Symmetry forcing

Finally, there is good news. It turns out that we may formulate a general iterative scheme which is able to produce various perturbation procedures, known and unknown in the literature. In addition the scheme has been designed by my nearest-neighbour colleagues (Jeziorski and Kolos). This scheme reads as  

At the end of the iterative scheme (convergence) the insertion of the operators T and g has no effect at all, but before that their presence may be crucial for the numerical convergence. This is the goal of symmetry forcing. 

This method of generating perturbation theories has been called by the authors the symmetry forcing method in symmetry adapted perturbation theory (SAPT). 

The corrections obtained in SAPT differ from those of the polarization perturba-tional method. The first-order energy correction is already different. 

To show the relation between the results of the two approaches, let us first introduce some new quantities. The first is an idempotent antisymmetrizer 

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Infrared reflectance spectroscopy provides information on the vibrational states in the interphase. It can be interpreted in terms of molecular symmetry, force constants and chemical bond lengths. The intensity of the spectral peaks of the adsorbed molecules is determined both by standard... 

For the simple cubic lattice, the very high symmetry forces further simplifications... 

Using a large basis set of GTO s, Meyer and Pulay68 obtained theoretical estimates of the symmetry force constants for CH4 which differed from the experimental values by amounts ranging from 0 to 12%. They concluded that the values obtained at the true Hartree-Fock limit would deviate by no more than 5% from the experimental ones. A similar approach, for use with extended Huckel and MINDO wavefunctions has been developed by Mclver and Komomicki.69... 

In the simplest symmetrized Rayleigh-Schrodinger (SRS) perturbation theory87-89 the symmetry forcing operator appears only in the energy expression. Hence, this formalism employs weak symmetry forcing and the operators v9, v9, and v9 are given by,... 

The symmetry forcing operator of the HS theory acts on a vector of / functions and gives, as the result, a single function VA It can be shown88 that... 

Various symmetry forcing schemes, convergence properties, and asymptotic behavior are summarized in Table 1-3. An inspection of this Table leads to quite pessimistic conclusions. All perturbation theories that are asymptotically consistent... 

Table 1-3. Summary of the symmetry forcing operators, convergence properties, and asymptotic correctness of various symmetry-adapted perturbation theories...

Jeziorski B, Chalasinski G, Szalewicz K (1978) Symmetry forcing and convergence properties of perturbation expansions for molecular interaction energies. Int J Quantum Chem 14 271-287... 

Jeziorski B, Kolos W (1977) On the symmetry forcing in the perturbation theory of weak intermolecular interactions. Int J Quantum Chem (Suppl. 1), 12 91-117... 

Patkowski K, Jeziorski B, Korona T, Szalewicz K (2002) Symmetry forcing procedure and convergence behavior of perturbation expansions for molecular interaction energies. J Chem Phys 117 5124-5134... 

Quadratic and cubic potential constants have been obtained from IR frequencies, isotopic shifts, inertial defects, Coriolis constants and centrifugal distortion, assuming the geometry from microwave data. The quadratic force field is characterized by four symmetry force constants F which are related to the inner force constants by the following equations... 

It is noteworthy that the 5 orbitals are always considered to be accessible in this analysis, and need not be considered further. Of course, this assumption is not always true, but clusters where the 5 orbitals need to be analyzed in detail will not be treated here. Furthermore, since the only inaccessible orbitals arise from the inwardly directed a and n interactions, the same analysis for symmetry-forced electron counts involving one additional occupied (or unoccupied) orbital holds. Closo, nido, and arachno relationships follow by analogy so long as no additional inaccessible orbitals arise. 

Theorem 1 immediately reveals that the single DSD processes in BsHs " and B9H9 are forbidden. The result follows from the same considerations of nonbonding orbitals and symmetry-forced deviations in electron count that were discussed in Section 5.3.2. The polar transition state in question must have a nonbonding e pair of Frontier Orbitals, which are absent in... 

A dilute solution of Ni(CO)4 in heptane shows a single band at 2045.7 cm in the v(CO) region. This is the t2 asymmetric stretch. The symmetric stretch (ui) is Raman-active only. The EFFF force constants required for Ni(CO)4 are kco and ki (the CO-stretching force constant and the CO,CO interaction constant respectively) and it would appear impossible to calculate two force constants from one v(CO) absorption. However, the symmetry force constant for the t2 mode is... 

We will, throughout this section, refer to symmetry force constants (K(F) where F is the symmetry of the vibrational mode) derived directly from observed frequencies. Since we will be dealing with all- C O molecules as a matter of course, these are simply found via a relationship analogous to equation (6) ... 

Before attempting to calculate a force field, it is best to ensure that the wavenumbers of the v(CO) bands are accurately known and the relative intensities are also known. A very usefiil quantity to have at hand at the start of the process that changes tfequencies into force constants is X, the sum of all the symmetry force constants (allowing for degeneracy). For example, K for Cr(CO)6 would be found by ... 

The calculated harmonic force field given in cartesian coordinates was transformed into a force field defined by symmetry coordinates. A normal coordinate analysis was performed using the geometry and the symmetry force constants from ab initio calculations, resulting in a description of the normal modes by their potential energy distribution (FED). Table 3 presents the potential energy distribution for v SiSi and v, SiCb for both rotamers, and Table 4 summarizes the SiSi, SiCl, and SiC stretching force constants. 

We shall not describe the internal coordinate system nor the symmetry coordinates, the symmetry force constants and the kinetic energy matrix, as they are detailed in McCullough s paper (/). Wilson s book, "Molecular vibrations , gives further explanations of the methods used (2). 

In the present paper we investigate the convergence properties of the RS, SRS, and HS perturbation series for He2 and HeH2 molecules, i.e. for the interaction of two ground-state two-electron systems. These perturbation formalisms correspond to none, weak, and strong symmetry forcing, respectively. In Sec. II the perturbation equations of the RS, SRS, and HS theories are briefly summarized. In Sec. Ill the computational details of the calculations are presented. The numerical results are presented and discussed in Sec. IV. 

In the SRS perturbation theory the symmetry forcing operator appears only in the energy expression. Thus, in this formalism VT = 1 and VQ = VA, where "A is the projection operator on the appropriate representation of the symmetric group 4. Moreover, the interpolation function "e(C) appearing in Eq. (6) is not calculated from Eq. (5) but is taken directly from the RS theory (10, 21). The SRS perturbation corrections to the interaction energy, "J srsi are given by,... 

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