Geminal wave functions

The geminal wave functions in eq. (2.60) in the SLG approximation are by definition obtained by diagonalizing the effective Hamiltonian for the m-th bond. These latter are as previously given by eq. (3.1) which can be recast to the form ... 

A.M. Tokmachev and A.L. TchougrdefT. Transferability of parameters of strictly local geminals wave function and possibility of sequential derivation of molecular mechanics, J. Comp. Chem., 26, 491, 2005. 

Though recent computational facilities make the use of larger basis sets possible, standard geminal wave functions can only be determined accurately for small molecules. For extended systems, including large molecules, polymers and solids, severe approximations have to be made. 

The spontaneous localization of geminals may also be important in extended systems, where long-range correlation effects may appear in the form of localization. For example, there is no way to describe the so called Wigner-crystal in a free electron gas (see e.g. [140]) at the HF level, while in principle it should be possible with a geminal wave function. 

In the calculations in terms of Gaussian geminal wave functions [10, 96] an interesting observation was made. While generally the DPT results converge much faster with extension of the basis than the results from the Breit-Pauli approximation, only small differences between the two ap-... 

In geminal space, G2 already is nearly diagonal, with only one pair of off-diagonal eTements, G 2zx 2xz Full diagonalization therefore requires merely a rotation about the 7 axis, not in the real space of the molecular framework (this degree of freedom is not available any more), but in the abstract space spanned by the geminal wave functions 5[x], S[y], and S[z]. 

The water molecule is represented by an antisymmetrized product of four geminals—two describing the chemical bonds 0-H and two describing the electron lone pairs. Interaction between molecules represented by geminal wave functions can be accounted for per-turbatively [24], but we employ a different scheme. The... 

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]. 

Partitioning of 4> among the N/2 geminals comprising the SOAGP wave-function. 

E. Rosta and P. R. Surjan, Interaction of chemical bonds. V. Perturbative corrections to geminal-type wave functions. Int. J. Quantum Chem. 80(2), 96-104 (2000). 

The second-order reduced density matrix in geminal basis is expressed by the parameters of the wave function [6-9]. The second-order reduced density matrix (3) is the kernel of the second-order reduced density operator. Quantities 0 are matrix elements of the second-order reduced density operator in the basis of geminals. In spite of this, the expression element of density matrix is usual. In this sense, in the followings 0 is called as element of second-order reduced density matrix. 

The components of vectors D k.=i..m are completely defined by the parameters of the underlying full-CI type wave function, and the index sets of Slater-determinants and their subdeterminants according to (13). The number of vectors D is ( ), and this is of course equal to the number of geminals g constructed over the M-dimensional one-particle basis Bm-... 

The generalized valence bond (GVB) method was the earliest important generalization of the Coulson—Fischer idea to polyatomic molecules (13,14). The method uses OEOs that are free to delocalize over the whole molecule during orbital optimization. Despite its general formulation, the GVB method is usually used in its restricted form, referred to as GVB SOPP, which introduces two simplifications. The first one is the perfect-pairing (PP) approximation, in which only one VB structure is generated in the calculation. The wave function may then be expressed in the simple form of Equation 9.1, as a product of so-called geminal two-electron functions ... 

Each geminal function is a singlet-coupled GVB pair (

associated with a particular bond or lone pair in the molecule. For example, CH4 will have the familiar Lewis structure and its wave function will involve a product of four geminal functions, each corresponding to a C-H bond. 

A semiempirical method can be developed for the arbitrary form of the trial wave function of electrons, which is predefined by the specific class of molecules to be described and by the physical properties and/or effects which have to be reproduced within its framework. Two characteristic examples will be considered in this section. One is the strictly local geminal (SLG) wave function the other is the somewhat less specified wave function of the GF form selected to describe transition metal complexes. 

In this section, we consider a family of semiempirical implementations of the antisymmetrized product of the strictly local geminals (SLG). Quite naturally, this approach applies only to compounds (largely organic) with well localized two-center two-electron bonds. It had been originally developed for an old-fashioned MINDO/3 type of parametrization of the molecular Hamiltonian and then extended to the more contemporary NDDO family of parametrizations. First, the description of the wave function is given in detail and then the energy functional is described and analyzed. Its variation provides the equilibrium values of the electronic structure variables (ESVs) relevant for this method. 

The wave function of electrons in the molecule is then taken as the antisymmetrized product of the geminals given by eqs. (2.60), (2.61) ... 

From this we see that the structure of the wave function allows the transfer of electrons between the one-electron states only within a geminal and the possible delocalization of electrons between the geminals is not taken into account. 

In this section we have considered a family of semiempirical methods of analysis of the electronic structure of molecules, using the trial wave function in the form of the antisymmetrized product of strictly local geminals. The studies performed on these methods allow us to conclude that ... 

The importance of the transferability of the geminals has been pointed out in [62], It was stated that the assumption of the transferability of the geminal amplitudes is a prerequisite for that of the bond energy. However, in [62] geminal transferability has not been proven and the authors concentrate on the statements equivalent to the transferability of the MM bond stretching force fields. Our proof of course strongly relies on the SLG form of the trial wave function. This may seem to be a very strong restriction on the proposed derivation scheme. However, it is not a restriction at all if... 

The fundamental reasons for the difficulties faced by the MM methods when metal (both transition and nontransition) complexes are involved can be understood if one does not consider the MM as a purely empirical scheme (as it is frequently done), but think about them as of some reflection of specific features of molecular electronic structure, formalized by the form of the trial wave function of that class of compounds where such a parameterization might be possible. As shown in Chapter 3, organic compounds for which the MM methods are known to demonstrate significant successes can be described by the QC method, which directly leads to local and transferable two-center bonds. It is shown in Chapter 3 that the derivation of the MM method from the QC description is possible due to a common background of the MM and SLG description, which consists in the physical presence of two-center, two-electron bonds in organic molecules (in strict terms of Section 1.7 - numbers of electrons in each of the geminals weakly fluctuate). 

A possible approximation to be used for the cls function can be chosen considering two ideas. In contrast to the directionality and saturability characteristic for organic covalent bonds, those formed by metal ions do not possess these properties. Thus there is no need to invoke the HO formation on the metal ion. At the infinite separation limit, the cls wave function must flow to the antisymmetrized product of the lone pair geminals of eq. (2.61). The latter is in fact a single determinant function with all lone pair HOs doubly filled. With these arguments, we arrive at the conclusion that the single determinant (HFR) wave function is an appropriate form... 

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