Diagonal Properties

Diagonal relationships are commonly observed between elements from the second and third series. This periodic trend is especially true for the following pairs of elements Li/Mg, Be/AI, and B/Si. While vertical periodic trends are still predominant, some properties match better along a diagonal. These diagonal periodic trends are no doubt related to the fact that the radius of an atom increases down and to the left in the periodic table, whereas I.E. and E.A increase up and to the right The diagonal nature of the metal-nonmetal line has already been discussed. 

The diagonal elements boron and silicon also have similar properties. The chlorides of both elements form network covalent solids that act as Lewis bases toward water, reacting according to Equations (5.11) and (5.12)  

The hydrides of B and Si are volatile and flammable compounds. The boranes, as the boron hydrides are termed, form many interesting hydrogen-bridged structures that are seldom observed with other main group elements. Both boron and silicon form numerous and complex oxygen-containing structures knovm as the borates 

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The inverse transformation of operators from Fourier space to Hilbert space can be performed straightforwardly. Using the fact that all operators (except the number operators in the Hamiltonian) satisfy the diagonal property... 

The special diagonal property of the canonical orbitals is exploited in the optimization of the Hartree-Fock state. The canonical orbitals of closed-shell states are obtained in an iterative process, where the Fock matrix in each iteration is rebuilt and diagonalized until self-consistency is achieved as described in Section 10.3.4 and in more detail in Section 10.6. The final electronic state may be obtained as an antisymmetrized product of the canonical spin orbitals... 

The properties of the head element of a main group in the periodic table resemble those of the second element in the next group. Discuss this diagonal relationship with particular reference to (a) lithium and magnesium, (b) beryllium and aluminium. 

The sum over eoulomb and exehange interaetions in the Foek operator runs only over those spin-orbitals that are oeeupied in the trial F. Beeause a unitary transformation among the orbitals that appear in F leaves the determinant unehanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to ehoose sueh a unitary transformation to make the 8i j matrix diagonal. Upon so doing, one is left with the so-ealled canonical Hartree-Fock equations ... 

Most ah initio calculations use symmetry-adapted molecular orbitals. Under this scheme, the Hamiltonian matrix is block diagonal. This means that every molecular orbital will have the symmetry properties of one of the irreducible representations of the point group. No orbitals will be described by mixing dilferent irreducible representations. 

The small size of lithium frequently confers special properties on its compounds and for this reason the element is sometimes termed anomalous . For example, it is miscible with Na only above 380° and is immiscible with molten K, Rb and Cs, whereas all other pairs of alkali metals are miscible with each other in all proportions. (The ternary alloy containing 12% Na, 47% K and 41% Cs has the lowest known mp, —78°C, of any metallic system.) Li shows many similarities to Mg. This so-called diagonal relationship stems from the similarity in ionic size of the two elements / (Li ) 76pm, / (Mg ) 72pm, compared with / (Na ) 102pm. Thus, as first noted by Arfvedson in establishing lithium as a new element, LiOH and LiiCOs are much less soluble than the corresponding... 

For computational purposes it is convenient to work with canonical MOs, i.e. those which make the matrix of Lagrange multipliers diagonal, and which are eigenfunctions of the Fock operator at convergence (eq, (3.41)). This corresponds to a specific choice of a unitary transformation of the occupied MOs. Once the SCF procedure has converged, however, we may chose other sets of orbitals by forming linear combinations of the canonical MOs. The total wave function, and thus all observable properties, are independent of such a rotation of the MOs. 

A further property which is shown on the psychrometric chart is the specific volume of the mixture, measured in cubic metres per kilogram. This appears as a series of diagonal lines, at intervals of 0.01 ml... 

The diagonal line or stairway that starts to the left of boron in the periodic table (Figure 2.7, page 31) separates metals from nonmetals. The more than 80 elements to the left and below that line, shown in blue in the table, have the properties of metals in particular, they have high electrical conductivities. Elements above and to the right of the stairway are nonmetals (yellow) about 18 elements fit in that category. 

Diagonally related pairs of elements often show similar chemical properties. 

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