Group theory is an extremely powerful method to obtain information like degeneracy o f eigeiüevels and the general form o f eigenfunctions by looking at the symmetry o f the problem involved. That such a link exist is not too surprising as the wavefunctions in a perfectly ordered material are not expected to be o f arbitrary shape. Group theory is set up in such a way that this sort o f information can be retrieved by just using character tables and symmetry groups (which will both be treated later on) without actually needing to solve the Hamiltonian.
This section is mainly concerned with outlining the fundamentals o f group theory and linking it to the physical problem o f solving the Hamiltonian for eigenvalues and wavefunctions. It is emphasised that this section does not contain a complete review o f group theory but is rather more focused on its application. More information about the specific use o f group theory in electron band calculations can for example be found in [Cor69,Bas75,Alt94].
Starting with the definition o f a group, a group 6 is defined as a set o f elem ents ... where • any product is itself an element o f 6
• (A B )C = A (B C ), i.e. the elements are associative • contains the identity element E, A E= EA
• for any element there is an inverse which is also an element o f 6 .
An example o f a group is the set o f transformations which leave the crystal lattice invariant Following [Cor69], a transformation is denoted by T and its effect on a vector r is given by
r' = [ R ( T ) \ t ( T ) ] r o r s h o r th a n d t = T r (2.12) where r and r' describe the same coordinate but in a different reference frame. Here, the new coordinate is obtained by a rotation R (T ) followed by a translation t(T ), Some examples o f a rotation, or point operation, R (T ) are :
-C ni : a proper rotation through In h i in the right hand sense about the /-axis -E : identity operation
- a : a reflection - I : an inversion
Every rotation R (T ) is partnered by an operator P(T). When followed by a scalar function yfr), P (T) performs the coordinate change given by R ( T ) '\ An example o f such an operator is : ‘replace
p ( T ) f i n = m
or (2.13)
P ( T ) f ( T r ) = f ( r ) .
The question now is to establish the set o f symmetry operations that leave the eigenvalue equation (2.2) or its one-electron equivalent (2.3) invariant. Obviously, the potential U (r) in the SchrOdinger equation obeys U ({R (T )lt(T ))j: )= U (r) if T is chosen such that it leaves the crystal invariant, and so do the other Hamiltonian elements. The group o f the Schrôdinger equation is therefore the space group o f the crystal, i.e. the symmetry transformations that leave the lattice invariant also leave the Hamiltonian invariant,
H ( T r ) = H ( r ) (2.14)
The Bloch theorem (2.9) that was introduced in sub-section 2.1.3 which states that every bulk wavefunction in a periodic structure can be written as a cell-periodic function u„jc multiplied by a plane wave is a direct consequence o f the presence o f translation symmetry. Translations symmetry is however not the only symmetry present in common crystalline structures but merely forms a subgroup o f the total symmetry group 6 . In addition, the point group Go which contains all the remaining transformations {R(T)fQ} is also a subgroup o f 6 .
The consequences o f the additional symmetry operations contained in Go are now investigated. Obviously, the presence o f rotational or inversion symmetry puts constraints on the general form o f the cell-periodic structure. To obtain these constraints, a link between abstract group theory and the physical problem o f solving the Hamiltonian for eigenfunctions and eigenvalues has to be made. First, this involves a more detailed look on the group theory without explicitly dealing with a particular problem (being an eigenvalue problem or otherwise). Secondly, an examination into the effects o f the presence o f a symmetry transformation on the physical problem has to be made, and finally both need to be connected.
The group concept was introduced in the previous paragraphs, and the elements o f the (sub)groups were visualised in terms o f rotations and translations. It now proves convenient to w oik in a more abstract way and represent these group elements by matrices which, naturally, have to satisfy all group postulates. It is straightforward to see that there exists an infinite number o f representations as there is no constraint on the order o f the matrices that one can choose. However, all these representations can be constructed from a number o f basis (so-called irreducible) representations. These irreducible representations consist o f a set of matrices F(T), one for each transformation, o f the low est possible order. As T (T ) is only uniquely determined within a similarity transformation,
Symmetry considerations
one would like to avoid working with an explicit form o f the matrices. Instead, an invariant quantity called the character o f T, which is the trace o f the matrix T (T ) and defined by
%(T) = , is used throughout group theory. i
Returning to the physical problem o f solving the Hamiltonian, it can be shown that the Hamiltonian H and transformation operator P(T) conunute when T corresponds to a symmetry transformation o f the lattice. From this it follow s that
H(.r) = E ^ , ( r ) => //(r){/> (7 0 (t),(r )} = £ { /> ( D ( l) ,( r ) }
which says that w hen (j)„ is an eigenfunction a t energy E, then P (T ) ())« is also an eigenfunction w ith the sam e eigenvalue, a very important observation. N ow consider an /-fold degenerate level o f the Schrôdinger equation, H(r)^n(L)=E^n(r)y / i =l .. /, for which the set ())« forms a set o f linearly independent eigenfunctions corresponding to this eigenvalue. It then follow s that P (T ) <}>„ must be a linear combination o f this set so that one can write
/ ’( r ) i|> ,( r ) = £ r { r ) „ „ ( i ) „ a ; ) (2.1 6) m=l
In words, (})« transforms under an operation P(T) as the row o f T (T ). This suggests that a se t o f b asis fu n c tio n s f o r a l-degenerate level fo r m s a set o f ba sis fu n c tio n s f o r an l-degenerate representation o f this subgroup o f the Schrodinger equation. The proof consists in showing that all group postulates are satisfied. But this means that a subgroup F o f order / corresponds to an /-degenerate energy level. The connection between group theory and the eigenvalue problem has now been made.
The latter observation has huge implications. Given a system, being a molecule or a crystal, it is not difficult to obtain its symmetry group. Group theory in turn, by various routines with which this work does not concern itself, can supply the appropriate character table. Using only this table and the symmetiy operations, one can extract elementary information about the degeneracy o f the various eigenlevels as well as the symmetry properties o f the eigenfunctions. The latter can be obtained using the projection operator which projects any function onto the group F
p ' ' = - Z % X n f ( T ) (2.17)
8 T
where T runs over all transformations, /r is the number o f basisfunctions in F and g is the total number o f symmetry operations.