Resolving perturbed wave functions into unperturbed Hamiltonian

Harriss, D. K.

Hirschfelder, J. O.

English

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019 OF A FEUMM6 WAVE. yt,RICTIOF
a ld K. Harriss
ember 1968
74
NA A
eTHE RESOLUTION OF A PERTURBED WAVE FUNCTION
INTO ITS SYMMETRY COMPONENTS
by
Joseph J. Hirschfelder and Donald K. Harriss+
University of Wisconsin Theoretical Chemistry Institute
•	 Madison, Wisconsin
ABSTRACT
The various orders of the wave functions for a perturbed
Hamiltonian can be resolved into components which transform in
accordance with the irreducible representations of the unperturbed
Hamiltonian. Each of these components satisfies a separate per-
turbation equation. It is hoped that this approach will be especially
useful in considering the perturbation of degenerate states.
This research was supported by the National Aeronautics and Space
Administration Grant NsG-275-62.
+ Present address: Department of Chemistry, University of Minnesota,
Duluth.
ial
1
Consider a perturbation problem in which the perturbation destroys
a part of the symmetry of the unperturbed state. The following treat-
ment serves to resolve the perturbed wave function into components each
of which transforms in accordance with irreducible representations of
the unperturbed Hamiltonian. Thus, for example, if the perturbation
destroys the spherical symmetry of a system, the perturbed wave function
can be expanded into spherical harmonics,
The zeroth order Hamiltonian Ho of a molecular system commutes
with a group G composed of g transformations R  o This group
has a set of s irreducible representations with matrix elements
D(.I) (Ri ) jk . The P-th irreducible representation is h  dimensional.
The Hilbert space of Ho is spanned by functions which transform
.;according to these irreducible representations.
If	 is an arbitrary function in this Hilbert space, then
+j - Afj §
	
transforms as the j-th row of the k-th irreducible
representationa l
 Here
2.
If G is a continuous, rather than a finite group, then the
summation over the R  must be replaced by integration over all of
the transformations using weight functions appropriate to the
parametrization. Thus, if H o has spherical symmetry, then
(^'R	 2'1f
A-01 = Y
Ij (0,0) y 0
	 J0	
Yij(A,O)* (	 ) sin 0 dO d8	 (3)
where the Y
Ij (0,0) are the normalized surface spherical harmonics.
Similarly, if Ho has cylindrical symmetry then
2 1T
A^ _ (1/211 ) exp( iiO ) j	 exp( - i1O) (	 ) dO	 (4)0
Now let us consider a perturbation problem where the Hamiltonian
for the perturbed system is H = H o + V . The usual Rayleigh-SchrUdinger
perturbation equations are
(H0 - Co) + 0 = 0
	 (S)
(H0 - E0) `I' 0 (1) + (V - C O (1) ) Y o = 0	 (6)
( H0 - Co) W0(2) + (V - E 0 (1) ) ^-?0 	 	 0	 7)
.....................................................
Since the zeroth order wavefunction 
T 
o is an eigenfunction of
H0
 it must belong to an irreducible representation of G . Further-
more, each of the k4-"- (n) can be resolved into a set of functions
AAi	 0(n) which transform in accordance with a particular row of a
3
particular irreducible representation,
YO In) = T— A T/ (n)
2,j
	
^ j 	0
The equation for AIj ^0 In)	 is obtained by multiplying the equation
for (n)Y0 by A'ej Thus,
(H0
	E0) AIj NP 0(1) + Aij[(v- e 0 (1) ) YO  = 0	 (g)
(Ho - 60) Aej q0(2) + A ej f (v- E 0 (1) ) x' 0 (1) ] = 6 0 (2) Aij %Po
(10)
(H0 - E0) AIj 
q.,
 In) + AIj [(v- E0(1)) kV0 In- 1)]
n-2T	 (n-k) A ^ (k)
k:0	 0	 ^j 	o	 , n> 1	 (11)
The normalization condition for the components of the perturbed
wavefunctions are given by
C A	 (n-k)	 (k)
k=O Ej ^o	 AIj	 o	 ^ = 0 , n ^ 0	 (12)
4 In the calculation of the terms contributing to a particular order
of the perturbed energy, the most important consideration is that the
direct product of all representations in the integrand must contain.
the identity representation D (0) (R) . A direct product of two
unitary irreducible representations D(k)(R)*xD (n)(R) can have a
component transforming as the identity representation2
 only if k-n .
(8)
For the three-dimensional rotationcgroup we have the Clebsch-Gordan
series
D (k) (R) * x D(n)(9)=D(k)(R)_x D(n)R)=D(fk-nJ)(R) + D(	 (R)
+...+ D(k+n)(R)
where a necessary condition  for a representation D (m) (R) to be
contained in the direct product D (k) (R) x D(n) (R) is that
Jk-n	 m S k+n . Let us suppose that 	 0 s Aoo %Vo ; then
the various orders of the perturbed energy are given by:
Eo(1) _ < To I Aoo [ V Vo l
 7
60(2)	 C T0 j Aoo [(V- 6 0 (1) )	 0(1 )1 >	 (14)
^0(3)	
,L <A18j	 0(1) ^A 8j I(V- X0(1))	 0(1) l 7
	
(15)
It is easy to discuss any specific case in terms of the properties of
the group under consideration.
The simplest illustration of this type of resolution corresponds
to	 o transforming as the identify representation and V trans-
:	 forming as the k-th irreducible representation (which has h 
dimensions). It follows from Eqs. (9) and (12) that	 (1) can have
non-zero projections only for Akj . Thus.,
h
2^ Akj `N (1)j=1
e
4
(13)
5and therefore, from Eq. (13),
	 'ol ) = 0	 We see from Eq. (14)
that if V is Hermitian
	 (2),	 Eo	 is not identically zero since
D (k)* (R) x D (k) (R) contains D (0) (R) . From Eq. (15) we see that
^^ 3) is identically zero unless D (k) (R) x D (k) (R) contains D (k) (R) .
From the second order equation, (10),
	 +(2) can have non-zero components
Aji	 (2) only for those irreducible representations DUnlyco -
tained in D (k) (R) x D (k) (R) . Similarly for the third order equation,
non-zero projections Aji Ike 
(3)
can exist only for those D(3)(R)
contained in D (k)* (R) x D (k) (R) x D (k) (R) .
A system exhibiting this type of behavior is the ground state
hydrogen at ,)m in a uniform electric field. In this example 	 V/0
is spherically symmetric, transforming as D (0) (R)	 For unit field
strength the perturbation may be taken as
	 -Z which has the symmetry
of Y10 (0,0) , transforming as D (1) (R) . Since Y10 is only one of
the three partners which form the basis for the representation D(1)(R)
the product Y10 x Y10 need not have components transforming as all of
the representations included in the Clebsch-Gordan series . Examination
of the products shows that they reduce as:
	
Y 10 x Y10	 __4
	
Y00'Y20
	
Y10 x Y10 x Y10	 !-'0	 Y10'Y30
	
Y 10 x Y10 x Y10 x Y10	 Y00'Y20'Y4O
..............................
This reduction is sufficient to show that all odd order energies are zero
and, since for the ground state 	 e(2n)
	
C	 0 , that the energy of
the perturbed system is a sum of negative numbers. It can also be
6e
seen that	 (n) can have (n+2)/2 components for n even,
(A00 Y (n) '
 A20 %p(n), ... , Ann W (n) ) , and (n+l)/2 components
for n odd, (A10 y(n) ,  A30 y (n) , ... , Ano W (n) ) . Each of the
components has known symmetry and the projected equations can be
treated as one-dimensional problems.


The resolution of a perturbed wave function into its symmetry components

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