Antisymmetry

The term antisymmetry has occurred several times above, and it is a whole new idea in our discussion. It is again a point where chemistry and other fields meet in a uniquely important symmetry concept. 

Not only a symmetry plane but also other symmetry elements may serve as antisymmetry elements. We have already seen the contour of the oriental symbol Yin/Yang representing twofold rotational 

Beside color change this symbol represents a whole array of opposites, such as night/day, hot/cold, male/female, young/old, etc. 

Color change is perhaps the simplest version of antisymmetry. The general definition of antisymmetry, at the beginning of this section, however, calls for a much broader interpretation and application. The relationship between matter and antimatter is a conspicuous example of antisymmetry. There is no limit to down-to-earth examples, as well as to abstract ones, especially if, again, symmetry is considered rather loosely. 

The above examples of antisymmetry may have implied at least as much abstraction as any chemical application. The symmetric and 

All the above examples applied to point groups. Such distinctions and further coloring, of course, may be introduced in space-group symmetries as 

The above examples of antisymmetry may have implied at least as much abstraction as any chemical application. The symmetric and antisymmetric behavior of orbitals describing electronic structure and vectors describing molecular vibrations may be perceived with greater ease after the preceding diversion. Before that, however, some more of group theory will be covered. 

Logo of a sporting goods store in Boston, Massachusetts. Photograph by the 

The state vector describing the system of identical particles is  

The many-particle state vector used to be constructed from simpler units— the electronic and nuclear wave functions—and the many-electron wave function consists of the one-electron wave functions, the spinorbitals pt). The general expression for the many-electron wave function in terms of the products of spinorbitals is 

Here the notation Pi(k)) means that the z th spinorbital, which is a function of the spatial coordinates and a spin variable, is occupied by the fcth electron. The above expression, however, should accommodate for  

The spin part, however, can adopt only two forms, denoted as 

for a two-electron system, four different antisymmetric functions are possible 

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The corresponding fiinctions i-, Xj etc. then define what are known as the normal coordinates of vibration, and the Hamiltonian can be written in tenns of these in precisely the fonn given by equation (AT 1.69). witli the caveat that each tenn refers not to the coordinates of a single particle, but rather to independent coordinates that involve the collective motion of many particles. An additional distinction is that treatment of the vibrational problem does not involve the complications of antisymmetry associated with identical fennions and the Pauli exclusion prmciple. Products of the nonnal coordinate fiinctions neveitlieless describe all vibrational states of the molecule (both ground and excited) in very much the same way that the product states of single-electron fiinctions describe the electronic states, although it must be emphasized that one model is based on independent motion and the other on collective motion, which are qualitatively very different. Neither model faithfully represents reality, but each serves as an extremely usefiil conceptual model and a basis for more accurate calculations. 

More general situations have also been considered. For example. Mead [21] considers cases involving degeneracy between two Kramers doublets involving four electronic components a), a ), P), and P ). Equations (4) and (5), coupled with antisymmetry under lead to the following identities between the various matrix elements... 

In these equations, J and M are quantum numbers associated with the angular momentum operators and J, respectively. The number II = 0, 1 is a parity quantum number that specifies the symmetry or antisymmetry of the column vector with respect to the inversion of the nuclei through G. Note that the same parity quantum number II appears for and Also, the... 

X molecular spin orbitals must be different from one another in a way that satisfies the Exclusion Principle. Because the wave function IS written as a determinan t. in torch an gin g two rows of Ihe determinant corresponds to interchanging th e coordin ates of Ihe two electrons. The determinant changes sign according to the antisymmetry requirement. It also changes sign when tw O col-uni n s arc in tcrch an ged th is correspon ds to in Lerch an gin g two spin orbitals. 

A determinant is the most convenient way to write down the permitted functional forms of a polv electronic wavefunction that satisfies the antisymmetry principle. In general, if we have electrons in spin orbitals Xi,X2, , Xn (where each spin orbital is the product of a spatial function and a spin function) then an acceptable form of the wavefunction is ... 

Strangely enough, the universe appears to be eomprised of only two kinds of paitieles, bosons and fermions. Bosons are symmetrical under exehange, and fermions are antisymmetrieal under exehange. This bit of abstiaet physies relates to our quantum moleeular problems beeause eleetions are femiions. 

In short, the Slater determinantal moleculai orbital and only the Slater determinantal moleculai orbital satisfies the two great generalizations of quantum chemistry, uncertainty (indistinguishability) and fermion exchange antisymmetry. 

One spin combination allowable in exeited state helium is ot(l)ot(2), which is symmetric. There are three others. What are they Indieate whieh are sym-metrie (s) and whieh are antisymmetrie (a). 

To satisfy the Pauli exelusion prineiple, the eleetronie wave funetion must be antisymmetrie. This eondition can be met in the exeited state of the helium atom by taking the produet of an antisymmetrie space part sueh as... 

Electronic Wavefuntions Must be Constructed to Have Permutational Antisymmetry Because the N Electrons are Indistinguishable Eermions... 

In particular, within the orbital model of eleetronie strueture (whieh is developed more systematieally in Seetion 6), one ean not eonstruet trial waveflmetions whieh are simple spin-orbital produets (i.e., an orbital multiplied by an a or P spin funetion for eaeh eleetron) sueh as lsalsP2sa2sP2pia2poa. Sueh spin-orbital produet funetions must be made permutationally antisymmetrie if the N-eleetron trial funetion is to be properly antisymmetrie. This ean be aeeomplished for any sueh produet wavefunetion by applying the following antisymmetrizer operator ... 

In summary, proper spin eigenfunetions must be eonstmeted from antisymmetrie (i.e., determinental) wavefunetions as demonstrated above beeause the total and total Sz remain valid symmetry operators for many-eleetron systems. Doing so results in the spin-adapted wavefunetions being expressed as eombinations of determinants with eoeffieients determined via spin angular momentum teehniques as demonstrated above. In... 

For all point, axial rotation, and full rotation group symmetries, this observation holds if the orbitals are equivalent, certain space-spin symmetry combinations will vanish due to antisymmetry if the orbitals are not equivalent, all space-spin symmetry combinations consistent with the content of the direct product analysis are possible. In either case, one must proceed through the construction of determinental wavefunctions as outlined above. 

Z (-1)) CSFs are, by no means, the true eleetronie eigenstates of the system they are simply spin and spatial angular momentum adapted antisymmetrie spin-orbital produets. In prineiple, the set of CSFs i of the same symmetry must be eombined to form the proper eleetronie eigenstates Fk of the system ... 

In similar fashion, the remaining five CSFs may be expressed in terms of atomie-orbital-based Slater determinants. In so doing, use is made of the antisymmetry of the Slater determinants... 

The Exclusion Principle is quantum mechanical in nature, and outside the realm of everyday, classical experience. Think of it as the inherent tendency of electrons to stay away from one another to be mutually excluded. Exclusion is due to the antisymmetry of the wave function and not to electrostatic coulomb repulsion between two electrons. Exclusion exists even in the absence of electrostatic repulsions. 

The symmetry species labels are conventional A and B indicate symmetry or antisymmetry, respectively, to C2, and the subscripts 1 and 2 indicate symmetry or antisymmetry, respectively, to n (xz). 

Fig. 33. Three-dimensional instanton trajeetories of a partiele in a symmetrie double well, interaeting with symmetrieally and antisymmetrieally eoupled vibrations with eoordinates and frequeneies q, to, and ru, respeetively. The curves are 1, ru, ru, P ojq (MEP) 2. to, (u, < (Oq (sudden approximation) 3. ru, < cOq, oj, P ojo 4. to, > (Oq, < (Oq.

Antisymmetry of a laminate requires (1) symmetry about the middle surface of geometry (i.e., consider a pair of equal-thickness laminae, one some distance above the middle surface and the other the same distance below the middle surface), but (2i some kind of a reversal or mirror image of the material properties [Qjjlk- In fact, the orthotropic material properties [Qjj], are symmetric, but the orientations of the laminae principal material directions are not symmetric about the middle surface. Those orientations are reversed from 0° to 90° (or vice versa) or from + a to - a (a mirror image about the laminate x-axis). Because the [Qjj]k are not symmetric, bending-extension coupling exists. 

The stiffnesses of an antisymmetric laminate of anisotropic laminae do not simplify from those presented in Equations (4.22) and (4.23). However, as a consequence of antisymmetry of material properties of generally orthotropic laminae, but symmetry of their thicknesses, the shear-extension coupling stiffness A.,6,... 

If the laminate is subjected to uniform axial extension on the ends X = constant, then all stresses are independent of x. The stress-displacement relations are obtained by substituting the strain-displacement relations, Equation (4.162), in the stress-strain relations. Equation (4.161). Next, the stress-displacement relations can be integrated under the condition that all stresses are functions of y and z only to obtain, after imposing symmetry and antisymmetry conditions, the form of the displacement field for the present problem ... 

Secondly, y must also be antisymmetric, meaning that it must change sign when two identical particles are interchanged. For a simple function, antisymmetry means that the following relation holds ... 

For an electronic wavefunction, antisymmetry is a physical requirement following from the fact that electrons are fermions. It is essentially a requirement that y agree with the results of experimental physics. More specifically, this requirement means that any valid wavefunction must satisfy the following condition ... 

Fermions are particles that have the properties of antisymmetry and a half-integral spin quantum number, among others. 

The exchange energy arising from the antisymmetry of the quantum mechanical wavefunction. 

In the Hartree-Fock model, where we take account of antisymmetry, it turns out that there is no correlation between the positions of electrons of opposite spin, yet,... 

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