Print publication date: 2014

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Published to Oxford Scholarship Online: April 2014

DOI: 10.1093/acprof:oso/9780199670888.001.0001

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# (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)
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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)
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and

(A11.3)
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where k is a constant. Linear operators possess several important properties:

For two linear operators with a functions f:

• (A11.4)
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• (A11.5)
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• (A11.6)
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• (A11.7)
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Except in special cases, $OiOj≠OjOi:$ 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 $O1=ddx,O2=x2,O3=d2dx2,O4=k,andf1=x3−2x+1,f2=2x2−3.$ Then,

1. (a) $O1f1=3x2−2$

2. (b) $O1kf1=6x2−4$

3. (c) $O1(f1+f2)=3x2+4x−2$

4. (d) $O1O2f1=5x4−6x2+2x$

5. (e) $O2O1f1=3x4−2x2(≠O1O2f)$

# (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)
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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)
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that moves r directly to $r''$.

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

(A11.10)
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For another function g in the same space as f, Eq. (A11.8) leads to

(A11.11)
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Since g is any function, let it be equal to $ORf$; then with Eq. (A11.8)

(A11.12)
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Applying this result to Eq. (A11.11),

(A11.13)
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which, from Eq. (A11.10), shows that

(A11.14)
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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

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and inverting these equations gives

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

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Substituting for x and r

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

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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

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Similarly,

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

(A11.15)
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For convenience, $px$, $py$ and $pz$ will be renamed here as $pi$, $i=1−3$, the D-matrix elements as $DRij$ and the unit vectors i, j, k as $ei$, $i=1−3$. The operations discussed in Section 7.5.2, in real space, may be now summarized as

(A11.16)
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Similarly, the p function transformation in function space is given by

(A11.17)
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which is a formalism of the equation in Example A11.2. Thus, by analogy with Eq. (A11.14),

(A11.18)
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From the nature of a scalar, the transformation operators leave the scalar product of two functions unchanged:

(A11.19)
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(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 $ε∗000ε0001$. 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)
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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)
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From the great orthogonality theorem, Eq. (7.37), the right hand side of Eq. (A11.21) may be equated to

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which is zero if $α≠β,$ 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)
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whereupon the projection operator $Pα,ij$ is given by

(A11.23)
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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)
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(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)
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Recalling that $∑iDα(R)ii∗=χα∗,$ and summing first over i,

(A11.26)
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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 $i=j,$

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Hence, for $α=β(andi=q)$

(A11.27)
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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)
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and

(A11.29)
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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:

$C3v$

$OE$

$OC3$

$OC32$

$Oσv$

$Oσv′$

$Oσv′′$

$α$

$α$

$α$

$α$

$α$

$α$

$α$

$β$

$β$

$β$

$β$

$β$

$β$

$β$

$μ$

$μ$

$μ1/2‾ν3/2‾$

$μ/2‾ν3/2$

$μ$

$μ1/2‾ν3/2$

$μ1/2‾ν3/2‾$

$ν$

$ν$

$μ3/2ν/2‾$

$μ3/2‾ν/2‾$

$−ν$

$μ3/2ν/2$

$μ3/2‾ν/2$

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

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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.