Electrons atomic orbitals

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... 

In the lithium atom, and for all other multi-electron atoms, orbitals in different energy sublevels differ in energy. 

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... 

The principal quantum number n is the most important determinant of the radius and energy of the electron atomic orbital. The orbital shape quantum number I determines the shape of the atomic orbital. When / = 1, the atomic orbital is called an s orbital there are two s orbitals for each value of n, and they are spherically symmetric in space around the nucleus. When I = 2, the orbitals are called the p orbitals there are six p orbitals, and they have a dumbbell shape of two lobes that are diametrically opposed. When I = 3 and 4, we have 10 d orbitals and 14 f orbitals. The orbital orientation quantum number m controls the orientation of the orbitals. For the simplest system of a single electron in a hydrogen atom, the most stable wave function Is has the following form ... 

Our approximations so far (the orbital approximation, LCAO MO approximation, 77-electron approximation) have led us to a tt-electronic wavefunction composed of LCAO MOs which, in turn, are composed of 77-electron atomic orbitals. We still, however, have to solve the Hartree-Fock-Roothaan equations in order to find the orbital energies and coefficients in the MOs and this requires the calculation of integrals like (cf. eqns (10-3.3)) ... 

To make the picture of 7T-electrons more intelligible the model of linear combinations of single electron atomic orbitals to molecular orbitals is helpful (Fig. 14). In this model one concentrates only on the outermost electrons or valence orbitals. Starting from the atomic wavefunctions the s, px and py atomic orbitals are combined in the (x,y)-plane to sp and sp2 orbitals. These sp and sp2 orbitals of the different atoms combine to molecular orbitals, building the molecular structure framework in the (x,y)-plane. The electrons in these molecular orbitals are called a-electrons and their wavefunctions are symmetric perpendicular to the (x,y)-plane extending only over two neighboring atoms. 

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). 

The total number of valence electron atomic orbitals equals the number of molecular orbitals. 

Hydrogen pursues to be surrounded by 2 electrons so that the valence electron atomic orbital Is will be filled up with electrons. 

Elements in the 2 period pursue to be surrounded by 8 electrons so that the valence electron atomic orbitals 2s and 3x2/)) will be filled up with electrons. This is referred to as the octet rule since octa means eight . 

In a multi-electron atom, orbitals with the same value of n but... 

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. 

Modern bonding theory treats the electron pairing in terms of the interaction of electron (atomic) orbitals and describes the covalent bond in terms of both bonding and anti-bonding molecular orbitals. 

Accurate representation of a many-electron atomic orbital (AO) requires a linear combination of several Slater-type orbitals. For rough calculations, it is convenient to have simple approximations for AO s- We might use hydrogenlike orbitals with effective nuclear charges, but Slater suggested an even simpler method to approximate an AO by a single function of the form (11-14) with the orbital exponent C taken as... 

From right to left in a row of the periodic systems, the ionization potential of the metal atom or workfunction of the corresponding metal decreases and the spatial extent of the d-valence electrons also increases. Moving down in a column of the periodic system, the metal workfunction tends to increase as well as the spatial extent of the d-valence electrons. The position of the d-valence band is also affected by the altered screening when f-electron atomic orbitals become occupied [9]. 

MOLECULAR ORBITAL and valence bond calculations of the IT—electron energies of unsaturated molecules customarily start with models in which appropriate atomic orbitals are assigned to each nucleus to provide a framework for -notions of the binding electrons. Atomic orbital representations of organic molecules are now very commonly used in the teaching of elementary organic chemistry, although there often seems to be confusion between atomic orbital and molecular orbital representations. 

An atomic orbital (AO) corresponds to a region of space about the nucleus of a single atom where there is a high probability of finding an electron. Atomic orbitals called s orbitals are spherical those called p orbitals are like two almost-tangent sph eres. Orbitals can hold a maximum of two electrons when their spins are paired. 

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. 

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