Atomic orbitals characterization

The Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers (n, , m, wrj. Another way of stating this principle is that each Hartree atomic orbital (characterized by a set of three quantum numbers, n, , and m) holds at most two electrons, one with spin up and the other with spin down. 

What has been learned thus far We have noted that in the context of our truncated basis, built from the atomic orbitals characterizing the free atoms, the... 

The next step towards increasing the accuracy in estimating molecular properties is to use different contributions for atoms in different hybridi2ation states. This simple extension is sufficient to reproduce mean molecular polarizabilities to within 1-3 % of the experimental value. The estimation of mean molecular polarizabilities from atomic refractions has a long history, dating back to around 1911 [7], Miller and Sav-chik were the first to propose a method that considered atom hybridization in which each atom is characterized by its state of atomic hybridization [8]. They derived a formula for calculating these contributions on the basis of a theoretical interpretation of variational perturbation results and on the basis of molecular orbital theory. 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

The energy, the phase, and the amphtude characterize sahent features of orbitals. This can be seen in atomic orbitals and bond orbitals (Sect. 1). 

An electron orbiting in an atom is a circular electric current that is surrounded by a magnetic field. This also can adopt only certain orientations in an external magnetic field according to quantum mechanics. The state of an electron in an atom is characterized by four quantum numbers ... 

Appendix 1 also shows how the periodic table of the elements (Appendix 5) can be built up from the known rules for filling up the various electron energy levels. The Bohr model shows that electrons can only occupy orbitals whose energy is fixed (quantized), and that each atom is characterized by a particular set of energy levels. These energy levels differ in detail between atoms of... 

The first kind of simplification exclusively concerns the size of the basis set used in the linear combination of one center orbitals. Variational principle is still fulfilled by this type of "ab initio SCF calculation, but the number of functions applied is not as large as necessary to come close to the H. F. limit of the total energy. Most calculations of medium-sized structures consisting for example of some hydrogens and a few second row atoms, are characterized by this deficiency. Although these calculations belong to the class of "ab initio" investigations of molecular structure, basis set effects were shown to be important 54> and unfortunately the number of artificial results due to a limited basis is not too small. 

There are many solutions to the Schrodinger equation, and each solution is called an atomic orbital with an energy E, and has a spatial distribution characterized by four quantum numbers ... 

According to quantum mechanics laws, electrons in free atoms occupy so-called atomic orbitals. Each orbital is characterized by its energy and is determined by quantum numbers n, I, and mg where n is the main quantum number, designated by numbers 1,2,3..., 1 is the orbital quantum number with 0,1,2,... (n - 1) values and m is the magnetic quantum number with -1,-1+ I,...0,...I- I,I values. 

The solutions of the Schrodinger equation show how j/ is distributed in the space around the nucleus of the hydrogen atom. The solutions for v / are characterized by the values of three quantum numbers and every allowed set of values for the quantum numbers, together with the associated wave function, strictly defines that space which is termed an atomic orbital. Other representations are used for atomic orbitals, such as the boundary surface and orbital envelopes described later in the chapter. 

Figure 4.2 is a diagram of the relative energies of the molecular orbitals of the E2 molecules, together with those of the atomic orbitals from which they were constructed. The sideways overlap involved in the production of k orbitals is not as effective as the end-on overlap which characterizes the production of o orbitals. For a given interatomic dis-... 

The quantum mechanical model proposed in 1926 by Erwin Schrodinger describes an atom by a mathematical equation similar to that used to describe wave motion. The behavior of each electron in an atom is characterized by a wave function, or orbital, the square of which defines the probability of finding the electron in a given volume of space. Each wave function has a set of three variables, called quantum numbers. The principal quantum number n defines the size of the orbital the angular-momentum quantum number l defines the shape of the orbital and the magnetic quantum number mj defines the spatial orientation of the orbital. In a hydrogen atom, which contains only one electron, the... 

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