Quantum Number Principle

To appreciate this point somewhat better it is useful to compare three types of Gaussian basis sets, (a) a set of Gaussians with common orbital exponents (for one 1) but a sequence of principle quantum-numbers... 

A contribution of 0.85 is added for each electron in an s or p orbital for which the principle quantum number is one less than that for the electron being described. For electrons in s or p orbitals which have an n value of two or more lower than that of the orbital for the electron being considered, a contribution of 1.00 is added for each electron. 

The angular momentum quantum number, , for a particular energy level as defined by the principle quantum number, n, depends on the value of n. I can take integral values from 0 up to and including (n - 1). 

The first quantum number is the principle quantum number (n) that describes the size of the orbital and relative distance from the nucleus. The possible values of n are positive integers (1,2,3,4, and so on). The smaller the value of n, the lower the energy, and the closer the orbital is to the nucleus. We sometimes refer to the principle quantum number as designating the shell the electron is occupying. 

Each shell contains one or more subshells, each with one or more orbitals. The second quantum number is the angular momentum quantum number (/) that describes the shape of the orbitals. Its value is related to the principle quantum number and has allowed values of 0 to (n-1). For example, if n = 4, then the possible values of / would be 0,1, 2, and 3 (= 4-1). 

The exceptions begin with the fourth energy level. The fourth energy level begins to fill before all the sublevels in the third shell are complete. More complications in the sequence appear as the value of the principle quantum number increases. The sequence of orbital filling, with complications, is Is, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, and so on. 

There are several ways of indicating the arrangement of the electrons in an atom. The most common way is the electron configuration. The electron configuration requires the use of the n and / quantum numbers along with the number of electrons. The principle quantum number, n, is represented by an integer (1,2,3. ..), and a letter represents the l quantum number (0 = s, 1 = p, 2 = d, and 3 = f). Any s-subshell can hold a maximum of two electrons, any p-subshell can hold up to six electrons, any d-subshell can hold a maximum of 10 electrons, and any f-subshell can hold up to 14 electrons. 

After the separation of the kinetic energy operator due to the center-of-mass motion from the Hamiltonian, the Hamiltonian describes the internal motions of electrons and nuclei in the system. These in the BO approximation can be separated into the vibrational and rotational motions of the nuclear frame of the molecule and the electronic motion that only parametrically depends on the instantenous positions of the nuclei. When the BO approximation is removed, the electronic and nuclear motions become coupled and the only good quantum numbers, which can be used to quantize the stationary states of the system, are the principle quantum number, the quantum number quantizing the square of the total (nuclear and electronic) squared angular momentum, and the quantum number quantizing the projection of the total angular momentum vector on a selected direction (usually the z axis). The separation of different rotational states is an important feamre that can considerably simplify the calculations. 

The first shell or energy level out from the nucleus is called the K shell or energy level and contains a maximum of two electrons in the s orbital— that is, K = s2, where the K represents the shell number (or principle quantum number), the s describes the orbital shape of the angular momentum quantum number, and the 2 is the maximum number of electrons that the s orbital can contain. This particular sequence is K = s2, which means K shell contains 2 electrons in the s orbital. This is the sequence for the element helium. Look up helium in the text for more information. 

Principle quantum number n Orbital angular momentum quantum number / Magnetic quantum number nil Spin quantum number s Atomic orbital designation... 

For a given angular momentum (d or f) the bands broaden as the principle quantum number increases. Thus 5d bands are much broader than the 3d bands. The reason for this is that states with principle quantum number n +1 must have an additional node (the orthogonality node) to the states with principle quantum number n. This pushes the wave function, relatively, further from the nucleus. Hence the 5 f wave functions are more extended than 4f wave functions which leads to their tendency to form bands. 

The principle quantum number, n, which can take integral values from 1 to infinity, describes the effective volume of an orbital. Chemists commonly use the word shell to refer to all orbitals with the same value of n, because each increasing value of n defines a layer of electron density that is farther from the nucleus. 

Fig. 22.5 The g factor of Sr 5snd states as a function of vp3/2, the effective principle quantum number measured relative to the 4d 2D3/2 ionization threshold at 60488.09 cm-1. The solid lines are the theoretical predictions, and the points correspond to the experimental measurements for the bound states designated by 5snd (from ref. 15).

The subshells and the types of orbitals for the first four energy levels (principle quantum numbers) are shown below. 

The azimuthal quantum number / may have integer values from 0 to n-1. / describes the angular momentum of an orbital. This determines the orbital s shape. Orbitals with the same value of n and / are in the same subshell, and each subshell may contain up to (41 + 2 electrons). Subshells are usually referred to by the principle quantum number followed by a letter corresponding to / as shown in the following table ... 

Configurations are also written with their principle quantum numbers together ... 

B. The energy of an electron in a hydrogen atom depends only on the principle quantum number. 

Fortunately, some of these troublesome uncertainties scale in a known way with the principle quantum number n, and they can be eliminated by combining measurements of different transitions. 

The hybrid orbital type d2sp3 refers to a case in which the d orbitals have a smaller principal quantum number than that of the s and p orbitals (e.g., 3d combined with 4s and 4p orbitals). The sp3d2 hybrid orbital type indicates a case where the s, p, and d orbitals all have the same principal quantum number (e.g., 4s, 4p, and 4d orbitals) in accord with the natural order of filling atomic orbitals having a given principle quantum number. Some of the possible hybrid orbital combinations will now be illustrated for complexes of first-row transition metals. 

Principal quantum number (n) The principle quantum number is the energy level of the electron given the designation n. The value of n can be any integer (1, 2, 3, 4. ..) and determines the energy of the orbitals. 

The valence electrons in transition metals are distributed over d(n -1) and s(n) and p n) electrons, n labeling the principle quantum-numbers. 

B is correct. The principle quantum number (n) represents the energy level of the electron. The lowest energy shell is n = 1. As n increases, the shells move farther from the nucleus and energy increases. 

The energy of an electron (E ) is inversely proportional to its radius from the nucleus. For a hydrogen atom, the principle quantum number determines the energy of an electron using the Rydberg constant Rh = 2.18x10 J) ... 

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