BioMaths  Lecture  7  

Solving  DFT  equa9ons:  Numerical  Methods  

Bulk  systems  

•  Before  aAemp9ng  to  solve  the  Kohn-­‐Sham  equa9on  let’s  consider  bulk  systems  

•  Periodic  boundary  condi9ons  

Periodic  Boundary  Condi9ons  

•  We  have:  

•  Where  L  is  the  repeat  length.  

•  Does  this  imply  that  

•  It  does  not.  €

V (r) =V (r + L)n(r) = n(r + L)

€

Ψ(r) =Ψ(r + L)?

(Felix)  Bloch’s  Theorem  

•  Statement:  eigenfunc9ons,  φ,  of  a  periodic  system,  V(r)=V(r+L)  can  be  wriAen  as  a  product  of  a  periodic  func9on,  u(r)=u(r+L),  and  a  phase  factor,  thus:  

€

φk (r) = uk (r)exp(ik⋅ L)

Proof  of  Bloch’s  Theorem  I  

•  We  have  

•  The  KS  equa9on  is  (in  shorthand  nota9on)  

•  Define  the  transla9on  operator  

€

V (r) =V (r + L)

€

ˆ H φ =−2

2m∇2 +V (r)

⎡

⎣ ⎢

⎤

⎦ ⎥ φ = εφ

€

ˆ T L f (r) = f (r + L)

Proof  of  Bloch’s  Theorem  II  

•  Since  V  and  hence  H  is  periodic  then  T  and  H  commute,  i.e.  

•  Therefore  eigenstates  of  H  must  be  eigenstates  of  T  

•  And  (obviously)  TL  and  TL’  commute:  

€

ˆ T L ˆ H (r)φ(r) = ˆ H (r + L)φ(r + L) = ˆ H (r)φ(r + L) = ˆ H (r) ˆ T Lφ(r)

€

ˆ T L ˆ T L ' = ˆ T L +L ' = ˆ T L '+L = ˆ T L 'ˆ T L€

ˆ H φ = εφˆ T Lφ = c(L)φ

Proof  of  Bloch’s  Theorem  III  

•  Therefore  the  eigenvalues  also  sa9sfy  periodicity  

•  This  is  sa9sfied  if  and  only  if  c  is  a  complex  number  of  radius  1  

•  So  we’ve  shown  that  

€

c(L + L') = c(L)c(L')

€

c(L) = exp(ik⋅ L)ki ⋅ L j = 2πδ ij

€

ˆ T Lφ(r) = φ(r + L) = c(L)φ(r) = exp(ik⋅ L)φ(r)

Implica9ons  

•  Bloch’s  theorem  tells  us  that  φ  is  not  quite  periodic  –  it’s  periodic  9mes  a  phase  factor.  

•  Consider  

•  So  u  is  strictly  periodic  u(r)=u(r+L)  •  Hence  the  wavefunc9on  of  a  periodic  system  is  of  the  form  

€

u(r) = exp(−ik⋅ L)φ(r)u(r + L) = exp(−ik⋅ [r + L])φ(r + L) = exp(−ik⋅ L)φ(r) = u(r)

€

φk (r) = u(r)exp(ik⋅ L)

Expand  periodic  u(r)  

•  A  periodic  func9on  can  be  expressed  as  a  Fourier  series  

•  And  combine  with  Bloch’s  theorem  gives  us  €

u(r) = cG exp(iG⋅ r)G∑

G =2πnL,n =1,2,3,...

€

φk (r) = cG exp(i[k +G]⋅ L)G∑

Our  goal  is  to  calculate  the  Fourier  coefficients,  cG  

Consequences  

•  Instead  of  solving  for  φ  over  all  space  – We  solve  for  φ  in  a  single  periodic  cell  

– φ  is  simply  a  Fourier  series  

•  So  we  know  the  Kohn-­‐Sham  equa9ons  and  we  know  the  form  of  the  solu9on  

•  Other  than  some  number-­‐crunching,  we’ve  just  solved  the  many-­‐par9cle  Schrodinger  equa9on!  

Computa9onal  considera9ons  

•  Well,  almost  solved  it!  •  There  are  an  infinite  number  of  terms  in  a  Fourier  series  

•  There  are  an  infinite  number  of  Bloch  ‘k-­‐points’  

•  Now  we’ll  consider  some  numerical  techniques  before  you  aAempt  it  yourself  

To  get  a  solu9on  

•  φ  is  a  func9on  of  Fourier  coefficients  cG – Therefore n(r) is a  func9on  of  Fourier  coefficients  cG, n[{cG}]

– Therefore Hamiltonian, H, is a  func9on  of  Fourier  coefficients  cG, H[{cG}]

– Therefore energy, E, is a  func9on  of  Fourier  coefficients  cG, E[{cG}]

•  Minimise E[{cG}] with respect to {cG}

E[{cG}]  Surface  

The  cut-­‐off  energy  

•  We  will  truncate  the  Fourier  series  (i.e.  the  number  of  G-­‐vectors  will  be  finite)  

•  Some  terminology:  instead  of  sta9ng  the  number  of  G-­‐vectors,  it  is  common  to  state  the  energy  of  the  highest  frequency  used  in  the  Fourier  series  

€

2

2mGn + k( )2 ≤ Ecut

Gn =2πnL

How  to  choose  Ecut  

•  HK-­‐II  implies  a  varia9onal  principle  •  As  the  Fourier  series  is  lengthened  the  energy  will  decrease  monotonically  to  obtain  eigenvalue,  E,  of  Schrodinger  equa9on  

•  So  converge  proper9es  of  interest  with  respect  to  the  Fourier  series  length  (Fourier  series  cut-­‐off  energy)  

Convergence  of  FS  

Fourier  (Reciprocal)  Space  

• Grid  of  points  

• |G|  <  Ecut  

• Maps  out  sphere  of  radius  |G|  

Bloch’s  Theorem:  k-­‐points  

•  Note  that  in  Bloch’s  Theorem  

•  a  wavevector  k  is  introduced  •  Observables  (density,  energy,  etc.)  are  calculated  by  integra9ng  over  this  quan9ty  €

φk (r) = uk (r)exp(ik⋅ L)

€

E =1V

E[nk (r)]dk∫

n(r) =1V

nk (r)dk∫

€

nk (r) = φi,k* (r)φi,k (r)

i=1

N

∑

Example  Integra9on  •  Turn  integral  into  finite  sum  

Convergence  w.r.t  k-­‐point  grid  

Summary  

•  For  each  electron  with  wavevector  k,  we  have  eigenstate  

•  where  i  labels  the  electron  •  k  labels  the  Bloch  k-­‐point  •  u  is  expanded  in  a  Fourier  series  (G-­‐wavevectors)  

•  Converge  result  w.r.t  k-­‐grid  and  Fourier  length  

€

φi,k (r) = uk (r)exp(ik⋅ r)

In  The  Next  Lecture…  

•  We  will  do  some  prac9cal  examples  of  geing  the  energy  eigenvalues  and  electron  densi9es  for  a  range  of  small  bio-­‐systems  

•  We  will  put  error  bounds  on  these  quan99es  due  to  numerical  considera9ons