Atomic orbitals nucleus

Population anaiysis methods of assigning charges rely on the LCAO approximation and express the numbers of electrons assigned to an atom as the sum of the populations of the AOs centered at its nucleus. The simplest of these methods is the Coulson analysis usually used in semi-empirical MO theory. This analysis assumes that the orbitals are orthogonal, which leads to the very simple expression for the electronic population of atom i that is given by Eq. (53), where Natomic orbitals centered... 

Ch em uses Slater atom ic orbitals to con struct sent i-em pirical molecular orbitals. I he complete set of Slater atomic orbitals is called the basis set. Core orbitals are assumed to be chemically inactive and arc not treated explicitly. Core orbitals and the atomic nucleus form the atomic core. 

Section 1 1 A review of some fundamental knowledge about atoms and electrons leads to a discussion of wave functions, orbitals, and the electron con figurations of atoms Neutral atoms have as many electrons as the num ber of protons m the nucleus These electrons occupy orbitals m order of increasing energy with no more than two electrons m any one orbital The most frequently encountered atomic orbitals m this text are s orbitals (spherically symmetrical) and p orbitals ( dumbbell shaped)... 

Thus for H and He, the basis set consists of one orbital, a Is atomic orbital. For atomsLi to Ne the 2 inner-shell electrons are combined with the nucleus and the basis set consists of 4 orbitals, the 2s, 2p, ... 

MOs around them - rather as we construct atomic orbitals (AOs) around a single bare nucleus. Electrons are then fed into the MOs in pairs (with the electron spin quantum number = 5) in order of increasing energy using the aufbau principle, just as for atoms (Section 7.1.1), to give the ground configuration of the molecule. 

The basis of constructing the MOs is the linear combination of atomic orbitals (LCAO) method. This takes account of the fact that, in the region close to a nucleus, the MO wave function resembles an AO wave function for the atom of which the nucleus is a part. It is reasonable, then, to express an MO wave function 1/ as a linear combination of AO wave functions Xi on both nuclei ... 

A convenient orbital method for describing eleetron motion in moleeules is the method of molecular orbitals. Molecular orbitals are defined and calculated in the same way as atomic orbitals and they display similar wave-like properties. The main difference between molecular and atomic orbitals is that molecular orbitals are not confined to a single atom. The crests and troughs in an atomic orbital are confined to a region close to the atomic nucleus (typieally within 1-2 A). The electrons in a molecule, on the other hand, do not stick to a single atom, and are free to move all around the molecule. Consequendy, the crests and troughs in a molecular orbital are usually spread over several atoms. 

The lowest energy molecular orbital of singlet methylene looks like a Is atomic orbital on carbon. The electrons occupying this orbital restrict their motion to the immediate region of the carbon nucleus and do not significantly affect bonding. Because of this restriction, and because the orbital s energy is very low (-11 au), this orbital is referred to as a core orbital and its electrons are referred to as core electrons. 

Atomic orbital Isa is associated with nucleus Ha and atomic orbital Isb with nucleus Hb. The first term in the electron density (is (r)) /(I -I-S ab) is taken to represent the amount of electron density associated with nucleus Ha- The corresponding term (ls (r)) /(I -I-S ab) represents the electron density associated with nucleus Hb and the remainder 2(lsA(r)lsB(r))/(l -t- AB) is taken to represent the amount shared by the two nuclei. Mulliken s first idea was to integrate these contributions, which gives the values 1/(1 -I-S ab), 1/(1 fi- AB) and (25ab)/(1 + ab). These values are assumed to contain some chemical information. Note that they sum to 2, the number of electrons in H2. 

In order to obtain an approximate solution to eq. (1.9) we can take advantage of the fact that for large R and small rA, one basically deals with a hydrogen atom perturbed by a bare nucleus. This situation can be described by the hydrogen-like atomic orbital y100 located on atom A. Similarly, the case with large R and small rB can be described by y100 on atom B. Thus it is reasonable to choose a linear combination of the atomic orbitals f00 and f00 as our approximate wave function. Such a combination is called a molecular orbital (MO) and is written as... 

To interpret the information in each atomic orbital, we need a way to identify the location of each point around a nucleus. It is most convenient to describe these locations in terms of spherical polar coordinates, in which each point is labeled with three coordinates ... 

To improve our model we note that s- and /7-orbitals are waves of electron density centered on the nucleus of an atom. We imagine that the four orbitals interfere with one another and produce new patterns where they intersect, like waves in water. Where the wavefunctions are all positive or all negative, the amplitudes are increased by this interference where the wavefunctions have opposite signs, the overall amplitude is reduced and might even be canceled completely. As a result, the interference between the atomic orbitals results in new patterns. These new patterns are called hybrid orbitals. Each of the four hybrid orbitals, designated bn, is formed from a linear combinations of the four atomic orbitals ... 

The two atomic orbitals that contribute to the antibonding orbital in Eq. 2 are each proportional to e r a°, where r is the distance of the point from its parent nucleus. Confirm that there is a nodal plane lying halfway between the two nuclei. 

The antibonding molecular orbital is obtained by taking the difference between two atomic orbitals that are proportional to e r,a". Halfway between the two nuclei, the distance from the first nucleus, ru is equal to the distance to the second nucleus, r2, and the antibonding orbital is proportional to vp oc e r a" — e r a" = 0. 

Here r,, and q> are polar coordinates used in the description of the orbitals of an atom, the nucleus being at the origin of the coordinate system. 

It should be noted that the weak energy dependence of the orbitals inside a finite volume around the nucleus has already been noted and used in different contexts the numerical determination of atomic orbitals (4) as well as the scattering of electrons by atoms (5). 

The experimentally observed quadrupole splitting AEq for Fe in inorganic compounds, metals, and solids reaches from 0 to more than 6 mm s [30, 32]. The range of AEq for other Mossbauer isotopes may be completely different because of the different nuclear quadrupole moment Q of the respective Mossbauer nucleus, and also because the EFG values may be intrinsically different due to markedly different radial distributions of the atomic orbitals (vide infra). As Q is constant for a given isotope, variations in the quadrupole coupling constants eQV can only arise from... 

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