Alpha electron

The UIIF wnive fimction can also apply to singlet molecules. F sn-ally, the results are the same as for the faster RHF method. That is, electron s prefer to pair, with an alpha electron sh arin g a m olecu lar space orbital with a beta electron. L se the L lIF method for singlet states only to avoid potential energy discontinuities when a covalent bond Is broken and electron s can impair (see Bond Breaking on page 46). 

The two sets of coeflicien ts, one for spin-up alpha electrons and the other for spin-down beta electrons, are solutions of iw O coupled matrix eigenvalue problems ... 

For systems with unpaired electrons, it is not possible to use the RHF method as is. Often, an unrestricted SCF calculation (UHF) is performed. In an unrestricted calculation, there are two complete sets of orbitals one for the alpha electrons and one for the beta electrons. These two sets of orbitals use the same set of basis functions but different molecular orbital coefficients. 

The spin multiplicity for a molecule is given by the equation 2S + 1, where S is the total spin for the molecule. Paired electrons contribute nothing to this quantity. They have a net spin of zero since an alpha electron has a spin of +Vi and a beta electron has a spin of -Vi. Each unpaired electron contributes +Vi to S. Thus, a singlet—a system with no unpaired electrons—has a spin multiplicity of 1, a doublet (one unpaired electron) has a spin multiplicity of 2, a triplet (two unpaired electrons of like spin) has a spin multiplicity of 3, and so on. 

The number of alpha electrons is NO A, and the number of beta electrons i... 

Rotational constants (GHZ) Isotopes 0-16,H-l,H-l 19 basis functions 5 alpha electrons... 

If the external magnetic field B(r), and m(r) have only a nonvanishing Z-component, B(r) = (0,0, B(r)) and m(r) = (0,0, m(r)), the universal functional F[p, m] may then be considered as a functional of the spin densities ps(r) and p(r), F[ps(r), p(r)], because the spin density is proportional to the z-component of the magnetization m(r) = p-bPsW P-b is the electron Bohr magneton. It is of worth mentioning that it is possible to define two spin densities that are the diagonal elements of the density matrix introduced by von Barth and Hedin [3]. These correspond to the spin-up (alpha) electrons density pT(r), and the spin-down (beta) electrons density p (r). In terms of these quantities, the electron and spin densities can be written as... 

Many nuclear processes may be indicated by a shorthand notation in which a light bombarding particle and a light product particle are represented by symbols in parentheses. These are located between the initial target nucleus and the final product nucleus. The symbols used are n, p, d, a, fi+, f, and y, representing the neutron, proton, deuteron (2H), alpha, electron, positron, and gamma rays, respectively. The atomic numbers are commonly left out in this notation because the symbol for any element implies its atomic number. Examples of the longhand and shorthand notations are as follows ... 

However, in an open-shell system involving unpaired alpha electrons, the electron environment depends on whether the electron is in an alpha or a beta orbital. Thus, we have also proposed that we retain canonical Fock orbitals but adopt spin-dependent orbital energies defined by... 

Ionizing radiation Particulate radiations (alpha, electron, neutron, proton) and electromagnetic radiations (X-rays and gamma rays) that have the ability to disrupt molecular bonds (ionize) when they interact with matter. 

These states can be depicted diagrammatically as lattice points in (w, my, m-) space. Thus, the number of states with energy less than is given approximately by the volume of an octant of a sphere of radius m. If we place n non-interacting alpha electrons into the cube, they will fill the lowest n states and mmax, the largest occupied m value, will therefore satisfy... 

In Table 11.1, the number of Slater determinants iVdet in the FCl wave function appears to grow by a constant factor of about 16 with each increment in k. To see how this exponential growth comes about, we note that iV et is the product of the number of determinants for the alpha and beta electrons separately - see Section 11.6 for details. Thus, for a system with n orbitals containing k alpha electrons and k beta electrons, the number of determinants is given by... 

For Na alpha electrons distributed among n orbitals, the number of alpha strings (i.e. the number of rows in C) is given by the binomial coefficient... 

As an example, consider a system with three alpha electrons and three beta electrons distributed among five orbitals. The (3) = 10 alpha and beta strings of this system are listed in Table 11.5. There are 10 x 10 = 100 Slater determinants in the wave function and the Cl coefficients may be arranged as a 10 x 10 matrix. 

The MOC algorithm performs poorly when the number of electrons of alpha spin is small (one or two). With a single alpha electron, the only string for which a apa I ) is nonzero is la = a a vac). The operation count for (11.8.42) then becomes comparable with the operation count for (11.8.43). Moreover, each element is only used once in (11.8.43) and the... 

An orbital configuration is characterized by the number of unpaired electrons - that is, by the number of orbital occupations equal to 1. For a given determinant in this configuration, the total number of unpaired electrons is equal to the number of unpaired alpha electrons n plus the number of unpaired beta electrons n ... 

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