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14
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84951349071
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The molecular structure associated with a given nuclear configuration X is stable if ρ(r,X) has a finite number of critical points and (a) each critical point is nondegenerate (of rank three)
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15
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84951349074
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the stable and unstable manifolds of any pair of critical points intersect transversely (Refs. 4-6)
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16
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84951349073
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An atomic property A is determined by an integration of the corresponding density [formula omitted] over the basin of the atom. The symbol [formula omitted] implies a summation over all spins and an integration over the spatial coordinates of all electrons but one whose coordinates are denoted by r. Thus [formula omitted] The quantity J is the single-particle vector current density
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21
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84951349076
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The Tr σ(r) may be expressed as [formula omitted] Even at, and in the neighborhood of a [formula omitted] or cage critical point where all three curvatures of ρ(r) are positive, [formula omitted]
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22
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33847801737
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An atomic core region may be defined in terms of an extremum principle as a spherical region of space centered on the nucleus bounded by a surface such that the contained Fermi correlation is maximized. This corresponds to finding, a region for which the fluctuation in the average electron population is a minimum and in which the electron number and pair densities are maximally localized. ]Such core regions in general exist and are found to possessan average population N̄ of approximately two electrons. Some values of N̄, radius r and % localization are 2.0, 0.53 a.u. and 88% for C; 2.0, 0.26 a.u. and 77% for Ne; 1.9, 0.12 a.u. and 73% for the K shell in Ar and 10.1, 0.75 a.u. and 80% for its combined K and L shells
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(1975)
J. Am. Chem. Soc
, vol.97
, pp. 7391
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Bader, R.F.W.1
Stephens, M.E.2
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23
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0000016323
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It has also been proposed that the first minimum in the radial distribution function for an atom be used to define a core radius [ ]. In general, these radii are slightly greater than those defined by minimizing the fluctuation in N̄ as described above
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(1976)
J. Chem. Phys
, vol.64
, pp. 4634
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Politzer, P.1
Parr, R.G.2
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24
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84951349075
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The relationship in Eq. (17) is found to be only approximately true for many-electron atoms using Hartree-Fock state functions in the calculations. In these systems the constant of proportionality equal to 0.541 in Eq. (17) exhibits a small Z dependence and its value decreases from 0.534 for He to 0.424 for Ne
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25
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84951349077
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In a one-electron atom the curvature of ρ parallel to a radial line [formula omitted] is always positive. In a many-electron atom [formula omitted] may become slightly negative over a very short range of radial distances r, near the inner boundary of the valence region, where [formula omitted] For example, in Ar, [formula omitted] for [formula omitted] In this situation the magnitude of [formula omitted] is less than the magnitudes of the perpendicular curvatures of ρ
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28
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84926268037
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have constructed a stress tensor in terms of the electronic and nuclear charge distributions applicable to one-electron and singlet ground states of a two-electron system within a single determinant approximation, i.e.,F(r)is conservative. They have studied the properties of this tensor in the hydrogen atom and have used it in a discussion of the binding in the [formula omitted] molecule.
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(1979)
Mol. Phys
, vol.38
, pp. 2069
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Deb1
Bamzai2
Deb, B.M.3
Bamzai, A.S.4
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29
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84951348843
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In a bound system, an atomic interaction line is called a bond path
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33
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84951348846
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Displays of the kinetic energy density G(r) for [formula omitted] and [formula omitted] are given in Ref. 26. To appreciate the dramatically low values attained by G(r) and in particular its parallel component in [formula omitted] the reader is referred to Fig. 1 of this reference for a profile of G(r) along the internuclear axis calculated using a correlated state function
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51
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84951348848
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Previous studies (Ref. 34) of the binding or antibinding nature of a molecular charge distribution have made use of density difference maps wherein superposed atomic densities are subtracted from the molecular density. Such studies suffer from a degree of arbitrariness in the choice of the atomic reference state and in the nonrealizable nature of the superposed atomic density distribution. The Laplacian distribution of ρ determines the regions of charge depletion and charge concentration without recourse to a reference state
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55
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84951348852
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The relative positionings and magnitudes of the regions where [formula omitted] for K in KF (Fig. 3) show it to be back polarized. One notes that the pattern of [formula omitted] for K in KF including its polarization, are similar to those for an argon atom in [formula omitted] as anticipated on the basis of the orbital shell model and the ionic model for KF
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56
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84942701884
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These results are calculated from a single determinant SCF function using a 6-31G** Gaussian basis at the optimized geometry. The geometries and binding energies compare favorably with previous calculations. For example, Tse et al. [ in a 6-31G* calculation of [formula omitted] calculate a hydrogen bond energy of 5.64 kcal/mol, an 0-0 separation of 2.98 Å and a value of 55° for the angle θ between the 0-0 vector and the plane of the proton acceptor. The corresponding values obtained in the present research are 5.50 kcal/mol, 2.99 Å and [formula omitted] The total energy of [formula omitted] is found to be [formula omitted] The energy of [formula omitted] at 6-31G** is [formula omitted] yielding a hydrogen bond energy of 5.88 kcal/mol, an F-F separation of 2.68 Å and an H-F…H angle of 115.5°. The F…H-F angle is calculated to be 170.4°. The calculated results compare favorably with experimental results (see, P. A. Kollman, Electronic Structure Theory, edited by H. F. Schaefer III (Plenum, New York, 1977), pp. 109-152): [formula omitted] hydrogen bond energy is [formula omitted] and 0-0 separation is 2.98 Å; [formula omitted] hydrogen bond energy is [formula omitted] and F-F separation is 2.80 Å.
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(1980)
Chem. Phys. Lett
, vol.75
, pp. 350
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Tse, Y.-C.1
Newton, M.D.2
Allen, L.C.3
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59
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84951348831
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Tse, Newton and Allen (Ref. 40) find the population of the hydrogen to decrease on hydrogen bond formation when the populations are determined using Mulliken’s method
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62
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0001215724
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These definitions of [formula omitted] [formula omitted] and E follow directly from the identification of the virial of the forces acting on a set of particles with their potential energy [ as is done in classical mechanics when the forces are conservative. Indeed, for a system with nonconservative forces the variation of the classical action integral [formula omitted] is replaced by a variation of the integral [formula omitted] i.e., if V\s undefined it is replaced by the virial of the forces acting on the particles. The variation of the integral /is also employed for the description of a system acted on by forces of constraint. This is the particular case of interest in quantum chemistry, for solving [formula omitted] in the fixed nucleus approximation is such a problem—external forces act on the nuclei to hold them in the fixed geometry. Thus the electronic energy [formula omitted] may be obtained by extremizing the variation of the electronic kinetic energy and the virtual work of the electrons. The Hamiltonian which yields the average energy [formula omitted] is [formula omitted] a sum of a kinetic energy operator and an effective single-particle potential energy operator for each electron in the system. This is a constrained variation in which the virtual work of the forces of constraint (the external forces acting on the nuclei) must vanish.
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(1975)
J. Chem. Phys
, vol.63
, pp. 3945
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Srebrenik, S.1
Bader, R.F.W.2
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64
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0000534419
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The in-plane ‘π-like’ nature of the CC bonds in cyclopropane is accounted for in molecular orbital theory through the choice of a particular orbital representation, the so-called Walsh orbitals. [
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(1947)
Nature
, vol.159
, pp. 712
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Walsh, A.D.1
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