(p.379) A11 Linear, unitary and projection operators
(p.379) A11 Linear, unitary and projection operators
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
then the operator O is clearly the differential operator There are many such operators, and a particular case is the linear operator. An operator is linear if, for any function f
where k is a constant. Linear operators possess several important properties:
For two linear operators with a functions f:
Except in special cases, this result is easily demonstrated, and may be compared with the product of two rotations RiRj where R is greater than 2.
(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 a transformation operator is associated with R according to
that is, the new function assigns the value of the original f to the new location thus, is related to r by an equation similar to Eq. (7.10).
Let R and be two symmetry operators in a point group, such that R moves a vector r to and then moves to From the definition of a group, there will be another symmetry operation in the group, given by
that moves r directly to .
The associated transformation operator moves a function f defined at r to the new location
For another function g in the same space as f, Eq. (A11.8) leads to
Since g is any function, let it be equal to ; then with Eq. (A11.8)
Applying this result to Eq. (A11.11),
which, from Eq. (A11.10), shows that
Consider a function space spanned by the three p orbital functions. From the matrix A in Appendix A3.4.11, an anticlockwise rotation moves to where and
and inverting these equations gives
Substituting for x and r
and since there are primes on both sides, neatness suggests the form
with similar equations for p and .
A11.3 Unitary operators
Since is a row vector, these equations form the matrix which shows a formal equivalence with Eq. (7.9)
For convenience, , and will be renamed here as , , the D-matrix elements as and the unit vectors i, j, k as , . The operations discussed in Section 7.5.2, in real space, may be now summarized as
Similarly, the p function transformation in function space is given by
From the nature of a scalar, the transformation operators leave the scalar product of two functions unchanged:
(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, in unitary form is . 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 that form a basis for the -dimensional irreducible representation in a given point group of order h. Then, from Eq. (A11.17)
Since the basis functions are orthonormal, the D-matrices are unitary. Pre-multiplication by 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
which is zero if but which otherwise gives because the sum over p is zero except for
Thus, Eq. (A11.21) becomes
whereupon the projection operator is given by
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
(p.383) so that operating on the function at location j reproduces that function multiplied by at a new location i.
Another projection operator 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
Recalling that and summing first over i,
where the coefficient of is the complex conjugate of the characters of R in the representation . Where the characters are real, is replaced by Whereas Eq. (A11.23) needs the complete D-matrix in order to define , may be obtained readily from the character table of the appropriate point group. From Eqs. (A11.21)–(A11.22), and with
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 may be applied to any combination of basis functions in the space Γα. Thus, from the linear combination
Thus, generates a sum of members of a basis set spanning the irreducible representation Γα.
Apply the projection operator to point group . The basis set of functions is (spanning A1), (spanning A2) and (spanning the two-dimensional E representation). Referring to the extended character table for , the effects of on each function follows they behave like row vectors:
(p.384) Apply the projection operators to the four functions using the above table:
so that operating in multiplies it by , but with A2 and E is annihilated. With the corresponding results are , and , respectively. The two parts of E together satisfy the requirement; multi-dimensional representations must be treated in this manner in order to achieve the correct result.