Presentation on theme: "Physics “Advanced Electronic Structure”"— Presentation transcript:
1Physics 250-06 “Advanced Electronic Structure” Earlier Electronic Structure MethodsContents:1. Solution for a Single Atom2. Solution for Linear Atomic Chain3. LCAO Method and Tight-Binding Representation4. Tight-Binding with BandLab
2Solving Schrodinger’s equation for solids Solution of differential equation is requiredProperties of the potential
3Properties of Solution for a Single Atom Atomic potential(spherically symmetric)
4Property of solution for symmetric potential Discrete set of levels is obtained. While degeneracy with respect to m remains,degeneracy with respect to l is now lifted since V V(r) is different from –Ze2/r . We can label the levels by main quantum number and orbital quantum number. For given l, n=l+1,l+2,... which is the property of the solution.For l=0 we have the states E1s, E2s,E3s,E4s,...For l=1 we have the states E2p, E3p,E4p,E5p,...For l=2 we have the states E3d, E4d,E5d,E6d,...No artificial degeneracy as in the hydrogen atom case: E2s=E2p,E3s=E3p=E3d,E4s=E4p=E4d=E4f…At some point, level Enl becomes above zero, i.e the spectrum changes from discrete to continuous.Note also that this spectrum of levels is a functional of V(r), i.e. it is different for agiven atom with a given number of electrons N.
5We can interpret different solutions as wave functions with different numbers of nodes. For example, for l=2, there are solutions of the equations with no nodes, with one node, with two nodes, etc. Number of nodes = n-l-1. If n runs from l+1, l+2, ..., nodes =0,1,2,3...
6Periodic table of elements The one-electron approximation is very useful as it allows to understand what happensif we have many electrons accommodated over different levels.Let us take atom with N electrons. Lets us find all discrete levels E1s<E2s<E2p<...Since electrons are the fermions they obey Fermi-Dirac statistics, i.e. they cannot bemore than two electrons (with opposite spins) occupying given non-degenerate level.If level is N-fold degenerate (for example, p level is 3 fold degenerate) then it can accomondate 2N electrons.So we can now fill various atomic shells with electrons E1s2,E2s2E2p6 and so on until we accomodate all N electrons within various slots.Thus we obtain a periodic table.For atom with given N we need to find discrete levels E1s,E2s,E3s,E2p,E3p,...We need to order them from lowest to highest.We need to fill the levels with electrons.In many cases, for a given number N ordering the levels is simple E1s<E2s<E2p<E3s...Therefore atoms of the periodic table have configurations which is easy to obtain.In some cases (since levels depend on Vscf(r) which depends on N) this rule is violated. This for example happens for later 3d elements.
7Hunds rulesFor a given l shell, the electrons will occupy the slots to maximize the total spinFor a given l shell, the electrons will maximize the total spin and total angular momentumHunds rules cannot be understood on the basis of one-electron self-consistent approximation.However, for atomic system, the many body problem for N electrons moving in theCoulomb potential can be solvedwhere En is the set of many body energy levels. Hunds rules follow from this Hamiltonian
8Bringing atoms together: what to expect? Properties of the solutions:Energy BandsCore Levels
9Solution for a Linear Atomic Chain Illustration: Periodic array of potential wells placed at distance L between the wells.Basic property of the potential V(x+l)=V(x), and of the Hamiltonian H(x+l)=H(x)Solving variationally:
10We obtain infinite system of equations If overlap is only between nearest wells, we simplify it to beThis is infinite set of equations. At first glance it seems that we cannot solve it. But we can do it indeed.
11introduce periodic boundary condition Since Hamitonian is periodical function, all wells are equivalent for the electron.That would mean that the probability to find the electron in each of the well should be the same.That is possible ifwhere u(x) is a periodical function:introduce periodic boundary conditionwhere n is any integer number and phase is frequently called the wave vector k_n
12obeys the periodicity condition automatically Once we understand the form for the wave function, we already see one problem with the representationEach term in this expansion is not a periodical function, only the entire sum is periodical. The questionis if we can construct another linear combinations of those basis functions so that each of the termsobeys the periodicity condition. In other words, we would like to have such combinations ofwith some coefficients c_n so that the combinationobeys the periodicity condition automatically
13where n can be any integer. For this to happenor, the coefficient is equal towhere n can be any integer.So we can label those combinations with index k_nwhich automatically obey the periodicity condition for wave function (Bloch theorem)
14What we want is to is to use those linear combinations in finding the solutions The advantage of this formulation is seen because all off-diagonal matrix elements disappear.This is seen becauseF(x) is periodical function, thereforeTherefore, we automatically obtain the diagonalized HamiltonianThat mean that the spectrum is known to usIn other words, we are able to solve the problem completely!
15Let us analyze the solutions We see that we can draw the solutions as a cosine centered around h as a function of wave number k.That is a band. Due to periodicity of cos, it is sufficient to draw it within –pi<kl<pi.We do not forget of course that k are discrete set of numbersHowever since we consider N to be very large number, we can deal with k as with continuous argument.The wave functions corresponding to each solutions are simply
16Summary: Bloch Property for a Local Basis Differential equation using expansionwhere is a basis set satisfying Bloch theoremTo force the Bloch property we now use instead of plane waves:Linear combinations of local orbitals
17Apply Variational Principle Variational principle leads us to solve matrix eigenvalueproblemwhereis hamiltonian matrixis overlap matrix
18LCAO Method Linear Combination of Atomic Orbitals (LCAO) Tailored to atomic potentialto be used in variationalprinciple
19Hamiltonian of LCAO Method: Hoppings between the orbitals
20Tight-Binding Parametrization In LCAO Method, the Hamitonain is parameterizedvia on site energies of the orbitals andnearest-neighbor hopping integrals betweenthe orbitals.In many situations symmetry plays an important role sincefor many orbitals hoppings intergrals between them areautomatically equal zero.
21Thinking of Cubic Harmonics - unitary transformation
22Hoppings between various orbitals ss,sp,sd hoppingspp,pd hoppingsdd hoppings
23Illustration, CuO2 plane It is clear that levels are non-bonding and are all occupied.The same is true for level of copper, and for orbitals. The active degrees of freedom here are orbitals which have hopping rate .Using the active degrees of freedom as the basis the hamiltonian has the form
24Let us introduce another basis set of bonding (b), antibonding (a), and non-bonding (n) orbitalsWithin this basis set, the Hamiltonian becomesThe bands are now seen as the bonding band below and theantibonding band above. The non--bonding band is also present.