Operations, symmetry

The arrow at the top in Fig. 5.7 has a twofold axis of symmetry. It is indistinguishable after rotation about this axis by 180°. The operation 2 reproduces the same arrow after half a turn. Naturally this is only true if the surface which is not seen in the 

A final, sixth, symmetry-element of the point groups is the center of inversion, i, marked by an open circle (o). Its operation involves a projection of each point of the object through its center to a point equally distant on the other side. The projection is illustrated by the letter A in Fig. 5.8. 

Note that the letter is different in front and back. The other symmetry elements of the letter A are also indicated. The six basic point-symmetry elements (1, 2, 3,4, 6, and i) can describe the crystal symmetry as it is macroscopically recognizable by inspection, if needed, helped by optical microscopy. 

The basic symmetry elements can also be carried out in sequence. This combination is called the product of the elements (see Appendix 14). Two fourfold rotations about the axis of symmetry are thus equal to one twofold rotation, two sixfold rotations, a threefold one, and three sixfold rotations, a twofold one. l en the element i is included in the products, one can generate the inversion axes 1, 2, 3, 

The invariance of an object or a structure with respect to some operation is called symmetry. A geometrical symmetry operation is the mapping of space onto itself. It transforms an object into itself without distortion. It is also called an isometry. 

In a wider sense, the terms symmetry and order are synonymous. Everything which is invariant or structured conveys the presence of symmetry, as, for example, the laws of conservation in physics. Symmetry and the lack of symmetry (asymmetry) play a major role in all artistic expression such as architecture, painting and music. 

A geometrical symmetry operation can be represented by an affine transformation of the type  

The matrix R transforming x into x as well as r into r is independent of the choice of origin for the coordinate system, however, it clearly depends on the 

If a structure is invariant with respect to two symmetry operations (P, tp) and (Q, (q), then it is clearly invariant with respect to the successive application of the two operations. We call this successive application the product. If we first apply (P, tp) and then (Q, Iq), the vector x is transformed into x = Qx + Iq = QPx + Qtp + tq. The multiplication of the matrices Q and P is thus carried out from right to left. We first apply P and then Q  

The emphasis here will be on practical computations, and no effort will be made to bring out the underlying theory. We shall use only two—fold symmetry axes. For three—fold 

Consider naphthalene. It has ten it—electron centers and three two—fold symmetry axes passing through the center of the molecule at 90° to one another  

Turning the formula 180 around the z axis (the c operation) changes the position of the numbers of the atomic orbital functions.centered on each atom  

The changes (l- 5, 2 - 6, etc. ) are usefully tabulated as follows (where E, the identity operation , does not change the numbers) f 

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In fignre A1.3.9 the Brillouin zone for a FCC and a BCC crystal are illustrated. It is a connnon practice to label high-synnnetry point and directions by letters or symbols. For example, the k = 0 point is called the F point. For cubic crystals, there exist 48 symmetry operations and this synnnetry is maintained in the energy bands e.g., E k, k, k is mvariant under sign pennutations of (x,y, z). As such, one need only have knowledge of (k) in Tof the zone to detennine the energy band tlnoughout the zone. The part of the zone which caimot be reduced by synnnetry is called the irreducible Brillouin zone. 

At this point the reader may feel that we have done little in the way of explaining molecular synnnetry. All we have done is to state basic results, nonnally treated in introductory courses on quantum mechanics, connected with the fact that it is possible to find a complete set of simultaneous eigenfiinctions for two or more commuting operators. However, as we shall see in section Al.4.3.2. the fact that the molecular Hamiltonian //coimmites with and F is intimately coimected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing tlirough the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the... 

Another distinction we make concerning synnnetry operations involves the active and passive pictures. Below we consider translational and rotational symmetry operations. We describe these operations in a space-fixed axis system (X,Y,Z) with axes parallel to the X, Y, Z) axes, but with the origin fixed in space. In the active picture, which we adopt here, a translational symmetry operation displaces all nuclei and electrons in the molecule along a vector, say. 

We now define the effect of a translational synnnetry operation on a fiinction. Figure Al.4.3 shows how a PHg molecule is displaced a distance A X along the X axis by the translational symmetry operation that changes Xq to X = Xq -1- A X. Together with the molecule, we have drawn a sine wave symbolizing the... 

Whenever a fiinction can be written as a product of two or more fiinctions, each of which belongs to one of the synnnetry classes, the symmetry of the product fiinction is the direct product of the syimnetries of its constituents. This direct product is obtained in non-degenerate cases by taking the product of the characters for each symmetry operation. For example, the fiinction xy will have a symmetry given by the direct product of the syimnetries of v and ofy this direct product is obtained by taking the product of the characters for each synnnetry operation. In this example it may be seen that, for each operation, the product of the characters for Bj and B2 irreducible representations gives the character of the representation, so xy transfonns as A2. 

CO, CO, co, and o, respectively. The integrals in Eqs. (E.9) and (E.IO) will then be different from zero only if the integrands are invariant under all symmetry operations allowed by the symmetry point group, in particular under C3. It is readily seen that the linear terms in Q+ and Q- vanish in and H In turn. 

SymApps converts 2D structures From the ChemWindow drawing program into 3D representations with the help of a modified MM2 force field (see Section 7.2). Besides basic visualization tools such as display styles, perspective views, and light source adjustments, the module additionally provides calculations of bond lengths, angles, etc, Moreover, point groups and character tables can be determined. Animations of spinning movements and symmetry operations can also he created and saved as movie files (. avi). 

Operators that eommute with the Hamiltonian and with one another form a partieularly important elass beeause eaeh sueh operator permits eaeh of the energy eigenstates of the system to be labelled with a eorresponding quantum number. These operators are ealled symmetry operators. As will be seen later, they inelude angular momenta (e.g., L2,Lz, S, Sz, for atoms) and point group symmetries (e.g., planes and rotations about axes). Every operator that qualifies as a symmetry operator provides a quantum number with whieh the energy levels of the system ean be labeled. 

Symmetry operators leave the eleetronie Hamiltonian H invariant beeause the potential and kinetie energies are not ehanged if one applies sueh an operator R to the eoordinates and momenta of all the eleetrons in the system. Beeause symmetry operations involve refleetions through planes, rotations about axes, or inversions through points, the applieation of sueh an operation to a produet sueh as H / gives the produet of the operation applied to eaeh term in the original produet. Henee, one ean write ... 

Because symmetry operators eommute with the eleetronie Hamiltonian, the wavefunetions that are eigenstates of H ean be labeled by the symmetry of the point group of the moleeule (i.e., those operators that leave H invariant). It is for this reason that one eonstruets symmetry-adapted atomie basis orbitals to use in forming moleeular orbitals. 

Beeause the total Hamiltonian of a many-eleetron atom or moleeule forms a mutually eommutative set of operators with S, Sz, and A = (V l/N )Zp Sp P, the exaet eigenfunetions of H must be eigenfunetions of these operators. Being an eigenfunetion of A forees the eigenstates to be odd under all Pij. Any aeeeptable model or trial wavefunetion should be eonstrained to also be an eigenfunetion of these symmetry operators. 

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... 

One more quantum number, that relating to the inversion (i) symmetry operator ean be used in atomie eases beeause the total potential energy V is unehanged when all of the eleetrons have their position veetors subjeeted to inversion (i r = -r). This quantum number is straightforward to determine. Beeause eaeh L, S, Ml, Ms, H state diseussed above eonsist of a few (or, in the ease of eonfiguration interaetion several) symmetry adapted eombinations of Slater determinant funetions, the effeet of the inversion operator on sueh a wavefunetion P ean be determined by ... 

Point groups in whieh degenerate orbital symmetries appear ean be treated in like fashion but require more analysis beeause a symmetry operation R aeting on a degenerate... 

Here g is the order of the group (the number of symmetry operations in the group- 6 in this ease) and Xr(R) is the eharaeter for the partieular symmetry T whose eomponent in the direet produet is being ealeulated. 

Here, Xr(R) is the eharaeter belonging to symmetry E for the symmetry operation R. Applying this projeetor to a determinental flinetion of the form ( )i( )j generates a sum of determinants with eoeffieients determined by the matrix representations Ri ... 

Let us eonsider the vibrational motions of benzene. To eonsider all of the vibrational modes of benzene we should attaeh a set of displaeement veetors in the x, y, and z direetions to eaeh atom in the moleeule (giving 36 veetors in all), and evaluate how these transform under the symmetry operations of D6h- For this problem, however, let s only inquire about the C-H stretehing vibrations. 

These veetors form the basis for a redueible representation. Evaluate the eharaeters for this redueible representation under the symmetry operations of the D h group. 

The ammonia moleeule NH3 belongs, in its ground-state equilibrium geometry, to the C3v point group. Its symmetry operations eonsist of two C3 rotations, C3, 3 ... 

These six symmetry operations form a mathematieal group. A group is defined as a set of objeets satisfying four properties. 

Note the refleetion plane labels do not move. That is, although we start with Hi in the <5y plane, H2 in ay", and H3 in ay", if Hi moves due to the first symmetry operation, ay remains fixed and a different H atom lies in the Gy plane. 

Symmetry tools are used to eombine these M objeets into M new objeets eaeh of whieh belongs to a speeifie symmetry of the point group. Beeause the hamiltonian (eleetronie in the m.o. ease and vibration/rotation in the latter ease) eommutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be "bloek diagonal". That is, objeets of different symmetry will not interaet only interaetions among those of the same symmetry need be eonsidered. 

To illustrate sueh symmetry adaptation, eonsider symmetry adapting the 2s orbital of N and the three Is orbitals of H. We begin by determining how these orbitals transform under the symmetry operations of the C3V point group. The aet of eaeh of the six symmetry operations on the four atomie orbitals ean be denoted as follows ... 

We ean likewise write matrix representations for eaeh of the symmetry operations of the C3v point group ... 

In faet, one finds that the six matriees, Df4)(R), when multiplied together in all 36 possible ways obey the same multiplieation table as did the six symmetry operations. We say the matriees form a representation of the group beeause the matriees have all the properties of the group. 

The importance of the characters of the symmetry operations lies in the fact that they do not depend on the specific basis used to form them. That is, they are invariant to a unitary or orthorgonal transformation of the objects used to define the matrices. As a result, they contain information about the symmetry operation itself and about the space spanned by the set of objects. The significance of this observation for our symmetry adaptation process will become clear later. 

The character of a class depends on the space spanned by the basis of functions on which the symmetry operations act. Above we used (Sn,S 1,82,83) as a basis. 

These six matrices form another representation of the group. In this basis, each character is equal to unity. The representation formed by allowing the six symmetry operations to act on the Is N-atom orbital is clearly not the same as that formed when the same six operations acted on the (8]s[,S 1,82,83) basis. We now need to learn how to further analyze the information content of a specific representation of the group formed when the symmetry operations act on any specific set of objects. 

These six matrices can be verified to multiply just as the symmetry operations do thus they form another three-dimensional representation of the group. We see that in the Ti basis the matrices are block diagonal. This means that the space spanned by the Tj functions, which is the same space as the Sj span, forms a reducible representation that can be decomposed into a one dimensional space and a two dimensional space (via formation of the Ti functions). Note that the characters (traces) of the matrices are not changed by the change in bases. 

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