Irreducible Bloch functions

The orbitals <]) j(k r) are Bloch functions labeled by a wave vector k in the first Brillouin zone (BZ), a band index p, and a subscript i indicating the spinor component. The combination of k and p. can be thought of as a label of an irreducible representation of the space group of the crystal. Thequantity n (k)is the occupation function which measures... 

In the case of a perfect crystal the Hamiltonian commutes with the elements of a certain space group and the wave functions therefore transform under the space group operations accorc g to the irreducible representations of the space group. Primarily this means that the wave functions are Bloch functions labeled by a wave vector k in the first Brillouin zone. Under pure translations they transform as follows... 

First, the irreducible part of the Brillouin zone now varies from k = 0 to k = Tr/d = tt/2d. Indeed, doubling the parameter of the unit cell in real space halves the size of the Brillouin zone (or the reciprocal-space unit cell). Second, recall that orbital interactions are additive and that the final MO diagram (or band structure) is just the result of the sum of all the orbital interactions. Within each individual H2 unit the interactions simply correspond to the bonding (a) and antibonding (a ) MOs of each individual H2 unit. There are three types of interactions involving the MOs of different H2 units interactions between all the a orbitals interactions between all the a orbitals and interactions between the a and the a orbitals. Since all the an orbitals are equivalent by translational symmetry, their interaction is described by the Bloch function ... 

In other words, k determines which irreducible representation we are dealing with (see Appendix C available at booksite.elsevier.com/978-0-444-59436-5 on p. el7). This means that k tells us which permitted rhythm is exhibited by the coefficients at atomic orbitals in a particular Bloch function (i.e., ensuring that the square has the symmetry of the crystal). There are a lot of such ihythms e.g., all the coefficients equal each other k = 0), or one node introduced, two nodes, etc. The FBZ represents a set of such k. which corresponds to all possible rhythms i.e., non-equivalent Bloch functions. In other words, the FBZ gives us all the possible symmetry orbitals that can be formed from an atomic orbital. 

To conclude, three remarks may be added first, the L3 — L2/ band gap projecting onto the S point is expected to support a Bi and an Ai surface state. The former has orbital representation Zsys and is identical to the one derived from the above consideration of the Wp point The Ai state has the orbital representation Zg — x. Second, the present discussion neglects the spin-orbit interaction. The latter wiU scramble the orbital representations to some extent Third, if one tries to explore not just the orbital character of a surface state with respect to the topmost surface atoms, but rather to reconstmct the orbital composition of the complete surface-state Bloch function (including the contributions from the deeper layers) one has to take into account also the translational symmetry. This is formally more complicated, as indicated in Ref [36], but again, once the appropriate operations have been carried out, the result is intuitively clear. In case of degenerate irreducible representations, transfer projection operators have to be applied, but this is a straightforward generahzation of the method outlined above. 

In this equation, N is equal to the number of unit cells in the crystal. Note how the function in Eq. 5.27 is the same as that of Eq. 5.19 for cyclic tt molecules, if a new index is defined ask = liij/Na. Bloch sums are simply symmetry-adapted linear combinations of atomic orbitals. However, whereas the exponential term in Eq. 5.19 is the character of the yth irreducible representation of the cychc group to which the molecule belongs, in Eq. 5.27 the exponential term is related to the character of the Mi irreducible representation of the cychc group of infinite order (Albright, 1985). This, in turn, may be replaced with the infinite linear translation group because of the periodic boundary conditions. It turns out that SALCs for any system with translational symmetry are con-stmcted in this same manner. Thus, as with cychc tt systems, there should never be a need to use the projection operators referred to earher to generate a Bloch sum. 

DOS = Density of states BO = Bloch orbital IBZ = Irreducible Brillouin zone BZ = Brillouin zone PZ = Primitive zone COOP = Crystal orbital overlap population CDW = Charge density wave MO = Molecular orbital DFT = Density functional theory HF = Hartree-Fock LAPW = Linear augmented plane wave LMTO = Linear muffin tin orbital LCAO = Linear combination of atomic orbitals. 

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