Electronic structure with Intel® Many Integrated Core (Intel MIC) with Elk W. Sco= Thornton Joint Ins@tute for Computa@onal Sciences Outline •  Introduc@on to the many-­‐body problem •  Density func@onal theory •  Computa@onal challenges for ground-­‐state calcula@on •  The APW method •  Results The Many-­‐body problem Ĥ = −
N
�
∇2
i
i=1
2
+
kine@c energy N
�
v (ri ) +
i
nuclear poten@al Born-­‐Oppenheimer Approxima@on Nuclei are considered fixed in space An@symmetric wavefunc@on Ψ = Ψ(r1 , r2 , · · · , rN )
Pauli principle dictates that with any interchange of the electron coordinates and spin, the sign of the wavefunc@on must reverse. N
�
i<j
1
|ri − rj |
e-­‐e interac@ons Density func@onal theory (DFT) Hohenberg-­‐Kohn theorem states that there is a one-­‐to-­‐one mapping between the the external poten@al, V, and the electronic density, n(r). v(r) ←→ n (r)
The total energy is a func@onal of the density. E [n] = F [n] +
�
drv (r) n (r)
v (r)
system dependent external poten@al F [n]
universal func@onal which includes the many-­‐
body kine@c energy and electron-­‐electron interac@on We don’t know F
[n]
Walter Kohn (www.nobelprize.org) Density func@onal theory (DFT) Kohn-­‐Sham scheme centers around using an auxillary system of non-­‐interac@ng electrons that give the same density as the interac@ng system. Kohn-­‐Sham equa@on �
electronic density �
1 2
− ∇ + Vks [n](r) φi (r) = �i φi (r)
2
n(r) =
�
i
|φi (r)|
2
Kohn-­‐Sham eff. poten@al Vks [n](r) = v(r) + Vh (r) + Vxc [n](r)
Hartree poten@al Vh (r) =
�
n(r� ) 3 �
d r
�
|r − r |
Exchange-­‐correla@on poten@al • local density approxima@on (LDA) • generalized gradient approxima@on (GGA) • LDA with Hubbard U (LDA+U) Density func@onal theory (DFT) (periodic systems) Periodic unit cell V (r + R) = V (r)
Bloch state φ(r) −→ ψnk (r)
ψnk (r) = eik·r unk (r)
c
unk (r) is periodic with the unit cell a
We have a k-­‐dependent Hamiltonian We can paralellize using k-­‐points. b
Computa@onal scope What about DFT calcula@ons makes them expensive? Sheer number of orbitals – nanopar@cles and molecules can have 100’s or 1000’s of atoms Large number of basis func@ons – full-­‐poten@al, all-­‐electron methods, orbitals span many length scales Pseudopoten@als Pseudopoten@als replace the core (nuclei and core orbitals) of each atom with an effec@ve poten@al Reduces the total number of orbitals for calcula@on Smooths the poten@al near the vicinity of each nuclei which reduces the number of required basis func@ons Mixed basis set Other ways of handling the span of length scales is to use a mixed basis set Augmented plane waves Mul@wavelet adap@ve mesh Quick diversion about APW methods In APW methods, space is divided into an atomic sphere region and an inters..al region In the muffin @n region, the Bloch states are composed of radial func@on and spherical harmonics ψnk (r) =
�
cnk
lm Rl (r)Ylm (θ, φ)
lm
In the inters@@al region, the Bloch states composed of plane waves ψnk (r) =
�
dn(k+G) ei(k+G)·r
lm
Inters@@al region Atomic sphere (Muffin @n) •
•
•
•
•
•
Full poten@al, all electron LAPW code Exci@ng EU Research and Training Network GNU General Public License Fortran 90 Paralellized with either MPI or OpenMP h=p://elk.sourceforge.net/ Kohn-­‐Sham equa@on �
�
1 2
− ∇ + Vks [n](r) φi (r) = �i φi (r)
2
!"#$%&'()*)+,-(
.'*/)&0(
!"#$%&'(
$"&'*+,-(
Kohn-­‐Sham eff. poten@al Vks [n](r) = v(r) + Vh (r) + Vxc [n](r)
!"*/&1%2&(3,/)/(
4%*2+"*/(
Hartree poten@al Vh (r) =
�
�
n(r ) 3 �
d r
�
|r − r |
electronic density n(r) =
�
i
|φi (r)|
5%)-.(
6,#)-&"*),*(
<=$")*&((
$,1,--'-)9,+"*(
7),8"*,-)9'(
6,#)-&"*),*(
2
!"*:'18'.;(
Results 30
actual
20
15
10
5
0
10
20
30
40
50
60
Number of OpenMP threads
70
actual
60
Speedup of Si (64 k-pts)
Ideal speedup equa@on Speedup of Si (27 k-pts)
ideal
25
ideal
50
40
30
20
10
0
10
20
30
40
50
60
Number of OpenMP threads
70
80
90
Speedup of Si (27 k-pts)
Ideal speedup equa@on 30
25
20
15
10
5
0
actual
ideal
10
20
30
40
50
60
Number of OpenMP threads
Speedup of Si (64 k-pts)
Results 70
60
50
40
30
20
10
0
actual
ideal
10 20 30 40 50 60 70 80 90
Number of OpenMP threads
Time dependent density func@onal theory (TDDFT) Runge-­‐Gross theorem extends the principles of DFT to the @me dependent case. One-­‐to-­‐one correspondence between the @me-­‐dependent density and a @me-­‐
dependent poten@al (up to an arbitrary purely @me-­‐dependent func@on. n(r, t) ←→ v(r, t) + c(t)
Time-­‐dependent Kohn-­‐Sham equa@on ∂
i φj (r, t) =
∂t
�
n(r, t) =
|φi (r, t)|
N
�
i=1
�
1 2
− ∇ + vks [n](r, t) φj (r, t) = �j φj (r, t)
2
2
vks [n](r, t) = v(r) + vh (r, t) + vxc [n](r, t)
TDDFT linear response Electronic density for the interac@ng system: n [vext ] (r, t) = n0 (r) +
�
�
�
� �
� � ∂n [vext ] (r, t) �
dr dt vext (r , t )
+ ···
�
�
�
∂vext (r , t ) vext [n0 ]
χ (r, r� ; t − t� )
Electronic density for the auxillary system: n [vks ] (r, t) = n0 (r) +
�
�
�
� �
� � ∂n [vks ] (r, t) �
dr dt vks (r , t )
+ ···
�
�
�
∂vks (r , t ) vks [n0 ]
χ0 (r, r� ; t − t� )
Rela@ng χ (r, r
�
; t − t� ) and χ0 (r, r� ; t − t� ) , we get: χ = χ0 + χ0 [v + fxc ] χ
TDDFT linear response For a periodic system, in the Bloch basis: χG,G� (q, ω) =
�
χ0G,G� (q, ω)
�
+
χ0G,G1 (q, ω)×
G1 ,G2
δG1 ,G2 v(q + G1 ) +
xc
fG
(q, ω)
1 ,G2
�
χG2 ,G� (q, ω)
non-­‐interac@ng response: BZ
1 ��
fj � ,k+q − fj,k
0
χG,G� (q; ω) =
×
Ω �
�j � ,k+q − �j,k − �(ω + iη)
jj
�j, k|e
i(G+q)·r
�
k
�
|j , k + q��j , k + q|e
−i(G� +q)·r
|j, k�
Applica@ons in electronic structure • GW Approxima@on (many-­‐body effects included in GS) • Constrained RPA • Dynamical Mean Field Theory • Quantum Monte Carlo