Convergence of DFT eigenvalues with cell volume and vacuum level

Transcription

1 Converence of DFT eienvalues with cell volume and vacuum level Sohrab Ismail-Beii October 4, 2013 Computin work functions or absolute DFT eienvalues (e.. ionization potentials) requires some care. Obviously, we as discussin systems with surfaces in contact with vacuum computin the absolute eienvalue for the interior of an infinite crystal is not possible since the alinment with vacuum depends on the dipoles on the surface which are not bein modeled. Therefore, what we really have in mind are 2D (slabs, sheets), 1D (polymers, nanotubes, nanowires) and 0D (molecules, atoms) where there are some periodic directions and some directions oin into the vacuum. et the lattice vector alon the periodic direction be a and the lattice vector alon the vacuum direction be. We will be investiatin the limit below. In eneral, the DFT Kohn-Sham Hamiltonian can be written as H = 2 2m + φ(r) + V xc(r) where φ is the total electrostatic potential from all chares, ions and electrons, φ(r) = d 3 r ρ(r ) r r = d 3 q 4π ρ(q)e iq r (2π) 3 q 2. Here, ρ(r) is the total (ion+electron) chare density and ρ(q) is its Fourier transform. Most of the problems with determinin absolute eneries and the vacuum level are because the Coulomb interaction is lon raned and so computin the potential in a periodic system of chares can be ill-defined. In could say it is about handlin the q = 0 diverence in periodic systems. Before continuin forward, there are two types of problems in determinin absolute eienvalues and potentials in a DFT calculation, and they are of different oriin. First, some DFT codes will set the averae potential to zero over the unit cell. This not only includes the electrostatic potential but also the exchane-correlation potential V xc. This means that the averae value of V xc over the volume of the unit cell Ω, 1 Ω Ω d 3 r V xc (r), is bein added or subtracted from the output potential. As the volume of the unit cell is chaned by increasin or decreasin the vacuum, this averae chanes as well. Whether a 1

2 iven code does this or not can only be determined by lookin inside. If one is usin a local or semi-local functional (e.. DA or GGA) and has enouh vacuum for the electron density to be localized on the system, then the interal is just some number and the averae will scale like 1/Ω. What this means is that the value of the potential in the vacuum reion could have such a contribution that depends on Ω and must be corrected for or extrapolated away. Second, all DFT codes run on periodic systems will have to do somethin with the diverence at q = 0 and thus adjust the averae potential in some way. This is what we are discussin below. Before we bein, we will need some eneral orientation and one mathematical result. It has to do with the fact that for a function of compact support with well-defined Fourier transform, interatin over the Fourier space or summin over the discrete reciprocal (Fourier) lattice ive exactly the same thin. We will do this in one dimension to keep the math simple. Consider two functions F (x) and G(x) whose product F ()G(x) is of compact support: it is non-zero only for 0 < x < a. We want the interal of their product (inner product) which can be done in real or reciprocal space I = where the Fourier convention is dx F (x) G(x) = F (x) = dq 2π F (q)e iqx. dq 2π F (q) G(q) Instead of interatin over all x, let s take a unit cell of lenth > a. Then the interal of F (x)g(x) over the finite rane 0 < x < is also I. Once we are workin on a finite interval of lenth, we can make F (x) and G(x) periodic without chanin I: F p (x) = n= F (x n) = 1 F ()e ix and similarly for G p (x). Here the reciprocal lattice vectors are = 2πn/ for inteers n. I is unchaned if we use the periodic functions What we found is that I = I = 0 dx F p (x) G p (x) = 1 F () G(). dq 2π F (q) 1 G(q) = F () G(). The discrete sum and the continuous interal are exactly the same: for a function of compact support, one can sample discrete or continuously and no information is lost (Nyquist theorem). We can write this in a slihtly different way which is more useful below. We can choose to write q = k + where π/ < k < π/ is in the first Brillouin zone (BZ) and a reciprocal lattice vector. Therefore I = π/ dk π/ 2π F (k + ) G(k + ). 2

3 Since the BZ has lenth 2π/ we chane replace the interal by the averae over the BZ and write I = 1 F (k + ) G(k + ) BZ where h BZ π/ π/ dk 2π/ h(k). Thus another way to write out main findin is I = 1 F () 1 G() = F (k + ) G(k + ) BZ. Now we bein by focusin on the slab case. We have to periodic directions in the xy plane and one direction oin into the vacuum. So if we let G xy denote the reciprocal lattice vectors in the xy lattice, all functions of interest have periodicity in the xy plane and have a Fourier series in the xy plane: for example, the chare density ρ(r) has Fourier representation ρ(x, y, z) = G xy dq z 2π ρ(g xy, q z ) e igxy rxy+iqzz. A The vector r xy = (x, y) is the position in the plane. A is the area of the periodic cell in xy. Instead of lookin at the potential φ(r) itself, consider the eienvalue E n for some bound stae ψ n (r): [ ] E n = d 3 r ψ n (r) 2 2m + φ(r) + V xc(r) ψ n (r). The part of interest is the electrostatic contribution Φ n = d 3 r φ(r) ψ n (r) 2. To see this, let s define the Fourier transform of ψ n (r) 2 to be p n (q). Then workin in Fourier space Φ n = dqz 4π ρ(g xy, q z ) p n (G xy, q z ) 2π A(G 2 G xy + q 2. xy z) Now, in an actual plane wave calculation, we have a periodic cell alon z of lenth. Both the bound state ψ n and the chare density ρ will decay exponentially into the vacuum outside the slab. Thus for practical purposes, they have compact support. So by choosin a lare enouh, the periodic imaes of these densities will not have any sinificant overlap so that the kinetic and exchane-correlation contributions to E n will be very well convered for any reasonable. It is aain the electrostatic part that ives us trouble. What the plane wave code actually calculates for Φ n uses the discrete reciprocal lattice vectors G z alon z instead of a continuous q z. The computed result from the periodic calculation is Φ p n and is Φ p n = G xy 4π ρ(g xy, G z ) p n (G xy, G z ) A(G 2 G xy + G 2 z z) where G xy = 0, G z = 0 is excluded from the sum. 3

4 (Most plane wave codes will write Ω = A the cell volume instead of A in the denominator.) We have to exclude the 0 wave vector contribution since the summand is very ill defined: the denominator oes to zero; while the numerator ρ(q) also oes to zero for q 0 for a neutral system, it may only o to zero linearly for a system with a dipole moment so the result is poorly defined. This missin term and what to do with it is the basic problem. Usin our mathematical result, for any G xy 0, all the interands or summands are quite smooth so we can use the equivalence of the continuous interal and discrete sum alon z to find dq z 4π ρ(g xy, q z ) p n (G xy, q z ) 2π A(G 2 xy + qz) 2 = 4π ρ(g xy, G z ) p n (G xy, G z ) A(G 2 G xy + G 2. z z) However, for G xy = 0, we can t just blindly apply the mathematical equivalence since the interand is actually diverent for G z = 0 while the interal is enerally well defined for physical systems (see below). Thus far, we have shown that all G xy 0 terms are identical between Φ n and Φ p n so that Φ n = Φ p dq z 4π ρ(q z ) p n (q z ) n + 2π Aqz 2 4π ρ(g z ) p n (G z ) AG 2. G z z 0 To clarify the difference further, we can write an arbitrary q z as the sum of the closest G z plus some k z that is in the first BZ q z = k z + G z as we did in our mathematical result above. We then have the equivalent expression Φ n = Φ p n + 4π ρ(kz ) p n (k z ) A BZ + 4π A [ ρ(kz + G z ) p n (k z + G z ) (k z + G z ) 2 ρ(g ] z) p n (G z ) BZ G 2. z G z 0 To proceed, we need to look at some of the physical characteristics of ρ and p n. We expand both in series ρ(k) = αk + βk 2 + O(k 3 ), p n (k) = 1 + α n k + β n k 2 + O(k 3 ). The expansion for ρ starts at dipolar order because our system is net neutral. The p n expansion starts at 1 since ψ n (r) 2 is normalized. The product will behave like This product divided by k 2 is then ρ(k) p n (k) = αk + Qk 2 + O(k 3 ) + O(k 4 ) + ρ(k) p n (k) k 2 = α k + Q + O(k) + O(k2 ) + This looks badly diverent, but the reciprocal lattice vectors and the BZ have inversion symmetry: for any valid G z, G z is also valid and for any k z in the BZ, so is k z. Thus the BZ averae as well as the sum over G z in the expression for Φ n are both samplin the diverent α/k term equally alon positive and neative values so its contribution actually vanishes around the oriin. Thus the effective summands are actually quite smooth here. What this means is that the sum over G z 0 terms above is very small since we are subtractin the averae of a smooth function over the BZ from the value at the center of the BZ; anyways, the 4

5 difference converes to zero as ets larer and the averain is over a narrower reion. Therefore, any type of discrepancy between Φ n and Φ p is really comin from the very first term: Φ n = Φ p n + 4π ρ(kz ) p n (k z ) A Usin the Taylor series in k from above, the desired averae is 4π ρ(kz ) p n (k z ) A BZ + terms converin very rapidly in. = 4π α BZ A k + Q + Rk + Sk2 + Only the even terms survive, and the averae of k 2 is proportional to (π/) 2. So we et 4π ρ(kz ) p n (k z ) A = C BZ + D 3 + E 5 + for some constants in the numerators. Therefore, we expect for the eienvalue E n () = E n ( ) + C 1 + D 3 + By performin a set of computations at various, we can fit E n () to the above form and extrapolate to = to et E n ( ). (This is an example of Richardson extrapolation.) This concludes the slab eometry. BZ. Considerin the other eometries (1D wire and 0D molecule) is mainly takin the above derivation and chanin some discrete sums to continuous ones. For example, for a wire eometry with periodic axis alon z, the x and y directions are those oin into the vacuum. Thus we will have a discrete sum over G z and interals over q x and q y ; when we put in a periodic box we et a discretized sum over G x and G y instead; the volume factor A is replaced by a 2 where a is the periodic lattice lenth alon z; the G z 0 contributions will be well behaved while G z = 0 will be problematic; for G z = 0, the contributions for (G x, G y ) 0 and those from averain over a BZ centered around (G x, G y ) will be very close; so the difference between Φ n and Φ p n will aain be due to the averain in the BZ around q = 0. The main difference is that now we have vector k so where α is a vector, Q is a matrix, etc., and ρ(k) p n (k) = α T k + k T Qk + O(k 3 ) ρ(k) p n (k) k 2 = αt ˆk k + ˆk T Qˆk + O(ˆkˆkk) + O(ˆkˆkkk) + where ˆk = k/ k is the unit vector. The averain proceeds as before notin that the odd power cancel by symmetry and the even terms ive for the averae of k 2 somethin of order (π/) 2. The overall volume prefactor is now 1/(a 2 ) so our expansion will look like E n () = E n ( ) + C 2 + D 4 + The 0D point eometry has vacuum in all directions so the q interals are continuous alon all three directions; the discretization is to replace q by a three vector G; the volume factor is Ω = 3 ; and we proceed as before. So the series starts with 3 and oes in powers of two from there. 5

6 Our final summary is thus that E n () E n ( ) = constant C 1 + D subleadin terms = E 2 + F 4 + Ω G 3 + H 5 + 2D slab case 1D wire case 0D point case. for constants C, D,... Therefore, we expect the eienvalue to convere to the limit as E n () = E n ( ) + C m + D m+2 + where m = 1 for 2D, m = 2 for 1D, and m = 3 for 0D. By performin a set of computations at various, we can fit E n () to the above form and extrapolate to = to et E n ( ). (This is an example of Richardson extrapolation.) 6