Wave function reduction

The full Cl expansion within the active space severely restricts the number of orbitals and electrons that can be treated by CASSCF methods. Table 4.3 shows how many singlet CSFs are generated for an [n, n]-CASSCF wave function (eq. (4.13)), without any reductions arising from symmetry. 

Boys, S. F., Proc. Roy. Soc. London) A207, 197, Electronic wave functions. V. Systematic reduction methods for all Schrodinger integrals of conventional systems of antisymmetric vector-coupled functions/ ... 

Electron-transfer proteins have a mechanism that is quite different from the conduction of electrons through a metal electrode or wire. Whereas the metal uses a continuous conduction band for transferring electrons to the centre of catalysis, proteins employ a series of discrete electron-transferring centres, separated by distances of I.0-I.5nm. It has been shown that electrons can transfer rapidly over such distances from one centre to another, within proteins (Page et al. 1999). This is sometimes described as quantum-mechanical tunnelling, a process that depends on the overlap of wave functions for the two centres. Because electrons can tunnel out of proteins over these distances, a fairly thick insulating layer of protein is required, to prevent unwanted reduction of other cellular components. This is apparently the reason that the active sites of the hydrogenases are hidden away from the surface. 

Therefore, the principal role of the inclusion of the ionic term in the wave function is the reduction of the kinetic energy from the value in the purely covalent wave function. Thus, this is the delocalization effect alluded to above. We saw in the last section that the bonding in H2 could be attributed principally to the much larger size of the exchange integral compared to the Coulomb integral. Since the electrical effects are contained in the covalent function, they may be considered a first order effect. The smaller added stabilization due to the delocalization when ionic terms are included is of higher order in VB wave functions. 

A second approach to achieve a reduction of the 4-component Hamiltonian to an electrons-only Hamiltonian is to introduce approximations by eliminating the small components of the wave function (41-53). Also here, different protocols have been successfully exploited in quantum chemistry. 

For a transition of Mott type we shall show in Chapter 4, Section 3, neglecting the discontinuity resulting from long-range forces, that the transition should occur when 2zl = U. Near the transition the energy needed to excite an electron into the upper Hubbard band is U — 2zl. The wave function of an electron then falls off as e-fltr, where a=2m(I7 — 2zI)1/2/fc2. Thus the amount of spin in the sphere surrounding each atom will be made up from electrons on many of the surrounding atoms, and will clearly go to zero as [Pg.88]

Tang et al. [20] have analyzed a reduction procedure using the Schrodinger representation of the dynamics of the n-state system. The molecular wave function of the n-state system can be written as a superposition... 

We shall not perform the somewhat elaborous calculation of the MC wave function in detail. A somewhat simpler example is the dissociation of a double bond and it is given as an exercise (exercise 2). Here we only note that the number of configuration state functions (CSF s) will increase very quickly with the number of active orbitals. In most cases we do not have to worry about the exact construction of the MC wave function that leads to correct dissociation. We simply use all CSFs that can be constructed by distributing the electrons among die active orbitals. This is the idea behind the Complete Active Space SCF (CASSCF) method. The total number of such CSFs is for N2 175 for a singlet wave function. A further reduction is obtained by imposing spatial symmetry. All these CSFs are not included in a wave... 

With jj coupling, the spin-angular part of the one-electron wave function (2.15) is obtained by vectorial coupling of the orbital and spin-angular momenta of the electron. Then the total angular momenta of individual electrons are added up. In this approach a shell of equivalent electrons is split into two subshells with j = l 1/2. The shell structure of electronic configurations in jj coupling becomes more complex, but is compensated for by a reduction in the number of electrons in individual subshells. 

For a single determinantal wave function k = 1 holds true (no reduction). 

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