One-electron atoms Atomic orbitals

Table 1.1. One-electron atoms atomic orbitals with principal quantum numbers n = 1,2 or 3. All functions are normalized and orthogonal. 

An important difference between the BO and non-BO internal Hamiltonians is that the former describes only the motion of electrons in the stationary field of nuclei positioned in fixed points in space (represented by point charges) while the latter describes the coupled motion of both nuclei and electrons. In the conventional molecular BO calculations, one typically uses atom-centered basis functions (in most calculations one-electron atomic orbitals) to expand the electronic wave function. The fermionic nature of the electrons dictates that such a function has to be antisymmetric with respect to the permutation of the labels of the electrons. In some high-precision BO calculations the wave function is expanded in terms of basis functions that explicitly depend on the interelectronic distances (so-called explicitly correlated functions). Such... 

Many of the principles and techniques for calculations on atoms, described in section 6.2 of this chapter, can be applied to molecules. In atoms the electronic wave function was written as a determinant of one-electron atomic orbitals which contain the electrons these atomic orbitals could be represented by a range of different analytical expressions. We showed how the Hartree-Fock self-consistent-field methods could be applied to calculate the single determinantal best energy, and how configuration interaction calculations of the mixing of different determinantal wave functions could be performed to calculate the correlation energy. We will now see that these technques can be applied to the calculation of molecular wave functions, the atomic orbitals of section 6.2 being replaced by one-electron molecular orbitals, constructed as linear combinations of atomic orbitals (l.c.a.o. method). 

As was emphasized before (cf. Chapter 3), a molecule is not simply a collection of its constituting atoms. Rather, it is a system of atomic nuclei and a common electron distribution. Nevertheless, in describing the electronic structure of a molecule, the most convenient way is to approximate the molecular electron distribution by the sum of atomic electron distributions. This approach is called the linear combination of atomic orbitals (LCAO) method. The orbitals produced by the LCAO procedure are called molecular orbitals (MOs). An important common property of the atomic and molecular orbitals is that both are one-electron wave functions. Combining a certain number of one-electron atomic orbitals yields the same number of one-electron molecular orbitals. Finally, the total molecular wave function is the... 

B. RAMACHANDRAN and P.C. KONG, Three-dimensional graphical visualization of one-electron atomic orbitals. J. Chem. Educ., 72, 406 (1995). 

Van Vleck introduced the pure precession approximation for diatomic molecules. The basic idea is that if the one-electron atomic orbital angular momentum is preserved in a diatomic molecule then the A-doubling and spin-rotation constants can be predicted, for example, in a 4pa(2l,+) and a 4p7r(2n) state. This pure precession picture can easily be generalized to... 

Thus, one might expect the one-electron atom orbitals (AOs, say, the familiar s, p, d etc. orbitals) to be the ones to use in expanding the orbitals in a singledeterminant model of the electronic structure of (at least) the rare-gas atoms which (presumably) have a central potential which is a combination of many interactions all described by Coulomb s law. The forms of these orbitals have been determined by the action of Coulomb s law and there are a huge number of them of all possible symmetry types, with which to msdce an expansion of apparently arbitrary accuracy. 

Escf is the self-consistent field energy, f is the electric field, a is a nuclear coordinate, is the one-electron atomic orbital integral, If is related to the derivative of the molecular orbital coefficients with respect to a by... 

The term alf in the above summations refers to all occupied and virtual molecular orbitals refers to doubly occupied orbitals such as those found in the ground state of a closed shell system. Terms such as Cf refer to the coefficients of the atomic orbital m in the i unperturbed molecular orbital, (see Yamaguchi et al. 1986, for a derivation and further explanation). Although the above expression is complicated, the main conclusion the reader should draw is that the dipole derivative depends on the derivatives of the one-electron atomic orbitals with respect to the electric field and nuclear coordinates and the derivative of the molecular orbitals with respect to nuclear coordinates multiplied by the derivative of the one-electron atomic orbitals with respect to the electric field. 

This way of looking at the molecule appeals to us for the same reason that we use the one-electron atomic orbitals to write electron configurations that approximate the much more complicated many-electron atomic wavefunctions. Reducing the number of coordinates, the dimensionality of the problem, makes the problem more tractable and can lead to a useful intuition about how these systems behave. 

Solving the One-Electron Atom Schrodinger Equation 104 BIOSKETCH Peter Beiersdorfer 116 The One-Electron Atom Orbital Wavefunctions 121 Electric Dipole Interactions 135... 

These atomic orbitals, called Slater Type Orbitals (STOs), are a simplification of exact solutions of the Schrodinger equation for the hydrogen atom (or any one-electron atom, such as Li" ). Hyper-Chem uses Slater atomic orbitals to construct semi-empirical molecular orbitals. The complete set of Slater atomic orbitals is called the basis set. Core orbitals are assumed to be chemically inactive and are not treated explicitly. Core orbitals and the atomic nucleus form the atomic core. 

Exact solutions to the electronic Schrodinger equation are not possible for many-electron atoms, but atomic HF calculations have been done both numerically and within the LCAO model. In approximate work, and for molecular applications, it is desirable to use basis functions that are simple in form. A polyelectron atom is quite different from a one-electron atom because of the phenomenon of shielding", for a particular electron, the other electrons partially screen the effect of the positively charged nucleus. Both Zener (1930) and Slater (1930) used very simple hydrogen-like orbitals of the form... 

Briefly, we write the atomie orbitals for a one-electron atom as... 

In order to retain the orbital model for a many-electron atom, Hartree assumed that each electron came under the influence of the nuclear charge and an average potential due to the remaining electrons. He therefore retained the form of the radial equation for a one-electron atom, equation 12.2, but assumed that the mutual potential energy U was the sum of... 

The rationale behind this choice of bond integrals is that the radical stabilizing alpha effect of such radicals are explained not by the usual "resonance form" arguments, but by invoking frontier orbital interactions between the singly occupied molecular orbital of the localized carbon radical and the highest occupied molecular orbital (the non-bonding electrons atomic orbital) of the heteroatom (6). For free radicals the result of the SOMO-HOMO interaction Ts a net "one-half" pi bond (a pi bond plus a one-half... 

It is also possible that orbitals of different kinds on the two atomic centers such as s-pz, p -d , d,z-p, etc. can combine to generate the MO for the diatomic molecules. As the one-center atomic orbitals are not orthogonal in molecules, for the depiction of electronic structure, the concept of hybridization is quite useful. 

A cr orbital can be formed either from two s atomic orbitals, or from one s and one p atomic orbital, or from two p atomic orbitals having a collinear axis of symmetry. The bond formed in this way is called a a bond. A n orbital is formed from two p atomic orbitals overlapping laterally. The resulting bond is called a n bond. For example in ethylene (CH2=CH2), the two carbon atoms are linked by one a and one n bond. Absorption of a photon of appropriate energy can promote one of the n electrons to an antibonding orbital denoted by n. The transition is then called Ti —> 7i. The promotion of a a electron requires a much higher energy (absorption in the far UV) and will not be considered here. 

The quantum numbers n and i. Multi-electron atoms can be characterized by a set of principal and orbital quantum numbers n, t which labels one-electron wave functions (orbitals). 

The deformation functions, however, must also describe density accumulation in the bond regions, which in the one-center formalism is represented by the atom-centered terms. They must be more diffuse, with a different radial dependence. Since the electron density is a sum over the products of atomic orbitals, an argument can be made for using a radial dependence derived from the atomic orbital functions. The radial dependence is based on that of hydrogenic orbitals, which are valid for the one-electron atom. They have Slater-type radial functions, equal to exponentials multiplied by r1 times a polynomial of degree n — l — 1 in the radial coordinate r. As an example, the 2s and 2p hydrogenic orbitals are given by... 

For an atom with many electrons, the first electron fills the lowest energy orbital, and the second electron fills the next lowest energy orbital, and so forth. For a one-electron atom or ion, the energy depends only on n, the principal quantum number but for a many-electron atom or ion, the value of I also plays a role in the energy. The order of atomic orbital energy is given by... 

In conclusion, the energies E that sahsfy Eq. (1.19) are associated to molecular electronic states. Since Eq. (1.19) is an equation of Nat order, we obtain Nat energy values E/ (/ = 1,. .., Nat), that is, as many molecular levels as atomic orbitals. In the simple example of H2 discussed in Sechon 1.1, Aat = 2 and both I5 atomic orbitals combine to form bonding ag and antibonding a MOs. In the case of N2 (see Fig. 1.1), neglechng I5 core electrons, the combinahon of two sp and one pz atomic orbitals per N atom leads to six MOs. 

Problem 8-15. A normalized hydrogenic Is orbital for a one-electron atom or ion of nuclear charge Z has the form ... 

The hydrogenic atom energy expression has no 1-dependence the 2s and 2p orbitals have exactly the same energy, as do the 3s, 3p, and 3d orbitals. This degree of degeneracy is only present in one-electron atoms and is the result of an additional symmetry (i.e., an additional operator that commutes with the Hamiltonian) that is not present once the atom contains two or more electrons. This additional symmetry is discussed on p. 77 of Atkins. 

To evaluate both the coupling and the magnitude of the moments, we consider first an array of one-electron atoms in non-degenerate states, and let 0 denote an atomic orbital for an electron on atom (i). Then the electron on atom (i) can overlap on to neighbouring atoms (/), so we may write its wave function in the form... 

It is assumed that each carbon atom contributes one 7r-electron and one 2pr atomic orbital to the system, so that... 

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