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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

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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.