Multideterminantal wave-function

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. 

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... 

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. 

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... 

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

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 ... 

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. 

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. 

To get insight in the parameters of the model Hamiltonian an ab initio calculation was performed giving the following multideterminantal wave functions 4>k and energies Ek. 

In the first place, the projected wave functions have to be expressed in terms of the 3d orbitals by substituting the expressions of [Pg.229]

Multideterminantal (MD) wave-function calculations are crucial to get correct Potential Energy Curves and a good description of bond breaking in many systems (for example, the H2 and F2 molecules). 

This idempotency property of particle number operators holds also in the more general case. Consider a many-electron multideterminantal wave function of the form ... 

The reader may easily investigate the idempotency property of Nj for this general case. It will turn out that the operator Nj is indeed idempotent for any multideterminantal wave function. 

The situation is not much more complicated for the particle number operator N corresponding to the total number of electrons. We have seen that single determinants are eigenfunctions of this operator with the eigenvalues of the total number of electrons N. Let us now study the way in which it acts on a multideterminantal wave function of the type given by Eq. (3.14) ... 

Any multideterminantal N-electron wave function is also an eigenfunction of the operator of the total number of electrons N. Operator N has an interesting property ... 

We considered only one-determinantal wave functions but the result of Eq. (4.27) holds for multideterminantal wave functions (linear combination of determinants) as well. 

The vast majority of quantum mechanical calculations are on the ground state of molecules. Calculations of excited states of molecules are still a research frontier but for small molecules very comphcated multideterminantal wave functions can be used to obtain accurate calculations of the eneigies of... 

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