Quantum numbers of electrons

For reasons we will discuss later, a fourth quantum number is required to completely describe a specific electron in a multielectron atom. The fourth quantum number is given the symbol ms. Each electron in an atom has a set of four quantum numbers n, l, mi, and ms. We will now discuss the quantum numbers of electrons as they are used in atoms beyond hydrogen. 

Valence Bond Theory theory of bonding based on overlapping valence orbitals Valence Electron Configuration quantum numbers of electrons that reside in the outermost shell of an atom Van Der Waal Force intermolecular force that can include dipole-dipole, ion-dipole, or London force... 

The table below shows the quantum numbers of electrons in the first four shells. 

Another important limitation on the quantum numbers of electrons in atoms, in addition to those listed in Table 4.1, is the Pauli exclusion principle. This principle states that no two electrons in an atom can have the same set of four quantum numbers. This is like the business law that states that no two tickets to a rock concert can have the same set of date and section, row, and seat numbers (Figure 4.7). The row number may depend on the section number, and the... 

In a multi-electronic atom, the following quanmm numbers can also be used to describe the energy levels, and the relationships between the quantum number of electrons are as follows. 

Here, Ms and Mi are the magnetic quantum numbers of electron and nuclear spins, respectively. 

Each electron in an atom is defined by four quantum numbers n, l, m and s. The principal quantum number (n) defines the shell for example, the K shell as 1, L shell as 2 and the M shell as 3. The angular quantum number (/) defines the number of subshells, taking all values from 0 to (n — 1). The magnetic quantum number (m) defines the number of energy states in each subshell, taking values —l, 0 and +1. The spin quantum number fsj defines two spin moments of electrons in the same energy state as + and —f The quantum numbers of electrons in K, L and M shells are listed in Table 6.1. Table 6.1 also gives the total momentum (J), which is the sum of (7 + s). No two electrons in an atom can have same set quantum numbers (n, /, m, s). Selection rules for electron transitions between two shells are as follows ... 

Table 8.1 Summary of Quantum Numbers of Electrons in Atoms...

The index for the orbital ( ). (r) can be taken to include the spin of the electron plus any other relevant quantum numbers. The index runs over the number of electrons, each electron being assigned a unique set of quantum... 

Photoelectron peaks are labelled according to the quantum numbers of the level from which the electron originates. An electron coming from an orbital with main quantum number n, orbital momentum / (0, 1, 2, 3,. .. indicated as s, p, d, f,. ..) and spin momentum s (+1/2 or -1/2) is indicated as For every orbital momentum / > 0 there are two values of the total momentum j = l+Ml and j = l-Ml, each state filled with 2j + 1 electrons. Flence, most XPS peaks come in doublets and the intensity ratio of the components is (/ + 1)//. When the doublet splitting is too small to be observed, tire subscript / + s is omitted. 

We can only determine and up to now. Later, we shall demonstrate that this equation is just the equations of motion of haimonic nucleai vibrations. The set of eigenstates of Eq. (43) can be written as IXBr). symbolizing that they are the vibrational modes of the nth electronic level, where v = (ui, 112,..., v ) if Q is N dimensional, and vi is the vibrational quantum number of the I th mode. 

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

Note. The maximum number of electrons that any quantum level can accommodate is seen to be given by the formula 2n where n is the number of the quantum level, for example = 3 the maximum number of electrons is therefore 18. 

Note. The electronic configuratioa of any element can easily be obtained from the periodic table by adding up the numbers of electrons in the various quantum levels. We can express these in several ways, for example electronic configuration of nickel can be written as ls 2s 2p 3s 3d 4s. or more briefly ( neon core ) 3d 4s, or even more simply as 2. 8. 14. 2... 

Except for the n = 1 quantum level the maximum number of electrons in the outermost quantum level ofany period isalwayseight. At this point the element concerned is one of the noble gases (Chapter 12). 

The table contains vertical groups of elements each member of a group having the same number of electrons in the outermost quantum level. For example, the element immediately before each noble gas, with seven electrons in the outermost quantum level, is always a halogen. The element immediately following a noble gas, with one electron in a new quantum level, is an alkali metal (lithium, sodium, potassium, rubidium, caesium, francium). 

The number of electrons in the outermost quantum level of an atom increases as we cross a period of typical elements. Figure 2.2 shows plots of the first ionisation energy for Periods 2 and 3,... 

Note that we are interested in nj, the atomic quantum number of the level to which the electron jumps in a spectroscopic excitation. Use the results of this data treatment to obtain a value of the Rydberg constant R. Compare the value you obtain with an accepted value. Quote the source of the accepted value you use for comparison in your report. What are the units of R A conversion factor may be necessary to obtain unit consistency. Express your value for the ionization energy of H in units of hartrees (h), electron volts (eV), and kJ mol . We will need it later. 

Organic molecules are the easiest to model and the easiest for which to obtain the most accurate results. This is so for a number of reasons. Since the amount of computational resources necessary to run an orbital-based calculation depends on the number of electrons, quantum mechanical calculations run fastest for compounds with few electrons. Organic molecules are also the most heavily studied and thus have the largest number of computational techniques available. 

In addition to being negatively charged electrons possess the property of spin The spin quantum number of an electron can have a value of either +5 or According to the Pauli exclusion principle, two electrons may occupy the same orbital only when... 

HyperChem quantum mechanics calculations must start with the number of electrons (N) and how many of them have alpha spins (the remaining electrons have beta spins). HyperChem obtains this information from the charge and spin multiplicity that you specify in the Semi-empirical Options dialog box or Ab Initio Options dialog box. N is then computed by counting the electrons (valence electrons in semi-empirical methods and all electrons in fll) mitio method) associated with each (assumed neutral) atom and... 

Previously we have considered the promotion of only one electron, for which Af = 1 applies, but the general mle given here involves the total orbital angular momentum quantum number L and applies to the promotion of any number of electrons. 

The overall form of each of these equations is fairly simple, ie, energy = a constant times a displacement. In most cases the focus is on differences in energy, because these are the quantities which help discriminate reactivity among similar stmctures. The computational requirement for molecular mechanics calculations grows as where n is the number of atoms, not the number of electrons or basis functions. Immediately it can be seen that these calculations will be much faster than an equivalent quantum mechanical study. The size of the systems which can be studied can also substantially ecHpse those studied by quantum mechanics. 

Liquid Helium-4. Quantum mechanics defines two fundamentally different types of particles bosons, which have no unpaired quantum spins, and fermions, which do have unpaired spins. Bosons are governed by Bose-Einstein statistics which, at sufficiently low temperatures, allow the particles to coUect into a low energy quantum level, the so-called Bose-Einstein condensation. Fermions, which include electrons, protons, and neutrons, are governed by Fermi-DHac statistics which forbid any two particles to occupy exactly the same quantum state and thus forbid any analogue of Bose-Einstein condensation. Atoms may be thought of as assembHes of fermions only, but can behave as either fermions or bosons. If the total number of electrons, protons, and neutrons is odd, the atom is a fermion if it is even, the atom is a boson. 

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