Subshells defined

By the rules of Section 5-15, each shell has an s subshell (defined by = 0) consisting of one s atomic orbital (defined >y m = 0). We distinguish among orbitals in different... 

Beginning with the second shell, each shell also contains a p subshell, defined by f = 1. Each of these subshells consists of a set of three p atomic orbitals, corresponding to the three allowed values of m (-1, 0, and +1) when f = 1. The sets are referred to as 2p, 3p, 4p, 5p,... orbitals to indicate the main shells in which they are found. Each set of atomic orbitals resembles three mutually perpendicular equal-arm dumbbells (see Eigure 5-22). The nucleus defines the origin of a set of Cartesian coordinates with the usual x, y, and z axes (see Eigure 5-2 3 a). The subscript x, y, or z indicates the axis along which each of the three two-lobed orbitals is directed. A set of three p atomic orbitals may be represented as in Eigure 5-2 3b. 

By the rules given in Section 4-16, each shell has an s subshell (defined by f = 0) consisting of one s atomic orbital (defined by m( = 0). We distinguish among orbitals in different principal shells (main energy levels) by using the principal quantum number as a coefficient Ir indicates the s orbital in the first shell, 2s is the s orbital in the second shell, 2p is a p orbital in the second shell, and so on (see Table 4-8). 

The numerical value of the quantity 25 + 1 (total multiplicity) is used as a left superscript with the corresponding term code to define the total atomic term symbol, e.g. for L = 2 and 5 = 1, the term symbol is ZD. Closed subshells make no contribution to L and 5 and may be ignored. 

Let us find expressions for the matrix elements of these operators for a subshell of equivalent electrons with respect to the relativistic wave functions, in a one-electron case defined by (2.15), and for a subshell, by (9.8). 

It is worth emphasizing that coefficients (20.29) and (20.30) do not depend on orbital quantum numbers. These numbers define only the parity of summation index k, following from the conditions of the nonvanishing of radial integrals in (20.27) and (20.28). For the direct term, parameter k acquires even values, whereas for the exchange part the parity of k equals the parity of sum h + h-If one or two subshells are almost filled, then the following equalities are valid ... 

Let us consider in a similar manner the matrix elements of the operators of magnetic Hm and retarding Hr interactions. For them also, a formula of the sort (20.6), where the necessary matrix elements in the case of one subshell of equivalent electrons are defined by equations (19.78) and (19.84), is valid. Retarding interactions exist only between subshells, while inside them they, according to (19.84), vanish. 

Here submatrix elements of the operator Tk are defined in accordance with (7.5). As in the non-relativistic case (25.23), only transitions described by one term in (26.10) take place, depending on which subshell an electron is jumping in. In this case the other summand must be considered as being equal to zero. 

Coefficients fk and gk are defined by (19.73) and (20.30), respectively, whereas Nxt denotes the number of electrons in subshell Quantities e, proportional to Lagrange multipliers, are in charge of orthonormality (2.17) of the radial functions. 

A shell is defined as a group of electrons in an atom all having the same principal quantum number. A subshell is defined as a group of electrons in an atom all having the same principal quantum number and also the same angular momentum quantum number. If two electrons in an atom have the same principal quantum number, the same angular momentum quantum number, and the same magnetic quantum number, the electrons are said to be in the same orbital. 

As discussed in section 2.8, relativistic effects on the valence electronic structure of atoms are dominated by spin-orbit splitting of (nl) states into (nlj) subshells, and stabilization of s- and p-states relative to d- and f-states. Here we examine consequences of relativistic interactions for cubo-octahedral metal-cluster complexes of the type [M6X8Xfi] where M=Mo, Nb, W and X=halogen, which have a well defined solution chemistry, and are building blocks for many interesting crystal structures. 

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 ... 

Which quantum number defines a shell Which quantum numbers define a subshell ... 

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