Antisymmetrized product of strongly

The antisymmetrized product of strongly orthogonal geminals is denoted by the acronym APSG. Properties of this wave function, its construction and use, will be discussed in the forthcoming sections. 

Since our wave function is most conveniently written in the form (69), i.e. as an antisymmetrized product of strongly orthogonal geminals i), the name APSG is now commonly used for this ansatz. 

The separated electron pair concept, which was first proposed by Hurley et al. [14] and which was later referred to as antisymmetrized product of strongly orthogonal geminals (APSG) [15], is also a special case of the group function concept. This kind of wave function is qualitatively correct at all internuclear distances and it can be improved either perturbationally [16, 17] or variationally [18]. 

The first pair theory was proposed as long ago as 1953 the antisymmetrized product of strongly orthogonal geminals (APSG) of Hurley et a ... 

The perturbation theory has widely been used in quantum chemistry to account for the dynamical electron correlation in single Slater [i.e., Hartree-Fock (HF)] and multideter-minantal states [1]. Suijdn has woiked on the perturbation theories for both HF and non-HF references. For non-HF reference functions, he and his cowoikers proposed a series of multiconfiguration perturbation (MCPT) theories [2-6]. Because the MCPT theories are applicable to any reference functions, they have occasionally been applied to the antisymmetric product of strongly orthogonal geminals wave functions [7-9]. 

The total wave function of a many-electron system can be constructed as an antisymmetrized product of individual geminals [see Eq. (3)]. Dealing with this product is substantially simpler if the geminals are kept orthogonal to each other in the strong sense, i.e. 

The first two terms of the r.h.s. of Eq.(20) are an antisymmetrized product of one-electron probabilities, therefore they can also be used as a reference in the absence of correlation in the 2-RDM [15-19], The main drawback, in considering these two terms as a reference, is that they do not form an N-representable 2-RDM. On the other hand, the 1-, 2-, 3-HF-RDM s as zero-correlation references are N-representable, well behaved RDM s, which is a strong argument in favor of taking them as references. 

We have in fact to compare the energy of an APG function (Antisymmetrized Product of Geminals, without strong orthogonality) with the sum of individual pair energies (without factorization of the Hilbert space). The difference... 

Eq. (20)), we find that the same product, ResR—Ch(NHj)COOH, is formed. In order to substantiate this observation, the IR spectra of GLY adsorbed on the strong-acid cation-exchange resin in the H -ion form were measured [16]. It was found that IR spectra of all samples prepared from aqueous solutions of GLY at a concentration level of 0.1 mol L and at pH values corresponding to 1.42, 4.88 and 5.64 exhibit the band at 1750-1752 cm that is assigned to the unionized carboxjd and the band at 1600 cm that is assigned to the antisymmetric vibration of carboxy-late ion. In spite of the presence of different predominant species in solutions and different mechanism encountered at different pH values, the IR spectra show that the amino adds adsorbed on ion exchanger are dissociated. 

Electronic spin exerts a strong influence on the energies associated with wavefunctions as a consequence of the Pauli principle. Consider the factored wavefunction, Equation 1.10. The product of two symmetric or two antisymmetric functions is symmetric, that of a symmetric and an antisymmetric function is antisymmetric. The spin functions for two-electron systems representing a singlet state (S = 0) are antisymmetric those representing a triplet state (5=1) are symmetric.18 Vibrational wavefunctions y(t/nuci) are invariant with respect to electron exchange, i.e. symmetric, as they do not depend on the electronic coordinates. Therefore, the electronic wavefunction for the triplet state must be antisymmetric in order to satisfy the Pauli principle. 

Now a strong band is seen at 1587 cm-1 and a weaker one at 1351 cm-1, which can be attributed to the antisymmetric and symmetric carbon-oxygen stretching vibrations. Hence, the adsorption of HCOOH on the nickel oxide must lead immediately and exclusively to the production of formate ions on the surface at — 60°C. The same difference in behavior between Ni and NiO was observed in the adsorption of butyric acid and acetic acid. The result for butyric acid at room temperature is given in Fig. 14. Here, again, a covalently bound acid is observed on the metal, whereas on the oxide obviously butyrate ions are present. 

These equations are expressed in the spin-orbital formalism and the products of orbitals are assumed to be antisymmetrized. The coefficients are the explicitly correlated analogues of the conventional amplitudes. The xy indices refer to the space of geminal replacements which is usually spanned by the occupied orbitals. The operator Q12 in Eq. (21) is the strong orthogonality projector and /12 is the correlation factor. In Eq. (18) the /12 correlation factor was chosen as linear ri2 term. It is not necessary to use it in such form. Recent advances in R12 theory have shown that Slater-type correlation factors, referred here as /12, are advantageous. Depending on the choice of the Ansatz of the wave function, the formula for the projector varies, but the detailed discussion of these issues is postponed until Subsection 4.2. The minimization of the Hylleraas functional... 

---