Atomic orbitals signs

The diagoital matrix elemeitts for an atomic orbital can be shown to bear a very close assoeialion wiLli the energy needed to ion i/ e (remove) an electron from th e orbital. Valen ce state ion-i /ation eti ergies (VSlKs) can be determ in ed from experiment and extended HUekel theory equates to the eon espon din g VSIH, with opposite sign ... 

The spins of two electrons are said to be paired if one is T and the other 1 (Fig. 1.43). Paired spins are denoted Tl, and electrons with paired spins have spin magnetic quantum numbers of opposite sign. Because an atomic orbital is designated by three quantum numbers (n, /, and mt) and the two spin states are specified by a fourth quantum number, ms, another way of expressing the Pauli exclusion principle for atoms is... 

To improve our model we note that s- and /7-orbitals are waves of electron density centered on the nucleus of an atom. We imagine that the four orbitals interfere with one another and produce new patterns where they intersect, like waves in water. Where the wavefunctions are all positive or all negative, the amplitudes are increased by this interference where the wavefunctions have opposite signs, the overall amplitude is reduced and might even be canceled completely. As a result, the interference between the atomic orbitals results in new patterns. These new patterns are called hybrid orbitals. Each of the four hybrid orbitals, designated bn, is formed from a linear combinations of the four atomic orbitals ... 

The raw output of a molecular structure calculation is a list of the coefficients of the atomic orbitals in each LCAO (linear combination of atomic orbitals) molecular orbital and the energies of the orbitals. The software commonly calculates dipole moments too. Various graphical representations are used to simplify the interpretation of the coefficients. Thus, a typical graphical representation of a molecular orbital uses stylized shapes (spheres for s-orbitals, for instance) to represent the basis set and then scales their size to indicate the value of the coefficient in the LCAO. Different signs of the wavefunctions are typically represented by different colors. The total electron density at any point (the sum of the squares of the occupied wavefunctions evaluated at that point) is commonly represented by an isodensity surface, a surface of constant total electron density. 

The in-phase and out-of-phase relations mean the same and opposite signs of the atomic orbitals in (j) and nearest to the charge. 

Figure 6.2 Polar graphs of the hydrogen 2p atomic orbitals. Regions of positive and negative values of the orbitals are indicated by + and — signs, respectively. The distance of the curve from the origin is proportional to the square of the angular part of the atomic orbital.

Figure 1.8 The overlapping of two hydrogen Is atomic orbitals with the same phase sign (indicated by their identical color) to form a bonding molecular orbital.

It is to be noted that o orbitals are formed not only by the overlap of two 5 orbitals but by the overlap of any of the atomic orbitals (s, p. d or f) whether same or different, but the two lobes must have the same sign. A + s orbital can form a bond only by overlapping with another positive 5 orbital or with a + lobe of a p, d or / orbital. 

Figure 3. MO diagram for the ai and a2u orbitals of porphine. The atomic orbital coefficients are proportional to the size of the circles solid or dashed circles indicate sign.

Although the formal method of describing orbitals is to use mathematical expressions, much understanding of orbital properties may be gained by the use of pictorial representations. The most useful pictorial representations of atomic orbitals are similar to boundary surfaces (which are based on V /2), but are based upon the distribution of jf values, with the sign of / being indicated in the various parts of the diagram. The shapes of these distributions are based upon the contours of jf within... 

The 2py orbital of the oxygen atom changes sign when the C2 operation is applied to it, and when it is reflected through the xz plane, but is symmetric with respect to the molecular plane, yz. The representation expressing the character of the 2py orbital is another member of the group ... 

The minus signs in the C2 and oh columns signify that, although the orbitals have not been moved by the symmetry operations, they have been inverted. This representation of the group orbitals may be converted to its constituent irreducible representations by considering that it is likely that they are identical with the representation to which the 2p, atomic orbital of the boron atom belongs, Le. a2". The subtraction of the characters of the a2" representation gives the result ... 

Problem 8.55 (a) Apply the MO theory to the C==C—C=C—C carbocation, considering the signs of the upper lobes of adjacent p atomic orbitals (f>) Indicate the relative energies of the molecular orbitals and state if they are bonding, nonbonding, or antibonding, (c) Show the distribution of the n electrons. ... 

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