A generalization of the Leman-Weaire-Thorpe theorem

LE JOURNAL DE TOME SEPTEMBRE 901PHYSIQUE 40, 1979,
Classification
AhstractsPhysics
- 71..25 02.10
A of the theoremLeman-Weaire-Thorpe generalization
O. Betbeder-Matibet and M. Hulin
de des Solides de l’Ecole Normale Groupe Physique Supérieure (*),
Université Paris 75221 Paris Cedex FranceVII, 2, place Jussieu, 05,
2 le mars révisé le 7 mai le 15 mai (Reçu 1979, 1979, accepté 1979)
Weaire et desRésumé. 2014 Nous avons un résultat, initialement démontré Leman, généralisé par Thorpe pour
semiconducteurs à un ensemble de situations très diverses où le désordre est letétracoordonnés, topologique
des constitués d’amasdes du Nous considérons matériaux électroniques système. principal responsable propriétés
reliés des chaînes d’atomes elles-mêmes et tels les interactions entre cesd’atomes identiques par identiques que
entités soient fonction de leur relative sur le associé à l’ensemble de la structure.uniquement position graphe
Nous montrons est alors de factoriser le Hamiltonien à liaisons on obtient ainsi une densitéfortes ; qu’il possible
« et de bandes celles-ci être uned’états en constituée de delta » déterminées, énergie pics d’énergie ; peuvent par
transformation à du de la matrice incidence associée au admet lesalgébrique simple, partir spectre graphe qui
où amas sommets et les chaînes comme arêtes. Cette est étendue au cas les amas sontpropriété pour atomiques
amas de différents. Pour donner un de deux les liaisons n’intervenant types, qu’entre types exemple d’application,
nous avons les calculs de structure de bande de Lannoo et et montré Bensoussan, repris GeSe2 publiés par qu’on
liantes et antiliantesaméliorer leur traitement en tenant du entre orbitales compte par exemple couplage peut
à un de même atome adjacentes germanium.
- for fourfold coordinated semiconductorsAbstract. A result of Weaire and Leman, Thorpe, initially proved
is to much wider situations where disorder be taken as the factor generalized topological may primary determining
the electronic of the We consider materials where identical clusters of atoms are linked throughproperties system.
interactions these entities on their relative identical atomic and where the between chains, only depend positions
on the the whole structure. We show that there is a factorization of the graph underlying possible tight-binding
these Hamiltonian which an of states made of delta and density up peaks energy bands; bands,yields energy
in can a from the of the matrixturn, be obtained, through simple algebraic manipulation, spectrum adjacency
of the which the clusters as vertices and the chains as This is further graph represents edges. property generalized
In to to the case where clusters are of two with chains between clusters of different order types. providetypes, only
of our we structure calculations for Lannooan illustrative band application formalism, repeat GeSe2 published by
and Bensoussan and show that more results can now be for into accountobtained, instance, precise easily by taking
the interaction between and orbitals to the same Ge atom.bonding antibonding adjacent
- the matrix of the the 1. Introduction. Several Weaire and atomsyears adjacency graph having ago,
rediscovered and a result as vertices and the covalent bonds between nearestThorpe [1, 2] generalized
« which had been obtained Leman as This Leman-Weaire-previously by [3] neighbours edges [4].
showed that the elec- well in a restricted context : theorem tothey Thorpe » (L.W.T.) equally applies
in a four fold-coordinated disordered materials and to tronic energy topologically crystals,spectrum,
covalent semiconductor described a for the can be used to show that for-and, former, by tight-binding
with four orbitals bidden bands remain when the Hamiltonian, belong- energy equivalent sp3 crystal perio-
of isolated is the disorder is to the same consists lost, atom, energy dicity provided ensuing purelying
levels contributions to the means that no (which bond(« (j-peak » energy density topological dangling
of the of the and that all atoms and bonds remain states) bands ; energies plus energy appears equi-
from the of valent to one latter can be obtained readily spectrum another).
The initial demonstration Weaire and by Thorpe
was somewhat and was anew in aclumsy, given
much form these authors as wellLaboratoire associé au Centre National de la Recherche(*) simplified by [5],
as some others Scientifique. Various to by [6, 7]. attempts gene-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004009090100902
- have also been for- orbitals ralize the L.W.T. theorem the interactions between atomic put belong-
Some of them make it to use to these clusters and chains.ward [8, 9]. possible ing
more this result in situations which are complicated We aim at H to the various matrices,relating
of view. Anfrom a structural and chemical point the matrix theC, including adjacency describing
the Ge-Se is by system [10] :example provided G. This is if :only graph possible
4-fold coordinated atoms there bemay germanium all clusters and all chains are as a(i) identical ; connected selenium chains of various lengths,by chain between two vertices mustconsequence any for someon the concentration ; depending be with to its middle symmetrical respect point :chains have critical all concentrations, essentially
ortherwise a would be introduced bet-dissymmetry the same and one comes close to a situa-length very
ween the vertices at both extremities of the chain ;tion of disorder.pure topological all interactions between orbitals to(ii) belonging as aThe aim of this is to complete paper provide these entities on the of theonly depend positions as of the L.W.T. result.generalization possible cluster or chain which orbitals within the they belongwithin the framework of a Keeping tight-binding and on the relative on the to, positions graph G,theone-electron we shall first treat Hamiltonian,
of these clusters chains.and/or can describedof a structure which be problem
Atomic orbitals will be distributed into two classes :which from some regular graph (in everystarting
- vertex in the same ofnumber, Z, « vertex orbitals » class participates say this (V.O.) : only
but where the vertices and the now contains orbitals that to atoms in vertexedges), edges belong
in the to entitiesmaterial, clusters and a role within thecorrespond, composite play totally isotropic
more-clusters or chains of several lone atoms ; structure including (for instance, pair orbitals, interacting
of atomic orbitals associated withsome the with all other orbitals involved in covalentover, identically
these atoms to orbitals (for we shall introduce n orbitals of this may correspond free bonds) ; type
in instance lone not involved covalent bonds. for vertex of pairs), G ;every
- A second situation will also be for whichexamined, « this which(E.O.) : class, edge-orbitals »
the is vertices are of two thekinds, all other atomic includes thosegraph bipartite : orbitals, gathers
dif-atomic clusters associated with atoms within the chains andcorresponding having possibly edge
and and in theferent structures to atoms in the vertex clustersproperties, bonding those belonging
between vertices of different which have a connection with onegraph only appears preferential
kinds. instance orbitals in a 4-fold coordinated(for edge sp3
These refinements offer the of using semiconductor will here be considered as possibility « edge
in a much widened ofthe L.W.T. theorem orbitals range »).
physical systems. the of which wechains, Actually, symmetry edge
Rather conditions on bond andstringent lengths of stressed the definition above, suggests symmetrical
of reduce the of topo-angles possibility appearance and combinations of the atomicantisymmetrical
for a covalent semi-conductor withdisorder logical E.O.’s still understood with(the symmetry being
one 4-fold coordinated atoms.of, only species say, of the to the middle To chain). respect point every
These conditions are much relaxed when bonding will thus E.O.’sedge correspond symmetrical ms
vertices be realized chains ofbetween may through and E.O.’s (S.E.O.) (A.E.O.).mA antisymmetrical
atoms which can be rotated and twisted in ·various = We shall take m + The S.E.O.’s ms MA, generalize
ways. withthe conventional orbitals associated bonding
more actual structuresThus, admitting complex covalent but also include lone bonds ; they may pair
or chains which are for the atomic clusters respectively orbitals to some atom at thepossibly belonging
associated with the vertices and of a edges graph, middle of the chain. The on theA.E.O.’s, point
the number of situations whereconsiderably enlarges orbitalsother are reminiscent of hand, antibonding
disorder be considered as the topological may pri- in Some must be intro-covalent coherence bonding.
electronic offactor the determining properties mary duced in their definition : all of the edges graph
a random system. should be in a oriented, completely arbitrary way.
will be For a A.E.O.’s systematicallygiven edge,
for of atomic orbitals defined any symmetricalpair
- 2. with one of vertices. 2. 1 DES-Systems type with to the middle therespect edge point, by taking
- OF THE CRIPTION SYSTEMS. A of the kindsystem orbital on the side of the vertex minus thestarting point
will we here be associated with a study regular graphon the side of the end and thenvertex,
and described a Hamiltonian G, by tight-binding H, normalizing.
the matrix elements of which are determinedentirely the conditions and conventionsAll preceding
once we know : us with the material to treat ourprovide necessary
- We shall consider cases where the structure of the atomic clusters and chains interac-only problem.
we associate with tion is restricted to orbitals to the vertices and respectively belonging adjacentedges
of the G : vertices atof G, components underlying graph 903
the extremities of some common or above. Within all of the M willE.O.’s, edge, edges groups edges
This will in a common vertex. the be taken the same order.sharing simplify
but does not to an actual with the matrix the exposition correspond Together adjacency C, underly-
limitation of the of our results. We shall G can be described two generality ing graph using vertex-edge,
x later indicate matrices :possible (N M) generalizations.
- all its elements are zero at the inter-except TS :
- 2.2 HAMILTONIAN STRUCTURE. Using adequate section of a row associated with some vertex and of a
we shall restrict consi-boundary conditions, periodic column associated to an this vertexedge admitting
deration to a set of N linked M vertices, by edges, as an the matrix element thenextremity, being
with :
to 1 ;equal
- same as that non zero matrixexcept TA : TS
elements are when the vertex is the (+ 1) starting
of the oriented and when at itsedge, (- 1) point
The hamiltonian H will be into fourdecomposed end point.
after the orbital basis for the submatrices, tight- with différent havematrices, notations, (These treatment of the has been orderedbinding system been in the same inconsidered, context, already in such a that V.O.’s and nextfirst, way appear ref. are well known in network [7] ; they analysis [12.])E.O.’s : Hl’ should into two be divided clearly submatrices,
for vertex-S.E.O. and vertex-A.E.O. interactions
respectively :
HVV is a x matrix which describes inter-(Nn Nn)
these matrices can be as directAgain expressed x action between HEE is a V.O.’s; (Mm Mm)
products :matrix for the interaction between HVE is aE.O.’s ;
x matrix for the interaction between V.O.’s(Nn Mm)
= and and HEV E.O.’s, (H VE) + (1).
Vertex-Vertex Hamiltonian Hvv. - It will be(i)
understood that all N V.O.’s to a belonging given
and are x and x (n (n Cs cA respectively ms) mA)to are taken so as to atype together correspond
matrices the interaction of V.O.’s describing belongingof N consecutive rows or columns in H group (and
to a with S.E.O.’s and A.E.O.’s givenvertex hence the of vertices to which theseHVV) ; ordering
to to this vertex. foredges adjacent TA appears V.O.’s are associated within such a is group repeated
A.E.O.’s instead because of an obvious changeof Ts, for all n of V.O.’s.identically types
of for these on the sign orbitals, depending edgeHvv is the sum of two which bematrices, may
from which are considered.extremity they written both as a direct (noted product [11] J by 0)
x x Hamiltonian HEE. - matrix a matrix (iii) of a (n n) (N N) (2) : Edge-Edge Distinguish-by
between S.E.O.’s and A.E.O.’s about aning brings
obvious submatrix decomposition :
where a is a x Hamiltonian which describes the(n n)
interaction between the different V.O.’s associated
x to a b’ is a matrix which describesvertex, (n n) given
withthe between V.O.’s to belonging neighbour-coupling
C the x matrixand is ing vertices, (N N) adjacency
of the G. The factorization in (3)underlying graph
results from the assumed of all verticesequivalence
and of all edges.
- Hamiltonian Hv’. We must,(ii) Vertex-Edge
for S.E.O.’s willorder E.O.’s as we did V.O.’s. first,
be taken M each first, grouped together by M, group The occurrence of such as products r s + rs, r r A,
to a A.E.O.’s are thencorresponding given type ; or to the fact that orbitalsr s + TA corresponds edge
in ordered the same after an orien-(arbitrary) way, can here interact when to only they belong edges
tation has been chosen for all as edges explained a common vertex : the matrixsharing non-diagonal
elements of these are different fromproducts actually
zero to ± when this condition is(and 1) equal only
thus a for matrix will stand for of A. realized, which convenient (’) A +, A, the provides any adjoint algebraic
¡(P), for will stand for the unit x matrix.(2) any integer p, (p p) of the (Dia-transcription topological prescription. 904
matrix elements also in and to the A.E.O.’s same gonal appear Ts rs edge belonging edge (ds), (dA),
and have thus to be cancelled suitable S.E.O.’s or to by belonging neighbouring edgesr r
M) S.E.O.’s from one andterms.) (fus and.fa respectively), edge
are x A.E.O.’s from a respectively (ms ms),s(A). dA(JA), JSA neighbouring edge (.fSA)’
x x matrices. describe the To sum the Hamiltonian has theThey up, tight-binding (mA mA), (ms mA)
to the samebetween S.E.O.’s form :belonging coupling
set for which has the same where : analogous rA property.
and these vec-Orthonormalizing gathering together
x tors as column-vectors of two matrices(M P)
and we write :Vs VA, may
3. of the Hamiltonian. - TheDiagonalization
relations following [7, 12] :
The column-vectors of consti-Vs (respectively VA)
that both and commute with C. tute a basis for a P-dimensional of theimply subspace rs Ts rA rÂ
We thus find a x matrix M-dimensional on which matrices (N N) U,may unitary space TS (respec-
such that are The N-dimensional tively TA) acting. comple-
be also an orthonormalmentary subspace may given
the vectors of which be asbasis, may again grouped
x the column of a matrix (M N) Ws (respec-
tively WA) :
x where are matrices.E, (N N) Fs, FA diagonal
x are both with(N M) matrices, TS and T’A
M > N.
Let
We H then transform the Hamiltonian into(9)
There exist P M-dimensional vectorsindependent
with :which sends into the null vector, and anotherT’S
the fundamental relation the usualThe transformation is (17) accomplished using connecting unitary readily
of and direct matricesproducts
for set of four matrices with suitable dimensions. H’ is found to have the forma, any fl, y, ô 905
notice that a first In order to for the We may immediately put, example, non-diagonaldiago-
in first row of into the nalization levels and levels submatrices the P ) (20) appears : (ms (mA P) proper
we could to to take withdraw from the rest of the cor- form, try, referring (12), Wsspectrum. They
to P-fold of and in the form :a therespond respectively repetition WA
and of matrices and Theseeigenvalues gs gA. ms mA
levels constitute a of thedegenerate generalization
in the electronic of states first noticed,£5-peaks density
for a Weaire and situation, by Thorpe [1, 2].simpler
we restrict ourselves to a Henceforth, may simpler x and are two matrices.where (N N) diagonal Gs GA Hamiltonian H derived from H’ theby suppressing Thanks to thus defined are ortho-(14), Ws and WA 3rd in and 5th rows and columns The (20). encourag- and matrices as to gonal Vs VA respectively, requiredfeature of H consists in the of sub-ing appearance the second set of relations Orthonormalizationby (16).
matrices like I(N) + b where the second(a ~ ~ E), of (21) yields :factor in the x direct is a (N N)products diagonal
matrix. if all submatrices in 77 shared theClearly,
same H could be factorized into N property, square
of To matrices dimension + obtain W matrices take final form :(n and the the ms + MA).
this we shall the we areresult, profit by flexibility
left the column vectors of matriceswith, concerning
and which are submitted to the condi-Ws WA, only
an orthonormal basis in sometions (16) of defining
vector The Hamiltonian fi now becomes :subspace.
x where all second factors in the direct are matrices (N N) diagonal (3). Thus, products coupling only appears
between rows columns index in the of and and the same various submatrices having p (24), corresponding
to a of C To matrix elements + and for (1 K N). (Z (Z - given eigenvalue p K correspond Fse. ëp e.) tp)
and + and for and (Z (Z - FA, (F S)1/2 (F A)1/2p )1/2 tp)1/2
rows and columns in so that the + + states associated with a (24), (n Reordering ms mA) given eigenvalue
of the matrix C be we factorize H into a succession of N subhamiltoniansadjacency brought together, may tp
withhp
- Let us sum the results obtained thus far. The + + bands each one with N(n up MA) energy mS
has :Hamiltonian these can be obtained the N(9) levels ; tight-binding by diagonalizing
subhamiltonians
- (25).two sets of P-fold eigenvalues (wheredegenerate
P the number M and Thus the of H involves threeis the difference between edge diagonalization
the vertex number these are thoseN) ; eigenvalues steps :
of the submatrix involves the(which (ms x ms) gs of matrices and which(i) diagonalization gs gA,
- interaction between bonding actually symmetrical - the contributions to the ofgives density b-peak
orbitals either within one or toedge, belonging states ;
and those of the x sub-adjacent edges), (mA mA) of the matrix (ii) C ;diagonalization adjacency
matrix the same role for (which gA plays antibonding are and no information(only eigenvalues necessary
- more orbitals) ;precisely antisymmetrical - on the is eigenvectors required) ;
for of (iii) C, every diagonalizationeigenvalue e.
of the subhamiltonian This is(25). corresponding
of course reminiscent of the Bloch transfor-highly Since the of C are confined to the interval(’) eigenvalues
mation for a where the real initial Hamil-( - Z, + the matrix elements and are periodic system, Z), positive,of FS FA
so that and are real and to their tonian be factorized into subhamil-(FS)I/2 (FA)I/2 equal adjoint. may k-dependent 906
in tonians. A considerable reduction computational
results in both cases.difficulty
the transformationThis factorization generalizes
of a Hamiltonian into a « 1 -band »« 4-band »
Weaire and forHamiltonian, proposed by Thorpe Atomic orbitals will be divided into twoagain
4-fold coordinated covalent semiconductors. categories :
- vertex orbitals of course 1-(V.O.), distinguishing
with V.O.’sand and 2-vertices, respectively, n, n2
per vertex,
- 4. with two of vertices and a orbitals are not(E.O.). However, Systems types bipartite edge edges
-
- 4. 1 HAMILTONIAN STRUCTURE. The now and the m E.O.’s of a graph. symmetrical given edge
as ideas the treatment cannot be characterized or anti-symmetrical general underlying previous
remain But the vertices fall’now, symmetrical.unchanged. graph
2 into two labelled 1 and andclasses, respectively, The hamiltonian H its tight-binding keeps general
the G links vertices to diffèrentonly graph belonging form the vertex-vertex Hamiltonian(2), but, now,
All vertices of 1 are withtypes. type identical, Hvv intodecomposes
and all vertices of 2 are also similardegree Zl, type
one to the with All fromother, degree Z2. edges,
If a 1-vertex to a are also identical. 2-vertex, Nj,
M of and are the numbers 1-vertices,respective N2
2-vertices and the relations hold : x edges, following and are andwhere al a2 respectively (ni ni)
x matrices the V.O.’s interactionsdescribing (n2 n2)
within a 1-vertex or a and b is a x 2-vertex, (ni n2)
matrix for the between V.O.’s on coupling adjacentand are taken to be than 2,Zi 1 Z2 strictly greater
vertices.
so that
The hamiltonian H" takes the formvertex-edge
We shall assume so that :Zl 1> Z2,
x where and are andm) ci c2 respectively (ni
x matrices the interactions betweenm) describing (n2
1-vertices first the Putting together gives graph V.O.’s and E.O.’s on vertices and adjacent edges.
matrix C the formadjacency the Hamiltonian HEE beFinally, edge-edge may
written as a sum of direct asproducts, essentially
in (8) :
x thewhere B is the matrix describing (N, N2)
to 2-vertices. In the same of 1-vertices way,linking
where the x matrices describe thematrix of the treatment now (m m) d, f l, the f2 previous rs
interactions between E.O.’s into belonging respectivelydecomposes
to the same to to the sameedge, edges adjacent
2-vertex.1-vertex and to to the same edges adjacent
In the we shall following put :
x where and are andM) F 1 T2 respectively (N,
x matrices. The relations hold : The whole initial Hamiltonian is thus :M) (N2 following
4.2 DIAGONALIZATION OF THE HAMILTONIAN. - as to devise a suitable transformationbefore, unitary
The main of the calculation will will transform be, which into a the sub-purpose following (36) matrix, 907
of which are all In a matrices the case similar issimultaneously diagonal. present diagonalization
In the the condition for such a transfor- thanks to The transfor-case, previous possible (31). proper unitary
to C mation was the fact and mation of H involves the matrix that, owing (11), (fg rs+) unitary (analogous
commuted. to (18)) :
where : with 1 and 0 with a( ± - obviously p % Nl, ep)
.of order Q ; degeneracy is a x matrix (a) U, unitary (Ni NI) diagonalizing
x is a matrix such that(c) Q ) (BB + ) : U3 (N2
is a matrix with (D positive elements) ;diagonal
= x M - withis a matrix (d) V (M P) (P N), a x matrix withis (b) U2 (N2 N,)
and thus :
x withW is a matrix (e) (M N)
E is a the of whichmatrix, diagonal eigenvalues 8p
will be taken > 0. The of C are H becomes(29) eigenvalues
- - We notice that m levels each of which is P-fold from the as inedge degenerate separate spectrum
the case. In the Hamiltonian the four submatrices in the left corner showfi, previous remaining upper already
the form. There remains the W matrix to in order to further in thedefine, proper diagonal bring diagonalization
submatrices of the 4th row and column of (45).
Let us take W as :
with :
Thanks to condition is and becomes(38)-(42B (43) satisfied, (44)
- LE JOURNAL DE T. N" SEPTEMBRE 1979PHYSIQUE. 40, 9, 908
R can be determined to a rotation in This matrix be taken as made of definedup N-space. may up properly
x x for submatrices :Q ) diagonal (N, 1 N,) (and (Q L)
A choice is (4) :simple possible
the final form of H is :
(51)(51)
is the of in the case. (24) Eq. (51) equivalent eq. previous
It should be remarked that a further set discrete levels + m) (now with Q-fold of (n2 degeneracy) appears ;
the mix E.O.’s and V.O.’s from the vertices of (here 2). The corresponding eigenvectors majority type eigen-
values are those of lro given by :
In a manner similar to that used in the we factorize the Hamiltoniansection, quite previous may remaining
into a subhamiltonians succession (1 of rank + + 2 each to anm), of N1 p NI) (n, n2 corresponding hp
of E to a of ± of the matrix (or pair opposite eigenvalues C) :adjacency eigenvalue Pp Ep
show that the of E are confined to theOne can (4) eigenvalues
interval + so that the matrix elements of(- IZ,-Z,) I-Z,-Z,,
real and these matrices are to their are equal adjoint.Jl, J2, Kl, K2 909
where
assume the same kind of us sum the results for this second class of here in orderLet approximation up
to as close as to their forThe Hamiltonian has :(36) keep possible treatment, systems.
the sake of To these atomic orbitals
- comparison. one set of m P-fold degenerate eigenvalues
thus two in the of states,correspond b-peaks density of the which are the eigenvalues edge-edge coupling
shown as « s » and « r » on 2.figure (35),matrix q
- one set of + m) eigen-(n2 Q-fold degenerate
those values, (52),of ho given by
- sets of 2 obtained+ + m) eigenvalues Ni (ni 1 n2
for each (53) by diagonalizing ep.
we have realized a of HAgain diagonalization
3 to that obtained in section (5).analogous
to 5. An of - In order showexample application.
in the how the formalism general presented foregoing
- 2. bands for without interaction(a) be worked out we shall now revert Fig. Energy GeSe2 : practically, may
between and orbitals with thisbonding antibonding (b) (cf. [10]) ; to the of one of vertex andsimpler example type
interaction taken into account.
the calculation of the electronic band structurerepeat
Lannoo and in Bensoussan by [lo].of GeSe2 presented
In the same we shall in retain, as way, only [10],In this a LCAO treatment is forward,paper, put
the matrix elements for the electronicfollowing which involves the atomic orbitals shown on 1 :figure
H :Hamiltonian
- Ge atom has 4 orbitalsequivalent every sp3
etc... for for atom 2 1, etc...) ;(0’ l’ u’, u
- Se atom bears two involvedevery p-orbitals
in covalent bonds with the Ge atomsneighbouring and
and one s-orbital, and another(p, P2), lone-pair
orbital labeled T.p-type lone-pair
we the atomic orbitals theFirst, replace along
Ge-Se bonds their and by bonding antibonding
linear and instead and combinations, AI BI of 0’ 1 pl,
with
and and instead and similarly A2 B2 of U2, p2, A’ B’ 1
instead of and b are etc. ; a ui, P’ given by :
1. - Atomic in orbitals Fig. Gese2.
all inter-Lannoo and Bensoussan have excluded
s and r Se-orbitals. We shallactions these involving
with
and
If 1- and 2-vertices are one show and that can be further(’) identical, may (by separating symmetrical antisymmetrical orbitals) (53)
factorized into two one identical to and one obtained in the of One Hamiltonians, thus recovers the results(25) by changing (25) sign 8p’
of the case.previous 910
We shall used also the numerical values in that iskeep [10],
orbitals on the one orbitals on the interact via the elements :hand, other, Bonding antibonding following matrix
These are the interactions that have been taken into account Lannoo and Bensoussan : only by bonding
and orbitals are not and thus be treated which antibonding coupled together may separately, brings great
into the some is into the this However, simplification problem. inconsistency brought picture by approximation,
since the Hamiltonian matrix elements between and orbitals are a non zero. Indeed,bonding antibonding priori
we find :
(60) Numerically, gives
While is of the same order of as the other matrix éléments (59),4,B is effectively negligible, L1 AB magnitude
and is worth into account. Thanks to the formalism we have at our this can be donetaking général disposal,
in a as we now shaH see.reasonably simple way,
All orbitals here hond-orhitals Let us S.E.O.’s asare (no introduce and A.E.O.’s such vertex-orbital).
with matrix elements
between orbitals the same Ge-Ge andbond, along
for orbitals two Ge-Ge bonds to the same Ge atom.along neighbouring adjacent