Quantum number, orbital

Absorption of a photon is accompanied by the excitation of an electron from a lower-energy atomic orbital to an orbital of higher energy. Not all possible transitions between atomic orbitals are allowed. For sodium the only allowed transitions are those in which there is a change of +1 in the orbital quantum number ) thus transitions from s—orbitals are allowed, but transitions from s d orbitals are forbidden. The wavelengths of electromagnetic radiation that must be absorbed to cause several allowed transitions are shown in Figure 10.18. 

The arrangement of electrons in an atom is described by means of four quantum numbers which determine the spatial distribution, energy, and other properties, see Appendix 1 (p. 1285). The principal quantum number n defines the general energy level or shell to which the electron belongs. Electrons with n = 1.2, 3, 4., are sometimes referred to as K, L, M, N,. .., electrons. The orbital quantum number / defines both the shape of the electron charge distribution and its orbital angular... 

H = di(Z—iy di are the potential parameters I is the orbital quantum number 3 characterizes the spin direction Z is the nuclear charge). Our experience has show / that such a model potential is convenient to use for calculating physical characteristics of metals with a well know electronic structure. In this case, by fitting the parameters di, one reconstructs the electron spectrum estimated ab initio with is used for further calculations. 

The modern theory of the behavior Of matter, called quantum mechanics, was developed by several workers in the years 1925-1927. For our purposes the most important result of the quantum mechanical theory is that the motion of an electron is described by the quantum numbers and orbitals. Quantum numbers are integers that identify the stationary states of an atom the word orbital means a spatial description of the motion of an electron corresponding to a particular stationary state. 

In addition to the magnetism due to the electron spin, the magnetism of the orbital motion has to be considered. For this purpose the magnetic quantum numbers of the electrons are added to a resultant orbital quantum number L = beginning with the highest magnetic quantum number. For example ... 

It can be shown, from wave-mechanical calculations, that the Is orbital (quantum numbers n = 1, Z = 0, m = 0, corresponding to the classical K shell) is spherically symmetrical about the nucleus of the atom, and that the 2s orbital (quantum numbers n = 2, Z = 0, m = 0) is similarly spherically symmetrical, but at a greater distance from the nucleus there is a region between the two latter orbitals where the probability of finding an electron approaches zero (a spherical nodal surface). 

As yet, this marks no radical departure from the classical picture of orbits, but with the 2p level (the continuation of the L shell) a difference becomes apparent. Theory now requires the existence of three 2p orbitals (quantum numbers n = 2, Z = 1, with m = +1,0, and... 

Atomic Size The associated Laguerre polynomial (x) is a polynomial of degree nr = n — l — 1, which has nr radial nodes (zeros). The radial distribution function therefore exhibits n — l maxima. Whenever n = l + 1 and the orbital quantum number, l has its largest value, there is only one maximum. In this case nT = 0 and from (14) follows... 

Any determinant changes sign when any two columns are interchanged. Moreover, no two of the product functions (columns) can be the same since that would cause the determinant to vanish. Thus, in all nonvanishing completely anti-symmetric wave functions, each electron must be in a different quantum state. This result is known as Pauli s exclusion principle, which states that no two electrons in a many-electron system can have all quantum numbers the same. In the case of atoms it is noted that since there are only two quantum states of the spin, no more than two electrons can have the same set of orbital quantum numbers. 

Energy level diagram of the sodium atom. The energy levels are denoted by the values for the principal quantum number , the orbital quantum number/, and the spin quantum number s. Levels with 1 = 0 are not split for / = 1 two separate levels are drawn (s = 1/2) for/> 1 the splitting is too small to be shown in the figure. Wavelengths of a few special transitions are given in nanometers. 

Table 10.1. Definition of electron orbitals in terms of the four orbital quantum numbers (n, l, mi, s).

The quantum numbers n and i. Multi-electron atoms can be characterized by a set of principal and orbital quantum numbers n, t which labels one-electron wave functions (orbitals). 

The magnetic quantum number m is related to the fact that only in an applied magnetic field it is possible to define a direction within the atom with respect to which the orbital can be directed. In general for a value of the orbital quantum number we have 21 + 1 possible values of the magnetic quantum number (which are 0, 1, 2,... up to ). To an s orbital, for instance, for which = 0 and is spherically symmetrical, only one value corresponds for the magnetic quantum number (m = 0). For p orbitals = 1) we have three possibilities (m = —1,0,+1) corresponding to three orientations (generally assumed as the x, y, z directions in Cartesian coordinates). Similarly we have five possibilities for d orbitals ( = 2) (that is m = —2, — 1,0, +1, +2), seven for/orbitals ( = 3), etc. 

Fig. 13.4 Logarithm of error(Eh) in the configuration interaction energy for the ground state of the helium atom as a function of maximum orbital quantum number, L, of the one-electron basis functions. The data were obtained in an...

Fig. 6a-c. Different ways of showing the nephelauxetic effect using the lanthanides for the formation of the chelate MA3 (HA=TTA) and the citric complex MHCit Cit2-. q = number of f electrons, L = orbital quantum number, a) double-double effect (23) b) tetrad effect (24) c) W inclined plot (25). 

Electronic parameters include the number of electrons, the number of valence electrons in the outer orbit, the orbital quantum numbers n, the azimuthal quantum number or spdf status, electron radius and energy, polarizability, dipole moment, quadrupole moment, and first ionization energy. 

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 third quantum number, mb describes the orientation of the electron orbital relative to an arbitrary direction. Because an external magnetic field (such as might be induced by a neighboring atom) provides a convenient reference direction, mt is usually called the magnetic orbital quantum number. It can take an integral value from —l to /. 

The wavefunction which fits the equation and leads to discrete values of V is the eigenfunction. The search for such eigenfunctions and eigenvalues can be a most demanding mathematical excercise, and need not be considered here. Let us note however that the solutions of the Schrodinger equation lead to the definitions of the orbital quantum numbers n, l and m. The quantum numbers of rotational and vibrational levels are also derived from the Schrodinger equation. 

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