Multideterminantal

Mj = Mjmax-this is the only case, which can, in principle, lead to a singledeterminant wave function. This situation may be encountered in some limited highly axial systems, such as Dy-0+ [37]. However, usual compounds do not have any site symmetry, which means that several projections of the /-manifold will be mixed by the low-symmetry components of the CF, leading to a multideterminantal character of the wave function. 

The accuracy/cost ratio of conventional multideterminantal methods depends on two factors. The first, the correlation or n-particle factor (discussed earlier), can be successfully reduced via the rapidly converging coupled-duster ansatz [see Eq. (13.4)]. The second, file one-partide factor, is the primary reason why predictions with spectroscopic accuracy using conventional expansion methods are hardly feasible even for the helium... 

To improve on the wave function one has to accept that the standard multideterminantal expansion [Eq. (13.3)] is unsuitable for near-exact but practical approximations to the electronic wavefunction. The problem is dear from a simple analysis of the electronic Hamiltonian in Eq. (13.2) singularities in the Coulomb potential at the electron coalescence points necessarily lead to irregularities in first and higher derivatives of the exact wave function with respect to the interpartide coordinate, rj 2. The mathematical consequences of Coulomb singularities are known as electron-electron (correlation) and electron-nuclear cusp conditions and were derived by... 

Geminal functional theory is a very promising research area. The different varieties of antisymmetrized products are very flexible and inherently handle difficult problems, like multideterminantal molecules. The computational effort is low compared to the quality of the solutions. The perturbation theoretical approach to SSG should essentially be possible for AGP and UAGP as well. The formal definition of GFT is a flexible framework that opens up many new opportunities for exploring the nature of solutions to the Schrodinger equation. 

Density functional theory. 249-300, 371 broken symmetry. 275-276. 545 multideterminantal, 276 overdelocalization, 279-280, 294, 330-331 predictions from, (see individual functional... 

The first term in the product is associated with the spatial part and the second with the spin labels. The letters ua and b stand for atomic orbitals centered in hydrogen atoms Ha and H respectively. To account for the indistinguishability of the electrons, spatial and spin factors appear in two products (configurations). Consequently, the VB approach is multideterminantal from the outset. This superposition of determinants causes the VB wave function, even in its most simple form, to maintain the indistinguishability of the electrons within the chemical bond. This effect is called exclusion correlation , a non-dynamical correlation effect. 

In the last few years, the polarizable continuum model for the study of solvation has been extended to consider multideterminantal wavefunctions. Such novel techniques allow the study of the most important solvent effects on chemical reactions. In this context, the valence bond theory provides a way to analyze such effects through the transcription of the, generally, complicated multiconfigurational wavefunctions into sums of few selected classical structures, which are, in fact, more useful to understand the electron distribution rearrangement along a reaction path. In this chapter, the valence bond analysis of CASSCF wavefunctions calculated for chemical reactions in solution is discussed in details. By way of example, the results for some basic chemical processes are also reported. 

Jeziorski B, Monkhorst HJ (1981) Coupled-cluster method for multideterminantal reference states. Phys Rev A 24 1668-1681... 

Pollet J, Savin A, Leininger T, Stoll H (2002a) Combining multideterminantal wave functions with density functionals to handle near-degeneracy in atoms and molecules, J Chem Phys, 116 1250-1258... 

Pad6 approximants to perturbation series have been considered by numerous authors,121 particularly those interested in effective interactions in nuclei. They have been used in studies of the intruder state problem which can arise when a multideterminantal function is used as a reference in a perturbative scheme.122... 

The zero-order and the internal multideterminantal wavefunctions, o and Xint/ respectively, are specified by those determinants that are created by the distribution of N electrons among M spin orbitals, and their number is fixed by the binomial term, (see Eqs. (3 and 9) of Ref. [77]). 

We have developed a software application, ATOMPLUS,i38 which can be used in a turnkey fashion to illustrate theoretical features of two electron atoms, such as basis set effects, correlation energy, virial ratio, variational principle, and multideterminantal wavefunction. The program is provided in executable format for either the IBM or Macintosh platform. The source code can be obtained from Project Seraphim and, since it is written in FORTRAN 77, it compiles readily on other platforms. 

Wesolowki, T. A. (2008). Embedding a multideterminantal wave function in an orbital-free environment, Phys. Rev. A. 77, 012504-012513. 

Usually, we define the correlation energy for the case of double occupancy of the molecular orbitals (the RHF method see p. 394). In the case of open shells, especially when the multideterminantal description is required, the notion of correlation energy still remains to be defined. These problems will not be discussed in this book. 

Within the closed-shell HE picture, molecular orbitals are occupied by either exactly two or exactly zero electrons represented by the variationally best one-determinant wave function. Correlated levels give a different electron density which cannot be represented by a single Slater determinant. A logical starting point to account for electron correlation is to expand a multideterminantal wave function with the HE wave function as a starting point ... 

The second and most important point of Table 3.16 is that the correct Hartree-Fock results are in qualitative disagreement with experiment. In the molecular orbital Hartree-Fock model, the l7r orbital is the highest occupied orbital, yet the lowest experimental ionization potential corresponds to the production of an ion with symmetry. This implies a breakdown of the simple orbital picture of ionization. The Hartree-Fock picture is an approximation. For the case of N2 this approximation is not sufficiently accurate for even a qualitative understanding of the ionization phenomena. As we shall see in Chapters 4 and 7, when the single determinant Hartree-Fock model is replaced by a multideterminantal model, with its associated inclusion of correlation effects, theoretical calculations and experiment ultimately agree on the ionization spectra of N2. 

Hartree-Fock dieory in special cases, such as for restricted open-shell wave functions, involves a multideterminantal wave function. Since we wiU be concerned only with unrestricted open-shell wave functions, all Hartree-Fock wave functions will be single determinants. 

Recent calculations by Alabart and Caballol [25] have extended the systems examined up to the third-row complex H3 As HBr, and have employed a multideterminantal zeroth-order wavefunction in the second-order perturbation theory approach to... 

It is obvious from the denominator in the second term that the matrix will become non-Hermitian when the zeroth-order energies of the determinants in So are not the same. Therefore, this recipe only works for (nearly-)degenerate states. One advantage of this "perturb-and-then-diagonalize approach is that the length of the wave function expansion remains of the dimension of the model space, and hence, especially suitable for analysis purposes. Multideterminantal perturbation schemes that follow the diagonalize-and-then-perturb approach are described in Sect.4.3.3. 

Ordering by spatial or spin part. In the notation of multideterminantal wave functions, one can either respect as much as possible the order of the spatial part in the different determinants, or strictly maintain the order of the spin part. Construct singlet and triplet functions for a two-electrons in two-orbitals case respecting (i) the order of the spatial part and (ii) the order of the spin part of the total wave function. 

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