interaction... 

In our early calculations we followed the procedure outlined in Quantum Chemistry. The wavefunctions were constructed from large basis sets of eigenfunctions of S, symmetry adapted to ) . More recently we have employed the spin-free formulation. Most of the states studied are the lower states of a particular symmetry type, in which case we minimize with respect... 

The Hubbard model, which has played a significant part in the earlier development of nonlinear optical response theory of molecules, has been revived by Shehadi et al.203 to explain the properties of small bridged metallic polymers. The Hubbard-Peierls hamiltonian has also been used by Shuai et al.,204 in conjunction with a symmetry adapted density matrix renormalization group formulation, to calculate a number of properties, including third harmonic generation in trans-octatetraene. 

Becke has proposed a novel approach that formulates the dispersion interaction in terms of the dipole moment that would be created when considering an electron and its exchange hole. " ° Like DFT-D, these methods appear to be more reliable than MP2 for noncovalent interactions. Alternatively, other workers " " have combined DFT with symmetry-adapted perturbation theory (SAPT) (discussed below). These DFT-SAPT approaches evaluate the dispersion term via the frequency-dependent density susceptibility functions of time-dependent DFT, an approach that appears to be theoretically sound. 

In the present context, the term symmetry refers to the permutational symmetry of electrons, that is to the Pauh principle. This problem plays a central role in perturbation theories of intermolecular interactions since the antisymmetry of the perturbed wave functions has to be ensured. The symmetry-adapted PT is called also as exchange-PT , because the antisymmetry results exchange interactions between molecules A and B. Several formulations of the exchange-PT have been developed (Van der Avoird, 1967, Amos Musher 1967, Hirschfedler 1967, Murrel Shaw 1967, Salewicz Jeziorski 1979) which will not be discussed in detail. In the spirit of the present treatment, we shall focus on the application of second quantization to this problem. This formalism eo ipso guarantees the proper antisymmetry of any wave function expressed in terms of anticommuting fermion operators, thus the symmetry adaptation is done automatically and it does not require any further discussion. 

Using the same formulation of the Hamiltonian as in Sec. VII [specifically Eqs. (67)—(70)], the two-step process makes use of five pairs of rovibrational states (specified explicitly below). The vibrational eigenstates correspond to the combined torsional and S-D asymmetric stretching modes. The rotational eigenfunctions are the parity-adapted symmetric top wave functions. Each eigenstate has additionally an Si A label denoting its symmetry with respect to inversion. Within the pairs used, the observable chiral states are composed as... 

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