Orbital hydrogenlike

Normalized radial functions for a hydrogenlike atom are given in Table A 1.1 and plotted graphically in Fig. A 1.1 for the first ten combinations of n and /. It will be seen that the radial functions for Is, 2p, 3d, and 4f orbitals have no nodes and are everywhere of... 

The natural orbitals %2v and %3p are, in contrast to the hydrogenlike functions, localized within approximately the same region around the nucleus as the Is orbital. This means that the polarization caused by the long-range interaction is associated mainly with an angular deformation of the electronic cloud on each atom. If %2p and %3p are expanded in the standard hydrogen-like functions, an appreciable contribution will again come from the continuum. 

The Fourier transforms of the hydrogenlike orbitals were shown by Fock [18] to be expressible in terms of 4-dimensional hyperspherical harmonics when momentum space is mapped onto the surface of a 4-dimensional unit hypersphere by the transformation ... 

With this transformation, as Fock was able to show, the Fourier-transformed hydrogenlike orbitals become ... 

The wave functions for a state of a hydrogenlike atom described by the quantum numbers n (total quantum number), l (azimuthal quantum number), and m (magnetic quantum number) are usually expressed in terms of the polar coordinates r, 8, and . The orbital wave function is a product of three functions, each depending on one of the coordinates ... 

Since H° is the sum of hydrogenlike Hamiltonians, the zeroth-order wave function is the product of hydrogenlike functions, one for each electron. We call any one-electron spatial wave function an orbital. To allow for electron spin, each spatial orbital is multiplied by a spin function (either a or 0) to give a spin-orbital. To introduce the required antisymmetry into the wave function, we take the zeroth-order wave function as a Slater determinant of spin-orbitals. For example, for the Li ground state, the normalized zeroth-order wave function is... 

Electrically neutral atoms with nuclear charge Z > I are not hydrogenlike, but have inure than a single orbital electron. With more than one electron in an atom, it is necessary to determine the relationship of one... 

Figure 6-3 Schematic diagram of the relative energies of the hydrogenlike atomic orbitals...

FIGURE 1.25 The radial wave-functions of the first three s-orbitals of a hydrogenlike atom. Note that the number of radial nodes increases (as n — l), as does the average distance of the electron from the nucleus. Because the probability density is given by ijr2, all s-orbitals correspond to a nonzero probability density at the nucleus. 

For the integrations in ab initio calculations we need the actual mathematical form of the spatial functions, and the hydrogenlike expressions are Slater functions [1]. For atomic and some molecular calculations Slater functions have been used [3]. These vary with distance from where they are centered as exp(-constant.r), where r is the radius vector of the location of the electron, but for molecular calculations certain integrals with Slater functions are very time-consuming to evaluate, and so Gaussian functions, which vary as exp(-constant.r2) are almost always used a basis set is almost always a set of (usually linear combinations of) Gaussian functions [4]. Very importantly, we are under no theoretical restraints about their precise form (other than that in the exponent the electron coordinate occurs as exp(-constant.r2)). Neither are we limited to how many basis functions we can place on an atom for example, conventionally carbon has one 1 s atomic orbital, one 2s, and three 2p. But we can place on a carbon atom an inner and outer Is basis function, an inner and outer 2s etc., and we can also add d functions, and even f (and g ) functions. This freedom allows us to devise basis sets solely with a view to getting... 

Concerning molecules, the wave function (molecular orbital) for a hydrogenlike molecule, for instance, is expanded in terms of hydrogen-like atomic orbitals Xaj(f) belonging to hydrogen-like atoms / = 1,2, respectively, as... 

Aufbau principle the principle stating that as protons are added one by one to the nucleus to build up the elements, electrons are similarly added to hydrogenlike orbitals. (12.13) Autoionization the transfer of a proton from one molecule to another of the same substance. (7.2)... 

The Bohr model can be readily extended to hydrogenlike ions, systems in which a single electron orbits a nucleus of arbitrary atomic number Z. Thus, Z = 1 for hydrogen, Z = 2 for He+, Z = 3 for Li++, and so on. The Coulomb potential (7.5) generalizes to... 

Verify that the 2)dxy orbital given in Table 7.1 is a normalized eigenfunction of the hydrogenlike Schrddinger equation. 

The simpler wavefunctions for helium atom, for example (8.5), can be interpreted as representing two electrons in hydrogenlike s orbitals, designated as a configuration. Pauli s exclusion principle, which states that no two electrons in an atom can have the same set of four quantum numbers, requires the two s electrons to have different spins one spin-up ora, the other spin-down or A product of an orbital with a spin function is called a spinorbital. For example, electron 1 might occupy a spinorbital which we designate... 

R.D. ALLENDOERFER, Teaching the shapes of the hydrogenlike and hybrid atomic orbitals. J. Chem. Educ., 67, 37 (1990). 

One of the very early triumphs of quantum theory was the exact solution of the wave equation for hydrogenlike atoms. It was therefore natural to try to use hydrogenlike orbitals to build up solutions to the Hartree-Fock equations for... 

The reader will recognize that this is just the wave equation obeyed by the familiar hydrogenlike orbitals, except that Z/n has been replaced by the constant k. Thus, if we start with a hydrogenlike orbital and replace Z/n everywhere by the constant k, we will have generated a set of Coulomb Sturmians. They have the form... 

The reader may verify that these become the familiar hydrogenlike orbitals if k is replaced by Z n, where Z is the nuclear charge and n is the principal quantum number. It can be shown [19] that the Coulomb Sturmians obey a set of potential-weighted orthonormality relations of the form ... 

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