1 Tight-Binding Model of Electronic Structures Consider a collection of N atoms. The electronic structure of this system refers to its electronic wave function and the description of how it is related to the binding energy that keeps the atoms together. In an independent electron approximation, a single electron time-independent Schrödinger equation, Hψν(!r ) =ενψν(!r ), (1) is solved to find the eigenstates, ψν (ν = 1, 2, ...), and the corresponding eigenenergies, εν, of the Hamiltonian operator, H = −12∇2+Vr( ), (2) where V (r ) is the potential energy operator and we have used the atomic unit. (For example, the density functional theory provides a framework to derive an effective single-electron potential energy operator, which incorporates the interaction among the many electrons [1-3].) §1. Tight-Binding Model In the tight-banding model of electronic structures, single-electron wave functions are expanded in terms of atomic orbitals [4,5], ψnlm(r,θ,ϕ) = Rnl(r)Ylm(θ,ϕ), (3) centered around each atom, where Rlm and Ylm are radial and spherical-harmonics functions in polar coordinates. (This scheme is also called the linear combination of atomic orbitals, or LCAO.) In Eq. (3), n, l and m are the principal, angular-momentum and magnetic quantum numbers, respectively. For example, 1s atomic orbital (n = 1 and l = m = 0) is spherically symmetric and is taken as positive definite (see the figure below). For p atomic orbitals, we work with the Cartesian representation, such that the three orthogonal p orbitals are along the x, y and z axes [4-6] (see the figure below), ψn1m(x, y, z) =34π1/2Rn1(r)x / ry / rz / r, (4) where r = x2+ y2+ z2.2 HOPPING INTEGRALS As a specific example, we consider a collection of silicon atoms, such as a silicon cluster. The electron configuration in a free silicon atom is 1s22s22p63s23p2, and thus there are four valance electrons (shown in bold) in the outer shell, which mainly contribute to the chemical bonding. We will represent the electronic structures of silicon clusters as a linear combination of four atomic orbitals per atom—one 3s orbital and three 3p orbitals, 3px, 3py and 3pz, centered around each silicon atom, ψr( )=i=1N∑ciαψαr −ri( )α∈{s, px, py, pz}∑. (5) (The effects of inner shell electrons can be effectively included using pseudopotential methods [7].) To solve the eigenvalue problem, Eq. (1), we need the Hamiltonian matrix elements between these atomic orbitals at different interatomic distances. In tight-binding methods, these so called hopping integrals are fitted as analytic functions of the interatomic distance, r. For the sp-bonding, there are only four nonzero hopping integrals as shown in the figure below, in which σ and π bondings are defined such that the axis of the involved p orbitals are parallel and normal to the interatomic vector, respectively. In this lecture, we adopt the tight-binding model of silicon by Kwon, et al. [8], in which the hopping integrals are fitted as hλ(r) =s1H s2λ= ssσs1H p2dλ= spσp1dH p2dλ= ppσp1nH p2nλ= ppπ!"###$###%&###'###= hλ(r0)r0r()*+,-nexp n −rrλ()*+,-nλ+r0rλ()*+,-nλ/0112344()**+,--, (6) whereas the diagonal Hamiltonian elements on each atom are given by s H s = EspxH px= pyH py= pzH pz= Ep. (7) In Eq. (6), p1d and p1n denote the p orbitals parallel and normal to the bonding axis, respectively, centered at the first atom. The parameters in Eqs. (6) and (7) are listed in the tables below.3 r0 (Å) n Es (eV) Ep (eV) 2.360352 2 -5.25 1.20 λ ssσ spσ ppσ ppπ hλ(r0) (eV) nλ rλ (Å) -2.038 9.5 3.4 1.745 8.5 3.55 2.75 7.5 3.7 -1.075 7.5 3.7 To convert the values into atomic units, divide all the lengths by the Bohr radius, aB = 0.5291772083 Å, and all the energies by the Hartree energy, EH = 27.2113834 eV. §2. Projection of Hopping Integrals In the tight-binding model presented in the previous section, the electronic wave functions are expanded in terms of the p orbitals along the Cartesian x, y and z axis, whereas the hopping integrals are parameterized for p orbitals that are parallel or normal to the bonding directions. To construct the Hamiltonian matrix elements, we need to decompose the Cartesian p orbitals into the bond-parallel and bond-normal p orbitals. PROJECTION OF S-P INTEGRALS Consider the Hamiltonian matrix element, 〈s|H|pα〉, between the s orbital, |s〉, on one atom and one of the p orbitals, |pα〉 (α = x, y, z), on another atom. Let ˆd be the unit vector along the bond from the first atom to the second. In the figure below, ˆa is the unit vector along one of the Cartesian (x, y or z) axes. We first decompose the p orbital along ˆa into two p orbitals that are parallel and normal to ˆd, respectively (see the figure below): pa=ˆa •ˆd pd+ˆa •ˆn pn, (8) where ˆn is the unit vector normal to ˆd within the plane spanned by ˆd and ˆa.4  Let θ be the angle between vectors ˆd and ˆa. Consider an arbitrary point in the 3D space, whose polar angle from the ˆd axis is χ. On this point, the value of the p basis function around the ˆa axis is given by pa= cosχ−θ( )= cosχcosθ+ sinχsinθ= cosθpd+ sinθpn=ˆa •ˆd pd+ cosπ2−θ"#$%&'pn=ˆa •ˆd pd+ˆa •ˆn pn, where we have used a trigonometric addition theorem to derive the second line from the first. // The Hamiltonian matrix element is then given by s H pa= s Hˆa •ˆd pd+ˆa •ˆn pn( )=ˆa •ˆd( )hspσr( ), (9) where hspσ(r) is the Hamiltonian matrix element in Eq. (6) evaluated at atomic distance r. Note the overlap between the |s〉 and |pn〉 orbitals is zero by symmetry. To obtain explicit formula in terms of px, py and pz centered around the first second atom, let us introduce the directional cosines along the x, y and z axes as ˆd = (dx, dy, dz). Then s1H p2 xs1H p2 ys1H p2 z= −p1xH s2p1yH s2p1xH s2=dxhspσr( )dyhspσr( )dzhspσr( ). (10) In Eq. (10), the matrix elements between the p orbitals on the first atom and the s orbital on the second atom are obtained simply by inverting the direction of the bonding unit vector,