Phase sign

Conservation of orbital symmetry is a general principle that requires orbitals of the same phase (sign) to match up in a chemical reaction. For example, if terminal orbitals are to combine with one another in a cyclixation reaction as in pattern. A, they must rotate in the same dii ection (conrotatory ovei lap). but if they combine according to pattern H. they must rotate in opposite directions (disrotatory). In each case, rotation takes place so that overlap is between lobes of the it orbitals that are of the same sign. 

Reinforce a crest meets a crest (waves of the same phase sign meet each other) => add together => resulting wave is larger than either individual wave. 

AOs of the same phase sign overlap => leads to reinforcement of the wave function => the value of is larger between the two nuclei => contains both electrons in the lowest energy state, ground state... 

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.

The regioselectivity in radical addition reactions to alkenes in general has successfully been interpreted by a combination of steric and electronic effects1815,47. In the absence of steric effects, regiochemical preferences can readily be explained with FMO theory. The most relevant polyene orbital for the addition of nucleophilic radicals to polyenes will be the LUMO for the addition of electrophilic orbitals it will be the HOMO. Table 10 lists the HOMO and LUMO coefficients (without the phase sign) for the first three members of the polyene family together with those for ethylene as calculated from Hiickel theory and with the AMI semiempirical method48. 

The different signs for si2 and Si3 show that the path from well 1 to well 2 has a Berry phase sign change and the path from well 1 to well 3 is Berry phase free. 

The easiest way to rationalize the stereospecificity in electrocyclic reactions is by examining the symmetry of the HOMO of the open (non-cyclic) molecule, regardless of whether it is the reactant or the product. For example, the HOMO of hexatriene is 3, in which orbital lobes (terminal) that overlap to make the new a-bond have the same phase (sign of the wave function). Thus, in this case, the new cr-bond between these two terminal orbital lobes can be formed only by the disrotation suprafacial overlap) (Fig. 8.45). If the terminal orbital lobes of HOMO of hexatriene were to close in a conrotatory antarafacial overlap) fashion, an antibonding interaction would result. 

Representation of the 2p orbitals, (a) The electron probability distribution for a 2p orbital. Generated from a program by Robert Allendoerfer on Project SERAPHIM disk PC 2402 reprinted with permission, (b) The boundary surface representations of all three 2p orbitals. Note that the signs inside the surface indicate the phases (signs) of the orbital in that region of space. 

The MOs of benzene are shown pictorially in Figure 1.5. The stability of a MO is related to the number of nodes it possesses that is to say, the number of times the wave function changes phase (sign) around the ring system. The most stable form has no nodes, when there is a bonding interaction between all six adjacent carbon atoms. 

FIGURE 8.23. Calculation of phase angles for a centrosymmetric crystal by the method of isomorphous replacement. Two isomorphous crystals have structure amplitudes I El I and T2. The replaceable atom M hcis calculated structure factors M = Ml -M2. From these it is possible to deduce relative phases (signs) for Fi and F2- In each ca.se the vector from Fi to F2 must be the same as the vector Fmi—Fm2-... 

Figure 9-1 An illustration of constructive and destructive interference of waves, (a) If the two identical waves shown at the left are added, they interfere constructively to produce the more intense wave at the right, (b) Conversely, if they are subtracted, it is as if the phases (signs) of one wave were reversed and added to the first wave. This causes destructive interference, resulting in the wave at the right with zero ampUtude that is, a straight hne.

Much of the attractivity of CIDNP for chemical investigations stems from the fact that simple qualitative rules can be formulated that relate the phases (signs) of the observed polarizations to the reaction mechanism and the magnetic parameters of the radical pairs. 

All these regularities are summed up by Kaptein s rule for the phase (sign) T/ of a CIDNP net effect (T/ = +1, enhanced absorption of nucleus i T/ = -1, emission),... 

The quantum numbers, 1 and m, define the angular properties of the electron distribution as the square of Tim(0, 0). The familiar icons for s, p and d atomic orbitals identify the phases (signs of the values of the functions) of the Y (9,4>) by including the signs or using shading to identify the contours of positive phase. Table 1.1 presents the detailed forms of the first few of these functions defined in Equation 1.5, while standard chemist s pictures of these orbitals are shown in Figure 1.2. 

In two-space dimensions, it is possible that there are particles (or quasiparticles) that have statistics intermediate between bosons and fermions. These particles are known as anyons for identical anyons the pgK wave function is not symmetric (a phase sign of-fl) or antisymmetric (a phase signof-1), but interpolates continuously between +1 and-l.y myons maybe involved in the fractional quantum Hall effect. 

The phase sign of a wave equation indicates whether the solution is positive or negative when calculated for a given point in space relative to the nucleus. 

Constructive interference occurs when wave functions with the same phase sign interact. There is a reinforcing effect and the amplitude of the wave function increases. 

All s orbitals are spheres. A Is orbital is a simple sphere. A 2s orbital is a sphere with an inner nodal surface tp = 0). The inner portion of the 2s orbital, ip2s, has a negative phase sign. 

The shape of orbital is like that of almost-touching spheres or lobes. The phase sign of a 2p wave function, >pip> positive in one lobe and negative in the other. A nodal plane separates the two lobes of a orbital, and the three p orbitals of a given energy level are arranged in space along the x, y, and z axes in a Cartesian coordinate system. 

FIGURE 1.25 (a) A wedge-dashed wedge formula for the sigma bonds in ethene and a schematic depiction of the overlapping of adjacent p orbitals that form the tt bond, (b) A calculated structure for ethene. The blue and red colors indicate opposite phase signs in each lobe of the tt molecular orbital. A ball-and-stick model for the o- bonds in ethene can be seen through the mesh that indicates the ir bond. 

Orbitals are described by the square of a wave function, and each orbital has a characteristic energy. The phase signs associated with an orbital may be + or —. 

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