Jump to ContentJump to Main Navigation
Symmetry of Crystals and Molecules$

Mark Ladd

Print publication date: 2014

Print ISBN-13: 9780199670888

Published to Oxford Scholarship Online: April 2014

DOI: 10.1093/acprof:oso/9780199670888.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 17 August 2017

(p.379) A11 Linear, unitary and projection operators

(p.379) A11 Linear, unitary and projection operators

Source:
Symmetry of Crystals and Molecules
Publisher:
Oxford University Press

A11.1 Linear operators

An operator has the property of changing one function into another function. Thus, if an operator is specified by O, and

(A11.1)
O(2x2+x)=4x+1,

then the operator O is clearly the differential operator ddx. There are many such operators, and a particular case is the linear operator. An operator is linear if, for any function f

(A11.2)
Okf=k(Of)

and

(A11.3)
O(f1+f2)=Of1+Of2

where k is a constant. Linear operators possess several important properties:

For two linear operators with a functions f:

  • (A11.4)
    (O1+O2)f=O1f+O2f

  • (A11.5)
    O1O2f=O1(O2f)

  • (A11.6)
    O1(O2+O3)=O1O2+O1O3

  • (A11.7)
    O1(O2O3)=(O1O2)O3

Except in special cases, OiOjOjOi: this result is easily demonstrated, and may be compared with the product of two rotations RiRj where R is greater than 2.

Example A11.1

Let O1=ddx,O2=x2,O3=d2dx2,O4=k,andf1=x32x+1,f2=2x23. Then,

  1. (a) O1f1=3x22

  2. (b) O1kf1=6x24

  3. (c) O1(f1+f2)=3x2+4x2

  4. (d) O1O2f1=5x46x2+2x

  5. (e) O2O1f1=3x42x2(O1O2f)

(p.380) A11.2 Operators in function space

Let a symmetry operation R be represented by a transformation operator OR. A transformation operator acts on the functions of function space and follows the rules for linear operators. A set of such operators for all symmetry operations R in a point group is homomorphic (Section 7.3) with the set of symmetry operations themselves. If an operation R moves a vector from a position r to a new position r, a transformation operator OR is associated with R according to

(A11.8)
ORf(r)=f(r)

that is, the new function ORf assigns the value of the original f to the new location r; thus, r is related to r by an equation similar to Eq. (7.10).

Let R and R be two symmetry operators in a point group, such that R moves a vector r to r, and then R moves r to r. From the definition of a group, there will be another symmetry operation R in the group, given by

(A11.9)
R=RR

that moves r directly to r''.

The associated transformation operator moves a function f defined at r to the new location r

(A11.10)
ORf(r)=f(r)

For another function g in the same space as f, Eq. (A11.8) leads to

(A11.11)
ORg(r)=g(r)

Since g is any function, let it be equal to ORf; then with Eq. (A11.8)

(A11.12)
g(r)=ORf(r)=f(r)

Applying this result to Eq. (A11.11),

(A11.13)
[OR(ORf)](r)=g(r)=(ORf)r=f(R)

which, from Eq. (A11.10), shows that

(A11.14)
OR=OROR

Thus, the relationship, Eq. (A11.9), for symmetry operators is paralleled by Eq. (A11.14) for the corresponding transformation operators.

Example A11.2

Consider a function space spanned by the three p orbital functions. From the matrix A in Appendix A3.4.11, an anticlockwise rotation Cθ moves rx,y,z to r(x,y,z) where r=r, and

x=xcosθysinθy=xsinθ+ycosθz=z

and inverting these equations gives

x=xcosθ+ysinθy=xsinθ+ycosθz=z

(p.381) Considering px and using a result from Appendix A8, the operator Oθ that corresponds to the symmetry operation Cθ has the property

Oθpx(x,y,z)=px(x,y,z)=f(r)x/r

Substituting for x and r

Oθpx(x,y,z)=f(r)(xcosθ+ysinθ)/r

and since there are primes on both sides, neatness suggests the form

Oθpx(x,y,z)=f(r)(xcosθ+ysinθ)/r

with similar equations for p and pz.

A11.3 Unitary operators

Again, from Appendix A8, the right hand side of Eq. (A11.14) may be recast as

Oθpx=pxcosθ+pysinθ+(pz×0)

Similarly,

Oθpy=pxsinθ+pycosθ+(pz×0)Oθpz=px×0+py×0+(pz×1)

Since Oθpx is a row vector, these equations form the matrix D(Cθ) which shows a formal equivalence with Eq. (7.9)

(A11.15)
D(Cθ)=cosθsinθ0sinθcosθ0001

For convenience, px, py and pz will be renamed here as pi, i=13, the D-matrix elements as DRij and the unit vectors i, j, k as ei, i=13. The operations discussed in Section 7.5.2, in real space, may be now summarized as

(A11.16)
Rei=j=13rjiei

Similarly, the p function transformation in function space is given by

(A11.17)
ORpi=j=13D(R)jipj

which is a formalism of the equation in Example A11.2. Thus, by analogy with Eq. (A11.14),

(A11.18)
D(R)=D(R)D(R)

From the nature of a scalar, the transformation operators leave the scalar product of two functions unchanged:

(A11.19)
ORf1ORf2=f1f2

(p.382) These operators are unitary operators and can be represented by unitary matrices (Appendix A3.4.11) which lead to unitary representations for point groups; thus, DC3 in unitary form is ε000ε0001. Representations are unitary if the basis functions are orthogonal and normalized to one and the same constant, usually unity. If orthonormal functions are employed, this condition holds implicitly.

A11.4 Projection operators

Consider a set of orthonormal functions ψn that form a basis for the βthnβ-dimensional irreducible representation Γβ in a given point group of order h. Then, from Eq. (A11.17)

(A11.20)
RDα(R)pqORψβ,q=p=1nβRDα(R)pqDβ(R)pqψβ,q

Since the basis functions are orthonormal, the D-matrices are unitary. Pre-multiplication by Dα(R)ij from a representation of the same symmetry R, followed by summation over all R and reversal of the order of summation on the right hand side, leads to

(A11.21)
RDα(R)ijORψβ,q=p=1nβRDα(R)ijDβ(R)pqψβ,q

From the great orthogonality theorem, Eq. (7.37), the right hand side of Eq. (A11.21) may be equated to

p=1nβhnαnβδαβδjqψβ,p

which is zero if αβ, but which otherwise gives (h/n)δjqψα,i because the sum over p is zero except for p=i.

Thus, Eq. (A11.21) becomes

(A11.22)
Pα,ijψβ,q=hnαδαβδjqψβ,i

whereupon the projection operator Pα,ij is given by

(A11.23)
Pα,ij=RDα(R)ijOR

and is a linear combination of operators OR with coefficients from the D-matrices of the Γα representation. For the non-zero case of Eq. (A11.22), with α=βandj=q,

(A11.24)
Pα,ijψα,j=hnαψα,i

(p.383) so that Pα,ij operating on the function ψα at location j reproduces that function multiplied by h/na at a new location i.

Another projection operator Pα may be defined that is a linear combination, having properties similar to those just described, by using the matrix elements from Eq. (A11.23) for the special case of i=j:

(A11.25)
Pα=i=1nαPα,ii=i=1nαRDα(R)iiOR

Recalling that iDα(R)ii=χα, and summing first over i,

(A11.26)
Pα=RχαOR

where the coefficient of OR is the complex conjugate of the characters of R in the representation Γα. Where the characters are real, χα(R) is replaced by χα(R). Whereas Eq. (A11.23) needs the complete D-matrix in order to define Pα,ij, Pα may be obtained readily from the character table of the appropriate point group. From Eqs. (A11.21)–(A11.22), and with i=j,

Pαψβ,q=i=1nαhnαδαβδiqψβ,i

Hence, for α=β(andi=q)

(A11.27)
Pαψα,i=hnαψα,i

The projection operator Pα acting on a function that is a member of the Γα function space reproduces that function multiplied by h/n, but any function not belonging to the Γα function space is annihilated. The operator Pα may be applied to any combination of basis functions in the space Γα. Thus, from the linear combination

(A11.28)
Φα=i=1nciψα,i

and

(A11.29)
PαΦα=hnαRχα(R)ORΦαi.

Thus, Pα generates a sum of members of a basis set spanning the irreducible representation Γα.

Example A11.2

Apply the projection operator to point group C3v. The basis set of functions is α (spanning A1), β (spanning A2) and μ,v (spanning the two-dimensional E representation). Referring to the extended character table for C3v, the effects of OR on each function follows they behave like row vectors:

C3v

OE

OC3

OC32

Oσv

Oσv

Oσv′′

α

α

α

α

α

α

α

β

β

β

β

β

β

β

μ

μ

μ1/2ν3/2

μ/2ν3/2

μ

μ1/2ν3/2

μ1/2ν3/2

ν

ν

μ3/2ν/2

μ3/2ν/2

ν

μ3/2ν/2

μ3/2ν/2

(p.384) Apply the projection operators to the four functions α,β,μandν, using the above table:

PA1α=α+α+α+α+α+α=6αPA2α=α+α+αααα=0PEα=2ααα=0

so that PA1 operating in α multiplies it by h/n, but with A2 and E α is annihilated. With β,μandν, the corresponding results are 6β, 3μ and 3ν, respectively. The two parts of E together satisfy the h/nα requirement; multi-dimensional representations must be treated in this manner in order to achieve the correct result.