Hartree-Fock atom density

The penetration contribution to the electrostatic potential at R, is evaluated by application of the general expression of Eq. (8.49) for per for the spherical density (lt = ml = 0). The point-charge term, proportional to 1/Rfj-, must subsequently be subtracted. Due to the rapid decrease of the penetration terms with increasing R j, convergence is quickly achieved. For spherically averaged Hartree-Fock atom densities, inclusion of penetration terms for atoms within 10 A of the point under consideration is more than adequate. 

Rauhut, G., Puyear, S., Wolinski, K., Pulay, P., 1996, Comparison of NMR Shielding Calculated from Hartree-Fock and Density Functional Wave Functions Using Gauge-Including Atomic Orbitals , J. Phys. Chem., 100,... 

P has been computed using Hartree-Fock atomic orbital wavefunctions and can be found in several published tabulations14 17 and in Appendix 1. Because of the (r 3) dependence of P, dipolar coupling of a nuclear spin with electron spin density on another atom is usually negligible. 

In the pseudobond method of Yang and coworkers [47] a pseudobond is formed with one free-valence atom with an effective core potential (optimized to reproduce the length and strength of the real bond). This core potential can be applied in Hartree-Fock and density functional calculations and is designed to be independent of the choice of the MM force field. 

As the wave function is not known analytically for systems larger than a hydrogen-like atom, suitable approximate wave functions have to be found and the accuracy of Eq. (1) depends of course on the level of approximation. A survey of the various quantum chemical methods to generate approximated wave functions can be found in Refs. (22,23). Here, we shall only present the foundations of Hartree-Fock and density functional theory (DFT) needed in later sections. 

Here P and Plm are monopole and higher multipole populations / , are normalized Slater-type radial functions ylm are real spherical harmonic angular functions k and k" are the valence shell expansion /contraction parameters. Hartree-Fock electron densities are used for the spherically averaged core and valence shells. This atom centered multipole model may also be refined against the observed data using the XD program suite [18], where the additional variables are the population and expansion/contraction parameters. If only the monopole is considered, this reduces to a spherical atom model with charge transfer and expansion/contraction of the valence shell. This is commonly referred to as a kappa refinement [19]. 

R. Lopez-Boada, R. Pino and E.V. Ludena. Hartree-Fock energy density functionals generated by local-scaling transformations Applications to first-row atoms. J. Chem. Phys. (submitted). 

The only atomic wave-functions that do not have a node at the nucleus are s-functions. The isotropic coupling constant is thus a measure of the s-character of the wave-function of the unpaired electron at the nucleus in question. The coupling constant for an atomic s-electron can be either measured experimentally or calculated from Hartree-Fock atomic wave-functions so that, to a first approximation, the s-electron density may be calculated from the ratio of the experimental and atomic coupling constants. Should the first-order s-character of the wave-function of the unpaired electron be zero, as for example in the planar methyl radical, then a small isotropic coupling usually arises from second-order spin-polarization effects. The ESR spectra of solutions show only isotropic hyperfine coupling. 

The Patterson or direct method solution will give a number of electron density peaks which can be identified as atoms of certain types. This is still a very crade model of the stracture, which should be optimized by the least squares (LS) refinement in the following way. Spherically symmetrical Hartree-Fock atoms are placed at the positions of the peaks and the coordinates (Section 2.2.2) and displacement parameters (Section 2.2.3) of these atoms are altered so as to minimize the function... 

Eq. 18 can be used to evaluate the total energy of the system for any pair of. /V-representable electron densities pa and pB- It is, therefore, worthwhile to recall the model proposed by Gordon and Kim49 in 1972 which can be seen as a particular application of Eq. 18. Gordon and Kim used the Hartree-Fock electron densities of isolated rare-gas atoms as pGK and p K in Eq. 18 to evaluate the energy of a dimer. The interaction energy can thus be expressed as ... 

Modem quantum-chemical methods can, in principle, provide all properties of molecular systems. The achievable accuracy for a desired property of a given molecule is limited only by the available computational resources. In practice, this leads to restrictions on the size of the system From a handful of atoms for highly correlated methods to a few hundred atoms for direct Hartree-Fock (HF), density-functional (DFT) or semiempirical methods. For these systems, one can usually afford the few evaluations of the energy and its first one or two derivatives needed for optimisation of the molecular geometry. However, neither the affordable system size nor, in particular, the affordable number of configurations is sufficient to evaluate statistical-mechanical properties of such systems with any level of confidence. This makes quantum chemistry a useful tool for every molecular property that is sufficiently determined (i) at vacuum boundary conditions and (ii) at zero Kelvin. However, all effects from finite temperature, interactions with a condensed-phase environment, time-dependence and entropy are not accounted for. 

When the application of Eq. (11) to a least squares analysis of x-ray structure factors has been completed, it is usual to calculate a Fourier synthesis of the difference between observed and calculated structure factors. The map is constructed by computation of Eq. (9), but now IFhid I is replaced by Fhki - F/f /, where the phase of the calculated structure factor is assumed in the observed structure factor. In this case the series termination error is virtually too small to be observed. If the experimental errors are small and atomic parameters are accurate, the residual density map is a molecular bond density convoluted onto the motion of the nuclear frame. A molecular bond density is the difference between the true electron density and that of the isolated Hartree-Fock atoms placed at the mean nuclear positions. An extensive study of such residual density maps was reported in 1966.7 From published crystallographic data of that period, the authors showed that peaking of electron density in the aromatic C-C bonds of five organic molecular crystals was systematic. The random error in the electron density maps was reduced by averaging over chemically equivalent bonds. The atomic parameters from the model Eq. (11), however, will refine by least squares to minimize residual densities in the unit cell. 

Assuming that an ab initio or semiempirical technique has been used to obtain p(r), we address the important question of how the calculated electrostatic potential depends on the nature of the wavefunction used for computing p(r). Historically, and today as well, most ab initio calculations of V(r) for reasonably sized molecules have been based on self-consistent-field (SCF) or near Hartree-Fock wavefunctions and therefore do not reflect electron correlation. Whereas the availability of supercomputers has made post-Hartree-Fock calculations of V(r) (which include electron correlation) a realistic possibility even for molecules with 5 to 10 first-row atoms, there is reason to believe that such computational levels are usually not necessary and not warranted. The M0l er-Plesset theorem states that properties computed from Hartree-Fock wavefunctions using one-electron operators, as is V(r), are correct through first order " any errors are no more than second-order effects. Whereas second-order corrections may not always be insignificant, several studies have shown that near-Hartree-Fock electron densities are affected to only a minor extent by the inclusion of correlation.The limited evidence available suggests that the same is true of V(r), ° ° as is indicated also by the following example. 

If you choose molecular mechanics as in Section 2, you may be asked to choose a data set. If you select a semi-empirical, ab initio Hartree-Fock or density-functional method you will need to choose a basis set for each type of atom. Many programs have a library of basis sets built in that you can simply select by name. 

Practical applications require an explicit form for and and a choice of ionic electron densities n (r). Early formulations of the theory used electronic charge densities derived from Hartree-Fock atomic codes. For a review of the various implementations of electron gas potentials we refer the reader to Wolf and Bukowinski [28], Gordon and LeSar [34] and Chizmeshya et al. [29]. In the remainder of this report we concentrate on topics relevant to applications of the VIB [28, 35] and MPIB [36] models. Both models... 

Basing on the first principles of Quantum mechanics as exposed in the previous chapters and sections, special chapters of quantum theory are here unfolded in order to further extend and caching the quantum information from free to observed evolution within the matter systems with constraints (boundaries). As such, the Feynman path integral formalism is firstly exposed and then applied to atomic, quantum barrier and quantum harmonically vibration, followed by density matrix approach, opening the Hartree-Fock and Density Functional pictures of many-electronic systems, with a worthy perspective of electronic occupancies via Koopmans theorem, while ending with a further generalization of the Heisenberg observability and of its first application to mesosystems. 

The upper two graphs in Figure 16.4 illustrate the difference between the four-component Dirac-Hartree-Fock reference density and the two-component ZORA density, which is only prominent in the inner core of the atoms in both complexes. The two plots in the middle of Figure 16.4 display the difference between the four-component density and the scalar-relativistic DKH density. The DKH density differs from the reference only close to the nucleus of the heavy atom Pt — spin-orbit effects are thus negligible for such a sixth-row atom. The lighter Ni analogue features differences only very close to the atomic nucleus. 

Figure 16.4 Total electronic densities of M(C2H2) with M=Ni,Pt from Hartree-Fock calculations with two-component ZORA, scalar-relativistic DKH10, and nonrelativistic Schrodinger one-electron operators subtracted from the four-component Dirac-Hartree-Fock reference densities (data taken from Ref. [880]). The molecular structure of the complexes is indicated by element symbols and lines positioned just below the atomic nuclei (top panel). Asymmetries in the plot are due to the discretization of the density on a cubic grid of points. The DKH densities have not been corrected for the picture-change effect and, hence, deviate from the four-component reference density in the closest proximity to the nuclei. But these effects can hardly be resolved on the numerical grid employed to represent the densities.

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