Spin-free density function

For example, suppose we have a two-electron system in a time-dependent state described by the wavefunction y, zi,a>i,x2, yi, 2,u)2,t)- The spin coordinates ol> would each be some combination of spin functions a and f. If we integrate over the spin coordinates of both electrons, we are left with a spin-free density function. Call it p(xi,yi,zi,x2,y2,Z2,t) = p vi,V2,t). We interpret p(vi,v2,t)dvidv2 as... 

In order to understand this, let us again consider the and S terms of the Is 2s configuration of a two-electron atom. In the framework of Slater s theory of atoms the electron density g(f) is the same for both states, and for an exact treatment we would expect (9) to differ very little between the two states. The pair density n(9i, 92) is, however, quite different for the IS and the states. If we call the two singly occupied real (orthogonal) orbitals a and b, we have (where stands for the spin-free wave function and where a (1) has the same meaning as a (fi)... 

This is plotted in the right-hand panel of Fig. 3.8 as a function of I/2 h. Remembering that h(R) - 0 as R - oo, we see that it shows the same square root distance-dependence as that displayed by the numerical self-consistent solution of the local spin density functional Schrddinger equation in Fig. 3.6. Thus, as the hydrogen molecule is pulled apart, it moves from the singlet state S = 0 at equilibrium to the isolated free atoms in doublet states with S = 2-... 

To arrive at significant insight for the biological reduction process, all theoretical studies should be taken into account since none of them is free of model-inherent approximations. We emphasize that none of the recent studies, which utilized DFT methods, discussed the theoretical complications with standard density functionals arising from different spin states, which are discussed in the Appendix. The authors thus have assumed that the dependence of the results on the chosen density functional is of little importance in... 

Wetmore et al. have achieved impressive results with the use of Density Functional Theory (DFT) calculations on the primary oxidation and reduction products observed in irradiated single crystals of Thymine [78], Cytosine [79], Guanine [80], and Adenine [81], The theoretical calculations included in these works estimated the spin densities and isotropic and anisotropic hyperfine couplings of numerous free radicals which were compared with the experimental results discussed above. The calculations involve a single point calculation on the optimized structure using triple-zeta plus polarization functions (B3LYP/6-31 lG(2df,p)). In many cases the theoretical and experimental results agree rather well. In a few cases there are discrepancies between the theoretical and experimental results. 

The SOF4 anion has been generated by y-irradiation of CsSOF5 and characterized by its isotropic EPR spectrum at 27°C [153]. The SOF4 anion has a /vew o-octahedral structure of C4v symmetry in which the equatorial positions are occupied by four equivalent fluorines, one axial position is occupied by a doubly bonded oxygen and the second axial position by the sterically active free valence electron. The structure and spin density of SOF4 have been analyzed by local density functional theory calculations and the isoelectronic POF42 radical anion has also been calculated. 

Energy is not the only property that is so determined by the electron density fragment pc. Since the (non-degenerate, ground state) local electron density pc(r) in any standard domain c fully determines the complete density pit), which in turn fully determines the molecular wavefunction P (up to a phase factor), all molecular properties P which can be expressed as expectation values of spin-free operators defined by the ground state wavefunction P are also determined by the local electron density pc(r) in the standard domain c. Consequently, any such property P is also a unique functional of the local electron density pc(r) within the standard domain c ... 

The electron density is closely related to a more general function, the so-called spin-free one-particle density matrix 6 7 8). Whereas the electron density is a function of the three coordinates x, y, z, the density matrix is a function of six coordinates, which are conventionally noted Xx,y, zi,xi,y, zx. In the case of a one-electron system, the density matrix is given by... 

These two auxiliary results, in fact, state all of the conclusions of the original Hohenberg-Kohn theorem the ground-state energy of a molecule, as well as the ground-state wavefunction T consequently, the expectation values of all spin-free observables are unique functionals of the ground-state electron density p(r). 

Eqs. (l)-(3), (13), and (19) define the spin-free CGWB-AIMP relativistic Hamiltonian of a molecule. It can be utilised in any standard wavefunction based or Density Functional Theory based method of nonrelativistic Quantum Chemistry. It would work with all-electron basis sets, but it is expected to be used with valence-only basis sets, which are the last ingredient of practical CGWB-AIMP calculations. The valence basis sets are obtained in atomic CGWB-AIMP calculations, via variational principle, by minimisation of the total valence energy, usually in open-shell restricted Hartree-Fock calculations. In this way, optimisation of valence basis sets is the same problem as optimisation of all-electron basis sets, it faces the same difficulties and all the experience already gathered in the latter is applicable to the former. 

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